diff --git a/data_all_eng_slimpj/shuffled/split2/finalzssv b/data_all_eng_slimpj/shuffled/split2/finalzssv new file mode 100644 index 0000000000000000000000000000000000000000..7f76b7168e53a0e9b45f9a57d7169d47fba19b26 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzssv @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe teleportation process, proposed by Bennett et al \n\\cite{1}, allows to transmit an unknown state of \na quantum system from a sender, traditionally named Alice, \nto receiver, or Bob, both are spatially separated.\nFor teleporting of a two-state particle, or qubit, \nit needs an EPR pair and a usual communication channel.\nThe large number of versions using two-particle \nentanglement has been considered \\cite{2}.\nQuantum teleportation of the photon polarized \\cite{4} and \na single coherent mode of field \\cite{40} has been demonstrated in optical \nexperiments.\n\nAs a source of EPR pair light can be used, particularly\nthe light of the optical parametric oscillator or down \nconversion as in \\cite{40}. \nHowever the physical nature of the particles may be \ndifferent, for instance one can choose the EPR-correlated atoms \nrealized experimentally in \\cite{41}. Indeed, \nthe particles of different nature are introduced \nin the scheme called the inter-space teleportation \\cite{42},\nwhere quantum state is transferred, for example, between the atom \nand light \\cite{43}. \n\nIn this work we consider teleportation of an entangled pair \nto two distant parties Bob and Claire with the use of the \ntriplet in the Greenberger-Horne-Zeilinger state (GHZ). Indeed\nthe GHZ triplet has been realized experimentally \\cite{5}.\nThe main problem is to find the three-particle projection \nbasis for \na joint measurement. By contrast the single qubit state \nteleportation, the maximally entangled basis does not \naccomplish the task. The obtained basis consist of a set of\nthe three-particle projection operators with the \nmaximally entanglement of two particles only.\nMeasuring allows both receivers to recover \nan unknown state of EPR pair, but each of them \ncan not do it separately. As it has been shown in ref. \\cite{6},\nwhere teleportation of a single qubit using \nthe GHZ triplet has been considered, \nonly one of the receivers and not both can recovered an unknown state.\nOur results are generalized for the \nteleportation of the N-particle entanglement as the EPR-nplet.\n\nOur work is organized as follows. In section 2 the main \nfeatures of the teleportation of a single qubit are given.\nThe initial states of the entangled pair and triplet are discussed \nin section 3. In section 4 \nthe basis for the joint measurement is found.\nThe teleportation protocol and network are presented in section 5, \nwhere the results are generalized for the N- particle entanglement.\n\n\\section{Teleportation}\nThe teleportation of an unknown quantum state between \ntwo parties spatially separated, Alice and Bob, includes the following \nsteeps \\cite{1}.\nLet Alice has a two level system or qubit prepared in an unknown \nstate \n\\begin{equation}\n|\\psi_{1}\\rangle=\\alpha|0\\rangle +\\beta|1\\rangle\n\\label{01}\n\\end{equation}\nwhere $|\\alpha|^{2}+|\\beta|^{2}=1$. \nLet Alice and Bob share a maximally entangled EPR pair\n$|\\Psi_{23}\\rangle =(|01\\rangle +|10\\rangle )\/\\sqrt{2}$, so that qubit 2 \nis for Alice and qubit 3 is for Bob.\nFirst, Alice performs a joint measurement of qubits 1 and 2 in the \nBell basis consisting of four projectors \n$\\Pi_{k}=|\\pi_{k}\\rangle \\langle \\pi_{k}|$, $k=1,\\dots 4$, \n$|\\pi_{1}\\rangle =|\\Phi ^{+}_{12}\\rangle$,\n$|\\pi_{2}\\rangle =|\\Phi ^{-}_{12}\\rangle$,\n$|\\pi_{3}\\rangle =|\\Psi ^{+}_{12}\\rangle$,\n$|\\pi_{4}\\rangle =|\\Psi ^{-}_{12}\\rangle$,\nwhere the Bell states are the maximally entanglement \nof two particles \n\\begin{equation}\n|\\Phi ^{\\pm}\\rangle =\\frac{1}{\\sqrt{2}}(|00\\rangle \\pm |11\\rangle )\n\\label{1}\n\\end{equation}\n\\begin{equation}\n|\\Psi ^{\\pm}\\rangle =\\frac{1}{\\sqrt{2}}(|01\\rangle \\pm |10\\rangle )\n\\label{2}\n\\end{equation}\n\nAs result of the joint measurement, the density operator of the \ncombined system \n$\\rho=|\\psi_{1}\\rangle \\langle \\psi_{1}|\n\\otimes |\\Psi_{23}\\rangle \\langle \\Psi_{23}|$, \nto be defined in the three-particle Hilbert space\n$H_{1}\\otimes H_{2}\\otimes H_{3}$, \nis projected into one of four Bell states.\nTwo point are important in this procedure, first, the k-th outcome \ndepends not on $\\psi_{1}$\nsecond, the reduced density matrix of the qubit 3 \n$\\rho_{3}(k)=Sp_{12}\\{\\Pi_{k}\\rho\\Pi_{k}^{\\dagger}\\}$ \nand unknown state both are connected by the unitary \ntransformation $U_{k}$\n\\begin{equation}\n\\rho_{3}(k)=U_{k}\\tilde \\rho_{1} U_{k}^{\\dagger}\n\\label{3}\n\\end{equation}\nwhere\n$\\tilde \\rho_{1}$ is the density operator of $H_{3}$, \nthat is the counterpart state\n$\\rho_{1}=|\\psi_{1}\\rangle \\langle \\psi_{1}|$, $U_{k}$ is the \nset of the Pauli matrices \n$U_{1}=\\sigma_{x}, U_{2}=-i\\sigma_{y}, U_{3}=1, \nU_{4}=\\sigma_{z}$. \nFinally Alice sends the outcomes of her measurement to Bob \nwho performs on his qubit 3 one of four unitary operations, \ncorresponding Alice' message and has his qubit in the \noriginal state \n$\\psi_{1}$. Teleportation is achieved.\n\n\\section{Initial states}\n\nTo teleport an EPR pair it needs a maximally entanglement of \nthree particles. From this fact let consider what initial \nstates would be used.\n\nThe wave function of an entangled pair can be chosen as\n\\begin{equation}\n|\\Psi_{12}\\rangle = \\alpha |00\\rangle +\\beta |11\\rangle\n\\label{0001}\n\\end{equation}\nwhere $|\\alpha|^{2}+|\\beta|^{2}=1$, or in the form of EPR-pair \n\\begin{equation}\n|\\Psi_{EPR}\\rangle = \\alpha |01\\rangle +\\beta |10\\rangle\n\\label{0002}\n\\end{equation}\nIt is possible to point eight states where three particles are maximally entangled. \nThey are\n\\begin{eqnarray}\n(|000\\rangle \\pm |111\\rangle )\/\\sqrt{2},&\\quad& (|001\\rangle \\pm |110\\rangle )\/\\sqrt{2},\n\\nonumber\n\\\\\n(|010\\rangle \\pm |101\\rangle )\/\\sqrt{2},&\\quad& (|100\\rangle \\pm |011\\rangle )\/\\sqrt{2}\n\\label{103}\n\\end{eqnarray}\nFrom the presented set of the initial states of particles without \nloss of generality we choose\n(\\ref{0002}) and triplet in the form of $GHZ$\n\\begin{equation}\n|\\Psi_{GHZ}\\rangle =\\frac{1}{\\sqrt{2}}(|000\\rangle +|111\\rangle )\n\\label{102}\n\\end{equation} \nNow we shall consider the combined system prepared initially \nin the state \n\\begin{equation}\n|\\Psi\\rangle =|\\Psi_{EPR}\\rangle \\otimes |\\Psi_{GHZ}\\rangle\n\\label{100}\n\\end{equation}\n\n\\begin{figure}\n\\unitlength=1.00mm\n\\special{em:linewidth 0.4pt}\n\\linethickness{0.4pt}\n\\begin{picture}(92.67,44.67)\n\\put(20.67,41.33){\\line(1,0){16.00}}\n\\put(20.67,35.00){\\line(1,0){15.67}}\n\\put(20.33,28.33){\\line(1,0){16.33}}\n\\put(37.33,41.33){\\circle{2.00}}\n\\put(37.00,35.00){\\circle{2.00}}\n\\put(37.33,28.33){\\circle{2.00}}\n\\put(32.67,24.33){\\framebox(9.67,20.33)[cc]{}}\n\\put(20.33,21.00){\\line(1,0){36.00}}\n\\put(56.33,21.00){\\line(0,0){0.00}}\n\\put(56.67,17.33){\\framebox(7.33,6.33)[cc]{B}}\n\\put(20.00,14.33){\\line(1,0){51.67}}\n\\put(71.33,10.67){\\framebox(7.33,6.33)[cc]{C}}\n\\put(64.33,21.00){\\line(1,0){28.00}}\n\\put(78.67,14.33){\\line(1,0){14.00}}\n\\put(42.67,34.33){\\rule{18.33\\unitlength}{1.00\\unitlength}}\n\\put(61.33,33.67){\\line(0,-1){10.00}}\n\\put(61.33,33.67){\\line(5,-2){14.00}}\n\\put(75.33,28.00){\\line(0,-1){11.00}}\n\\put(11.67,21.00){\\line(1,0){11.33}}\n\\put(11.33,21.33){\\line(4,3){9.33}}\n\\put(11.33,21.00){\\line(4,-3){9.00}}\n\\put(11.67,38.00){\\line(5,2){8.67}}\n\\put(12.00,38.00){\\line(3,-1){9.33}}\n\\put(61.33,34.67){\\circle*{2.67}}\n\\put(11.00,38.00){\\circle{2.00}}\n\\put(10.33,21.33){\\circle{2.00}}\n\\put(10.00,44.00){\\makebox(0,0)[cc]{$\\Psi_{EPR}$}}\n\\put(9.67,28.67){\\makebox(0,0)[cc]{GHZ}}\n\\put(27.00,44.33){\\makebox(0,0)[cc]{1}}\n\\put(27.00,38.33){\\makebox(0,0)[cc]{2}}\n\\put(27.00,31.33){\\makebox(0,0)[cc]{3}}\n\\put(27.00,24.67){\\makebox(0,0)[cc]{4}}\n\\put(26.67,17.33){\\makebox(0,0)[cc]{5}}\n\\put(46.00,44.67){\\makebox(0,0)[cc]{A}}\n\\put(87.67,27.00){\\makebox(0,0)[cc]{$\\Psi_{EPR}$}}\n\\end{picture}\n\\caption{Teleportation of EPR pair of qubits 1 and 2 using GHZ triplet. Alice, Bob and Claire\nshare qubits 3,4 and 5 of GHZ. Alice sends outcome of a joint measurement to \nBob and Claire who recover an unknown EPR-state.}\n\\end{figure}\nThe scheme to teleport an unknown state of EPR pair of \nqubit 1 and 2 with the use of the GHZ triplet of \nqubit 3,4 and 5 is presented in fig. 1. \nHere three parties spatially separated, Alice and two receivers \nBob and Claire share the GHZ particles 3,4 and 5. \nAlice sends outcomes of her joint measurement of qubits 1,2 \nand 3 to both receivers by classical channel. To perform the joint \nmeasurement it needs \neight projection operators to form a complete set\non which the initial wave function $|\\Psi\\rangle $\ncan be decomposed. The choice of such basis is the main moment in the solution \nof the problem. \n\n\\section{The projection basis}\n\nIt would be possible to imagine that the set of the projection \noperators $\\Pi_{k}$ for joint measurement will consist of the maximally\n entangled states (\\ref{103}). Let denote this basis as\n$\\pi_{(123)}$. However one can find that it is not true. The reason is \nthat in series expansion of the initial wave function $|\\Psi\\rangle $ \nits projections into four vectors \n$(|010\\rangle \\pm |101\\rangle)\/\\sqrt{2}$, \n$(|100\\rangle \\pm |011\\rangle)\/\\sqrt{2}$ of the basis $\\pi_{(123)}$\nare equal to zero.\nIt is impossible to recover unknown state of EPR pair by the such \noutcomes using unitary transformation. Therefore the maximally entangled tree- particle basis does not\nsolve the task. \n\nThe basis required turns out to be composed from the states \nwhere only two particle are maximally entangled, say 1,3 or 1,2.\nHowever the pair entangled is only the necessary condition.\n\nTo consider realization of operators $\\Pi_{k}$ we introduce classification\nwhere one of tag will be number of the particles to be maximally entangled, \nsay two or three in our case. As all complete set of vectors are \nconnected among themselves by unitary transformation \none can take an initial basis. Let the initial basis be $\\pi_{123}$\n\\begin{equation}\n|\\pi_{123}\\rangle =|ijk\\rangle \\quad i,j,k =0,1\n\\label{p123}\n\\end{equation}\nwhere each of eight elements is the state of three \nindependent or non-correlated particles. Any element of the other \nbasis can be presented by a linear superposition of\n$s\\leq 8$ vectors of the set $\\pi_{123}$. Further let suppose the \nnumber s be common for the given basis and we use it for \nclassification. So it can be introduced the set \n$\\pi_{1(23)}(s)$ consisting of the maximally entanglement of \ntwo particles 2 and 3. \nFor $s=2$ it has the form \n\\begin{equation}\n|\\pi_{1(23)}(2)\\rangle =\\{|i\\rangle |\\Phi^{\\pm}_{23}\\rangle ; \n|i\\rangle |\\Psi^{\\pm}_{23}\\rangle \\} \\quad i=0,1\n\\label{p}\n\\end{equation}\nwhere each of eight vectors, for example\n$|0\\rangle |\\Phi^{\\pm}_{23}\\rangle =(|000\\rangle \\pm |011\\rangle )\/\\sqrt{2}$, \nis presented by two elements of \n$\\pi_{123}$. \nFor the case $s=4$\n\\begin{equation}\n|\\pi_{1(23)}(4)\\rangle =\\{\n|\\pi_{1}^{\\pm}\\rangle |\\Phi^{\\pm}_{23}\\rangle ;\n|\\pi_{1}^{\\pm}\\rangle |\\Psi^{\\pm}_{23}\\rangle \\}\n\\label{pp}\n\\end{equation}\nwhere pair of vectors generating a complete single-particle set\nlooks like\n\\begin{equation}\n|\\pi_{1}^{\\pm}\\rangle =\\frac{1}{\\sqrt{2}}\n(|0\\rangle \\pm \\exp(i\\varphi)|1\\rangle )\n\\label{ppp}\n\\end{equation}\n\nThe sets presented here are complete and orthogonal, however the basis\n$\\pi_{1(23)}(2)$ does not solve the problem. The reason is that \nthe outcomes of the joint measurement depend on the wave function to be \nteleported so that the unitary transformation \nthat Bob and Claire have to perform \nat their qubits will depend on an unknown state. For base \n$\\pi_{(12)3}(4)$ the situation is similar to \n$\\pi_{(123)}(2)$, where half of projections of $\\Psi$ into the \nbasis states is equal to zero.\n\nFor telepoting EPR pair two sets are useful for which $s=4$. \nThere are $\\pi_{1(23)}(4)$ ore $\\pi_{(13)2}(4)$, \nwhere the particles 2,3 \nor 1,3 are maximally entangled. The structure of the initial \nstate, projection basis $\\pi_{1(23)}(4)$ and the total wave function \nare presented in fig 2. \n\\begin{figure}[ht]\n\\unitlength=1.00mm\n\\special{em:linewidth 0.4pt}\n\\linethickness{0.4pt}\n\\begin{picture}(130.00,52.00)\n\\put(11.00,43.00){\\circle{5.66}}\n\\put(11.00,31.00){\\circle{5.66}}\n\\put(24.00,29.00){\\circle{5.66}}\n\\put(41.00,37.00){\\circle{5.66}}\n\\put(41.00,17.00){\\circle{5.66}}\n\\put(11.00,43.00){\\makebox(0,0)[cc]{1}}\n\\put(11.00,31.00){\\makebox(0,0)[cc]{2}}\n\\put(24.00,29.00){\\makebox(0,0)[cc]{3}}\n\\put(41.00,37.00){\\makebox(0,0)[cc]{4}}\n\\put(41.00,17.00){\\makebox(0,0)[cc]{5}}\n\\put(3.00,8.00){\\framebox(28.00,44.00)[cc]{}}\n\\put(33.00,8.00){\\framebox(17.00,44.00)[cc]{}}\n\\put(40.00,20.00){\\line(0,1){14.00}}\n\\put(40.00,34.00){\\line(-3,-1){14.00}}\n\\put(59.00,44.00){\\circle{5.66}}\n\\put(59.00,30.00){\\circle{5.66}}\n\\put(72.00,30.00){\\circle{5.66}}\n\\put(59.00,44.00){\\makebox(0,0)[cc]{1}}\n\\put(59.00,30.00){\\makebox(0,0)[cc]{2}}\n\\put(72.00,30.00){\\makebox(0,0)[cc]{3}}\n\\put(91.00,42.00){\\circle{5.66}}\n\\put(91.00,29.00){\\circle{5.66}}\n\\put(104.00,28.00){\\circle{5.66}}\n\\put(121.00,36.00){\\circle{5.66}}\n\\put(121.00,16.00){\\circle{5.66}}\n\\put(91.00,42.00){\\makebox(0,0)[cc]{1}}\n\\put(91.00,29.00){\\makebox(0,0)[cc]{2}}\n\\put(104.00,28.00){\\makebox(0,0)[cc]{3}}\n\\put(121.00,36.00){\\makebox(0,0)[cc]{4}}\n\\put(121.00,16.00){\\makebox(0,0)[cc]{5}}\n\\put(83.00,7.00){\\framebox(28.00,44.00)[cc]{}}\n\\put(113.00,7.00){\\framebox(17.00,44.00)[cc]{}}\n\\put(121.00,19.00){\\line(0,1){14.00}}\n\\put(8.00,13.00){\\makebox(0,0)[cc]{A}}\n\\put(46.00,42.00){\\makebox(0,0)[cc]{B}}\n\\put(46.00,19.00){\\makebox(0,0)[cc]{C}}\n\\put(88.00,12.00){\\makebox(0,0)[cc]{A}}\n\\put(127.00,41.00){\\makebox(0,0)[cc]{B}}\n\\put(127.00,18.00){\\makebox(0,0)[cc]{C}}\n\\put(27.00,29.00){\\line(3,-2){13.00}}\n\\put(22.00,1.00){\\makebox(0,0)[cc]{a)}}\n\\put(68.00,1.00){\\makebox(0,0)[cc]{b)}}\n\\put(108.00,1.00){\\makebox(0,0)[cc]{c)}}\n\\put(54.00,25.00){\\framebox(24.00,25.00)[cc]{}}\n\\put(11.00,40.00){\\line(0,-1){6.00}}\n\\put(61.00,30.00){\\line(1,0){8.00}}\n\\put(94.00,28.00){\\line(1,0){7.00}}\n\\end{picture}\n\\caption{The entanglement structure of the states. \na) The initial state, where\nqubits of EPR pair 1,2 and qubits 3,4,5 of GHZ are entangled.\nb) Projection basis $\\pi_{1(23)}(4)$. \nc) The state after measuring.}\n\\end{figure}\n \n\\section{Teleportation of EPR pair}\nUsing the obtained set $\\pi_{1(23)}$, where we put \n$\\varphi =0$, the initial wave function can be rewritten as\n\\begin{eqnarray}\n|\\Psi\\rangle&=&\n|\\pi_{1}^{+}\\rangle |\\Phi^{+}_{23}\\rangle |1\\rangle +\n|\\pi_{1}^{+}\\rangle |\\Phi^{-}_{23}\\rangle |2\\rangle \n\\nonumber\n\\\\ \n&+&\n|\\pi_{1}^{-}\\rangle |\\Phi^{+}_{23}\\rangle |3\\rangle +\n|\\pi_{1}^{-}\\rangle |\\Phi^{-}_{23}\\rangle |4\\rangle \n\\nonumber\n\\\\\n&+&\n|\\pi_{1}^{+}\\rangle |\\Psi^{+}_{23}\\rangle |5\\rangle +\n|\\pi_{1}^{+}\\rangle |\\Psi^{-}_{23}\\rangle |6\\rangle \n\\nonumber\n\\\\\n&+&\n|\\pi_{1}^{-}\\rangle |\\Psi^{+}_{23}\\rangle |7\\rangle +\n|\\pi_{1}^{-}\\rangle |\\Psi^{-}_{23}\\rangle |8\\rangle \n\\label{201}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n|1,2\\rangle &=& \\beta|00\\rangle \\pm \\alpha |11\\rangle \n\\nonumber\n\\\\\n|3,4\\rangle &=&-(\\beta|00\\rangle \\mp \\alpha |11\\rangle )\n\\nonumber\n\\\\\n|5,6\\rangle &=&\\beta|11\\rangle \\pm \\alpha |00\\rangle \n\\nonumber\n\\\\\n|7,8\\rangle &=&-(\\beta|11\\rangle \\mp \\alpha |00\\rangle )\n\\label{203}\n\\end{eqnarray}\nthat for each outcome the reduced density matrix of qubit 4 and 5 \nIt follows from equation (\\ref{203}) \n$\\rho_{45}(k)=Sp_{123}\\{\\Pi_{k}|\\Psi\\rangle \\langle \\Psi|\\Pi_{k}\\}$ \nis connected to the density matrix of EPR pair by unitary transformation\n\\begin{equation}\n\\rho_{45}(k)=U_{k}|\\tilde\\Psi_{EPR}\\rangle \\langle \\tilde\\Psi_{EPR}|\nU^{\\dagger}_{k}\n\\quad k=1, \\dots 8\n\\label{204}\n\\end{equation}\nwhere \n$\\tilde\\Psi_{EPR}$ is the wave function of Hilbert space\n$H_{4}\\otimes H_{5}$ and counterpart of $\\Psi_{EPR}$.\nThe unitary operator from (\\ref{204}) has the form\n$U_{1}=\\sigma_{x4}\\otimes I_{5}$,\n$U_{2}=-U_{3}=i\\sigma_{y4}\\otimes I_{5}$, \n$U_{4}=-U_{1}$,\n$U_{5}=I_{4}\\otimes \\sigma_{x5}$, \n$U_{6}=-U_{7}= I_{4}\\otimes (-i\\sigma_{y5})$,\n$U_{8}=-U_{5}$. \nwhere the Pauli operators $\\sigma_{\\gamma j}$\n$\\gamma =x,y,z$\nand identity $I_{j}$ \naffect the particle $j=4,5$.\n\nTeleportation of an unknown EPR state can be reached by the following \nprotocol:\n\\begin{enumerate}\n\\item \nAlice performs the joint measurement of qubits 1,2 and 3 \nin basis $\\pi_{1(23)}(4)$ and sends her outcomes to Bob and Claire. \n\\item \nFor outcomes $k=1 - 4$ \nBob have to rotate his qubit by local operations\n$\\sigma_{x}$, $i\\sigma_{y}$, $-i\\sigma_{y}$, $-\\sigma_{x}$\nand Claire does nothing. As result Bob and Claire has EPR pair\nin the state $\\Psi_{EPR}$.\n\\item \nTo recover an unknown EPR state for outcomes $k=5 - 6$\nBob does nothing and Claire performs unitary transformation \n$\\sigma_{x}$, $-i\\sigma_{y}$, $i\\sigma_{y}$,\n$-\\sigma_{x}$ on her qubit.\n\\end{enumerate}\nIn the presented protocol in half of cases only one of receivers \naffects on his particle while another acts on his particle by \nunity operator or does nothing. This version is not unique \nbecause unknown state can be recovered by different way. \nFor instance, the wave vector \n$|2\\rangle $ from (\\ref{203}) \ncan be obtained by two ways \n\\begin{equation}\n\\beta |00\\rangle -\\alpha|11\\rangle \n=i\\sigma_{y4}\\otimes I_{5}\n|\\tilde\\Psi_{EPR}\\rangle \n=\\sigma_{x4}\\otimes \\sigma_{z5}\n|\\tilde\\Psi_{EPR}\\rangle \n\\label{205}\n\\end{equation}\nThe expression (\\ref{205}) means that for outcome \n$k=2$ both receivers Bob and Claire should simultaneously affect \non their qubits (as in the above protocol) by unitary operations \n$\\sigma_{x4}$ and $\\sigma_{z5}$ (instead of \n$i\\sigma_{y4}$ and identity operator).\nThese differences however do not change the result. The \nmain feature of the teleportation procedure considered here is \npresence of two receivers which one can not accomplish the \ntask separately.\n\\begin{figure}\n\\unitlength=1.00mm\n\\special{em:linewidth 0.4pt}\n\\linethickness{0.4pt}\n\\begin{picture}(147.67,91.00)\n\\put(7.67,78.00){\\line(1,0){20.33}}\n\\put(136.67,77.33){\\circle{5.37}}\n\\put(32.33,77.34){\\circle*{1.33}}\n\\put(32.99,62.34){\\circle{5.37}}\n\\put(7.33,62.67){\\line(1,0){18.33}}\n\\put(46.67,57.00){\\framebox(9.00,10.00)[cc]{H}}\n\\put(55.67,63.00){\\line(1,0){9.67}}\n\\put(65.00,57.33){\\framebox(9.00,10.00)[cc]{H}}\n\\put(7.33,47.67){\\line(1,0){9.67}}\n\\put(17.33,42.00){\\framebox(9.00,10.00)[cc]{H}}\n\\put(60.33,63.00){\\circle*{1.33}}\n\\put(60.33,47.67){\\circle{5.37}}\n\\put(26.33,47.67){\\line(1,0){31.33}}\n\\put(63.33,47.67){\\line(1,0){75.67}}\n\\put(74.00,62.67){\\line(1,0){65.33}}\n\\put(32.67,76.67){\\line(0,-1){12.00}}\n\\put(60.33,62.67){\\line(0,-1){12.67}}\n\\put(30.00,47.33){\\circle*{1.33}}\n\\put(40.00,47.67){\\circle*{1.33}}\n\\put(30.00,33.33){\\circle{5.37}}\n\\put(7.33,33.33){\\line(1,0){20.00}}\n\\put(40.00,17.67){\\circle{5.37}}\n\\put(7.00,17.67){\\line(1,0){30.33}}\n\\put(30.00,46.33){\\line(0,-1){10.33}}\n\\put(40.33,47.00){\\line(0,-1){26.67}}\n\\put(69.33,47.00){\\circle*{1.33}}\n\\put(69.33,33.33){\\circle{5.37}}\n\\put(32.67,33.33){\\line(1,0){34.00}}\n\\put(76.00,28.33){\\framebox(9.00,10.00)[cc]{H}}\n\\put(90.33,34.00){\\circle{5.37}}\n\\put(123.67,12.67){\\framebox(9.00,10.00)[cc]{H}}\n\\put(118.00,33.33){\\circle{5.37}}\n\\put(126.33,33.67){\\circle{5.37}}\n\\put(137.00,17.67){\\circle*{1.33}}\n\\put(118.33,17.67){\\circle*{1.33}}\n\\put(96.67,28.00){\\framebox(9.00,10.00)[cc]{H}}\n\\put(110.33,48.00){\\circle*{1.33}}\n\\put(110.33,17.33){\\circle{5.37}}\n\\put(128.00,78.00){\\line(1,0){6.00}}\n\\put(139.33,78.00){\\line(1,0){8.00}}\n\\put(139.33,62.67){\\line(1,0){7.67}}\n\\put(139.33,47.67){\\line(1,0){8.00}}\n\\put(147.33,47.67){\\line(0,0){0.00}}\n\\put(72.00,33.33){\\line(1,0){4.33}}\n\\put(85.00,33.33){\\line(1,0){3.33}}\n\\put(93.00,33.33){\\line(1,0){4.00}}\n\\put(105.67,33.33){\\line(1,0){9.33}}\n\\put(121.00,33.33){\\line(1,0){2.67}}\n\\put(129.33,33.33){\\line(1,0){18.33}}\n\\put(42.67,17.67){\\line(1,0){65.00}}\n\\put(112.67,17.67){\\line(1,0){5.33}}\n\\put(132.33,17.67){\\line(1,0){15.33}}\n\\put(119.00,17.67){\\line(1,0){4.67}}\n\\put(126.00,78.00){\\circle*{1.33}}\n\\put(90.33,62.67){\\circle*{1.33}}\n\\put(69.33,47.00){\\line(0,-1){11.33}}\n\\put(90.33,62.33){\\line(0,-1){25.33}}\n\\put(110.33,47.67){\\line(0,-1){27.33}}\n\\put(118.33,30.33){\\line(0,-1){13.00}}\n\\put(126.33,77.67){\\line(0,-1){41.33}}\n\\put(137.00,74.33){\\line(0,-1){56.67}}\n\\put(30.33,33.00){\\makebox(0,0)[cc]{$\\times$}}\n\\put(40.33,17.33){\\makebox(0,0)[cc]{$\\times$}}\n\\put(60.67,47.67){\\makebox(0,0)[cc]{$\\times$}}\n\\put(69.33,33.33){\\makebox(0,0)[cc]{$\\times$}}\n\\put(90.33,34.00){\\makebox(0,0)[cc]{$\\times$}}\n\\put(110.33,17.33){\\makebox(0,0)[cc]{$\\times$}}\n\\put(118.33,33.33){\\makebox(0,0)[cc]{$\\times$}}\n\\put(126.33,33.67){\\makebox(0,0)[cc]{$\\times$}}\n\\put(136.67,77.33){\\makebox(0,0)[cc]{$\\times$}}\n\\put(8.00,80.00){\\makebox(0,0)[lc]{$1$}}\n\\put(11.33,79.00){\\makebox(0,0)[lb]{$\\alpha|0\\rangle+\\beta|1\\rangle$}}\n\\put(27.67,78.00){\\line(1,0){98.67}}\n\\put(43.00,62.67){\\line(1,0){3.33}}\n\\put(7.67,65.33){\\makebox(0,0)[lc]{2}}\n\\put(7.67,50.33){\\makebox(0,0)[lc]{3}}\n\\put(7.67,36.00){\\makebox(0,0)[lc]{4}}\n\\put(7.67,20.33){\\makebox(0,0)[lc]{5}}\n\\put(14.00,66.67){\\makebox(0,0)[cb]{$|1\\rangle$}}\n\\put(12.67,51.33){\\makebox(0,0)[cb]{$|0\\rangle$}}\n\\put(13.00,36.67){\\makebox(0,0)[cb]{$|0\\rangle$}}\n\\put(13.33,20.67){\\makebox(0,0)[cb]{$|0\\rangle$}}\n\\put(142.67,81.67){\\makebox(0,0)[cb]{$|1\\rangle$}}\n\\put(147.34,66.34){\\makebox(0,0)[cb]{$|0\\rangle+|1\\rangle$}}\n\\put(147.67,51.01){\\makebox(0,0)[cb]{$|0\\rangle+|1\\rangle$}}\n\\put(24.67,62.67){\\line(1,0){5.67}}\n\\put(35.67,62.67){\\line(1,0){11.00}}\n\\put(32.67,62.33){\\makebox(0,0)[cc]{$\\times$}}\n\\put(5.00,57.67){\\framebox(33.00,33.33)[cc]{}}\n\\put(22.00,87.67){\\makebox(0,0)[cc]{$EPR$}}\n\\put(5.00,5.00){\\framebox(39.33,51.00)[cc]{}}\n\\put(20.00,9.00){\\makebox(0,0)[cc]{$GHZ$}}\n\\put(126.33,78.00){\\line(1,0){3.00}}\n\\put(146.00,26.00){\\makebox(0,0)[cc]{$|\\Psi_{EPR}\\rangle$}}\n\\end{picture}\n\\caption{Network for teleportation of EPR pair}\n\\end{figure}\n\nThe network presented in fig. 3 illustrates the teleportation \nprocedure of an EPR state. It is built similarly to the \none-particle teleportation \\cite{7}, \nand includes set of logical operations C-NOT (controlled- not) and \nHadamard transformation H. In the unit $EPR$ operation C-NOT $C_{12}$\nacting qubit 1,2 produces the entanglement of particles 1,2 of the \nform of EPR state.\n$C_{12}$ flips the second qubit (target) if and only if \nthe first (control) is logical 1. \nThe unit $GHZ$ prepares three-particle entanglement by the Hadamard \ntransformation H acting qubit 3 \n($H|0\\rangle=(|0\\rangle +|1\\rangle )\/\\sqrt{2}$, \n$H|1\\rangle=(|0\\rangle -|1\\rangle )\/\\sqrt{2})$ \nand two operations c-NOT \n$C_{34}, C_{35}$. On the end of the scheme the joint state of qubits \n4 and 5 being independent of others is turned out to be \n$\\Psi_{EPR}$. Indeed, the above network can be used for teleporting \nthe entangled pair of the form (\\ref{0001}).\n\nConsider the more general case of teleportation of the N-particle \nentanglement as EPR-nplet\n\\begin{equation}\n|\\Psi_{N}\\rangle =\\alpha |0\\rangle ^{N}\n+\\beta |1\\rangle ^{N}\n\\label{210}\n\\end{equation}\nusing $N+1$ qubits in the maximally entangled state as GHZ\n\\begin{equation}\n|\\Psi_{(N+1)}\\rangle = \\frac{1}{\\sqrt{2}}\n\\left(|0\\rangle ^{N+1} +\n|1\\rangle ^{N+1}\\right)\n\\label{211}\n\\end{equation} \nwhere $|i\\rangle ^{N}= |i\\rangle \\otimes \\dots|i\\rangle$, \n$i=0,1$.\nIn this procedure, that includes \n2N+1 qubits, a sender Alice and N receivers share the entanglement \nof the form (\\ref{211}).\nThe combined wave function defined in the Hilbert space \n$H_{1}\\otimes \\dots H_{2N+1}$ \nis product\n$|\\Psi\\rangle =|\\Psi_{N}\\rangle \\otimes |\\Psi_{(N+1)}\\rangle $. \n\nFor joint measuring of\nN+1 ($1,2,\\dots N,N+1$) particles it needs a complete set of \n$2^{N+1}$ projectors describing states of any maximally \nentangled pair $M,N+1$, $M=1,\\dots N$. \nFor $M=N$ \nthe required basis has the form\n\\begin{equation}\n\\pi_{1,...N-1(N,N+1)}(s)=\n\\{|\\pi_{1,...N-1}\\rangle \\otimes |\\Phi^{\\pm}_{N,N+1}\\rangle ;\n |\\pi_{1,...N-1}\\rangle \\otimes |\\Psi^{\\pm}_{N,N+1}\\rangle \\}\n\\label{212}\n\\end{equation}\nwhere the particles\n$N,N+1$ are entangled, \n$|\\pi_{1,...N-1}\\rangle $ are the vectors from \n$H_{1}\\otimes \\dots H_{N-1}$. As we note before the \npresence of the entangled pair is only \nthe necessary condition for the required basis. The sufficient \ncondition is magnitude of parameter s which one together with vectors \n$\\pi_{1,...N-1}$ \ncan be obtained by expanding the combined wave function over \nset of (\\ref{212}). It can be written as\n\\begin{eqnarray}\n|\\Psi \\rangle &=&\n\\{ P_{N-1} \\alpha |0\\rangle ^{N}\\pm Q_{N-1}\\beta |1\\rangle ^{N}\\}\n|\\pi_{1,...N-1}\\rangle |\\Phi^{\\pm}_{N,N+1}\\rangle\n\\nonumber\n\\\\\n&+&\n\\{ P_{N-1}\\alpha |1\\rangle ^{N}\\pm Q_{N-1}\\beta |0\\rangle ^{N}\\}\n|\\pi_{1,...N-1}\\rangle |\\Psi^{\\pm}_{N,N+1}\\rangle\n\\label{213}\n\\end{eqnarray}\nwhere\n\\begin{equation}\nP_{N-1}=\\langle \\pi_{1,...N-1}|0\\rangle ^{N-1}\n\\qquad\nQ_{N-1}=\\langle \\pi_{1,...N-1}|1\\rangle ^{N-1}\n\\label{214}\n\\end{equation}\nProcess teleportation needs the following condition \n\\begin{equation}\nP_{N-1}\\neq 0, Q_{N-1}\\neq 0\n\\label{215}\n\\end{equation}\nIt means that all terms of the series expansion of $\\Psi$ \nhave to involve a linear superposition of\n$\\alpha |i\\rangle ^{N}$ and $\\beta|j\\rangle ^{N}$, $i\\neq j=0,1$, \nwhich one can be retrieved from \n(\\ref{210}) by unitary transformation not dependent from an unknown \nstate. The following set of vectors obeys \n(\\ref{215})\n\\begin{equation}\n|\\pi_{1,...N-1}\\rangle =\\{|\\pi^{\\pm}_{1}\\rangle ^{N-1}\\}\n\\label{216}\n\\end{equation}\nwhere $\\pi^{\\pm}_{1}$ is the one-particle set defined by\n(\\ref{ppp}). It can be easily established, noting that the set \n(\\ref{216}) consists of \n$2^{N-1}$ elements each of which contains two terms \n$|i\\rangle ^{N-1}, i=0,1$. \n\nFor the obtained basis defined by \n(\\ref{212}) and (\\ref{216})\nthe value $s$ is equal to $2^{N}$. Note, that all cases with \n$s<2^{N}$, where there are bases inclusive more than one pair of \nthe entangled particles (two pairs or triplet) does not accomplish \nthe task. \n\nAs result, {\\em for teleporting an $N$-particle entangled state as EPR -nplet \nusing the $N+1$ particle entanglement it needs the set of the\nprojection operators with one pair of the maximally entangled particles. \nEach element of this set has to be presented by $2^{N}$ \nvectors corresponding the $N+1$-independent particle state}. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nAlong with the increasing demand for the quality of communication service, future wireless systems are required to support a peak rate of thousands of megabits per second and service density of hundreds of multi-antenna devices per square meter \\cite{ZBM20}. To this end, a large number of wireless terminals and base stations (BSs) will be deployed for greater data traffic \\cite{WSN17}. However, these ubiquitous communication services are often accompanied by great challenges, in not only technical implementations but also environmental considerations. On the one hand, the deployment of plenty of BSs with a large number of radio frequency (RF) chains will immensely increase the power consumption and impose a great burden on the shape design of antenna arrays \\cite{ZWX17,ZZW19}. On the other hand, the surge in the number of network connections engenders the substantial growth of electromagnetic (EM) radiation that is nonnegligible to the public health \\cite{JHB20}. \n\nTo reduce the hardware cost of deploying large-scale antennas at the BS, we resort to the dynamic metasurface antenna (DMA). DMA is proposed as a brand-new concept for the realization of antenna arrays, where metamaterials are used in the aperture antenna design \\cite{SAI21,YXA21}. A simple DMA-based array is composed of several microstrips in parallel, each of which consists of a set of subwavelength and frequency selective resonant metamaterial elements \\cite{HDM13}. They are capable of tailoring the beams and processing signals in a dynamically configurable way \\cite{SIX16}. The application of DMAs enables a large number of adjustable elements, which are reconfigurable owing to the introduction of solid-state switchable components in each metamaterial, to be set in a small physical area \\cite{SDE19}. In addition, the number of RF chains required for DMA assisted communication is equal to that of microstrips, which is usually far less than that required in conventional antenna systems \\cite{YXA21}. Hence, both the physical size and the power consumption will be greatly reduced.\nInstead of a passive configurable metasurface that only reflects the signals, DMA array performs as an active transceiver that inherently implements signal processing techniques such as analog beamforming and combining \\cite{WSE21}.\nThe flexible architecture of metasurface as an active antenna array makes DMAs attractive for multiple-input multiple-output (MIMO) transceivers in future wireless networks \\cite{SAI21}. \n\nHowever, the application of DMAs usually restricts the achievable system spectral efficiency (SE) due to the reduction of the RF chains at the BS. To compensate for this defect, adopting reconfigurable intelligent surface (RIS) is proposed as an effective method to improve the system SE with low hardware complexity \\cite{ZLP21}. \nUnlike DMAs which perform as active metamaterials equipped at BS, RIS is a two-dimensional metamaterial surface composed of ultra-thin composite material layers that can programmatically reflect the incident EM waves to the desired directions \\cite{LNT18,AZW20}.\nRIS contains a plurality of reflecting elements that are usually constituted by positive-intrinsic-negative diodes to tune the phase of the incident signal in a software defined manner \\cite{CAZ21}. With the reconfigurable intelligent property, RIS can superimpose the incident waves by adjusting the phase shifts and then reflect them to the appropriate directions, which provides excellent flexibility for cellular mobile communications with complex propagating scenes \\cite{MA21,CMH21,ZDS21}. Consequently, by optimizing the phase shifts of RIS reflecting elements in the system design, it is possible to enhance the designed signal power while suppressing interference so as to improve the system SE. In addition, with low hardware footprints, RIS assisted communication has become a valuable wireless transmission strategy in the next generation network \\cite{ZLP21,CAZ21}. \n\nAs for the concern that the public is vulnerable to the increasing EM radiation, EM exposure is quantified at the user side and specified at a low level by communication regulatory agencies, which calls for new transmission strategy designs for MIMO uplink \\cite{CLL20,C04}. \nEM exposure refers to the radiation exposure generated by the propagation of EM waves, which usually comes from the power electronic devices and various kinds of artificial and natural lights \\cite{JHB20}. Recently, the swift development of wireless networks and the gradual maturity of the Internet of Things technology have made EM exposure a critical issue \\cite{AGM15}. Therefore, many government departments require that the EM radiation emitted by qualified electronics be kept at a low dose. To this end, specific absorption rate (SAR), which denotes the absorbed power per unit mass of human tissues with the unit W\/kg, has become a standard metric to measure the exposure level of the public \\cite{C04}. According to the Federal Communications Commission (FCC) standard, for wireless devices with frequencies in the range of 100 kHz to 6 GHz, the peak SAR on partial-body EM exposure should be limited to 1.6 W\/kg \\cite{SARstandard}. \nAs a time-averaged quadratic metric, SAR mainly relates to the near-field of transmitting antenna in uplink communication, where the peak value of SAR is much higher than the average \\cite{HLY13}. Then, SAR is considered to comply with the worst-case. In single antenna cases, SAR can be naturally complied for the worst-case by reducing the transmit power. However, for multi-antenna systems, it is inefficient to ensure the worst-case compliance in the same way, which brings the demand for the transmitter adaptive design that is actively satisfying different SAR constraints \\cite{YLH15,JYW22}.\n\nIn the fifth-generation (5G) cellular systems, the prevalence of wireless handsets with multiple antennas has evoked the EM aware optimization design for the transmission SE. For example, authors in \\cite{HLY14} proposed the matrix constraint form of EM radiation in the uplink transmission design with multi-antenna user terminals. Then, the SAR constrained sum-rate maximization precoding at users was investigated in \\cite{YLH17} for the uplink multiuser MIMO systems. Recently, due to the proposal of the controllable intelligent radio environment \\cite{HHA20}, RIS and DMA have been paid enormous attention to the next generation wireless networks. In \\cite{YXH21,YXN21}, the phase shifts of RIS and transmit precoding at BS were jointly optimized to obtain the maximum energy efficiency (EE) and resource efficiency of the downlink multiuser MIMO system. In addition to the RIS assisted system, which can dynamically adjust the propagation environment, DMAs can be adopted at the BS to reduce the energy consumption and implementation cost of massive antenna arrays. The impact of DMAs on the capacity of wireless systems was investigated in \\cite{SDE19,YXA21}, where the corresponding weights of DMAs at the BS are optimized to maximize SE and EE, respectively.\nNote that although most studies focus on the power allocation algorithms of RIS or DMA assisted systems under power constraints, the introduction of EM exposure constraints poses new challenges to the optimization. Furthermore, the fast time-varying channel is a common scenario in wireless communication systems, where instantaneous channel state information (CSI) is difficult to obtain and becomes outdated easily \\cite{GJL09,WMJ13}. Compared with instantaneous CSI, the statistical CSI, e.g., the spatial correlation and channel mean, is more likely to be stable for a longer period, thus bringing the lower cost for acquisition. In this case, efficient utilization of statistical CSI is promising for transmission design.\nIn addition, the CSI is usually not perfectly available in practical systems, which might degrade the transmission performance. Robust transmission design which incorporates the imperfect CSI effect is of practical interest \\cite{NCN21}.\n\nIn this paper, we investigate the EM aware SE maximization design in RIS and DMA assisted multiuser MIMO uplink transmission. Specifically, the transmit precoding, RIS phases, and DMA weights are jointly designed to maximize the system SE under both power and SAR constraints at users. We consider the practical scenario where RIS and DMAs are statically deployed, and hence full CSI between RIS and DMA is available in the considered system. On the other hand, both full CSI and partial CSI cases from users to RIS are considered in the optimization design. We intend to figure out the impact of EM exposure on the SE performance of hybrid RIS and DMA assisted systems by comparing with conventional systems, then compare our proposed algorithm with the conventional backoff algorithms. The main contributions of this paper are outlined as follows:\n\n\n\\begin{itemize}\t\n\t\\item[$\\bullet$] We investigate the hybrid RIS and DMA assisted multiuser MIMO system for practical interest, where RIS and DMA are actually complementary in wireless transmissions. In particular, RIS is adopted to dynamically adjust the propagation environment. Meanwhile, DMA is adopted as a new form of BS antennas to reduce the energy consumption and implementation cost. \n\t\\item[$\\bullet$] \n\tThe SAR constraints are taken into account to protect users from the high dose of EM radiation in hybrid RIS and DMA assisted transmissions. To address the EM aware problem, we propose a modified water-filling algorithm to optimize the transmit covariances, apply the minimum mean square error (MMSE), block coordinate descent (BCD) and minorization-maximization (MM) methods to optimize RIS phase shifts in closed form, and design the DMA weights by approaching the performance of unconstrainted DMA problems.\n\t\\item[$\\bullet$] The partial CSI case is studied in the transmission scenario. To reduce the complexity of the Monte Carlo method in dealing with partial CSI, we apply the deterministic equivalent (DE) method to asymptotically approximate SE. Then, we propose the AO-based SE maximization algorithm with the utilization of the channel eigenmode coupling matrices from users to RIS.\n\\end{itemize}\n\nThe rest of the paper is organized as follows: \\secref{sec:system model} elaborates the model of RIS and DMA assisted system, specifies the representation of EM exposure, and formulates the problem of SE maximization for two cases of available CSI. Based on the AO framework, \\secref{sec:opt_SE_design_PCSI} provides SE maximization algorithms of the optimization variables separately under full CSI scenario. \\secref{sec:opt_SE_design_SCSI} address the same problems with partial CSI. \\secref{sec:analysis} analyzes the convergence and complexity of the overall AO-base algorithm. In \\secref{sec:numerical_results}, numerical results are presented to analyze the performance of the proposed algorithms. Lastly, \\secref{sec:conclusion} concludes the study.\n\n\\emph{Notations}: Suppose matrix $\\mathbf{A} = \\mathrm{diag}\\{\\mathbf{A}_k\\}_{k=1}^{K}$ is the block diagonal matrix composed of $K$ sub-matrices, and the element on the $k$th diagonal block sequenced from the upper left is $\\mathbf{A}_k$. $\\mathbf{A}_{[1:k]}$ is the matrix obtained by truncating the first $k$ column vectors of matrix $\\mathbf{A}$. $\\mathbf{A}_{m,n}$ denotes the element located in row $m$ and column $n$ of matrix $\\mathbf{A}$. $\\mathbb{E}\\{\\cdot\\}$ means calculating the expected value. $\\mathbf{0}$ denotes zero vector. $\\mathbb{C}$, $\\mathbb{R}$, $\\mathbb{R}^+$ represent complex, real and positive real number sets, respectively. $\\odot$ denotes the Hadamard product. $\\tr{\\cdot}$ means the trace. $(x)^+=\\max\\{x,0\\}$. $\\Re\\{\\cdot\\}$ means taking the real part of a complex number. $||\\cdot||_\\mathrm{F}^2$ is the Frobenius norm. $\\jmath=\\sqrt{-1}$ is the imaginary unit.\n\n\\section{System Model}\\label{sec:system model}\nConsider the hybrid RIS and DMA assisted multiuser MIMO uplink transmission with $K$ users in the single cell transmitting signals to a $M$-antenna BS simultaneously, as shown in \\figref{fig:RISDMAmodel}. Denote the user set as $\\mathcal{K}=\\{1,...,K\\}$, and the number of antennas for user $k\\in\\mathcal{K}$ is $N_k$. We assume that the encoded transmit signal from user $k$ is $\\mathbf{x}_k\\in\\mathbb{C}^{N_K\\times 1}$, which is zero mean and independent of signals from other users, i.e., $\\mathbb{E}\\{\\mathbf{x}_k\\}=\\mathbf{0}, \\forall k\\in\\mathcal{K}$ and $\\mathbb{E}\\{\\mathbf{x}_i\\mathbf{x}_j^H\\}=\\mathbf{0},\\forall i \\neq j \\in \\cal{K}$. Denote the covariance matrix corresponding transmit signal $\\mathbf{x}_k$ as $\\mathbf{Q}_k\\triangleq\\mathbb{E}\\{\\mathbf{x}_k\\mathbf{x}_k^H\\},\\forall k \\in \\cal{K}$. As the elements of $\\mathbf{x}_k$ are spatially correlated, $\\mathbf{Q}_k$ is essentially a non-diagnoal matrix. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{Visio-RIS-DMA-model.eps}\n\t\\caption{The hybrid RIS and DMA assisted multiuser MIMO system.}\n\t\\label{fig:RISDMAmodel}\n\\end{figure}\n\n\\subsection{RIS Assised Model}\nIn \\figref{fig:RISDMAmodel}, the signals from transmitters are reflected in the channel with the deployment of RIS consisting of $N_R$ reflecting elements, each of which can tune the phase of the incident signal separately by applying the programmable controller \\cite{CMH21, AZW20}. \nAs the enormous increase of the RF weakens the diffraction and scattering effect, electromagnetic waves are prone to blockage by obstacles such as buildings in urban areas \\cite{HZA19}. In this paper, we assume the typical model that the direct channel from users to the BS is blocked, i.e., only the paths that users to the BS via RIS are considered in our system \\cite{HZA19}.\nIn addition, the cases that signals experience multiple reflections by RIS are ignored due to the tremendous path loss \\cite{PRW20,AZW20}. We suppose the channel matrix from user $k$ to RIS as $\\bH_{2,k}\\in\\mathbb{C}^{N_R\\times N_k}$, and that from the RIS to BS as $\\mathbf{H}_1\\in\\mathbb{C}^{M\\times N_R}$. Then, the received signal gathered at the BS side can be formulized as\n\\begin{align}\\label{equ:received_y1}\n\\mathbf{y}=\\sum\\limits_{k=1}^{K}\\mathbf{H}_1\\bm{\\Phi}\\bH_{2,k}\\mathbf{x}_k+\\mathbf{n} \\in \\mathbb{C}^{M\\times1},\n\\end{align}\nwhere $\\mathbf{n}$ is the additive noise following $\\mathcal{CN}(0,\\sigma^{2}\\mathbf{I}_{M})$, and $\\bm{\\Phi}=\\mathrm{diag}\\{\\phi_1,...,\\phi_{N_R}\\}$ denotes the phase shift matrix of RIS, whose diagonal elements are the reflection coefficients.\n\nSuppose that RIS can achieve total reflection, which means for any $n\\in\\{1,...,N_R\\}$, the reflection coefficients can be written as $\\phi_n=\\mathrm{e}^{\\jmath\\theta_n}$, where $\\theta_n$ is the phase shift introduced by the $n$th element of RIS. In addition, we adopt the ideal assumption that the reflecting elements can perform continuous phase shift, i.e., \\cite{WLW20}\n\\begin{align}\\label{equ:phase shift}\n\\phi_n\\in\\mathcal{F}_1\\triangleq\\{\\phi|\\phi=\\mathrm{e}^{\\jmath\\theta},\\theta\\in\\left[\\right.0,2\\pi\\left.\\right)\\},\n\\end{align}\nwhere $\\mathcal{F}_1$ denotes the feasible set of the reflection coefficients.\n\n\\subsection{DMA Assised Model}\nSuppose that the DMA array is equipped at the BS consisting of $M$ metamaterial units. These DMAs are composed of $S$ microstrips, each of which contains $L$ metamaterial units, i.e., $M = S \\cdot L$. In practice, DMA array can be regarded as a two-dimensional antenna array composed of a set of one-dimensional microstrips, and its configurable weight matrix ${\\boldsymbol\\Xi}\\in\\mathbb{C}^{S\\times M}$ can be written as \\cite{SDE19}\n\\begin{align}\\label{equ:Xi_structure}\n{\\boldsymbol\\Xi}_{s_1,(s_2-1)L+l}=\n\\left\\{\n\\begin{array}{cccc}\n\\xi_{s_1,l}\\in \\mathcal{F}_2 &s_1=s_2 \\\\\n0 &s_1\\neq s_2 \\\\\n\\end{array}\n\\right. \\in \\mathcal{F}_3^{S\\times M},\n\\end{align}\nwhere $s_1,s_2\\in\\{1,...,S\\}$, $l\\in\\{1,...,L\\}$, $\\{\\xi_{s_1,l}\\}_{\\forall s_1,l}$ are the weights \\iffalse(gains)\\fi of the DMA elements and the feasible set of weight matrices is denoted as $\\mathcal{F}_3^{S\\times M}$. Please note that $\\xi_{s_1,l}$ often satisfies certain constraints, e.g., its feasible set represented by $\\mathcal{F}_2$ may have the following forms \\cite{SYM17}:\n\\begin{itemize}\n\t\\item[(1).] $\\mathcal{F}_2= \\mathbb{C}$ for unconstrained DMA weights.\n\t\\item[(2).] $\\mathcal{F}_2=[x,y]$ where $0}[rr]^{i} & & D \\ar@{->}[dl]^{j}\\\\\n\t\t& E \\ar@{->}[ul]^{k}&\n\t}\n\t$$\n\twhich is exact at each vertex.\n\\end{dfn}\n\nThe morphism $d$ defined as the composition $jk$ is a differential, i.e. $d^2=0$. Set $E'=\\ker(d)\/\\Ima(d)$ and $D'=\\Ima(i)$. By some standard arguments, one can define morphisms $j':D' \\rightarrow E'$ and $k':E' \\rightarrow D'$ respectively by $j'(i(x))=j(x)$ and $k'([y])=k(y)$ for any $x \\in D$ and $y \\in \\ker(d)$. The triangle obtained in this way\n$$\n\\xymatrix{\n\tD' \\ar@{->}[rr]^{i} & & D' \\ar@{->}[dl]^{j'}\\\\\n\t& E' \\ar@{->}[ul]^{k'}&\n}\n$$\nis again an exact couple, called the derived couple. Reiterating this construction, one gets a sequence of objects $E_r$ in $\\A$, endowed with differentials $d_r$, each of which is the homology of the previous one. More precisely, we can give the following definition.\n\n\\begin{dfn}\n\tThe sequence $\\{E_r,d_r\\}_{r \\geq 1}$ constructed inductively by\n\t$$E_r=\\ker(d_{r-1})\/\\Ima(d_{r-1})$$\n\tis called the spectral sequence associated to the exact couple in Definition \\ref{ec}.\n\\end{dfn}\n\nIn practical situations, one often encounters bigraded exact couples, which naturally give rise to bigraded spectral sequences $\\{E_r^{s,t},d_r^{s,t}\\}_{r \\geq 1}$. We now provide the definition of the limit page of a bigraded spectral sequence.\n\n\\begin{dfn}\n\tLet $\\{E_r^{s,t},d_r^{s,t}\\}_{r \\geq 1}$ be a bigraded spectral sequence and suppose there is an integer $r(s,t)$ such that $E_r^{s,t} \\cong E_{r(s,t)}^{s,t}$ for any $r \\geq r(s,t)$, then we say that the spectral sequence abuts to $E_{\\infty}^{s,t}=E_{r(s,t)}^{s,t}$.\n\\end{dfn}\n\nAt this point, let us recall some notions about filtrations and covergence of spectral sequences from \\cite{boardman}.\n\nAn increasing filtration of an object $G$ in $\\A$ is a diagram of the following shape \n$$\\dots \\hookrightarrow F^1 \\hookrightarrow F^2 \\hookrightarrow \\dots \\hookrightarrow F^m \\hookrightarrow F^{m+1} \\hookrightarrow \\dots \\hookrightarrow G.$$\n\n\\begin{dfn}\n\tThe increasing filtration $\\{F^m\\}_{m \\in \\Z}$ of $G$ is said to be:\\\\\n\t1) exhaustive if $G=\\varinjlim_m F^m$;\\\\\n\t2) Hausdorff if $\\varprojlim_m F^m=0$;\\\\\n\t3) complete if $\\varprojlim^1_m F^m=0$.\n\\end{dfn}\n\nIn practice, one is often interested in filtrations of graded objects. So, for any $F^s$ in the filtration we denote by $F^{s,t}$ its graded component in degree $t$.\n\n\\begin{dfn}\n\tA spectral sequence associated to an exact couple is called:\\\\\n\t1) weakly convergent to $G$ if there exists an increasing filtration of $G$ which is exhaustive and such that $E^{s,t}_{\\infty} \\cong F^{s,t}\/F^{s-1,t}$ for any $s$;\\\\\n\t2) convergent to $G$ if it is weakly convergent and the filtration of $G$ is Hausdorff;\\\\\n\t3) strongly convergent to $G$ if it is weakly convergent and the filtration of $G$ is complete Hausdorff.\n\\end{dfn}\n\nWe now recall the definition of Postnikov system in a triangulated category (see \\cite{gelfand.manin}). \n\n\\begin{dfn}\\label{ps}\n\tA Postnikov system for an object $X$ in $\\mathcal{C}$ is a diagram\n\t$$\n\t\\xymatrix{\n\t\t\\dots \\ar@{->}[r] & X_{i+1} \\ar@{->}[r] \\ar@{->}[d]\t &X_i \\ar@{->}[r] \\ar@{->}[d] & \\dots \\ar@{->}[r] & X_2 \\ar@{->}[r] \\ar@{->}[d]\t & X=X_1 \\ar@{->}[d] \\\\\n\t\t& Y_{i+1} \\ar@{->}[ul]^{[1]} &\tY_i \\ar@{->}[ul]^{[1]} & & Y_2 \\ar@{->}[ul]^{[1]} &\tY_1 \\ar@{->}[ul]^{[1]} \n\t}\n\t$$\n\twhere all the triangles are distinguished triangles in $\\mathcal{C}$.\n\\end{dfn}\n\nAssociated to a Postnikov system one can always construct an exact couple by applying a cohomological functor $H$. More precisely, we have the following bigraded exact couple\n$$\n\\xymatrix{\n\tD \\ar@{->}[rr]^{i} & & D \\ar@{->}[dl]^{j}\\\\\n\t& E \\ar@{->}[ul]^{k}&\n}\n$$\nwhere $D^{s,t}=H(X_s[-t])$, $E^{s,t}=H(Y_s[-t])$ and the morphisms $i:D^{s,t} \\rightarrow D^{s+1,t}$, $j:D^{s,t} \\rightarrow E^{s-1,t+1}$ and $k:E^{s,t} \\rightarrow D^{s,t}$ are induced by the morphisms in the Postnikov system.\n\nAs usual, we obtain an increasing filtration of the object $H(X)$ in $\\A$ given by \n$$F^1 \\hookrightarrow F^2 \\hookrightarrow \\dots \\hookrightarrow F^m \\hookrightarrow F^{m+1} \\hookrightarrow \\dots \\hookrightarrow H(X)$$\nwhere $F^m=\\ker(H(X) \\rightarrow H(X_m))$ and the morphism $H(X) \\rightarrow H(X_m)$ is the one induced by the Postnikov system. Moreover, observe that the filtration $\\{F^m\\}_{m \\geq 1}$ just introduced is complete Hausdorff, since it is bounded from below, but not necessarily exhaustive.\tAnyway, we have the following result that guarantees the strong convergence of the spectral sequence just constructed, provided that a certain condition holds.\n\n\\begin{thm}\\label{SC}\n\tIf $\\varinjlim_m H(X_m) \\cong 0$, then the spectral sequence associated to the Postnikov system in Definition \\ref{ps} is strongly convergent to $H(X)$.\n\\end{thm}\n\\begin{proof}\n\tSee \\cite[Theorem 6.1]{boardman}.\n\\end{proof}\n\n\\section{Motives over a bisimplicial scheme}\n\nFor technical reasons, in this paper we need to work over bisimplicial schemes. To this end, we need a triangulated category of motives over a bisimplicial scheme. The triangulated category of motives over a simplicial scheme was introduced and studied in \\cite{voevodsky.simplicial}. We point out that the constructions and results we need from \\cite{voevodsky.simplicial} extend to the bisimplicial case in a straightforward way. In this section we briefly summarise them. \n\nLet $Y_{\\bullet,\\bullet}$ be a smooth bisimplicial scheme over $k$. Following \\cite[Section 2]{voevodsky.simplicial}, we define $Sm\/Y_{\\bullet,\\bullet}$ in the following way.\n\n\\begin{dfn}\nDenote by $Sm\/Y_{\\bullet,\\bullet}$ the category whose objects are triples $(U,i,h)$, where $i$ and $h$ are non-negative integers and $U$ is a smooth scheme over $Y_{i,h}$, and whose morphisms from $(U,i,h)$ to $(V,j,k)$ are triples $(f,\\phi,\\psi)$, where $\\phi:[j] \\rightarrow [i]$ and $\\psi:[k] \\rightarrow [h]$ are simplicial maps and $f:U \\rightarrow V$ is a morphism of schemes such that the square\n$$\n\\xymatrix{\n\tU \\ar@{->}[r]^{f} \\ar@{->}[d] & V \\ar@{->}[d]\\\\\n\tY_{i,h} \\ar@{->}[r]_{Y_{\\phi,\\psi}} & Y_{j,k}\n}\n$$\ncommutes.\n\\end{dfn}\n\nWe can also define presheaves on $Y_{\\bullet,\\bullet}$ following \\cite[Definition 2.1]{voevodsky.simplicial}.\n\n\\begin{dfn}\n\tA presheaf of sets (respectively with transfers) on $Y_{\\bullet,\\bullet}$ consists of a collection $\\{F_{i,h}\\}_{i,h \\geq 0}$ of presheaves of sets (respectively with transfers) on $Sm\/Y_{i,h}$ together\n\twith a morphism of presheaves of sets (respectively with transfers) $F_{\\phi,\\psi}: Y_{\\phi,\\psi}^*(F_{j,k}) \\rightarrow F_{i,h}$ for any simplicial maps $\\phi:[j] \\rightarrow [i]$ and $\\psi:[k] \\rightarrow [h]$, such that $F_{id,id} =id$ and $F_{\\phi\\alpha,\\psi\\beta} :Y_{\\phi\\alpha,\\psi\\beta}^*(F_{m,n}) \\rightarrow F_{i,h}$ is equal to the composition of $Y_{\\phi,\\psi}^*F_{\\alpha,\\beta} : Y_{\\phi,\\psi}^*Y_{\\alpha,\\beta}^*(F_{m,n}) \\rightarrow Y_{\\phi,\\psi}^*(F_{j,k})$ and $F_{\\phi,\\psi} : Y_{\\phi,\\psi}^*(F_{j,k}) \\rightarrow F_{i,h}$, where $\\alpha : [m] \\rightarrow [j]$ and $\\beta : [n] \\rightarrow [k]$ are simplicial maps.\n\\end{dfn}\n\nDenote by $PShv(Y_{\\bullet,\\bullet})$ the category of presheaves of sets on $Y_{\\bullet,\\bullet}$ and by $PST(Y_{\\bullet,\\bullet},R)$ the abelian category of presheaves with transfers on $Y_{\\bullet,\\bullet}$.\nNote that $PShv(Y_{\\bullet,\\bullet})$ is nothing but the category of contravariant functors from $Sm\/Y_{\\bullet,\\bullet}$ to $Sets$.\n\nIf $F = \\{F_{i,h}\\}_{i,h \\geq 0}$ is a presheaf of sets on $Y_{\\bullet,\\bullet}$, then $R_{tr}F = \\{R_{tr}F_{i,h}\\}_{i,h \\geq 0}$ is a presheaf with transfers on $Y_{\\bullet,\\bullet}$. In particular, denote by $R_{tr}(U,i,h)$ the presheaf with transfers associated to the representable presheaf of sets corresponding to $(U,i,h)$.\n\nLet $SmCor(Y_{\\bullet,\\bullet},R)$ be the full subcategory of $PST(Y_{\\bullet,\\bullet},R)$ whose objects are direct sums of objects of the form $R_{tr}(U,i,h)$.\n\n\\begin{lem}\n\tThe category $PST(Y_{\\bullet,\\bullet},R)$ is naturally equivalent to the category of $R$-linear contravariant functors from $SmCor(Y_{\\bullet,\\bullet},R)$ to the category of $R$-modules that preserve coproducts.\n\\end{lem}\n\\begin{proof}\n\tSee \\cite[Lemma 2.3]{voevodsky.simplicial}.\n\t\\end{proof}\n\nThe previous result allows, as usual, to construct left resolutions $Lres(F)$ consisting of representable presheaves with transfers for any $F$ in $PST(Y_{\\bullet,\\bullet},R)$.\n\nFor any non-negative integers $i$ and $h$ denote by $r_{i,h} : SmCor(Y_{i,h},R) \\rightarrow SmCor(Y_{\\bullet,\\bullet},R)$ the functor that sends $U$ to $R_{tr}(U,i,h)$. These functors induce in the standard way pairs of adjoint functors\n\\begin{align*}\n\tPST&(Y_{\\bullet,\\bullet},R)\\\\\n\tr_{i,h,\\#} \\uparrow & \\downarrow r_{i,h}^*\\\\\n\tPST&(Y_{i,h},R).\n\\end{align*}\n\nThere are similar functors $r_{i,\\bullet} : SmCor(Y_{i,\\bullet},R) \\rightarrow SmCor(Y_{\\bullet,\\bullet},R)$ sending $R_{tr}(U,h)$ to $R_{tr}(U,i,h)$, which induce pairs of adjoint functors\n\\begin{align*}\n\tPST&(Y_{\\bullet,\\bullet},R)\\\\\n\tr_{i,\\bullet,\\#} \\uparrow & \\downarrow r_{i,\\bullet}^*\\\\\n\tPST&(Y_{i,\\bullet},R),\n\\end{align*}\n\nand $r_{\\bullet,h} : SmCor(Y_{\\bullet,h},R) \\rightarrow SmCor(Y_{\\bullet,\\bullet},R)$ sending $R_{tr}(U,i)$ to $R_{tr}(U,i,h)$, which induce pairs of adjoint functors\n\\begin{align*}\n\tPST&(Y_{\\bullet,\\bullet},R)\\\\\n\tr_{\\bullet,h,\\#} \\uparrow & \\downarrow r_{\\bullet,h}^*\\\\\n\tPST&(Y_{\\bullet,h},R).\n\\end{align*}\n\nFinally, we can also consider the diagonal functor $d : SmCor(d(Y_{\\bullet,\\bullet}),R) \\rightarrow SmCor(Y_{\\bullet,\\bullet},R)$ that sends $R_{tr}(U,i)$ to $R_{tr}(U,i,i)$. As usual, this functor induces a pair of adjoint functors\n\\begin{align*}\n\tPST&(Y_{\\bullet,\\bullet},R)\\\\\n\td_{\\#} \\uparrow & \\downarrow d^*\\\\\n\tPST&(d(Y_{\\bullet,\\bullet}),R).\n\\end{align*}\n\nAs in \\cite[Section 3]{voevodsky.simplicial}, we can define the tensor product of presheaves with transfers $F$ and $G$ on $Y_{\\bullet,\\bullet}$ in the following way\n$$(F \\otimes G)_{i,h} = (F_{i,h} \\otimes G_{i,h}) = h_0(Lres(F_{i,h}) \\otimes Lres(G_{i,h} )).$$\nLet $D(Y_{\\bullet,\\bullet},R)$ be the derived category of complexes on $PST(Y_{\\bullet,\\bullet},R)$ bounded from above. Then, the tensor product just defined induces a tensor triangulated structure on $D(Y_{\\bullet,\\bullet},R)$ given by\n$$K \\stackrel{L}{\\otimes} L = Lres(K) \\otimes Lres(L)$$\nfor all complexes of presheaves with transfers $K$ and $L$.\n\nThe unit of this tensor structure is the constant presheaf with transfers denoted also by $R$ whose components are the constant presheaves with transfers on each $Y_{i,h}$.\n\n\\begin{lem}\\label{tens}\n\tConsider the bisimplicial object $LR_{\\bullet,\\bullet}$ in $SmCor(Y_{\\bullet,\\bullet},R)$ with terms \n\t$$LR_{i,h} =R_{tr}(Y_{i,h},i,h)$$\n\tand the obvious structure morphisms. Let $LR_*$ be the total complex of the corresponding double complex. Then there is a natural quasi-isomorphism\n\t$$LR_* \\rightarrow R.$$\n\\end{lem}\n\\begin{proof}\n\tSee \\cite[Lemma 3.9]{voevodsky.simplicial}.\n\t\\end{proof}\n\nLet $W_{i,h}^{el}(Y_{\\bullet,\\bullet},R)$ be the class of complexes on $PST(Y_{\\bullet,\\bullet},R)$ obtained as $r_{i,h,\\#}(W^{el}(Y_{i,h},R))$, where $W^{el}(Y_{i,h},R)$ is defined in \\cite[Section 4]{voevodsky.simplicial}.\n\nDenote by $W(Y_{\\bullet,\\bullet},R)$ the smallest localizing subcategory of $D(Y_{\\bullet,\\bullet},R)$ containing all $W_{i,h}^{el}(Y_{\\bullet,\\bullet},R)$. A morphism in $D(Y_{\\bullet,\\bullet},R)$ is called an $A^1$-equivalence if its cone lives in $W(Y_{\\bullet,\\bullet},R)$.\n\n\\begin{dfn}\nThe triangulated category $\\DM^-_{eff}(Y_{\\bullet,\\bullet},R)$ of motives over $Y_{\\bullet,\\bullet}$ is the localization of $D(Y_{\\bullet,\\bullet},R)$ with respect to $A^1$-equivalences.\n\\end{dfn}\n\nWhat follows consists of a bunch of properties of the restriction functors whose simplicial analogues can be found in \\cite[Sections 3 and 4]{voevodsky.simplicial}.\n\nThe family $\\{r_{i,h}^*\\}_{i,h \\geq 0}$ induces a family of restriction functors from $D(Y_{\\bullet,\\bullet},R)$ to $D(Y_{i,h},R)$, with respective left adjoints $Lr_{i,h,{\\#}}$, that respect $A^1$-equivalences. Hence, we get a family of restriction functors $\\{r_{i,h}^*\\}_{i,h \\geq 0}$ from $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$ to $\\DM_{eff}^-(Y_{i,h},R)$ that is moreover conservative. The same is true also for the families of functors $\\{r_{i,\\bullet}^*\\}_{i \\geq 0}$ and $\\{r_{\\bullet,h}^*\\}_{h \\geq 0}$.\n\nThe diagonal functor $d^*$ also induces a functor from $D(Y_{\\bullet,\\bullet},R)$ to $D(d(Y_{\\bullet,\\bullet}),R)$, with left adjoint $Ld_{\\#}$, respecting $A^1$-equivalences. Therefore, we get a diagonal restriction functor $d^*$ from $\\DM_{eff}^-(Y_{\\bullet,\\bullet})$ to $\\DM_{eff}^-(d(Y_{\\bullet,\\bullet}))$.\n\nThe tensor product on $D(Y_{\\bullet,\\bullet},R)$ respects $A^1$-equivalences, making $\\DM^-_{eff}(Y_{\\bullet,\\bullet},R)$ into a tensor triangulated category. All the restriction functors introduced above respect this tensor structure.\n\n There are also standard functoriality properties. If $p:Y_{\\bullet,\\bullet} \\rightarrow Y'_{\\bullet,\\bullet}$ is a morphism of smooth bisimplicial schemes, then we get a pair of adjoint functors\n\\begin{align*}\n\t\\DM_{eff}^-&(Y_{\\bullet,\\bullet},R)\\\\\n\tLp^* \\uparrow & \\downarrow Rp_*\\\\\n\t\\DM_{eff}^-&(Y'_{\\bullet,\\bullet},R).\n\\end{align*}\nIf $p$ is smooth, then $Lp^*=p^*$ and there is also the following adjunction\n\\begin{align*}\n\t\\DM_{eff}^-&(Y_{\\bullet,\\bullet},R)\\\\\n\tLp_{\\#} \\downarrow & \\uparrow p^*\\\\\n\t\\DM_{eff}^-&(Y'_{\\bullet,\\bullet},R).\n\\end{align*}\nIn particular, we have a pair of adjoint functors\n\\begin{align*}\n\t\\DM_{eff}^-&(Y_{\\bullet,\\bullet},R)\\\\\n\tLc_{\\#} \\downarrow & \\uparrow c^*\\\\\n\t\\DM_{eff}^-&(k,R)\n\\end{align*}\nwhere $c:Y_{\\bullet,\\bullet} \\rightarrow Spec(k)$ is the projection to the base. \n\n\\begin{dfn}\n\tA Tate object $T(q)[p]$ in $\\DM^-_{eff}(Y_{\\bullet,\\bullet},R)$ is defined as $c^*(T(q)[p])$. In general, for any motive $M$ in $\\DM^-_{eff}(k,R)$, we also denote by $M$ its image $c^*(M)$ in $\\DM^-_{eff}(Y_{\\bullet,\\bullet},R)$.\n\\end{dfn}\n\nA smooth bisimplicial scheme $Y_{\\bullet,\\bullet}$ induces a bisimplicial object $R_{tr}(Y_{\\bullet,\\bullet})$ in $SmCor(k,R)$. Then, one can define the motive $M(Y_{\\bullet,\\bullet})$ of $Y_{\\bullet,\\bullet}$ in $\\DM^{-}_{eff}(k,R)$ as the total complex of the double complex $R_{tr}(Y_{*,*})$ associated to $R_{tr}(Y_{\\bullet,\\bullet})$. It is an immediate consequence of Lemma \\ref{tens} that $$M(Y_{\\bullet,\\bullet}) \\cong Lc_{\\#}T.$$\nThe latter definition naturally extends to the bisimplicial case the definition of the motive of a simplicial scheme given in \\cite[Section 5]{voevodsky.simplicial}. Note that, by the Eilenberg-Zilber theorem, $M(Y_{\\bullet,\\bullet})$ and $M(d(Y_{\\bullet,\\bullet}))$ are isomorphic in $\\DM^{-}_{eff}(k,R)$. In particular, they have the same motivic cohomology.\n\nThe most important result that we need in the following sections is the following.\n\n\\begin{prop}\n\tLet $Y_{\\bullet,\\bullet}$ be a smooth bisimplicial scheme. Then, there is an isomorphism\n\t$$\\Hom_{\\DM^{-}_{eff}(Y_{\\bullet,\\bullet},R)}(T(q')[p'],T(q)[p]) \\cong \\Hom_{\\DM^{-}_{eff}(k,R)}(M(Y_{\\bullet,\\bullet})(q')[p'],T(q)[p])$$\n\tfor all integers $q$, $q'$, $p$ and $p'$.\n\\end{prop}\n\\begin{proof}\n\tSee \\cite[Proposition 5.3]{voevodsky.simplicial}.\n\t\\end{proof}\n\n\\section{A Serre spectral sequence for motivic cohomology} \n\nThe main purpose of this section is to construct Postnikov systems in a suitable triangulated category of motives and to study the associated spectral sequences. More precisely, we set our triangulated category $\\mathcal{C}$ to be $\\DM^-_{eff}(Y_{\\bullet,\\bullet},R)$, our abelian category $\\A$ to be the category of $H^{**}(Y_{\\bullet,\\bullet},R)$-modules and our cohomological functor $H$ to be motivic cohomology $H^{**}(-,R)$.\n\nFor all $i \\geq 0$ denote simply by\n$$r_i^*:\\DM_{eff}^-(Y_{\\bullet,\\bullet},R) \\rightarrow \\DM_{eff}^-(Y_{i,\\bullet},R)$$\nthe restriction functors $r_{i,\\bullet}^*$ introduced in the last section, and by $Lr_{i,\\#}$ the respective left adjoint functors. The image of a motive $N$ in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$ under $r_i^*$ is simply denoted by $N_i$.\n\nNow let us recall some facts about coherence from \\cite{smirnov.vishik} and adapt them to the bisimplicial case we are interested in. \n\n\\begin{dfn} \n\tA smooth coherent morphism of smooth bisimplicial schemes is a smooth morphism $\\pi:X_{\\bullet,\\bullet} \\rightarrow Y_{\\bullet,\\bullet}$ such that there is a cartesian square of simplicial schemes\n\t$$\n\t\\xymatrix{\n\t\tX_{j,\\bullet} \\ar@{->}[r]^{\\pi_j} \\ar@{->}[d]_{X_ {\\theta}} & Y_{j,\\bullet} \\ar@{->}[d]^{Y_ {\\theta} }\\\\\n\t\tX_{i,\\bullet} \\ar@{->}[r]_{\\pi_{i}} & Y_{i,\\bullet} \n\t}\n\t$$\n\tfor any simplicial map $\\theta:[i] \\rightarrow [j]$.\n\\end{dfn} \n\n\\begin{dfn}\n\tA motive $N$ in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$ is said to be coherent if all simplicial morphisms $\\theta:[i] \\rightarrow [j]$ induce structural isomorphisms $N_ \\theta :LY_ {\\theta} ^*(N_i) \\rightarrow N_j$ in $\\DM_{eff}^-(Y_{j,\\bullet},R)$. The full subcategory of $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$ consisting of coherent motives is denoted by $\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$.\n\\end{dfn}\n\n\\begin{rem}\\label{cohloc}\n\t\\normalfont\n Note that $\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$ is a localizing subcategory of $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$. Since $L\\pi_{\\#}$ maps coherent motives to coherent ones for any smooth coherent morphism $\\pi$, we have that $M(X_{\\bullet,\\bullet} \\xrightarrow{\\pi} Y_{\\bullet,\\bullet})$ is an object in $\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$, where $M(X_{\\bullet,\\bullet} \\xrightarrow{\\pi} Y_{\\bullet,\\bullet})$ is the image $L\\pi_{\\#}(T)$ of the unit Tate motive in $\\DM_{eff}^-(X_{\\bullet,\\bullet},R)$.\n\\end{rem}\n\n\\begin{prop}\\label{filtr}\n\tFor any motive $N$ in $\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$ there exists a functorial increasing filtration \n\t$$(N)_{\\leq 0} \\rightarrow (N)_{\\leq 1} \\rightarrow \\dots \\rightarrow (N)_{\\leq n-1} \\rightarrow (N)_{\\leq n} \\rightarrow \\dots \\rightarrow N$$\n\twith graded pieces $(N)_n=Cone((N)_{\\leq n-1} \\rightarrow (N)_{\\leq n}) \\cong Lr_{n,\\#}r^*_n(N)[n]$ which converges in the sense that\n\t$$\\bigoplus_n (N)_{\\leq n} \\xrightarrow{id-sh} \\bigoplus_n (N)_{\\leq n} \\rightarrow N$$\n\textends to a distinguished triangle, where $sh:(N)_{\\leq n-1} \\rightarrow (N)_{\\leq n}$ is the map from the filtration.\n\\end{prop}\n\\begin{proof}\n\tThe proofs of \\cite[Propositions 3.1.6 and 3.1.8]{smirnov.vishik} extend verbatim to the bisimplicial case. \n\\end{proof}\n\nThe next proposition is a generalisation of \\cite[Proposition 3.1.5]{smirnov.vishik}. Indeed, it allows to construct Postnikov systems for coherent motives with simplicial components which are direct sums of Tate motives satisfying some specific conditions. The proof follows the guidelines of \\cite[Proposition 3.1.5]{smirnov.vishik} and essentially reproduces the same arguments in our more general context. Before proceeding, we need to define a strict order relation on the bidegrees $(q)[p]$.\n\n\\begin{dfn}\n\tWe set $(q)[p] \\prec (q')[p']$ if and only if one of the following two conditions is satisfied:\\\\\n\t1) $q}[r] & N^{j+1} \\ar@{->}[r] \\ar@{->}[d]\t &N^j \\ar@{->}[r] \\ar@{->}[d] & \\dots \\ar@{->}[r] & N^1 \\ar@{->}[r] \\ar@{->}[d]\t &N=N^0 \\ar@{->}[d] \\\\\n\t\t& T^{j+1} \\ar@{->}[ul]^{[1]} &\tT^j \\ar@{->}[ul]^{[1]} & & T^1 \\ar@{->}[ul]^{[1]} &\tT^0 \\ar@{->}[ul]^{[1]} & \n\t}\n\t$$\n\tsuch that the simplicial components $N^j_i$ are isomorphic to the direct sum $\\bigoplus_{k \\geq j} T^k$ and the morphisms $r_i^*(N^j \\rightarrow T^j)$ are the natural projections $\\bigoplus_{k \\geq j} T^k \\rightarrow T^j$ in $\\DM_{eff}^-(Y_{i,\\bullet},R)$.\n\\end{prop}\n\\begin{proof}\n\tTo construct the aimed Postnikov system we just need to produce morphisms $N^j \\rightarrow T^j$ where each $N^j$ is defined as the cone of the previous morphism, namely $N^j=Cone(N^{j-1} \\rightarrow T^{j-1})[-1]$. We proceed by induction.\n\t\n\tNotice that each simplicial component of $N$ is isomorphic to $\\bigoplus_{j \\geq 0} T^j$ and $T^0$ is the direct sum of possibly infinite $T(q_0)[p_0]$ such that $(q_0)[p_0] \\prec (q_j)[p_j]$ for any $j \\geq 1$ by hypothesis. By applying the triangulated functor $Lc_{\\#}$ to the filtration of Proposition \\ref{filtr}, one gets a filtration $(Lc_{\\#}N)_{\\leq n}$ for $Lc_{\\#}N$ with graded pieces $(Lc_{\\#}N)_n \\cong \\bigoplus_{j \\geq 0} \\bigoplus_{I_j} M(Y_{n,\\bullet})(q_j)[p_j+n]$. Following the lines of the proof of \\cite[Proposition 3.1.5]{smirnov.vishik}, we denote by $(Lc_{\\#}N)_{>n}$ the cone $Cone((Lc_{\\#}N)_{\\leq n} \\rightarrow Lc_{\\#}N)$ and by $(Lc_{\\#}N)_{m \\geq *>n}$ the cone $Cone((Lc_{\\#}N)_{\\leq n} \\rightarrow (Lc_{\\#}N)_{\\leq m})$ for any $m>n$. Now, note that \n\t$$(Lc_{\\#}N)_{>n} \\cong Cone(\\bigoplus_{m>n}(Lc_{\\#}N)_{m \\geq *>n} \\xrightarrow{id-sh} (Lc_{\\#}N)_{m \\geq *>n} )$$ \n\tand moreover $(Lc_{\\#}N)_{m \\geq *>n}$ is an extension of $(Lc_{\\#}N)_k$ for $n0},T^0) \\cong 0,$$\n\t$$\\Hom_{\\DM^{-}_{eff}(k,R)}((Lc_{\\#}N)_{>1},T^0) \\cong 0,$$\n\t$$\\Hom_{\\DM^{-}_{eff}(k,R)}((Lc_{\\#}N)_{>1},T^0[1]) \\cong 0,$$\n\tsince $\\Hom_{\\DM^{-}_{eff}(k,R)}(M(Y_{n,\\bullet}),T(q)[p]) \\cong 0$ for any $n \\geq 0$ and any $(q)[p] \\prec (0)[0]$. We deduce from these remarks and by applying the cohomological functor $\\Hom_{\\DM^{-}_{eff}(k,R)}(-,T^0)$ to the distinguished triangle\n\t$$(Lc_{\\#}N)_0 \\rightarrow Lc_{\\#}N \\rightarrow (Lc_{\\#}N)_{>0} \\rightarrow (Lc_{\\#}N)_0[1]$$\n\tthat there exists an exact sequence\n\t$$0 \\rightarrow \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#}(N),T^0) \\rightarrow \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#,0}(N_0),T^0)$$\n\t$$\\rightarrow \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#,1}(N_1),T^0).$$\n\tRepeating the same arguments for $T^0$ in $\\DM_{coh}^-(Y_{\\bullet,\\bullet},R)$ one gets a similar sequence \n\t$$0 \\rightarrow \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#}(T^0),T^0) \\rightarrow \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#,0}(T^0),T^0)$$\n\t$$\\rightarrow \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#,1}(T^0),T^0).$$\n\tIn order to produce an isomorphism between $\\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#}(N),T^0)$ and $\\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#}(T^0),T^0)$ we need to identify the last morphisms of the two exact sequences. Since in the exact sequences only the $0$th and the $1$st simplicial components appear, it is enough to get a compatibility between the coherent system $(N_i,N_{\\theta})$ and the one of $T^0$ for $i=0,1$ and simplicial maps $\\theta:[0] \\rightarrow [1]$, where $N_{\\theta}$ is the structural isomorphism $LY^*_{\\theta}(N_0) \\rightarrow N_1$ in $\\DM_{eff}^-(Y_{1,\\bullet},R)$. In other words, we want a commutative diagram\n\t$$\n\t\\xymatrix{\n\t\tN_1 \\cong \\bigoplus_{j \\geq 0} T^j \\ar@{->}[r]^{\\omega^N} \\ar@{->}[d] & N_1 \\cong \\bigoplus_{j \\geq 0} T^j \\ar@{->}[d]\\\\\n\t\tT^0 \\ar@{->}[r]_{id} & T^0.\n\t}\n\t$$\n Indeed, we have such a commutative diagram since by hypothesis $\\omega^N_0$ is trivial. Hence, the two exact sequences above coincide. Then, we have that\n$$\\Hom_{\\DM^{-}_{eff}(Y_{\\bullet,\\bullet},R)}(N,T^0) \\cong \\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#}(N),T^0) \\cong$$\n$$\\Hom_{\\DM^{-}_{eff}(k,R)}(Lc_{\\#}(T^0),T^0) \\cong \\Hom_{\\DM^{-}_{eff}(Y_{\\bullet,\\bullet},R)}(T^0,T^0)$$\nby adjunctions and the identity of $T^0$ provides the pursued morphism $N \\rightarrow T^0$ whose restriction on each simplicial component is given by the natural projection $\\bigoplus_{k \\geq 0}T^k \\rightarrow T^0$. It follows that $N^1_i$ is isomorphic to $\\bigoplus_{k \\geq 1}T^k$ for any $i$. This proves the induction basis.\n\t\nNow, suppose we have a morphism from $N^k$ to $T^k$ for any $0 \\leq k \\leq j-1$, where each $N^k$ is defined as $Cone(N^{k-1} \\rightarrow T^{k-1})[-1]$. We denote by $N^j$ the cone $Cone(N^{j-1} \\rightarrow T^{j-1})[-1]$. Notice that the simplicial components of $N^j$ are all isomorphic to $\\bigoplus_{l \\geq j} T^l$ and $T^j$ is the direct sum of possibly infinite $T(q_j)[p_j]$ such that $(q_j)[p_j] \\prec (q_l)[p_l]$ for any $l \\geq j+1$ by hypothesis. Therefore, by applying the same arguments of the induction basis to $N^j$, using the fact that $\\omega^{N^j}_0=\\omega^N_j$ is trivial by hypothesis, there exists a morphism $N^j \\rightarrow T^j$. This completes the proof.\n\\end{proof}\n\n\\begin{rem}\n\t\\normalfont\n\tWe point out that, in the case it exists an integer $k$ such that $I_j$ is empty for all $j \\geq k$, the previous result provides a finite Postnikov system for $N$ in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$.\n\\end{rem}\n\nWe want to apply Proposition \\ref{post} to produce a spectral sequence for morphisms having motivically cellular fibers, i.e. fibers whose motives are direct sums of Tate motives satisfying certain conditions. First, we need to construct suitable Postnikov systems. The next result is a generalisation of \\cite[Proposition 4.2]{tanania.b}.\n\n\\begin{prop} \\label{Serre}\n\tLet $\\pi:X_{\\bullet,\\bullet} \\rightarrow Y_{\\bullet,\\bullet}$ be a smooth coherent morphism of smooth bisimplicial schemes over $k$, $A_{\\bullet}$ a smooth simplicial $k$-scheme and, for any $j \\geq 0$, $T^j$ the possibly infinite direct sum of Tate motives $\\bigoplus_{I_j} T(q_j)[p_j]$ in $\\DM_{eff}^-(k,R)$ such that $(q_j)[p_j] \\prec (q_{j+1})[p_{j+1}]$. Moreover, suppose the following conditions hold:\\\\\n\t1) over the $0$th simplicial component $\\pi$ is isomorphic to the projection $Y_{0,\\bullet} \\times A_{\\bullet} \\rightarrow Y_{0,\\bullet}$;\\\\\n\t2) $\\omega^{M(X_{\\bullet,\\bullet} \\rightarrow Y_{\\bullet,\\bullet})}_j$ is trivial for any $j \\geq 0$;\\\\\n\t3) $M(A_{\\bullet}) \\cong \\bigoplus_{j \\geq 0} T^j \\in \\DM_{eff}^-(k,R)$.\\\\\n\tThen, there exists a Postnikov system in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$\n\t$$\n\t\\xymatrix{\n\t\t\\dots \\ar@{->}[r] & N^{j+1} \\ar@{->}[r] \\ar@{->}[d]\t &N^j \\ar@{->}[r] \\ar@{->}[d] & \\dots \\ar@{->}[r] & N^2 \\ar@{->}[r] \\ar@{->}[d]\t &N^1 \\ar@{->}[r] \\ar@{->}[d] &M(X_{\\bullet,\\bullet} \\xrightarrow{\\pi} Y_{\\bullet,\\bullet})=N^0 \\ar@{->}[d]\\\\\n\t\t& T^{j+1} \\ar@{->}[ul]^{[1]} &\tT^j \\ar@{->}[ul]^{[1]} & & T^2 \\ar@{->}[ul]^{[1]} &\tT^1 \\ar@{->}[ul]^{[1]} & T^0 \\ar@{->}[ul]^{[1]}\n\t}\n\t$$\n\tsuch that the simplicial components $N^j_i$ are isomorphic to the direct sum $\\bigoplus_{k \\geq j} T^k$ and the morphisms $r_i^*(N^j \\rightarrow T^j)$ are the natural projections $\\bigoplus_{k \\geq j} T^k \\rightarrow T^j$ in $\\DM_{eff}^-(Y_{i,\\bullet},R)$.\n\\end{prop}\n\\begin{proof}\n\tBy coherence of $\\pi$, we have that $\\pi_i:Y_{i,\\bullet} \\times A_{\\bullet} \\cong X_{i,\\bullet} \\rightarrow Y_{i,\\bullet}$ is the projection onto the first factor for any $i$. It follows that the coherent motive $N^0$ (see Remark \\ref{cohloc}) has simplicial components given by $N^0_i \\cong M(A_{\\bullet})$ in $\\DM_{eff}^-(Y_{i,\\bullet},R)$ for any $i$. Therefore, Proposition \\ref{post} implies the existence of the aimed Postnikov system in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$, and the proof is complete. \n\\end{proof} \n\nRecall from Section \\ref{spec} that, once constructed a Postnikov system in a triangulated category and considered a suitable cohomological functor, one can obtain a spectral sequence which may converge if some extra requirements are met. The following theorem just states the existence of a strongly convergent spectral sequence related to the Postnikov system of Proposition \\ref{Serre}.\n\n\\begin{thm}\\label{Serre2}\n\tLet $\\pi:X_{\\bullet,\\bullet} \\rightarrow Y_{\\bullet,\\bullet}$ be a smooth coherent morphism of smooth bisimplicial schemes over $k$ and $A_{\\bullet}$ a smooth simplicial $k$-scheme satisfying all conditions of Proposition \\ref{Serre}. Moreover, for any bidegree $(q)[p]$, suppose there is an integer $l$ such that $(q)[p] \\prec (q_l)[p_l]$. Then, there exists a strongly convergent spectral sequence\n\t$$E_1^{p,q,s}=\\prod_{I_s} H^{p-p_s,q-q_s}(Y_{\\bullet,\\bullet},R) \\Longrightarrow H^{p,q}(X_{\\bullet,\\bullet},R).$$\n\\end{thm}\n\\begin{proof}\n\tWe start by applying the construction of the exact couple associated to a Postnikov system of Section \\ref{spec} to the cohomological functor $\\Hom_{\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)}(-,T(q))$, for any $q$. This way, we get a spectral sequence with $E_1$-page given by \n\t$$E_1^{p,q,s}=\\Hom_{\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)}(T^s,T(q)[p]) \\cong \\prod_{I_s} H^{p-p_s,q-q_s}(Y_{\\bullet,\\bullet},R).$$\n\tThe filtration we are considering is defined by $F^m=\\ker(H^{**}(X_{\\bullet,\\bullet},R) \\rightarrow H^{**}(N^m,R))$. In order to get the strong convergence we need to check that $\\varinjlim_m H^{**}(N^m,R) \\cong 0$. Since all the $N^m$ are coherent motives, by Proposition \\ref{filtr} we have filtrations $(N^m)_{\\leq n}$ with graded pieces $(N^m)_n \\cong Lr_{n,\\#}r^*_n(N^m)[n]$. Hence, we have filtrations $(Lc_{\\#}N^m)_{\\leq n}$ with graded pieces\n\t$$(Lc_{\\#}N^m)_n \\cong \\bigoplus_{k \\geq m} \\bigoplus_{I_k} M(Y_{n,\\bullet})(q_k)[p_k+n].$$ \n\tNow, fix a bidegree $(q)[p]$, then by hypothesis there exists an integer $l$ such that $(q)[p] \\prec (q_l)[p_l]$, from which it follows that\n\t$$\\Hom_{\\DM_{eff}^-(k,R)}((Lc_{\\#}N^l)_n,T(q)[p]) \\cong 0$$\n\tfor any $n$. Therefore,\n\t$$\\Hom_{\\DM_{eff}^-(k,R)}(Lc_{\\#}(N^l),T(q)[p]) \\cong 0$$ \n\tfrom which we deduce by adjunction that $H^{p,q}(N^l,R) \\cong 0$ that implies, in particular, the triviality of $\\varinjlim_m H^{**}(N^m,R)$. Hence, by Theorem \\ref{SC} we obtain the result.\n\\end{proof}\n\nThe next result assures that the spectral sequence just constructed is functorial.\n\n\\begin{prop}\\label{Serre 3}\n\tLet $\\pi:X_{\\bullet,\\bullet} \\rightarrow Y_{\\bullet,\\bullet}$ and $\\pi':X'_{\\bullet,\\bullet} \\rightarrow Y'_{\\bullet,\\bullet}$ be smooth coherent morphisms of smooth bisimplicial schemes over $k$ and $A_{\\bullet}$ a smooth simplicial $k$-scheme that satisfies all conditions from Proposition \\ref{Serre} with respect to $\\pi'$ and such that the following square is cartesian with all morphisms smooth\n\t$$\n\t\\xymatrix{\n\t\tX_{\\bullet,\\bullet} \\ar@{->}[r]^{\\pi} \\ar@{->}[d]_{p_X} & Y_{\\bullet,\\bullet} \\ar@{->}[d]^{p_Y}\\\\\n\t\tX'_{\\bullet,\\bullet} \\ar@{->}[r]_{\\pi'} & Y'_{\\bullet,\\bullet}\n\t}\n\t$$\n\tThen, the induced square of motives in the category $\\DM_{eff}^-(Y'_{\\bullet,\\bullet},R)$ extends uniquely to a morphism of Postnikov systems where, for any $j \\geq 0$, $Lp_{Y\\#}T^j \\rightarrow T^j$ is given by $\\bigoplus_{I_j} M(p_Y)(q_j)[p_j]$.\n\\end{prop}\n\\begin{proof}\n\tWe denote by $N^j$ the objects from the Postnikov system of $\\pi$ and by $N'^j$ the ones from the Postnikov system of $\\pi'$. \n\t\n\tFirst, recall that, by Proposition \\ref{filtr}, there is a filtration of $Lc_{\\#}N^j$ with graded pieces \n\t$$(Lc_{\\#}N^j)_n \\cong \\bigoplus_{k \\geq j} \\bigoplus_{I_k} M(Y_{n,\\bullet})(q_k)[p_k+n].$$ \n\tIt follows that $\\Hom_{\\DM_{eff}^-(k,R)}((Lc_{\\#}N^j)_n,T^{j-1}[-1]) \\cong 0$ for any $n$ since, for any $k \\geq j$, we have that $(q_{j-1})[p_{j-1}] \\prec (q_k)[p_k]$ by hypothesis. Therefore, \n\t$$\\Hom_{\\DM_{eff}^-(Y'_{\\bullet,\\bullet},R)}(Lp_{Y\\#}N^j,T^{j-1}[-1]) \\cong \\Hom_{\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)}(N^j,T^{j-1}[-1]) \\cong$$\n\t$$\\Hom_{\\DM_{eff}^-(k,R)}(Lc_{\\#}N^j,T^{j-1}[-1]) \\cong 0$$\n\tfrom which we deduce that there are no non-trivial morphisms from $Lp_{Y\\#}N^j $ to $T^{j-1}[-1]$. It follows that there exist unique morphisms $Lp_{Y\\#}N^j \\rightarrow N'^j$ fitting into a morphism of Postnikov systems in $\\DM_{eff}^-(Y'_{\\bullet,\\bullet},R)$\n\t$$\n\t\\xymatrix{\n\t\t\\dots \\ar@{->}[r] &Lp_{Y\\#}N^j \\ar@{->}[rr] \\ar@{->}[dd]& & Lp_{Y\\#}N^{j-1} \\ar@{->}[ld] \\ar@{->}[dd] \\ar@{->}[r] & \\dots\\\\\n\t\t& & Lp_{Y\\#}T^{j-1} \\ar@{->}[ul]^{[1]} \\ar@{->}[dd]& &\\\\\n\t\t\\dots \\ar@{->}[r] & N'^j \\ar@{->}[rr] & & N'^{j-1} \\ar@{->}[ld] \\ar@{->}[r] & \\dots\\\\\n\t\t& & T^{j-1} \\ar@{->}[ul]^{[1]}& &\n\t}\n\t$$\n\tIf we restrict our previous diagram to the $0$th simplicial component we obtain in $\\DM_{eff}^-(Y'_{0,\\bullet},R)$ the following morphism of Postnikov systems\n\t$$\n\t\\xymatrix{\n\t\t\\dots \\ar@{->}[r] &\\bigoplus_{k \\geq j}Lp_{Y_0\\#}T^k \\ar@{->}[rr] \\ar@{->}[dd]& & \\bigoplus_{k \\geq j-1}Lp_{Y_0\\#}T^k \\ar@{->}[ld] \\ar@{->}[dd] \\ar@{->}[r] & \\dots\\\\\n\t\t& & Lp_{Y_0\\#}T^{j-1} \\ar@{->}[ul]^{[1]} \\ar@{->}[dd]& &\\\\\n\t\t\\dots \\ar@{->}[r] & \\bigoplus_{k \\geq j} T^k \\ar@{->}[rr] & & \\bigoplus_{k \\geq j-1} T^k \\ar@{->}[ld] \\ar@{->}[r] & \\dots\\\\\n\t\t& & T^{j-1} \\ar@{->}[ul]^{[1]}& &\n\t}\n\t$$\n\twhere each triangle is split. By hypothesis, the morphism $Lp_{Y_0\\#}T^{j-1} \\rightarrow T^{j-1}$ in the previous diagram is basically given by $\\bigoplus_{I_{j-1}} M(p_{Y_0})(q_{j-1})[p_{j-1}]$, while the map $Lp_{Y_0\\#}N_0^{j-1} \\rightarrow N_0'^{j-1}$ is given by $\\bigoplus_{k \\geq {j-1}}\\bigoplus_{I_k} M(p_{Y_0})(q_k)[p_k]$.\n\t\n\tNow, note that the morphisms $Lp_{Y\\#}T^j \\rightarrow T^j$ and $\\bigoplus_{I_j} M(p_Y)(q_j)[p_j]$ are both in \n\t$$\\Hom_{\\DM_{eff}^-(Y'_{\\bullet,\\bullet},R)}(Lp_{Y\\#}T^j,T^j) \\cong \\Hom_{\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)}(T^j,p_Y^*T^j) \\cong$$\n\t$$\\Hom_{\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)}(T^j,T^j) \\cong \\prod_{I_j} \\bigoplus_{I_j} H^{0,0}(Y_{\\bullet,\\bullet},R)$$\n\tand, for the same reason, $(Lp_{Y\\#}T^j \\rightarrow T^j)=\\bigoplus_{I_j} M(p_{Y_0})(q_j)[p_j]$ is in \n\t$$\\Hom_{\\DM_{eff}^-(Y'_{0,\\bullet},R)}(Lp_{Y_0\\#}T^j,T^j) \\cong \\Hom_{\\DM_{eff}^-(Y_{0,\\bullet},R)}(T^j,p_{Y_0}^*T^j)\n\t\\cong$$\n\t$$\\Hom_{\\DM_{eff}^-(Y_{0,\\bullet},R)}(T^j,T^j) \\cong \\prod_{I_j} \\bigoplus_{I_j} H^{0,0}(Y_{0,\\bullet},R).$$\n\tRecall that $H^{0,0}(Y_{\\bullet,\\bullet},R)$ is the free $R$-module with rank equal to the number of connected components of $Y_{\\bullet,\\bullet}$ and, analogously, $H^{0,0}(Y_{0,\\bullet},R)$ is the free $R$-module with rank equal to the number of connected components of $Y_{0,\\bullet}$. Since, as in the argument at the end of \\cite[Proposition 3.4]{tanania.a}, the homomorphism\n\t$$r_0^*:H^{0,0}(Y_{\\bullet,\\bullet},R) \\rightarrow H^{0,0}(Y_{0,\\bullet},R)$$\n\tis injective, we deduce that $Lp_{Y\\#}T^j \\rightarrow T^j$ and $\\bigoplus_{I_j} M(p_Y)(q_j)[p_j]$ are identified, which completes the proof.\n\\end{proof}\n\n\\begin{rem}\n\t\\normalfont\n\tIn particular, Proposition \\ref{Serre 3} guarantees that the Postnikov system of Proposition \\ref{Serre} is essentially unique.\n\\end{rem}\n\nNow, suppose for simplicity that $T^j \\cong T(q_j)[p_j]$ for any $j \\geq 0$ and consider the diagonal map $\\Delta: X_{\\bullet,\\bullet} \\rightarrow X_{\\bullet,\\bullet} \\times_{Y_{\\bullet,\\bullet}} X_{\\bullet,\\bullet}$. The latter induces a morphism on motives $N \\rightarrow N \\otimes N$ in $\\DM_{eff}^-(Y_{\\bullet,\\bullet},R)$ that, in turn, extends to a morphism of Postnikov systems $N^k \\rightarrow (N \\otimes N)^k$, which on the cones looks like\n$$T^k \\rightarrow \\bigoplus_{J_k} T^i \\otimes T^j$$\nwhere $J_k$ is the set of all couples $(i,j)$ such that $(q_i+q_j)[p_i+p_j]=(q_k)[p_k]$.\nThe induced homomorphism in motivic cohomology\n$$\\bigoplus_{J_k} H^{**}(T^i,R) \\otimes H^{**}(T^j,R) \\rightarrow H^{**}(T^k,R)$$\nsends the element $1 \\otimes 1 \\in H^{p_i,q_i}(T^i,R) \\otimes H^{p_j,q_j}(T^j,R) \\cong H^{0,0}(Y_{\\bullet,\\bullet},R) \\otimes H^{0,0}(Y_{\\bullet,\\bullet},R)$ to $1 \\in H^{p_k,q_k}(T^k,R) \\cong H^{0,0}(Y_{\\bullet,\\bullet},R)$. In general, if $u$ is an element in $H^{**}(T^i,R)$ and $v$ is an element in $H^{**}(T^j,R)$, then we denote by $u \\cdot v$ the image of $u \\otimes v$ under the previous homomorphism.\n\nNote that the differential on the $E_1$-page $d_1^{p,q,s}:H^{p,q}(T^s,R) \\rightarrow H^{p+1,q}(T^{s-1},R)$ respects the $H^{**}(Y_{\\bullet,\\bullet},R)$-module structure. Hence, it is completely determined by $d_1^{2s,s,s}(1)$ that is the composition $T^{s-1} \\rightarrow N^s[1] \\rightarrow T^s[1]$. These observations imply the following Leibniz rule where, for brevity, we write only the last degree of the differential, i.e. the one related to the filtration.\n\n\\begin{prop}\\label{mult}\n\tWith the notations just introduced we have that\n\t$$d_1^k(a\\cdot b) = d_1^i(a)\\cdot b + a \\cdot d_1^j(b)$$\n\twhere $a$ and $b$ are respectively classes in $H^{**}(T^i,R)$ and $H^{**}(T^j,R)$. \n\\end{prop}\n\\begin{proof}\n\tSince the differential $d_1^k$ is a homomorphism of $H^{**}(Y_{\\bullet,\\bullet},R)$-modules, it is enough to prove the Leibniz formula only in the case $a=b=1$.\n\t\n\tFrom the morphism of Postnikov systems induced by the diagonal map we can extract a commutative diagram\n\t$$\\xymatrix\n\t{\n\t\tT^{k-1}\t\\ar@{->}[r] \\ar@{->}[d] & N^{k}[1] \\ar@{->}[r] \\ar@{->}[d] & T^{k}[1] \\ar@{->}[d]\\\\\n\t\t\\bigoplus_{J_{k-1}} T^i \\otimes T^j \\ar@{->}[r] & (N \\otimes N)^{k}[1] \\ar@{->}[r] & \\bigoplus_{J_k} T^i \\otimes T^j[1]}\n\t$$\n\twhere the composition of the top horizontal morphisms is $d_1^k(1)$. On the other hand, by degree reasons, for any $(i,j) \\in J_k$ the composition of the bottom horizontal morphisms restricts to\n\t$$(T^{i-1} \\otimes T^j) \\oplus (T^i \\otimes T^{j-1}) \\rightarrow T^i \\otimes T^j[1]$$\n\twhich is nothing but $d_1^i(1) \\otimes 1 + 1 \\otimes d_1^j(1)$. Thus, the result follows immediately.\n\\end{proof}\n\nWe would like to finish this section by establishing a comparison between the spectral sequence here presented and the Serre spectral sequence associated to a fiber bundle in topology. Recall that in topology for a fibre sequence\n$$F \\rightarrow E \\rightarrow B$$\nwith $\\pi_1(B)$ acting trivially on $H^*(F)$ one has a spectral sequence converging to $H^*(E)$\n$$E^{s,t}_2=H^s(B,H^t(F)) \\Longrightarrow H^*(E)$$\ncalled Serre spectral sequence (see for example \\cite[Theorem 15.27]{switzer}).\n\nAnalogously, our spectral sequence allows to reconstruct somehow the cohomology of the total bisimplicial scheme from the cohomology of the base and of the fiber, provided that the fiber is motivically cellular. Moreover, the triviality condition on the $\\omega^{M(X_{\\bullet,\\bullet} \\rightarrow Y_{\\bullet,\\bullet})}_j$ for any $j \\geq 0$ is reminiscent of the topological condition on the triviality of the action of $\\pi_1(B)$ on $H^*(F)$. On the other hand, the main difference between the two spectral sequences resides in how they are obtained. In fact, while the topological Serre spectral sequence is classically achieved by filtering the base, our spectral sequence is instead realized by filtering the fiber.\n\n\\section{The case of $BPGL_n$}\n\nIn this section we want to apply the spectral sequence of Theorem \\ref{Serre2} to approach the computation of the motivic cohomology of the Nisnevich classifying space of $PGL_n$. From now on we assume that the base field $k$ has characteristic not dividing $n$.\n\nFirst, we recall a few definitions. For a simplicial algebraic group $G_{\\bullet}$ denote by $EG_{\\bullet}$ the weakly contractible bisimplicial scheme defined by \n$$(EG_{\\bullet})_{i,h} = G_{h}^{i+1}$$\nwith standard face and degeneracy maps. Note that $EG_{\\bullet}$ has a right free $G_{\\bullet}$-action. Denote by $BG_{\\bullet}$ the bisimplicial scheme obtained as a quotient of $EG_{\\bullet}$ by this action. On each simplicial component $BG_{\\bullet}$ looks like\n$$(BG_{\\bullet})_{i,h} = G_{h}^{i}.$$\nBy abuse of notation, we denote by $EG_{\\bullet}$ and $BG_{\\bullet}$ also the diagonals of the respective bisimplicial schemes.\n\n\\begin{dfn}\nThe simplicial scheme $BG_{\\bullet}$ is called the Nisnevich classifying space of $G_{\\bullet}$ (\\cite[Example 1.11]{morel.voevodsky}).\n\\end{dfn}\n\n\\begin{rem}\n\t\\normalfont\nNote that the map of bisimplicial schemes $EG_{\\bullet} \\rightarrow BG_{\\bullet}$ is smooth coherent and over the 0th simplicial component is the projection $G_{\\bullet} \\rightarrow Spec(k)$. This is the reason why we need to work with bisimplicial models. Indeed, these maps provide a rich source of examples where it is possible to apply the spectral sequence in Theorem \\ref{Serre2}. In fact, if $G$ is a commutative algebraic group, then for any $n \\geq 1$ there are simplicial commutative algebraic groups $B^nG$ each of which is the (diagonal of the) classifying space of the previous one. So, we get coherent morphisms of bisimplicial schemes $EB^nG \\rightarrow B^{n+1}G$ with fiber $B^nG$. In particular, if $G=\\Gm$, then $B^nG$ is a motivic Eilenberg-MacLane space $K(\\Z,n+1,1)$, and we know from \\cite{voevodsky.em} that their motives are cellular, i.e. direct sum of Tate motives. Hence, we get Serre spectral sequences for the motivic cohomology of Eilenberg-MacLane spaces of this type.\n\\end{rem}\n\nFrom the short exact sequence of algebraic groups\n\\begin{align}\\label{cenext}\n\t1 \\rightarrow \\Gm \\rightarrow GL_n \\rightarrow PGL_n \\rightarrow 1\n\t\\end{align}\none gets the following fiber sequence\n$$B\\Gm \\rightarrow BGL_n \\rightarrow BPGL_n.$$\nFor our purposes, consider $EB\\Gm \\times BGL_n$ as a bisimplicial model for $BGL_n$ and $(EB\\Gm \\times BGL_n)\/B\\Gm$ as a bisimplicial model for $BPGL_n$ (see \\cite[Example 9.11]{jardine}). The smooth coherent morphisms of bisimplicial schemes\n$$EB\\Gm \\times BGL_n \\rightarrow (EB\\Gm \\times BGL_n)\/B\\Gm$$\nand \n$$EB\\Gm \\rightarrow BB\\Gm$$\nare both trivial projections over the 0th simplicial component with fiber $B\\Gm$. Hence, we get a cartesian square of bisimplicial schemes\n\t\\begin{align}\\label{square}\n\\xymatrix{\n\tEB\\Gm\\times BGL_n \\ar@{->}[r] \\ar@{->}[d] & (EB\\Gm \\times BGL_n)\/B\\Gm \\ar@{->}[d]\\\\\n\tEB\\Gm \\ar@{->}[r] & BB\\Gm.\n}\n\\end{align}\n \nRecall from \\cite[Proposition 3.7]{morel.voevodsky} that $B\\Gm$ is $A^1$-homotopy equivalent to $P^{\\infty}$ whose motive is cellular. Indeed, we have that $M(P^{\\infty}) \\cong \\bigoplus_{j=0}^{\\infty} T(j)[2j]$. Therefore, we can apply the spectral sequence constructed in Theorem \\ref{Serre2} to the $BPGL_n$ case, which leads to the following result.\n\n\\begin{thm}\\label{bpg}\n\tThere exists a strongly convergent spectral sequence\n\t$$E_1^{p,q,s}= H^{p-2s,q-s}(BPGL_n) \\Longrightarrow H^{p,q}(BGL_n)$$\n\twith differentials $d_r^{p,q,s}:E_r^{p,q,s} \\rightarrow E_r^{p+1,q,s-r}$. Moreover, the differential\n\t$$d_1^{p,q,s}:H^{p-2s,q-s}(BPGL_n) \\rightarrow H^{p-2s+3,q-s+1}(BPGL_n)$$\n\t is the multiplication by $s\\cdot d_1^{2,1,1}(1)$.\n\\end{thm}\n\\begin{proof}\n\tApplying Theorem \\ref{Serre2} to the coherent morphism $EB\\Gm \\times BGL_n \\rightarrow (EB\\Gm \\times BGL_n)\/B\\Gm$ gives the strongly convergent spectral sequence. The description of the first differential follows easily by induction on $s$. In fact, suppose $d_1^{2s-2,s-1,s-1}(1)=(s-1) \\cdot d_1^{2,1,1}(1)$, then from Proposition \\ref{mult} we deduce that\n\t$$d_1^{2s,s,s}(1)=d_1^{2s-2,s-1,s-1}(1)+d_1^{2,1,1}(1)=s \\cdot d_1^{2,1,1}(1)$$\n\twhich concludes the proof.\n\\end{proof}\n\nSince the motivic cohomology of $BGL_n$ is known, i.e. $H^{**}(BGL_n) \\cong H^{**}(k)[c_1,\\dots,c_n]$, we can ``reverse-engineer\" the previous spectral sequence in order to obtain information about the motivic cohomology of $BPGL_n$.\n\nBefore proceeding, recall that the Chern class $c_i$ is in bidegree $(i)[2i]$ for any $i$, so $H^{p,q}(BGL_n) \\cong 0$ for $p > 2q$.\n\n\\begin{cor}\\label{triv}\n\tFor all $p \\geq 3q+1$ we have that $H^{p.q}(BPGL_n) \\cong 0$.\n\\end{cor}\n\\begin{proof}\n\tWe proceed by induction on $q$. For $q=0$, it follows from an easy inspection of the spectral sequence that $H^{p,0}(BPGL_n) \\cong H^{p,0}(BGL_n) \\cong 0$ for all $p \\geq 1$, which provides the induction basis.\n\t\n\tNow, suppose that the statement holds for all motivic weights less than $q$. The $E_1$-page of the spectral sequence is\n\t$$E_1^{p,q,s}= H^{p-2s,q-s}(BPGL_n).$$\n\tConsider $p \\geq 3q+1$, then $p-2s \\geq 3q+1-2s \\geq 3(q-s) +1$, Hence, by induction hypothesis $E_1^{p,q,s} \\cong 0$ for all $s \\geq 1$. It follows that the only piece of the spectral sequence that contributes for $p \\geq 3q+1$ to $H^{p,q}(BGL_n) \\cong 0$ comes from $E_1^{p,q,0}= H^{p,q}(BPGL_n)$. But the differentials $d_r: E_r^{p-1,q,r} \\rightarrow E_r^{p,q,0}$ are all trivial since $E_1^{p-1,q,r} \\cong H^{p-1-2r,q-r}(BPGL_n) \\cong 0$ as $p-1-2r \\geq 3q-2r \\geq 3(q-r)+1$.\n\t\n\tTherefore, \n\t$$H^{p,q}(BGL_n) \\cong E_{\\infty}^{p,q,0} \\cong H^{p,q}(BPGL_n)$$\n\tfor $p \\geq 3q+1$ that concludes the proof.\n\t\\end{proof}\n\nRecall from \\cite[Example 9.11]{jardine} that $BB\\Gm$ is the Eilenberg-MacLane space $K(\\Gm,2)$, so by adjunction there is a canonical element $\\chi$ in $H^{3,1}(BB\\Gm) \\cong H_{Nis}^2(BB\\Gm,\\Gm) \\cong [BB\\Gm,BB\\Gm]$ corresponding to the identity $BB\\Gm \\rightarrow BB\\Gm$ (here by $[-,-]$ we mean hom-sets in ${\\mathcal H}_s(k)$).\n\n\\begin{lem}\\label{bb}\n\tWe have that\n\t$$H^{3,1}(BB\\Gm) \\cong \\Z$$\n\tgenerated by $\\chi$.\n\\end{lem}\n\\begin{proof}\n\tWe can apply Theorem \\ref{Serre2} to the coherent morphism $EB\\Gm \\rightarrow BB\\Gm$ with fiber $B\\Gm$. Note that for $p=q=0$ the differentials are all trivial and we get\n\t$$H^{0,0}(EB\\Gm) \\cong E_{\\infty}^{0,0,0} \\cong H^{0,0}(BB\\Gm)$$\n\tfrom which it follows that $H^{0,0}(BB\\Gm) \\cong \\Z$.\n\t\n\tIn order to compute $H^{3,1}(BB\\Gm)$, the part of the $E_1$-page we need consists of the groups $E_1^{3,1,0} \\cong H^{3,1}(BB\\Gm)$ and $E_1^{2,1,1} \\cong H^{0,0}(BB\\Gm) \\cong \\Z$ linked by the differential $d_1^{2,1,1}:H^{0,0}(BB\\Gm) \\cong \\Z \\rightarrow H^{3,1}(BB\\Gm)$. Hence, we obtain \n\t$$0\\cong H^{3,1}(EB\\Gm) \\cong E_{\\infty}^{3,1,0} \\cong H^{3,1}(BB\\Gm)\/\\Ima(d_1^{2,1,1}),$$\n\t$$E_{\\infty}^{2,1,0} \\cong H^{2,1}(BB\\Gm)$$\n\tand\n\t$$E_{\\infty}^{2,1,1} \\cong \\ker(d_1^{2,1,1}).$$\n\tTherefore, from the short exact sequence\n\t$$0 \\rightarrow E_{\\infty}^{2,1,0} \\rightarrow H^{2,1}(EB\\Gm) \\cong 0 \\rightarrow E_{\\infty}^{2,1,1} \\rightarrow 0$$\n\tone gets that $d_1^{2,1,1}$ is an isomorphism, which completes the proof.\n\t\\end{proof}\n\nThe right vertical map in \\ref{square} induces in ${\\mathcal H}_s(k)$ a class of $[BPGL_n,BB\\Gm] \\cong H^{3,1}(BPGL_n)$ that classifies the central extension \\ref{cenext} (see \\cite[Theorem 1.2]{rolle}). Denote by $x$ this canonical element. Note that $x$ is nothing but the image of $\\chi$ under the induced homomorphism $H^{3,1}(BB\\Gm) \\rightarrow H^{3,1}(BPGL_n)$.\n\n\\begin{thm}\\label{comp}\n\tIn motivic weights 0, 1 and 2 the following isomorphisms hold\n\t\\begin{align*}\n\t\tH^{p,0}(BPGL_n) &\\cong \n\t\t\\begin{cases}\n\t\t\t\\Z & p=0\\\\\n\t\t\t0 & otherwise\n\t\t\t\\end{cases}\\\\\n\t\tH^{p,1}(BPGL_n) &\\cong \n\t\t\\begin{cases}\n\t\t\tk^* & p=1\\\\\n\t\t\t\\Z\/n \\cdot x & p=3\\\\\n\t\t\t0 & otherwise\n\t\t\\end{cases}\\\\\n\tH^{p,2}(BPGL_n) &\\cong \n\t\\begin{cases}\n\t\tH^{p,2}(k) & p\\leq2\\\\\n\t\t\\mu_n(k) & p=3\\\\\n\t\tk^*\/n \\cdot x \\oplus \\Z & p=4\\\\\n\t\t\\Z\/2 \\cdot x^2 & p=6 \\: and \\: n \\: even\\\\\n\t\t0 & otherwise.\n\t\\end{cases}\n\t\t\\end{align*}\n\\end{thm}\n\\begin{proof}\n\tThe result follows from the spectral sequence in Theorem \\ref{bpg}. \n\t\n\tLet us start from the case $q=0$. Then, the only possibly non-trivial groups in the $E_1$-page are $E_1^{p,0,0} \\cong H^{p,0}(BPGL_n)$ for $p \\geq 0$. In this case differentials are all trivial and we get\n\t$$H^{p,0}(BGL_n) \\cong E_{\\infty}^{p,0,0} \\cong H^{p,0}(BPGL_n)$$\n\tfrom which it follows the motivic weight $0$ case.\n\t\n\tFor the case $q=1$, the non-trivial part of the $E_1$-page possibly consists of the groups $E_1^{p,1,0} \\cong H^{p,1}(BPGL_n)$ for $p \\geq 1$ and $E_1^{2,1,1} \\cong H^{0,0}(BPGL_n) \\cong \\Z$. There is only one non-zero differential $d_1^{2,1,1}:H^{0,0}(BPGL_n) \\cong \\Z \\rightarrow H^{3,1}(BPGL_n)$. Hence, we obtain \n\t$$H^{p,1}(BGL_n) \\cong E_{\\infty}^{p,1,0} \\cong H^{p,1}(BPGL_n)$$\n\tfor $p \\neq 2,3$,\n\t$$0\\cong H^{3,1}(BGL_n) \\cong E_{\\infty}^{3,1,0} \\cong H^{3,1}(BPGL_n)\/\\Ima(d_1^{2,1,1}),$$\n\t$$E_{\\infty}^{2,1,0} \\cong H^{2,1}(BPGL_n)$$\n\tand\n\t$$E_{\\infty}^{2,1,1} \\cong \\ker(d_1^{2,1,1}).$$\n \tTherefore, from the short exact sequence\n \t$$0 \\rightarrow E_{\\infty}^{2,1,0} \\rightarrow H^{2,1}(BGL_n) \\rightarrow E_{\\infty}^{2,1,1} \\rightarrow 0$$\n \tone gets the exact sequence\n \t$$0 \\rightarrow H^{2,1}(BPGL_n) \\rightarrow \\Z \\rightarrow \\Z \\xrightarrow{d_1^{2,1,1}} H^{3,1}(BPGL_n) \\rightarrow 0.$$\n At this point, we only need to understand the homomorphism in the middle $\\Z \\rightarrow \\Z$. Note that the latter is just the homomorphism $H^{2,1}(BGL_n) \\rightarrow H^{2,1}(N^1)$ induced by the Postnikov system generating the spectral sequence. Recall that $H^{2,1}(BGL_n)$ is generated by the first Chern class $c_1$ while $H^{2,1}(N^1) \\cong H^{2,1}(B\\Gm)$ is generated by the Chern class $c$. Since the map $B\\Gm \\rightarrow BGL_n$ factors through $(B\\Gm)^n$ we have that the previous homomorphism maps $c_1$ to $nc$. It follows that $H^{2,1}(BPGL_n) \\cong 0$ and $H^{3,1}(BPGL_n) \\cong \\Z\/n$ is generated by $x=d_1^{2,1,1}(1)$, by Lemma \\ref{bb} and the functoriality of the spectral sequence.\n \n For the case $q=2$, we have $E_1^{p,2,0} \\cong H^{p,2}(BPGL_n)$, $E_1^{3,2,1} \\cong H^{1,1}(BPGL_n) \\cong k^*$, $E_1^{5,2,1} \\cong H^{3,1}(BPGL_n) \\cong \\Z\/n$ and $E_1^{4,2,2} \\cong H^{0,0}(BPGL_n) \\cong \\Z$. The possibly non-trivial differentials on the $E_1$-page are $d_1^{4,2,2}$, $d_1^{3,2,1}$ and $d_1^{5,2,1}$. Note that, by Theorem \\ref{bpg}, $d_1^{4,2,2}$ is the multiplication by $2x$ and $d_1^{5,2,1}$ is surjective since $H^{6,2}(BGL_n)$ is trivial.\n \n From the short exact sequence\n $$0 \\rightarrow E_{\\infty}^{5,2,0}\\rightarrow H^{5,2}(BGL_n) \\rightarrow E_{\\infty}^{5,2,1} \\rightarrow 0$$\n and since $H^{5,2}(BGL_n) \\cong 0$ we get that $E_{\\infty}^{5,2,0} \\cong H^{5,2}(BPGL_n)\/ \\Ima(d_2^{4,2,2})$ and $E_{\\infty}^{5,2,1} \\cong \\ker(d_1^{5,2,1})\/ \\Ima(d_1^{4,2,2})$ are both trivial. In particular, $d_2^{4,2,2}$ is surjective and the complex\n $$H^{0,0}(BPGL_n) \\xrightarrow{\\cdot 2x} H^{3,1}(BPGL_n) \\xrightarrow{\\cdot x} H^{6,2}(BPGL_n) \\rightarrow 0$$\n is exact. The first homomorphism of the latter complex is $\\Z \\xrightarrow{\\cdot 2} \\Z\/n$. Hence, when $n$ is odd, it is surjective and $H^{6,2}(BPGL_n) \\cong 0$, while, when $n$ is even, its image is $\\Z\/({\\frac n 2})$ and $H^{6,2}(BPGL_n) \\cong \\Z\/2$ generated by $x^2$. \n \n From the short exact sequence\n $$0 \\rightarrow E_{\\infty}^{3,2,0}\\rightarrow H^{3,2}(BGL_n) \\rightarrow E_{\\infty}^{3,2,1} \\rightarrow 0$$\n we get the exact sequence\n $$0 \\rightarrow H^{3,2}(BPGL_n) \\rightarrow k^* \\xrightarrow{\\cdot n} k^* \\rightarrow k^*\/n \\rightarrow 0.$$\n Hence, $H^{3,2}(BPGL_n) \\cong \\mu_n(k)$.\n \n Finally, from the short exact sequence\n $$0 \\rightarrow E_{\\infty}^{4,2,0}\\rightarrow H^{4,2}(BGL_n) \\rightarrow E_{\\infty}^{4,2,2} \\rightarrow 0$$\n we get the exact sequence\n $$0 \\rightarrow H^{4,2}(BPGL_n)\/\\Ima(d_2^{3,2,1}) \\rightarrow \\Z \\oplus \\Z \\rightarrow E_{\\infty}^{4,2,2} \\rightarrow 0.$$\n Note that $E_{\\infty}^{4,2,2}$ is a subgroup of $H^{4,2}(N^2)\\cong \\Z$. The latter is generated by $c^2$ and the homomorphism $H^{4,2}(BGL_n) \\rightarrow E_{\\infty}^{4,2,2}$ maps $c_1^2$ to $n^2c^2$ and $c_2$ to ${\\frac{n(n-1)}2} c^2$. At this point we want to prove that $d_2^{4,2,2}$ is trivial. To this end, it is enough to prove that the homomorphism $H^{4,2}(BGL_n) \\rightarrow H^{4,2}(N^1)$ is surjective. First, note that since $H^{2,1}(BPGL_n) \\cong 0$ then the homomorphism $H^{4,2}(N^1) \\rightarrow H^{4,2}(N^2)$ induced by the Postnikov system is injective. Hence, $H^{4,2}(N^1) \\cong \\Z$ is generated by an element $z$ mapping to $nc^2$, if $n$ is odd, and to ${\\frac n 2}c^2$, if $n$ is even, in $H^{4,2}(N^2)$. But $c_1^2-2c_2$ in $H^{4,2}(BGL_n)$ maps to $nc^2$ in $H^{4,2}(N^2)$ if $n$ is odd, while ${\\frac n 2}c_1^2-(n+1)c_2$ maps to ${\\frac n 2}c^2$ if $n$ is even. Therefore, $d_2^{4,2,2}$ is trivial and surjective, so $H^{5,2}(BPGL_n) \\cong 0$. It immediately follows that $H^{4,2}(BPGL_n) \\cong k^*\/n \\cdot x \\oplus \\Z$ where the generator of $\\Z$ maps to $(n-1)c_1^2-2nc_2$ if $n$ is even, and to ${\\frac {n-1} 2}c_1^2-nc_2$ if $n$ is odd. This concludes the proof. \n\t\\end{proof}\n\nThe next result tells us that, as expected, the interesting part of $H^{**}(BPGL_n)$ is $n$-torsion.\n\n\\begin{prop}\n\tThere are isomorphisms of $H^{**}(k,\\Z[{\\frac 1 n}])$-algebras\n\t$$H^{**}(BPGL_n,\\Z[{\\tfrac 1 n}]) \\cong H^{**}(BSL_n,\\Z[{\\tfrac 1 n}]) \\cong H^{**}(k,\\Z[{\\tfrac {1} {n}}])[c_2,\\dots,c_n].$$\n\\end{prop}\n\\begin{proof}\n The second isomorphism is well-known (already with $\\Z$-coefficients), so we only need to show the first one. \n \n Since the standard morphism $GL_n \\rightarrow PGL_n$ is a $\\Gm$-torsor we have a cartesian square\n \t$$\n \\xymatrix{\n \tGL_n \\times \\Gm \\ar@{->}[r]^{\\pi} \\ar@{->}[d]_{\\alpha} & GL_n \\ar@{->}[d]\\\\\n \tGL_n \\ar@{->}[r] & PGL_n\n }\n $$\n where $\\pi$ is the projection and $\\alpha$ is the $\\Gm$-action. The latter induces in turn a cartesian square\n\t$$\n\\xymatrix{\n\tSL_n \\times \\Gm \\ar@{->}[r]^{\\pi} \\ar@{->}[d]_{\\tilde{\\alpha}} & SL_n \\ar@{->}[d]\\\\\n\tGL_n \\ar@{->}[r] & PGL_n\n}\n$$\nwhere the morphism $SL_n \\rightarrow PGL_n$ (factoring through $GL_n$) is the usual $\\mu_n$-torsor.\n\nNote that $\\tilde{\\alpha}$ induces a homomorphism on the motivic cohomology of the respective classifying spaces $H^{**}(BGL_n) \\rightarrow H^{**}(BSL_n) \\otimes_{H^{**}(k)} H^{**}(B\\Gm)$ that maps $c_i$ to $c_i$ for $i \\geq 2$ and $c_1$ to $nc$. Hence, we get an isomorphism \n$$B\\tilde{\\alpha}^{*}:H^{**}(BGL_n,\\Z[{\\tfrac 1 n}]) \\cong H^{**}(BSL_n,\\Z[{\\tfrac 1 n}]) \\otimes_{H^{**}(k,\\Z[{\\frac 1 n}])}H^{**}(B\\Gm,\\Z[{\\tfrac 1 n}]).$$\n\nNow we want to prove by induction on the motivic weight that the homomorphism \n$$H^{**}(BPGL_n,\\Z[{\\tfrac 1 n}]) \\rightarrow H^{**}(BSL_n,\\Z[{\\tfrac 1 n}])$$ \nis an isomorphism. We use the functoriality of the Postnikov systems provided by Proposition \\ref{Serre 3}. For $q=0$, our spectral sequence implies that $H^{p,0}(BPGL_n,\\Z[{\\frac 1 n}]) \\cong H^{p,0}(BGL_n,\\Z[{\\frac 1 n}]) \\cong H^{p,0}(BSL_n,\\Z[{\\frac 1 n}])$ for all $p$, which provides the induction basis. Suppose that $H^{p,q'}(BPGL_n,\\Z[{\\frac 1 n}]) \\cong H^{p,q'}(BSL_n,\\Z[{\\frac 1 n}])$ for all $q' < q$ and all $p$. Since both $H^{p,q}(BPGL_n,\\Z[{\\frac 1 n}])$ and $H^{p,q}(BSL_n,\\Z[{\\frac 1 n}])$ can be reconstructed respectively from compatible extensions of $H^{p,q'}(BPGL_n,\\Z[{\\frac 1 n}])$ and $H^{p,q'}(BSL_n,\\Z[{\\frac 1 n}])$ for $q' < q$ and $H^{p,q}(BGL_n,\\Z[{\\frac 1 n}])$, five lemma implies that \n$$H^{p,q}(BPGL_n,\\Z[{\\tfrac 1 n}]) \\rightarrow H^{p,q}(BSL_n,\\Z[{\\tfrac 1 n}])$$\nis an isomorphism that is what we aimed to show.\n\t\\end{proof}\n\n\\section{The motive of a Severi-Brauer variety}\n\nThe purpose of this section is to apply previous results to obtain a description of the motive of a Severi-Brauer variety.\n\nLet $A$ be a central simple algebra of degree $n$ and $\\check C(\\seb(A))$ be the \\v{C}ech simplicial scheme of the respective Severi-Brauer variety $\\seb(A)$, i.e. $\\check C(\\seb(A))_n=\\seb(A)^{n+1}$ with face and degeneracy maps given respectively by partial projections and diagonals. Moreover, denote by $\\X_A$ the motive of $\\check C(\\seb(A))$ in $\\DM_{eff}^{-}(k)$.\n\nLet $X_A$ be the $PGL_n$-torsor associated to $A$, i.e. $X_A = {\\mathrm Iso}\\{A \\leftrightarrow M_n(k)\\}$. Note that $X_A$ is a form of $PGL_n$. Then, the scheme $(X_A \\times P^{n-1})\/PGL_n$ is a Severi-Brauer variety for $A$, i.e.\n$$\\seb(A) \\cong (X_A \\times P^{n-1})\/PGL_n.$$\n\n\\begin{prop}\\label{pssb}\n\tThere exists a Postnikov system in $\\DM_{eff}^-(k)$\n\t$$\n\t\\xymatrix{\n\t\t \\X_A(n-1)[2n-2] \\ar@{->}[r] &M^{n-2} \\ar@{->}[r] \\ar@{->}[d] & \\dots \\ar@{->}[r] & M^2 \\ar@{->}[r] \\ar@{->}[d]\t &M^1 \\ar@{->}[r] \\ar@{->}[d] &M(\\seb(A)) \\ar@{->}[d]\\\\\n\t &\t\\X_A(n-2)[2n-4] \\ar@{->}[ul]^{[1]} & & \\X_A(2)[4] \\ar@{->}[ul]^{[1]} &\t\\X_A(1)[2] \\ar@{->}[ul]^{[1]} & \\X_A \\ar@{->}[ul]^{[1]}\n\t}.\n\t$$\n\\end{prop}\n\\begin{proof}\n\tIt follows from \\cite[2.3.11 and Proposition 2.3.14]{smirnov.vishik} that $\\check C(\\seb(A)) \\cong (X_A \\times EPGL_n)\/PGL_n$ in ${\\mathcal H}_s(k)$. Therefore, by restricting the Postnikov system for the motive $M(BGL_n \\rightarrow BPGL_n)$ along the functor $\\DM_{eff}^-(BPGL_n) \\rightarrow \\DM_{eff}^-(\\check C(\\seb(A)))$ and then applying the forgetful functor to $\\DM_{eff}^-(k)$, one obtains a Postnikov system for the motive of $(X_A \\times EGL_n)\/GL_n$\n\t$$\n\t\\xymatrix{\n\t\t\\dots \\ar@{->}[r] & N^{j+1} \\ar@{->}[r] \\ar@{->}[d]\t &N^j \\ar@{->}[r] \\ar@{->}[d] & \\dots \\ar@{->}[r] \t &N^1 \\ar@{->}[r] \\ar@{->}[d] &M((X_A \\times EGL_n)\/GL_n) \\ar@{->}[d]\\\\\n\t\t& \\X_A(j+1)[2j+2] \\ar@{->}[ul]^{[1]} &\t\\X_A(j)[2j] \\ar@{->}[ul]^{[1]} & &\t\\X_A(1)[2] \\ar@{->}[ul]^{[1]} & \\X_A \\ar@{->}[ul]^{[1]}\n\t}.\n\t$$\n\tNow, note that \n\t$$(X_A \\times EGL_n)\/GL_n \\cong ((X_A \\times EGL_n)\/\\Gm)\/(GL_n\/\\Gm) \\cong (X_A \\times (EGL_n\/\\Gm))\/PGL_n \\cong (X_A \\times B\\Gm)\/PGL_n.$$\n\tSince $B\\Gm$ is $A^1$-homotopy equivalent to $P^{\\infty}$, we have a map $\\seb(A) \\rightarrow (X_A \\times EGL_n)\/GL_n$ in ${\\mathcal H}(k)$ that induces a morphism on motives. From the latter we can construct a Postnikov system for $M(\\seb(A))$ that is compatible with the one for $M((X_A \\times EGL_n)\/GL_n)$. The Postnikov system for $\\seb(A)$ is actually finite since $M^j$ vanishes when restricted to a splitting field of $A$ for any $j \\geq n$.\n\t\\end{proof}\n\nLet us denote $\\ker(H^p_{\\acute{e}t}(k,\\mu_n^{\\otimes p-1}) \\rightarrow H^p_{\\acute{e}t}(k(\\seb(A)),\\mu_n^{\\otimes p-1}))$ simply by $\\ker_p$. Also, in the following results, we denote by $H^{**}_{\\acute et}(-)$ the \\'etale motivic cohomology. So, in particular, we have that \n$$H^{p,q}_{\\acute et}(-,\\Z\/n) \\cong H^p_{\\acute{e}t}(-,\\mu_n^{\\otimes q}).$$\n\n\\begin{prop}\nWe have the following isomorphisms: \n$$H^{p,q}(\\X_A) \\cong H^{p,q}(k)$$ \nfor all $p \\leq q$. Moreover,\n$$H^{p,p-1}(\\X_A) \\cong 0$$\nand \n$$H^{p+1,p-1}(\\X_A) \\cong \\ker_p$$\nfor all $p$.\n\\end{prop}\n\\begin{proof}\nBy Bloch-Kato conjecture (see \\cite[Theorems 6.16, 6.17 and 6.18]{voevodsky.motivic}), one has that $H^{p,q}(\\X_A) \\cong H^{p,q}_{\\acute{e}t}(\\X_A)$ and $H^{p,q}(k) \\cong H^{p,q}_{\\acute{e}t}(k)$ for $p \\leq q+1$. Since $H^{p,q}_{\\acute{e}t}(\\X_A) \\cong H^{p,q}_{\\acute{e}t}(k)$, we get the first two isomorphisms of the statement.\n\nRegarding the last one, again by Bloch-Kato conjecture we have that\n$$H^{p,p-1}(\\X_A,\\Z\/n) \\cong \\ker_p$$\n(see \\cite[Remark 5.3]{voevodsky.motivic}). The short exact sequence\n$$0 \\rightarrow \\Z \\xrightarrow{\\cdot n} \\Z \\rightarrow \\Z\/n \\rightarrow 0$$\ninduces a long exact sequence in motivic cohomology\n$$ \\dots \\rightarrow H^{p,p-1}(\\X_A) \\rightarrow H^{p,p-1}(\\X_A,\\Z\/n) \\rightarrow H^{p+1,p-1}(\\X_A) \\xrightarrow{\\cdot n} H^{p+1,p-1}(\\X_A) \\rightarrow \\dots.$$ \nTherefore, since $H^{p,q}(\\X_A)$ is $n$-torsion for $p \\geq q+1$ and $H^{p,p-1}(\\X_A) \\cong 0$, one obtains\n$$H^{p,p-1}(\\X_A,\\Z\/n) \\cong H^{p+1,p-1}(\\X_A)$$\nthat conludes the proof.\n\\end{proof}\n\nRecall that there is a natural homomorphism $\\alpha_A^*:H^{**}(BPGL_n) \\rightarrow H^{**}(\\X_A)$ induced by the map $\\alpha_A: (EPGL_n \\times X_A)\/PGL_n \\rightarrow BPGL_n$.\n\n\\begin{prop}\\label{xA}\n\t We have that \n\t $$\\alpha_A^*(x)=[A],$$\n\t where $x$ is the canonical class in $H^{3,1}(BPGL_n)$ from Theorem \\ref{comp} and $[A]$ is the Brauer class of $A$ in $H^{3,1}(\\X_A) \\cong \\ker_2$.\n\\end{prop}\n\\begin{proof}\nFirst note that, since $H^{3,1}(BPGL_n)$ and $H^{3,1}(\\X_A)$ are both $n$-torsion, they are respectively isomorphic to $H^{2,1}(BPGL_n,\\Z\/n)$ and $H^{2,1}(\\X_A,\\Z\/n)$. \n\nThe change of topology from Nisnevich to \\'etale gives a commutative square\n\t$$\n\\xymatrix{\n\tH^{2,1}(BPGL_n,\\Z\/n) \\ar@{->}[r] \\ar@{->}[d] & H_{\\acute{e}t}^{2,1}(BPGL_n,\\Z\/n) \\cong H^2_{\\acute{e}t}(BPGL_n,\\mu_n) \\ar@{->}[d]\\\\\n\tH^{2,1}(\\X_A,\\Z\/n) \\cong \\ker_2 \\ar@{->}[r] & H_{\\acute{e}t}^{2,1}(\\X_A,\\Z\/n) \\cong H^2_{\\acute{e}t}(k,\\mu_n)\n}\n$$\nwhere the bottom horizontal morphism is the inclusion of $\\ker_2$ in the Brauer group of $k$. By \\cite[Theorem 1.2]{rolle}, the right vertical morphism maps the central extension\n$$1 \\rightarrow \\mu_n \\rightarrow SL_n \\rightarrow PGL_n \\rightarrow 1$$\n(that is the image of $x$ under the top horizontal homomorphism: see Lemma \\ref{ce} below for more details) to the class $[A]$ in the Brauer group. Hence, we deduce that the left vertical morphism does the same, as we aimed to show.\n\\end{proof}\n\n\\begin{thm}\\label{sba}\n\tThere exists a strongly convergent spectral sequence\n\t$$E_1^{p,q,s}= \n\t\\begin{cases}\n\tH^{p-2s,q-s}(\\X_A) & 0 \\leq s \\leq n-1\\\\\n\t0 & otherwise\n\t\\end{cases}\n\t\\Longrightarrow H^{p,q}(\\seb(A))$$\n\twith differentials $d_r^{p,q,s}:E_r^{p,q,s} \\rightarrow E_r^{p+1,q,s-r}$. Moreover, the differential\n\t$$d_1^{p,q,s}:H^{p-2s,q-s}(\\X_A) \\rightarrow H^{p-2s+3,q-s+1}(\\X_A)$$\n\tis the multiplication by $s[A]$ for $1 \\leq s \\leq n-1$.\n\\end{thm}\n\\begin{proof}\nThe spectral sequence is obtained by applying motivic cohomology to the Postnikov system in Proposition \\ref{pssb}. The first differential is computed by using Theorem \\ref{bpg}, Proposition \\ref{xA} and the functoriality of the spectral sequence.\n\\end{proof}\n\n\\begin{cor}\n\tFor all $p \\geq 3q+1$ we have that $H^{p.q}(\\X_A) \\cong 0$.\n\\end{cor}\n\\begin{proof}\n\tThe proof is the same as Corollary \\ref{triv}.\n\\end{proof}\n\nAs an immediate consequence of the spectral sequence for the Severi-Brauer variety we obtain a description of the Chow group $CH^2(\\seb(A))$.\n\n\\begin{prop}\n\tThere is a short exact sequence\n\t$$0 \\rightarrow \\coker(k^* \\xrightarrow{\\cdot [A]}\\ker_3) \\rightarrow CH^2(\\seb(A)) \\rightarrow \\Z \\rightarrow 0.$$\n\\end{prop}\n\\begin{proof}\n\tWe use the spectral sequence of Theorem \\ref{sba}. In this case the $E_1$-page is given by\n\t$$E_1^{4,2,0} \\cong H^{4,2}(\\X_A) \\cong \\ker_3,$$\n\t$$E_1^{4,2,1} \\cong H^{2,1}(\\X_A) \\cong 0,$$\n\t$$E_1^{4,2,2} \\cong H^{0,0}(\\X_A) \\cong \\Z.$$\n\tIn order to compute the $E_2$-page we also need\n\t$$E_1^{3,2,1} \\cong H^{1,1}(\\X_A) \\cong k^*,$$\n\t$$E_1^{5,2,1} \\cong H^{3,1}(\\X_A) \\cong \\ker_2.$$\n\tNote that $E_2^{4,2,2}$ is the kernel of the differential $d_1^{4,2,2}: E_1^{4,2,2} \\rightarrow E_1^{5,2,1}$, i.e. $$E_2^{4,2,2} \\cong \\ker(\\Z \\xrightarrow{\\cdot 2[A]} \\ker_2) \\cong \\Z,$$\n\twhile $E_2^{4,2,0}$ is the cokernel of $d_1^{3,2,1}:E_1^{3,2,1} \\rightarrow E_1^{4,2,0}$, i.e. the cokernel of the homomorphism $k^* \\xrightarrow{\\cdot [A]} \\ker_3$. Since $E_2^{5,2,0} \\cong E_1^{5,2,0} \\cong H^{5,2}(\\X_A)$ is $n$-torsion, we have that $E_{\\infty}^{4,2,2} \\cong E_3^{4,2,2} \\cong \\Z$. Moreover, $E_{\\infty}^{4,2,0} \\cong E_2^{4,2,0}$ and we get a filtration\n\t$$F^{4,2,0} \\hookrightarrow F^{4,2,1} \\hookrightarrow F^{4,2,2} \\cong H^{4,2}(\\seb(A))$$\n\tsuch that $E_{\\infty}^{4,2,0} \\cong F^{4,2,0}$, $E_{\\infty}^{4,2,1} \\cong F^{4,2,1}\/F^{4,2,0}$ and $E_{\\infty}^{4,2,2} \\cong F^{4,2,2}\/F^{4,2,1}$. Since $E_{\\infty}^{4,2,1} \\cong 0$, we obtain a short exact sequence\n\t$$0 \\rightarrow E_{\\infty}^{4,2,0} \\rightarrow F^{4,2,2} \\rightarrow E_{\\infty}^{4,2,2} \\rightarrow 0$$\n\tthat is exactly the one we aimed to get.\n\\end{proof}\n\nThe previous result was already obtained by Peyre in \\cite{peyre} by using different techniques. We have reported this new proof anyways as an example of a possible approach to the computation of Chow groups (and, more generally, motivic cohomology groups) of Severi-Brauer varieties by means of the spectral sequence in Theorem \\ref{sba}. Of course, in order to get any information on the torsion of $CH^i(\\seb(A))$ for $i \\geq 3$ by using our spectral sequence one should first compute $H^{p,q}(\\X_A)$ for $p \\geq q+3$, which are generally unknown, at the best of our knowledge.\n\n\\section{Torsion classes in $H^{**}(BPGL_n)$}\n\nIn this section, following \\cite{gu2} and \\cite{gu3}, we find torsion classes in the motivic cohomology of $BPGL_n$ . This allows also to generalise some results about the Chow groups of $B_{\\acute et}PGL_n$ from the complex numbers (see \\cite[Theorem 1.1]{gu2} and \\cite[Theorem 1]{gu3}) to more general fields. Indeed, we only require that the base field $k$ has characteristic not dividing $n$ and contains a primitive $n$th root of unity.\n\nFirst, let $n=p$ be an odd prime and consider the finite subgroup $C_p \\times \\mu_p$ of $PGL_p$ described in \\cite[Section 5]{vistoli}. Recall that $C_p$ is the subgroup of the symmetric group $S_p \\subset PGL_p$ generated by the cycle $\\sigma = (1 \\: 2 \\: \\dots \\: p)$ and $\\mu_p$ is the subgroup of $PGL_p$ generated by the diagonal matrix $\\rho =[\\omega,\\dots,\\omega^{p-1},1]$, where $\\omega$ is a primitive $p$th root of unity. Note that $\\rho \\sigma= \\omega \\sigma \\rho$ in $GL_p$, so the two generators commute in $PGL_p$. The inclusion $\\iota: C_p \\times \\mu_p \\rightarrow PGL_p$ induces a homomorphism $B\\iota^*:H^{**}(BPGL_p,\\Z\/p) \\rightarrow H^{**}(B(C_p \\times \\mu_p),\\Z\/p)$. \n\nRecall from Theorem \\ref{bpg} that $H^{2,1}(BPGL_p)\\cong 0$ and $H^{3,1}(BPGL_p) \\cong \\Z\/p$, so the Bockstein homomorphism $\\bock: H^{2,1}(BPGL_p,\\Z\/p) \\rightarrow H^{3,1}(BPGL_p)$ is an isomorphism. Let $z$ be the class in $H^{2,1}(BPGL_p,\\Z\/p)$ such that $x=\\bock(z)$. By \\cite[Theorem 1.1]{rolle}, we know that $H^{2,1}_{\\acute et}(BPGL_p,\\Z\/p) \\cong H^2_{\\acute et}(BPGL_p,\\mu_p)$ is the group of central extensions of $PGL_p$ by $\\mu_p$. \n\nBefore proceeding note also that the change of topology homomorphisms\n$$H^{2,1}(-,\\Z\/p) \\rightarrow H^{2,1}_{\\acute et}(-,\\Z\/p)$$\nand \n$$H^{2,1}(-) \\rightarrow H^{2,1}_{\\acute et}(-)$$\nare respectively a monomorphism and an isomorphism for all simplicial schemes by \\cite[Theorems 6.17 and 6.18]{voevodsky.motivic}.\n\n\\begin{lem}\\label{ce}\nThe change of topology homomorphism $H^{2,1}(BPGL_p,\\Z\/p) \\rightarrow H^{2,1}_{\\acute et}(BPGL_p,\\Z\/p)$ sends $z$ to the central extension\n$$1 \\rightarrow \\mu_p \\rightarrow SL_p \\rightarrow PGL_p \\rightarrow 1.$$\n\\end{lem}\n\\begin{proof}\n\tWe have a commutative square\n\t\t$$\n\t\\xymatrix{\n\t\tH^{2,1}(BPGL_p,\\Z\/p) \\ar@{->}[r] \\ar@{->}[d]_{\\bock} & H_{\\acute{e}t}^{2,1}(BPGL_p,\\Z\/p) \\cong H^2_{\\acute et}(BPGL_p,\\mu_p) \\ar@{->}[d]^{\\bock}\\\\\n\t\tH^{3,1}(BPGL_p) \\ar@{->}[r] & H^{3,1}_{\\acute et}(BPGL_p)\\cong H^2_{\\acute et}(BPGL_p,\\Gm) .\n\t}\n\t$$\n\tNote that $H_{\\acute{e}t}^{2,1}(BPGL_p) \\cong H^{2,1}(BPGL_p) \\cong 0$, so the Bockstein on the right is a monomorphism. Now, the statement immediately follows from the fact that $x=\\bock(z)$ maps to the central extension\n\t$$1 \\rightarrow \\Gm \\rightarrow GL_p \\rightarrow PGL_p \\rightarrow 1$$\n\tin $H^{3,1}_{\\acute et}(BPGL_p) \\cong H^2_{\\acute et}(BPGL_p,\\Gm)$.\n\t\\end{proof}\n\nIt follows from \\cite[Lemma 2.3]{rolle} that $$H^{**}(B(C_p \\times \\mu_p),\\Z\/p) \\cong H^{**}(k)[a,b,u,v]\/(a^2=b^2=0)$$\nwith $a$ and $b$ in bidegree $(0)[1]$, $u$ and $v$ in bidegree $(0)[2]$, such that $\\beta(a)=u$ and $\\beta(b)=v$ (where $\\beta$ is the reduction mod $p$ of $\\bock$).\n\n\\begin{lem}\n\tWe have that\n\t$$B\\iota^*(z)=\\lambda \\tau ab,$$\n\twhere $\\lambda $ is a non-zero element in $\\Z\/p$ and $\\tau$ is the class in $H^{0,1}(k,\\Z\/p) \\cong \\mu_p(k)$ corresponding to the primitive $p$th root of unity $\\omega$.\n\\end{lem}\t\n\\begin{proof}\n Note that $B\\iota^*(z)$ is the class in $H^{2,1}(B(C_p \\times \\mu_p),\\Z\/p)$ that maps via the change of topology homomorphism to the central extension\n$$1 \\rightarrow \\mu_p \\rightarrow G \\rightarrow C_p \\times \\mu_p \\rightarrow 1$$\n\tinduced by the one in Lemma \\ref{ce}. The class $B\\iota^*(z)$ is non-zero since the previous extension is non-split, but restricts to a split extension both of $C_p$ and of $\\mu_p$.\n\t\n\tBy degree reasons $B\\iota^*(z)$ has the following general form\n\t$$B\\iota^*(z)=\\lambda \\tau ab +\\lambda_u \\tau u +\\lambda_v \\tau v + \\{r_a\\} a + \\{r_b\\} b$$\n\twhere $\\lambda$, $\\lambda_u$ and $\\lambda_v$ are in $\\Z\/p$ and $\\{r_a\\}$ and $\\{r_b\\}$ are in $K^M_{1}(k)\/p$. Since $B\\iota^*(z)$ restricts to zero both in $H^{2,1}(BC_p,\\Z\/p)$ and in $H^{2,1}(B\\mu_p,\\Z\/p)$, we deduce that $\\lambda_u=\\lambda_v=0$ and $\\{r_a\\}=\\{r_b\\}=0$. Therefore, $B\\iota^*(z)= \\lambda \\tau ab$ that concludes the proof.\n\t\\end{proof}\n\n\\begin{prop}\n\tThere are non-trivial classes $z_{p,k}$ in $H^{2p^{k+1}+1,p^{k+1}}(BPGL_p,\\Z\/p)$ for all $k \\geq 0$.\n\\end{prop}\n\\begin{proof}\nFor all $k \\geq 0$, define classes\n$$z_{p,k}=\\Pa^{p^k}\\Pa^{p^{k-1}} \\cdots \\Pa^p\\Pa^1\\beta(z)$$\nwhere $\\Pa^i$ are the motivic Steenrod $p$th power operations constructed in \\cite{voevodsky.reduced}.\n\nSince $\\beta(z)$ is mapped by $B\\iota^*$ to $\\lambda \\tau(ub-av)$, by Cartan formula we have that\n$$B\\iota^*(z_{p,k})= \\lambda \\tau (u^{p^{k+1}}b-av^{p^{k+1}})$$\nfor any $k\\geq 0$. Hence, $z_{p,k}$ is non-trivial for all $k$ that is what we aimed to show.\n\\end{proof}\n\n\\begin{prop}\n\tThere are non-trivial $p$-torsion classes $y_{p,k}$ in $H^{2p^{k+1}+2,p^{k+1}}(BPGL_p)$ for all $k \\geq 0$.\n\\end{prop}\n\\begin{proof}\n\tDefine $y_{p,k}$ as $\\bock(z_{p,k})$ where $\\bock:H^{**}(BPGL_p,\\Z\/p) \\rightarrow H^{**}(BPGL_p)$ is the Bockstein homomorphism. Note that the reduction mod $p$ of $y_{p,k}$ is nothing but $\\beta(z_{p,k})$ which is non-trivial since maps to $\\lambda \\tau (u^{p^{k+1}}v-uv^{p^{k+1}})$ via $B\\iota^*$. This finishes the proof.\n\t\\end{proof}\n\nNote that the classes $z$, $\\beta(z)$, $z_{p,k}$ and $\\beta(z_{p,k})$ are not $\\tau$-torsion.\n\nRecall from \\cite{morel.voevodsky} that the \\'etale classifying space $B_{\\acute et}G$ is defined as the object ${\\mathrm R}\\pi_*\\pi^*(BG)$ in ${\\mathcal H}_s(k)$, where $(\\pi^*,{\\mathrm R}\\pi_*)$ is the couple of adjoint functors induced by the morphism of sites $\\pi:(Sm\/k)_{\\acute et} \\rightarrow (Sm\/k)_{Nis}$. \n\n\\begin{prop}\n\tThere are non-trivial $p$-torsion classes ${\\upsilon}_{p,k}$ in $CH^{p^{k+1}+1}(B_{\\acute et}PGL_p)$ for all $k \\geq 0$.\n\\end{prop}\n\\begin{proof}\n\tBy \\cite[Theorem 6.17]{voevodsky.motivic} we have an isomorphism $H^{2,2}(B_{\\acute et}PGL_p,\\Z\/p) \\rightarrow H^{2,2}(BPGL_p,\\Z\/p)$. Let $\\zeta$ be the class in $H^{2,2}(B_{\\acute et}PGL_p,\\Z\/p)$ lifting $\\tau z$ and define\n\t$${\\upsilon}_{p,k}=\\bock \\Pa^{p^k}\\Pa^{p^{k-1}} \\cdots \\Pa^p\\Pa^1\\beta ({\\zeta}).$$\n\tThe classes ${\\upsilon}_{p,k}$ are non-trivial since their reductions mod $p$ map to $\\tau \\beta(z_{p,k})$.\n\t\\end{proof}\n\n\tLet $p$ be an odd prime dividing $n$. Then, the diagonal map $\\Delta: PGL_p \\rightarrow PGL_n$ induces a homomorphism $H^{**}(BPGL_n) \\rightarrow H^{**}(BPGL_p)$ that maps $x$ to $x$. Since the classes $z_{p,k}$, $y_{p,k}$ and $\\upsilon_{p,k}$ for $BPGL_p$ are constructed starting from $\\beta(z)$ (that is the reduction mod $p$ of $x$), we can define in the same way classes for $BPGL_n$. This immediately implies the following result.\n\t\n\t\\begin{cor}\\label{gentor}\n\t\tFor any odd prime $p$ dividing $n$ and $k \\geq 0$, there are non-trivial $p$-torsion classes:\n\t\t\n\t\t1) $z_{p,k}$ in $H^{2p^{k+1}+1,p^{k+1}}(BPGL_n,\\Z\/p)$;\n\t\t\n\t\t2) $y_{p,k}$ in $H^{2p^{k+1}+2,p^{k+1}}(BPGL_n)$;\n\t\t\n\t\t3) ${\\upsilon}_{p,k}$ in $CH^{p^{k+1}+1}(B_{\\acute et}PGL_n)$.\n\t\\end{cor}\n \n\n\\footnotesize{\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nIn this paper we consider biased positional games played on the edge set of the complete graph $K_n$ in which the winning sets are spanning subgraphs. Biased positional games were introduced by Chv\\'atal and Erd\\H{o}s~\\cite{ce1978} in 1978 and form a central part of positional games, see, for example, the monograph by Beck~\\cite{BeckBook}, or~\\cite{hkss2014, k2014} for a more recent treatment. \n\nLet $X$ be a finite set and let ${{\\mathcal F}}\\subseteq 2^X$ be a family of subsets. The set $X$ is called the {\\em board} and ${{\\mathcal F}}$ is referred to as the family of {\\em winning sets}. In the $b$-biased \\emph{Maker--Breaker} game $(X,{{\\mathcal F}})$, two players called Maker and Breaker play in rounds. In every round Maker claims one previously unclaimed element of $X$ and Breaker responds by claiming $b$ previously unclaimed elements of $X$. Maker wins if she claims all elements of some $F\\in {{\\mathcal F}}$, otherwise Breaker wins the game. By definition a draw is impossible and thus exactly one player has a winning strategy since Maker--Breaker games are perfect information games. \n\nA certain class of games that received particular attention are Maker--Breaker games played on the edge set of the complete graph on $n$ vertices, denoted by $K_n$, in which case $X$ is the set of all unordered 2-element subsets of $K_n$, denoted by $\\binom{[n]}{2}$. In the {\\em connectivity game}, the {\\em perfect matching game}, the {\\em Hamiltonicity game}, and the {\\em triangle game}, for example, the winning sets are the edge sets of all spanning trees, all perfect matchings, all Hamilton cycles, and all copies of $K_3$, respectively. When $n$ is large enough these games are heavily in favour of Maker in the {\\em unbiased} version when $b=1$. Chv\\'atal and Erd\\H{o}s~\\cite{ce1978} therefore examined the biased variant for these games. Define the threshold bias $b^*$ of a game~$(X,{\\mathcal F})$ to be the largest integer $b$ such that Maker wins the $b$-biased Maker--Breaker game~$(X,{\\mathcal F})$. Note that Maker--Breaker games are bias-monotone, that is Maker wins for every $b\\le b^*$ and Breaker wins for every $b>b^*$. \n\nChv\\'atal and Erd\\H{o}s found that the threshold bias $b^*$ is of the order $n\\ln n$ for the connectivity, the perfect matching, and the Hamiltonicity game; and of order $\\sqrt{n}$ for the triangle game. The order of the threshold bias for an $H$-game, the game in which winning sets correspond to copies of $H$ in $K_n$, was later determined by Bednarska and {\\L}uczak~\\cite{bl2000} for any fixed graph $H$.\nExcept for the connectivity game, all the aforementioned games \ncan be cast in the following common form. Given a graph $H=H_n$, what is the threshold bias $b^*$ of the Maker--Breaker game played on $K_n$ in which all winning sets are copies of $H_n$? In the case of Hamiltonicity we simply have $H_n = C_n$, a cycle of length $n$, and in the perfect matching game $H_n$ is a collection of $n\/2$ vertex disjoint edges. \n\nThere are choices of $H_n$ for which Maker cannot win even if $b = 1$. A trivial such example is $H_n = K_n$, however even for $H_n$ being a complete graph with only $2 \\log n$ vertices Maker cannot win the $H_n$-game \\cite[Theorem 6.4]{BeckBook}. It turns out that this can be avoided if we restrict our attention to graphs with maximum degree some constant $\\Delta$, and let $n$ be sufficiently large. Furthermore, rather than asking for the threshold bias for a specific $H_n$, we seek a \\emph{universal} upper bound: given $\\Delta$ and $n$, what is the largest $b_\\Delta = b_\\Delta(n)$ such that, on the one hand, for every graph $H_n$ with at most $n$ vertices and maximum degree at most $\\Delta$ Maker can win a $b$-biased $H_n$-game with $b \\le b_\\Delta$, and on the other hand there exists at least one such $H_n$ for which Breaker can win with bias $b = b_\\Delta$ + 1?\n\nRecently, Allen, B\\\"ottcher, Kohayakawa, Naves, and Person~\\cite{abknp2017} showed that $b_\\Delta(n)$ is of order at least $\\Omega((n\/\\log n)^{1\/\\Delta})$\\footnote{All asymptotic statements refer to $n$, the number of vertices, tending to $\\infty$.}. The triangle-preventing strategy for Breaker due to Chv\\'atal and Erd\\H{o}s~\\cite{ce1978} shows that this is tight up to a factor of $\\sqrt{\\log n}$ when $\\Delta = 2$. Furthermore, when $\\Delta = 3$ the authors of~\\cite{abknp2017} show that Breaker can win a \\emph{$K_4$-factor} game for some $b = \\Omega(n^{1\/3})$, which shows (almost) optimality in this case as well. Here the $K_4$-factor and, in general, a $K_r$-factor, corresponds to a graph $H_n$ which consists of $\\lfloor n\/r \\rfloor$ vertex-disjoint copies of $K_r$. However, the authors of~\\cite{abknp2017} have expressed a belief that their lower bound of $\\Omega((n\/\\log n)^{1\/\\Delta})$, in general, is not optimal. We provide evidence for this feeling by determining the order of the threshold bias for the $K_{\\Delta + 1}$-factor game for all $\\Delta \\ge 3$. For $\\Delta = 3$, the threshold bias matches the upper bound in~\\cite{abknp2017}, while for $\\Delta \\ge 4$ the exponent of $n$ of the threshold bias is strictly larger than $1\/\\Delta$.\n\n\\begin{theorem} \\label{thm:main}\n For any integer $r \\ge 4$ there exist $c, C > 0$ such that the following holds for every $n \\in r \\mathbb{Z}$.\n \\begin{enumerate}[label={(\\roman*)}]\n \\item\\label{main:M} If $b < c n^{2\/(r+2)}$ then Maker has a winning strategy in the $b$-biased $K_r$-factor game played on the edge set of $K_n$. \n\n \\item\\label{main:B} If $b > C n^{2\/(r+2)}$ then Breaker has a winning strategy in the $b$-biased $K_r$-factor game played on the edge set of $K_n$. \n \\end{enumerate}\n\\end{theorem}\n\\begin{remark}\nBy taking $c$ and $C$ to be sufficiently small and large, respectively, we have that the theorem vacuously holds for all $n < n_0$ for any chosen $n_0$. Therefore, we assume throughout the paper that $n$ is as large as needed for the calculations to be correct.\n\\end{remark}\n\nFor $b \\ge C n^{2\/(r+2)}$ we show that Breaker has a strategy to `isolate' one particular vertex from being in a copy of $K_r$, which clearly prevents Maker's graphs from containing a $K_r$-factor. Somewhat surprisingly, though not uncommon in extremal and probabilistic combinatorics, this turns out to be Breaker's best strategy: as soon as he cannot achieve this Maker is able to build a $K_r$-factor. \n\nTheorem \\ref{thm:main} suggests the following. \n\\begin{conjecture}\nFor all $\\Delta \\ge 3$, $b_\\Delta = \\Theta(n^{2\/(\\Delta + 3)}).$\n\\end{conjecture}\nIn other words, we believe that it is not significantly harder for Maker to build any other graph of maximum degree $\\Delta$ than a $K_{\\Delta+1}$-factor. \nWe take justification for this assumption from two similar settings in extremal graph theory and in random graph theory. The celebrated theorem of Hajnal and Szemer\\'edi~\\cite{hs1970} states that every graph $G$ of minimum degree at least $(1-1\/(\\Delta+1))n$ contains a $K_{\\Delta + 1}$-factor, and that condition is tight. \nBollob\\'as and Eldridge~\\cite{be1978}, and independently Catlin~\\cite{c1976}, conjectured that the condition $\\delta(G)\\ge (1-1\/(\\Delta + 1))n$ is in fact sufficient to contain every graph $H$ with $n$ vertices and maximum degree $\\Delta$. \nA similar assumption is made on the threshold bias $p^*$ for the random graph $G(n,p)$ to contain a certain graph $H_n$. Johansson, Kahn and Vu~\\cite{jkv2008} showed that $p^*(n)=(n^{-1}\\log^{1\/\\Delta}n)^{2\/(\\Delta+1)}$ is a threshold function for $G(n,p)$ to contain a $K_{\\Delta + 1}$-factor. It is folklore belief that, for every graph $H_n$ on at most $n$ vertices and of maximum degree $\\Delta$, the function $p^*(n)$ is in fact an upper bound on the threshold functions for $G(n,p)$ to contain $H_n$, see for example Conjecture~1.3 in~\\cite{fln2017}. Supporting evidence towards this conjecture is given by Ferber, Luh and Nguyen~\\cite{fln2017} who prove it when $H_n$ is almost-spanning, that is when $H_n$ occupies at most $(1-\\ensuremath{\\varepsilon})n$ vertices. \n\n\n\n\\medskip\n\\noindent\n{\\bf Structure of the paper.} \\\\\nIn Section 2 we take a little detour and discuss the \\emph{probabilistic intuition}, also called the {\\em Erd\\H{o}s paradigm}. While this paradigm in its basic form does not apply to the problem we consider here, a variation of it due to Allen et al.~\\cite{abknp2017} (Theorem \\ref{thm:maker_rg}) turns out to give the correct answer. This result will also serve us to provide further intuition why the threshold bias in Theorem~\\ref{thm:main} is of the order $n^{2\/(r+2)}$, or, more precisely, why Breaker is not able to isolate a single vertex from being in a copy of $K_r$ for $b < cn^{2\/(r+2)}$. In Section 3 we fix notation and state preliminary results. In Section 4, we provide Maker's strategy and prove Theorem~\\ref{thm:main}~\\ref{main:M}. Section 5 is devoted to Breaker's strategy, i.e.~Theorem~\\ref{thm:main}~\\ref{main:B}. \n\n\n\\section{Probabilistic Intuition Revised} \nChv\\'atal and Erd\\H{o}s~\\cite{ce1978} found a surprising connection between biased positional games and random graphs. Replace Maker and Breaker by RandomMaker and RandomBreaker, respectively, who choose their edges uniformly at random from all unclaimed edges. At the end of the game, the graph of RandomMaker has the same distribution as $G(n,m)$, a graph with $m$ edges chosen uniformly at random from all $\\binom{n}{2}$ possible edges, where $m$ is roughly $\\binom{n}{2}\/(b+1)$ (we omit floor and ceiling signs unless crucial). It is well known~\\cite{JLRbook} that $G(n,m)$ is (a) connected, (b) has a perfect matching, or (c) has a Hamilton cycle with probability tending to 0 if $m\\ll n \\ln n$, and with probability tending to 1 if $m\\gg n \\ln n$. That is, the threshold biases of the random version of the connectivity, the perfect matching, and the Hamiltonicity game are of the order $n\\ln n$. The results in~\\cite{ce1978} imply that the threshold bias $b^*$ in the game with clever players is of the same order of magnitude for the connectivity, the perfect matching, and the Hamiltonicity game. This phenomenon is often called the {\\em random graph intuition}, or the {\\em Erd\\H{o}s paradigm}. In fact, it turns out that the threshold biases for the random and the clever game are asymptotically equal in the connectivity game~\\cite{gs2009} and in the Hamiltonicity game~\\cite{k2011}. \n\nIt is one of the central questions in positional games to classify games for which the random graph intuition applies. A game which does very much not obey the random graph intuition is the above-mentioned triangle game or, more generally, an $H$-game for a fixed graph $H$ which contains a cycle. It is well-known that the threshold for the appearance of a triangle in $G(n,m)$ is of the order $\\Theta(n)$ (see, e.g.,~\\cite{JLRbook}). Chv\\'atal and Erd\\H{o}s~\\cite{ce1978}, however, showed that Breaker can prevent a triangle in Maker's graph when playing with a bias $b =\\Theta(\\sqrt{n})$. \n\nIt follows from Beck's winning criterion for Breaker~\\cite{b1982}, a generalisation of the classical Erd\\H{o}s-Selfridge criterion to biased games, that Breaker can {\\em always} play at least as good as RandomBreaker against RandomMaker. A result by Chv\\'atal and Erd\\H{o}s~\\cite{ce1978} shows that in some cases Breaker can play in a smarter way than just claiming edges at random. Bednarska and \\L{}uczak~\\cite{bl2000} verified that this is also the case for any $H$-game. However, the main message of their paper is not that the probabilistic intuition completely fails in these games, but rather that it has to be slightly adjusted. \n\nAs mentioned before, if both players play at random then Maker's graph is distributed as a random graph $G(n, m)$ for $m = \\binom{n}{2} \/ (b + 1)$. If Breaker does not play at random then by Maker still playing uniformly at random from the set of all \\emph{available} elements we lose control over the distribution of its graphs. To circumvent this, Bednarska and \\L{}uczak~\\cite{bl2000} suggested the following strategy for Maker: choose a next element uniformly at random from the set of \\emph{all} elements (even those that have been previously claimed) and take it only if it forms a valid move, i.e.\\ if it has not been previously claimed. Observe that Maker's graph obtained following this strategy is not a random graph but rather a subgraph obtained from a random graph after deleting a few edges. Thus, even though we might not have a fine control over the actual Maker's graph, knowing that it is a subgraph of a random graph turns out to give sufficient information to win an $H$-game. In particular, they show that when $b$ is not too large, the random graph $G(n,\\binom{n}{2}\/(b+1))$ is {\\em globally robust} with respect to containing a copy of $H$, which in turn implies that Maker has a winning strategy. That is, even after removing any small proportion of the edges the plucked random graph still contains a copy of $H$. For a precise definition of robustness we refer the reader to~\\cite{sv2008} where a systematic study of this concept was initiated. \n\nThe next step in explaining a connection between Maker--Breaker games and random graphs was done by Ferber, Krivelevich and Naves~\\cite{fkn2015}. While the strategy of playing purely at random works well in the case of $H$-games for graphs $H$ of fixed size, it fails when the winning sets are spanning subgraphs of $K_n$ as Breaker can isolate a vertex before Maker is likely to claim an edge incident to that vertex. To manifest the connection between Maker--Breaker games and random graphs for these spanning-graph games, Ferber, Krivelevich and Naves~\\cite{fkn2015} provided a {\\em local-resilience} analogue to the theorem in~\\cite{bl2000} and showed that in a $b$-biased game played on $K_n$, Maker can claim a subgraph of $G(n,m)$ for $m=\\Theta(n^2\/b)$ such that each vertex is incident to $\\Omega(n\/b)$ Maker's edges. Thus, if Maker tries to achieve a graph property ${\\mathcal P}$ that cannot be destroyed by deleting a fixed proportion of edges at each vertex then the strategy in~\\cite{fkn2015} yields a winning strategy for Maker. In particular, lower bounds on the threshold bias for several games like the perfect matching game, the connectivity and the Hamiltonicity game could be re-established this way, though with a sub-optimal constant factor. \n\nHowever, as the reader could guess, the approach via local resilience does not work for all spanning-structure Maker--Breaker games on $K_n$. The property of containing a $K_3$-factor, for example, is not locally resilient as all triangles in $G(n,m)$ containing a fixed vertex $v$ can be destroyed by removing a vanishing proportion of edges incident to every vertex, see e.g.~\\cite{huang2012bandwidth}. For the same reason, the property of containing a $K_r$-factor, $r\\ge 4$, is not locally resilient and the approach in~\\cite{fkn2015} is not applicable. \nCircumventing the short-coming of the resilience-type approaches, Allen, B\\\"ottcher, Kohayakawa, Naves, and Person~\\cite{abknp2017} finally show that Maker can also assume not only that its graph is a subgraph of a random graph with minimum degree of order $n\/b$, but also that the neighbourhood of each vertex has sufficiently many edges. The following theorem makes this precise. For a real $p \\in [0, 1]$ and an integer $n$, we write $\\Gamma \\sim G(n,p)$ if $\\Gamma$ is formed by starting with an empty graph on $n$ vertices and adding each possible edge with probability $p$, independently of all other edges. Furthermore, $\\Gamma\\sim G(n,p)$ satisfies a certain property ${\\mathcal P}$ {\\em asymptotically almost surely (a.a.s)} if the probability that $\\Gamma$ satisfies ${\\mathcal P}$ tends to 1 as $n\\to\\infty$.\n\n\\begin{theorem} \\label{thm:maker_rg}\n For every $n$, $\\gamma = \\gamma(n) \\in (0, 1)$, $p \\ge 10^8 \\gamma^{-2} n^{-1\/2}$, and $b \\le 10^{-24} \\gamma^6 p^{-1}$ the following holds. In the $b$-biased Maker-Breaker game played on $K_n$, for any fixed strategy of Breaker, if Maker draws a random graph $\\Gamma \\sim G(n,p)$ then a.a.s.~$\\Gamma$ is such that Maker can claim a spanning subgraph $G$ of $\\Gamma$ with $\\delta(G) \\ge (1 - \\gamma)np$ and $e_G(N_\\Gamma(v)) \\ge (1 - \\gamma)p^3 n^2 \/ 2$ for every $v \\in V(\\Gamma)$.\n\\end{theorem}\n\nUsing Theorem \\ref{thm:maker_rg} in combination with a sparse blow-up lemma from~\\cite{abhkp2016}, Allen et al.\\ \\cite{abknp2017} show that, for some $b =\\Omega ((n\/\\log n)^{1\/\\Delta})$, these {\\em neighbourhood properties} are enough for $G$ to contain all graphs of maximum degree $\\Delta$ on at most $n$ vertices. \n\n\nFor which $p$ can we guarantee that the neighbourhood properties given by Theorem \\ref{thm:maker_rg} guarantee that every vertex of $G$ is contained in a copy of $K_r$? \nThe neighbourhood $N_G(v)$ of a vertex $v$ in $G$ has size roughly $pn$, and the graph induced on $N_G(v)$ is a subgraph of $G(n,p)$ that still contains about $(1-\\ensuremath{\\varepsilon})n^2p^3\/2$ edges, i.e.~all but a small proportion of edges of $G(n,p)$ in $N_G(v)$ are also edges of $G$. That is, the subgraph of $G$ induced by $N_G(v)$ has roughly the same distribution as the random graph $G(np,p)$, and for the latter to robustly contain a copy of $K_{r-1}$ it is enough to have $p > C(np)^{-2\/r}$ for some constant $C$, which translates to $p>Cn^{-2\/(r+2)}$. It turns out that this is the main obstacle for Maker to create a $K_r$-factor.\n\n\n\n\\section{Preliminaries}\n\nWe use standard graph-theoretic notation. All considered graphs are finite and simple. Given a graph $G$, we let $e(G)$ and $v(G)$ denote its number of edges and vertices, respectively. Given a set $X \\subseteq V(G)$, let $e_G(X)$ denote the number of edges of $G$ with both endpoints in $X$. Similarly, for disjoint subsets $X, Y \\subseteq V(G)$ we let $e_G(X, Y)$ denote the number of edges of $G$ with one endpoint in $X$ and the other in $Y$. Given a vertex $v \\in V(G)$, we let $N_G(v)$ denote its neighbourhood, and for a set $X$ let $N_G(X) = \\bigcup_{v \\in X} N_G(v)$. When $G$ is clear from the context, we omit the subscript. For brevity we also omit floors and ceilings, keeping in mind that all the calculations leave enough margin to accumulate all the rounding errors. \nAll asymptotic statements refer to $n$, the number of vertices, tending to $\\infty$. \nFollowing standard asymptotic notation we write in particular, $f\\ll g$ when $f\/g \\to 0$ as $n\\to\\infty$, and $f\\gg g$ if $g\\ll f$. \n\n\\subsection{Properties of random graphs}\n\nThe following well-known estimates on the likely discrepancy of edges and the concentration of degrees in random graphs follow immediately from Chernoff's inequality and the union bound. \n\\begin{lemma} \\label{lemma:disc}\nLet $p = p(n)$ be such that $n^{-1} \\le p \\le 0.99$. Then a.a.s.~$\\Gamma \\sim\\ensuremath{G(n,p)}$ satisfies the following properties: \n \\begin{itemize}\n \\item For all disjoint subsets $X, Y \\subseteq V(\\Gamma)$ such that $|X| \\le |Y|$ we have\n $$\n e(X, Y) = |X||Y|p \\pm O\\left( |Y| \\sqrt{|X| p \\log (n\/|Y|)} \\right);\n $$\n\n \\item For every subset $X \\subseteq V(\\Gamma)$ we have\n $$\n e(X) = |X|^2p\/2 \\pm O\\left( |X|\\sqrt{|X|p \\log (n\/|X|)} \\right);\n $$\n\n \\item For every vertex $v \\in V(\\Gamma)$ we have \n $$\n |N(v)| = np \\pm O(\\sqrt{np \\log n})\n $$\n \\end{itemize}\n\\end{lemma}\n\nIn order to state the second result we first need some preparation. Given a graph $G$ and $\\ensuremath{\\varepsilon} \\in [0,1]$, we say that a pair of disjoint subsets $V_1, V_2 \\subseteq V(G)$ forms an \\emph{$(\\ensuremath{\\varepsilon})$-regular pair} if for $i=1,2$ and for every $V_i' \\subseteq V_i$ of size $|V_i'| \\ge \\ensuremath{\\varepsilon} |V_i|$ we have\n$$\n \\left| e(V_1', V_2') - |V_1'||V_2'|p \\right| \\le \\ensuremath{\\varepsilon} |V_1'||V_2'| p,\n$$\nwhere $p = e(V_1, V_2)\/|V_1||V_2|$. \nNote that Lemma \\ref{lemma:disc} implies that a.a.s.~every pair of subsets of $\\ensuremath{G(n,p)}$ of size, say, at least $\\log n \/ p$, forms an $(\\ensuremath{\\varepsilon})$-regular pair for every fixed $\\ensuremath{\\varepsilon}>0$. \n\nLet $H$ be a graph with vertex set $\\{1, \\ldots, k\\}$. We denote by ${\\mathcal G}(H, n, m, \\ensuremath{\\varepsilon})$ the collection of all graphs $G$ obtained in the following way: (i) The vertex set of $G$ is a disjoint union $V_1 \\cup \\ldots \\cup V_k$ of sets of size $n$; (ii) For each edge $ij \\in E(H)$, we add to $G$ an $(\\ensuremath{\\varepsilon})$-regular bipartite graph with $m$ edges between the pair $(V_i, V_j)$. Let ${\\mathcal G}^*(H, n, m, \\ensuremath{\\varepsilon})$ denote the family of all graphs $G \\in {\\mathcal G}(H, n, m, \\ensuremath{\\varepsilon})$ which do not contain a copy of $H$. The following result, originally conjectured by Kohayakawa, \\L uczak, and R\\\"odl \\cite{kohayakawa1997onk}, was proven by Balogh, Morris, and Samotij \\cite{balogh2015independent} and, independently, Saxton and Thomason \\cite{saxton2015hypergraph}.\n\n\\begin{theorem} \\label{thm:KLR}\n Let $H$ be a fixed graph and $\\beta > 0$. Then there exist $C, \\ensuremath{\\varepsilon} > 0$ and a positive integer $n_0$ such that \n $$\n \\left| {\\mathcal G}^*(H, n, m, \\ensuremath{\\varepsilon}) \\right| \\le \\beta^m \\binom{n^2}{m}^{e(H)}\n $$\n for every $n \\ge n_0$ and every $m \\ge Cn^{2 - 1\/m_2(H)}$, where\n $$\n m_2(H) = \\max\\left\\{ \\frac{e(H') - 1}{v(H') - 2} \\colon H' \\subset H, \\; v(H) \\ge 3 \\right\\}.\n $$\n\\end{theorem}\n\nTheorem \\ref{thm:KLR} states that a random element from ${\\mathcal G}(H, n, m, \\ensuremath{\\varepsilon})$ is highly unlikely to be $H$-free. Even more, it implies that the random graph $G(n,p)$ is unlikely to contain any graph from ${\\mathcal G}^*(H, \\tilde n, m, \\ensuremath{\\varepsilon})$ for appropriate $\\tilde n$ and $p$. This is made precise in the following lemma.\n\n\n\\begin{lemma} \\label{lemma:KLR_rg}\nLet $H$ be a graph such that $m_2(H) \\ge 2$. Then there exist $\\ensuremath{\\varepsilon}, B > 0$ such that for $n^{-1\/m_2(H)} \\le p = p(n) \\le \\ln^{-2} n$, the graph $\\Gamma \\sim \\ensuremath{G(n,p)}$ a.a.s.~has the property that, for every $\\tilde n \\ge Bp^{-m_2(H)}$, every $m \\ge \\tilde n^2 p \/ 2$ and \nevery graph $G'\\in{\\mathcal G}(H, \\tilde n, m, \\ensuremath{\\varepsilon})$, if $G' \\subseteq \\Gamma$ then $G'$ contains a copy of $H$. \n\\end{lemma}\n\n\\begin{proof}\nLet $\\ensuremath{\\varepsilon}$, $C > 0$ be as given by Theorem~\\ref{thm:KLR} for $H$ and $\\beta = (1\/(2e^2))^{e(H)}$, and set $B = (2C)^{m_2(H)}$. Let $\\Gamma\\sim\\ensuremath{G(n,p)}$. In order to prove the lemma it suffices to show that $\\mu$ vanishes, where $\\mu$ is the expected number of subgraphs of $\\Gamma$ that are isomorphic to an element in ${\\mathcal G}^*(H, \\tilde n, m, \\ensuremath{\\varepsilon})$. \nNote that \n \\begin{align*}\n \\mu &= \\sum_{\\tilde n \\ge B p^{-m_2(H)}} \\sum_{m \\ge \\tilde n^2 p\/2} \\sum_{G\\in {\\mathcal G}^*} \\Pr(G\\se \\Gamma),\n\\end{align*} \nwhere ${\\mathcal G}^* ={\\mathcal G}^*(H, \\tilde n, m, \\ensuremath{\\varepsilon})$. \nNow \n$$\n \\Pr(G\\se \\Gamma)\\le \\binom{n}{\\tilde n k} (\\tilde n k)! p^{e(H) m}\n$$ \nwhere $k=v(H)$ for brevity. \nFurthermore, $m \\ge C \\tilde n^{2 - 1\/m_2(H)}$ follows from $m \\ge \\tilde n^2 p\/2$ and $\\tilde n \\ge B p^{-m_2(H)}$. Thus we can apply the bound on $|{\\mathcal G}^*(H, \\tilde n, m, \\ensuremath{\\varepsilon})|$ given by Theorem \\ref{thm:KLR}. \nWe therefore have that \n \\begin{align} \\label{aux332} \n \\mu &\\le \\sum_{\\tilde n \\ge B p^{-m_2(H)}} \\sum_{m \\ge \\tilde n^2 p\/2}\n \\beta^m \\binom{\\tilde n^2}{m}^{e(H)} \\; \\binom{n}{\\tilde n k } (\\tilde n k)! p^{e(H) m} \\nonumber\\\\\n &\\le \\sum_{\\tilde n \\ge B p^{-m_2(H)}} \\sum_{m \\ge \\tilde n^2 p\/2} \\binom{n}{\\tilde nk} (\\tilde n k)! \n \\beta^m \\left( \\frac{{\\tilde n}^2 e}{m} \\right)^{e(H) m} p^{e(H) m} \\nonumber\\\\\n &\\le \\sum_{\\tilde n \\ge B p^{-m_2(H)}} \\sum_{m \\ge \\tilde n^2 p\/2} n^{2 \\tilde nk}\n \\left( \\beta^{1\/e(H)} 2 e \\right)^{e(H) m} \\nonumber\\\\\n &\\le \\sum_{\\tilde n \\ge Bp^{-m_2(H)}} \\sum_{m \\ge \\tilde n^2 p \/ 2} \n \\exp\\left(2 k\\tilde n \\ln n - m \\right),\n \\end{align}\n where the third inequality follows from $m\\ge \\tilde n^2p\/2$, and the last inequality follows from our choice of $\\beta$ and $e(H)\\ge 1$ since $m_2(H)\\ge 2$. Now, for sufficiently large $n$, \n $$\n 2 k \\tilde n \\ln n - m \\le \\tilde n (2k \\ln n - \\tilde n p \/2) \\le \\tilde n (2k \\ln n - Bp^{-1}\/2) < - \\ln^2 n,\n $$\nfrom the lower bound on $m$, the lower bound on $\\tilde n$ and $m_2(H)\\ge 2$, and from the upper bound on $p$, respectively. \nThus the final expression in~\\eqref{aux332} tends to 0 as $n\\to \\infty$. The assertion of the lemma follows from Markov's Inequality.\n\\end{proof}\n\nWe remark that the condition $m_2(H) \\ge 2$ is purely for convenience, and in fact $m_2(H) > 1$ would work as well (having an impact only on the upper bound on $p$). It should be noted that the previous lemma could also be derived from a result of Conlon, Gowers, Samotij, and Schacht \\cite{conlon2014klr}. Finally, to apply the previous result in our proof we make use of the following lemma (see, e.g., \\cite[Lemma 4.3]{gerke_steger_2005}).\n\n\n\\begin{lemma} \\label{lemma:exact_m_edges}\n Given a positive $\\ensuremath{\\varepsilon} < 1\/6$, there exists a constant $C$ such that any $(\\ensuremath{\\varepsilon})$-regular graph $B = (V_1 \\cup V_2, E)$ contains a $(2\\ensuremath{\\varepsilon})$-regular subgraph $B = (V_1 \\cup V_2,E')$ with $|E'| = m$ edges for all $m$ satisfying $C |V(B)| \\le m \\le |E(B)|$.\n\\end{lemma}\n\n\n\n\n\\subsection{System of disjoint hyperedges}\n\nGiven a hypergraph $H$, we denote by $\\tau(H)$ the size of a smallest \\emph{vertex cover} of $H$, that is the size of a smallest subset $X \\subseteq V(H)$ that intersects all the edges of $H$. Note that if $H$ and $H'$ are hypergraphs on the same vertex set then $\\tau(H \\cup H') \\le \\tau(H) + \\tau (H')$. We make use of the following generalisation of Hall's theorem due to Haxell~\\cite{haxell1995condition}. \n\n\\begin{theorem} \\label{thm:haxell}\n Let $H_1, \\ldots, H_t$ be a family of $r$-uniform hypergraphs on the same vertex set. If for every $I \\subseteq [t]$ we have $\\tau(\\bigcup_{i \\in I} H_i) \\ge 2 r |I|$ then one can choose a hyperedge $h_i \\in E(H_i)$ for each $i \\in [t]$ such that $h_i \\cap h_j = \\emptyset$ for distinct $i, j \\in [t]$.\n\\end{theorem}\n\nThe theorem from \\cite{haxell1995condition} gives a slightly better bound than $2 r |I|$, however for our purposes this is sufficient. \n\n\\section{Maker's strategy} \\label{sec:Maker}\n\nOur proof strategy is to show that Maker can build a graph which has certain properties and then show that these properties imply the existence of a $K_r$-factor. The properties we need are summarised in the following definition.\n\n\\begin{definition}\n Given $\\alpha, \\beta, p \\in [0,1]$ and $r \\in \\mathbb{N}$, we say that a graph $G$ with $n$ vertices is \\emph{$(\\alpha, \\beta, p, r)$-neat} if it has the following properties:\n \\begin{enumerate}[label={(P\\arabic*)}]\n \\item \\label{prop:expand}\n For every $v \\in V(G)$ we have $|N_G(v)| \\ge np\/2$ and for all disjoint $X, Y \\subseteq V(G)$ of size $|X| \\ge \\log n\/p$ and $|Y| \\ge \\alpha n$ there exists a vertex $v \\in X$ with at least $|Y|p\/2$ neighbours in $Y$; \n\n \\item \\label{prop:in_nbr}\n For every $v\\in V(G)$ and every subset $X \\subseteq N_G(v)$ of size $|X| \\ge \\alpha np$ the induced subgraph $G[X]$ contains a copy of $K_{r-1}$;\n\n \\item \\label{prop:chain}\n For all disjoint subsets $V_1, \\ldots, V_{r+1} \\subseteq V(G)$ of size $|V_i| \\ge n^{1 - \\beta}$ each, there exists a copy of $K_{r+1}^{-}$ with one vertex in each $V_i$, where $K_{r+1}^{-}$ is a graph obtained by removing an edge from a complete graph with $r+1$ vertices.\n \\end{enumerate}\n\\end{definition}\n\nThe next lemma is the heart of the proof of Theorem \\ref{thm:main}~\\ref{main:M}. It shows that neat graphs contain $K_r$-factors under certain mild conditions on the parameters. \n\n\\begin{lemma} \\label{lemma:neat_factor}\n For any integer $r \\ge 4$ and a positive $\\beta$ there exists positive $\\alpha$ and $n_0 \\in \\mathbb{N}$ such that any $(\\alpha, \\beta, p, r)$-neat graph with $n_0 < n \\in r \\mathbb{Z}$ vertices and $p \\ge n^{-1\/3}$ contains a $K_r$-factor.\n\\end{lemma}\n\nThe proof of Lemma \\ref{lemma:neat_factor} follows an approach from \\cite{nenadov18triangle} and we postpone it for the next subsection. We now show how Lemma \\ref{lemma:neat_factor} implies the first part of Theorem \\ref{thm:main}. \n\n\\begin{proof}[Proof of Theorem \\ref{thm:main} (i)]\nLet $\\alpha > 0$ be as given by Lemma~\\ref{lemma:neat_factor} for $\\beta = 1\/(5 (r+2)^2(r-1))$, let $K=K(\\alpha)$ be a sufficiently large integer, and suppose $n \\in r \\mathbb{Z}$ is sufficiently large. Let $p= K n^{-2\/(r+2)}$. We show that Maker can build an $(\\alpha, \\beta, p, r)$-neat graph in the $b$-biased Maker--Breaker game where $b= cp^{-1}$ for some small constant $c$. Such a graph contains a $K_r$-factor by Lemma~\\ref{lemma:neat_factor}.\n\nMaker plays two games in parallel: she plays Game 1 in every odd round and Game 2 in every even round, where Game 1 and Game 2 are defined below. Thus both games can be viewed as $(2b)$-biased Maker--Breaker games and can be played completely independently. For the rest of the argument we assume that Breaker has some fixed strategy, that is, for every disjoint pair $(E_M, E_B)$ of subsets of $E(K_n)$, which represents the current set of Maker's and Breaker's edges, he has some fixed rule what to claim next. If we can show that Maker has a winning strategy against an arbitrary such \\emph{rulebook}, then she can win regardless of what Breaker plays. In Game 1, Maker's goal is to build a graph $G_1$ that satisfies \\ref{prop:expand} and \\ref{prop:in_nbr}, and in Game 2 she builds a graph $G_2$ that satisfies \\ref{prop:chain}. Overall, this implies that $G_1 \\cup G_2$ is an $(\\alpha, \\beta, p, r)$-neat graph.\n\n\n \\paragraph{Game 1.} \nLet $\\gamma > 0$ be a constant that we specify later, and let $\\Gamma$ be a graph on $n$ vertices that has the following properties. \n \\begin{enumerate}[label={($\\Gamma$\\arabic*)}]\n \n \\item \\label{game1:maker} \n In the $(2b)$-biased Maker--Breaker game on $K_n$, Maker has a strategy to claim a spanning subgraph \n $G \\subseteq \\Gamma$ with $\\delta(G) \\ge (1 - \\gamma)np$ and \n $e_G(N_\\Gamma(v)) \\ge (1 - \\gamma)p^3n^2\/2$ for every $v \\in V(\\Gamma)$.\n\n \\item \\label{game1:disc} \n $\\Gamma$ satisfies the assertion of Lemma~\\ref{lemma:disc}.\n\n \\item \\label{game1:r_1} \n For every $\\tilde n\\ge Bp^{-r\/2}$, every $m\\ge \\tilde n^2 p\/2$ and every graph $G'\\in {\\mathcal G}(K_{r-1},\\tilde n, m,\\ensuremath{\\varepsilon})$, \n if $G'\\se \\Gamma$ then $G'$ contains $K_{r-1}$ as a subgraph, \n where $B=B(K_{r-1})$ and $\\ensuremath{\\varepsilon}=\\ensuremath{\\varepsilon}(K_{r-1})$ are the constants from Lemma~\\ref{lemma:KLR_rg} applied to $H=K_{r-1}$.\n\n\n \\end{enumerate}\nWe argue briefly that such a graph $\\Gamma$ exists. Let $\\Gamma \\sim \\ensuremath{G(n,p)}$. \nThen $\\Gamma$ satisfies the assertion of~\\ref{game1:maker} a.a.s.~by Theorem~\\ref{thm:maker_rg} if we choose $K=K(\\gamma)$ large enough and $c=c(\\gamma)$ small enough. \nFurthermore, $\\Gamma$ satisfies \\ref{game1:disc} a.a.s.~by Lemma~\\ref{lemma:disc}, and it satisfies~\\ref{game1:r_1} by Lemma~\\ref{lemma:KLR_rg} applied to $H=K_{r-1}$ where we note that $p= K n^{-2\/(r+2)} > n^{-2\/r} = n^{-1\/m_2(K_{r-1})}$ and $m_2(K_{r-1}) \\ge 2$. Therefore, we can choose one particular graph $\\Gamma$ which has these properties. \n\nLet $G \\subseteq \\Gamma$ be a spanning subgraph guaranteed by \\ref{game1:maker}. We show that $G$ satisfies \\ref{prop:expand} and \\ref{prop:in_nbr}. \n\nFor \\ref{prop:expand} note that $G$ can be obtained from $\\Gamma$ by removing at most $2 \\gamma np$ edges touching each vertex since the maximum degree of $\\Gamma$ is at most $(1 + \\gamma)np$, by \\ref{game1:disc}, and since $\\delta(G) \\ge (1 - \\gamma)np$, by \\ref{game1:maker}. \nFurthermore, let $X, Y\\se V(G)$ be disjoint subsets of size $|X|\\ge \\log n\/p$ and $|Y|\\ge \\alpha n$, respectively. Then $e_\\Gamma(X, Y) \\ge (1 - \\gamma)|X||Y|p$ by \\ref{game1:disc}. But then at most $|X| \\cdot 2\\gamma np$ of those edges are not present in $G$ by the preceding observation. \nBy choosing $\\gamma < \\alpha \/ 8$, we have \n $$\n e_G(X, Y) \\ge (1 - \\gamma)|X||Y|p - |X| \\cdot 2 \\gamma np > |X||Y|p\/2.\n $$\n Therefore there exists a vertex $v \\in X$ with at least $|Y|p\/2$ neighbours in $Y$.\n\nFor \\ref{prop:in_nbr} we show that for every $v\\in V(G)$, every subset $X\\se N_G(v)$ of size $|X|\\ge \\alpha np$ hosts a copy of some $G'\\in {\\mathcal G}(K_{r-1},\\tilde n, m,\\ensuremath{\\varepsilon})$, for suitable $\\tilde n$, $m$ and $\\ensuremath{\\varepsilon}$, which contains a copy of $K_{r-1}$ by~\\ref{game1:r_1}. Fix $v\\in V(G)$ and note that we have $|N_{\\Gamma}(v)| = (1\\pm \\gamma) np\\gg \\log n \/p$ by \\ref{game1:disc} and assumption on $p$. Thus, again by \\ref{game1:disc},\n$$ e_\\Gamma(N_\\Gamma(v)) \\le (1 + \\gamma) |N_\\Gamma(v)|^2p\/2 \\le (1 + \\gamma)^3 n^2p^3 \/ 2 < (1 + 4 \\gamma) n^2 p^3 \/ 2, $$ \nwhere in the last inequality we assumed that $\\gamma$ is sufficiently small. From \\ref{game1:maker} we conclude that $G[N_\\Gamma(v)]$ is `missing' at most $5 \\gamma n^2 p^3 \/ 2$ edges. For brevity, let us upper bound this by $3 \\gamma n^2 p^3$. More precisely, there exists a graph $R_v$ on the vertex set $N_\\Gamma(v)$ such that $e(R_v) \\le 3 \\gamma n^2p^3$ and $G[N_\\Gamma(v)] = \\Gamma[N_\\Gamma(v)] \\setminus R_v$. Therefore, for all disjoint $X, X' \\subseteq N_\\Gamma(v)$ we have \n $$\n e_\\Gamma(X, X') - 3 \\gamma n^2 p^3 \\le e_G(X, X') \\le e_\\Gamma(X, X'). \n $$\nAdditionally, if $|X|,|X'|\\gg \\log n \/p$ then $e_\\Gamma(X, X') = (1\\pm \\gamma)|X||X'|p$ by~\\ref{game1:disc}. Let $\\ensuremath{\\varepsilon}' = \\ensuremath{\\varepsilon}(K_{r-1})\/4$, where $\\ensuremath{\\varepsilon}(K_{r-1})$ is given in~\\ref{game1:r_1}. \nIt follows that for any two disjoint subsets $X,X'\\se N_{\\Gamma}(v)$ of size at least $\\ensuremath{\\varepsilon}'\\cdot (\\alpha n p\/r)$ we have \n \\begin{align*}\n e_G(X, X') &\\ge (1 - \\gamma)|X||X'| p - 3 \\gamma n^2 p^3 \n \\ge \\left(1- \\ensuremath{\\varepsilon}'\\right) |X||X'| p\n \\end{align*}\n and \n$$ e_G(X, X') \\le (1+\\gamma) |X||X'| p \\le \\left(1+\\ensuremath{\\varepsilon}'\\right) |X||X'| p,$$ \nif we choose $\\gamma$ small enough in terms of $\\ensuremath{\\varepsilon}'$, $\\alpha$ and $r$. \nThis implies that any two disjoint subsets $V_1, V_2 \\subseteq N_\\Gamma(v)$ of size $\\alpha np\/r$ form a $(2 \\ensuremath{\\varepsilon}')$-regular pair.\n\n\nLet now $X \\subseteq N_G(v) \\subseteq N_\\Gamma(v)$ be of size $\\alpha np$. Arbitrarily choose $r-1$ disjoint subsets $V_1, \\ldots, V_{r-1} \\subseteq X$ of size $ \\tilde n = \\alpha np \/ r$. Note that \n$ \\tilde n \\ge B p^{-r\/2} = B p^{-m_2(K_{r-1})}$, where $B=B(K_{r-1})$ is the constant given by~\\ref{game1:r_1}, since $p \\ge K n^{-2\/(r+2)}$ and $K$ is a sufficiently large constant. As previously observed, every $(V_i, V_j)$ forms a $(2\\ensuremath{\\varepsilon}')$-regular pair with $m_{i,j} = (1 \\pm \\ensuremath{\\varepsilon}') \\tilde n^2 p$ edges,\n thus we can apply Lemma \\ref{lemma:exact_m_edges} to each pair $(V_i, V_j)$ in order to obtain a subset $E_{ij} \\subseteq E_G(V_i, V_j)$ of size exactly \n $$\n m = \\tilde n^2 p \/2\n $$\n such that $(V_i, V_j)$ is $(4\\ensuremath{\\varepsilon}')$-regular, i.e.~$(\\ensuremath{\\varepsilon})$-regular, with respect to $E_{ij}$. This gives us a graph $G' \\in \\ensuremath{\\mathcal G}(K_{r-1}, \\tilde n, m, \\ensuremath{\\varepsilon})$. As $G'$ is a subgraph of $\\Gamma$, from \\ref{game1:r_1} we conclude that it contains a copy of $K_{r-1}$. Thus, $G$ satisfies~\\ref{prop:in_nbr}.\n\n\n \\paragraph{Game 2.} The properties \\ref{prop:in_nbr} and \\ref{prop:chain} are achieved in a fairly similar way. Thus, the analysis of Game 2 follows along the lines of the second part of Game 1. There are some crucial differences in the choice of parameters though. \nRecall that $\\beta = 1\/(5(r+2)^2(r-1))$ and that for property \\ref{prop:chain} we want to find a copy of $K_{r+1}^-$ in certain sets of size $n^{1-\\beta}$. Let $H=K_{r+1}^-$, let now $\\gamma = n^{-3\\beta}$ and let $q \\in (0,1)$ satisfy \n \\begin{equation}\\label{qBounds}\nn^{-\\frac{1-\\beta}{m_2(H)}}\\ll q \\ll n^{-\\frac{2}{r+2}} \\gamma^6.\n\\end{equation} \nNote that this is possible since $m_2(K_{r+1}^-)= \\frac{r+2}{2}-\\frac{1}{r-1}$ and by choice of $\\beta$. Delicate choices for parameters $\\gamma$ and $q$ will become apparent soon. We claim that a random graph $\\Gamma \\sim \\ensuremath{G(n,q)}$ has the following properties with high probability. \n \\begin{enumerate}[label={($\\Gamma$\\arabic*)}]\n \\item \\label{game2:maker} In the $(2b)$-biased Maker--Breaker game on $K_n$, Maker has a strategy to claim a spanning subgraph $G \\subseteq \\Gamma$ with $\\delta(G) \\ge (1 - \\gamma)nq$ and $e_G(N_\\Gamma(v)) \\ge (1 - \\gamma)q^3n^2\/2$ for every $v \\in V(\\Gamma)$.\n\n \\item \\label{game2:disc} $\\Gamma$ satisfies the assertion of Lemma \\ref{lemma:disc};\n\n \\item \\label{game2:r_1} \n For every $\\tilde n\\ge Bq^{-m_2(H)}$, every $m\\ge \\tilde n^2 q\/2$ and every graph $G'\\in {\\mathcal G}(H,\\tilde n, m,\\ensuremath{\\varepsilon})$, \n if $G'\\se \\Gamma$ then $G'$ contains $H$ as a subgraph, \n where now $B=B(H)$ and $\\ensuremath{\\varepsilon}=\\ensuremath{\\varepsilon}(H)$ are the constants from Lemma~\\ref{lemma:KLR_rg} applied to $H=K_{r+1}^-$.\n \n \\end{enumerate}\nLet us verify that $\\Gamma$ indeed has these properties~a.a.s. For~\\ref{game2:maker} let us verify that the conditions of \nTheorem~\\ref{thm:maker_rg} hold. Firstly, $q\\ge 10^{8}\\gamma^{-2}n^{-1\/2}$ is implied by the lower bound in~\\eqref{qBounds} since $r\\ge 4$ and since $\\beta< 1\/50$, say. Secondly, recall that $b= cp^{-1}=O(n^{2\/(r+2)})$. This together with the upper bound in~\\eqref{qBounds} implies that $\\gamma^{6} q^{-1} \\gg b$, so that \\ref{game2:maker} holds a.a.s.~by Theorem~\\ref{thm:maker_rg}.\nJust as in Game 1, $\\Gamma$ satisfies~\\ref{game2:disc}~a.a.s.~by Lemma~\\ref{lemma:disc}. \nFinally, note that the lower bound in~\\eqref{qBounds} implies in particular that $q\\ge n^{-1\/m_2(H)}$. Thus~\\ref{game2:r_1}\nholds a.a.s.~by Lemma~\\ref{lemma:KLR_rg} applied to $H=K_{r+1}^-$. \n \nFix $\\Gamma$ with these three properties and let $G \\subseteq \\Gamma$ be a spanning subgraph satisfying \\ref{game2:maker}. Crucially, we have sacrificed the value of $q$, which is now significantly smaller than $n^{-2\/(r+2)}$, in order to get a smaller error term $\\gamma$. Note that in order to guarantee that $G$ satisfies property \\ref{prop:in_nbr} in Game 1 we needed $p = \\Omega(n^{-2\/(r+2)})$. Here it will turn out that a smaller $p$ (which we denote by $q$) suffices provided $\\gamma$ is sufficiently small. We now make this precise.\n\nFor $\\ensuremath{\\varepsilon}' = \\ensuremath{\\varepsilon}(H) \/4 >0$ consider disjoint subsets $X, X' \\subseteq V(G)$ of size at least $\\ensuremath{\\varepsilon}' n^{1 - \\beta}$. \nFirst note that \n \\begin{align*}\n e_{\\Gamma} (X,X') = (1\\pm \\gamma)|X||X'|q, \n \\end{align*}\n by~\\ref{game2:disc} since $\\gamma^2n^{1-\\beta}q\\gg \\log n$ by choice of $\\beta$ being small enough and~\\eqref{qBounds}. \nAs before, we have that $G$ is obtained from $\\Gamma$ by removing at most $2 \\gamma nq$ edges touching each vertex, which sums to at most $\\gamma n^2 q$ removed edges in total. This, together with the above estimate on $ e_{\\Gamma} (X,X')$, implies that\n \\begin{align*}\n (1 + \\gamma)|X||X'| q \\ge e_G(X, X') &\\ge (1 - \\gamma)|X||X'|q - \\gamma n^2 q \\\\\t&\\ge (1 - \\ensuremath{\\varepsilon}')|X||X'|q, \n \\end{align*}\n since $\\gamma < \\ensuremath{\\varepsilon}'\/2$, say, and $\\gamma n^2 = n^{2-3\\beta}\\ll \\ensuremath{\\varepsilon}' |X||X'|\/2$. Note that it was crucial here that $\\gamma \\ll n^{- 2\\beta}$, that is, $\\gamma$ polynomially depends on $n$. Therefore, every pair of disjoint subsets $X, Y \\subseteq V(G)$ of size $n^{1 - \\beta}$ forms a $(2\\ensuremath{\\varepsilon}')$-regular pair. \n\n The rest of the argument is the same as in the previous case. Consider some disjoint $V_1, \\ldots, V_{r+1} \\subseteq V(G)$, each of size $\\tilde n = n^{1 - \\beta}$. As observed, each pair $(V_i, V_j)$ forms a $(2\\ensuremath{\\varepsilon}')$-regular pair with $m_{ij} \\ge (1 - \\ensuremath{\\varepsilon}')\\tilde n^2 q$ edges. By Lemma \\ref{lemma:exact_m_edges}, there exists a subset $E_{ij} \\subseteq E_G(V_i, V_j)$ of size exactly\n $$\n m = \\tilde n^2 q \/ 2\n $$\n such that $(V_i \\cup V_j, E_{ij})$ is $(4\\ensuremath{\\varepsilon}')$-regular, i.e.~$(\\ensuremath{\\varepsilon})$-regular. This gives us a subgraph $G' \\subseteq G$ which belongs to $\\ensuremath{\\mathcal G}(K_{r+1}^-, \\tilde n, m, \\ensuremath{\\varepsilon})$. From the lower bound in~\\eqref{qBounds} we infer that $\\tilde n \\ge B(H) q^{-m_2(H)}.$ Thus \\ref{game2:r_1} implies that $G'$ contains a copy of $K_{r+1}^-$ with one vertex in each $V_i$. \n\\end{proof}\n\n\n\n\n\\subsection{Neat graphs contain $K_r$-factors (Lemma \\ref{lemma:neat_factor})}\n\nThe proof of Lemma \\ref{lemma:neat_factor} closely follows ideas from \\cite{nenadov18triangle} which are, in turn, based on ideas of Krivelevich \\cite{krivelevich1997triangle}. Recall that $K_{r+1}^-$ denotes the graph obtained from $K_{r+1}$ by removing an edge. The main building block in the proof is an \\emph{$(r, \\ell)$-chain}, the graph obtained by sequentially `gluing' $\\ell \\ge 0$ copies of $K_{r+1}^-$ on a vertex of degree $r-1$ (see Figure \\ref{fig:chain}). We define the $(r, 0)$-chain to be a single vertex. A graph is an \\emph{$r$-chain} if it is isomorphic to an $(r, \\ell)$-chain, for some integer $\\ell \\ge 0$.\n\n\\begin{figure}[h!] \n \\centering\n \\begin{tikzpicture}[scale = 0.6]\n \\tikzstyle{blob} = [fill=black,circle,inner sep=1.7pt,minimum size=0.5pt]\n \\tikzstyle{sq} = [fill=black,rectangle,inner sep=2.5pt,minimum size=2.5pt]\n\n \n \\foreach \\x in {1,...,4}{\n \\node[sq] (f\\x) at (0 + 5 * \\x, 0) {};\n \\foreach \\c\/\\y\/\\z in {1\/2.5\/1, 2\/2.5\/-1, 3\/2\/0.3, 4\/3\/-0.3}\n \\node[blob] (v\\x\\c) at (0 + 5 * \\x + \\y, \\z) {}; \n }\n \\node[sq] (f5) at (25, 0) {};\n\n \\foreach \\x\/\\d in {1\/1,2\/2,3\/3,4\/4}{\n \\foreach \\c in {1,...,4}\n \\draw (f\\x) -- (v\\x\\c);\n\n \\draw (v\\x1) -- (v\\x2) -- (v\\x3) -- (v\\x4) -- (v\\x1);\n \\draw (v\\x1) -- (v\\x3); \\draw (v\\x2) -- (v\\x4);\n }\n\n \\foreach \\x\/\\d in {2\/1,3\/2,4\/3,5\/4}{\n \\foreach \\c in {1,...,4}\n \\draw (f\\x) -- (v\\d\\c); \n }\n\n \\node [below=2pt of f4] {$v$};\n\n \\foreach \\x in {1,...,3}\n \\draw[dashed, rounded corners] (-0.5 + 5 * \\x, 1.5) rectangle (0.5 + 5 * \\x + 3, -1.5);\n\n \\draw[dashed, rounded corners] (-0.5 + 5 * 4 + 2, 1.5) rectangle (0.5 + 5 * 5, -1.5);\n \\end{tikzpicture}\n \\caption{The $(5, 4)$-chain with a $K_5$-factor after removing vertex $v$.} \n \\label{fig:chain} \n\\end{figure}\n\nAn $(r, \\ell)$-chain contains $\\ell + 1$ vertices such that removing either of them (but exactly one!) results in a graph which contains a $K_r$-factor. We call such vertices \\emph{removable}. If a graph $H$ is an $r$-chain then we use $R(H)$ to denote the set of its removable vertices. We repeatedly use the following observation.\n\n\\begin{observation} \\label{obs:canonical}\n Let $G$ be a graph and $C_1, \\ldots, C_r \\subseteq G$ be vertex disjoint $r$-chains. If there exists a copy of $K_r$ in $G$ which intersects each $R(C_i)$ then the subgraph of $G$ induced by $\\bigcup_{i \\in [r]} V(C_i)$ contains a $K_r$-factor.\n\\end{observation}\n\nThe following lemma together with property \\ref{prop:chain} ensures the existence of large $(r, \\ell)$-chains. \n\\begin{lemma} \\label{lemma:long_chain}\nLet $G$ be a graph with $n$ vertices such that for every disjoint $X, Y \\subseteq V(G)$, each of size at least $\\alpha n$, there exists a copy of $K_{r+1}^-$ in $G$ with one vertex of degree $r-1$ in $X$ and all other vertices in $Y$. Then $G$ contains an $(r, \\ell$)-chain for every $\\ell < (1 - (r+2)\\alpha) n\/r$. \n\\end{lemma}\nIn the case when $r=3$ this is Lemma~3.1 in~\\cite{nenadov18triangle}. Trivial adjustments to that proof give Lemma~\\ref{lemma:long_chain}. We omit the proof.\n\nThe following, somewhat technical looking lemma provides a crucial \\emph{absorbing} property of a collection of $r$-chains that we exploit in the proof of Lemma \\ref{lemma:neat_factor}.\n\n\\begin{lemma} \\label{lemma:absorbing}\n Let $G$ be a graph with $n$ vertices which satisfies \\ref{prop:chain} for some $\\beta > 0$ and $r$, where $n \\ge n_0(\\beta, r)$ is sufficiently large. Let $W \\subseteq V(G)$ be a subset of size $|W| \\ge n\/8$, and let $\\ell, t \\in \\mathbb{N}_0$ and $\\ell' \\ge \\ell, t' \\in \\mathbb{N}$ be such that:\n \\begin{itemize}\n \\item $(\\ell + 1) t' > n^{1 - \\beta\/2}$, and\n \\item $(t + t')(r \\ell + 1) < |W|\/2$.\n \\end{itemize}\n Suppose we are given disjoint $(r, \\ell')$-chains $C_1', \\ldots, C'_{t'} \\subset V(G) \\setminus W$. Then there exist disjoint $(r, \\ell)$-chains $C_1, \\ldots, C_t \\subset G[W]$ with the following property: for every $L \\subseteq [t]$ there exists $L' \\subseteq [t']$ such that the subgraph of $G$ induced by\n $$\n \\left( \\bigcup_{i \\in L} V(C_i) \\right) \\cup \\left( \\bigcup_{i \\in L'} V(C'_i) \\right)\n $$\n contains a $K_r$-factor.\n\\end{lemma}\n\n\n\nBefore we prove Lemma \\ref{lemma:absorbing} it is instructive to first see how it is used to derive Lemma \\ref{lemma:neat_factor}. \n\n\\begin{proof}[Proof of Lemma \\ref{lemma:neat_factor}]\n Consider an equipartition $V(G) = V_1 \\cup V_2$ chosen uniformly at random. As each vertex has $np\/2 \\gg \\log n$ neighbours (follows from \\ref{prop:expand} and the bound on $p$), by Chernoff's inequality and union-bound we have with high probability that every vertex has at least $np\/8$ neighbours in $V_1$. Therefore there exists a partition for which this holds. \n\n Without loss of generality we may assume that $\\beta = 1\/k$ for some integer $k \\ge 2$. For each $i \\in \\{1, \\ldots, 4k-1\\}$ set $\\ell_i = n^{1 - (4k-i)\/4k}$ and $t_i = n \/ (32 k (r \\ell_i + 1))$. Note that \n \\begin{equation} \\label{eq:ell_i_t_i}\n (\\ell_i + 1) t_{i+1} = \\Theta(n^{1 - \\beta\/4}).\n \\end{equation}\n\n By repeated application of Lemma \\ref{lemma:long_chain} we can find a collection $C_1^{4k-1}, \\ldots, C_{t_{4k-1}}^{4k-1} \\subseteq G[V_2]$ of pairwise vertex-disjoint $(r, \\ell_{4k-1})$-chains. Let us elaborate briefly why this is indeed possible. Such chains occupy $t_{4k - 1} \\cdot (r \\ell_{4k - 1} + 1) < n \/ 32$ vertices. Thus if we greedily choose them one by one the set $W \\subseteq V_2$ of unoccupied vertices in $V_2$ after every step is of size at least, say, $|W| \\ge n\/4$. Therefore, by \\ref{prop:chain} we have that $G[W]$ satisfies the assumption of Lemma \\ref{lemma:long_chain} for any constant $\\alpha > 0$, and consequently it contains an $(r, \\ell)$-chain for $\\ell < (1 - (r+2)\\alpha) |W| \/ r$. As $\\ell_{4k-1} = o(n)$, this proves our claim.\n\n Let $U_{4k - 1} = \\bigcup_{i \\in [t_{4k - 1}]} V(C_i^{4k - 1})$. For each $i = 4k - 2, \\ldots, 1$, iteratively, let $C_1^i, \\ldots, C_{t_i}^i \\subset G[V_2] \\setminus U_{i+1}$ be disjoint $(r, \\ell_i)$-chains given by Lemma \\ref{lemma:absorbing} for $C_1^{i+1}, \\ldots, C_{t_{i+1}}^{i+1}$ (as $C_1', \\ldots, C'_{t'}$) and $W_i = V_2 \\setminus U_{i+1}$, and set $U_i = U_{i+1} \\cup \\bigcup_{j \\in [t_i]} V(C_j^i)$. Let us verify that the conditions of Lemma \\ref{lemma:absorbing} are met. First, $|W_i| = n\/2 - |U_{i+1}|$ and\n $$\n |U_{i+1}| = \\sum_{j = i+1}^{4k-1} t_j \\cdot (r \\ell_j + 1) < n\/4.\n $$\n From \\eqref{eq:ell_i_t_i} we have $(\\ell_i + 1) t_{i+1} > n^{1 - \\beta\/2}$, and as\n $$\n (t_i + t_{i+1})(r \\ell_i + 1) < 2 t_i (r \\ell_i + 1) < n \/ 16 < |W|\/2\n $$\n we can indeed apply Lemma \\ref{lemma:absorbing} in each iteration. \n\n Finally, let $W_0 = V_2 \\setminus U_1$. Apply Lemma \\ref{lemma:absorbing} one last time with $\\ell_0 = 0$, $t_0 = |W_0|\/4$ and $C_1^1, \\ldots, C_{t_1}^1$ (as $C_1', \\ldots, C_{t'}')$. This is justified as $t_1=\\Theta(n^{1-1\/4k})$ and $t_0+t_1=o(n)$. The obtained $0$-chains are then just a set of vertices $C_0 \\subseteq W_0$ with the property that for every $L_0 \\subseteq C_0$ there exists a subset $L_1' \\subseteq [t_1]$ such that the subgraph of $G$ induced by\n $$\n L_0 \\cup \\left( \\bigcup_{j \\in L_1'} V(C_j^1) \\right)\n $$ \n contains a $K_r$-factor. \n\n Next, we show that the set $C_0 \\cup U_1$ has a strong absorbing property.\n \\begin{claim} For any subset $L_0 \\subseteq C_0$ such that $|L_0| + |U_1| \\in r \\mathbb{Z}$, the induced subgraph $G[L_0 \\cup U_1]$ contains a $K_r$-factor.\n \\end{claim}\n \\begin{proof}\n Consider one such $L_0$ and let $L_1' \\subseteq [t_1]$ be a subset such that\n $$\n L_0 \\cup \\left(\\bigcup_{j \\in L_1'} V(C_j^1) \\right)\n $$\n contains a $K_r$-factor. We further take $L_1 = [t_1] \\setminus L_1'$ and use the property guaranteed by Lemma \\ref{lemma:absorbing} to obtain a subset $L_2' \\subseteq [t_2]$ such that the subgraph of $G$ induced by \n $$\n \\left( \\bigcup_{j \\in L_1} V(C_j^1) \\right) \\cup \\left( \\bigcup_{j \\in L_2'} V(C_j^2) \\right)\n $$\n contains a $K_r$-factor. Continuing this way, we obtain a subset $L_{4k-1}' \\subseteq [t_{4k - 1}]$ such that the subgraph of $G$ induced by \n $$\n L_0 \\cup \\bigcup_{i = 1}^{4k - 2} \\left( \\bigcup_{j \\in [t_i]} V(C_j^i) \\right) \\cup \\left( \\bigcup_{j \\in L_{4k-1}'} V(C_j^{4k - 1}) \\right) = (L_0 \\cup U_1) \\setminus \\bigcup_{j \\in L_{4k - 1}} V(C_j^{4k - 1})\n $$\n contains a $K_r$-factor, where $L_{4k - 1} = [t_{4k - 1}] \\setminus L_{4k - 1}'$. As $|V(C_j^{4k - 1})| \\equiv 1 (\\textrm{mod } r)$ and $|L_0| + |U_1| \\in r \\mathbb{Z}$ we necessarily have $|L_{4k - 1}| \\in r \\mathbb{Z}$. Therefore, to complete a $K_r$-factor in $G[L_0 \\cup U_1]$ it suffices to partition $L_{4k - 1}$ into groups of size $r$ and for each such group $\\{i_1, \\ldots, i_r\\}$ find a copy of $K_r$ with one vertex in each $R(C_{i_1}^{4k - 1}), \\ldots, R(C_{i_r}^{4k - 1})$ (see Observation \\ref{obs:canonical}). The existence of such $K_r$ follows from \\ref{prop:chain} and $|R_j^{4k - 1}| = \\ell_{4k - 1} + 1 > n^{1 - 1\/4k} > n^{1 - \\beta}$. \n \\end{proof}\n\n We now use this absorbing property to find a $K_r$-factor in $G$. First, let $B \\subseteq V_1 \\cup (W_0 \\setminus C_0)$ be the set of all vertices which are not part of chains and such that they have less than $|C_0|p\/2$ neighbours in $C_0$. As $|C_0| \\ge \\alpha n$, we have $|B| < \\log n \/ p \\ll np$, by~\\ref{prop:expand} and the lower bound on $p.$ By \\ref{prop:in_nbr} and the assumption that every vertex has at least $np\/8$ neighbours in $V_1$, we can iteratively take one vertex $v \\in B$ at a time and find a copy of $K_r$ which contains $v$ and has all other vertices in $V_1 \\setminus B$. This takes care of $B$. Furthermore, we can continue covering the remaining vertices in $V_1 \\cup (W_0 \\setminus C_0)$ (i.e.\\ those which are not part of previously chosen $K_r$'s) with disjoint copies of $K_r$ as long as there are still at least $r n^{1 - \\beta}$ vertices, by~\\ref{prop:chain}. Let us denote the set of remaining vertices by $L$. With the absorbing property of $C_0 \\cup U_1$ in mind, to find a $K_r$-factor of $G$ it now suffices to find vertex-disjoint copies of $K_r$, each of which contains one vertex from $L$ and the others from $C_0$. Whatever we are left with in $C_0$ is guaranteed to form a $K_r$-factor with $U_1$, thus we are done. Note that this is very similar with how we took care of $B$, however the main difference is that $L$ is significantly larger than $B$ and a simple greedy strategy might not work. Instead, we find the desired copies of $K_r$ using Haxell's matching theorem (Theorem~\\ref{thm:haxell}). \n\n For each $v \\in L$ create an $(r-1)$-uniform hypergraph $H_v$ on the vertex set $C_0$ such that $\\{v_1, \\ldots, v_{r-1}\\}$ forms a hyperedge if and only if $\\{v, v_1, \\ldots, v_{r-1}\\}$ form $K_r$ in $G$. If we can find for each $v \\in L$ a hyperedge $h_v \\in E(H_v)$ such that all these hyperedges are pairwise vertex-disjoint, then we are done.\n To show that such edges exist it suffices to verify Haxell's criterium:\n \\begin{equation} \\label{eq:verify_haxell}\n \\tau(\\bigcup_{v \\in I} H_v) \\ge 2(r-1)|I|\n \\end{equation}\n for every $I \\subseteq L$. \n Equivalently, for all subsets $I\\se L$ and all $Z \\subseteq C_0$ of size $|Z| \\le 2(r-1)|I|$ there exists a copy of $K_r$ with one vertex in $I$ and all other vertices in $C_0 \\setminus Z$.\n\n We consider two cases. Consider first the case when $|I| \\le \\log n \/p$ and let $Z$ be some subset of $C_0$ of size at least $2r \\log n \/ p$. As $L \\cap B = \\emptyset$, every vertex $v \\in I$ has at least $|C_0|p\/2 > np \/ 32$ neighbours in $V_1$, thus the subset $X = (N_G(v) \\cap C_0) \\setminus Z$ is of size at least $np\/16$ (we used $np \\gg \\log n \/p$ which follows from the lower bound on $p$). By \\ref{prop:in_nbr} there exists a copy of $K_{r-1}$ in $X$. Suppose now that $|L| \\ge |I| > \\log n \/p$ and consider a subset $Z \\subseteq C_0$ of size $2r |L| < n\/32$. The set $Y = C_0 \\setminus Z$ is then of size at least $n\/16$. Thus, there exists a vertex $v \\in I$ with at least $|Y|p\/2 \\ge np\/32$ neighbours in $Y$, by \\ref{prop:expand}. By \\ref{prop:in_nbr} such a neighbourhood contains a copy of $K_{r-1}$, which gives us a desired copy of $K_r$. This finishes the proof.\n\\end{proof}\n\nIt remains to prove Lemma \\ref{lemma:absorbing}.\n\n\\begin{proof}[Proof of Lemma \\ref{lemma:absorbing}]\n By repeated application of Lemma \\ref{lemma:long_chain} we can find a collection of $t + t'$ disjoint $(r, \\ell)$-chains $C_1, \\ldots, C_{t + t'} \\in G[W]$. Clearly, for this we could have allowed $W$ to be much smaller than $n\/8$, thus this constraint is only for convenience. For each $i \\in [t + t']$ we create an auxiliary $(r-1)$-uniform hypergraph $H_i$ on the vertex set $V' = [t']$ by adding a hyperedge $\\{j_1, \\ldots, j_{r-1}\\}$ if and only if there exists a copy of $K_r$ in $G$ with one vertex in each $R(C_i), R(C_{j_1}'), \\ldots, R(C_{j_{r-1}}')$. Note that for every such hyperedge the subgraph of $G$ induced by \n $$\n V(C_i) \\cup V(C_{j_1}') \\cup \\ldots V(C_{j_{r-1}}')\n $$\n contains a $K_r$-factor (see Observation \\ref{obs:canonical}).\n\n We first show that there exists a subset $B \\subseteq [t + t']$ of size at most $|B| \\le t'$ such that for every subset $J \\subseteq [t + t'] \\setminus B$ of size $|J| \\le t'\/8r$ we have\n \\begin{equation} \\label{eq:2rJ}\n \\tau(\\bigcup_{i \\in J} H_i) \\ge 2r|J|.\n \\end{equation}\n Initially, set $q = 0$ and $B = \\emptyset$. As long as $|B| < t'\/8r$ and there exists a subset $J \\subseteq [t + t'] \\setminus B$ of size $|J| \\le t'\/8r$ that violates \\eqref{eq:2rJ} set $B = B \\cup J$, $J_{q+1} = J$ and increase $q$ by 1. Suppose towards a contradiction that for some $q$, $|B| \\ge t'\/8r$, and let $q$ be the smallest such index. Then $|B| \\le t'\/4r$ as $|J_q| \\le t'\/8r$. Moreover, we have\n $$\n \\tau(\\bigcup_{i \\in B} H_i) \\le \\sum_{j = 1}^q \\tau(\\bigcup_{i \\in J_j} H_i) < \\sum_{j = 1}^q 2 r |J_j| = 2r|B| \\le t'\/2.\n $$\n This implies that there exists a set $\\tilde B \\se V'$ of size at most $t'\/2$ such that every hyperedge $h\\in\\bigcup_{i\\in B}H_i$ intersects $\\tilde B$. In other words, there exists $B' \\subseteq V'$ of size $|B'| \\ge t'\/2$ such that there is no copy of $K_r$ in $G$ with one vertex in $\\bigcup_{i \\in B} R(C_i)$ and the others in each $R(C_{j_2}'), \\ldots R(C_{j_{r}}')$ for some distinct $j_2, \\ldots, j_{r} \\in B'$. Split $B'$ arbitrarily into $r-1$ sets of nearly equal size, denoted by $B_2', \\ldots, B_{r}'$, each of size at least $t'\/2r$, and set $X_j = \\bigcup_{i \\in B'_j} R(C_i')$ for $j=2,\\ldots,r$. Then each such $X_j$ is of size at least\n $$\n (\\ell' + 1) \\frac{t'}{2r} > (\\ell + 1) \\frac{t'}{2r}.\n $$\n On the other hand, $X_1$ defined as $\\bigcup_{i \\in B} R(C_i)$ is of size at least \n $$\n |X_0| \\ge (\\ell + 1) |B| \\ge (\\ell + 1) \\frac{t'}{8r}.\n $$\n Thus we have $|X_i| \\ge n^{1 - \\beta}$ by the assumption of the lemma, with room to spare. By \\ref{prop:chain} there exists a copy of $K_r$ intersecting each $X_i$, which is a contradiction. Therefore we have that there exists a set $|B|$ of size less than $t'\/8r$ and every subset $J \\subseteq [t + t'] \\setminus B$ of size $|J| \\le t' \/ 8r$ satisfies \\eqref{eq:2rJ}.\n\n Take an arbitrary $t$-subset $I \\subseteq [t + t'] \\setminus B$ and relabel $\\{C_i\\}_{i \\in I}$ as $\\{C_i\\}_{i \\in [t]}$. We show that such $(r, \\ell)$-chains have the desired property. Consider some $L \\subseteq [t]$. First, let $S \\subseteq L$ be a smallest subset such that the subgraph of $G$ induced by\n $$\n \\bigcup_{i \\in L \\setminus S} V(C_i)\n $$\n contains a $K_r$-factor. We claim that $|S| < t'\/8r$. Suppose towards a contradiction that $|S| \\ge t' \/ 8r$. Consider an equipartition $S = S_1 \\cup \\ldots \\cup S_r$. Then each set $X_i = \\bigcup_{j \\in S_i} R(C_j)$ is of size \n $$\n |X_i| \\ge (\\ell + 1) \\frac{t'}{8r^2} > n^{1 - \\beta}\n $$\n thus by \\ref{prop:chain} there exist a copy of $K_r$ intersecting each $X_i$. Therefore there exists distinct $i_1, \\ldots, i_r \\in S$ and a copy of $K_r$ intersecting each $R(C_{i_j})$. By Observation \\ref{obs:canonical} this is a contradiction with the minimality of $S$. Finally, as $|S| \\le t'\/8r$ we have that every subset $J \\subseteq S$ satisfies \\eqref{eq:2rJ} thus we can choose $h_i \\in H_i$ for each $i \\in S$ such that these edges are pairwise vertex disjoint, by Theorem~\\ref{thm:haxell}. Let $L' = \\bigcup_{i \\in S} h_i$. The construction of such hyperedges implies that the subgraph of $G$ induced \n $$\n \\bigcup_{i \\in L} V(C_i) \\cup \\bigcup_{i \\in L'} V(C_i')\n $$\n contains a $K_r$-factor, as desired.\n\\end{proof}\n\n\n\n\\section{Breaker's strategy}\n\nThe idea behind the proof is that Breaker prevents a fixed vertex $v$ from being in a copy of $K_r$ in Maker's graph. To illustrate why this could be possible, fix a vertex $v\\in [n]$ and assume for now that Maker at first only claims edges incident to $v$, as long as there is at least one such unclaimed edge. Breaker responds by claiming $b$ edges incident to $v$ in every round as well, so that at the end of this first stage of the game the set of neighbours of $v$ in Makers graph, denoted by $N_M(v)$, has size roughly $n\/b$. For the rest of the game, Breaker only needs to prevent Maker from claiming a copy of $K_{r-1}$ in $N_M(v)$, which is possible if $b\\ge C (n\/b)^{2\/r}$ for some constant $C$ which is independent of $n$, by the result of Bednarska and \\L uczak~\\cite{bl2000}; or equivalently if $b\\ge C n^{2\/(r+2)}$ (with a different constant $C$). \n\nIf Maker indeed first claims as many edges incident to $v$ as possible, this would be the end of the proof. Of course, we cannot rely on this assumption. The way to counterfeit it is to divide the attention of Breaker into two: the first $b\/2$ claimed edges are incident to $v$, thus preventing its neighbourhood in the Maker's graph from becoming larger than $2n\/b$; the second $b\/2$ claimed edges lie inside its current neighbourhood and prevent a copy of $K_{r-1}$. Crucially, the board of the game where we want to use the strategy ${\\mathcal S}$ from~\\cite{bl2000} will be revealed over time only (as the neighbourhood of $v$ in Maker's graph increases). It turns out that the proof of a static version of the game (where the whole board is `visible') can be turned into a proof of a suitable dynamic version (where the board is revealed over time). Unfortunately, none of the ingredients of the proof is black-boxable so we need to dig into each part. \n\nLet us introduce necessary notation in a bit more generality than needed for our application. Let ${\\mathcal H}$ be a given hypergraph, say on vertex set $V({\\mathcal H})$ and edge set $E({\\mathcal H})$, and let $m$ and $b$ be integers. We define the {\\em dynamic-board $({\\mathcal H},m,b)$-game} as follows. \nLet $V_0 = \\emptyset$. The two players Maker and Breaker play in rounds, with Maker going first. For $i\\ge 0$, suppose that $i$ rounds have been played and that $V_i\\se V({\\mathcal H})$ is defined. In round $i+1$, Maker may play either according to {\\em Option (a)} in which she claims up to $m$ elements of $V_i$ and sets $V_{i+1}=V_i$ (in case there are less than $m$ elements she claims all of them), or according to {\\em Option (b)} in which she chooses elements $v_1,\\ldots,v_{\\ell}$ (for some $\\ell\\ge 1$) from $V({\\mathcal H}) \\sm V_i$ and sets $V_{i+1} = V_i\\cup\\{v_1,\\ldots,v_{\\ell}\\}$ (but Maker does not claim edges in a round when she enlarges the board). In case Option (a) is not possible, Maker is forced to play Option (b), unless it is the end of the game. Afterwards, Breaker claims (up to) $b$ elements in $V_{i+1}$. In case there are less than $b$ unclaimed elements in $V_{i+1}$, Breaker claims all of them. Maker wins if at the end of the game she has claimed all elements of some hyperedge $H\\in E({\\mathcal H})$. Otherwise, Breaker wins. \n\nGiven a (fixed) graph $H$ and a complete graph $K_n$, we define a {\\em dynamic $b$-biased $H$-game} as the $({\\mathcal H}, 1, b)$-game where the vertex set of ${\\mathcal H}$ are the edges of $K_n$, and the hyperedges of ${\\mathcal H}$ correspond to edge sets of $K_n$ which form a copy of $H$. The following theorem is a generalisation of the mentioned result by Bednarska and \\L uczak \\cite{bl2000} to the dynamic setting. For the definition of $m_2(H)$, see Theorem \\ref{thm:KLR}.\n\n\\begin{theorem}\\thlab{ourDynamicBreaker}\nFor every graph $H$ which contains at least three non-isolated vertices there exists a constant $C>0$ such that Breaker has a winning strategy in the dynamic $b$-biased $H$-game played on $K_n$ if $b\\ge Cn^{1\/m_2(H)}$.\n\\end{theorem}\n\n\nWe may take $C$ sufficiently large such that the theorem statement is true for small $n$, so that in the proof we can safely assume that $n$ is as large as needed. The proof of Theorem~\\ref{ourDynamicBreaker} proceeds along the lines of~\\cite{bl2000}. We sketch the argument in the next section, leaving out calculations that are identical to those in~\\cite{bl2000}. \n\nTheorem \\ref{ourDynamicBreaker} is all we need to describe Breaker's strategy for isolating a vertex $v$ from being in a copy of $K_r$. \n\n\\begin{proof}[Proof of Theorem \\ref{thm:main} (ii)]\nLet $r\\ge 4$, let $C$ be a large enough constant, let $n$ be an integer and let $b\\ge Cn^{2\/(r+2)}$. Let $v$ be a fixed vertex of $K_n$. We show that Breaker has a strategy in the $b$-biased Maker--Breaker game played on the edge set of $K_n$ to prevent Maker from claiming a copy of $K_r$ that contains the vertex $v$. Consequently, Maker's graph does not contain a $K_r$-factor. Before we present the strategy of Breaker we describe an auxiliary game that Breaker simulates in parallel. \n\nLet $T$ be a set of size $2n\/b$, disjoint from $V(K_n)$. By \\thref{ourDynamicBreaker}, if $b\/2\\ge C'(2n\/b)^{1\/m(K_{r-1})}$ then Breaker has a winning strategy ${\\mathcal S}$ in the dynamic $(b\/2)$-biased $K_{r-1}$-game played on $K_T$, the complete graph on the vertex set $T$. Equivalently, $b\\ge Cn^{2\/(r+2)}$ for suitable $C$. \n\nWe now describe the strategy of Breaker in the $b$-biased Maker--Breaker game played on the edge set of $K_n$. Suppose that $i \\ge 0$ rounds have been played already. Let $M$ and $B$ denote the graphs formed by Maker's edges and by Breaker's edges, respectively (we suppress dependence on $i$ for clarity of presentation). Breaker maintains the property that every vertex $w\\in N_M(v)$ has a (unique) corresponding vertex $t_w\\in T$ such that an edge $uw$ in $N_M(v)$ belongs to Maker's (Breaker's) graph if and only if $t_ut_w$ belongs to Maker's (Breaker's) graph in the auxiliary dynamic $b$-biased $K_{r-1}$-game played on $K_T$. Clearly, this is the case before the first round of the game, and we show that Breaker can maintain such a correspondence throughout the game. Set $T_{i} = \\{t_w:w\\in N_M(v)\\}$. \n\nLet $xy$ denote the edge that Maker claims in round $i+1$. Then Breaker claims up to $b\/2$ edges incident to $v$ including $xv$ or $yv$ if those edges are not claimed yet by either of the players. If Breaker has claimed $b' < b\/2$ edges and there are no more unclaimed edges incident to $v$, then he claims $b\/2 - b'$ arbitrary edges (note that additional edges do not hurt Breaker). For the remaining $b\/2$ edges in round $i+1$ we distinguish between three cases (where the latter two are similar). In Case~1, assume that $x\\not\\in N_M(v) \\cup \\{v\\}$ or $y\\not\\in N_M(v) \\cup \\{v\\}$. Then Breaker claims $b\/2$ arbitrary edges. In Case~2.1, assume that $x=v$ (the case $y=v$ is analogous). Let $t\\in T\\sm T_{i}$ and set $t_y= t$. In the auxiliary dynamic $K_{r-1}$-game, Breaker pretends that (the auxiliary) Maker plays according to Option (b) and adds the elements $\\{t_y t_w: w\\in N_M(v)\\}$ to the board (recall that the vertices in the hypergraph corresponding to that game are the edges of $K_T$). In Case~2.2, assume that $x,y\\in N_M(v)$. Then Breaker pretends that in the auxiliary dynamic $K_{r-1}$-game Maker plays according to Option (a) and claims the edge $t_xt_y$. In either of Case~2.1 or~2.2, the strategy ${\\mathcal S}$ in the auxiliary game gives $b\/2$ edges $e_1,\\ldots, e_{b\/2} \\in E(K_T)$ for Breaker to claim in the auxiliary board. Let $f_1,\\ldots,f_{b\/2}$ be the corresponding edges in $N_M(v)$, that is $f_i$ is the edge with endpoints $w_i$ and $u_i$ such that $e_i$ has endpoints $t_{w_i}$ and $t_{u_i}$. Breaker then claims $e_1,\\ldots, e_{b\/2}$ in the auxiliary game and $f_1,\\ldots,f_{b\/2}$ in the real game. \n\nWe claim that this is indeed a winning strategy. First note that $|N_M(v)|\\le 2n\/b$ since Breaker claims $b\/2$ of the $n-1$ total edges incident to $v$ in every round. Thus, the set $T$ is large enough so that Breaker can indeed maintain an injective map $w\\mapsto t_w$ for $w\\in N_M(v)$. Furthermore, it is clear from the strategy description that a Maker\/Breaker edge in $N_M(v)$ corresponds to a Maker\/Breaker edge in the auxiliary game in $T$. Finally, since ${\\mathcal S}$ is a strategy for Breaker to prevent Maker in the auxiliary $b\/2$-biased game to claim a copy of $K_{r-1}$ this implies that Breaker can indeed prevent Maker from claiming a copy of $K_{r-1}$ in $N_M(v)$, i.e.~the vertex $v$ is not in a copy of $K_r$ in Maker's graph.\n\\end{proof}\n\n\n\n\n\\subsection{Proof of Theorem \\ref{ourDynamicBreaker} (sketch)}\n\\label{proof:ourLemma5}\n\nThe following is a dynamic-board variant of \\cite[Lemma 5]{bl2000}. We switch notation from $m$ to $p$ and from $b$ to $q$ for the bias of Maker and Breaker, respectively, to be consistent with the literature. \n\n\\begin{lemma}\\thlab{ourLemma5} In every dynamic-board $({\\mathcal H},p,q)$-game Breaker has a strategy such that at the end of the game at most $(1+q) f({\\mathcal H},p,q)$ edges of the hypergraph ${\\mathcal H}$ have all their vertices claimed by Maker, where $f({\\mathcal H},p,q)=\\sum_{H\\in E({\\mathcal H})}(1+q)^{-|H|\/p}.$ \n\\end{lemma}\n\nThe proof is a simple adaptation of the potential function technique as introduced by Erd\\H{o}s and Selfridge~\\cite{es1973} that was generalised by Beck~\\cite{b1982} to biased Maker--Breaker games. We are unaware of such a dynamical-board variant thus the full proof follows. We follow notation and strategy of the proof of \\cite[Theorem 20.1]{BeckBook}. \n\n\\begin{proof}[Proof of \\thref{ourLemma5}]\nLet ${\\mathcal H}$, $p$, $q$ be as in the lemma and let $\\mu$ be defined by $1+\\mu = (1+q)^{1\/p}$. \nGiven two disjoint subsets $M$ and $B$ of the board $V=V({\\mathcal H})$ and an element $z\\in V$ set \n\\begin{align*}\n\\Phi(M,B) &= \\sum_{H\\in {\\mathcal H}: H\\cap B =\\emptyset} (1+\\mu)^{-|H\\sm M|},\\text{ and}\\\\\n\\Phi(M,B,z) &= \\sum_{z\\in H\\in {\\mathcal H}: H\\cap B =\\emptyset} (1+\\mu)^{-|H\\sm M|}\n\\end{align*}\nand note straight away the following inequalities:\n\\begin{align}\n\\Phi(M\\cup\\{e\\},B,z)&\\le (1+\\mu)\\Phi(M,B,z),\\lab{T1}\\\\\n\\Phi(M,B\\cup\\{e\\},z)&\\le \\Phi(M,B,z).\\lab{T2}.\n\\end{align}\nFor integers $r$ and $j$, let \n$b_r^{(j)}$ be the $j^{\\mathrm{th}}$ element that Breaker picks in round $r$, and let $m_r^{(j)}$ be the $j^{\\mathrm{th}}$ element that Maker picks in round $r$ if she decides to play according to Option (a) and pick elements in $V_{r-1}$ rather than enlarging $V_r$ (Option (b)). \nFurthermore, let $M_r$ and $B_r$ be the set of all elements of Maker and of Breaker, respectively, {\\em after} round $r$, \nand let \n$M_{r,j}=M_r\\cup \\{m_{r+1}^{(1)},\\ldots,m_{r+1}^{(j)}\\}$ and \n$B_{r,j}=B_r\\cup \\{b_{r+1}^{(1)},\\ldots,b_{r+1}^{(j)}\\}$ (assuming that Maker\/Breaker has claimed at least $j$ elements in round $r+1$). \n\nWe now describe Breaker's strategy in round $r$. If there are less than $q$ unclaimed elements in $V_{r+1}$, then Breaker claims all of them. Otherwise, for every $1\\leq j\\le q$, sequentially, Breaker calculates $\\Phi(M_r,B_{r-1,j-1},z)$ for every unclaimed element $z\\in V_r\\sm(M_r\\cup B_{r-1,j-1})$ and claims the element $b_r^{(j)}$ which maximises this expression. Note that here we chose the element $b_r^{(j)}$ in $V_r$, and not in the whole board $V$. If, for some $j$, there are no unclaimed elements, then it is the end of Breaker's turn.\n\nThe crucial part of the potential function technique in positional games is to show that the {\\em potential} $\\Phi(M_{r+1},B_r)$ is decreasing (if evaluated after Makers move). But this is now straight-forward along the lines of the proof in \\cite{BeckBook}. The only thing we have to notice is that in round $r+1$, if Maker chooses to claim elements in $V_r$, then their choices are on the same sub-board where Breaker claimed their elements in round $r$. \n\n\\begin{claim}\nFor all $r\\ge 1$, \n$\\Phi(M_{r+1},B_r)\\le \\Phi(M_r,B_{r-1})$.\n\\end{claim}\n\n\\begin{proof} \nIf Maker has played according to Option (b) in round $r+1$, then $\\Phi(M_{r+1}, B_r) = \\Phi(M_r, B_r) \\le \\Phi(M_r, B_{r-1})$, where the inequality follows from \\eqref{T2}. Therefore, if Breaker was not able to claim $q$ elements in round $r$, then in round $r+1$ Maker is forced to play Option (b) and the claim follows. For the rest of the proof we can assume that Breaker is able to claim $q$ elements in round $r$. Without loss of generality, we can also assume that Maker claims $p$ elements in round $r+1$ (claiming fewer than $p$ elements only makes it easier for the desired inequality to hold). Let us denote these elements by $b_r^{(1)}, \\ldots, b_r^{(q)}$ and $m_{r+1}^{(1)}, \\ldots, m_{r+1}^{(p)}$, respectively.\n\nWe first note that $\\Phi(M_r,B_{r-1,j+1})=\\Phi(M_r,B_{r-1,j})-\\Phi(M_r,B_{r-1,j},b_r^{(j+1)})$ for all $0\\le j100$\\,MeV) & 8 and 11 & $16-20$, $37-53$ \\\\\nBAT & 15--18 & $88.2-253.2$ \\\\\nXRT & from bin 12 & $>58.5$ \\\\\n\n\\hline\n\\end{tabular}\n\\label{tobs}\n\\end{table}\n\n \n\\begin{figure*}\\centering\n\\includegraphics[width=\\textwidth]{lc.pdf} \n\\caption{Light curve of GRB 151006A extracted from the {\\it Fermi}~, Astrosat\/CZTI and {\\it Swift}~\n at different energies (labeled in the respective panels).\n The time intervals chosen for our analysis are shown by dashed lines. \n We also mark the GBM trigger time with dot-dashed red line (at $t=0$\\,s)\n and the first LAT photon detection time with a vertical magenta line \n with horizontal dashes (at $t=17.5$\\,s). The time intervals for polarization\n measurements (0--16\\,s and 16--33\\,s) are marked in the CZTI-veto panel \n with dotted brown line.}\n\\label{lc}\n\\end{figure*}\n\n\nThe CZT Imager (CZTI), on-board the recently launched multi-wavelength Indian mission {\\it Astrosat}, \nprovides detection in the energy range 20 -- 200\\,keV and becomes an open detector above 100\\,keV, \nthereby enabling it to detect GRB events. The Veto detector when augmented, raises the detection energy \nlevel to 600 keV. Thus, when analysed along with the BAT data, the CZTI + Veto will become crucial in \nconstraining the spectral peak energies, which is otherwise not generally possible with the BAT \ndata alone (\\citealt{Raoetal_2016}). The CZTI also possesses X-ray polarization detection capabilities\nin the energy range, 100 -- 380\\,keV (\\citealt{Vadawale_etal_2015}). Thus, with its wide field \nof view, good spectral resolution and polarimetry capability, the CZTI data will be a key addition\nto the existing observatories like the {\\it Swift} and {\\it Fermi}. \n\n\n\nIn this paper, we present a detailed study of spectral evolution of \nGRB 151006A, the first detected GRB by the CZTI. We follow the strategy of multi-instrument\nanalysis using the detectors on-board \\emph{Fermi} and \\emph{Swift} at different \nphases of GRB emission. This is further complemented by \nthe polarization data from the CZTI. The paper is organized \nas follows. Section 2 presents the observation by various instruments,\nthe methodology of the data analysis and the spectral models used; section 3 presents \nthe results of the spectral fits \nand the polarization measured by CZTI, finally followed by conclusions and a discussion in Section 4.\n\n\n\n\\section{Observations and Data Analysis}\nOn 2015 October 06 at 09:54:57.83 UT, the {\\it Fermi}\/GBM (\\citealt{151006A_gcn_gbm}) triggered on GRB 151006A.\nThe burst also triggered many other detectors including the {\\it Swift}\/BAT at 09:55:01 UT (\\citealt{151006A_gcn_bat}), \nthe \\emph{Astrosat}\/CZTI (\\citealt{151006A_gcn_czti}) at 09:54:57.825 UTC, Konus-Wind at 09:54:57.7 UT \n(\\citealt{Konuswind_151006A}) and CALET at 09:54:59.97 UT (\\citealt{CALET_151006A}). \n\nIn the current paper, we present the spectral analysis of {\\it Fermi} and {\\it Swift} data, and the\npolarization measurement obtained from CZTI data.\nFigure \\ref{lc} shows a composite count rate light curve of various detectors on-board the {\\it Fermi},\n{\\it Swift}, and the CZTI of {\\it Astrosat} arranged from higher to lower energy bands. These are \nLAT (P8$\\_$SOURCE class events, $>100$ MeV), LAT Low energy \nevents (LLE, 30 MeV--100 MeV), GBM\/BGO detector (300 keV--30 MeV), \nGBM\/NaI detectors (100--300\\,keV),\nCZTI-veto (100--300\\,keV),\nGBM\/NaI detectors (8--100 keV), followed by the\n{\\it Swift} BAT in 50--100\\,keV, 25--50\\,keV, 15--25\\,keV and \nthe {\\it Swift} XRT (0.3--10\\,keV).\nThe GBM light curve shows a single pulse having a fast rise and an exponential decay (FRED) \nwith a duration ($T_{90}$) of $\\sim84$\\,s (50--300\\,keV) (\\citealt{151006A_gcn_gbm}). \nThe LLE emission of the burst is coincident with the GBM trigger and peaks simultaneously\nwith the BGO emission at $ \\sim 1.4$ s. The LLE emission is quite significant with a detection \nsigma of 54, and extends for about $\\sim 30$ s (\\citealt{151006A_LAT_gcn}). The emission consists\nof a narrow pulse accompanied by two other smaller pulses towards the end. Thus, LLE emission \ndoes not show the typical delay that is observed in its onset with respect to the GBM observations \n(\\citealt{Ackermannetal_2013_LAT}). The LAT LLE data bridges the gap between the BGO and LAT \ndetections, and thereby helps in constraining high MeV spectral peaks \n(\\citealt{Axelsson_etal_2012,Moretti&Axelsson2016}). However, the LAT\n(P8R2$\\_$source class events, $>100$ MeV) events are observed to arrive \nonly $17.5$ s after the GBM trigger. The LAT emission is less significant with\nonly 5 photons detected in total. The top panel of Fig. \\ref{lc} shows the photons\nto be associated to the GRB with probability $ > 0.9$ in red filled circle, \notherwise in open circles (using {\\sc gtsrcprob} of {\\it Fermi} science \ntool\\footnote{http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/scitools\/overview.html}). \nThe delayed onset of the LAT events with respect to GBM emission is consistent\nwith that observed for long bursts as reported in \\cite{Ackermannetal_2013_LAT}.\nWe note that high energy emission in the energy range, 100 keV -- 30 MeV \nand 30 MeV -- 100 MeV, peaks nearly simultaneously at $\\sim 1.4$ s, whereas \nthe low energy emission in the energy range, 8 keV - 100 keV, peaks later at $\\sim 5.4$ s. \nThe BAT light curve also shows a single FRED like pulse with a $T_{90}$ (15 -350 keV) \nof $203.9 \\pm 41.6$ s (\\citealt{151006A_refined_BAT}). The XRT started observations nearly \n$48.6$ s after the BAT trigger and located an X-ray source at RA = $147.43 \\,\\rm d$ and DEC = $70.5 \\,\\rm d$. \nThe XRT observed in Window Timing (WT) mode during 55.3--570.8\\,s, and in photon counting mode \nafterwards until 114\\,ks (both the time counted from the BAT trigger time). The long term XRT light curve \nis best fitted with a double-broken power law with slopes of $-0.5_{-2}^{+0.2}$ until 91\\,s, $-2.1\\pm0.4$ until\n134\\,s and $-1.39\\pm0.02$ afterwards (XRT repository; \\citealt{Evansetal_2009}). In the current analysis we have included \nthe XRT data only until $\\sim 600\\, \\rm s$, i.e the part with WT mode observation and nearly coincident \nwith the observed BAT emission.\\footnote{For a full light curve of the XRT observations, please refer to the \nonline repository in the link \\url{http:\/\/www.swift.ac.uk\/xrt_curves\/00657750\/}}\n\nGRB 151006A was also the first GRB detected by the \\emph{Astrosat} CZTI, on it's first day of operation\n(\\citealt{151006A_gcn_czti, Raoetal_2016}). \nBoth CZTI and Veto detector light curves also show a single FRED \nlike pulse in both the energy ranges 50--200\\,keV and 100 - 500 keV, see Figure 2 in \\cite{Raoetal_2016}. \n\nFor the spectral study with the {\\it Fermi}\/GBM, we choose three NaI detectors having the highest count rate, namely \nn0, n1 and n3, where the numbering of the NaI detectors follows the usual convention, i.e., `nx' with $x=0-11$. \nAs the number of all the NaI detectors are within $x\\le5$, we choose the BGO number b0, where `by' \ndenotes a BGO detector with $y=0-1$. \nWe then use Fermi Burst Analysis GUI v 02-01-00p1 \n({\\sc gtburst}\\footnote{\\url {http:\/\/fermi.gsfc.nasa.gov\/ssc\/data\/analysis\/scitools\/gtburst.html}}) \nto extract the spectrum. As the n3 has the \nhighest count rate among the chosen NaI detectors, we use it to define the time intervals for time resolved\nspectroscopy. We apply a signal-to-noise ratio (S\/N) of 20 and find 14 time bins in the interval \n$-2.0-88.1$\\,s. The LAT LLE data is also extracted in these time bins until $\\sim 27$\\,s, following\nthe standard procedure described in \\cite{Ackermannetal_2013_LAT}. \nThe LAT P8R2$\\_$source class events, $>100$ MeV, events were selected within a $12 \\deg$\nregion centred around the {\\it Swift}~XRT position. Among the 5 detected LAT photons, only 4 \narrive during the GBM $T_{\\rm 90}$ of the burst. Thus, for the temporal analysis, \ndepending on the availability of the data, the LAT spectra is extracted in the energy \nrange, $100 \\,\\rm MeV - 1\\,\\rm GeV$, only for time intervals: 16.3 - 20.5 s (bin 8) and 37 -53 s (bin 11).\n\nThe first $88$\\,s of the {\\it Fermi}~ data is augmented by the observation with the {\\it Swift}\/BAT ($88-253$\\,s) and {\\it Swift}\/XRT \nat later times ($> 58.5$\\,s). \nThe data of the BAT was extracted following the standard procedure. The data is calibrated using the task\n{\\sc bateconvert}, followed by constructing a detector plane image using {\\sc batbinevt}. The known\nbad detectors and the noisy pixels are eliminated by {\\sc batdetmask} and {\\sc bathotpix}. It is then mask\nweighted by {\\sc batmaskwtevt}. Finally, the spectrum is extracted with {\\sc batbinevt} in a specified \ntime interval. The spectrum is corrected by ray-tracing with {\\sc batupdatephakw} and the response \nmatrix is generated using {\\sc batdrmgen}. The XRT data was extracted using the standard tools provided \nby the UK Swift Science Data Centre (\\citealt{Evansetal_2009})\\footnote{\\url {http:\/\/www.swift.ac.uk\/burst_analyser\/}}.\nWe extract the spectrum from the WT data with a pileup and exposure map correction. \n\nAs for the spectral model, we first choose Band function (\\citealt{Bandetal_1993}), which \nis a broken power law with two photon indices, $\\alpha$ and $\\beta$ and a peak, $E_{\\rm p}$ \nin the $\\nu F_\\nu$ representation. A majority of GRB \ndata is consistent with this function (e.g., \\citealt{Gruberetal_2014, Goldstein_etal_2013}).\nAs we will show in Section~\\ref{spectral_evolution}, the evolution of the $E_{\\rm p}$ of the \nBand function with time shows a sudden jump. As GRB spectrum has been found to have multiple spectral components, this \nsudden jump can as well represent another peak in the spectrum which is not captured \nby the single peak Band function. In order to check that the spectral variation is real, we then \nuse two models, Band function + Blackbody (Band + BB), e.g., \n\\cite{Guiriecetal_2011, Axelsson_etal_2012, Guiriecetal_2013} and \nTwo blackbodies + power law (2BB + PL), e.g., \\cite{Basak_Rao_2014_MNRAS,\nBasak_Rao_2015_090618, Iyyanietal_2015}. In addition, at the later phase\nwhere the data does not allow to put a constrain on the high energy power law,\nwe use instead a blackbody + power law (BB + PL) model.\n\nThe spectral analysis is carried out in {\\sc xspec} version: 12.9.0. For the analysis involving {\\it Fermi}\/GBM \nand LAT data, PG-Statistic is used (\\citealt{Greiner_etal_2016}) and that involving {\\it Swift}\/BAT and XRT data, \n$\\chi^2$ statistic\\footnote{\\url {http:\/\/swift.gsfc.nasa.gov\/analysis\/bat_swguide_v6_3.pdf}} is used.\nAll the errors on the fit parameters are quoted at $1\\sigma$ (nominal 68\\% confidence). \n\n\n\n\\begin{table*}\\centering\n\\caption{Parameters of time resolved spectral fitting with Band model. Bin 8 and 11 have a \nsimultaneous coverage with the LAT data marked $^{(a)}$\n\\begin{tabular}{ccccccccc}\n\n\\hline\nBin \\# & Time interval (s) & $\\alpha$ & $\\beta$ & $E_{\\rm peak}$\\,(keV) & $N_{\\rm Band}\\,(10^{-3})$ & PG-Stat (dof) \\\\ \n\\hline\n\\hline \n$1$ &$-2.0$ -- $ 2.7$ & $ -0.90_{ -0.06}^{+ 0.05}$ & $ -2.7_{-0.2}^{+0.1}$ & $ 3082_{ -631}^{+ 1110}$ & $ 3.4_{-0.2}^{+ 0.2}$ & $ 489.8\\,(477)$ \\\\\n$2$ &$ 2.7$ -- $ 4.5$ & $ -1.13^{+0.04}_{-0.05}$ & $ -2.4^{ +0.1}_{- 0.2}$ & $ 2325^{+3475}_{-620}$ & $ 10.9_{- 0.5}^{ +0.4}$ & $ 457.2\\,(477)$ \\\\\n$3$ &$ 4.5$ -- $ 6.3$ & $ -1.10^{+0.11}_{-0.04}$ & $ -2.5^{ +0.2}_{- 0.2}$ & $ 921^{ +9036}_{- 416}$ & $ 12.3^{ +2.0}_{- 0.6}$ & $ 532.3\\,(477)$ \\\\\n$4$ &$ 6.3$ -- $ 8.2$ & $ -1.14^{+ 0.29}_{- 0.09}$ & $ -2.3^{+ 0.2}_{- 0.2}$ & $ 625^{+670}_{- 424}$ & $ 13.3^{+ 7.7}_{- 1.5}$ & $ 499.2\\,(477)$ \\\\\n$5$ &$ 8.2$ -- $10.3$ & $ -1.25^{+ 0.10}_{-0.03}$ & $ -2.5^{+ 0.1}_{- 0.2}$ & $ 1292^{+ 669}_{- 596}$ & $ 10.5^{+ 1.4}_{- 0.4}$ & $ 507.8\\,(477)$ \\\\\n$6$ &$10.3$ -- $13.2$ & $ -1.18^{+0.09}_{- 0.08}$ & $ -2.4^{ +0.1}_{- 0.2}$ & $ 582^{ +370}_{- 191}$ & $ 9.9^{+ 1.3}_{-1.1}$ & $ 471.2\\,(477)$\\\\\n$7$ &$13.2$ -- $16.3$ & $ -1.16^{+ 0.11}_{-0.09}$ & $ -2.3^{+ 0.1}_{-0.1}$ & $ 411^{ +237}_{- 135}$ & $ 10.1^{ + 1.8}_{- 1.3}$ & $ 529.5\\,(477)$\\\\ \n$8^{(a)}$ &$16.3$ -- $20.6$ & $ -1.21^{+ 0.13}_{-0.12}$ & $ -2.5^{+ 0.1}_{-0.2}$ & $ 492^{+ 619}_{-204}$ & $ 7.6^{+ 1.7}_{-1.3}$ & $ 574.9\\,(487)$\\\\\n$9$ &$20.6$ -- $27.5$ & $ -1.08^{+0.22}_{- 0.30}$ & $ -2.2^{+ 0.1}_{-0.3}$ & $ 208^{+950}_{-86}$ & $7.3^{+3.3}_{-2.9}$ & $ 612.8\\,(477)$\\\\\n$10$ &$27.5$ -- $37.1$ & $ -1.34^{+ 0.16}_{-0.10}$ & $ -1.6^{+0.1}_{-0.1}$ & $ 580^{+6619}_{- 397}$ & $3.1^{+0.8}_{-0.4}$ & $ 640.5\\,(469)$\\\\\n$11^{(a)}$ &$37.1$ -- $53.2$ & $ -1.34^{+ 0.06}_{- 0.06}$ & $ -3.1^{+0.4}_{- 0.3}$ & $ 8318^{+7462}_{-4801}$ & $1.6^{+0.1}_{-0.1}$ & $ 747.4\\,(479)$\\\\ \n\n\\hline\n\\end{tabular}\n\n\\label{t_band}\n\\end{table*}\n\n\n\n\n\n\\subsection{Effective area correction:} \nThe different detectors used for the spectral analysis are expected to have some differences\nin the calibration. We multiply a constant factor for each detector used for a spectral fitting\nfor the effective area correction (EAC). However, as each time-resolved data has a limited number \nof counts, the EAC constant may not be well constrained. Hence, we adopt the following procedure.\n\nFirst, note that it is sufficient to determine the correction factor of all the detectors relative\nto one of the detectors. Hence, we freeze the EAC constant value corresponding to the detector having \nthe highest count rate to 1, while that of the other detectors are made free. We then load the time-resolved \ndata of all the detectors in all the time bins simultaneously. Then the EAC constant parameter of \nthe individual detectors are linked throughout all the time bins. For example, let us assume that \nwe have three detectors and four time bins, with detector 1 having the highest count rate. \nWe load $3\\times4=12$ data groups simultaneously. Group 1--3 correspond to time bin 1, group 4--6 \ncorrespond to time bin 2 and so on. Now, the constant parameter of group 1 is fixed to 1, while that of \ngroup 2 and 3 are free. From the next time bin onward this parameter is linked to the \ncorresponding value of time bin 1. Thus, the constant parameter of the group 4 is \nset equal to that of the group 1, that of the group 5 is set equal to that of the group 2,\nand so on. Note that this procedure is allowed as the EAC of the detectors cannot vary with time.\nOn the other hand, linking this parameter throughout all the time bins enable us to determine \nthe value with much higher accuracy than that obtained by individual fits.\n\n\nIn the present study, as the n3 detector has the highest count rate, we freeze the EAC constant \nof this to 1. As the LAT-LLE data is available until bin 9, we use the procedure until this time \nbin. For the various models we use, we find that the EAC constant of all the detectors of GBM give quite \nsimilar values. These are --- n3: 1 (fixed), n0: $0.95\\pm0.03$, n1: $0.93\\pm0.03$, b0: $1.0\\pm0.1$.\nThe EAC constant corresponding to the LAT-LLE was found to widely differ between models. The value \nis sometimes unrealistically low ($<0.3$). Based on the current understanding of the different \ndetectors of the {\\it Fermi},~ the EAC constant factor is not expected to differ by more than $30\\%$.\nWe find that the Band + BB gives the most realistic value of this factor \nin the range $0.40-1.02$. Note that due to much less number of spectral bins, the \nconstant factor is not well constrained even after using the above mentioned procedure. As the value is \nconsistent with having no correction, we freeze the EAC constant factor corresponding to the LAT-LLE \nand the LAT data ($>100$\\,MeV) \nto 1 for all spectral fitting. For the XRT analysis, we find the constant factor in \nthe range 0.8--1.2, and hence, we freeze it to 1. We do not apply any cross-calibration between the XRT \nand the BAT.\n\n\\section{Results}\n\\label{results}\n\n\\subsection{Spectral evolution during the prompt emission}\n\\label{spectral_evolution}\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{band_parm-eps-converted-to.pdf} \n}\n\\caption{Evolution of the parameters of the Band model fitted to the time-resolved \\emph{Fermi}\/GBM,\nLAT and XRT data. For Bin 12--14, we show the parameters of the power law model fits in orange \nsymbols. Top to bottom --- Panel 1: $E_{\\rm p}$; Panel 2: Photon index $\\alpha$ \n(red filled boxes), $\\beta$ (black filled circles) of the Band function, and the \npower law index, $\\Gamma$ (orange filled boxes); \nPanel 3: Energy flux in units of $10^{-7}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$. The light curve in energy range,\n8 keV -- 100 keV, is shown in gray in Panel 1.\nThe detection of the first LAT photon at 17.5\\,s is marked by a vertical dashed line in Panel 1.\nIn Panel 2, the synchrotron fast cooling photon index of 3\/2 and the slow cooling photon index of 2\/3 are marked by \nhorizontal dashed lines.\n}\n\\label{band_parm}\n\\end{figure}\n\n\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{wide_spectrum-eps-converted-to.pdf} \n}\n\\caption{Wide band spectrum of Bin 8 fitted with the Band function. Markers\n(and colours) used for different detectors are shown in the legend.\n}\n\\label{wide_spectrum}\n\\end{figure}\n\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{babb_parm-eps-converted-to.pdf} \n}\n\\caption{Evolution of the parameters of the Band + BB model fitted to the time-resolved \\emph{Fermi}\/GBM,\nLAT and XRT data. For Bin 10 --14, the parameters of the BB + PL model fits are shown in orange \nsymbols. From top to bottom --- Panel 1: $E_{\\rm p}$ of the Band (black filled circles) and $kT$ \nof the blackbody (red open circles); Panel 2: Photon index $-\\alpha$ (red filled boxes), \n$-\\beta$ (black filled circles) of the Band function, and the power law index, $\\Gamma$ (orange filled boxes);\nPanel 3: Energy flux in units of $10^{-7}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$\nfor Band (black filled circles), BB (red open circles) and power law (orange filled circles);\nPanel 4: $\\cal{R}$ in units of $10^{-21}$. The light curve in energy range, 8 keV - 100 keV, is shown in grey in Panel 1.\nThe detection of the first LAT photon at 17.5\\,s is marked by a vertical dashed line in Panel 1.\nIn Panel 2, the synchrotron fast cooling photon index of 3\/2 and the slow cooling photon index of 2\/3 are marked by \nhorizontal dashed lines.}\n\\label{babb_parm}\n\\end{figure}\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{2bbpl_parm-eps-converted-to.pdf} \n}\n\\caption{Evolution of the parameters of the 2BB + PL and BB + PL model fitted to the time-resolved \\emph{Fermi}\/GBM,\nLAT and XRT data are shown. The parameters of the high energy blackbody, low energy blackbody \nand the power law component of the 2BB + PL model\nare shown in black filled circles, red open circles and violet open boxes respectively.\nThe parameters of BB + PL model fits are shown in symbols of orange colour.\nFrom top to bottom --- Panel 1: Temperature of the blackbodies; Panel 2: Power law photon index, $\\Gamma$; \nPanel 3: Energy flux of each component in units of $10^{-7}$\\,erg\\,cm$^{-2}$\\,s$^{-1}$;\nPanel 4: $\\cal{R}$ in units of $10^{-21}$. The light curve in energy range, 8 keV - 100 keV, is shown in grey in Panel 1. \nThe detection of the first LAT photon at 17.5\\,s is marked by a vertical dashed line in Panel 1.\n}\n\\label{2bbpl_parm}\n\\end{figure}\n\n \nThe results of the Band fits are shown in Table~\\ref{t_band} and Fig.~\\ref{band_parm}.\nAn example of Band fit is shown for Bin 8, where the wide band data including that of \nthe LAT is available, see Fig.~\\ref{wide_spectrum}.\nThe parameters could be constrained until Bin 11, and afterwards the Band turns to be a \nsingle power law. Note that the $E_{\\rm p}$ evolution in Panel 1 of Fig.~\\ref{band_parm} shows a \nhard-to-soft (HTS) evolution until Bin 9 and then the value starts to increase and reaches a few MeV. \nWhen the spectral evolution is compared with the 8 keV --100\\,keV light curve, shown in the background, the\nevolution does not seem to show an intensity-tracking (IT) trend either. Note from Fig.~\\ref{lc} that none \nof the light curves in any energy band show any new pulse at this phase. \nHowever, interestingly, we note that the first LAT photon ($ > 100\\, \\rm MeV$) arrives at 17.5\\,s,\nwhich is marked by a dashed line in both Fig.~\\ref{lc} and Fig.~\\ref{band_parm}. \nThe observed change in the spectral peak of the Band function\ntakes place during this time. This thereby \nsuggests either an onset of a second hard pulse related to the prompt emission \nor the beginning of the afterglow phase (see Section~\\ref{second_pulse} and \nSection~\\ref{disc} for a detailed discussion).\n\n\n\nIn the second panel of Fig.~\\ref{band_parm}, we show the evolution of the photon indices of the Band \nfunction as well as that of the power law. Negative values of $\\alpha$ and $\\beta$\nare shown in order to match the convention of the photon index $\\Gamma$ of the power law model i.e., \n$N(E)\\propto E^{-\\Gamma}$. \nThe $\\alpha$ values are found to be softer than the line of death of synchrotron emission (\\citealt{Preeceetal_1998})\ni.e $\\alpha = -0.67$, corresponding to slow cooling synchrotron emission,\nthroughout the burst duration. In accordance to what is typically observed \n(\\citealt{Kanekoetal_2006}), $\\alpha$ is found to get softer with time, tending to values, \n$\\alpha = -1.5$, consistent with fast cooling synchrotron emission (marked as pink horizontal dash\nline in Fig.~\\ref{band_parm} ). The value of $\\beta$ also decreases gradually over time. \nAs a result of which, at later times, it becomes difficult to constrain the spectral peak and \nspectrum is then modelled by a simple power law. The power law index, $\\Gamma$ has values equal\nto $1.5$. Such hard values of $\\Gamma < 2$ suggest either a spectral peak or cutoff to lie beyond \nthe observed energy window, e.g \\cite{Gonzalezetal_2003}. \n\n\nIn the third panel we show the evolution of bolometric energy flux in 0.1\\,keV -- 100\\,GeV.\nWe however note that the bolometric \nvalue inherently assumes that the same model can be extended in both lower and higher energies. While an observed \nflux would not have such assumption, it can underestimate the powerlaw flux. Also, as we have used different \ndetectors at different phases of the GRB, the bolometric flux automatically provides a uniform band for all\nobservation. The flux at any desired energy band can be easily calculated using the parameters given in the tables.\nThe Band flux smoothly decreases until bin 9.\nNote that the power law model as well as in case of the Band function (bin10), where the Band nearly tends \nto be a power law ($\\beta$ is nearly equal to $\\alpha$), show an increased flux in comparison to the \notherwise smooth evolution. This is due to the fact that in other bins, the Band function has a \nsteeper slope at higher energies than the power law which does not have a break. \nConsequently, in these bins the fluxes are over estimated. \n\n\nThe apparent jump in $E_{\\rm p}$ evolution as found above calls for a detailed study. This is solely a spectral \nvariation as the underlying lightcurve does not show any change. It is known that GRB spectrum can \nhave multiple components with two peaks e.g., Band + BB, or 2BB + PL. If one of the \ncomponents is dominant, it is possible that our single component Band function picks up that one\nleading to an artificial $E_{\\rm p}$ variation. Hence, in order to check that the variation is \nphysical we use these two models. The corresponding\nspectral evolutions are shown in Fig.~\\ref{babb_parm} for Band + BB and Fig.~\\ref{2bbpl_parm}\nfor 2BB + PL model. Whenever the curvature at high energies cannot be constrained,\nwe use the simpler BB + PL model, which these two models would converge to. \nFor the parameters of all the fits, see Appendix~\\ref{table_spectral}.\nFor bins 12--14, we include simultaneous XRT observation as well. \nAs soft X-rays suffer from absorption at the source and the Galaxy, we include two absorption \nterms ({\\sc TBabs} in {\\sc Xspec}) in the model. The equivalent hydrogen column density ($n_{\\rm H}$) of the \nGalactic term is fixed to $7.74\\times10^{20}$\\,atoms cm$^{-2}$. Then we link the $n_{\\rm H}$ of the \nsource absorption term between the time bins. We obtain $n_{\\rm H}=(5.1\\pm0.6)\\times10^{21}$\\,atoms cm$^{-2}$.\nFor subsequent fitting with the XRT data, we freeze the source $n_{\\rm H}$ at this value as this cannot \nevolve with time and the linking of the parameter gives good confidence on the obtained value.\n\nFrom the upper panels of Fig.~\\ref{babb_parm} and Fig.~\\ref{2bbpl_parm} we see a very similar \nvariation of the spectral peak as before. This indeed shows that the sudden jump in the peak \nenergy is not an artefact of the adopted model. \nThe next two panels of the Figures are same as \nFig.~\\ref{band_parm}. In the fourth panel, we also show the \nevolution of the parameter $\\cal{R}$$\\equiv(F_{\\rm BB}\/\\sigma T^4)^{1\/2}$ of the blackbody component. \n${\\cal{R}}$ represents the effective transverse size of the emitting region (i.e photosphere) \nprovided the bulk Lorentz factor of the outflow, $\\Gamma_{\\rm B} \\gg 1\/\\theta_{\\rm j}$ where\n$\\theta_{\\rm j}$ is the jet opening angle (\\citealt{Ryde_Pe'er_2009}).\nFor Band + BB note that $\\cal{R}$ does not show a linear increase in contrast to what is typically observed\n(\\citealt{Ryde_Pe'er_2009,Ryde_2004,Ryde_2005}), instead only exhibits an overall increment throughout.\nOn the other hand, the variation of $\\cal{R}$ for both the thermal components of 2BB + PL \nmodel exhibit a similar jump corresponding to the evolution of $E_{\\rm p}$.\n\n\n\n\n\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{2bbpl_late_evolution-eps-converted-to.pdf} \n}\n\\caption{Long time evolution of the parameters of the 2BB + PL model are shown. \nThe symbols used are the same as in Fig.~\\ref{2bbpl_parm}. The data after 88\\,s \ncorrespond to Table~\\ref{t_late_2bbpl}. The 0.3--10 keV XRT flux\n(erg\\,cm$^{-2}$\\,s$^{-1}$) is shown in the background of Panel 1\nand the corresponding vertical scale is shown on the right.\nThe value of $\\cal{R}$ of the lower-temperature blackbody is not shown for \nthe last time interval, as due to a very low temperature, we obtained an \nunrealistically high value.\n}\n\\label{2bbpl_late_evolution}\n\\end{figure}\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{babb_late_evolution-eps-converted-to.pdf} \n}\n\\caption{Long time evolution of the parameters of the Band+BB model (Bin 1--9) and \nBB+PL model (Bin 10--18) are shown.\nThe symbols used are the same as in Fig.~\\ref{babb_parm}. \nThe 0.3--10 keV XRT flux\n(erg\\,cm$^{-2}$\\,s$^{-1}$) is shown in the background of Panel 1\nand the corresponding vertical scale is shown on the right.\n}\n\\label{babb_late_evolution}\n\\end{figure}\n\n\n\n\n\\subsection{Spectral evolution at late times ($> 88 \\,\\rm s$)}\n\\label{late_phase}\n\n\n\n\n\n\\begin{table*}\\centering\n\\caption{Parameters of time resolved spectral fitting of the joint BAT and XRT data at late times.\nWe also show the $\\chi^2$ of the models, including that of the Band function fits.\n\\begin{tabular}{cccccccccc}\n\n\\hline\nBin \\# & Interval (s) & Model & $kT_{\\rm h}$\\,(keV) & $N_{\\rm h}$\\,($10^{-1}$) & $kT_{\\rm l}$ or $kT$\\,(keV) & $N_{\\rm l}$\\,($10^{-1}$) & $\\Gamma$ & $N_{\\Gamma}$\\,($10^{-1}$) & $\\chi^2$\\,(dof) \\\\\n\\hline\n\\hline\n$15$ &$88.2-94.2$ & PL & & & & & $ 1.39_{ -0.05}^{+ 0.05}$ & $ 5.4_{ -0.4}^{+ 0.5}$ & $ 113.4\\,(75)$ \\\\\n & & BBPL & & & $ 12_{ -2}^{+ 2}$ & $ 2.1_{ -0.3}^{+ 0.3}$ & $ 1.97_{ -0.12}^{+ 0.13}$ & $ 8.4_{ -0.8}^{+ 0.8}$ & $ 86.9\\,(73)$ \\\\\n & & 2BBPL & $ 38_{ -13}^{+ 52}$ & $ 3.8_{ -1.4}^{+ 13}$ & $ 6_{ -1}^{+ 1}$ & $ 1.4_{ -0.3}^{+ 0.3}$ & $ 2.21_{ -0.16}^{+ 0.17}$ & $ 8.9_{ -0.8}^{+ 0.8}$ & $ 70.8\\,(71)$ \\\\\n&&Band&&&&&&&103.1 (73)\\\\\n \\hline\n$16$ &$94.2-104.2$ & PL & & & & & $ 1.40_{ -0.04}^{+ 0.04}$ & $ 4.6_{ -0.3}^{+ 0.3}$ & $ 88.9\\,(75)$ \\\\\n & & BBPL & & & $ 11_{ -2}^{+ 2}$ & $ 1.1_{ -0.3}^{+ 0.3}$ & $ 1.64_{ -0.10}^{+ 0.11}$ & $ 5.4_{ -0.5}^{+ 0.5}$ & $ 77.8\\,(73)$ \\\\\n & & 2BBPL & $ 17_{ -3}^{+ 4}$ & $ 1.6_{ -0.3}^{+ 0.3}$ & $ 4_{ -1}^{+ 1}$ & $ 0.8_{ -0.2}^{+ 0.2}$ & $ 2.14_{ -0.22}^{+ 0.23}$ & $ 6.2_{ -0.5}^{+ 0.5}$ & $ 70.4\\,(71)$ \\\\\n&&Band&&&&&&&82.8 (73)\\\\\n\\hline\n$17$ &$104.2-137.2$ & PL & & & & & $ 1.43_{ -0.03}^{+ 0.04}$ & $ 3.0_{ -0.1}^{+ 0.1}$ & $ 89.5\\,(75)$ \\\\\n & & BBPL & & & $ 14_{ -5}^{+ 8}$ & $ 0.4_{ -0.2}^{+ 0.2}$ & $ 1.55_{ -0.07}^{+ 0.07}$ & $ 3.2_{ -0.2}^{+ 0.2}$ & $ 84.9\\,(73)$ \\\\\n & & 2BBPL & $ 25_{ -5}^{+ 10}$ & $ 1.1_{ -0.3}^{+ 0.5}$ & $ 4.3_{ -0.4}^{+ 0.6}$ & $ 0.5_{ -0.1}^{+ 0.1}$ & $ 1.93_{ -0.14}^{+ 0.15}$ & $ 3.6_{ -0.2}^{+ 0.2}$ & $ 72.8\\,(71)$ \\\\\n&&Band&&&&&&&77.8 (73)\\\\\n\\hline\n$18$ &$137.2-253.2$ & PL & & & & & $ 1.38_{ -0.04}^{+ 0.04}$ & $ 0.92_{ -0.04}^{+ 0.04}$ & $ 96.8\\,(75)$ \\\\\n & & BBPL & & & $ 7_{ -1}^{+ 1}$ & $ 0.3_{ -0.1}^{+ 0.1}$ & $ 1.59_{ -0.08}^{+ 0.09}$ & $ 1.00_{ -0.06}^{+ 0.06}$ & $ 75.8\\,(73)$ \\\\\n & & 2BBPL & $ 7_{ -1}^{+ 1}$ & $ 0.2_{ -0.1}^{+ 0.1}$ & $ 0.07_{ -0.01}^{+ 0.02}$ & $ 0.2_{ -0.1}^{+ 0.3}$ & $ 1.51_{ -0.07}^{+ 0.08}$ & $ 0.93_{ -0.06}^{+ 0.06}$ & $ 65.8\\,(71)$ \\\\\n&&Band&&&&&&& $96.2 (73)$\\\\\n\\hline\n$19$ &$253.2-573.2$ & PL & & & & & $ 1.50_{ -0.06}^{+ 0.06}$ & $ 0.03_{ -0.02}^{+ 0.02}$ & $ 14.3\\,(17)$ \\\\\n\\hline\n\n\n\\end{tabular}\n\n\\label{t_late_2bbpl}\n\\end{table*}\n\nThe BAT and XRT spectrum at late times are extracted \nwhen the S\/N of the {\\it Fermi}~ does not allow any further time division. The XRT data in WT mode\nand the BAT data are available until 570\\,s and $\\sim250$\\,s, respectively. Note that the \nXRT light curve has two breaks one at 91\\,s and another at 134\\,s, both counted with respect \nto the BAT trigger time. Hence, in choosing the time intervals, we respect these break times \nand choose roughly logarithmic bins i.e., the interval length increases roughly as a\ngeometric progression. We obtain 5 bins: 85--91\\,s, 91--101\\,s, 101--134\\,s, 134--250\\,s\nand 250--570\\,s (from BAT trigger time). The simultaneous BAT data is extracted in the first \nfour intervals.\n\nThe time-resolved data is analysed with a power law (PL), BB + PL, 2BB + PL and Band function. As \nbefore, two absorption terms --- the Galactic $n_{\\rm H}$ is set to \n$7.74\\times10^{20}$\\,atoms cm$^{-2}$ and the source $n_{\\rm H}$ to $5.1\\times10^{21}$\\,atoms cm$^{-2}$, are used.\nThe result of the fits are shown in Table~\\ref{t_late_2bbpl}. We find that the PL \ndoes not give a good fit (see Table~\\ref{t_late_2bbpl}) and shows large residual. The spectra are much better fitted with \nthe BB+ PL model and even better with the 2BB + PL model. This shows that there is\na curvature in the spectrum. Hence, this phase most probably corresponds to the prompt \nemission phase. We found that the Band function shows an equally bad fit as the PL model \nand the $E_{\\rm p}$ could not be constrainred. The 2BB + PL model seems to be the best fit at the \nlate time. For the last bin, as the BAT data is no more available we could not constrain the \nparameters of more complicated models other than the PL in the limited bandwidth of the XRT.\nHence, only the results of PL fit is reported for this bin. \n\nIn Fig.~\\ref{2bbpl_late_evolution}, we show the evolution of the parameters of the \n2BB + PL model taking all the observations together. The panels are same as in Fig.~\\ref{2bbpl_parm}\nthat refers to the 2BB + PL model fitted at early times. In Fig.~\\ref{babb_late_evolution}, \nsimilar plot is shown for the Band + BB model. In both cases, the evolution at the late phase \nis consistent with the previous evolution. The late time spectral evolution together with a \nsignificantly better reduced $\\chi^2$ in the high resolution data of the XRT \nsuggests that the 2BB + PL model probably most consistently captures the \noverall spectral evolution of the burst. \n\n\n\n\n\\subsection{Evidence for a second hard pulse}\\label{second_pulse}\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{LC_HR-eps-converted-to.pdf} \n}\n\\caption{\\emph{Upper panel:} Bayesian blocks shown on the light curve of combined NaI 0, 1 and 3 in 8--900\\,keV.\nThe blocks are -1--1\\,s, 1--10\\,s, 10--15\\,s and 15--24\\,s. clearly, a new block starts near the time of first \ndetected LAT photon at $\\sim 18$\\,s. \\emph{Lower panel:} Hardness ratio defined as the ratio of count between \n500\\,keV--5\\,MeV and 200\\,keV--400\\,keV band of BGO 0. A sudden increment of hardness is apparent at $\\sim 18$\\,s.\n}\n\\label{lc_hr}\n\\end{figure}\n\n\nIn case of GRBs with multiple but separable pulses, it is frequently observed that the first pulse shows \na HTS evolution, followed by a jump in the peak energy during the onset of a second pulse,\nand then again showing a HTS evolution in the falling part of that pulse (see e.g., \n\\citealt{Ghirlandaetal_2010, Luetal_2012}). \nA very similar behaviour is seen for GRB 151006A, though the presence of a second pulse is not \nseen in the lightcurve. The late time spectral evolution being consistently HTS since the time of sudden jump\n($\\gtrsim 18$\\,s, see Section~\\ref{late_phase}) indicates that the second phase is probably a second pulse \nof the prompt emission hidden in the data. This is not readily evident probably because it is expected to be a hard pulse, \nwhere the signal is weak. In order to have a better look at the phenomenon we perform the following analysis. \n\nWe first construct the Bayesian blocks for the combined count rate data of NaI 0, 1 and 3 in 8--900\\,keV\n(see Fig.~\\ref{lc_hr}, upper panel). The Bayesian block approach finds the best possible way to represent a \ntime-series data as a series of blocks or segments such that the signal underlying each block is \nconstant within the observational error (\\citealt{scargel2013}). We use the dynamical programing \nalgorithm of \\cite{scargel2013} to construct these blocks which are then over-plotted on the \ncount rate lightcurve in Fig.~\\ref{lc_hr}, upper panel. During the interval shown in the Figure, \nwe find the Bayesian blocks as -1--1\\,s, 1--10\\,s, 10--15\\,s and 15--24\\,s. With only the NaI 3 \ndetector these blocks are -1--1\\,s, 1--14\\,s and 14-24\\,s. We note that a new block starts \nat $\\sim14-15$\\,s which shows that the count rate flux of this bin is statistically different from the \nBayesian blocks on either side on the time axis. But, as the flux level of the consecutive blocks \naround this block decreases monotonically, there is no evidence for a new pulse in the data. \nHowever, we note from Table~\\ref{t_bandbb} that the peak energy at the transition time reaches a very \nhigh value $\\sim 5$\\,MeV, and hence, it is unlikely that such a change will give rise to any\nsignificant pulse profile in the NaI detectors. On the other hand, the flux level of BGO \ndetector is quite poor to carry out such analysis. \n\nGiven the high value of the peak energy, and no pulse profile with Bayesian blocks in the NaI detector,\nwe are left with only one possibility that the pulse is hidden in the BGO data and we investigate \nthat possibility. Before proceeding, we note that there are two competing factors here:\na lightcurve constructed in a wide energy \nband will smear out the small variations that we expect at the high energies. On the other hand, a \nlightcurve in a limited bandwidth suffers from the poor statistics of photon count, more so \nfor the BGO detector and that too at a time when the photon flux is already low. This is why we used \nthe full band of the NaI detector for the Bayesian block analysis in the first place.\nA more robust way to investigate it further is to study the evolution of hardness ratio (HR)\nrather than the count rate lightcurve in a limited band. As the HR is a ratio of count between \ntwo energy bands, the small changes in the photon count subjected to the change in spectral \npeak will be amplified. More importantly, if there is a smooth pulse presumably a hard one, which is \nnot seen in the lightcurve otherwise, the HR should track that pulse profile. We use the BGO \ndetector (the BGO 0) rather than the NaI since the peak energy reaches high values covered by the BGO energy band.\nOn the other hand, we cannot use the LAT LLE data as it does not provide the low energy band required for this analysis.\nWe choose the hard band as $H=\\rm 500\\,keV-5\\,MeV$ and the soft band as $S=\\rm 200\\,keV-400\\,keV$,\nand then the Hardness Ratio $=H\/S$. \nWe then stick to the same definition throughout.\nThe evolution is shown in Fig.~\\ref{lc_hr}, lower panel.\nThe first phase shows a HTS evolution starting with a hardness ratio of $\\sim1$, \nreaching down to a value close to 0 until $\\sim15$\\,s and then shows a sudden jump again \nreaching a hardness ratio of $\\sim1$, followed by another HTS evolution. The errors \nin the hardness ratio are larger at the later phase due to the lower flux level, but the evidence \nof the jump in hardness lightcurve, and a smooth pulse profile afterwards is very clear. \nThis changeover time remarkably coincides with the onset of the unusual spectral evolution.\nThis is exactly when the second phase of emission begins \nand the first LAT photon ($>100$\\,MeV) is detected. This analysis, combined with the fact that the \nsecond phase shows a very similar HTS evolution as seen in other GRBs, points towards \nthe fact that this new energy injection is due to a second hard pulse of the prompt emission \nitself.\n\n\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{pol-eps-converted-to.pdf} \n}\n\\caption{Modulation curves for GRB 151006A in two phases: 0--16\\,s (upper panel) and \n16--33\\,s (lower panel). The modulation curve is obtained\nafter geometrically correcting the raw azimuthal angle histogram by \nnormalizing with respect to a simulated unpolarized distribution for a \nsimilar spectra and off-axis detection angle (\\citealt{Raoetal_2016}). The blue solid line\nis the $\\cos{\\phi}^2$ fit to the modulation curve. The fitted modulation\nfactor, polarization fraction ($\\Pi$) and polarization angle (P. A.) are \nshown in the inset of the respective panels, with errors \nestimated at 68\\% confidence level (see text for details).\n}\n\\label{pol}\n\\end{figure}\n\n\\subsection{Polarization measurement -- further evidence for the second pulse of prompt emission}\n\nAnother piece of information about these two emission phases can be obtained from \npolarization measurement. If the second emission phase is a hard pulse, and we attribute \npolarization to the non thermal processes that produce harder spectrum, then we expect to have \nan enhancement of the degree of polarization in the second phase.\nCZTI being a pixelated detector and having a significant Compton scattering \nprobability at energies beyond 100 keV, essentially works as a \nCompton polarimeter at \nthese energies. The double pixel events within the photon tagging time\nwindow of 40 $\\mu$s which satisfy the Compton kinematics are identified\nas valid Compton events. The validity of the Compton event selection \nand the polarization measurement capability of CZTI have been established\nby detailed simulation and experimental studies during ground calibration\nof CZTI (\\citealt{chattopadhyay14,Vadawale_etal_2015}). The first onboard validation \nof CZTI polarimetry was obtained with the detection of polarization of\nGRB 160131A (\\citealt{vadawale16}) and GRB 151006A (\\citealt{Raoetal_2016}). GRB151006A, though\nis moderately bright for X-ray polarization measurement, a hint of \npolarization is seen with an estimated polarization degree $> 90\\%$ with\na detection significance of 1.5$\\sigma$ (68$\\%$ confidence level with \n1 parameter of interest). Since our spectroscopic analysis shows that there\ncould be two distinct phases of emission, we tried to explore polarization\nmeasurement in these two sectors using the CZTI data. Fig.~\\ref{pol} shows the\nmodulation curves for these two time bins in 100 $-$ 350 keV, where the blue\nsolid line is the sinusoidal fit to the observed distribution of \nazimuthal angle of scattering.\n\n\nIn order to estimate the polarization fractions, we did detailed \nGeant 4 simulations (\\citealt{agostinelli03}) using \\emph{Astrosat} mass model \nwhich includes all the instruments of \\emph{Astrosat} along with the complete satellite structure. We employ\nthe fitted spectroscopic models to simulate the energy distribution of the\nincident photons in Geant 4 to obtain the modulation factors for \n100$\\%$ polarized beam ($\\mu_{100}$) which are then used to estimate the polarization fractions\n($P = \\mu\/\\mu_{100}$) in two phases, 0--16\\,s and 16--32\\,s. Estimated polarization\nfractions are 77$\\%$ and 94$\\%$ with $1\\sigma$ detection significance\nat polarization angles 334$^\\circ$ and 325$^\\circ$ respectively in these \ntwo phases. The detection significance of polarization\nhas reduced significantly compared to the initial report of\npolarization for GRB 151006A (\\citealt{Raoetal_2016}) which is\nexpected due to reduced number of events in the individual phases. Though\nfraction and angle of polarization are poorly constrained in both the time \nsectors, the measured values do not show any decrement of polarization.\nIn the multi-component models with thermal and non thermal parts, it is \nsuggested that the thermal emission dominates in the first part while the \nnon thermal processes become important at the later stage (e.g., \\citealt{Gonzalezetal_2003}).\nAttributing the polarization to the non thermal processes,\nthe above evolution is consistent with the expected behaviour for such models. \nHence, there is a hint of multi-component spectra in the data.\n\nIt is important to note that the polarization degree in the afterglow phase found \nby optical measurements so far show a pretty low value ($\\lesssim10\\%$) and it is \nexpected to reduce further as the afterglow proceeds,\nsee e.g., \\cite{greiner2003, mundell2007}. This appears to be due to the fact that \nthe magnetic field during the afterglow phase is mostly generated by turbulence \nand therefore has a random orientation, having only a small coherence length. This is in contrast \nwith the prompt emission phase, where measurements show a high degree of polarization ($\\sim40-80\\%$) which \ncan be achieved by an ordered magnetic field, see \\cite{Waxman2003}. Though a different \nmechanism is possible to achieve a high degree of polarization of the prompt emission,\nthe important point is that such high values are not seen in the afterglow phase.\nOur measurement is consistent with a high value, and though the statistics is \npoor the measured values in the two phase do not contradict the interpretation\nthat we are most probably observing the prompt emission extending out to the second phase. Hence, the \ndata is indicative of a second hard pulse rather than the onset of an afterglow phase.\n\n\\subsection{Comparison between the models}\n\n\\begin{figure}\\centering\n{\\vspace{-0.1in}\n\\includegraphics[width=\\columnwidth]{pgstat_dof_allmodels-eps-converted-to.pdf} \n}\n\\caption{The PGStat\/dof of all the models are shown.\n}\n\\label{pgstat}\n\\end{figure}\n\n\n\\begin{figure}\\centering\n{\n\\includegraphics[width=\\columnwidth]{late_spectrum-eps-converted-to.pdf} \n}\n\\caption{Spectral fitting of the joint BAT and XRT data with PL, BB + PL, Band and 2BB + PL models \nfor the late time bin 15. \nThe data and residual are shown by grey filled circles. The high energy blackbody, \nlow energy blackbody (or the blackbody for the BB + PL model), the power law and\nthe total model, are shown by red dot-dashed line, orange dashed line, violet dotted\nline and thick black line, respectively.\n}\n\\label{late_spectrum}\n\\end{figure}\n\nWe then compare the different models based on their goodness of fit.\nThe PG-Stat\/dof of all the models in different time bins are shown in Fig.~\\ref{pgstat}.\nWe note that all the models have very similar PG-Stat. Thus, making the judgment of the \nbest model is quite indecisive. We then use Bayesian inference criteria (BIC) to see \nwhether the more complicated models are indeed required by statistics \n(e.g., \\citealt{wang2017}). The BIC \nis defined as $-2\\ln \\cal{L}$$+ k\\ln(\\nu+k)$, where $-2\\ln \\cal{L}$ is log likelihood,\n$k$ is the number of free parameters and $\\nu$ is the degrees of freedom. In Table~\\ref{bic}, \nwe show the BIC values and the preferred models.\n\n\n\\begin{table}\\centering\n\\caption{BIC values and preferred model\n\\begin{tabular}{ccccc}\n\n\\hline\n\\# & \\multicolumn{4}{c}{$-2\\ln \\cal{L}$ (BIC)}\\\\\n & Band & Band+BB & 2BB+PL & Preferred \\\\\n\\hline\n\\hline\n1 & 489.8 (514.5) & 488.1 (525.1) & 501.7 (538.7) & Band \\\\\n2 & 457.2 (481.9) & 450.4 (525.1) & 478.3 (515.3) & Band \\\\\n3 & 532.3 (557.0) & 517.0 (554.0) & 527.4 (564.4) & Band+BB \\\\\n4 & 499.2 (523.9) & 485.5 (522.5) & 483.8 (520.8) & 2BB+PL \\\\\n5 & 507.8 (532.5) & 506.7 (543.7) & 515.8 (552.8) & Band \\\\\n6 & 471.2 (495.9) & 468.3 (505.3) & 471.1 (508.1) & Band \\\\\n7 & 529.5 (554.2) & 528.6 (565.6) & 534.1 (571.1) & Band \\\\\n8 & 574.9 (599.7) & 572.5 (609.7) & 578.1 (615.3) & Band \\\\\n9 & 612.8 (637.5) & 601.9 (638.9) & 604.2 (641.2) & Band \\\\\n10 & 640.5 (665.1) & & & Band \\\\\n11 & 747.4 (772.1) & & 751.1 (788.2) & Band \\\\\n\\hline\n & BBPL & BBCPL & PL \\\\\n \\hline\n12 & 1497 (1526) & 1498 (1534) & 1505 (1520) & PL \\\\\n13 & 1571 (1600) & 1566 (1602) & 1573 (1587) & PL \\\\\n14 & 1162 (1191) & 1162 (1199) & 1165 (1179) & PL \\\\\n\\hline\n & BBPL & 2BB+PL & PL \\\\\n \\hline\n15 & 86.9 (104.3) & 70.8 (96.9) & 113.4 (122.1) & 2BB+PL \\\\\n16 & 77.8 (95.2) & 70.4 (96.5) & 88.9 (97.6) & BBPL\/2BBPL \\\\\n17 & 84.9 (102.3) & 72.8 (98.9) & 89.5 (98.2) & PL\/2BBPL \\\\\n18 & 75.8 (93.2) & 65.8 (91.9) & 96.8 (105.5) & 2BB+PL \\\\ \n\\hline\n\\end{tabular}\n\\label{bic}\n\\end{table}\n\nWe note that in the initial bins where we use the {\\it Fermi}~ data, the Band function is the\npreferred model. This signifies that the data is consistent with the Band function \nand a more complicated model is not statistically required. There are only two exceptions,\nBin 3 and 4. In the former Band + BB is the preferred model, while in the later the \n2BB + PL is the preferred model. At the later phase, we see even the PL is the \npreferred model and the other complicated models are not statistically required. \nOur result clearly shows the limitations of a background dominated low resolution \ndetector. \n\nHowever, in the late phase, though we have lower count rate, the signal to noise of\nthe data is improved due to both BAT and XRT and we have good resolution of the XRT. \nWe see a preference for the 2BBPL model in the data at late phase.\nFrom Table~\\ref{t_late_2bbpl}, we also see that the Band function is not \npreferred. In fact, we could not constrain the peak energy, and it signifies that the \nspectral shape is not a Band function. \n\n\nIn Fig.~\\ref{late_spectrum}, we show the spectral fit and residuals of the PL, BB + PL, Band and 2BB + PL model fit\nfor bin 15. Note that for the PL fit the slope at lower and higher energies do not match which is \nclearly indicated by the deviation of the residual in the opposite direction of the zero line.\nFor the BBPL model, this is improved, but then we find large residual on the both sides of the \nblackbody peak. The Band function fares no better than the PL model and has a similar residual.\nAll the curvatures in the spectrum are consistently taken care by the 2BB + PL model. \n\n\n\\section{Conclusions and Discussion}\n\\label{disc}\n\n\n\n\nSingle pulse bursts have been carefully studied using time resolved spectroscopy, \nas it is generally assumed to alleviate the issue of pulse overlapping which hinders \nour understanding of spectral evolution of GRB with time. GRB 151006A is a single\npulse burst in different energy bands and thereby is an ideal \none for time resolved spectroscopy. In general, such bursts exhibit a HTS\nspectral peak evolution. However, the current analysis of GRB 151006A \nshows an unexpected behaviour of change in trend from HTS\nevolution to increasing spectral peak with time, after a few seconds from \nthe burst trigger. Coincidently, this rise is observed after the arrival \nof LAT photons ($> 100 \\, \\rm MeV$). Such a dramatic change in the spectral\nbehaviour for single pulse burst is observed for the first time. \nThis new injection of energy does not show any additional pulse\nin the counts light curve. Thus, this cautions us that a single \npulse need not always suggest a HTS spectral peak evolution and\nthat there may be other pulses hidden in the light curve profile.\n\n\n\nIt is important to notice that high energy emission is often delayed \nand generaly modelled with powerlaw appearing in the spectrum at late times\n(e.g., \\citealt{Gonzalezetal_2003, Ackermannetal_2013_LAT}). This can potentially \nchange the behaviour of the spectral evolution we report here. However, we have \ntested that a powerlaw with Band function is not statistically required\nfor Bin 8 and 11. This is apparent from the spectral shape in Fig.~\\ref{wide_spectrum}.\nGRB 151006A is not among the high-fluence LAT class \n(cf. \\citealt{Ackermannetal_2013_LAT}), and for several such cases the spectrum \nis found to be well fitted with Band function only and no powerlaw is required. \n\n\n\n\nIn order to show that the $E_{\\rm p}$ variation is physical and not due \nto single peak Band function, we also tried models with two peaks i.e., Band + BB \nand 2BB + PL, and re-confirm the spectral variation.\nThe analysis also presents one of the key issues in current GRB spectral \nanalysis i.e., a preferred model cannot be decided based only on statistics.\nThough in the later part the 2BB + PL model seems to be a better fit.\nIn \\cite{Basak_Rao_2015_090618} the spectral components of this model is proposed to\noriginate in a spine-sheath jet (\\citealt{Ramirez-Ruizetal_2002, Zhangetal_2003, Zhangetal_2004},\nsee \\citealt{Iyyanietal_2015} for an alternative explanation).\nThe blackbody radiations are produced at the two photospheres.\nIn addition, photons crossing the boundary layer of the spine-sheath structure are \non an average Compton up-scattered and hence form a high energy power-law spectrum. \nFurther non thermal processes can occur due to \nsynchrotron emission at a higher radial distance. These photons are naturally delayed\nwith respect to the photons produced by thermal and non thermal processes stated above. \nHence, the second phase in this scenario is probably an onset of the delayed emission \nphase. We note here that though the second phase starts after $\\sim18$\\,s, the pulse \ncan have a much lower start time, which is hidden in the falling part of the first pulse.\nHence, an actual delay of this phase can be much smaller, though remains undetermined. \n\n\n\n\n\n\n\n\n\n\nRecently, \\cite{Moretti&Axelsson2016} reported a rise in the high energy spectral peak with time, \nfor the first LAT detected GRB 080825C, a multi-pulse burst (\\citealt{Abdo_etal_2009_080825C}), \nwhen it was re-analysed including the LLE data. \nThis points out the significance of LLE emission \ndetections which can be crucial in constraining the high energy \nspectral behaviour of GRB spectra. The interesting part of our \nobservation is that the increasing beahaviour is seen in an otherwise \nsingle pulse. The new injection of energy\nmay be associated with a second pulse of emission in the prompt\nphase which brings about a dynamical change in the behaviour of\nthe radiation process, or it may be an overlap of the onset of \nafterglow (emission from a different region) with the prompt \nemission, which then significantly dominates the high energy \nand thereby making the prompt emission indiscernible in the spectrum. \nWe note that the $\\alpha$ values of Band function to be consistent \nwith slow cooling synchrotron radiation in the beginning of the burst ($< -0.67$) \nand gets softer with time approaching the fast cooling index \n($\\alpha = -1.5$). If the new energy injection is due to a second pulse \nin the prompt emission, we may be observing a transition in the microphysical\nparameters related to the synchrotron radiation, such as strength or structure \nof magnetic fields, fraction of energy injected into electrons, radiation\nefficiency etc (\\cite{Daigne&Mochkovitch1998}), causing the transformation \nfrom slow to fast cooling of electrons, see \\cite{Iyyani_etal_2016}. \nSince this transformation is observed to be sudden, it perhaps indicate the\nonset of a second pulse with a different source parameter.\nFor the afterglow scenario,\nsuch emission has been mostly modelled using a power law\n(\\citealt{Ackermannetal_2013_LAT, Kumar2009, Ghisellini_etal_2010}).\nHowever, we note a definite spectral curvature at late times and \nthe detection of LAT emission \nis less significant and not many photons are detected.\nOur analysis with Bayesian block and evolution study of the hardness \nratio also points towards the former scenario i.e., a new hard pulse hidden in the data.\n\n\n\n\nFinally, the inability of determining the best model based on statistics \nalso summons for constraining observations like polarization. This can potentially\ncharacterize various radiation processes leading to the observed emission\nas well as in revealing the structure and strength of magnetic fields of emitting region.\nThis can in turn be an invaluable input in enhancing our understanding of \nshock physics as well as the content of GRB jets. \nA significant polarization detection requires high photon\nstatistics and lack of this has actually prevented the CZTI instrument onboard \n\\emph{Astrosat} in constraining the polarization measurement for GRB 151006A. \nThe current measurement hints a moderately high polarization, however, with low \nsignificance, and thereby is consistent with nearly all model predictions.\nHowever, the capability demonstrated by CZTI, offers a promising era of \npolarization detections above $100 \\, \\rm keV$ and also its time dependent study, in case of brighter GRBs. \n\n\n\n\n\\section*{Acknowledgments} \nWe gratefully acknowledge the referee for comments on reshaping \nthe paper. This publication uses data from the \\emph{Astrosat} mission of the\nIndian Space Research Organisation (ISRO), archived at the\nIndian Space Science Data Centre (ISSDC). CZT-Imager is built\nby a consortium of Institutes across India including Tata Institute\nof Fundamental Research, Mumbai, Vikram Sarabhai Space\nCentre, Thiruvananthapuram, ISRO Satellite Centre, Bengaluru,\nInter University Centre for Astronomy and Astrophysics, Pune,\nPhysical Research Laboratory, Ahmedabad, Space Application\nCentre, Ahmedabad: contributions from the vast technical team\nfrom all these institutes are gratefully acknowledged.\nThis research has made use of data obtained through the\nHEASARC Online Service, provided by the NASA\/GSFC, in support of NASA High Energy\nAstrophysics Programs. RB is a stipendiary of START program of the Polish \nScience Foundation (2016) and supported by \nPolish National Science Centre grants \n2013\/08\/A\/ST9\/00795,\n2013\/10\/M\/ST9\/00729 and\n2015\/18\/A\/ST9\/00746.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzusb b/data_all_eng_slimpj/shuffled/split2/finalzusb new file mode 100644 index 0000000000000000000000000000000000000000..7988c573e21343513a6fbaa9bfd418939e9e42ee --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzusb @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction. The EFT Lagrangian}\nFlavour-changing neutral currents (FCNC) are absent at lowest order in the SM, and are significantly suppressed through the Glashow--Iliopoulos--Maiani \nmechanism~\\cite{Glashow:1970gm} at higher orders.\nVarious rare decays of $\\ensuremath{{K}}$, $\\ensuremath{{D}}$, and $\\ensuremath{{B}}$ mesons, as well as \nthe oscillations in $\\ensuremath{\\mathrm{K^0}}\\PAKz$, $\\ensuremath{{D^0}}\\PADz$, and $\\ensuremath{{{B}^0}}\\PABz$\nsystems, strongly constrain FCNC interactions involving the first two \ngenerations and the \\cPqb\\ quark~\\cite{Agashe:2014kda}.\nHowever, FCNC involving the top quark are significantly less constrained.\nIn the SM, the FCNC couplings of the top quark are predicted to be very \nsmall and not detectable at current experimental sensitivity.\nHowever, they can be significantly enhanced in various SM extensions. \n\nThe FCNC interactions of the top quark with the quarks from the first two generations can be encoded in an effective field theory through dimension six gauge-invariant\n operators as indicated using slightly different notations in~\\cite{Buchmuller:1985jz}~(see, eq.~3.61) and in~\\cite{Malkawi:1995dm,AguilarSaavedra:2008zc,Grzadkowski:2010es,Willenbrock:2014bja,AguilarSaavedra:2018nen}.\n The gauge-invariant effective Lagrangian for the tug FCNC interactions has\n the following form in the notations~\\cite{AguilarSaavedra:2018nen}:\n\\begin{eqnarray}\n\\label{effective-lgrn}\n{\\cal L_{\\rm EFT}} \\,= \\,\n\\frac{C^{13}_{uG}}{\\Lambda^2}\\,\n\\left( \\bar u_L \\bar d_L\\right)\\,\\sigma^{\\mu\\nu}\\,t^a\\,t_R\\, \\Phi^c\\, \nG^a_{\\mu\\nu}\\,+\\,\\rm h.c.\\nonumber\\\\\n \\,+\\, \n\\frac{C^{31}_{uG}}{\\Lambda^2}\\,\n\\left( \\bar t_L \\bar b_L\\right) \\,\\sigma^{\\mu\\nu}\\,t^a \\,u_R \\,\\Phi^c \\,\nG^a_{\\mu\\nu}\\,+\\,\\rm h.c. ,\n\\end{eqnarray}\nwhere $\\Lambda$ is the scale of new physics (${\\gtrsim} 1$ TeV), \n$\\left( \\bar u_L \\bar d_L\\right)$ and $\\left( \\bar t_L \\bar b_L\\right)$ are the left quark doublets, \n$C^{13}_{uG}$ and $C^{31}_{uG}$ are Wilson coefficients being complex in general, $t^a$ are \nthe generators of the SU(3) color gauge group, $G^{\\rm a}_{\\mu\\nu}$ is a\n gluon field strength tensor, and $\\Phi^c$ is the conjugated Higgs field doublet.\nThe effective Lagrangian for the tcg FCNC interactions has the same form as\n (\\ref{effective-lgrn}) with\nobvious replacement of the $u$ and $d$ quarks by $c$ and $s$ quarks, and\n $C^{13}_{uG}$ and $C^{31}_{uG}$ by $C^{23}_{cG}$ and $C^{32}_{cG}$ respectively.\n\nIn the unitary gauge the conjugated Higgs field doublet has a well known \nform \n$$ \\Phi^c = \\colvec{2}{\\frac{v+h}{\\sqrt{2}}}{0},$$\nwhere $h$ is the Higgs boson field and $v$ is the Higgs vacuum expectation value.\nIt is easy to see that in the unitary gauge the Lagrangian (\\ref{effective-lgrn}) \ngets the following form:\n\\begin{eqnarray}\n\\label{lgrn_in_unigauge}\n{\\cal L_{\\rm EFT}}= \\frac{v+h}{\\sqrt{2}}\\,\\frac{1}{\\Lambda^2}\\,\n\\big[\\,\n\\bar q \\,\\sigma^{\\mu\\nu}\\,t^a \\,\n\\big(\\,C^{i3}_{qG}\\,\\frac{1+\\gamma^5}{2}\\,+\\,(C^{3i}_{qG})^*\n \\,\\frac{1-\\gamma^5}{2}\\,\\big)\\, t \\nonumber \\\\\n\\,+\\, \\bar t\\, \\sigma^{\\mu\\nu}\\,t^a\n\\big(\\,(C^{i3}_{qG})^*\\,\\frac{1-\\gamma^5}{2}\\,+\\,C^{3i}_{qG} \n\\, \\frac{1+\\gamma^5}{2}\\,\\big)\\, q \\big]\\, \nG^a_{\\mu\\nu}, \n\\end{eqnarray}\nwhere q refers to either $u$ ($i = 1$) or $c$ ($i = 2$) quarks.\n\nIn case of $C^{i3}_{uG}=(C^{3i}_{uG})^*$ \nthe Lagrangian (\\ref{lgrn_in_unigauge}) becomes:\n\\begin{eqnarray}\n{\\cal L_{\\rm EFT}}= \\frac{v+h}{\\sqrt{2}}\\,\\,\\frac{1}{\\Lambda^2}\\,\n\\big(\\,C^{i3}_{qG}\\,\\bar q \\,\\sigma^{\\mu\\nu}\\,t^a\\, t \n \\,+ \\,\n(C^{i3}_{qG})^*\\,\\bar t\\, \\sigma^{\\mu\\nu}\\,t^a\\, q\\,\\big)\\, G^a_{\\mu\\nu}.\n\\label{Lagrangian}\n\\end{eqnarray}\n One should note that \nthe Lagrangian (\\ref{Lagrangian})\ncontains not only the FCNC vertex of the top quark interaction with the u quark \nand gluon (tqg) but also the four-point FCNC vertex involving two \ngluons (tqgg). The latter is needed to preserve gauge invariance. The\nLagrangian (\\ref{Lagrangian}) also includes the FCNC top quark interactions involving \nthe Higgs boson\n(tqgh) and (tqggh)\\footnote[1]{The Lagrangian describing the tree point \n(tqg) and the four-point (tqgg) interaction vertices (tqg) as well as \ncorresponding Feynman rules were presented in~\\cite{Malkawi:1995dm}. The \nLagrangian describing not only the tree point vertex (tqg), but also the \ninteraction vertex with the Higgs boson (tqgh) as follows from the \ndimension 6 operators was worked out in~\\cite{AguilarSaavedra:2008zc}.}.\nbut also the four-point FCNC vertex involving two gluons (tqgg). The latter is needed to preserve gauge invariance.\n The Lagrangian (\\ref{Lagrangian}) also includes the FCNC top quark interactions involving the Higgs boson (tqgh) and (tqggh). But corresponding processes are \nobviously out of reach at the LHC and will be not considering. \nThe experimental limits~\\cite{Aad:2015gea,Khachatryan:2016sib,Aaltonen:2008qr,Abazov:2010qk} are given in terms of \\ensuremath{\\lvert\\kappa_{\\rm tug}\\rvert\/\\Lambda}\\xspace and \\ensuremath{\\lvert\\kappa_{\\rm tcg}\\rvert\/\\Lambda}\\xspace couplings~\\cite{Beneke:2000hk} which can be rewritten in the form of $C^{i3}_{qG}$ coefficients as follows:\n\\begin{eqnarray}\n\\label{couplings}\n\\lvert\\kappa_{\\rm tqg}\\rvert\/\\Lambda = \\frac{1}{g_s}\\frac{v}{\\sqrt{2}}\\frac{C^{i3}_{qG}}{\\Lambda^2},\n\\end{eqnarray}\nwhere $g_s$ is the coupling constant of the strong interaction. \n\n\\section{Numerical estimation of the FCNC tqg contributions }\nThe representative set of Feynman diagrams for the top quark production in FCNC interactions described by Lagrangian~(\\ref{Lagrangian}) are shown in Fig.~\\ref{diag}. In this paper we consider $pp\\to tj$ process, where j means a jet originated from either a light flavor quark or a gluon. Based on the Lagrangian~(\\ref{Lagrangian}) necessary Feynman rules were implemented to the CompHEP~\\cite{Pukhov:1999gg,Boos:2004kh} and all numerical calculations are performed by means of this package\\footnote[2]{Note that the four-point vertex of the gluon--gluon--top~ quark--$u$~quark interaction from (\\ref{Lagrangian}) was implemented in CompHEP using an auxiliary color octet field with spin two $t_{\\mu\\nu}^a$, denoted as $G.t$ in CompHEP, with the propagator defined by the Lagrangian $-\\frac{1}{2}\\,t_{\\mu\\nu}^a \\, t^{\\mu\\nu}_a$. It should be noted that the four-point vertex is necessary for gauge invariance in calculating the contributions with the initial states $gg$ and $gu$.}. Calculations are performed in Feynman gauge.\n\nThe total cross section at 14 TeV collision energy is about 120 pb for some particular value of the \nparameter \\ensuremath{\\lvert\\kappa_{\\rm tug}\\rvert\/\\Lambda}\\xspace=0.03 TeV$^{-1}$ and requirements of transverse momenta and pseudorapidity of the final light flavor quark or gluon to be $P_T^{q,g}>20$ GeV, $|\\eta^{q,g}|<4$. The cross section depends quadratically on the \\ensuremath{\\lvert\\kappa_{\\rm tug}\\rvert\/\\Lambda}\\xspace and can be recalculated for all other values of the coupling. \nThe partial contribution of the processes with different initial states are \n$gg$ (2\\%) (Fig.~\\ref{diag}a),\n$qg$ (89\\%) (Fig.~\\ref{diag}b),\n$q\\bar q$ (0.03\\%) (Fig.~\\ref{diag}c),\n$uq^\\prime$ (7.8\\%) (Fig.~\\ref{diag}d) and\n$u\\bar u$ (1.2\\%) (Fig.~\\ref{diag}e)\nfor the LHC collider energy $\\sqrt{s}=14$ TeV. \nThe destructive contributions of the last diagrams with four-point vertex in~Figs.~\\ref{diag}a,~\\ref{diag}b decrease the cross section of $gg$ part by 60\\% and $qg$ part by 4\\%, while practically not affecting kinematic properties. \n\nIn order to analyze properties related to unitarity we need to check the behavior of FCNC contribution, introduced in EFT approach, with increasing the $\\hat s$. The most transparent way is to exclude convolution with Proton Density Functions (PDF) and check the dependence of the total cross section of the parton level processes on the $\\sqrt{\\hat s}$. The Fig.~\\ref{gg-shat} demonstrates dependence of the total cross section of the parton level process with $gg$ initial states (Fig.~\\ref{diag}a) on the $\\sqrt{\\hat s}$, without integration over PDF. The same dependencies are shown in Fig.~\\ref{gu-shat} for the $gu$ initial state (Fig.~\\ref{diag}b) and in Fig.~\\ref{uu-shat} for the $uu$ initial state (Fig.~\\ref{diag}d). The required cutoff was applied as $|\\cos (p_i,p_f)|<0.95$, where $p_i$ and $p_f$ are momenta of the initial and final partons. The wide region of possible $\\sqrt{\\hat s}$ was tested, started from the threshold about 173 GeV and up to the 100 TeV. For the better clarity only the region with visible changes are shown in Figs.~\\ref{gg-shat}--\\ref{uu-shat} and the cross section becomes constant with increasing the collision energy and without convolution with PDF. The overall behavior demonstrates the absence of the increasing of the cross section with the energy up to the 100 TeV of $\\sqrt{\\hat s}$. This observation is confirmed by the direct symbolic computation of asymptotic behavior of the cross section at large $\\sqrt{\\hat s}$ which is demonstrated in the next section.\n\n\\section{Unitary limit }\n\nEffective operators lead to growing contributions with energies that violate unitarity. In order for our calculations to be self-consistent, we have to check that we do not consider kinematic regions where perturbative unitarity is violated. To estimate the allowed region of parameters, we apply optical theorem, which follows from the unitarity of the S-matrix. The optical theorem says that the imaginary part of the forward scattering amplitude is proportional to the total cross section of the process.\n\\begin{align}\\label{sigma1}\n\\sigma = \\frac{1}{s}{\\rm Im}\\left(A(\\theta=0)\\right)=\\frac{16\\pi}{s}\\sum\\limits^{\\infty}_{l=0}(2l+1)|a_l|^2,\n\\end{align}\nwhere $a_l$ is the partial-wave amplitude. Hence, Im$a_l=|a_l|^2$, which means that: \n\\begin{align}\\label{unitarity}\n|a_0| < \\frac{1}{2}.\n\\end{align}\nThe complete expression for the amplitude of the process $uu\\to ut$ has the form:\n\n\\begin{align}\\label{amplitude1}\nA = 2g_s^2\\frac{\\lvert\\kappa_{\\rm tqg}\\rvert}{\\Lambda}\\cdot\\sum\\limits_{spin}&~\\big[~\\bar{u}_u(p_3)(-i t^a\\gamma^{\\mu})u_u(p_1)\\left(\\frac{-i g_{\\mu\\nu}}{(p_4-p_2)^2}\\right)\\bar{u}_t(p_4)\\left(t^a\\sigma^{\\nu k}(p_4-p_2)_k\\right) u_u(p_2)\\\\\n\\nonumber &\n+\\bar{u}_t(p_4)\\left(t^a\\sigma^{\\mu k}(p_4-p_1)_k\\right) u_u(p_1)\\left(\\frac{-i g_{\\mu\\nu}}{(p_4-p_1)^2}\\right)\\bar{u}_u(p_3)(-i t^a\\gamma^{\\nu})u_u(p_2)~\\big].\n\\end{align}\n\nAfter the convolution of Lorentz indices and summation over all spin states, as well as after the expression of scalar products via Mandelstam variables, the amplitude takes the form:\n\\begin{align}\\label{amplitude2}\nA = 8\\cdot g_s^2\\cdot \\frac{\\lvert\\kappa_{\\rm tqg}\\rvert}{\\Lambda}\\cdot\\sqrt{s\\cdot t\\cdot u}\\left(\\frac{1}{t} + \\frac{1}{u}\\right).\n\\end{align}\nHere we assume that $\\sqrt{s}\\gg M_{\\rm top}$.\nUsing the substitution $(u = -s-t)$ and integrating over the variable $t$, we obtain the partial-wave amplitude $a_0$:\n\\begin{align}\\label{a0}\n|a_0| = \\frac{1}{16\\pi s}\\left| \\int\\limits^{0}_{-s}dt\\cdot A \\right | = 2\\pi\\alpha_s\\cdot \\frac{\\lvert\\kappa_{\\rm tqg}\\rvert}{\\Lambda}\\cdot\\sqrt{s}.\n\\end{align}\nThe unitarity condition, following from the optical theorem ($|a_0|<\\frac{1}{2}$)\nand recent experimental limits ($\\ensuremath{\\lvert\\kappa_{\\rm tug}\\rvert\/\\Lambda}\\xspace\\leqslant 0.004\\ \\rm TeV^{-1}$~\\cite{Khachatryan:2016sib}, $\\alpha_s\\approx 0.1$~\\cite{Agashe:2014kda}) on FCNC coupling parameter leads to the following estimation:\n\\begin{align}\\label{estimation}\n\\sqrt{s} < \\frac{1}{4\\pi\\alpha_s}\\cdot\\frac{\\Lambda}{\\lvert\\kappa_{\\rm tqg}\\rvert} \\approx 200~\\rm TeV.\n\\end{align}\nThus our estimation shows that the current limit value of FCNC coupling does not violate the unitary behavior of the amplitudes of the studied processes at energies available at the present and future colliders.\n\nPerforming the calculation of the matrix element square, averaging over the initial spin and color \nstates, and integrating over the $\\cos\\theta$ in the range from \n($-1+\\epsilon$) to ($1-\\epsilon$), we obtain the following expression \nfor the cross section of the process $uu\\to ut$\nin the limit of $\\sqrt{s}\\gg M_{\\rm top}$ as a function of the energy \n$\\sqrt{s}$ and the cutoff parameter $\\epsilon$:\n\\begin{align}\\label{estimation2}\n\\sigma(\\epsilon) = \\frac{64}{9}\\pi\\alpha_s^2\\cdot\\frac{\\kappa^2_{\\rm \ntqg}}{\\Lambda^2}\\cdot\\left(\\ln\\left|\\frac{2}{\\epsilon}-1\\right|-\\frac{11}{12}(1-\\epsilon)\\right).\n\\end{align}\nFormula (\\ref{estimation2}) explicitly demonstrates\nthe constant asymptotic behavior at large $\\sqrt{s}$ in case of the \nscattering angular cut is applied.\n\nIf one integrates the matrix element square over the transverse momentum \nof the t-quark in the range from the cut parameter $\\delta p_T$ to the \nkinematic limit $\\sqrt{s}\/2$ one gets for the cross section at \n$\\sqrt{s}\\gg M_{top}$:\n\\begin{align}\\label{estimation3}\n\\sigma(\\delta p_T,s) = \\frac{64}{9}\\pi\\alpha_s^2\\cdot\\frac{\\kappa^2_{\\rm \ntqg}}{\\Lambda^2}\\cdot\\left(2\\cdot\\ln\\left|\\frac{\\sqrt{s}}{2\\cdot\\delta \np_T}\\left(1+\\beta\\right)\\right|-\\frac{11}{12}\\beta\\right),\n\\end{align}\nwhere: $\\beta=\\sqrt{1-\\frac{4\\cdot\\delta p_T^2}{s}}$.\n\nThe formula (\\ref{estimation3}) shows the logarithmic dependence of the\ncross section at high energies\n$\\ln\\left|\\frac{\\sqrt{s}}{\\delta p_T}\\right|$\nin case of a cut on $p_T$.\n\nThis behavior of the scattering cross section is confirmed by numerical simulations performed in the previous section and is in a good agreement with the Froissart bound~\\cite{Froissart:1961ux} (the cross section does not increase faster than the square of the logarithm of energy), which confirms the correctness of the investigated approach.\n\n\\section{Conclusion }\nIn the paper we investigate limitation of EFT approach to describe possible FCNC processes. Numerical and analytic calculations demonstrate constant asymptotic behavior of the total cross section with the increasing of the energy. It is shown in the paper, that the EFT approach for the introducing of the possible FCNC contribution does not violate the pertubative unitary limit and Froissart bound, and can be used to setup the corresponding experimental limits on the FCNC couplings or Wilson coefficients at the present~\\cite{Aad:2015gea,Khachatryan:2016sib} and future~\\cite{CMS:2018kwi,CYRM950,Dudko:2643278,Abada:2019lih} colliders.\n\n\\section*{Acknowledgements}\n\\label{sec:acknownlegements}\n\nThe work was supported by grant no.~16-12-10280 of the Russian Science Foundation.\n\n\\section*{References}\n\\nocite{*}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n\\label{sec:introduction}\n\nA remarkable property of our universe is that is seems to be accelerating during the present epoch \\cite{acceleration}. The precise cause of cosmic acceleration is presently unknown but might rest\nin the presence of the cosmological constant or some other form of\n`Dark Energy' (DE) capable of violating the strong energy condition.\nAlternatively, cosmic acceleration might arise due to modifications to the\ngravitational sector of the theory, such as $f(R)$ gravity, Braneworld models,\netc. However, as the simple example of a non-minimally coupled\nscalar field shows, these two possibilities are only extreme particular cases of a more general notion of DE both including new non-gravitational physical fields and modifying gravity, too. Many of these possibilities have been extensively examined in recent years\n\\cite{DE_review} but firm conclusions as to the nature of dark energy\nare still to be drawn.\nOne reason for this is that current observational data sets, despite steady\nimprovement, are still hampered by uncertainties both of a statistical\nas well as systematic nature \\cite{systematics}.\n\nIndeed, in order for firm and robust conclusions to be drawn about the nature of\nDE one will need to (a) minimize statistical uncertainties by increasing the\ndepth and quality of observational data sets,\n(b) understand (and model better) the nature of systematics in the different kinds\nof data sets used to explore cosmic acceleration.\n\nThe main evidence for cosmic acceleration currently comes from two types of\ndata sets: \n\n(i) Those probing the luminosity distance, $d_L$, by observing the flux, $\\cal F$, of type Ia supernovae of given luminosity $L$ through\n\\begin{equation}\n{\\cal F} = \\frac{L}{4\\pi d_L^2}~,\n\\end{equation}\nwhere\n\\begin{equation}\n\\frac{H_0 d_L}{1+z} = \\int\\frac{dz}{h(z)}~, \\hspace{10 mm} h(z)=\\frac{H(z)}{H_0},\n\\label{eq:lum_dis}\n\\end{equation}\n\nin a spatially flat universe where $H(t)=\\frac{\\dot a}{a}$ and $a(t)$ is the FRW scale factor.\n\n(ii) Those based on the angular diameter distance, $d_A$ to a source of spatial size $d$ via the relation\n\\begin{equation}\n\\Delta \\theta = \\frac{d}{d_A}~.\n\\end{equation}\n\nRemarkably, for a wide range of cosmological models, the two distances are\nrelated through the {\\em cosmic duality relation} (CDR)\n\\begin{equation}\nd_L = (1+z)^2 d_A~.\n\\label{eq:duality}\n\\end{equation}\n\n\nIn recent years numerous studies have devoted themselves to establishing whether or not the\nCDR holds in practice \\cite{duality}. The reason for this largely stems from the hope that\na violation of the equality in (\\ref{eq:duality}) might signal the presence of new physics.\nSuch a violation may occur, for instance, through photon number non-conservation\neither through axion-photon mixing \\cite{csaki}, or because of photon absorption enroute to\nthe observer \\cite{ChenKantowski09a}, or due to an incorrect modelling of the\n ultra-narrow beams from point sources such as type Ia supernovae \\cite{Clarkson2012}.\n\nHowever, since (\\ref{eq:duality}) follows simply: (i) from the requirement that\nsources and observers be connected via null geodesics in a Riemannian\nspace-time, (ii) the phase-space conservation of photons;\ntherefore the CDR remains valid for a very wide class of spatially homogeneous\n(and even inhomogeneous) cosmologies \\cite{eth33,bk04}.\nTherefore it could well be that the cosmic duality relation is an {\\em exact principle\nin nature}. If this is indeed the case, then a violation of the CDR would no longer imply\nnew physics, but would signal instead to the presence of hitherto undetected systematics in\ndata sets relating either to $d_L$ or to $d_A$ (or to both).\nSince the actual cosmological model of the present Universe is not known and there exist many models of \ndark energy which provide an alternative to an exact cosmological constant, it is interesting to investigate \nif one can compare different observational data and look for systematics in them without making any theoretical \nassumptions regarding the cosmological model.\n\nThe purpose of the present paper is to show how the presence of a systematic increase in the value of the cosmological distance modulus (as an observable)\n\\begin{equation}\n\\mu \\equiv m-M = 5\\log_{10}{d_L} + 25\n\\label{eq:mu}\n\\end{equation}\n\nshows itself as an apparent violation of the cosmic duality relation.\n In this paper we use the idea of the Bayesian interpretation of Crossing Statistic \\cite{Crossing,Crossing_B,Crossing_C} along with the smoothing method \\cite{Shafieloo06,Shafieloo07,Shafieloo10} to compare cosmic distances from different data sets in a purely model independent manner.\nWe shall show that even a small change in $\\mu$ can be detected\nusing the CDR and cosmological data sets which shall soon become available.\nUsing this approach we are able in fact to disclose differences between the two data sets without a need to make any assumptions about the nature of dark energy or to set any priors on cosmological parameters.\n\n\nIn the following, we first discuss the method that we use to compare different data sets. \nThe smoothing method and the Bayesian interpretation of Crossing statistic will be discussed briefly and we\nshall explain how we combine these methods to look for consistency between different data sets. This idea has been discussed briefly in \\cite{Crossing_C} and in this work we explore the application of this method in greater detail\n and in a different context. In this approach we use the smoothing method together with the\n Crossing Statistic to reconstruct the cosmic distances from the two different datasets and set confidence limits on the reconstructed Crossing hyperparameters. In the absence of\n systematics in either of the data sets, the confidence limits of the Crossing hyperparameters should \nhave a reasonable overlap. On the other hand, if the Crossing hyperparameters do not overlap nicely, \nthen this would imply an inconsistency between data sets which could\n be interpreted as the existence of some sort of systematics. We shall\n apply our method on future simulated data where in one case we assume that there are no systematics in the data and in\nthe other we assume that there exists some form of systematics. Finally, we present our results\n and show how our method can discern the presence of systematics without any prior assumption on the \ncosmological model. \nWe should note that the method we use in this paper can be generally used to compare different data sets in a model independent manner and this work is an application of the method to compare angular diameter distance data with luminosity distance data assuming the cosmic duality relation. \n\n\n\\section{Method and Analysis} \n\\label{meth}\n\n\\subsection{Smoothing Method}\n\n\nThe smoothing method is a completely model independent approach to derive the $d_L(z)$ relation directly from the data, without any assumptions other than the introduction of a smoothing scale. The only parameter used in the smoothing method is the smoothing width $\\Delta$, which is constrained only by the quality and quantity of the data, and has nothing to do with any cosmological model. The smoothing method is an iterative procedure with each iteration typically giving a better fit to the data. It has been shown in \\cite{Shafieloo06,Shafieloo07,Shafieloo10} that the final reconstructed results are independent of the assumed initial guess, $d_L(z_i)^g$. \n\nThe modified smoothing method (error-sensitive) can be summarised by the following equation~\\cite{Shafieloo10}:\\\\\n\n\\begin{eqnarray}\n\\label{eq:bg}\n&&\\ln d_L(z,\\Delta)^{\\rm s}=\\ln\n\\ d_L(z)^g \\nonumber\\\\\n&& +N(z) \\sum_i \\frac{\\left [ \\ln d_L(z_i)- \\ln\n\\ d_L(z_i)^g \\right]}{\\sigma^2_{d_L(z_i)}} \n\\ {\\rm exp} \\left [- \\frac{\\ln^2 \\left\n( \\frac{1+z_i}{1+z} \\right ) }{2 \\Delta^2} \\right ], \\nonumber \\\\\n&&N(z)^{-1}=\\sum_i {\\rm exp} \\left\n[- \\frac{\\ln^2 \\left ( \\frac{1+z_i}{1+z} \\right ) }{2 \\Delta^2} \\right ] \\frac{1}{\\sigma^2_{d_L(z_i)}} ~,\n\\end{eqnarray}\n\n\\noindent where $d_L(z)$ is the data, $N(z)$ is the normalization factor, $d_L(z_i)^g$ is the initial guess model and $\\Delta$ is the width of smoothing. \n\nThe absolute brightness of type Ia supernovae is degenerate with $H_0$ since the observed quantity is the distance modulus $\\mu(z)$. The outcome of the smoothing method is therefore $H_0d_L(z)\/c\\equiv d_L^{rec}(z)=(1+z)D(z)$. In this paper we set $\\Delta=0.30$ which\n is similar to the value used in~\\cite{Shafieloo10}. Complete explanation of the relations between $\\Delta$, the number of data points, quality of the data and results of the reconstruction exercise can be found in~\\cite{Shafieloo06,Shafieloo07}. It has earlier been shown that the smoothing method is a promising approach to reconstruct the expansion history of the universe. However, setting confidence limits has been an issue and in earlier work\nthe bootstrap approach was used to set the confidence limits. In this paper, the reconstructed form of \n$d_L(z)$ will be used as a mean function in the full reconstruction process~\\cite{Crossing_C}\nwhich includes the idea of Bayesian interpretation of Crossing Statistic as explained in the next section.\n \n\\subsection{Reconstructing the Expansion History of the Universe using the Crossing Statistic }\n\nThe main idea behind the crossing statistic lies in the fact that the actual model of the universe and the \nreconstruction using smoothing would have one or two mild crossings: \nthe distance modulus $\\mu(z)_{\\rm fiducial}$ of the fiducial cosmological model and the reconstructed \n$\\mu(z)_{Smooth}$ \nwould cross each other at one or two points in the redshift range defined by the data \\cite{Crossing}. \n\nFurthermore, in a FRW universe the distance modulus monotonically increases with redshift. Consequently\nany two cosmological models become virtually indistinguishable if the distance modulus of one of them \nis multiplied by a suitable function of the redshift. The coefficients of this function (multiplying $\\mu$) constitute\nthe Crossing hyperparameters and the functions themselves will be called Crossing functions following \n\\cite{Crossing_B}.\nIn our case the Crossing functions multiply the smoothed distance modulus reconstructed from the data\nusing the method described in the previous section. We shall refer to the smoothed functions as the\n{\\em mean} functions since they accurately describe the mean value of the function that is being smoothed,\nwhich in this particular case is $\\mu$. From the preceding argument, and that in \\cite{Crossing_B}, it is clear that\nthe crossing functions, multiplied by the mean functions,\nshould virtually coincide with the actual model of the universe. \\\\\n\nThe reconstruction of the expansion history of the universe using Bayesian interpretation of Crossing statistic~\\footnote{The Bayesian interpretation of Crossing Statistic is hidden in two prior assumptions that 1) Cosmic distances increases by redshift monotonically for all cosmological models hence there are no high frequency fluctuations in $\\mu(z)$ and 2) since the distance modules of all cosmological models increases by redshift, $\\mu(z)$ of any two cosmological models can become so close to each other at all redshifts up to an indistinguishable level if we multiply $\\mu(z)$ of one of them to a suitable smooth mean function of degree $n$ with some particular values for the coefficients~\\cite{Crossing_B}.} is therefore a combination of a non-parametric (smoothing) method with a parametric approach (using a Crossing function) to define and set the confidence limits on the cosmic expansion history. The crossing function is defined by \nChebyshev polynomials~\\cite{Crossing_B,Crossing_C}:\n\n\\begin{equation}\nT_{II}(C_1,C_2,z)=1+C_1(\\frac{z}{z_{max}})+C_2[2(\\frac{z}{z_{max}})^2-1],\n\\label{eq:Cheb}\n\\end{equation}\n\nand we fit \n\n\\begin{equation}\n\\mu_{Smooth}^{T_{II}}(z)=\\mu_{Smooth}(z) \\times T_{II}(C_1,C_2,z)\n\\label{eq:main}\n\\end{equation}\n\n\\noindent to the data and find the best fit point $C^{best}_1,C^{best}_2$ in the hyperparameter space as well as the $C_1,C_2$ parameters related to the $1\\sigma$, $2\\sigma$ and $3\\sigma$ confident limits. Each \n$T_{II}(C_1,C_2,z)$ (where $C_1,C_2$ pairs are within hyperparameter confidence contours) multiplied by\n $\\mu_{Smooth}(z)$ represents a reconstruction of the expansion history of the universe consistent with the data.\n\nTheoretically one can use higher orders of Chebichev polynomial as well but this will result to more degrees of freedom and larger confidence limits which would eventually limit us from distinguishing between cases. It has been shown in~\\cite{Crossing_B} that for the current and near future data, using up to second order of Chebishev polinomyals would be sufficient for the purpose of reconstruction. It may also look like that limiting the analysis up to using only second order of Chebichev polynomials might not make our method sensitive to the possible systematics of higher order but this is in fact true only if we use very smooth mean functions (usually suggeested by parameteric forms). In this work we use the well developed smoothing method to derive the mean function and smoothing method basically recover all detectable features of the data. The Crossing hyperparameters only generates smooth variations around this mean function. So if either of the datasets suffer from a kind of local systematic, lets say data is shifted up or down in a short range of redshift like a bump, this bump would be detected by the smoothing method and would be present in the mean function. In this paper we simulated the data only based on one type of systematics to show how the method works but for different kinds of systematics that affect the data in a similar way (changing the cosmic distances systematically) the method would be applicable. \n\nIt has been shown that this method works very well in reconstructing the expansion history of the universe and \nin determining cosmological quantities such as the\n Hubble parameter $h(z)$, Om diagnostic $Om(z)$ and the deceleration parameter $q(z)$ \\cite{Crossing_C},\n since by reconstructing $d_L(z)$ one can derive $h(z)$, $Om(z)$ and $q(z)$ using: \n\\begin{eqnarray}\nH(z) =\\left[\\frac{d}{dz}\\left(\\frac{d_L(z)}{1+z}\\right)\\right]^{-1}\\\\\nq(z)=(1+z)\\frac{H'(z)}{H(z)}-1\\\\\nOm(z)=\\frac{h^2(z)-1}{(1+z)^3-1}\n\\label{eq:h}\n\\end{eqnarray}\n\n\\noindent where derivatives are respect to the redshift and $h(z)=H(z)\/H_0$. Its important\n to note that in applying this method to derive the cosmological quantities in\n(\\ref{eq:h}) one does not require a prior knowledge \nof the matter density. In fact $Om(z)$ can be used to falsify the $\\Lambda$CDM model\nwithout any prior knowledge of $\\Omega_{0m}$\n since $Om(z)$ is a constant at all redshifts in $\\Lambda$CDM ~\\cite{sahni08,zunckel08}. \n\n\\subsection{Comparing Data Sets}\n\nWe shall apply our method to compare the consistency of the luminosity distance (obtained\nfrom type Ia supernovae) and the angular size distance (obtained\nfrom X-ray and SZ clusters).\nTo compare these two data sets one need not reconstruct $h(z)$, $Om(z)$, etc.,\n instead one can simply compare the confidence limits of the Crossing hyperparameters, $C_1$ and $C_2$,\n derived from these different data sets. In order to perform this comparison, we first apply the smoothing method on supernovae data \nto derive $\\mu_{Smooth}(z)$ and then use this $\\mu_{Smooth}(z)$ \n{\\em to fit both luminosity distance data from supernovae as well as\n angular diameter distance data from X-ray and SZ clusters.}\nThis is done using Chebyshev Crossing functions in (\\ref{eq:Cheb})\n so that one derives the Crossing hyperparameters in each case. \nIn other words, after deriving $\\mu_{Smooth}(z)$ by smoothing supernovae data,\n we fit supernovae data using (\\ref{eq:main}) and place constraints on $C_1^{SN}$ and $C_2^{SN}$. \nSince the mean function is derived from supernovae data itself we expect the confidence limits of \n$C_1^{SN}$ and $C_2^{SN}$ to be centred around the $(0,0)$ point in the hyperparameter space. \n\n\n\nNext we again use (\\ref{eq:main}) (with the same $\\mu_{Smooth}(z)$ from supernovae data) but replace supernovae data by cluster data to determine $C_1^{Cluster}$ and $C_2^{Cluster}$. If the confidence contour of\n$C_1^{Cluster}$ and $C_2^{Cluster}$ has a significant overlap with the confidence contour of \n$C_1^{SN}$ and $C_2^{SN}$, then we can conclude that the two data sets (SNIa \\& Cluster) are in concordance \nwith each other. Otherwise we are observing an inconsistency most probably due to the \npresence of systematics in either of the data sets. The reader should note that in fitting both supernovae and cluster data, and to determine crossing hyperparameters, we use the same smooth mean function, $\\mu_{Smooth}(z)$ (Above it was derived from supernovae data).\n In fact the crossing hyperparameters in each case represent the deviation from the mean function \nsuggested by the data. If two data sets imply different deviations from a given mean function, $\\mu_{Smooth}(z)$, then this might suggest an inconsistency between the different data sets. \n\nWe apply this method to a simulated future SNIa data set assumed to have about 2300 supernovae \nwith $0.015 2\\sigma$ CL)\nand marginally\nso in the right panel (at $1\\sigma$ CL) . \nThe systematics is reflected in a redshift dependent increase in the distance modulus (1.5) and is\ndescribed by (2.7) with $\\alpha=0.01$ (left panel) and $\\alpha=0.002$ (right panel).}\n\\label{fig:sys}\n\\end{figure*}\n\n\n\\begin{figure*}[!t]\n\\centering\n\\begin{center}\n\\vspace{.in}\n\\centerline{\\mbox{\\hspace{0.in} \\hspace{2.1in} \\hspace{2.1in} }}\n$\\begin{array}{@{\\hspace{-0.3in}}c@{\\hspace{0.3in}}c@{\\hspace{0.3in}}c}\n\\multicolumn{1}{l}{\\mbox{}} &\n\\multicolumn{1}{l}{\\mbox{}} \\\\ [-2.8cm]\n\\hspace{-0.5in}\n\\includegraphics[scale=0.45, angle=0]{SN_CL_NoSysData.pdf}\n\\hspace{-1.5in}\n\\includegraphics[scale=0.45, angle=0]{SN_CL_p002SysData.pdf}\n\\hspace{-.1in}\n\\vspace{0.8in}\n\\end{array}$\n\\vspace{-1.3in}\n\\end{center}\n\\caption {\\small The distance modulus for the SNIa and cluster data is compared in these figures. \nThe distance modulus for cluster data has been obtained after applying the cosmic duality relation\n(1.4).\nThe left panel alludes to data with no systematics. The systematic errors in the right panel are\ndescribed by $\\alpha=0.002$ in (2.7).\nIt clearly seems difficult to distinguish between the two panels by eye, whereas the crossing statistic\nshown in\nthe right panel of the previous figure accomplishes this.}\n\\label{fig:data}\n\\end{figure*}\n\n\\begin{figure*}[!t]\n\\hspace{0.in}\n\\includegraphics[scale=0.50, angle=0]{C1C2_Sys_LCDM266_p002_H069_73.pdf}\n\\vspace{-0.5in}\n\\caption {\\small Confidence contours of the Crossing hyperparameters using simulated future supernovae data \n($1\\sigma$ CL in magenta, $2\\sigma$ CL in blue) and simulated future SZ angular diameter distance data \n($1\\sigma$ CL in green, $2\\sigma$ CL in red) assuming \n the presence of small systematic errors in supernovae data, namely $\\alpha=0.002$ in (2.7).\nWe assume no precise knowledge of $H_0$ choosing instead a flat prior on \n$0.68 1$ written in lowest terms and define $\\Lambda = \\Lambda_f$ as in\n\\eqref{eq:Lambda}. Then\n\\begin{enumerate}[(i)]\n\\item\nFor $\\tau \\in \\operatorname{Aut}({\\mathbb{B}}_N)$, the $\\Lambda$-function for $f$ and $\\tau \\circ f$ are\nidentical,\n$\\Lambda_f = \\Lambda_{\\tau \\circ f}$.\n\\item\nIf $\\psi \\in \\operatorname{Aut}({\\mathbb{B}}_n)$, then\n$\\Lambda_f \\circ \\psi = C \\Lambda_{f \\circ \\psi}$ for\na constant~$C$.\n\\item\n$\\Lambda$ is a strictly plurisubharmonic exhaustion function for ${\\mathbb{B}}_n$, that is,\n$\\Lambda$ is strictly plurisubharmonic and $\\Lambda(z)$ goes to $+\\infty$ as $z \\to \\partial {\\mathbb{B}}_n$.\n\\item\n$\\Lambda$ has a unique critical point (a minimum) in ${\\mathbb{B}}_n$.\n\\end{enumerate}\n\\end{samepage}\n\\end{thm}\n\nThe fourth item is the key to proving Theorem~\\ref{thm:normdenom}. \nThe normal form is attained by\nan automorphism $\\psi$ that takes the unique critical point of $\\Lambda$\nto the origin. Via the uniqueness, the map is normalized up to\nthe unitary group. The $\\sigma_k$s are \nfound by diagonalizing the quadratic part of the denominator via unitaries.\nThe main step in the normalization is solving $\\nabla \\Lambda = 0$,\nand hence the normalization is computable provided we can solve for\nroots of the polynomials involved.\n\nIf $0 < \\sigma_1 < \\sigma_2 < \\cdots < \\sigma_n$, then\nthe only possible $V$ such that $G_2 \\circ V = G_2$ is the diagonal\nmatrix with $1$s and $-1$s on the diagonal.\nThat is, any $z_j$ can be replaced by $-z_j$, and the\nset of such $V$ forms\na finite subgroup of $U(n)$ of size $2^n$.\nWe can also put $P$ into a normal form.\nFix an ordering on the $n$-variable multiindices $\\alpha$\nand write $P(z) = \\sum_{\\alpha} c_{\\alpha} z^\\alpha$, where $c_{\\alpha} \\in {\\mathbb{C}}^N$.\nChoose the unitary matrix $U$ so that the matrix $[c_\\alpha]$,\nwhose columns are $c_\\alpha$ ordered as given, is in row echelon form with\npositive pivots.\nIf $N$ is the embedding dimension for $f$, then $[c_\\alpha]$ has full\nrank and $U$ is unique.\nThat is, when the $\\sigma_k$s are distinct and nonzero,\nwe have a normal form up to a small finite subgroup of the unitary\ngroup.\n\nWhen $\\sigma_1 = \\cdots = \\sigma_n=0$ and $G \\equiv 1$, the map is\npolynomial and takes $0$ to $0$.\nD'Angelo proved \\cite{DAngelo:book} that if two polynomial maps that fix\nthe origin are equivalent, then they are equivalent up to unitaries.\nTherefore, Theorem~\\ref{thm:normdenom} extends D'Angelo's observation to rational maps.\nThere do exist degree 3 maps not spherically equivalent to a polynomial,\nfor example, a product of three Blaschke factors is in general not equivalent to\na polynomial proper map of discs, that is, to $z^3$.\nFor an example from ${\\mathbb{B}}_2$ to ${\\mathbb{B}}_4$ see\nalso Faran--Huang--Ji--Zhang~\\cite{FHJZ}.\nIn section~\\ref{section:poly}, we prove that if the \nembedding dimension of the map is maximal for its degree, then there exists\na linear fractional $\\tau$ such that $\\tau \\circ f$ is a polynomial\nproper map to the ball ${\\mathbb{B}}_N$, the generalized ball ${\\mathbb{B}}_{1,N-1}$, or\nthe Heisenberg realization of the ball ${\\mathbb{H}}_N$. So $f$ is equivalent\nto a polynomial if we take possibly a different representation of the\nball and do not assume that $f$ takes the origin to the origin.\n\nWhen considering the normal form,\nit is better to dispense with the target automorphism\ngroup via Lemma~\\ref{lemma:targetautform} and consider the normal form for\nthe polynomials $r(z,\\bar{z}) = \\sabs{g(z)}^2-\\snorm{f(z)}^2$. For example,\nwhen $0 < \\sigma_1 < \\cdots < \\sigma_n$, then if we put the map, and\ntherefore $r$, into the form from the theorem, then $r$ is normalized up to\nswapping the signs of the $z_k$s.\n\nWhen $d < 3$, then $G$ is of degree 2 or less, that is, $\\sigma_1=\\cdots=\\sigma_n=0$ and $G\\equiv 1$.\nIn other words, the theorem reproves the result that all degree 2\nproper rational maps of balls are spherically equivalent to a polynomial map\ntaking the origin to the origin.\n\nIf degree $d=3$, then the denominator is normalized to\n\\begin{equation}\nG(z) = 1 + \\sum_{k=1}^n \\sigma_k z_k^2.\n\\end{equation}\nIn this case the denominator is completely in normal form:\nIf $F=\\frac{P}{G}$ and $\\Phi=\\frac{\\Pi}{\\Gamma}$\nare spherically equivalent and in the form above, then $G=\\Gamma$\nand $\\Pi = U \\circ P \\circ V$, where $U$ and $V$ are unitary matrices\nand $G \\circ V = G$.\n\nAs an example, let us study the case $n=N=1$ and $d=3$. That is,\nwe find the normal form for proper rational maps\n$f \\colon {\\mathbb{B}}_1 \\to {\\mathbb{B}}_1$ of degree 3.\nA proper map of discs is a finite Blaschke product,\nin this case a product of 3 factors.\nAs we require $f(0) =0$, one of these factors is $z$. We also\nrequire the denominator to be $1+ \\sigma z^2$ ($\\sigma \\geq 0$ and $\\sigma < 1$\nso that the denominator does not vanish on the ball).\nThus the other two factors are the Blachke factors\ntaking zero to $\\pm i\\sqrt{\\sigma} \\in {\\mathbb{B}}_1$.\nSo the normal form is\n\\begin{equation} \\label{eq:deg3n1normform}\nf(z) =\nz \\left(\n\\frac{i\\sqrt{\\sigma}-z}{1-i\\sqrt{\\sigma}\\, z}\n\\right)\n\\left(\n\\frac{-i\\sqrt{\\sigma}-z}{1+i\\sqrt{\\sigma}\\, z}\n\\right)\n=\n\\frac{\\sigma z + z^3}{1+ \\sigma z^2} ,\n\\qquad\n0 \\leq \\sigma < 1 .\n\\end{equation}\nIf $\\sigma = 0$, the map is simply $z^3$.\nThe map is in normal form as we said above and switching the sign of $z$\n(the only unitary allowed on the source if $\\sigma > 0$)\ncan be counteracted by a unitary on the target disc.\nThus \\eqref{eq:deg3n1normform} is a complete normal form of third degree proper maps\nof the disc up to spherical equivalence. \nThe corresponding exhaustion function $\\Lambda$ is\n\\begin{equation}\n\\Lambda(z,\\bar{z})\n=\n\\frac{\n\\sabs{1+\\sigma z^2}^2-\\sabs{\\sigma z + z^3}^2\n}{\n(1-\\sabs{z}^2)^3\n} .\n\\end{equation}\nThis $\\Lambda$ has a unique critical point (a minimum) at the\norigin.\nMoreover, the only degree 3 polynomial proper map of ${\\mathbb{B}}_1$ to ${\\mathbb{B}}_1$\nis $e^{i\\theta} z^3$. Hence, the only map \\eqref{eq:deg3n1normform}\nspherically equivalent to a polynomial is the one where $\\sigma = 0$.\n\nAs we mentioned above, D'Angelo observed that\nany polynomial not zero on the closed ball\nis the denominator of a rational proper map written in lowest terms.\nBy tensoring with the identity\nwe ensure that the map takes the origin to the origin.\nThat is, for any\ndenominator in normal form $G(z) = 1 + \\sum_{k=1}^n \\sigma_k z_k^2 + E(z)$ that is not\nzero on the closed ball, a numerator exists giving a proper rational map of\nballs (in lowest terms) taking the origin to the origin, hence one in normal form.\nThe degree of the numerator depends on the $\\sigma_k$ and $E$.\nD'Angelo (see Chapter 2 section 14 of \\cite{DAngelo:spheresbook})\nhas worked out the following rather interesting example\\footnote{D'Angelo\nuses $1-\\lambda z_1 z_2$, which we put into our normal form.}:\nGiven $\\sigma \\geq 0$, a polynomial in normal form\n\\begin{equation}\nG(z) = 1 + \\sigma z_1^2 + \\sigma z_2^2\n\\end{equation}\nis nonzero on the closed\nunit ball whenever $\\sigma < 1$, and consequently \nit is the denominator of a proper rational map\n${\\mathbb{B}}_2$ to ${\\mathbb{B}}_N$. However the degree $d$ of this map necessarily goes to\ninfinity as $\\sigma$ approaches $1$. A third degree map exists if $\\sigma <\n\\frac{\\sqrt{3}}{2}$.\nA natural question given our normal form is, therefore, which possible denominators of\ndegree 2 in normal form,\n$G(z) = 1 + \\sum_{k=1}^n \\sigma_k z_k^2$,\nare denominators of third degree maps taking the origin to the origin.\nD'Angelo~\\cite{DAngelo:preprint}\nrecently made a systematic study of how the denominator is\ndetermined by the numerator, and in particular gives a method\nfor constructing the precise equations needed, working out\nthe degree 3 case in detail.\nFor our purposes, we content ourselves with proving in section~\\ref{section:exist},\nthat a degree 3 numerator taking the origin to the origin\ncorresponding to the denominator (in lowest terms)\nexists for all small enough $\\sigma_k$.\n\nFinally, we remark that it is common to switch back and forth\nbetween rational proper maps of balls ${\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ and rational maps\nof spheres $S^{2n-1} \\to S^{2N-1}$. When $n > 1$, the only difference\nbetween these two setups is the existence of a constant sphere map not\ncorresponding to a proper map of balls. If $n=1$, then there are\nmany sphere maps that are not proper maps of balls. For example,\n$\\frac{1}{z}$ takes the circle to the circle, but not the unit disc to the\nunit disc. The results of this paper hold for any $n$ as long as they are\nstated for proper maps of balls. They hold for rational proper maps of spheres\nif $n > 1$.\n\nThe author would like to thank John D'Angelo, Dusty Grundmeier,\nand Han Peters for many discussions on this and related topics, which led to \nthe present result.\n\n\n\\section{Preliminaries on proper maps of balls}\n\nWhen \n$\\frac{p}{g} \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ is a proper rational map written\nin lowest terms, we consider\nthe real polynomial $\\sabs{g(z)}^2-\\snorm{p(z)}^2$.\nWe will call this real polynomial the \\emph{underlying form} of\n$\\frac{p}{g}$.\nNote that normally\n$\\snorm{p(z)}^2-\\sabs{g(z)}^2$ is considered in the literature,\nbut its negative will be more convenient for us here.\nAs $f$ maps to the ball, the underlying form is positive on ${\\mathbb{B}}_n$,\nand as $f$ is proper it is zero on the sphere. That is,\n\\begin{equation}\n\\sabs{g(z)}^2- \\snorm{p(z)}^2 = 0 \\quad \\text{on} \\quad \\snorm{z}^2 = 1 ,\n\\end{equation}\nor in other words, there exists a polynomial $q(z,\\bar{z})$ such that\n\\begin{equation}\n\\sabs{g(z)}^2- \\snorm{p(z)}^2 = q(z,\\bar{z}) \\bigl( 1-\\snorm{z}^2 \\bigr) .\n\\end{equation}\n\nThe techniques in this paper are based on the following well-known idea.\nA real polynomial $r$ can be written as\n\\begin{equation}\nr(z,\\bar{z})=\n\\sum_{\\alpha,\\beta} c_{\\alpha,\\beta} z^\\alpha \\bar{z}^\\beta\n=\\snorm{h(z)}^2\n\\end{equation}\nfor a holomorphic polynomial\n$h \\colon {\\mathbb{C}}^n \\to {\\mathbb{C}}^N$ if and only if the matrix of coefficients\n$[c_{\\alpha,\\beta}]$ is\npositive semidefinite, and the\nrank of $[c_{\\alpha,\\beta}]$ is the\nleast $N$ necessary.\nMore generally, if $[c_{\\alpha,\\beta}]$ has $a$ positive and $b$ negative\neigenvalues, then $r$ can be written as a difference\n$\\snorm{h(z)}^2 - \\snorm{u(z)}^2$, where $h$ has $a$ components and $u$ has\n$b$ components. This fundamental decomposition follows from elementary linear algebra.\nWe write the matrix of coefficients as a sum of rank one matrices,\neach of which is $\\pm$ an outer product of a vector with itself.\nSee~\\cites{HornJohnson}.\n\nSuppose $\\frac{p}{g} \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ is a proper rational map, and\nsuppose that the components of $p$ together with $g$ are linearly independent, that is,\nthe image of $\\frac{p}{g}$ is not contained in a complex hyperplane.\nIf it were, a linear fractional automorphism of the ball can move this\nhyperplane to the hyperplane $\\{ z_N = 0 \\}$. To make a long story short,\nafter composing with an automorphism, we may remove any\nzero components and we can assume that $(p_1,\\ldots,p_N,g)$ are linearly\nindependent.\nThis minimal $N$ is called the \\emph{embedding dimension} for\nthe map.\nWe also remark that fixing $d$ we find that the maximal possible embedding\ndimension for a degree $d$ map is one less (to account for $g$) than\nthe dimension of the space of polynomials of degree $d$.\nIf $N$ is the\nembedding dimension of the rational proper map $\\frac{p}{g}$,\nthen the matrix of coefficients of\nthe underlying form\n$\\sabs{g(z)}^2 - \\snorm{p(z)}^2$ has\n$1$ positive and $N$ negative eigenvalues.\n\nThe underlying form \n$\\sabs{g(z)}^2 - \\snorm{p(z)}^2$ can be rescaled by a real positive constant\nwithout changing the map $\\frac{p}{g}$ as such rescaling can be done by\nrescaling $p$ and $g$ by that same constant.\nThe value $\\sabs{g(0)}^2-\\snorm{p(0)}^2$ is some positive quantity,\ntherefore, by such a rescaling we may assume\n$\\sabs{g(0)}^2-\\snorm{p(0)}^2 = 1$.\nOnce we normalize the polynomial in this way, we ask what does it mean\nif two proper maps have the same underlying form.\nIn \\cite{Lebl:normal}, the author proved that the underlying forms are equal\n(with the value at zero normalized as above) if and only if \nthe maps themselves differ by a target automorphism.\n\n\\begin{lemma} \\label{lemma:targetautform}\nSuppose\n$\\frac{p}{g} \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ \nand $\\frac{P}{G} \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ are proper rational maps\nwritten in lowest terms such that\n$\\sabs{g(0)}^2-\\snorm{p(0)}^2 = 1$ and\n$\\sabs{G(0)}^2-\\snorm{P(0)}^2 = 1$.\nThen there exists a $\\tau \\in \\operatorname{Aut}({\\mathbb{B}}_N)$ such that\n\\begin{equation}\n\\tau \\circ \\frac{p}{g} = \\frac{P}{G}\n\\qquad\n\\text{if and only if}\n\\qquad\n\\sabs{g(z)}^2-\\snorm{p(z)}^2 = \n\\sabs{G(z)}^2-\\snorm{P(z)}^2 .\n\\end{equation}\n\\end{lemma}\n\nThe lemma holds in more generality, the target can be an arbitrary\nhyperquadric, see \\cite{Lebl:normal} or \\cite{DX}.\nThe power of this lemma is that it reduces the problem of\nclassification of proper maps up to pre and post composition by automorphisms\nto classification of the underlying forms by precomposition with an\nautomorphism. We have eliminated the target automorphism group from the\nproblem.\n\nSince $\\operatorname{Aut}({\\mathbb{B}}_N)$ is transitive, for\nany proper ball map $\\frac{p}{g}$, there exists\na $\\tau \\in {\\mathbb{B}}_N$ such that \n$\\tau \\circ \\frac{p}{g} = \\frac{P}{G}$, where $\\frac{P}{G}$\nis a proper ball map with $P(0) = 0$, $G(0) = 1$.\nBy Lemma~\\ref{lemma:targetautform},\n$\\abs{g(z)}^2-\\snorm{p(z)}^2 = \n\\abs{G(z)}^2-\\snorm{P(z)}^2$ (as long as\n$\\abs{g(0)}^2-\\snorm{p(0)}^2 = 1$).\nInstead of looking for $\\tau$, we simply read off the $G$\nby looking at the coefficients of the\npolynomial $r(z,\\bar{z}) = \\sabs{g(z)}^2-\\snorm{p(z)}^2$.\n\n\\begin{lemma} \\label{lemma:Gfrommatrix}\nSuppose\n$\\frac{p}{g} \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ \nis a proper rational map of degree $d$\nwritten in lowest terms such that $p(0) = 0$ and $g(0)=1$.\nLet $r(z,\\bar{z}) = \\sabs{g(z)}^2-\\snorm{p(z)}^2$\nand\n$r(z,\\bar{z}) = q(z,\\bar{z}) \\bigl(1-\\snorm{z}^2\\bigr)$.\nThen $g$ is of degree $d-1$ or less, and\n$g(z) = r(z,0) = q(z,0)$.\n\\end{lemma}\n\nIn particular, the coefficients of the denominator $g$ for the $p$ that takes the\norigin to the origin are given by the row or column of the\ncoefficient matrix of $r$ or $q$ corresponding to pure holomorphic or\nantiholomorphic terms.\nSo we can read off~$g$ from the coefficient matrix of~$r$ or~$q$.\nAnother way to think about it is\nthat the harmonic terms of $r$ equal\n$g(z) + \\overline{g(z)} -1$.\n\n\\begin{proof}\nThat $g$ is of degree at most $d-1$ is the result of D'Angelo we mentioned\nabove.\nThat $g(z) = r(z,0) = q(z,0)$ follows by complexifying the\n$\\sabs{g(z)}^2-\\snorm{p(z)}^2$ and plugging in $\\bar{z}=0$ using the fact\nthat $p(0)=0$ and $g(0)=1$.\n\\end{proof}\n\nEvery automorphism of the unit ball ${\\mathbb{B}}_n$ is written as $U\n\\varphi_\\alpha$,\nwhere $U$ is a unitary matrix and\n\\begin{equation} \\label{eq:varphia}\n\\varphi_{\\alpha}(z) =\n\\frac{{\\alpha}-L_{\\alpha} z}{1-\\langle z,{\\alpha}\\rangle},\n\\qquad\nL_{\\alpha}z = \\left(1-\\sqrt{1-\\snorm{{\\alpha}}^2}\\right)\n\\frac{\\langle z,{\\alpha}\\rangle}{\\snorm{{\\alpha}}^2} {\\alpha} \n+\n\\sqrt{1-\\snorm{{\\alpha}}^2} z,\n\\end{equation}\nwhere if ${\\alpha}=0$, then $L_0=I$.\nNote that $L_\\alpha$ is a linear map.\nThe automorphism $\\varphi_\\alpha$ is the one where\n$\\varphi_\\alpha(0) = \\alpha$ and $\\varphi_\\alpha(\\alpha) = 0$. In fact,\n$\\varphi_\\alpha$ is an involution.\nNote that the coefficients of the numerator and denominator of the\nmap $\\varphi_\\alpha$ are continuous in ${\\alpha} \\in {\\mathbb{B}}^n$.\n\n\nWe need a version of a result of\nCima--Suffridge~\\cite{CimaSuffridge} (or Chiappari~\\cite{Chiappari}) that the\ndenominator does not vanish, although they consider mainly $n \\geq 2$.\nThe one dimensional case is rather simple, so for completeness, we give a proof.\nSee also~\\cite{DHX}.\n\n\\begin{lemma} \\label{lemma:chiappari}\nSuppose\n$f = \\frac{p}{g} \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ \nis a proper rational map written in lowest terms.\nThen $g$ never vanishes on $\\partial {\\mathbb{B}}_n$.\n\\end{lemma}\n\n\\begin{proof}\nWhen $n > 1$, it is the result of\nCima--Suffridge~\\cite{CimaSuffridge}, so suppose $n=1$.\nIf $g(z_0)=0$ for some $z_0 \\in \\partial {\\mathbb{B}}_1 = S^1$,\nthen, as $f$ is in lowest terms, at least one component \nof $f$ has a pole at~$z_0$. But\n$\\snorm{f(z)} < 1$ for all $z \\in {\\mathbb{B}}_1 = {\\mathbb{D}}$. So for any\ncomponent $f_j(z)$ we also have $\\sabs{f_j(z)} < 1$, so $f_j$\ncannot have a pole on the closure of ${\\mathbb{D}}$.\n\\end{proof}\n\nThe lemma has the following stronger corollary about the quotient\npolynomial, which is the result that we will require.\n\n\\begin{lemma} \\label{lemma:qpositive}\nSuppose\n$\\frac{p}{g} \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ \nis a proper rational map written in lowest terms.\nLet $q$ be the quotient polynomial $q(z,\\bar{z}) =\n\\frac{\\sabs{g(z)}^2-\\snorm{p(z)}^2}{1-\\snorm{z}^2}$ as above.\nThen $q > 0$ on the sphere $S^{2n-1}$,\nand hence on $\\overline{{\\mathbb{B}}_n}$.\n\\end{lemma}\n\nIt is not difficult to see that $q > 0$ on the ball ${\\mathbb{B}}_n$:\nAfter all, $\\frac{p}{g}$ is a proper map and so for all\n$z \\in {\\mathbb{B}}_n$, $\\sabs{g(z)}^2-\\snorm{p(z)}^2 > 0$ and $1-\\snorm{z}^2 > 0$.\nThe point of the lemma is that $q$ does not vanish on the boundary.\n\n\\begin{proof}\nWrite $f = \\frac{p}{g}$. By Lemma~\\ref{lemma:chiappari},\nthe function $f$ is holomorphic on a neighborhood of $\\overline{{\\mathbb{B}}_n}$.\nIn particular, the denominator is nonzero in a neighborhood of\n$\\overline{{\\mathbb{B}}_n}$. Thus we write\n\\begin{equation}\n1-\\snorm{f(z)}^2 = \\frac{q(z,\\bar{z})}{\\sabs{g(z)}^2}\n\\bigl( 1-\\snorm{z}^2 \\bigr) .\n\\end{equation}\nTaking the radial derivative $\\frac{\\partial}{\\partial r}$ for a point $z_0$\non the sphere we find\n\\begin{equation}\n\\left.\\frac{\\partial}{\\partial r}\\right|_{z=z_0}\\Bigl[\\snorm{f(z)}^2\\Bigr] =\n2 \\frac{q(z_0,\\bar{z}_0)}{\\sabs{g(z_0)}^2} .\n\\end{equation}\nThe function\n$\\snorm{f(z)}^2$ is plurisubharmonic, and hence by the Hopf lemma, we find that\nits radial derivative on the sphere must be positive, and hence\n$q(z_0,\\bar{z}_0) > 0$, and in particular, not zero.\n\\end{proof}\n\n\n\n\\section{The $\\Lambda$-function}\n\nFor a proper ball map $f = \\frac{p}{g}$ written in lowest terms,\nwe define the \\emph{$\\Lambda$-function corresponding to $f$} as before:\n\\begin{equation} \\label{eq:Lambda2}\n\\Lambda(z,\\bar{z}) = \\Lambda_f(z,\\bar{z}) =\n\\frac{\\sabs{g(z)}^2-\\snorm{p(z)}^2}{(1-\\snorm{z}^2)^d} .\n\\end{equation}\n\nThe $\\Lambda$ does not change at all when postcomposing $f$ with an automorphism.\n\n\\begin{lemma} \\label{lemma:Lambda1}\nSuppose $f \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ is a rational proper map and\n$\\tau \\in \\operatorname{Aut}({\\mathbb{B}}_N)$.\nThen $\\Lambda_f = \\Lambda_{\\tau \\circ f}$.\n\\end{lemma}\n\n\\begin{proof}\nThe proof follows from Lemma~\\ref{lemma:targetautform},\nas the underlying form $r$ is the same for both $f$ and $\\tau \\circ f$.\n\\end{proof}\n\nFor precomposing with an automorphism, the $\\Lambda$, up to a constant,\ntransforms by simply precomposing with the same automorhism.\nThe point is that when precomposing both $r$ and $(1-\\snorm{z}^2)^d$, when\nwe try to clear denominators we multiply by the same function, that is,\nfor $\\Lambda$ there is no need to clear the denominators.\n\n\\begin{lemma} \\label{lemma:Lambda2}\nSuppose $f \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ is a rational proper map and\n$\\psi \\in \\operatorname{Aut}({\\mathbb{B}}_n)$.\nThen $\\Lambda_f \\circ \\psi = C \\Lambda_{f \\circ \\psi}$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $r(z,\\bar{z})=\\sabs{g(z)}^2-\\snorm{p(z)}^2$ is the underlying form\nof $f$, and $R(z,\\bar{z}) = \\sabs{G(z)}^2-\\snorm{P(z)}^2$ is the underlying form\nof $f \\circ \\psi$.\n\nFirst suppose $\\psi$ is a unitary matrix. Then\n$G(z) = g(\\psi(z))$ and \n$P(z) = p(\\psi(z))$, as precomposing with a unitary does not introduce any\ndenominators and does not change the value at 0.\nFurthermore $(1-\\snorm{\\psi(z)}^2)^d = (1-\\snorm{z}^2)^d$.\nThus,\nin this case $\\Lambda_f \\circ \\psi = \\Lambda_{f\\circ \\psi}$.\n\nNow suppose that $\\psi = \\varphi_\\alpha$ from \\eqref{eq:varphia} for some $\\alpha \\in {\\mathbb{B}}_n$.\nLet us first figure out how the underlying form $r$ transforms if we compose with $\\varphi_{\\alpha}$.\nWe compose $r$ and $\\varphi_\\alpha$ and then clear the denominators by multiplying by\n$\\sabs{1-\\langle z,\\alpha \\rangle}^{2d}$. To keep $r(0,0) = 1$ we also\nmultiply by a constant,\n$\\frac{1}{r(\\alpha,\\overline{\\alpha})}$.\nThat is,\n$r$ transforms to\n\\begin{equation}\nR(z,\\bar{z}) =\n\\frac{1}{r(\\alpha,\\overline{\\alpha})} \\sabs{1-\\langle z,\\alpha \\rangle}^{2d}\nr\\bigl(\n\\varphi_{\\alpha}(z),\n\\bar{\\varphi}_{\\alpha}(\\bar{z})\n\\bigr)\n\\end{equation}\nSimilarly,\n$1-\\snorm{z}^2$ transforms to\n\\begin{equation}\n\\frac{1}{1-\\snorm{\\alpha}^2} \\sabs{1-\\langle z,\\alpha \\rangle}^{2}\n\\left(\n1-\\norm{\n\\varphi_{\\alpha}(z)\n}^2\n\\right)\n=\n1-\\snorm{z}^2 .\n\\end{equation}\nThat is, after clearing denominators,\n$1-\\snorm{z}^2$ is untouched by composing with an automorphism up\nto a constant.\n\nTherefore,\n\\begin{equation}\n\\Lambda \\circ \\varphi_{\\alpha}\n=\n\\frac{\nr\\bigl(\n\\varphi_{\\alpha}(z),\n\\bar{\\varphi}_{\\alpha}(\\bar{z})\n\\bigr)\n}{\n(1-\\snorm{\\varphi_\\alpha(z)}^2)^d\n}\n=\n\\left(\n\\frac{r(\\alpha,\\overline{\\alpha})}{ (1-\\snorm{\\alpha}^2)^d}\n\\right)\n\\,\n\\frac{\n\\sabs{G(z)}^2-\\snorm{P(z)}^2\n}{\n(1-\\snorm{z}^2)^d\n} .\n\\end{equation}\n\\end{proof}\n\nWhen the degree of the map is $d=1$, then $f = \\tau \\circ (z \\oplus 0)$ for\nsome $\\tau \\in \\operatorname{Aut}({\\mathbb{B}}_N)$, so $r(z,\\bar{z}) = 1-\\snorm{z}^2$ and\n$\\Lambda(z,\\bar{z}) \\equiv 1$. In particular, $\\Lambda$ is not an exhaustion\nfunction, nor is it strictly plurisubharmonic. Therefore, $d > 1$\nis required for the next two results.\n\n\\begin{lemma} \\label{lemma:Lambda3}\nSuppose $f \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ is a rational proper map\nof degree $d > 1$. Then\n$\\Lambda = \\Lambda_f \\colon {\\mathbb{B}}_n \\to {\\mathbb{R}}$ is a strictly plurisubharmonic\nfunction such that\n$\\Lambda(z)$ goes to $+\\infty$ as $z \\to \\partial {\\mathbb{B}}_n$.\n\\end{lemma}\n\n\\begin{proof}\nBy the transitivity of $\\operatorname{Aut}({\\mathbb{B}}_n)$,\nas composing $\\Lambda$ with $\\varphi_{\\alpha}$ preserves strict plurisubharmonicity, and it\nalso transforms to another $\\Lambda$-function for another proper map by\nLemma~\\ref{lemma:Lambda2}, we\nsimply need to prove strict pluriharmonicity at the origin.\nBy Lemma~\\ref{lemma:Lambda1} we can assume that $f = \\frac{p}{g}$ where $p(0)=0$\nand $g(0) = 1$.\nA function is strictly plurisubharmonic if its restriction to every complex line\nis strictly subharmonic.\nWe restrict to a\ncomplex line through the origin, and so we can, without loss of generality,\nassume $n=1$.\nWe follow the argument of the typical proof of Schwarz's lemma:\nFirst, $f(z) = z F(z)$, as each component is divisible by $z$.\nAs $F$ is not constant ($d > 1$),\nthen $F(z)$ still takes the ball to the ball by the maximum principle.\nIn particular, $\\snorm{F(z)} < 1$ for all $z \\in {\\mathbb{B}}_1$. Then\n$\\snorm{f'(0)} = \\snorm{F(0)} < 1$.\nUsing the fact that $p(0)=0$ and $g(0)=1$ and the quotient rule we have\n$f'(0) = p'(0)$, so $\\snorm{p'(0)} < 1$.\n\nNow compute the Laplacian and notice it is strictly positive (using $d > 1$)\n\\begin{multline}\n\\frac{\\partial^2}{\\partial z \\partial \\bar{z}}\\Big|_{z=\\bar{z}=0} \\Lambda(z,\\bar{z})\n=\nd \\, \\sabs{g'(0)}^2 - d \\, \\snorm{p'(0)}^2 + \\sabs{g'(0)}^2-\\snorm{p'(0)}^2\n\\\\\n=\nd + \\sabs{g'(0)}^2-\\snorm{p(0)}^2\n>\nd + \\sabs{g'(0)}^2-1 > 0 .\n\\end{multline}\n\nTherefore, $\\Lambda$ is strictly plurisubharmonic, we need to show that it\nis an exhaustion function.\nConsider the quotient polynomial $q(z,\\bar{z}) =\n\\frac{r(z,\\bar{z})}{1-\\snorm{z}^2}$. Lemma~\\ref{lemma:qpositive} says that\n$q > 0$ on the closed ball $\\overline{{\\mathbb{B}}_n}$, so\n\\begin{equation}\n\\Lambda(z,\\bar{z}) = \n\\frac{\\sabs{g(z)}^2-\\snorm{p(z)}^2}{(1-\\snorm{z}^2)} \\,\n\\frac{1}{(1-\\snorm{z}^2)^{d-1}} =\nq(z,\\bar{z})\n\\frac{1}{(1-\\snorm{z}^2)^{d-1}} .\n\\end{equation}\nThe result follows,\nagain using that $d > 1$.\n\\end{proof}\n\n\\begin{prop} \\label{prop:g}\nSuppose $g(z) = 1 + g_2 z^2 + g_3 z^3 + \\cdots + g_k z^k$ is a polynomial\nin one variable with no zeros on $\\overline{{\\mathbb{D}}}$. Then $\\sabs{g_2} < \\frac{k}{2}$.\n\\end{prop}\n\n\\begin{proof}\nWrite $g(z) = (1-r_1 z)(1-r_2 z) \\cdots (1-r_k z)$ where $\\sabs{r_j} < 1$\nfor all $j$. We also have $r_1 + r_2 + \\cdots + r_k = 0$.\nWrite\n\\begin{equation}\n0 =\n\\left( \\sum_{j=1}^k r_j \\right)^2 = \n\\sum_{j=1}^k r_j^2\n+2\n\\left(\n\\sum_{1 \\leq j < \\ell \\leq k} r_j r_\\ell\n\\right) .\n\\end{equation}\nUsing this inequality and\nexpanding $g_2$ we find\n\\begin{equation}\n\\sabs{g_2}\n=\n\\abs{\n\\sum_{1 \\leq j < \\ell \\leq k} r_j r_\\ell \n}\n=\n\\frac{1}{2}\n\\abs{\n\\sum_{j=1}^k r_j^2\n}\n\\leq\n\\frac{1}{2}\n\\left(\n\\sum_{j=1}^k \\sabs{r_j}^2\n\\right)\n< \\frac{k}{2} .\n\\end{equation}\n\\end{proof}\n\n\\begin{lemma} \\label{lemma:Lambda4}\nSuppose $f \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ is a rational proper map\nof degree $d > 1$. Then \n$\\Lambda = \\Lambda_f \\colon {\\mathbb{B}}_n \\to {\\mathbb{R}}$ has\na unique critical point (a minimum) in ${\\mathbb{B}}_n$.\n\\end{lemma}\n\n\\begin{proof}\nThere must clearly be at least one critical point,\nand so after composing with an automorphism we can assume it is at the\norigin. We restrict to an arbitrary complex line through the origin.\nIf we show that the determinant of the Hessian of this restricted function\nat the origin is strictly positive, then since the Laplacian (the trace of the Hessian)\nis also strictly positive by Lemma~\\ref{lemma:Lambda3},\nthe critical point is a minimum. Hence, without loss of generality\nassume that $n=1$.\n\nAgain, by Lemma~\\ref{lemma:Lambda1}, we can assume that $f = \\frac{p}{g}$ where $p(0)=0$\nand $g(0) = 1$. Write\n\\begin{equation}\ng(z) = 1 + g_1 z + g_2 z^2 + \\cdots + g_{d-1} z^{d-1} .\n\\end{equation}\nLet us expand $\\Lambda$ at the origin up to the second order:\n\\begin{multline}\n\\Lambda(z,\\bar{z})\n=\n1+g_1 z + \\bar{g}_1 \\bar{z} + g_2 z^2 + \\bar{g}_2 \\bar{z}^2\n+ (d- \\snorm{p'(0)}^2+\\sabs{g_1}^2 ) z \\bar{z}\n\\\\\n+\n\\text{ (higher order terms).}\n\\end{multline}\nAs the origin is a critical point, we notice that $g_1 = 0$.\nAs in Lemma~\\ref{lemma:Lambda3}, we have $\\snorm{p(0)} < 1$.\nThe Hessian determinant is\n\\begin{equation}\n4 (d- \\snorm{p'(0)}^2)^2\n-\n16 \\sabs{g_2}^2\n>\n4 (d- 1)^2\n-\n16 \\sabs{g_2}^2 .\n\\end{equation}\nWith $g_1=0$, the function $g$ satisfies the hypothesis of\nProposition~\\ref{prop:g}. Hence $\\sabs{g_2} < \\frac{d-1}{2}$,\nand therefore, the Hessian determinant of $\\Lambda$ at the origin\nis positive.\n\\end{proof}\n\nThe four items of Theorem~\\ref{thm:Lambda} are simply\nlemmas\n\\ref{lemma:Lambda1},\n\\ref{lemma:Lambda2},\n\\ref{lemma:Lambda3}, and\n\\ref{lemma:Lambda4}.\n\n\n\\section{Normalizing the denominator}\n\nThe key point in Theorem~\\ref{thm:normdenom} is that we can zero out the\nlinear terms of the denominator while making sure the map takes the origin\nto the origin.\n\n\\begin{lemma} \\label{lemma:alphaiscrit}\nSuppose $f \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ is a rational proper map\nof degree $d > 1$.\nThen for some\n$\\tau \\in \\operatorname{Aut}({\\mathbb{B}}_N)$, the map\n$\\tau \\circ f \\circ \\varphi_\\alpha$\ntakes the origin to the origin, and its denominator (when written in lowest terms)\nhas no linear terms\nif and only if $\\alpha \\in {\\mathbb{B}}_n$\nis a critical point of the corresponding $\\Lambda$-function.\n\\end{lemma}\n\nThe map $\\tau$ is simply the automorphism that takes $f(\\alpha)$ to $0$.\n\n\\begin{proof}\nLet $r(z,\\bar{z}) = \\sabs{g(z)}^2-\\snorm{p(z)}^2$.\nConsider\n\\begin{equation}\nR(z,\\bar{z}) =\n\\sabs{1-\\langle z,\\alpha \\rangle}^{2d}\nr\\bigl(\n\\varphi_{\\alpha}(z),\n\\bar{\\varphi}_{\\alpha}(\\bar{z})\n\\bigr) ,\n\\end{equation}\nthat is, consider the transformed form up to a constant that we forget for simplicity.\nSo up to a constant, by Lemma~\\ref{lemma:Gfrommatrix}, we have that the\nnew denominator is (up to a constant) $R(z,0)$. The new denominator has\nzero linear coefficients if for all $j$\n\\begin{equation}\n\\frac{\\partial R}{\\partial z_j} \\Big|_{z=\\bar{z}=0} = 0 .\n\\end{equation}\n\n\nWe differentiate \n\\begin{equation}\n\\begin{split}\n\\frac{\\partial}{\\partial z_j}\nR(z,\\bar{z}) = &\nd(-\\bar{\\alpha}_j)\\bigl(1-\\langle z, \\alpha \\rangle\\bigr)^{d-1}\n\\bigl(1-\\langle \\alpha, z \\rangle\\bigr)^{d}\nr\\bigl(\n\\varphi_{\\alpha}(z),\n\\bar{\\varphi}_{\\alpha}(\\bar{z})\n\\bigr)\n\\\\\n&\n+\n\\bigl(1-\\langle z, {\\alpha} \\rangle\\bigr)^{d}\n\\bigl(1-\\langle {\\alpha}, z \\rangle\\bigr)^{d}\n\\,\n\\nabla_z r\\big|_{(\n\\varphi_{\\alpha}(z),\n\\bar{\\varphi}_{\\alpha}(\\bar{z})\n)}\n\\cdot\n\\frac{\\partial \\varphi_{\\alpha}}{\\partial z_j}\n.\n\\end{split}\n\\end{equation}\nBy $\\nabla_z$ we mean the gradient in the $z$ variables\n(and not the $\\bar{z}$ variables).\nTo find the $z$ coefficients of $R$, we set $z=0$ and $\\bar{z}=0$ above,\nand then set the whole expression to zero:\n\\begin{equation} \\label{eq:maineqforalpha}\nd(-\\bar{\\alpha}_j)\nr(\\alpha,\\bar{\\alpha})\n+\n\\nabla_z r \\big|_{(\\alpha,\\bar{\\alpha})}\n\\cdot\n\\frac{\\partial \\varphi_{\\alpha}}{\\partial z_j}\\Big|_{z=0}\n=0 .\n\\end{equation}\nWe now wish to show that\nthis solution is precisely the critical point of\n\\begin{equation}\n\\Lambda(z,\\bar{z}) =\n\\frac{r(z,\\bar{z})}{(1-\\snorm{z}^2)^d} .\n\\end{equation}\n\nFor every unitary matrix $U$, we have $\\varphi_{U\\alpha}(Uz) = U \\varphi_{\\alpha}(z)$.\nSo if there is a solution $\\alpha$ to \\eqref{eq:maineqforalpha},\nwe rotate our map by $U$\nand assume that $\\alpha = (\\alpha_1,0,\\ldots,0)$.\nWe differentiate $(\\varphi_\\alpha)_k$, the $k$th component of\n$\\varphi_\\alpha$.\n\\begin{equation}\n\\frac{\\partial (\\varphi_\\alpha)_k}{\\partial z_j} \\Big|_{z=0}\n=\n\\begin{cases}\n-(1-\\sabs{\\alpha_1}^2) & \\text{if } k=j=1 , \\\\\n\\sqrt{1-\\sabs{\\alpha_1}^2} & \\text{if } k=j \\text{ and } k\\not=1 , \\\\\n0 & \\text{else.}\n\\end{cases}\n\\end{equation}\nFor any $j$,\n\\begin{equation}\n\\frac{\\partial \\Lambda}{\\partial z_j} =\n\\frac{(1-\\snorm{z}^2)^d \\frac{\\partial r}{\\partial z_j} - d(-\\bar{z}_j)\n(1-\\snorm{z}^2)^{d-1} r}{(1-\\snorm{z}^2)^{2d}}\n.\n\\end{equation}\nIf $j= 1$, then \\eqref{eq:maineqforalpha} becomes\n\\begin{equation}\nd(-\\bar{\\alpha}_1)\nr(\\alpha,\\bar{\\alpha})\n-\n(1-\\sabs{\\alpha_1}^2)\n\\frac{\\partial r}{\\partial z_1} \\Big|_{(\\alpha,\\bar{\\alpha})}\n=0 .\n\\end{equation}\nAnd that is equivalent to\n$\\frac{\\partial \\Lambda}{\\partial z_1}\\big|_{(\\alpha,\\bar{\\alpha})} = 0$.\n\nSimilarly when $j > 1$, then\n\\eqref{eq:maineqforalpha} becomes\n\\begin{equation}\n\\sqrt{1-\\sabs{\\alpha_1}^2}\n\\,\n\\frac{\\partial r}{\\partial z_j} \\Big|_{(\\alpha,\\bar{\\alpha})}\n=0 .\n\\end{equation}\nAnd that is also equivalent to\n$\\frac{\\partial r}{\\partial z_j} \\Big|_{(\\alpha,\\bar{\\alpha})}=0$ and\nhence\n$\\frac{\\partial \\Lambda}{\\partial z_j}\\big|_{(\\alpha,\\bar{\\alpha})} = 0$\n(noting that $\\alpha_j = 0$).\n\nRepeating the argument by differentiating with respect to $\\bar{z}_j$,\nwe obtain that the conjugate of \\eqref{eq:maineqforalpha} is\nequivalent to $\\frac{\\partial \\Lambda}{\\partial\n\\bar{z}_j}\\big|_{(\\alpha,\\bar{\\alpha})} = 0$.\n\\end{proof}\n\nWe now prove Theorem~\\ref{thm:normdenom}.\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:normdenom}]\nIf $d = 1$, the theorem is trivial, so assume the degree $d > 1$.\nWrite $\\Lambda$ as before. By Theorem~\\ref{thm:Lambda}, \n$\\Lambda$ has a unique critical point $\\alpha$.\nBy Lemma~\\ref{lemma:alphaiscrit}, the critical point gives\nthe unique automorphism $\\varphi_\\alpha$ such that if we\nlet $\\tau$ be an automorphism that takes $f(\\alpha)$ to $0$, and we write\n$\\tau \\circ f \\circ \\varphi_\\alpha = \\frac{P}{G}$ in lowest terms, where\n$G(0)=1$, then $P(0) = 0$ and $G$ has no linear terms. That is,\n\\begin{equation}\nG(z) = 1 + G_2(z) + G_3(z) + \\cdots + G_{d-1}(z) ,\n\\end{equation}\nwhere $G_j$ are homogeneous of degree $j$.\nBy the uniqueness of the critical point of $\\Lambda$, the only\nautomorphisms that we can precompose with to keep $G$ in this form are\nunitary matrices.\nIf we are only allowed to precompose with unitaries, the only\nautomorphisms on the target are also unitaries as they must fix the origin.\n\nConsider $z$ as a column vector, then\n\\begin{equation}\nG_2(z) = z^t C z\n\\end{equation}\nfor some complex symmetric matrix $C$ (the $z^t$ is the regular transpose).\nApplying a unitary we find that\n$G_2$ transforms to\n\\begin{equation}\nz^t U^tCU z .\n\\end{equation}\nIt is an elementary result in linear algebra (see e.g.~\\cite{HornJohnson}\nCorollary 4.4.4 part c)\nthat a symmetric matrix can be\ndiagonalized by congruence with unitary matrices,\nwhere the diagonal entries are all nonnegative and sorted according size.\nThe diagonal entries are the singular values (moduli of the eigenvalues) of\nthe matrix $C$.\nIn other words, after such a unitary\n\\begin{equation}\nG(z) = 1 + \\sum_{k=1}^n \\sigma_k z_k^2 + \\text{ (higher order terms)},\n\\end{equation}\nwhere $0 \\leq \\sigma_1 \\leq \\cdots \\leq \\sigma_n$.\nBy restricting to $z_1=\\cdots=z_{n-1}=0$, we obtain a polynomial in \none variable of degree $d-1$ or that satisfies the hypotheses of\nProposition~\\ref{prop:g}. Thus $\\sigma_n \\leq \\frac{d-1}{2}$.\nThe only unitary matrices that can still be applied\nthat preserve the normal form are those that preserve this quadratic part.\n\\end{proof}\n\n\n\\section{Existence of third degree maps} \\label{section:exist}\n\nAs we mentioned, any polynomial function $g(z)$ that is\nnon-zero on the closed ball $\\overline{{\\mathbb{B}}_n}$ is the\ndenominator of a proper map of balls written in lowest terms, see\nD'Angelo~\\cite{DAngelo:spheresbook}. The degree of the resulting map\ndepends not only on the degree of $g$, but also on the coefficients.\nSo a natural question is: Given\n\\begin{equation}\ng(z) = 1 + \\sum_{k=1}^n \\sigma_k z_k^2 ,\n\\end{equation}\ndoes there exist a degree 3 polynomial map $p$ such that $p(0)=0$ and $f = \\frac{p}{g}$\nis a rational proper map of balls written in lowest terms?\nWe show that for small enough\n$\\sigma_k$, such a map always exists.\n\nFor $\\frac{p}{g}$ to be a proper map of balls, $g$ cannot be\nzero on $\\overline{{\\mathbb{B}}_n}$, and so Proposition~\\ref{prop:g} gives a crude upper\nbound for $\\sigma_k$, that is, $\\frac{d-1}{2}$.\nFor $d=3$, we find that $\\sigma_k < 1$.\nBut as mentioned in the introduction,\nfor $g(z) = 1+\\sigma z_1^2 + \\sigma z_2^2$,\nthe degree of the numerator\nrequired for a proper map to exist goes to infinity as $\\sigma$ approaches 1.\nFor a degree 3 numerator to exist, the actual inequalities are\ncomplicated except in the simplest case of $n=1$.\n\nFor $n=1$, the problem is easy to solve.\nIf $f \\colon {\\mathbb{B}}_1 \\to {\\mathbb{B}}_N$ is a rational proper map with\ndenominator $g(z) = 1 + \\sigma z^2$, then $\\sigma < 1$ as mentioned above.\nTo show existence it is enough to consider $N=1$. In the introduction,\nwe computed the normal form for degree 3 proper rational maps\n$f \\colon {\\mathbb{B}}_1 \\to {\\mathbb{B}}_1$, and we showed one exists with\ndenominator $1+\\sigma z^2$ for every $0 \\leq \\sigma < 1$.\n\nFor $n > 1$,\nthe exact inequalities on $\\sigma_k$ are more complicated.\nFrom the proof below, these are inequalities guaranteeing a\ncertain matrix with some extra variables is positive semidefinite.\nFor any particular $n$ it is possible\nto constructively write down the particular inequalities.\nWe dispense with attempting to\nwrite down the exact inequalities and simply show that for\nsmall enough $\\sigma_k$, a degree 3 numerator $p$ always exists\nmaking $\\frac{p}{g}$ a proper map of balls written in lowest terms.\nWe refer the reader to a recent new preprint by\nD'Angelo~\\cite{DAngelo:preprint} for more precise information.\n\nRecall that the $N$ required (after having fixed an $n$) is finite.\nThat is, the maximal $N$ is the number of\nnonconstant degree 3 or lower monomials in $n$ variables.\nThat is, $N+1$ is the size of the matrix of coefficients \nof the underlying form $r(z,\\bar{z})$ corresponding to degree~3 maps.\n\n\\begin{prop} \\label{prop:exist}\n\\pagebreak[2]\nGiven $n$, there exists an $\\epsilon > 0$ such that whenever\n$0 \\leq \\sigma_1 \\leq \\cdots \\leq \\sigma_n < \\epsilon$,\nthen there exists a degree 3 (or lower) polynomial $p \\colon {\\mathbb{C}}^n \\to {\\mathbb{C}}^N$\n(where $N$ depends only on~$n$)\nsuch that $p(0) = 0$ and\n\\begin{equation}\nz \\mapsto\n\\frac{p(z)}{1+\\sum_{k=1}^n \\sigma_k z_k^2}\n\\end{equation}\nis a rational proper map of ${\\mathbb{B}}_n$ to ${\\mathbb{B}}_N$ written in lowest terms.\n\nMoreover, the set of $\\sigma_1,\\ldots,\\sigma_n$ for which \na degree $p$ numerator exists as above is a semialgebraic set.\n\\end{prop}\n\nWe recall that a semialgebraic set is a subset of ${\\mathbb{R}}^m$ given by a finite\nset of polynomial equalities and inequalities. A key result on semialgebraic sets is\nthe Tarski--Seidenberg theorem, which says that a projection of a\nsemialgebraic set onto a subspace is itself semialgebraic.\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:exist}]\nGiven a denominator $g(z) = 1+\\sum_{k=1}^n \\sigma_k z_k^2$,\nwe have a solution $p$ if $\\norm{\\frac{p(z)}{g(z)}}^2 = 1$ on the\nsphere. That is, if $\\sabs{g(z)}^2-\\snorm{p(z)}^2 = 0$ when\n$\\snorm{z}=1$. In other words, $p$ is a solution when there is a\npolynomial $q(z,\\bar{z})$ such that\n\\begin{equation}\n\\sabs{g(z)}^2-\\snorm{p(z)}^2 = q(z,\\bar{z}) ( 1-\\snorm{z}^2 ) .\n\\end{equation}\nA degree 3 polynomial $p$ as in the\ntheorem exists if and only if there exists a $q$ of bidegree $(2,2)$ (degree\n2 in $z$ and 2 in $\\bar{z}$) such that\n\\begin{equation} \\label{eq:tobepositive}\n\\sabs{g(z)}^2\n-\nq(z,\\bar{z}) ( 1-\\snorm{z}^2 )\n\\end{equation}\nhas a positive semidefinite matrix of coefficients and therefore is a sum of\n(hermitian) squares. If the matrix also has no harmonic terms, the\nresulting polynomial map $p$ will take the origin to the origin.\n\nLet $[q_{\\alpha,\\beta}]$ be the matrix of coefficients of $q$. Let\n$[c_{\\alpha,\\beta}]$ be the matrix of coefficients of\n\\eqref{eq:tobepositive}.\nThe matrix\n$[c_{\\alpha,\\beta}]$\nbeing positive definite is a set of inequalities on the elements\n$c_{\\alpha,\\beta}$,\nwhich are polynomials in the various $q_{\\alpha,\\beta}$ and the $\\sigma_k$,\nand these inequalities give a semialgebraic set.\nThe set of possible $\\sigma$s is the projection of this\nsemialgebraic set onto $(\\sigma_1,\\ldots,\\sigma_n)$. By\nTarski--Seidenberg this is a semialgebraic set and we have proved the\n``Moreover.''\n\nWe move to the main conclusion of the proposition. First, there\nexists a third degree polynomial proper map of balls where the matrix\nfor $\\snorm{p(z)}^2$ in the setup above is diagonal, that is,\nwhere each component of the map $p$ is a single monomial.\nFurthermore, we wish all the monomials be used, except the\nconstant. One possibility for the form corresponding to $\\snorm{p(z)}^2$ is\n\\begin{equation}\n\\frac{1}{3} (\\snorm{z}^6 + \\snorm{z}^4 + \\snorm{z}^2 ) .\n\\end{equation}\nExcept for the diagonal term corresponding to the constant, all\nthe diagonal terms of matrix of coefficients are of size at least $\\frac{1}{3}$.\nNote that\n\\begin{equation}\n1-\n\\frac{1}{3} (\\snorm{z}^6 + \\snorm{z}^4 + \\snorm{z}^2 )\n=\nq(z,\\bar{z}) ( 1-\\snorm{z}^2 ) .\n\\end{equation}\nThe trick now is to perturb this form (perturb the matrix) while keeping it\nof the form $\\sabs{g(z)}^2-\\snorm{p(z)}^2$ (for a different $p$ of course)\nand making sure that it is still divisible by $\\snorm{z}^2-1$.\n\nConsider\n\\begin{equation}\nr(z,\\bar{z}) =\n1-\\frac{1}{3} (\\snorm{z}^6 + \\snorm{z}^4 + \\snorm{z}^2 )\n+\n\\sum_{k=1}^n \\sigma_k (z_k^2 +\\bar{z}_k^2) ( 1-\\snorm{z}^2 ) .\n\\end{equation}\nClearly $r = 0$ when $\\snorm{z}=1$. It remains to show that it can be\nwritten as $\\sabs{g(z)}^2-\\snorm{p(z)}^2$ where $g$ is the given\ndenominator.\n\nIn the matrix of coefficients for $\\sabs{g(z)}^2 =\ng(z)\\bar{g}(\\bar{z})$, \nthe column and row corresponding to the constant monomial is the same\nas $r(z,\\bar{z})$. That is because if we plug in $\\bar{z}=0$\n\\begin{equation}\nr(z,0) = 1\n+ \\sum_{k=1}^n \\sigma_k z_k^2 = g(z) = g(z) \\bar{g}(0) .\n\\end{equation}\nThe matrix of coefficients of\n$-r(z,\\bar{z})+\\sabs{g(z)}^2$ has zeros in the corresponding column and row.\nWe need to show that $-r(z,\\bar{z})+\\sabs{g(z)}^2$ can be written\nas $\\snorm{p(z)}^2$ for some $p \\colon {\\mathbb{C}}^n \\to {\\mathbb{C}}^N$, where $p(0)=0$.\nThe entries corresponding to (nonharmonic) mixed terms\nin the matrix are of two types.\nThe diagonal entries come\nfrom $\\frac{1}{3} (\\snorm{z}^6 + \\snorm{z}^4 + \\snorm{z}^2 )$ and are \nall of size~$\\frac{1}{3}$ or greater, and they do not depend on $\\sigma_k$.\nThe off-diagonal entries are all multiples of $\\sigma_k$ for various $k$.\nIn particular, if $\\sigma_k$ are all small enough, then \nthe coefficient matrix for $-r(z,\\bar{z})+\\sabs{g(z)}^2$ is positive\nsemidefinite, the matrix has zeros in the pure (harmonic) terms, and\nthe $N \\times N$ submatrix of mixed (nonharmonic) terms is positive\ndefinite.\nTherefore,\n$-r(z,\\bar{z})+\\sabs{g(z)}^2$ can be written as a sum of $N$ hermitian\nsquares, that is, there exists a polynomial map $p$ such that\n\\begin{equation}\nr(z,\\bar{z}) = \\sabs{g(z)}^2-\\snorm{p(z)}^2 .\n\\end{equation}\nSince the matrix we are decomposing to get $p$ is the $N \\times N$\nmatrix of the mixed terms, all the components of $p$ vanish at the origin.\nIt remains to show that $\\frac{p}{g}$ is in lowest terms. If $g$ and all\ncomponents of $p$ have a common factor $h$, then dividing by $\\sabs{h}^2$\nwould result in a form corresponding to a proper map of balls of\ndegree 2 or 1. In such a case, the rank of the coefficient matrix of\n$\\frac{r}{\\sabs{h}^2}$ is necessarily the same as the rank of $r$,\nas the number of linearly independent squares in the expansion does\nnot change. But for small\nenough $\\sigma_k$s, $r$ has a matrix that is of full rank (for bidegree\n$(3,3)$).\nIf such an $h$ existed, the matrix\nfor $\\frac{r}{\\sabs{h}^2}$ is necessarily of strictly lower rank as\nthe bidegree is lower and all the coefficients for monomials of degree 3\nare zero. Thus no $h$ exists and $\\frac{p}{g}$ is in lowest terms.\n\\end{proof}\n\n\n\\section{Rational and polynomial maps of maximal embedding dimension}\n\\label{section:poly}\n\nThe normal form does not necessarily answer the question of when is\na rational proper ball map equivalent to a polynomial. In a certain\nsense, most maps are equivalent to a polynomial map although that map\nmay not take the origin to the origin, and may involve the ball in different\ncoordinate patch of the projective space.\nAs we mentioned before, once the degree is 3 or higher,\nthere do exist maps that are not spherically equivalent to a polynomial.\n\nRecall that for a fixed degree~$d$, there is a maximal embedding dimension for\na rational proper map of balls.\nThis dimension is exactly one less than the dimension of the space\nof holomorphic polynomials of degree $d$,\nas the embedding dimension for $\\frac{p}{g}$ is\nthe dimension of the span of $(p_1,\\ldots,p_N,g)$.\nIn a certain sense, the ``generic'' degree~$d$ map\nhas the maximal embedding dimension: It is given by a generic real polynomial\n$r(z,\\bar{z})$ of bidegree $(d,d)$ that is divisible by $1-\\snorm{z}^2$\nand has 1 negative eigenvalue, and a generic such form is of full rank.\n\nWe do not, however, get polynomial maps to the standard model of the ball.\nConsider the following two other models of the unit ball,\nthat are both equivalent to ${\\mathbb{B}}_n$\nvia a linear fractional transformation. First, the hyperquadric or\ngeneralized ball with $1$ positive and $N-1$ negative eigenvalues:\n\\begin{equation}\n{\\mathbb{B}}_{1,N-1} = \\left\\{\nz \\in {\\mathbb{C}}^N : \\sabs{z_1}^2 - \\sabs{z_2}^2 - \\cdots - \\sabs{z_{N-1}}^2 < 1\n\\right\\} .\n\\end{equation}\nSecond, the Heisenberg model of the ball\n\\begin{equation}\n{\\mathbb{H}}_{N} = \\left\\{\nz \\in {\\mathbb{C}}^N : \\Re z_N > \\sabs{z_1}^2 + \\cdots + \\sabs{z_{N-1}}^2 \\right\\} .\n\\end{equation}\nThe linear fractional maps that go between ${\\mathbb{B}}_n$, ${\\mathbb{B}}_{1,N-1}$, and\n${\\mathbb{H}}_N$ also transform the defining forms\n$1-\\snorm{z}^2$ for ${\\mathbb{B}}_N$,\n$\\sabs{z_N}^2 - 1 - \\sabs{z_1}^2 - \\cdots - \\sabs{z_{N-1}}^2$ for\n${\\mathbb{B}}_{1,N-1}$,\nand\n$\\Re z_N - \\sabs{z_1}^2 - \\cdots - \\sabs{z_{N-1}}^2$ for ${\\mathbb{H}}_N$.\nTherefore, \nLemma~\\ref{lemma:targetautform} holds with these forms instead of\n$1-\\snorm{z}^2$ if we replace the target ball by ${\\mathbb{B}}_{1,N-1}$ or ${\\mathbb{H}}_N$.\n\n\n\\begin{prop} \\label{prop:poly}\n\\pagebreak[2]\nSuppose $f \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ is a rational proper map of degree $d$\nand $N$ is the maximal embedding dimension for degree $d$.\nThen there exists a linear fractional map $\\tau$ on ${\\mathbb{C}}^N$ such that we get\none of the three following conclusions:\n\\begin{enumerate}[(i)]\n\\item\n$\\tau \\circ f \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ is a polynomial proper map.\n\\item\n$\\tau \\circ f \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_{1,N-1}$ is a polynomial proper map.\n\\item\n$\\tau \\circ f \\colon {\\mathbb{B}}_n \\to {\\mathbb{H}}_{N}$ is a polynomial proper map.\n\\end{enumerate}\n\\end{prop}\n\nTo decide which conclusion holds, write $f = \\frac{p}{g}$, and\n$r(z,\\bar{z}) = \\sabs{g(z)}^2-\\snorm{p(z)}^2$ and normalize so that $r(0,0)\n= 1$. If the coefficient matrix of the mixed terms, that is,\n$r(z,\\bar{z}) - r(z,0) - r(0,\\bar{z}) + 1$,\nhas rank less than $N$, then item (iii) holds. Otherwise, we will see\nbelow that $r(z,\\bar{z})-\\gamma$ is of rank $N$ for some $\\gamma$.\nItem (i) holds if $\\gamma > 0$ and item (ii) holds if $\\gamma < 0$.\n\n\\begin{proof}\nLet $f = \\frac{p}{g}$ as usual, normalize so that $g(0) = 1$.\nBy Lemma~\\ref{lemma:targetautform}, to prove (i) the proposition we must show that\n\\begin{equation}\nr(z,\\bar{z})=\\sabs{g(z)}^2 - \\snorm{p(z)}^2 = \\gamma (1 - \\snorm{P(z)}^2)\n\\end{equation}\nfor some constant $\\gamma > 0$ and some polynomial map $P(z)$ to ${\\mathbb{C}}^N$.\nTo prove (ii), we need to show that\n\\begin{equation}\nr(z,\\bar{z}) = \\gamma(\\sabs{G(z)}^2 - 1 - \\snorm{P(z)}^2)\n\\end{equation}\nfor some polynomial $G(z)$ and a polynomial map $P(z)$ to ${\\mathbb{C}}^{N-1}$.\nTo prove (iii), we need to show that\n\\begin{equation}\nr(z,\\bar{z}) = \\Re G(z) - \\snorm{P(z)}^2\n\\end{equation}\nfor some polynomial $G(z)$ and\nsome polynomial map $P(z)$ to ${\\mathbb{C}}^{N-1}$.\nThe proof lies in showing that one of the three possibilities holds.\n\nLet $C$ be the hermitian coefficient matrix of the form $r(z,\\bar{z})$.\nThat $N$ is the maximal embedding dimension for this degree,\nmeans that $C$ has full rank.\nThe decomposition $\\sabs{g(z)}^2 - \\snorm{p(z)}^2$\nis simply writing $C$ as a sum of hermitian rank one matrices. That is\nif ${\\mathcal{Z}} = (z_1^d,\\ldots,1)^t$ is the column vector of all monomials\nof degree $d$ or less. Then for a column vector $v$ composed of\nthe complex conjugates of the coefficients of $g$, we have $g(z) = v^* {\\mathcal{Z}}$\n(where $v^*$ is the complex conjugate transpose), and\n$\\sabs{g(z)}^2 = {\\mathcal{Z}}^* v v^* {\\mathcal{Z}}$.\nHence, if $w_j$ is the vector of complex conjugates of the coefficients of\n$p_j(z)$, then\n\\begin{equation}\nC = v v^* - w_1 w_1^* - \\cdots - w_N w_N^* .\n\\end{equation}\nNote that $C$ is a full rank matrix of size $(N+1) \\times (N+1)$.\nLet $J$ be the rank $1$ matrix that is the coefficient matrix of the constant\nform $1$. That is, $J$ is a matrix of all zeros except a~1 in the entry\ncorresponding to the constant.\nThus to find a decomposition giving (i) or (ii) we need to find \na $\\gamma$ such that $C - \\gamma J$ is of rank $N$ (it can only be of rank\n$N+1$ or $N$).\n\nWe are looking for zeros of $\\det(C-\\gamma J)$. This function is constant\nand has no zeros if and only if the matrix of coefficients of the mixed\nterms $r(z,\\bar{z}) - r(z,0) - r(0,\\bar{z}) + 1$ is singular. As\n$r(z,0) + r(0,\\bar{z}) - 1$ has rank 2, we find that in this case the matrix\nof mixed terms must be of rank $N-1$ exactly. But then since \n$r(z,0) + r(0,\\bar{z}) - 1$ has one positive and one negative eigenvalue,\nthe matrix of mixed terms must be negative semidefinite, so write\n\\begin{equation}\nr(z,\\bar{z}) - r(z,0) - r(0,\\bar{z}) + 1 = -\\snorm{P(z)}^2 ,\n\\end{equation}\nwhere $P$ is a polynomial map to ${\\mathbb{C}}^{N-1}$. Then write\n\\begin{equation}\nr(z,0) + r(0,\\bar{z}) - 1 = \\Re G(z) ,\n\\end{equation}\nwhere $G(z) = 2r(z,0) - 1$. And so we obtain (iii).\n\nSo suppose that $\\det(C-\\gamma J)$ is not constant.\nIt is an affine linear function, so there must exist\na $\\gamma$ where it vanishes, that is,\nwhere $C-\\gamma J$ is of rank $N$,\nand $C-\\gamma J$ is a sum of $N$ rank 1 matrices.\nThe signature of $C-\\gamma J$ depends on the sign of $\\gamma$.\nIf $\\gamma < 0$, then $r(z,\\bar{z}) + (-\\gamma)$ must have one positive, $N-1$ negative\neigenvalues, so we could rewrite $r$ as\n\\begin{equation}\nr(z,\\bar{z}) = \\snorm{G(z)}^2 -\\sabs{\\sqrt{-\\gamma}}^2 -\n\\snorm{P(z)}^2\n\\end{equation}\nfor some polynomial $G(z)$ and a polynomial map $P(z)$ to ${\\mathbb{C}}^{N-1}$. That gives (ii).\n\nFinally, suppose $\\gamma > 0$.\nThen $C-\\gamma J$ must have $N$ negative eigenvalues.\nBy decomposing this matrix into rank 1 matrices we find the representation\n\\begin{equation}\nr(z,\\bar{z}) = \\sabs{\\sqrt{\\gamma}}^2-\\snorm{P(z)}^2\n\\end{equation}\nfor some polynomial $P$. Part (i) follows.\n\\end{proof}\n\nIt is good to remark why the idea of the proof does not work if the\nembedding dimension is not the maximum possible:\nIf $C$ is not of full rank,\nthen the rank of \n$C-\\gamma J$ may never drop, but the rank of \nthe matrix of mixed terms could still be of rank $N$ rather than $N-1$.\n\nThe set of proper maps equivalent to polynomial maps to ${\\mathbb{B}}_N$\n(resp.\\ ${\\mathbb{B}}_{1,N-1}$)\nof maximal embedding dimension is open.\nWe say a rational proper map $f \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$\nis \\emph{target equivalent to a polynomial proper map to ${\\mathbb{B}}_N$\n(resp. ${\\mathbb{B}}_{1,N-1}$)} if there exists a linear fractional transformation\n$\\tau$ such that $\\tau \\circ f$ is a polynomial proper map to ${\\mathbb{B}}_N$ (resp.\n${\\mathbb{B}}_{1,N-1}$).\nBelow,\ndenote $\\snorm{f}_{{\\mathbb{B}}_n} = \\sup_{z \\in {\\mathbb{B}}_n} \\snorm{f(z)}$.\n\n\\begin{prop}\nSuppose $f \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$\nis a rational proper map of degree $d$\nof maximal embedding dimension for degree $d$\nthat is target equivalent to a polynomial proper map to ${\\mathbb{B}}_N$\n(resp.\\ ${\\mathbb{B}}_{1,N-1}$). Then there exists an $\\epsilon > 0$\nsuch that if $F \\colon {\\mathbb{B}}_n \\to {\\mathbb{B}}_N$ is a rational proper map \nof degree $d$ where\n$\\snorm{f-F}_{{\\mathbb{B}}_n} < \\epsilon$, then\n$F$ is target equivalent to a polynomial proper map to ${\\mathbb{B}}_N$\n(resp.\\ ${\\mathbb{B}}_{1,N-1}$).\n\\end{prop}\n\n\\begin{proof}\nWrite $f = \\frac{p}{g}$ and $F=\\frac{P}{G}$ and write\nthe underlying forms\n\\begin{equation}\nr(z,\\bar{z})=\\sabs{g(z)}^2 - \\snorm{p(z)}^2\n\\qquad \\text{and}\\qquad\nR(z,\\bar{z})=\\sabs{G(z)}^2 - \\snorm{P(z)}^2 ,\n\\end{equation}\nwhere we assume $r(0,0)=R(0,0)=1$ as usual,\nwhich\nalso normalizes $\\sabs{g(0)}$ and $\\sabs{G(0)}$.\nWe further assume $g(0)$ and $G(0)$ to be positive.\nThe condition $\\snorm{f-F}_{{\\mathbb{B}}_n} < \\epsilon$\nis equivalent to\nthe coefficients of the Taylor series at the origin up to any\nfixed degree being within some~$\\epsilon'$.\nFrom that we find, as both $f$ and $F$ are of the same degree $d$,\nthat this condition is equivalent to \nthe coefficients of the coefficient matrix of $r$ and $R$ being close.\nThe argument in the proof of Proposition~\\ref{prop:poly}\nmeans that we can choose either a positive or negative $\\gamma$ such that\n$\\det(C-\\gamma J)=0$.\nA small enough perturbation of the underlying matrix also allows the choice of\nsuch a $\\gamma$ of the same sign, so the proposition follows.\n\\end{proof}\n\nIn particular, any rational proper map that is a small enough perturbation \nof a polynomial map of maximal embedding dimension for a given degree\nis spherically equivalent to a polynomial proper map.\n\n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\n\\bib{Alexander}{article}{\n author={Alexander, H.},\n title={Proper holomorphic mappings in $C^{n}$},\n journal={Indiana Univ. 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Amer. Math. Soc.},\n volume={145},\n date={2017},\n number={6},\n pages={2649--2660},\n issn={0002-9939},\n review={\\MR{3626518}},\n doi={10.1090\/proc\/13425},\n}\n\n\n\\bib{DL:homotopies}{article}{\n author={D'Angelo, John P.},\n author={Lebl, Ji\\v{r}\\'{\\i}},\n title={Homotopy equivalence for proper holomorphic mappings},\n journal={Adv. Math.},\n volume={286},\n date={2016},\n pages={160--180},\n issn={0001-8708},\n review={\\MR{3415683}},\n doi={10.1016\/j.aim.2015.09.007},\n}\n\n\\bib{DX}{article}{\n author={D'Angelo, John P.},\n author={Xiao, Ming},\n title={Symmetries in CR complexity theory},\n journal={Adv. Math.},\n volume={313},\n date={2017},\n pages={590--627},\n issn={0001-8708},\n review={\\MR{3649233}},\n doi={10.1016\/j.aim.2017.04.014},\n}\n\n\\bib{Dor}{article}{\n author={Dor, Avner},\n title={Proper holomorphic maps between balls in one co-dimension},\n journal={Ark. Mat.},\n volume={28},\n date={1990},\n number={1},\n pages={49--100},\n issn={0004-2080},\n review={\\MR{1049642}},\n doi={10.1007\/BF02387366},\n}\n\n\\bib{Ebenfelt13}{article}{\n author={Ebenfelt, Peter},\n title={Partial rigidity of degenerate CR embeddings into spheres},\n journal={Adv. Math.},\n volume={239},\n date={2013},\n pages={72--96},\n issn={0001-8708},\n review={\\MR{3045142}},\n doi={10.1016\/j.aim.2013.02.011},\n}\n\n\n\\bib{Faran:B2B3}{article}{\n author={Faran, James J.},\n title={Maps from the two-ball to the three-ball},\n journal={Invent. Math.},\n volume={68},\n date={1982},\n number={3},\n pages={441--475},\n issn={0020-9910},\n review={\\MR{669425}},\n doi={10.1007\/BF01389412},\n}\n\n\\bib{Faran:firstgap}{article}{\n author={Faran, James J.},\n title={The linearity of proper holomorphic maps between balls in the low\n codimension case},\n journal={J. Differential Geom.},\n volume={24},\n date={1986},\n number={1},\n pages={15--17},\n issn={0022-040X},\n review={\\MR{857373}},\n}\n\n\n\\bib{FHJZ}{article}{\n author={Faran, James},\n author={Huang, Xiaojun},\n author={Ji, Shanyu},\n author={Zhang, Yuan},\n title={Polynomial and rational maps between balls},\n journal={Pure Appl. Math. Q.},\n volume={6},\n date={2010},\n number={3, Special Issue: In honor of Joseph J. Kohn.},\n pages={829--842},\n issn={1558-8599},\n review={\\MR{2677315}},\n doi={10.4310\/PAMQ.2010.v6.n3.a10},\n}\n\n\\bib{Forstneric}{article}{\n author={Forstneri\\v{c}, Franc},\n title={Extending proper holomorphic mappings of positive codimension},\n journal={Invent. Math.},\n volume={95},\n date={1989},\n number={1},\n pages={31--61},\n issn={0020-9910},\n review={\\MR{969413}},\n doi={10.1007\/BF01394144},\n}\n\n\n\\bib{Hamada05}{article}{\n author={Hamada, Hidetaka},\n title={Rational proper holomorphic maps from $\\bold B^n$ into $\\bold\n B^{2n}$},\n journal={Math. Ann.},\n volume={331},\n date={2005},\n number={3},\n pages={693--711},\n issn={0025-5831},\n review={\\MR{2122546}},\n doi={10.1007\/s00208-004-0606-2},\n}\n\n\n\\bib{HornJohnson}{book}{\n author={Horn, Roger A.},\n author={Johnson, Charles R.},\n title={Matrix analysis},\n edition={2},\n publisher={Cambridge University Press, Cambridge},\n date={2013},\n pages={xviii+643},\n isbn={978-0-521-54823-6},\n review={\\MR{2978290}},\n}\n\n\n\\bib{Huang:firstgap}{article}{\n author={Huang, Xiaojun},\n title={On a linearity problem for proper holomorphic maps between balls\n in complex spaces of different dimensions},\n journal={J. Differential Geom.},\n volume={51},\n date={1999},\n number={1},\n pages={13--33},\n issn={0022-040X},\n review={\\MR{1703603}},\n}\n\n\\bib{HJX06}{article}{\n author={Huang, Xiaojun},\n author={Ji, Shanyu},\n author={Xu, Dekang},\n title={A new gap phenomenon for proper holomorphic mappings from $B^n$\n into $B^N$},\n journal={Math. Res. Lett.},\n volume={13},\n date={2006},\n number={4},\n pages={515--529},\n issn={1073-2780},\n review={\\MR{2250487}},\n doi={10.4310\/MRL.2006.v13.n4.a2},\n}\n\n\\bib{HJY14}{article}{\n author={Huang, Xiaojun},\n author={Ji, Shanyu},\n author={Yin, Wanke},\n title={On the third gap for proper holomorphic maps between balls},\n journal={Math. Ann.},\n volume={358},\n date={2014},\n number={1-2},\n pages={115--142},\n issn={0025-5831},\n review={\\MR{3157993}},\n doi={10.1007\/s00208-013-0952-z},\n}\n\n\\bib{Lebl:normal}{article}{\n author={Lebl, Ji\\v{r}\\'{\\i}},\n title={Normal forms, Hermitian operators, and CR maps of spheres and\n hyperquadrics},\n journal={Michigan Math. J.},\n volume={60},\n date={2011},\n number={3},\n pages={603--628},\n issn={0026-2285},\n review={\\MR{2861091}},\n doi={10.1307\/mmj\/1320763051},\n}\n\n\n\\end{biblist}\n\\end{bibdiv}\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{intro}\nIn Receding Horizon Control (RHC), the control action, at each time $t$ in $[0,\\infty)$,\nis derived from the solution of an optimal control\nproblem defined over a finite future horizon $[t,\\, t+T]$.\nThe RHC strategy establishes a feedback law which, under\ncertain conditions, can ensure asymptotic stability of the controlled system.\nThis control strategy has been successfully developed over\nthe last twenty years for systems described by deterministic equations.\nIn this context RHC is also well known as Model Predictive Control (MPC)\nand has proven to be very successful in dealing with non-linear and constrained\nsystems, see e.g.~\\cite{Mayne-et-al-00,Maciejowski-02,Magni-Scattolini-04}.\nThe extension of RHC from deterministic to stochastic systems is the objective of current research.\nRHC schemes for the control of discrete-time\nstochastic systems have been proposed recently in\n\\cite{Primbs-et-Sung-09,Cannon-et-al-09,Chatterjee-et-al-11,Hokayem-et-al-12}.\n\nIn this note, we discuss RHC for systems described by continuous-time non-linear\nstochastic differential equations (SDEs). To the extent of our knowledge,\nthe RHC strategy has not yet been considered in this context.\nIn order to study the stability of RHC for continuous-time SDEs,\nwe formulate conditions under which the value function of the associated\nfinite-time optimal control problem can be used as Lyapunov function\nfor the RHC scheme.\nThis is a well established approach for studying the stability\nof RHC schemes, which here is extended using Lyapunov criteria for\nstochastic dynamical systems \\cite{Kushner-65}.\nWe illustrate this contribution with a simple example of\nan optimal investment problem.\nOptimal investments problems are well suited to be tackled by stochastic\ncontrol methods, see e.g.~\\cite{Primbs-09,Pola-et-Pola-09}.\nIn our example, we design an investment strategy to repay a debt.\nHaving negative wealth due to an initial debt, the investor has the option\nto increase his\/her current debt in order to buy a risky asset.\nThe asymptotic stability of the adopted RHC scheme\nguarantees that the wealth of the investor tends to zero,\nso that the initial debt is eventually repaid.\n\n\n\\section{Problem statement}\n\nLet $(\\Omega,\\mathcal{F},\\mathbb{P})$ be a complete probability space equipped with the natural filtration $(\\mathcal{F}_{t})_{t\\ge0}$\ngenerated by a standard Wiener process $W:[0,\\infty)\\times\\Omega\\to\\mathbb{R}^{d}$ on it.\nWe consider a controlled time-homogeneous SDE for a process $X:[0,\\infty)\\times\\Omega\\to\\mathbb{R}^{n}$,\n\\begin{eqnarray}\\label{eq:sde1}\ndX^{0,x_{0},u}_{t}&=&b(X^{0,x_{0},u}_{t},u_{t})dt+\\sigma(X^{0,x_{0},u}_{t},u_{t})dW_{t},\\\\\nX^{0,x_{0},u}_{0}&=&x_{0},\\nonumber\n\\end{eqnarray}\nwhere $x_{0}\\in\\mathbb{R}^{n}$; $b:\\mathbb{R}^{n}\\times\\mathbb{R}^{m}\\to\\mathbb{R}^{n}$ and $\\sigma:\\mathbb{R}^{n}\\times\\mathbb{R}^{m}\\to\\mathbb{R}^{n\\times d}$\nare measurable functions and satisfy\n$$\n|b(x,u)|+|\\sigma(x,u)|\\le C(1+|x|),\\forall(x,u)\\in\\mathbb{R}^{n}\\times U,\\mbox{(linear growth)},\n$$\nand\n$$\n|b(x,u)-b(y,u)|+|\\sigma(x,u)-\\sigma(y,u)|\\le C|x-y|,\\forall(x,y,u)\\in\\mathbb{R}^{n}\\times\\mathbb{R}^{n}\\times U,\\mbox{(Lipschitz)},\n$$\nfor some constant $C$; and $u_{(\\cdot)}$ is an admissible control process\n$$u_{(\\cdot)}\\in\\mathcal{U}:=\\left\\{u:[0,\\infty)\\times\\Omega\\to U:\\mbox{ progressively measurable and }\\mathbb{E}\\int_{0}^{\\infty}|u_{t}(\\omega)|^{2}dt<\\infty\\right\\},$$\nwith the set $U\\subset\\mathbb{R}^{m}$ compact.\nHere the superscripts of $X^{0,x_{0},u}$ mean that the initial value of the process at time $0$ is $x_{0}$\nand the involved control process is $u_{(\\cdot)}$.\nIn this paper we are concerned with the conditions under which there exists a control process that drives the stochastic system $X$\nto the origin $0\\in\\mathbb{R}^{n}$ and guarantees asymptotic stability of the controlled process.\nHere, the following definition of stability is adopted \\cite{Kushner-65}:\n\\begin{defn}\\label{def:stab}\\upshape\nGiven a stochastic continuous-time process $X: \\mathbb{R}_{+}\\times\\Omega\\to\\mathbb{R}^{n}$, where $\\mathbb{R}_{+}:=[0,\\infty)$, with $X_{0}=x_{0}\\in\\mathbb{R}^{n}$\n\\begin{itemize}\n\\item[(S1)] The origin is stable almost surely\nif and only if, for any $\\rho>0$, $\\epsilon>0$, there is a $\\delta(\\rho,\\epsilon)>0$ such that,\nif $|x_{0}|\\le\\delta(\\rho,\\epsilon)$,\n $$\n \\mathbb{P}\\left[\\sup_{t\\in\\mathbb{R}_{+}}|X_{t}|\\ge\\epsilon\\right]\\le\\rho.\n $$\n\\item[(S1')]An equivalent definition to (S1) is:\nLet $h(\\cdot):\\mathbb{R}_{+}\\to\\mathbb{R}_{+}$ be a scalar-valued, nondecreasing,\nand continuous function of $|x|$. Let $h(0)=0$, $h(r)>0$ for $r\\ne0$.\nThen the origin is stable almost surely if and only if, for any\n$\\rho>0$, $\\lambda>0$, there is a $\\delta(\\rho,\\lambda)>0$ such that,\nfor $|x_{0}|\\le\\delta(\\rho,\\lambda)$,\n $$\n \\mathbb{P}\\left[\\sup_{t\\in\\mathbb{R}_{+}}h(|X_{t}|)\\ge\\lambda\\right]\\le\\rho.\n $$\n\\item[(S2)] The origin is asymptotically stable almost surely if and only if\nit is stable a.s., and $X_{t}\\to0$ a.s. for all $x_{0}$ in some neighborhood $R$ of the origin.\nIf $R=\\mathbb{R}^{n}$ then we add `in the large'.\n\\end{itemize}\n\\end{defn}\n\n\n\\section{Main results}\\label{sec:main}\n\nLet $T>0$. As a preliminary step we consider the SDE\nfor $X:[t,T]\\times\\Omega\\to\\mathbb{R}^{n}$ starting from the point\n$x\\in\\mathbb{R}^{n}$ at the time $t\\in[0,T]$\n\\begin{eqnarray}\\label{eq:sde2}\ndX^{t,x,u}_{s}&=&b(X^{t,x,u}_{s},u_{s})ds+\\sigma(X^{t,x,u}_{s},u_{s})dW_{s},\\\\\nX^{t,x,u}_{t}&=&x.\\nonumber\n\\end{eqnarray}\nLet $f:\\mathbb{R}^{n}\\times\\mathbb{R}^{m}\\to\\mathbb{R}_{+}$ and $g:\\mathbb{R}^{n}\\to\\mathbb{R}_{+}$ be measurable nonnegative functions.\nNow we consider the problem of minimizing the following cost functional, $\\forall(t,x)\\in[0,T]\\times\\mathbb{R}^{n}$,\n\\begin{equation}\\label{eq:cost}\nJ[t,x;T;u_{(\\cdot)}]:=\\mathbb{E}\\left[\\int_{t}^{T}f(X^{t,x,u}_{s},u_{s})ds+g(X^{t,x,u}_{T})\\right]\n\\end{equation}\nover the set $\\mathcal{U}$ of admissible control processes.\nWe define the value function as\n\\begin{equation}\\label{eq:valuefunc}\nv(t,x;T):=\\inf_{u_{(\\cdot)}\\in\\mathcal{U}}J[t,x;T;u_{\\cdot}]=\\inf_{u_{(\\cdot)}\\in\\mathcal{U}}\\mathbb{E}\\left[\\int_{t}^{T}f(X^{t,x,u}_{s},u_{s})ds+g(X^{t,x,u}_{T})\\right].\n\\end{equation}\nand denote $u^*_s(t,x;T)$, $t\\leq s\\leq T$, the optimal control process if it exists.\nIn particular, when $t=0$ we denote $V(x;T):=v(0,x;T)$.\n\nStandard stochastic optimal control theories (see, for instance, \\cite{Fleming-et-Rishel-75,Fleming-et-Soner-06})\nabout the controlled SDE \\eqref{eq:sde1} tell us that\nthe Hamilton-Jacobi-Bellman (HJB) equation for the value function\n$v(\\cdot,\\cdot;T)$ is, $\\forall(t,x)\\in[0,T]\\times\\mathbb{R}^{n}$,\n\\begin{eqnarray}\\label{eq:hjb1}\n-\\partial_{t}v(t,x;T)&=&\\inf_{u\\in U}\\left[\\frac{1}{2}\\mbox{tr}[\\sigma\\sigma^{*}(x,u)D^{2}v(t,x;T)]+\\langle b(x,u),Dv(t,x;T)\\rangle+f(x,u)\\right],\\\\\nv(T,x;T)&=&g(x).\\nonumber\n\\end{eqnarray}\nHereafter we use the notations\n$$\\partial_{t}v:=\\frac{\\partial v}{\\partial t},\\\nDv=\\left(\\begin{array}{c}\n\\frac{\\partial v}{\\partial x_{1}}\\\\\\vdots\\\\\\frac{\\partial v}{\\partial x_{n}}\n\\end{array}\\right),\n\\mbox{ and }\nD^{2}v=\n\\left(\\begin{array}{cccc}\n\\frac{\\partial^{2}v}{\\partial x_{1}^{2}}&\\frac{\\partial^{2}v}{\\partial x_{1}\\partial x_{2}}\n&\\cdots&\n\\frac{\\partial^{2}v}{\\partial x_{1}\\partial x_{n}}\\\\\n\\vdots&\\vdots&\\cdots&\\vdots\\\\\n\\frac{\\partial^{2}v}{\\partial x_{n}\\partial x_{1}}&\\frac{\\partial^{2}v}{\\partial x_{n}\\partial x_{2}}\n&\\cdots&\n\\frac{\\partial^{2}v}{\\partial x_{n}^{2}}\n\\end{array}\\right).\n$$\nSuppose this HJB equation has a solution and that the infimum in the equation is attained by $\\tilde{u}(t,x;T)$\nfor every $(t,x)\\in[0,T]\\times\\mathbb{R}^{n}$, i.e.,\n$$\n-\\partial_{t}v(t,x;T)=\\frac{1}{2}\\mbox{tr}[\\sigma\\sigma^{*}(x,\\tilde{u}(t,x;T))D^{2}v(t,x;T)]+\\langle b(x,\\tilde{u}(t,x;T)),Dv(t,x;T)\\rangle+f(x,\\tilde{u}(t,x;T)),\n$$\nthen we construct the optimal control process for the SDE \\eqref{eq:sde2} as\n\\begin{equation}\\label{eq:ustar}\nu^{*}_{s}(t,x;T):=\\tilde{u}(s,X^{t,x,\\tilde{u}}_{s};T),\\ t\\le s\\le T.\n\\end{equation}\nAs a consequence, the value function turns out to be\n$$\nv(t,x;T)=\\mathbb{E}\\left[\\int_{t}^{T}f(X^{t,x,u^{*}}_{s},u^{*}_{s})ds+g(X^{t,x,u^{*}}_{T})\\right]=\\mathbb{E}\\left[\\int_{t}^{T}f(X^{t,x,\\tilde{u}}_{s},\\tilde{u}(s,X^{t,x,\\tilde{u}}_{s};T))ds+g(X^{t,x,\\tilde{u}}_{T})\\right].\n$$\nNow, for all the states $x\\in\\mathbb{R}^{n}$ all the time, we apply the specifically designed feedback law\n\\begin{equation}\\label{eq:RHC}\nu^{c}(x;T):=\\tilde{u}(0,x;T)\n\\end{equation}\nto the stochastic system \\eqref{eq:sde1}.\nIn other words, for the state $X^{0,x_{0},u^{c}}_{t}$, at any time $t\\ge0$,\nwe apply only the initial optimal control\n$$\nu^{c}(X^{0,x_{0},u^{c}}_{t};T)=\\tilde{u}(0,X^{0,x_{0},u^{c}}_{t};T)=u^{*}_{0}(0,X^{0,x_{0},u^{c}}_{t};T)\n$$\nto the system.\nIn particular, for $t=0$,\n$u^c(X^{0,x_{0},u^{c}}_{0};T)=u^{c}(x_{0};T)=\\tilde{u}(0,x_{0};T)$.\nWe call $u^{c}(\\cdot;T)$ the\ncontinuous receding horizon control\nprocess with the receding horizon $T$ \\mbox{for the controlled SDE \\eqref{eq:sde1}.}\n\nWhen $t=0$, using the HJB equation and \\eqref{eq:RHC}, we obtain\n$$\n-\\partial_{t}v(t,x;T)|_{t=0}-f(x,u^{c}(x;T))=\\frac{1}{2}\\mbox{tr}[\\sigma\\sigma^{*}(x,u^{c}(x;T))D^{2}V(x;T)]+\\langle b(x,u^{c}(x;T)),DV(x;T)\\rangle.\n$$\nLet us denote\n\\begin{equation}\\label{eq:phi}\n\\phi(x;T):=\\partial_{t}v(t,x;T)|_{t=0}+f(x,u^{c}(x;T))\n\\end{equation}\nfor all $x\\in\\mathbb{R}^{n}$.\nLet us assume that\n\\begin{equation}\\label{A1}\nV\\in C^{2,1}(\\mathbb{R}^{n}\\times\\mathbb{R}_{+},\\mathbb{R}),\\tag{A1}\n\\end{equation}\ni.e.~the set of functions $\\psi:\\mathbb{R}^{n}\\times\\mathbb{R}_{+}\\to\\mathbb{R},\\ (x,t)\\mapsto\\psi(x,t)$\nthat are twice continuously differentiable in $x$ and once continuously differentiable in $t$; and that\n\\begin{equation}\\label{A2}\n\\phi(x;T)>0,\\ \\forall x\\in\\mathbb{R}^{n}\\setminus\\{0\\},\\mbox{ and }\\phi(0;T)=0.\\tag{A2}\n\\end{equation}\nIn particular, note that $\\phi(0;T)=0$ can be achieved if we suppose that\n\\begin{equation}\\label{A2.1}\nb(0,0)=0, \\mbox{ and } \\sigma(0,0)=0;\\tag{A2.1}\n\\end{equation}\nand that\n\\begin{equation}\\label{A2.2}\nf(0,0)=0,\\, f(x,u)>0,\\, \\forall x\\in\\mathbb{R}^{n}\\setminus\\{0\\},\\ \\forall u\\in U,\n\\,\\mbox{ and }\\,\ng(0)=0,\\, g(x)>0,\\,\\forall x\\in\\mathbb{R}^{n}\\setminus\\{0\\}.\\tag{A2.2}\n\\end{equation}\nWe now state one of our main results and then discuss Assumptions (\\ref{A1}-\\ref{A2})\nin more detail.\n\\begin{prop}[Convergence]\\label{th:converge}\\upshape\nSuppose $b(\\cdot,\\cdot)$ and $\\sigma(\\cdot,\\cdot)$ are of linear growth and Lipschitz.\nSuppose the HJB equation \\eqref{eq:hjb1} has a unique classical solution.\nUnder the assumptions \\eqref{A1}, and \\eqref{A2},\nwe have that almost all the trajectories of the stochastic system \\eqref{eq:sde1}\ndriven by the continuous receding horizon control $u^{c}(\\cdot;T)$ defined in \\eqref{eq:RHC}\nconverge to the origin.\n\\end{prop}\n\n\n\\begin{prf}\nTemporarily we denote $X_{t}:=X^{0,x_{0},u^{c}}_{t}$ for simplicity.\nBy It\\^{o}'s formula and the Hamilton-Jacobi-Bellman equation we have\n\\begin{eqnarray*}\ndV(X_{t};T)&=&\\langle DV(X_{t};T),dX_{t}\\rangle+\\frac{1}{2}\\mbox{tr}[\\sigma\\sigma^{*}(X_{t},u^{c}(X_{t};T))D^{2}V(X_{t};T)]dt\\\\\n&=&\\langle DV(X_{t};T),b(X_{t},u^{c}(X_{t};T))\\rangle dt+\\langle DV(X_{t};T),\\sigma(X_{t},u^{c}(X_{t};T))dW_{t}\\rangle\\\\\n&& +\\,\\,\\frac{1}{2}\\mbox{tr}[\\sigma\\sigma^{*}(X_{t},u^{c}(X_{t};T))D^{2}V(X_{t};T)]dt\\\\\n&=&-\\phi(X_{t};T)dt+\\langle DV(X_{t};T),\\sigma(X_{t},u^{c}(X_{t};T))dW_{t}\\rangle.\n\\end{eqnarray*}\nTherefore by the assumption \\eqref{A2} we get that, for $0\\le s\\le t<\\infty$,\n\\begin{equation}\\label{V-supermart}\n\\mathbb{E}[V(X_{t};T)|\\mathcal{F}_{s}]-V(X_{s};T)=-\\mathbb{E}\\left[\\int_{s}^{t}\\phi(X_{\\xi};T)d\\xi\\Big|\\mathcal{F}_{s}\\right]\\le0.\n\\end{equation}\nIn particular when $s=0$, $\\mathbb{E} \\left[V(X_{t};T)\\right]\\le V(X_{0};T)=V(x_{0};T)<\\infty$.\nHence, $\\{V(X_{t};T),\\mathcal{F}_{t}\\}_{0\\le t<\\infty}$ is a nonnegative supermartingale.\nLet us recall the supermartingale convergence theorem, see, for instance, \\cite[pag.~18]{Karatzas-et-Shreve-06}:\nSuppose $\\{M_{t},\\mathcal{F}_{t}\\}_{0\\le t<\\infty}$ is a right-continuous, nonnegative supermartingale.\nThen $M_{\\infty}(\\omega):=\\lim_{t\\to\\infty}M_{t}(\\omega)$ exists for $\\mathbb{P}$-a.e. $\\omega\\in\\Omega$,\nand $\\{M_{t},\\mathcal{F}_{t}\\}_{0\\le t\\le\\infty}$ is a supermartingale.\nHence, there exists a random variable $V_{\\infty}$, integrable, such that\n$V(X_{t};T)\\to V_{\\infty},\\mbox{ a.s.}$.\nNow, equation \\eqref{V-supermart} implies particularly that\n\\begin{equation}\n\\mathbb{E} \\left[V_{\\infty}\\right]-V(x_{0};T)=-\\int_{0}^{\\infty}\\mathbb{E}\\left[\\phi(X_{t};T)\\right]dt,\n\\end{equation}\nwhich in turn gives\n$\\lim_{t\\to\\infty}\\mathbb{E}\\left[\\phi(X_{t};T)\\right]=0$.\nTherefore we obtain that\n$$\n\\lim_{t\\to\\infty}\\phi(X_{t};T)=0,\\mbox{ a.s.,}\n$$\nbecause $\\phi\\ge0$. Denote the set $\\Phi:=\\{x\\in\\mathbb{R}^{n}:\\phi(x;T)=0\\}$ and from the assumption \\eqref{A2}\nwe know that $\\Phi=\\{0\\}$ and it is closed. Thus by the continuity of $\\phi$ we learn that almost all\nthe trajectories of the process $X_{t}$ converge to the origin in $\\mathbb{R}^{n}$.\\smallskip\n\\end{prf}\n\n\nIn \\eqref{A1}, continuous twice differentiability of $V$ is required for\nthe applicability of It\\^{o}'s formula.\nIn order to illustrate Assumption (\\ref{A2}) note that, since \\eqref{eq:sde2}\nis time-homogeneous, we have\n$$\n\\partial_{t}v(t,x;T)|_{t=0} = - \\partial_{H}v(0,x;H)|_{H=T} = - \\partial_{H} V(x;H)|_{H=T}.\n$$\nHence, $\\phi(x;T)$ can be equivalently expressed as:\n\\begin{equation}\n\\phi(x;T) = - \\partial_{H} V(x;H)|_{H=T} + f(x,u^{c}(x;T)).\n\\end{equation}\nNote that, here, $\\partial_{H} V(x;H)|_{H=T}$ is the rate at which the optimal cost\nincreases with increments in the horizon $T$.\nNote that $\\phi(x;T)>0$ is implied by\n\\begin{equation}\\label{eq:neg}\n\\partial_{H} V(x;H)|_{H=T} \\leq 0\\, .\n\\end{equation}\nHere, we recover the condition of monotonic decrease of\nthe value function with the length of the horizon, which is\na well-studied condition for the stability of receding horizon\ncontrol schemes in a deterministic setting, see e.g.~\\cite{Magni-Scattolini-04,Chen-Shaw-82,Bertsekas-05}.\nFinally, note that Assumption (\\ref{A2.1}) requires that the\nstate, at the origin, is not affected by the noise.\nThis is an unavoidable assumption for obtaining\nasymptotic stability in the sense of Definition \\ref{def:stab}.\nIf this assumption is not met, then one has to resort to other notions\nof stability such as, for example,\\mbox{ mean square boundeness,\nsee e.g.~\\cite{Chatterjee-et-al-11,Hokayem-et-al-12,Chatterjee-et-al-12}.}\n\n\nUnder additional assumptions, asymptotic stability according to Definition \\ref{def:stab} can be obtained.\nLet us assume in addition that for any $\\epsilon>0$, there exists a $\\delta(\\epsilon)>0$ such that\n\\begin{equation}\\label{A3}\nV(x;T)<\\epsilon \\quad\\mbox{if } |x|\\le\\delta(\\epsilon); \\tag{A3}\n\\end{equation}\nand that there exists\na continuous, nondecreasing function $h:\\mathbb{R}_{+}\\to\\mathbb{R}_{+}$, satisfying\n$$\nh(0)=0,\\ h(r)>0, \\forall r\\ne0,\n$$\nsuch that\n\\begin{equation}\\label{A4}\nV(x;T)\\ge h(|x|),\\ \\forall x\\in\\mathbb{R}^{n}.\\tag{A4}\n\\end{equation}\n\\begin{cor}[Asymptotic stability]\\upshape\\label{th:corr}\nSuppose $b(\\cdot,\\cdot)$ and $\\sigma(\\cdot,\\cdot)$ are of linear growth and Lipschitz.\nUnder the assumptions \\eqref{A1}, \\eqref{A2}, \\eqref{A3}, and \\eqref{A4},\nwe have that the origin in $\\mathbb{R}^{n}$ is asymptotically stable.\n\\end{cor}\n\\begin{prf}\nLet us recall the supermartingale inequality, see, \\cite[pag.~13]{Karatzas-et-Shreve-06}:\nSuppose $\\{M_{t},\\mathcal{F}_{t}\\}_{0\\le t<\\infty}$ is a supermartingale whose every path is right-continuous.\nLet $\\lambda>0$ and $[a,b]$ be a subinterval of $[0,\\infty)$. Then, we have\n$$\n\\mathbb{P}\\left[\\sup_{a\\le t\\le b}M_{t}\\ge\\lambda\\right]\\le\\frac{\\mathbb{E}[M^{+}_{a}]}{\\lambda}\\, .\n$$\nHence, for any $\\lambda>0$, we have\n$$\n\\mathbb{P}\\left[\\sup_{0\\le t<\\infty}V(X_{t};T)\\ge\\lambda\\right]\\le\\frac{\\mathbb{E}[V(x_{0};T)]}{\\lambda}=\\frac{V(x_{0};T)}{\\lambda}\\, .\n$$\nUsing assumption \\eqref{A3}, we obtain that for any $\\lambda>0$, $\\rho>0$, there\nexists $\\delta(\\rho,\\lambda)>0$ such that\n$$\n\\mathbb{P}\\left[\\sup_{0\\le t<\\infty}V(X_{t};T)\\ge\\lambda\\right]\\le\\rho \\quad\\mbox{if } |x_{0}|<\\delta(\\rho,\\lambda).\n$$\nThen, by assumption \\eqref{A4}, we immediately obtain\n$$\\mathbb{P}\\left[\\sup_{0\\le t<\\infty} h(X_{t})\\ge\\lambda\\right]\\le\\mathbb{P}\\left[\\sup_{0\\le t<\\infty}V(X_{t};T)\\ge\\lambda\\right]\\le\\rho,$$\nwhich entails the asymptotic stability of the origin in $\\mathbb{R}^{n}$.\n\\end{prf}\n\nIn the following section, we present a simple illustrative example in which\nAssumptions (\\ref{A1}-\\ref{A4}) can be verified explicitly.\n\n\n\n\\section{Example: repayment of a debt}\\label{sec:example}\n\nIn this example we consider a variant of the\nMerton's portfolio problem, see e.g.~\\cite{Fleming-et-Rishel-75,Fleming-et-Soner-06}.\nSuppose in a complete financial market there are only two assets, one asset being risk free such as, for instance,\na bank deposit or a bond, and the other one being a risky asset such as, for instance, a stock.\nThe assets obey the following price process, $t\\ge0$,\n\\begin{eqnarray}\\label{eq:exprices}\ndP^{1}_{t}&=&rP^{1}_{t}dt,\\\\\ndP^{2}_{t}&=&bP^{2}_{t}dt+\\sigma P^{2}_{t}dW_{t},\\nonumber\n\\end{eqnarray}\nwhere $b>r>0$, and $\\sigma\\ne0$ are given constants. Here $r$ is the risk free interest rate for the bank deposit,\n$b$ is the drift, or average, rate of the stock's return, and $\\sigma$ is the volatility of the stock's return.\nSuppose an agent's total wealth at time $t$ is $X_{t}$, comprising the risk\nfree part $\\Pi^{1}_{t}P^{1}_{t}$ and the risky part $\\Pi^{2}_{t}P^{2}_{t}$,\nwhere $\\Pi^{1}_{t}$ and $\\Pi^{2}_{t}$ are the quantity of the assets, respectively.\nThe control variable $u_t$ is the portion of the total wealth $X_{t}$ invested by the agent on\nthe risky asset at time $t$. In a continuous-time setting, it is assumed that the allocation\nof wealth takes place instantaneously. Hence, for $t\\ge0$, we have:\n\\begin{eqnarray}\\label{eq:exassets}\nX_{t}&=&\\Pi^{1}_{t}P^{1}_{t}+\\Pi^{2}_{t}P^{2}_{t},\\nonumber \\\\\n\\Pi^{1}_{t}P^{1}_{t}&=&(1-u_{t})X_{t},\\\\\n\\Pi^{2}_{t}P^{2}_{t}&=& u_{t}X_{t}.\\nonumber\n\\end{eqnarray}\nWe assume that the investment obeys the self financing condition.\nThat is, starting from an initial wealth $x_{0}$, the agent can only sell or buy these\ntwo assets but is not allowed to borrow money from outside or consume his or her wealth.\nWritten down as a differential equation, this is $P^{1}_{t}d\\Pi^{1}_{t}+P^{2}_{t}d\\Pi^{2}_{t}=0$.\nUsing \\eqref{eq:exprices} and \\eqref{eq:exassets} we obtain\n\\begin{eqnarray*}\ndX_{t}&=&\\Pi^{1}_{t}dP^{1}_{t}+\\Pi^{2}_{t}dP^{2}_{t}+P^{1}_{t}d\\Pi^{1}_{t}+P^{2}_{t}d\\Pi^{2}_{t}\\\\\n&=&\\Pi^{1}_{t}dP^{1}_{t}+\\Pi^{2}_{t}dP^{2}_{t}\\\\\n&=&[r+(b-r)u_{t}]X_{t}dt+\\sigma u_{t}X_{t}dW_{t}.\n\\end{eqnarray*}\nTherefore, the wealth satisfies the stochastic differential equation\n\\begin{eqnarray}\\label{eq:exwealth}\ndX^{0,x_{0},u}_{t}&=&[r+(b-r)u_{t}]X^{0,x_{0},u}_{t}dt+\\sigma u_{t}X^{0,x_{0},u}_{t}dW_{t},\\\\\nX^{0,x_{0},u}_{0}&=&x_{0},\\nonumber\n\\end{eqnarray}\nwhich is called the wealth process.\n\nIn this example, we assume that the agent's initial wealth is negative and his or\nher aim is to repay any debt eventually.\nHence, we have $x_{0}<0$ and investigate the asymptotic stability of the origin $0\\in\\mathbb{R}$.\nIn this case, the wealth process will always be negative until the time it reaches zero.\nHowever, note that $X_{t}<0$ does not necessarily mean that the\nagent has only debt without any money to invest or stock to sell.\nIf negative values of $u_{t}$ in \\eqref{eq:exassets} are allowed then\nthe investor is able to increase his or her debt in order to buy stocks.\nLet us explain the financial meanings of all possible values\nof $u_{t}$ when $X_{t}<0$:\n\\begin{itemize}\n\\item If $u_t=0$, then $\\Pi^{1}_{t}P^{1}_{t}= X_{t}$ and $\\Pi^{2}_{t}P^{2}_{t}= 0$.\nIn this case, the investor just owns a debt with the bank equal to his or her negative wealth.\n\\item If $u_t < 0$, then $\\Pi^{1}_{t}P^{1}_{t} < X_{t}$ and $\\Pi^{2}_{t}P^{2}_{t} > 0$.\nIn this case, the investor is borrowing additional funds from the bank and is using\nit to buy stocks.\nThe investor owns a debt with the bank equal to $(1-u_t)X_t$ and\na positive quantity of stocks whose value is $u_t X_{t}$.\n\\item If $u_t > 0$, then $\\Pi^{1}_{t}P^{1}_{t} > X_{t}$ and $\\Pi^{2}_{t}P^{2}_{t} < 0$.\nIn this case, the investor is borrowing stocks (short selling) and is using this additional\nwealth to reduce his or her debt with the bank. However, in general, this is not desirable\nbecause a debt in stocks is more risky than a debt with the bank.\n\\end{itemize}\nNote that if $\\Pi^{1}_{t}P^{1}_{t} <0$ (i.e. when the investor has a debt with the bank)\nthen $r$ is the rate at which interests are payed to the the bank when owing debt.\nHere, in order to simplify the exposition of the problem, we assume that $r$ is the same\nwhether $\\Pi^{1}_{t}P^{1}_{t} <0$ or $\\Pi^{1}_{t}P^{1}_{t} >0$.\nHowever, it will be shown that this issue is immaterial. In fact, the derived control process\nwill be constantly negative until the wealth reaches zero. In turn, this means that\n$\\Pi^{1}_{t}P^{1}_{t}<0$ and, therefore, $r$ will not change meaning throughout.\nFinally, we consider the constraints $u_{t}\\in[c_{1},c_{2}]$ with $c_{1}<0$ and $c_{2}\\geq 0$.\nThe constraint $c_{1}$ means that the investor is not allowed to increase the\ndebt with the bank by more than $(1+|c_{1}|)$ times his or her current (negative) wealth.\nSimilarly, $c_{2}$ is a constrain on borrowing stocks. If short selling is not allowed\nthen $c_{2}=0$.\n\n\\subsection{Running cost}\\label{sec:exrun}\n\nWe consider cost function \\eqref{eq:cost} with\n\\begin{equation}\\label{eq:excost}\nf(x,u):=\\begin{cases}\n(-x)^{\\beta},&x\\le0,\\\\\n0,&x>0,\n\\end{cases}\\quad\\mbox{ and }\\quad\ng(x):=0,\\,\\forall x,\n\\end{equation}\nwith $\\beta>2$ being a given constant.\nIn this case, the value function $v(t,x;T)$\nsatisfies the HJB equation, $(t,x)\\in[0,T]\\times(-\\infty,0]$,\n\\begin{eqnarray}\\label{eq:ex1HJB}\n-\\partial_{t}v(t,x;T)&=&\\inf_{u\\in[c_{1},c_{2}]}\\left[\\frac{1}{2}(\\sigma ux)^{2}D^{2}v(t,x;T)+(r+(b-r)u)xDv(t,x;T)+(-x)^{\\beta}\\right],\\\\\nv(T,x;T)&=&0.\\nonumber\n\\end{eqnarray}\nIf $D^{2}v(t,x;T)>0$ then a necessary condition for $\\tilde{u}$ to be a minimiser is\n$$\n\\sigma^{2}\\tilde{u}x^{2}D^{2}v(t,x;T)+(b-r)xDv(t,x;T)=0,\n$$\nthat is,\n\\begin{equation}\\label{eq:ex1utilde}\n\\tilde{u}=-\\frac{(b-r)Dv(t,x;T)}{\\sigma^{2}xD^{2}v(t,x;T)}.\n\\end{equation}\nIn order to solve the HJB equation \\eqref{eq:ex1HJB}, we try to find a value function in the form\n\\begin{equation}\\label{eq:ex1w}\nv(t,x;T)=\\begin{cases}\n(-x)^{\\beta}w(t),&x\\le0,\\\\\n0,&x>0,\n\\end{cases}\n\\end{equation}\nwith $w$ to be determined.\nFor $x\\le0$, by substituting \\eqref{eq:ex1w} into \\eqref{eq:ex1utilde},\nwe obtain\n\\begin{equation}\\label{eq:utilde}\n\\tilde{u}=-\\frac{(b-r)(-\\beta)(-x)^{\\beta-1}}{\\sigma^{2}x\\beta(\\beta-1)(-x)^{\\beta-2}}=-\\frac{b-r}{(\\beta-1)\\sigma^{2}}.\n\\end{equation}\nNote that the so-obtained $\\tilde{u}$ has the same expression as\nin the classical Merton's problem (although here we assumed $\\beta>2$ instead of $\\beta<1$),\nsee e.g. \\cite[pag.~160-161]{Fleming-et-Rishel-75}, or \\cite[pag.~168-169]{Fleming-et-Soner-06}.\nUsing \\eqref{eq:ex1w} and \\eqref{eq:utilde},\nthe HJB equation becomes\n\\begin{equation*}\n-(-x)^{\\beta}w'(t)=\\frac{\\beta(b-r)^{2}}{2(\\beta-1)\\sigma^{2}}x^{2}(-x)^{\\beta-2}w(t)-\\left(\\beta r-\\frac{\\beta(b-r)^{2}}{(\\beta-1)\\sigma^{2}}\\right)(-x)^{\\beta-1}xw(t)+(-x)^{\\beta}\n\\end{equation*}\nthat is,\n\\begin{equation}\\label{ex1:wdiff}\nw'(t)=\\left[\\frac{\\beta(b-r)^{2}}{2(\\beta-1)\\sigma^{2}}-\\beta r\\right]w(t)-1\n\\end{equation}\nand we obtain that the solution is\n\\begin{equation}\\label{eq:ex1wsol}\nw(t)=\\frac{1}{\\eta}\\left(1-e^{\\eta(t-T)}\\right),\n\\end{equation}\nwhere\n\\begin{equation}\\label{eq:ex1eta}\n\\eta:=\\frac{\\beta(b-r)^{2}}{2(\\beta-1)\\sigma^{2}}-\\beta r\\, .\n\\end{equation}\nHence, the value function \\eqref{eq:ex1w} is actually given by\n\\begin{eqnarray}\\label{eq:ex1value}\nv(t,x;T)=\\begin{cases}\n(-x)^{\\beta}\\frac{1}{\\eta}\\left(1-e^{\\eta(t-T)}\\right),&x\\le0,\\\\\n0,&x>0.\n\\end{cases}\n\\end{eqnarray}\nHere we assume $\\eta\\ne0$.\nNote that whether $\\eta<0$ or $\\eta>0$ it always holds that $w(t)>0$.\nThus $D^{2}v=\\beta(\\beta-1)(-x)^{\\beta-2}w(t)>0$ as we expected.\n(However, in the next subsection, we will narrow the requirement to be $\\eta>0$.)\n\nEventually, we obtain that the corresponding receding horizon\ncontrol process is\n\\begin{equation}\\label{eq:ex1uc}\nu^{c}(X_{t};T)=\\tilde{u}(0,X_{t};T)=-\\frac{b-r}{(\\beta-1)\\sigma^{2}}.\n\\end{equation}\nThe stochastic system \\eqref{eq:exwealth} under the receding horizon control process\n\\eqref{eq:ex1uc} becomes\n\\begin{eqnarray}\ndX^{0,x_{0},u}_{t}&=&\\left[r-\\frac{(b-r)^{2}}{(\\beta-1)\\sigma^{2}}\\right]X^{0,x_{0},u}_{t}dt-\\frac{b-r}{(\\beta-1)\\sigma}X^{0,x_{0},u}_{t}dW_{t},\\\\\nX^{0,x_{0},u}_{0}&=&x_{0}.\\nonumber\n\\end{eqnarray}\nIn this case the solution turns out to be a geometric Brownian motion that can be\nwritten down explicitly (see e.g.\\cite[pag.~349-50]{Karatzas-et-Shreve-06})\n\\begin{eqnarray}\\label{eq:ex1sol}\nX^{0,x_{0},u}_{t}&=&\nx_{0}\\exp\\left\\{\\left[r-\\frac{(2\\beta-1)(b-r)^{2}}{2(\\beta-1)^{2}\\sigma^{2}}\\right]t-\\frac{b-r}{(\\beta-1)\\sigma}W_{t}\\right\\}.\n\\end{eqnarray}\nThe asymptotic properties near the origin can be seen directly from the explicit solution \\eqref{eq:ex1sol}.\nHowever, in the following subsection we will use Proposition \\ref{th:converge} and Corollary \\ref{th:corr}\nto asses the stability of the system.\nNote that the derived control policy \\eqref{eq:ex1uc} is in fact a constant process with negative value.\nThis implies that the agent is always advised to borrow money from the bank and buy stocks with it.\nHe or she will repay the debt in the end when his or her total wealth reaches zero by making profit\nfrom investment in stocks.\n\n\\subsection{Verification of assumptions}\n\nHere we verify when Assumptions (\\ref{A1}-\\ref{A4}) are met by the receding horizon control process \\eqref{eq:ex1uc}.\nNote that, for $v(0,x;T)$ given by $\\eqref{eq:ex1value}$ we have $V(x;T)=v(0,x;T)<\\infty$ for all $(x,T)\\in\\mathbb{R}^{n}\\times\\mathbb{R}_{+}$.\nIn addition, since $\\beta>2$, we learn that\n$$D^{2}V(x;T)=\\begin{cases}\n\\frac{\\beta(\\beta-1)(-x)^{\\beta-2}}{\\eta}\\left(1-e^{-\\eta T}\\right),&x\\le0,\\\\\n0,&x>0,\n\\end{cases}$$\nis continuous. This, combined with the continuity of\n$$\\partial_{T}V(x;T)=\\begin{cases}\n(-x)^{\\beta}e^{-\\eta T},&x\\le0,\\\\\n0,&x>0,\n\\end{cases}$$\nensures that $V\\in C^{2,1}(\\mathbb{R}^{n}\\times\\mathbb{R}_{+},\\mathbb{R})$; thus Assumption \\eqref{A1} is satisfied.\nFor\n$$\n\\phi(x)=\\partial_{t}v(t,x;T)|_{t=0}+f(x,u^{c}(x;T))=\n\\begin{cases}\n(-x)^{\\beta}\\left(1-e^{-\\eta T}\\right),&x\\le0,\\\\\n0,&x>0.\n\\end{cases}\n$$\nto be positive when $x<0$ we require $\\eta>0$. Hence Assumption \\eqref{A2} is satisfied when $\\eta>0$.\nIn light of the continuity of\n$V$ we know that for any $\\epsilon>0$ there is $\\delta>0$ such that $V(x;T)<\\epsilon$ for $-\\deltaV(x_{2};T)>0$\nthus Assumption \\eqref{A4}\nis satisfied if we choose $h$ to be $V(\\cdot;T)$ itself.\n\nIn conclusion, for given $r$, $b$ and $\\sigma$, we obtain that\nAssumptions (\\ref{A1}-\\ref{A4}) are satisfied for all choices of $\\beta$ such that:\n\\begin{equation}\\label{ex:betarange}\n2\\,<\\,\\beta\\,<\\, 1+\\frac{1}{2}\\frac{1}{r}\\frac{(b-r)^2}{\\sigma^2},\n\\end{equation}\nwhere the inequality on the right-hand side corresponds to the condition $\\eta>0$.\nThus, a necessary condition to have a stabilizing control process is that the right-hand side\nof the above inequality is greater that the left-hand side; that is $(b-r)^2>2r\\sigma^2$.\nFinally, it is easy to se that the constraint $u\\in[c_{1},c_{2}]$ can be met\nprovided that it is possible to choose\n\\begin{equation}\\label{ex:betaconst}\n\\beta\\,>\\, 1+\\frac{(b-r)}{|c_1|\\sigma^2},\n\\end{equation}\nwhich, taking into account \\eqref{ex:betarange}, can be done if $|c_1|>2r\/(b-r)$.\n\n\\subsubsection{Terminal cost}\\label{sec:exterm}\n\nIt is also possible to consider a cost function in the form\n\\begin{equation}\nf(x,u):=0,\\,\\forall x,\n\\quad\\mbox{ and }\\quad\ng(x):=\\begin{cases}\n(-x)^{\\beta},&x\\le0,\\\\\n0,&x>0,\n\\end{cases}\n\\end{equation}\nThe derived controlled process turns out to be the same as in \\eqref{eq:ex1uc}.\nHowever, in this case, the value function is given by\n\\begin{eqnarray}\\label{eq:ex2value}\nv(t,x;T)=\\begin{cases}\n(-x)^{\\beta}e^{\\eta(t-T)},&x\\le0,\\\\\n0,&x>0.\n\\end{cases}\n\\end{eqnarray}\nwhere $\\eta$ is given again by \\eqref{eq:ex1eta}.\nThrough similar steps, one eventually obtains the same\nstability conditions of the previous case.\\\\\n\n\\subsection{Numerical illustration}\\label{sec:exsim}\n\nWe present a simulation example where:\n$r=0.03$, $b=0.1$, $\\sigma = 0.15$, and $x_0 = -100$.\\linebreak\nHere we illustrate the behaviour of the wealth process under the\nreceding horizon control process \\eqref{eq:ex1uc}\nfor three different choices of $\\beta$ in cost function \\eqref{eq:excost}: $\\beta=2.1$, $\\beta=4.5$ and $\\beta=8$.\nThe corresponding Monte Carlo simulations are displayed in\n Figures \\ref{fig:b21}-\\ref{fig:b80} respectively.\nNote that, according to \\eqref{ex:betarange}, for the given values of $r$, $b$ and $\\sigma$,\nwe have that the wealth process is asymptotically stable for $\\beta\\in(2,\\,4.6)$.\nBy inspecting the figures, it can be seen that for $\\beta=2.1$\nthe wealth process is clearly asymptotically stable\nbut there is a significant risk of a large initial undershoot.\nFor $\\beta=4.5$ the process converges much slower but the risk\nof a initial undershoot is reduced.\nFor $\\beta=7.8$ the wealth process is not\nasymptotically stable (note the different time scale in the figure).\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.78\\columnwidth]{ex121.png}\n\\caption{Wealth process for $\\beta=2.1$ (100 simulations)}\n\\label{fig:b21}\n\\end{figure}\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.78\\columnwidth]{ex145.png}\n\\caption{Wealth process for $\\beta=4.5$ (100 simulations)}\n\\label{fig:b45}\n\\end{figure}\n\\clearpage\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.78\\columnwidth]{ex178b.png}\n\\caption{Wealth process for $\\beta=7.8$ (100 simulations)}\n\\label{fig:b80}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nIn this note, we have discussed the RHC strategy for systems\ndescribed by continuous-time SDEs.\nWe have obtained conditions on the associated\nfinite-horizon optimal control problem\nwhich guarantee\nthe asymptotic stability of the RHC law.\nWe have shown that these conditions recall their\ndeterministic counterpart.\nWe have illustrated the results with\na simple example in which the RHC law\ncan be obtained explicitly.\nIn current work, we are addressing\nthe problem of implementation in more realistic applications.\nFor this purpose, it will be necessary to formulate\nconditions on the control problem \\eqref{eq:cost}\nwhich can guarantee that Assumptions (\\ref{A1}-\\ref{A4})\nhold true and which can be imposed or verified easily.\nFor problems where the dimension of the\nstate space is not prohibitive,\nit is possible to solve the associated finite-horizon\noptimal control problem with numerical methods \\cite{Pham-05,Borkar-05}.\nThe other possibility is is to approach the\nfinite-horizon problem \\eqref{eq:cost}\nby direct on-line optimization.\nIn this case, one considers a parameterized class\nof feedback policies and iteratively optimizes\nthe parameter of the feedback policy, at regular time intervals,\nconditioned on the value of the current state.\nIn this context simulation-based optimization\nmethods have shown to be promising tools, see e.g.~\\cite{Maciejowski-et-al-05,Lecchini-et-al-06,Kantas-et-al-10}.\\\\\n\n\\noindent{\\bf References}\n\n\n\\bibliographystyle{model1-num-names}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{sec:introduction}}\n\nHigh-energy heavy-ion collisions at the \nRelativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider \n(LHC) probe nuclear matter at extreme densities and temperatures.\nOne of the primary goals of heavy-ion research is to study the properties of \nthe new phase of deconfined matter created in such collisions: the Quark-Gluon Plasma (QGP).\n\nSophisticated multistage models, in which the QGP evolution is described by\nviscous relativistic hydrodynamics, have been remarkably successful in\ndescribing the soft hadronic observables measured in heavy-ion collisions~\\cite{Heinz:2013th,Teaney:2009qa,Luzum:2013yya,Gale:2013da,deSouza:2015ena}.\nComprehensive model-to-data comparisons have been made to quantify\nsystematically the constraints provided by measurements on the transport\ncoefficient of the QGP, and to understand the impact of different observables\non these constraints~\\cite{Niemi:2015qia,Bernhard:2015hxa,Sangaline:2015isa}.\n\nConsiderable progress has been made to increase the predictive power of hydrodynamic simulations and to fully understand the assumptions built into such models. \nThese advances include an evolving understanding of the conditions necessary\nfor hydrodynamics to be\napplicable~\\cite{Denicol:2014tha, Heller:2015dha, Heller:2013fn,Strickland:2017kux,Romatschke:2017acs}, \nand a more consistent treatment of the transition from hydrodynamics to hadronic kinetics at late times~\\cite{Molnar:2014fva}.\nProgress is also being made on understanding the kinetics of early stages of the collision\nand the subsequent transition to hydrodynamics, which is the topic of this work.\n\n\nThe initial conditions of hydrodynamic models remain one of the major sources of uncertainty in phenomenological studies of heavy ion collisions.\nWe will provide a practical way to propagate the energy-momentum tensor in a far-from-equilibrium initial state to a time when viscous hydrodynamics becomes applicable. \nOur goal is to have consistent overlapping descriptions of the early time\ndynamics, and to limit the dependence of the hydrodynamic model on ad hoc\nparameters such as the hydrodynamic initialization\ntime $\\tau_\\text{hydro}$~\\cite{vanderSchee:2013pia,Kurkela:2016vts}.\n\n\n\n\n\nOne approach to the initial conditions is simply to \nparameterize the initial \nenergy density and its fluctuations, sidestepping the thermalization process with additional parameters. Glauber-based models are\ncommonly used for this purpose, and provide the energy density at $\\tau_\\text{hydro}$\n~\\cite{Miller:2007ri,Moreland:2014oya,Blaizot:2014nia,Gronqvist:2016hym}.\nBesides the energy density, the remaining hydrodynamic fields (such as the\nflow velocity and the shear and bulk tensors) must also be parameterized,\nleading to an uncomfortable growth in the number of free parameters.\nPhysically\nmotivated models such as free streaming~\\cite{Broniowski:2008qk,Liu:2015nwa}\nand the gradient\nexpansion~\\cite{Vredevoogd:2008id,vanderSchee:2013pia,Romatschke:2015gxa} have\nbeen used to relate the initial energy density to the full stress tensor which is ultimately needed to\nstart the hydrodynamic simulation.\n\n\n\n\nAt weak coupling significant progress has been made in constructing a\ncomplete picture of the early time dynamics before $\\tau_\\text{hydro}$. Since the relevant\ndegrees of freedom change as a function of time, a consistent theoretical\ndescription of the pre-equilibrium stages requires a combination of different\nweak-coupling methods. Based on the Color Glass Condensate framework (CGC),\nthe initial state immediately after the collision is characterized by strong\ncolor fields, whose dynamics is essentially non-perturbative and best-described\nin terms of classical-statistical field\ntheory~\\cite{Iancu:2002xk,Iancu:2003xm,Gelis:2010nm, Gelis:2007kn,\nLappi:2011ju}. After a short period of time $\\sim1\/Q_s$ the system becomes\nincreasingly dilute. Genuine quantum effects can then be no longer neglected,\nand the subsequent dynamics is better described in terms of QCD effective\nkinetic theory~\\cite{Arnold:2002zm}.\nSeveral studies (which include some of the present authors) have investigated\nthe various stages of the equilibration process in detail, including the early\ntime dynamics using classical-statistical real-time lattice\ntechniques~\\cite{Berges:2013fga,Gelis:2013rba,Berges:2014yta,\nSchenke:2015aqa}, as well as the subsequent approach towards\nlocal thermal equilibrium using effective kinetic theory simulations~\\cite{ Xu:2004mz,El:2007vg,Kurkela:2015qoa,Keegan:2016cpi}, i.e.\\ the\n``bottom-up\" thermalization\nscenario~\\cite{Baier:2000sb}. \n\nAlthough the output of classical field simulations has been used \nto initialize hydrodynamic codes~\\cite{Gale:2012rq}, \na consistent treatment at weak coupling would pass the classical\noutput through the kinetic theory simulation to \ndetermine the initial conditions for the subsequent hydrodynamic evolution.\nIn this paper we provide a concrete realization of this set of steps,\nallowing for an event-by-event description\nof the early time dynamics of high-energy heavy-ion collisions which\nsmoothly approaches hydrodynamics.\n\nElaborating on\nthe ideas formulated in Ref.~\\cite{Keegan:2016cpi}, we describe the pre-equilibrium\ndynamics macroscopically in terms of non-equilibrium response functions of the\nenergy-momentum tensor. Specifically, linearized energy and momentum\nperturbations are propagated on top of a boost-invariant and locally\nhomogeneous background, and the energy-momentum tensor is evolved\nfrom a non-equilibrium initial state to a later time when viscous hydrodynamics becomes applicable. We demonstrate that the pre-equilibrium\nevolution smoothly matches onto hydrodynamics, and\nthe subsequent hydrodynamic evolution becomes essentially independent of the\nmatching time $\\tau_\\text{hydro}$.\n\nIn order to obtain a smooth transition from kinetic description to realistic\nviscous hydrodynamic evolution with typical shear viscosity over entropy ratio\n$\\eta\/s\\sim0.16$, the coupling constant $\\lambda=N_c g^2$ (the single parameter\nof kinetic theory) has to be extrapolated to large values of\n$\\lambda=10\\text{--}25$. For such values of $\\lambda$, the entire\nnon-equilibrium kinetic evolution is very well described by universal functions\nof scaled evolution time $\\tau T\/(\\eta\/s)$. The scalability of background and\nlinear response functions greatly simplifies practical application, since the\nkinetic pre-equilibrium evolution needs only to be calculated once and then can\n be applied to any event-by-event hydrodynamic simulations. In order to\nfacilitate the use of our results in phenomenological description of\nevent-by-event high-energy heavy-ion collisions, we make public the linearized\nkinetic theory response functions and our implementation of the linear\npre-equilibrium propagator \\kompost~\\cite{kompost_github}.\n\n\n\n\n\n\n\n\n\nThe paper is organized as follows. We introduce a general macroscopic description of local pre-equilibrium evolution based on linear response theory out of equilibrium in \\Sec{sec:macro}, and discuss how to obtain the relevant inputs from an underlying microscopic description in effective kinetic theory in \\Sec{sec:ekt}. We further study the equilibration of a locally uniform background in \\Sec{subsec:background} and derive local hydrodynamization time for realistic initial conditions in \\Sec{sec:hydtime}. In \\Sec{sec:generalresponse} we provide the general decomposition of energy-momentum tensor response functions to initial energy and momentum perturbations and in \\Sec{subsec:ektresponse} we discuss their realization in effective kinetic theory. The implementation of the kinetic theory pre-equilibrium phase for hydrodynamic models of heavy-ion collisions is detailed in \\Sec{sec:implementation}, and then applied to two types of initial conditions, MC-Glauber and IP-Glasma initial conditions, in Sections~\\ref{sec:glauber} and \\ref{sec:ipglasma} respectively. We conclude with a compact summary of our findings and discussion of future directions in \\Sec{sec:summary}. Several appendices provide details on the background scaling functions (\\App{app:parameters}), determination of the kinetic response functions (\\App{app:Tmunu_decompose}), free streaming response functions (\\app{sec:freestreaming}), hydrodynamic response functions (\\app{sec:hydroresp}) and kinetic response in the low-$k$ limit (\\app{app:lowk}).\n\n\n\\section{Pre-equilibrium evolution}\n\n\\subsection{Macroscopic description of equilibration\\label{sec:macro}}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{ipglasma3d_v2}\n\n\\caption{The transverse energy density distribution for boost invariant IP-Glasma initial conditions at the start of the kinetic theory pre-equilibrium evolution at $\\tau_\\text{EKT}=0.2\\,\\text{fm}$ and after the linearized kinetic evolution given by \\Eq{one} at the hydrodynamic initialization time \n$\\tau_\\text{hydro}=1.2\\,\\text{fm}$. \nThe white circle indicates the size of the causal neighbourhood in the transverse plane, \\Eq{eq:causalnb}.\\label{fig:glasma3d}}\n\\end{figure}\n\nAlthough the initial state\nshortly after the collision \nof two heavy\nnuclei \nis presumably very complicated, many of the microscopic details \nwash out\n over the first $\\sim\n1\\,\\text{fm\/c}$ \nas the Quark Gluon Plasma (QGP)\nequilibrates and becomes a hydrodynamically expanding fluid.\nSince the late time hydrodynamic behavior is fully characterized by the\nenergy and momentum densities, it is\nconceivable that the most important features of the pre-equilibrium evolution\nalso can be characterized by the energy-momentum tensor $T^{\\mu\\nu}$.\nFollowing the computational strategy\nof Ref.~\\cite{Keegan:2016cpi},\n we will use QCD kinetics as a microscopic\ntheory to determine the non-equilibrium\nevolution of $T^{\\mu\\nu}$. \nThis evolution propagates the non-equilibrium stress\nfrom an initial time $\\tau_{\\scriptscriptstyle \\text{EKT}}\\, {\\sim}\\, 0.1\\,\\text{fm}$\nto a time when\nhydrodynamics becomes applicable $\\tau_\\text{hydro}\\,{\\sim}\\,1\\,\\text{fm}$,\n and provides a map of the form\n\\begin{equation}\n\\left.T^{\\mu\\nu}(\\tau_{\\scriptscriptstyle \\text{EKT}},\\mathbf{x})\\right|_\\text{out-of-equilibrium} \\longrightarrow \nT^{\\mu\\nu}(\\tau_\\text{hydro},\\mathbf{x}),\n\\end{equation} \nas illustrated by \\Fig{fig:glasma3d}.\n\n\nWe first note that, by virtue of causality, all contributions to the energy-momentum tensor at a given space-time point $(\\tau_\\text{hydro},\\mathbf{x})$ are fully determined by the initial conditions at earlier time $\\tau_{\\scriptscriptstyle \\text{EKT}}$ in the causal neighborhood of point $\\mathbf{x}$\n\\begin{equation}\n|\\mathbf{x}'-\\mathbf{x}|1$. We therefore define\na hydrodynamization time, $\\tau_\\text{hydro}$, as the boundary of \napplicability, i.e.\\ $\\tau_\\text{hydro} T_\\text{id.}\/(4\\pi \\eta\/s)=1$.\nSubstituting the definition of $T_\\text{id.}$, we can express \n$ \\tau_\\text{hydro}$ in terms of $\\eta\/s$\n\\begin{equation}\n \\tau_\\text{hydro} \\equiv \\left( \\frac{4\\pi\\eta}{s}\\right)^\\frac{3}{2} \n \\frac{1}{\\Lambda_T}\\label{eq:tauhydro}\\,,\n\\end{equation}\nwhere $\\Lambda_{T}$ is an energy scale defined \nthrough the asymptotic temperature \\Eq{eq:TId}, or alternatively to the asymptotic\nenergy or entropy densities of the system. \nSpecifically for a Bjorken expansion the temperature, energy density, and\nentropy density have the asymptotic forms:\n\\begin{subequations}\n\t\\begin{align}\n \\lim_{\\tau\\rightarrow\\infty} (\\tau T^3) =& \\Lambda_{T}^2\\, , \\label{eq:Tasymptotics}\\\\\n \\lim_{\\tau\\rightarrow\\infty} (\\tau e^{3\/4}) =& \\Lambda_{E}^2 \\, , \\\\\n \\lim_{\\tau\\rightarrow\\infty} (\\tau s) =& \\Lambda_{S}^2 \\, .\n\\end{align}\n\\end{subequations}\nwhere $\\Lambda_{T}$, $\\Lambda_E$, and $\\Lambda_S$ are all related to each other by the equation of state. We will parametrize all equations of state with an effective number of degrees of freedom \n\\begin{equation}\n\\label{eq:Landau}\ne = \\nu_{\\rm eff}(T) \\frac{\\pi^2}{30} T^4 \\, .\n\\end{equation}\nwhere $\\nu_{\\rm eff}(T)=\\nu_g=2(N_c^2-1)=16$ for the gluon gas used in simulations, while for a three flavor gas of quarks and gluons $\\nu_{\\rm eff}(T)= 47.5$. Finally, \n at $T\\sim 0.4\\,{\\rm GeV}$ lattice QCD simulations give $\\nu_{\\rm eff}\\sim 40$~\\cite{Bazavov:2014pvz,Borsanyi:2016ksw}. Using the definition \\Eq{eq:Landau} and thermodynamic identities, it is straightforward to show that the relation between the integration constants is\\footnote{In \\Eq{eq:LambdaS} we used that $(e+p)\/e=4\/3$, which is true within 5\\% even for lattice equation of state at $T\\sim 0.4\\,{\\rm GeV}$ ~\\cite{Bazavov:2014pvz,Borsanyi:2016ksw}.}\n \\begin{align}\n \\Lambda_T^2 &\\approx 0.3 \\left( \n \\frac{16}{\\nu_{\\rm eff}} \\right)^{3\/4} \\Lambda_E^2 \\, , \\\\\n \\Lambda_S^2 &\\approx 2.0 \\left( \\frac{\\nu_{\\rm eff}}{16} \\right)^{1\/4} \\Lambda_E^2 \\, .\\label{eq:LambdaS}\n \\end{align}\n\nHydrodynamic simulations usually adjust the energy density $\\Lambda_{E}$ (or entropy density $\\Lambda_S$) at equilibrium time\nto reproduce the multiplicity in the event. For central $\\text{Pb}+\\text{Pb}$ events at $\\sqrt{s_{NN}}=2.76\\,\\text{TeV}$ discussed later in \\Sec{sec:ipglasma} we estimated the average value of $\\Lambda_E^2$ to be\n\\begin{equation}\n \\left\\langle \\tau e^{3\/4} \\right\\rangle \\approx 1.6\\,{\\rm GeV}^2,\n\\end{equation}\nwhich is consistent with other hydrodynamic simulations with a realistic equation of state (i.e. $\\nu_{\\rm eff}=40$) where $\\left\\langle \\tau s \\right\\rangle \\approx 4.1\\, {\\rm GeV}^2$~\\cite{Keegan:2016cpi}. Based on this estimate and using the hydrodynamization condition \\Eq{eq:tauhydro}, we find that the hydrodynamization time of a boost invariant homogenous plasma is given by\n\\begin{equation}\n\\label{eq:hydrotime}\n \\tau_\\text{hydro}\\approx 0.8\\,{\\rm fm} \\, \\left( \\frac{4\\pi(\\eta\/s)}{2} \n \\right)^\\frac{3}{2} \\left( \\frac{ \\left\\langle \\tau e^{3\/4}\\right\\rangle }{1.6 \\, {\\rm GeV}^2 } \\right)^{-1\/2} \\left( \\frac{\\nu_{\\rm eff}} {16}\\right)^{3\/8},\n\\end{equation}\nwhich provides a realistic bound\n for the applicability of relativistic viscous hydrodynamics (for a constant value of $\\eta\/s=2\/(4\\pi)$). Changing the number of degrees of freedom from $\\nu_{\\rm eff}=16$ for a gluon gas to the more realistic $\\nu_{\\rm eff}=40$ increases the hydrodynamization time to $1.1\\,{\\rm fm}$.\n\n\n\n\n\n\n\n\n\n\n\n\nOne additional consequence of the pre-equilibrium evolution is a rather rapid entropy production associated with the increase of the gluon number density per rapidity. With the full gluon distribution at our disposal we can immediately calculate the Boltzmann entropy $s$ and the particle number $n$\n\\begin{align}\ns(\\tau) &= -\\nu_g\\int \\frac{d^3\\mathbf{p}}{(2\\pi)^3}\\big[ \\bar f(\\tau, \\mathbf{p}) \\ln \\bar f(\\tau, \\mathbf{p})\\nonumber\\\\\n&\\qquad\\qquad-(1+\\bar f(\\tau, \\mathbf{p}))\\ln (1+\\bar f(\\tau,\\mathbf{p}))\\big]\\;, \\\\\nn(\\tau) &=~ \\nu_g\\int \\frac{d^3\\mathbf{p}}{(2\\pi)^3} \\bar f(\\tau,\\mathbf{p}) \\;. \n\\end{align}\nOur results for the non-equilibrium entropy $(s)$ and particle number $(n)$ production \n are summarized in \\Fig{fig:entropya}. \n We find that, independently of the coupling constant (or effective $\\eta\/s$), over $80\\%$ of total entropy per rapidity is produced by the end of the pre-equilibrium stage $\\tau T_\\text{id.}\/(4\\pi \\eta\/s)\\sim 1$. One observes that the non-equilibrium production can be very well reproduced for later times $\\tau T_\\text{id.}\/(4\\pi \\eta\/s) \\gtrsim 1$ in second order hydrodynamics, provided a non-equilibrium entropy definition $s_\\text{non-eq}(\\tau)=s_\\text{eq}(\\tau)\\,(1-\\frac{\\tau_\\pi}{4T\\eta\/s}\\pi^{\\mu\\nu}\\pi_{\\mu\\nu})$ is used~\\cite{Baier:2007ix}.\nFinally, the entropy and gluon number densities roughly doubles from the start of kinetic evolution to the time $\\tau_\\text{hydro}$ (see \\Fig{fig:entropya}). \nClearly, the rapid generation of entropy during the approach to equilibrium is very important in relating the initial state energy or gluon number density to the experimentally measured charged particle multiplicities (see Fig.~4 in Ref.~\\cite{Aamodt:2010pb} and Fig.~3 in Ref.~\\cite{Adam:2015ptt}, and references therein).\nConsequently, such large gluon multiplication factor needs to be taken into account in the correct estimation the properties of the initial state, e.g. the saturation scale $Q_s$ in Color Glass Condensate picture, from the measured multiplicities.\n\n\n\n\n\n\n\n\n\n\n\\section{Response functions\\label{sec:response}}\n\n\\subsection{General decomposition of macroscopic response functions}\n\\label{sec:generalresponse}\nContinuing the discussion of \\Sec{sec:macro}, we now look into the general properties of response functions for the linearized energy-momentum perturbations evolving on top of the out-of-equilibrium background.\n We consider only boost \ninvariant perturbations in the transverse plane, and focus on the energy-momentum response to \nperturbations of the conserved charges---initial energy density $\\delta \nT^{\\tau\\tau}$ and initial momentum density $\\delta T^{\\tau i}$.\n\n By normalizing \nthe perturbations to the background energy density $\\overline{T}^{\\tau\\tau}_\\mathbf{x}(\\tau)$, \nthe evolution of energy-momentum perturbations can be compactly summarized as\n\\begin{equation}\n\\frac{\\delta \nT^{\\mu\\nu}(\\tau,\\mathbf{x})}{\\overline{T}^{\\tau\\tau}_\\mathbf{x}(\\tau)}=\\frac{1}{\\overline{T}^{\\tau\\tau}_\\mathbf{x} \n(\\tau_0)}\\int \\!\nd^2\\mathbf{x}_0 \\,G^{\\mu\\nu}_{\\alpha\\beta}\\Big(\\mathbf{x},\\mathbf{x}_0,\\tau,\\tau_0\\Big) {\\delta \nT^{\\alpha\\beta}_\\mathbf{x}(\\tau_0,\\mathbf{x}_0)}\\;.\\label{eq:dTmunuconvolv}\n\\end{equation}\nSince we consider perturbations on top of a (locally) homogenous and boost invariant background, translation invariance guarantees that the response functions depend only on the difference $\\mathbf{x}-\\mathbf{x}_0$ and it is often more convenient to work in Fourier space, where we define the Fourier transformed response function $\\tilde{G}^{\\mu\\nu}_{\\alpha\\beta}$ according to\\footnote{Note that vectors $\\mathbf{x}$ and $\\mathbf{k}$ are both confined to the transverse ($x\\text{-}y$) plane.}\n\\begin{align}\n{G}^{\\mu\\nu}_{\\alpha\\beta}&\\Big(\\mathbf{x}-\\mathbf{x}_0,\\tau,\\tau_0,\\overline{T}_\\mathbf{x}^{\\tau\\tau}(\\tau_0)\\Big)=\\nonumber\\\\\n&\\int\\!\\frac{d^2\\mathbf{k}}{(2\\pi)^2}~\\tilde{G}^{\\mu\\nu}_{\\alpha\\beta}\\Big(\\mathbf{k},\\tau,\\tau_0,\n\\overline{T}_\\mathbf{x}^{\\tau\\tau}(\\tau_0)\\Big)~e^{i\\mathbf{k}\\cdot (\\mathbf{x}-\\mathbf{x}_0)}\\;.\\label{eq:Fourier}\n\\end{align}\nBased on rotational symmetry in the transverse plane, one can further decompose \nthe response functions into a tensor basis. For (scalar) energy perturbations \nthe response function $\\tilde{G}^{\\mu\\nu}_{\\tau\\tau}(\\mathbf{k},\\tau,\\tau_0, \n\\overline{T}^{\\tau\\tau}_\\mathbf{x}(\\tau_0))$ has only four independent structures\n\\begin{align}\n\\tilde{G}^{\\tau\\tau}_{\\tau\\tau}(\\mathbf{k})&=\\tilde{G}_s^s(|\\mathbf{k}|)\\;, \\quad \n\\tilde{G}^{\\tau i}_{\\tau\\tau}(\\mathbf{k})=-i\\tilde{G}_s^v(|\\mathbf{k}|) \n\\frac{\\mathbf{k}^i}{|\\mathbf{k}|}\\;, \\nonumber \\\\\n\\tilde{G}^{ij}_{\\tau\\tau}(\\mathbf{k})&= \\tilde{G}_s^{t,\\delta}(|\\mathbf{k}|) \\delta^{ij} \n+\\tilde{G}_s^{t,k}(|\\mathbf{k}|) \\frac{~\\mathbf{k}^i \\mathbf{k}^j}{|\\mathbf{k}|^2}\\;, \\nonumber \\\\\n\\tilde{G}^{\\tau\\eta}_{\\tau\\tau}(\\mathbf{k})&=0\\;, \\qquad \n\\tilde{G}^{i\\eta}_{\\tau\\tau}(\\mathbf{k})=0\\;, \\label{eq:G_energy_decomp}\n\\end{align}\nwhile for (vector) momentum perturbations the decomposition reads\n\\begin{align}\n \\tilde{G}^{\\tau\\tau}_{\\tau k}(\\mathbf{k})&=-i\\tilde{G}_{v}^{s}(|\\mathbf{k}|) \n \\frac{\\mathbf{k}^k}{|\\mathbf{k}|}\\;,\\nonumber \\\\\n \\tilde{G}^{\\tau i}_{\\tau k}(\\mathbf{k})&=\\tilde{G}_v^{v,\\delta}(|\\mathbf{k}|)\\delta^{ik} \n +\\tilde{G}_v^{v,k}(|\\mathbf{k}|) \\frac{~\\mathbf{k}^i \\mathbf{k}^k}{|\\mathbf{k}|^2} \\;,\\nonumber \\\\\n \\tilde{G}^{ij}_{\\tau k}(\\mathbf{k})&= -i\\tilde{G}_v^{t,\\delta}(|\\mathbf{k}|) \\delta^{ij} \n \\frac{~\\mathbf{k}^k}{|\\mathbf{k}|} -i \\tilde{G}_{v}^{t,m}(|\\mathbf{k}|) \\frac{ \n \\delta^{ik}\\mathbf{k}^{j}+\\delta^{jk}\\mathbf{k}^{i}}{2|\\mathbf{k}|} \\nonumber \\\\\n &\\qquad \\qquad -i\\tilde{G}_v^{t,k}(|\\mathbf{k}|) \\frac{~\\mathbf{k}^i \\mathbf{k}^j \\mathbf{k}^k}{|\\mathbf{k}|^3}\\;, \n \\nonumber \\\\\n\\qquad \\tilde{G}^{\\tau\\eta}_{\\tau k}(\\mathbf{k})&=0\\;, \\qquad \n\\tilde{G}^{i\\eta}_{\\tau \nk}(\\mathbf{k})=0\\;. \\label{eq:G_momentum_decomp}\n\\end{align}\nSince the longitudinal pressure components $\\tilde{G}^{\\eta\\eta}_{\\alpha\\beta}$ \nare uniquely determined by the tracelessness of the energy-momentum tensor, one \nis then left with a total of ten independent response functions, which need to be determined by a particular microscopic model.\n\nSimilarly to the discussion in $\\mathbf{k}$-space, the coordinate space response $G^{\\mu\\nu}_{\\alpha\\beta}$ can be \ndecomposed in tensors constructed from the radial vector\n$\\mathbf{r}=\\mathbf{x}-\\mathbf{x}_0$. In practice, we first compute the response in $\\mathbf{k}$-space and then do the reverse Fourier transform~\\cite{Keegan:2016cpi} . The relations between momentum and \ncoordinate space Green's functions are detailed in \\app{app:Tmunu_decompose}.\n\n\n\n\\subsection{Non-equilibrium response functions from effective kinetic \ntheory\\label{subsec:ektresponse}}\n\nThe independent components of macroscopic response functions $G_{\\alpha \\beta}^{\\mu\\nu}$ in \\Eqs{eq:G_energy_decomp} and \\eq{eq:G_momentum_decomp} need to be calculated by a particular microscopic theory. In this section we discuss the numerical realization of linear response in QCD kinetic theory around the non-equilibrium background presented in \\Sec{subsec:background}. At late times and close to thermal equilibrium, kinetic response functions are bound to approach the hydrodynamic limit, which is studied in \\App{sec:hydroresp}. Similarly, the early time dynamics can be profitably compared to the analytic results of collision-free evolution, discussed in \\App{sec:freestreaming}.\n\n We follow the methodology of Ref.~\\cite{Keegan:2016cpi} and linearize the \n phase-space distribution function $f_{\\mathbf{x},\\mathbf{p}}$ around the background \n $\\bar{f}_\\mathbf{p}$, which is spatially \n homogeneous, but \n anisotropic in momentum space $\\mathbf{p}=(p^x,p^y,p^z)$. We consider only boost \n invariant $\\delta f$\n perturbations, which can be decomposed into a Fourier integral of plane wave \n perturbation $\\delta f_{\\mathbf{k},\\mathbf{p}}$ labelled by the wavenumber $\\mathbf{k}$ \n in \n the transverse plane\n\\begin{equation}\nf_{\\mathbf{x},\\mathbf{p}} = \\bar{f}_\\mathbf{p}+\\int \\frac{d^2 \\mathbf{k}}{(2\\pi)^2}\\,\\delta f_{\\mathbf{k},\\mathbf{p}}~e^{i\\mathbf{k}\\cdot\\mathbf{x}} .\n\\end{equation}\nTo linear order in perturbations, $\\delta f_{\\mathbf{k},\\mathbf{p}}$ evolves according to coupled Boltzmann equations\n\\begin{align}\n\\label{eq:bolz1}\n\\left(\\partial_\\tau - \\frac{p_z}{\\tau} \\partial_{p_z}\\right) \\bar f_{\\mathbf{p}} &= - \n\\mathcal{C}[\\bar f],\\\\\n\\left(\\partial_\\tau - \\frac{p_z}{\\tau} \\partial_{p_z} + \\frac{i \\mathbf{p}\\cdot \n\\mathbf{k}}{p} \\right) \\delta f_{\\mathbf{k},\\mathbf{p}} &= - \\delta\\mathcal{C}[\\bar f, \\delta f] ,\\label{eq:bolz2}\n\\end{align}\nwhere the collision kernel $ \\delta\\mathcal{C}[\\bar f, \\delta f]+\\mathcal{O}(\\delta f^2)= \\mathcal{C}[\\bar f+ \\delta f]-\\mathcal{C}[\\bar f]$ is linearized in $\\delta f$. Since the evolution for different values of $\\mathbf{k}$ on top of a homogenous background $\\bar{f}_\\mathbf{p}$ decouples from each other~\\cite{Keegan:2016cpi}, the linearized Boltzmann equation can be solved independently for each wavenumber $\\mathbf{k}$ in the transverse plane.\n\nEven though the effective kinetic description of the pre-equilibrium dynamics \nrequires the knowledge of the phase-space distribution $\\delta f_{\\mathbf{k},\\mathbf{p}}$ \nat the initial time $\\tau_0$, one naturally expects the \noccurrence of memory loss during the evolution, so that the details of the initial \nphase-space distribution become irrelevant as the system approaches local \nthermal equilibrium. Since our ambition is merely to extract the \nenergy-momentum tensor, a representative choice of $\\delta f_{\\mathbf{k},\\mathbf{p}}$ to \ncharacterize initial energy and momentum perturbations should be sufficient \nto describe the non-equilibrium evolution.\n\n\n\nBased on a weak-coupling picture of the initial state, where the properties of \nthe background distribution $\\bar{f}(\\tau_0,\\mathbf{p})=\\bar{f}_{0}(|\\mathbf{p}|\/Q_s,\\theta)$ are determined by a \nsingle dimensionful scale $Q_s$,\nit is natural to associate perturbations of \nthe energy-momentum tensor $\\delta T^{\\mu\\nu}(\\mathbf{x},\\tau_0)$ with local \nfluctuations of the scale $Q_s(\\mathbf{x})=\\bar{Q}_s+\\delta Q_s(\\mathbf{k})e^{i\\mathbf{k}\\cdot \n\\mathbf{x}}$. Hence one can motivate the initial phase-space distribution of \nscalar perturbations to be of the form\n\\begin{equation}\n\\delta f_{\\mathbf{k},\\mathbf{p}}^{(\\text{Energy})}=\\frac{\\delta Q_s(\\mathbf{k})}{\\bar Q_s} ~\\delta f_{\\mathbf{k},\\mathbf{p}}^{(s)}\n\\end{equation}\nwhere $\\delta Q_s(\\mathbf{k})\/\\bar Q_s=\\delta T^{\\tau\\tau}\/\\bar T^{\\tau\\tau}$ denotes the amplitude of the perturbation and the spectral shape\n\\begin{equation}\n\\delta f_{\\mathbf{k},\\mathbf{p}}^{(s)}=\\frac{1}{4} \\bar Q_s \\partial_{\\bar Q_s} \\bar{f}_{0}\\left(\\frac{|\\mathbf{p}|}{\\bar Q_s},\\theta\\right)\\;\\label{eq:dfe}\n\\end{equation}\nis determined from the variation of the background distribution, \\Eq{eq:init_cond}, with respect to the scale $\\bar Q_s$.\n\n\n\n\nSimilarly, considering that gradients of $Q_s(\\mathbf{x})$ will lead to an initial \nvelocity perturbation, e.g.\\ $v_{i}(\\tau_0) \\propto -\\tau_0 \\frac{\\partial_{i} \nQ_s(\\mathbf{x})}{\\bar Q_s}$, we can motivate vector perturbations of the form\n\\begin{equation}\n\t\\delta f_{\\mathbf{k},\\mathbf{p}}^{(\\text{Momentum})} = v^i(\\mathbf{k}) \\delta f_{\\mathbf{k},\\mathbf{p}}^{(v),i}\\;,\n\\end{equation}\nwhere $v^i(\\mathbf{k}) = \\delta T^{\\tau i}_{\\mathbf{k}, (v)}\/\\bar T^{\\tau \\tau}$ denotes the amplitude of the initial velocity perturbation and the spectral shape\n\\begin{equation}\t\n\\delta f_{\\mathbf{k},\\mathbf{p}}^{(v),i}=\\frac{1}{2} \\partial_{v^{i}}\\!\\left. \\bar{f}_{0}\\left(\\frac{|\\mathbf{p}-\\mathbf{v} |\\mathbf{p}||}{\\bar Q_s},\\theta \\right) \\right|_{\\mathbf{v}=0}\\label{eq:dfg}\n\\end{equation}\nis determined by a linearized velocity boost of the background momentum distribution. We note that in order to compute all response functions in the tensor decomposition in \\Eq{eq:G_momentum_decomp} it is important that we keep track of the independent components (labeled by the index $i$) of the momentum response.\n\nEven though at the level of the linearized Boltzmann equation, the \nactual magnitude of the perturbations is irrelevant in computing the response \nfunctions (as long as it remains sufficiently small to justify the linearized \napproximation at the relevant momentum scale), we find it convenient to choose an appropriate normalization. Defining the moments of the distribution function\n\\begin{subequations}\n\t\\label{eq:dTmunudef}\n\\begin{align}\n\\delta T^{\\mu\\nu}_{\\mathbf{k},(s)}(\\tau)&= \\nu_{g} \\int \n\\frac{d^3\\mathbf{p}}{(2\\pi)^3}~\\frac{p^{\\mu}p^{\\nu}}{p}~\\delta f^{(s)}_{\\mathbf{k},\\mathbf{p}}\\;, \n\\\\\n\\delta T^{\\mu\\nu,i}_{\\mathbf{k},(v)}(\\tau)&= \\nu_{g} \\int \n\\frac{d^3\\mathbf{p}}{(2\\pi)^3}~\\frac{p^{\\mu}p^{\\nu}}{p}~\\delta \nf^{(v),i}_{\\mathbf{k},\\mathbf{p}}\\;, \n\\end{align}\n\\end{subequations}\nthe corresponding energy and momentum perturbations associated with $\\delta f_{\\mathbf{k},\\mathbf{p}}^{(s)}$ and respectively $\\delta f_{\\mathbf{k},\\mathbf{p}}^{(v),i}$ at initial time are normalized such that for a highly oblate distribution ($\\xi \\gg 1$)\n\\begin{subequations}\n\t\\begin{align}\n\t\\frac{\\delta T^{\\tau \\tau}_{\\mathbf{k},(s)}(\\tau_0) }{\\overline{T}^{\\tau \n\t\\tau}(\\tau_0)}&= 1\\,,\n\t\t\\\\\n\\frac{\\delta T^{\\tau k,i}_{\\mathbf{k},(v)}(\\tau_0)}{\\overline{T}^{\\tau \n\\tau}(\\tau_0)}&= \\delta^{ki}\\;. \n\t\\end{align}\n\\end{subequations}\n\n\n\n\n\n\n\n\nGiven the above explicit form of the initial perturbations one can then \ndetermine the response of the energy-momentum tensor at any later time by \nnumerically solving the kinetic equations, \\Eqs{eq:bolz1} and \\eq{eq:bolz2}. Once the solution to the linearized \nBoltzmann equation is calculated numerically, the response functions \n$\\tilde{G}^{\\mu\\nu}_{\\tau\\tau}(\\mathbf{k},\\tau,\\tau_0, \n\\overline{T}_\\mathbf{x}^{\\tau\\tau}(\\tau_0))$ can be directly constructed from the moments of the distribution function. For example, the energy response to \nenergy or momentum perturbations is determined by the ratios\n\\begin{align}\n\\label{eq:Gss}\n\\tilde{G}_{s}^{s}(\\tau,\\tau_0,|\\mathbf{k}|)&= \\left. \\frac{\\delta \nT^{\\tau\\tau}_{\\mathbf{k},(s)}(\\tau)} {\\overline{T}^{\\tau\\tau}(\\tau)} \\right\/ \\frac{\\delta \nT^{\\tau\\tau}_{\\mathbf{k},(s)}(\\tau_0) }{ \\overline{T}^{\\tau\\tau}(\\tau_0)}\\;, \\\\\n\\tilde{G}_{v}^{s}(\\tau,\\tau_0,|\\mathbf{k}|)&= \\left. \\frac{ \n\\frac{i\\mathbf{k}_{i}}{|\\mathbf{k}|} \\delta T^{\\tau\\tau, i }_{\\mathbf{k},(v)}(\\tau)} {\\overline{T}^{\\tau\\tau}(\\tau)} \\right\/ \n\\frac{ \\frac{1}{2} \\delta_{jk} \\delta T^{\\tau j, k}_{\\mathbf{k},(v)}(\\tau_0) }{ \\overline{T}^{\\tau\\tau}(\\tau_0)}\\;.\\label{eq:Gvs}\n\\end{align}\nSimilarly, all the other components can be constructed by linear combinations \nof different components of $\\delta T^{\\mu\\nu}$ as described in \n\\app{app:Tmunu_decompose}\n\n\n\n\n\n\\subsubsection{Scaling of response functions\\label{sec:Gscaling}}\n\n\\begin{figure*}%\n\\centering\n\\subfig{a}{\\includegraphics[width=0.4\\linewidth]{{plot_scalingGks_s_xs1.0}.pdf}}\\qquad\n\\subfig{b}{\\includegraphics[width=0.4\\linewidth]{{plot_scalingGkv_s_xs1.0}.pdf}}\n\\caption{\n (a) The universal scaling function $\\tilde{G}_s^s$ (see \\Eq{eq:scalable})\n for the energy response to an initial energy perturbation as a function of\n the phase $|\\mathbf{k}|(\\tau -\\tau_0)$ at a fixed scaling time, $\\frac{\\tau \\TId}{\\eta\/s}$. The\n different curves correspond to different coupling strengths (or $\\eta\/s$)\n which collapse onto\n a universal curve.\n The response\n functions at different scaling times $\\tauT_\\text{id.}\/(4\\pi\\eta\/s)=0.5,1.5,2.0$\n exhibit an equally good overlap (not shown). (b) The analogous plot for the\n energy response to an initial momentum perturbation, $\\tilde{G}_s^v$.\n}\n\\label{fig:rescaledresponse19de}\n\\end{figure*}\n\nWe now present our numerical results for the non-equilibrium response \nfunctions calculated in effective kinetic theory. Even though generally \nthe response functions \n$\\tilde{G}^{\\mu\\nu}_{\\alpha\\beta}\\big(\\mathbf{k},\\tau,\\tau_0,\\overline{T}_\\mathbf{x}^{\\tau\\tau}(\\tau_0)\\big)$\n depend separately on the wavenumber $\\mathbf{k}$, the initial and final times $\\tau$ \nand $\\tau_0$, the energy scale $\\overline{T}_\\mathbf{x}^{\\tau\\tau}(\\tau_0)$, and the \ncoupling constant $\\lambda=g^2N_c$, we expect that in analogy to the evolution \nof the background the number of independent variables can be drastically \nreduced by identifying appropriate scaling variables. Based on our analysis of \nthe background evolution in \\Sec{subsec:background}, the \nnatural candidate variables are the scaled evolution time $\\tau T_\\text{id.} \/ \n(\\eta\/s)$ and phase $|\\mathbf{k}| (\\tau-\\tau_0)$.\n\nIndeed we find that in the relevant range \nof parameters the postulated scaling property holds and the response functions \ncan be compactly expressed in terms of a universal function of the scaling \nvariables such that, for example\n\\begin{align}\n\\label{eq:scalable}\n\\tilde{G}_{s}^{s}\\Big(|\\mathbf{k}|,\\tau,\\tau_0,\\overline{T}_\\mathbf{x}^{\\tau\\tau}(\\tau_0),\\lambda\\Big)= \\tilde{G}_{s}^{s,{\\rm univ}}\\left(\\frac{\\tau \\TId}{\\eta\/s},|\\mathbf{k}| (\\tau-\\tau_0) \n\\right)\\;.\n\\end{align}\nEven though this behavior is expected to emerge in the hydrodynamic limit\\footnote{This hydrodynamic regime is discussed in more details in \\app{sec:hydrolimit}.} of sufficiently \nlate evolution times $T_\\text{id.} \\tau \/ (\\eta\/s) \\gg 1$ and for small wave-numbers $|\\mathbf{k}| \n(\\tau-\\tau_0) \\ll 1$, it is remarkable to observe that the scaling \nproperty holds across a much wider range of evolution times and wavelengths. \nAs an illustrative example of the scaling, in \\Fig{fig:rescaledresponse19de} we present our results for the response \nfunctions $\\tilde{G}_{s}^{s}$ and $\\tilde{G}_{v}^{s}$ given by \\Eqs{eq:Gss} and \\eq{eq:Gvs} at scaled evolution time $\\tau T_\\text{id.}\/(4\\pi \\eta\/s)\\approx 1.0$.\n Different curves in \\Fig{fig:rescaledresponse19de} \ncorrespond to simulations performed at different values of $\\lambda$, which not only correspond to different effective $\\eta\/s$, but also amount to variations of the initial energy scale $\\overline{T}_\\mathbf{x}^{\\tau\\tau}(\\tau_0)$, as seen from \\Eq{eq:init_cond}.\n Other components of the response function in the decomposition given by \\Eqs{eq:G_energy_decomp} and \\eq{eq:G_momentum_decomp} also show a good scaling with $\\tau T_\\text{id.} \/ \n(\\eta\/s)$ and phase $|\\mathbf{k}| (\\tau-\\tau_0)$ (not shown).\n\nBecause of the scaling of the response functions with $\\eta\/s$---the same kinetic theory response function computed for one set of initial conditions, can be used for different values of $\\eta\/s$ and different values of the initial energy density to map the early out-of-equilibrium energy and momentum perturbations to the hydrodynamized energy-momentum tensor perturbations at later times. This is the procedure adopted in the pre-equilibrium propagator \\kompost{}.\n\nFinally, the coordinate space response functions used in the propagation formula \\Eq{eq:dTmunuconvolv} are obtained by Fourier transforming $|\\mathbf{k}|$-space components, e.g.\\ \\Fig{fig:rescaledresponse19de}, according to \\Eq{eq:Fourier}. The details of the procedure are given in \\app{app:Tmunu_decompose} and also discussed in Ref.~\\cite{Keegan:2016cpi}. Here we only point to the final result, i.e. the complete set of coordinate space response functions summarized in Figures \\ref{fig:plot_grgss} and \\ref{fig:plot_grgvs} in the \\app{app:Tmunu_decompose}. Because the momentum space response functions are, to a good approximation, universal functions of scaling variables $\\frac{\\tau \\TId}{\\eta\/s}$ and $|\\mathbf{k}| (\\tau-\\tau_0)$, the coordinate space response functions are also universal when expressed in terms of the scaling variables $\\frac{\\tau \\TId}{\\eta\/s}$ and $|\\mathbf{x}-\\mathbf{x}_0| \/ (\\tau-\\tau_0)$.\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Hydrodynamic constitutive equations}\n\\begin{figure*}\n\t\\centering\n\\subfig{a}{\t\\includegraphics[width=0.4\\linewidth]{rescaledResponse_Gstd_const}}\n\t\t\\subfig{b}{\\includegraphics[width=0.4\\linewidth]{rescaledResponse_Gstk_const}}\n \\caption{Comparison of the response functions for $\\delta T^{ij}$ from an initial energy perturbation (see $\\tilde G_s^{t,\\delta}$ and $\\tilde G_s^{t,k}$ in \\Eq{eq:G_energy_decomp})\n with the constitutive relations of first and second order hydrodynamics (\\Eqs{eq:Gstk_const} and \\eq{eq:Gtds_const}).\n }\n\t\\label{fig:rescaledresponsegstdconst}\n\\end{figure*}\n\nAt late times when the system starts behaving hydrodynamically, the different \ncomponents of response function decomposition \\Eqs{eq:G_energy_decomp} and \\eq{eq:G_momentum_decomp}, are no longer independent, but are related by hydrodynamic constitutive equations. In second order hydrodynamics and for small wavenumber perturbations, the spatial part of energy momentum tensor $\\delta T^{ij}$ can be written as a sum of energy $\\delta T^{\\tau\\tau}$ and momentum $\\delta T^{\\tau i}$ density perturbations, with prefactors depending on first and second order hydrodynamic transport coefficients, i.e.\\ $\\eta,\\tau_\\pi$ and $\\lambda_1$ ~\\cite{Keegan:2016cpi}. By comparing constitutive equations with the \ndecomposition in \\Eq{eq:G_energy_decomp}, we find the following relation between response function components for initial energy perturbations\n\\begin{align}\n&\\tilde G_s^{t,k}(\\tau,\\tau_0,|\\mathbf{k}|)=\n\\frac{3}{2}c_s^2\\tau_\\pi \\eta \\frac{\\tilde G_s^s(\\tau,\\tau_0,|\\mathbf{k}|) |\\mathbf{k}|^2 \n}{\\overline{T}^{\\tau\\tau}}\\nonumber\\\\\n&+2(-\\eta+\\frac{1}{3}\\eta \\tau_\\pi \n\\frac{3c_s^2+1}{\\tau}\n-\\frac{4}{3\\tau}\\lambda_1 \n)\\frac{|\\mathbf{k}|\\tilde \n\tG_s^v(\\tau,\\tau_0,|\\mathbf{k}|)}{\\overline{T}^{\\tau\\tau}+\\frac{1}{2}\\overline{T}^k_k},\\label{eq:Gstk_const}\\\\\n&\\tilde G_s^{t,\\delta} (\\tau,\\tau_0,|\\mathbf{k}|)=\\nonumber\\\\ \n&\\left(p+\\frac{\\eta}{2\\tau}+\\frac{1}{3}(1-c_s^2)\\frac{\\tau_\\pi\n\t\\eta-{\\lambda_1}}{\\tau^2}-\\frac{1}{2}c_s^2 \\tau_\\pi \\eta k^2\n\t\\right)\\!\\frac{\\tilde \n\tG_s^s \n(\\tau,\\tau_0,|\\mathbf{k}|)}{\\overline{T}^{\\tau\\tau}}\\nonumber\\\\\n&+\\left(\\frac{2}{3} \n\\eta-\\frac{2}{9} \\eta \n\\tau_\\pi \n\\frac{3c_s^2+2}{\\tau}\n+\\frac{16}{9\\tau}\\lambda_1 \\right)\\frac{|\\mathbf{k}| \\tilde G_s^v(\\tau,\\tau_0,|\\mathbf{k}|) \n}{\\overline{T}^{\\tau\\tau}+\\frac{1}{2}\\overline{T}^k_k}\\label{eq:Gtds_const}.\n\\end{align}\nIn \\Fig{fig:rescaledresponsegstdconst} we explicitly test the first and second order constitutive relations, \\Eqs{eq:Gstk_const} and \\eq{eq:Gtds_const}, in the kinetic evolution at scaled time $\\tau T_\\text{id.}\/(4\\pi \\eta\/s)\\approx 2$. Indeed for small wavenumbers $|\\mathbf{k}|(\\tau-\\tau_0)< 6$ the second order constitutive equations are well satisfied, demonstrating \nthat the low wavenumber perturbations approach hydrodynamic regime at \nsufficiently late times ${\\tau T_\\text{id.}}\/({4\\pi\\eta\/s})>1$.\n\nSimilar constitutive relations can be derived for momentum response components and are given in \\Eq{eq:G_momentum_const}.\nThe complete derivation is summarized in \\App{sec:hydroresp}.\n\n\n\n\n\n\n\n\n\\section{Practical implementation: \\kompost \\label{sec:implementation}}\n\n\nBased on the general formalism and results of linearized effective kinetic \ntheory presented in the \nprevious sections, we will now describe a practical implementation of the \npre-equilibrium evolution for hydrodynamic modeling of heavy-ion collisions---\\kompost. \nStarting from a given profile of the energy-momentum tensor \n$T^{\\mu\\nu}(\\tau_{\\scriptscriptstyle \\text{EKT}},\\mathbf{x})$ at an early time $\\tau_{\\scriptscriptstyle \\text{EKT}}$, for example from the \nIP-Glasma model, we follow the procedure outlined below to calculate the energy \nmomentum tensor $T^{\\mu\\nu}(\\tau_{\\textrm{hydro}},\\mathbf{x}_{0})$ at a later time \n$\\tau_\\text{hydro}>\\tau_{\\scriptscriptstyle \\text{EKT}}$ when the system is sufficiently \nclose to local thermal equilibrium for viscous hydrodynamics to become \napplicable\\footnote{Equilibration is not necessarily achieved everywhere at \nthe same time $\\tau$, and the initial conditions could, in theory, be provided on a \nmore complex $(\\tau,x,y,\\eta)$ hypersurface. However, it is the common practice \nto initialize hydrodynamic simulations on a constant $\\tau$ hypersurface and we \nwill follow this procedure in our present work.}.\\\\\n\n\\paragraph{ Decomposition in Background \\& Perturbations}\nWe first split the initial energy-momentum tensor $T^{\\mu\\nu}$ at \\kompost{} initialisation time $\\tau_{\\scriptscriptstyle \\text{EKT}}$ in the background and perturbations\n\\begin{equation}\n T^{\\mu\\nu}(\\tau,\\mathbf{x}) = \n \\underbrace{\\overline{T}^{\\mu\\nu}_{\\mathbf{x}_0}(\\tau)}_\\text{background}+\\underbrace{T^{\\mu\\nu}(\\tau,\\mathbf{x})-\\overline{T}^{\\mu\\nu}_{\\mathbf{x}_0}(\\tau)}_{\\equiv\\delta\n T_{\\mathbf{x}_0}^{\\mu\\nu}(\\tau,\\mathbf{x})}\\;,\\label{eq:split}\n\\end{equation}\nwhere for linear evolution the decomposition into the background \n$\\overline{T}^{\\mu\\nu}_{\\mathbf{x}_0}$ and perturbations $\\delta T^{\\mu\\nu}$ is arbitrary, as \nlong as the perturbations are sufficiently small.\nAs discussed in \\Sec{subsec:background} we consider locally homogeneous boost invariant background energy-momentum tensor $\\overline{T}_{\\mathbf{x}_0}^{\\mu\\nu}=\\text{diag}\\,(e, P_T,P_T,\\tfrac{1}{\\tau^2}P_L)$ with only one independent component $e(\\tau)$. In order to obtain a smooth energy density profile from a discrete grid of input $T^{\\tau\\tau}$ values, we define the local background energy density as a \nGaussian weighted average around the point $\\mathbf{x}_0$ of interest \n\\begin{equation}\n\\overline{T}^{\\tau\\tau}_{\\mathbf{x}_0}(\\tau)\\equiv \\int d^2\\mathbf{x}'\\, \\frac{1}{2\\pi \\sigma^2} \ne^{\\frac{-(\\mathbf{x}'-\\mathbf{x}_0)^2}{2\\sigma^2}} T^{\\tau\\tau}(\\tau ,\\mathbf{x}')\\;,\\label{eq:avgBG}\n\\end{equation}\nwhere the Gaussian width is taken to be $\\sigma=\\Delta \\tau\/2$\n\\footnote{Note that changing the smearing \nwidth $\\sigma$ changes the decomposition into background and perturbations. We \nhave \nchecked by varying $\\sigma$ by a factor of two that the sum of background \nand perturbations after the kinetic evolution remains remarkably invariant everywhere, \nexcept for edges of the fireball, where the linearized \ntreatment breaks down.}. Here $\\Delta \\tau = (\\tau_\\text{hydro}-\\tau_{\\scriptscriptstyle \\text{EKT}})$ is the duration of kinetic evolution, i.e.\\ the causal circle radius discussed in \\Sec{sec:macro}. \nOnce the homogeneous diagonal background energy-momentum tensor $\\overline{T}^{\\mu\\nu}_{\\mathbf{x}_0}$ is determined in the neighbourhood of point $\\mathbf{x}_0$, the perturbation tensor $\\delta T^{\\mu\\nu}_{\\mathbf{x}_0}(\\tau,\\mathbf{x})$ is obtained according to \\Eq{eq:split}. In particular, the (small) initial off-diagonal components of energy-momentum tensor $T^{\\mu\\nu}$ are completely absorbed in the perturbation tensor, e.g.\\ $\\delta T^{\\tau i}_{\\mathbf{x}_0}=T^{\\tau i}$.\nIn accordance with the discussion in \\Sec{sec:generalresponse} we only \nconsider the response to initial energy $\\delta T_{\\mathbf{x}_0}^{\\tau \\tau}(\\tau_{\\scriptscriptstyle \\text{EKT}},\\mathbf{x})$ and \nmomentum perturbations $\\delta T^{\\tau i}_{\\mathbf{x}_0}(\\tau_{\\scriptscriptstyle \\text{EKT}},\\mathbf{x})$. The initial perturbations in the shear-stress part of the energy-momentum tensor, i.e. $\\delta T^{ij}_{\\mathbf{x}_0}$ are not taken into account in this work\\footnote{\nNote that in this implementation the diagonal components of the background energy-momentum tensor are entirely given by $T^{\\tau\\tau}$.\nEven though the initial diagonal components of energy-momentum tensor, $T^{xx}$, $T^{yy}$ and $T^{\\eta\\eta}$, are not used directly to determine the background energy-momentum tensor $T^{\\mu\\nu}_{\\mathbf{x}_0}$, it is sufficient to assume that (on average) the system is highly anisotropic in the longitudinal direction, but approximately isotropic in the transverse plane, so that $T^{\\tau\\tau}$ can be used to estimate the diagonal components, i.e.\\ $\\tau^2 T^{\\eta\\eta}\\ll T^{xx}\\approx T^{yy}\\approx T^{\\tau\\tau}\/2$. This assumption is justifiable at early times in central heavy ion collisions.}.\\\\\n\n\n\n\\paragraph{ Background evolution \\& scale parameter}\nOnce the decomposition in background and perturbations is determined at each \npoint $\\mathbf{x}_0$ in the transverse plane of the collision, we proceed to calculate the evolution of the background components\nof the energy-momentum tensor. Since in the relevant range of parameters the \neffective kinetic theory evolution exhibits a universal behavior in terms of \nthe scaling variable $\\frac{\\tau \\TId}{\\eta\/s}$ (see \\Sec{subsec:background}), we can \nimmediately obtain the evolution for a specified values of $\\eta\/s$ by matching \nthe point associated with the initial energy density \n$e(\\tau_{\\scriptscriptstyle \\text{EKT}})=\\overline{T}^{\\tau\\tau}_{\\mathbf{x}_0}(\\tau_{\\scriptscriptstyle \\text{EKT}})$ at time $\\tau_{\\scriptscriptstyle \\text{EKT}}$ to \nthe universal scaling curve. Specifically, we determine the scale parameter \n$\\Lambda_T$ (which fixes the temperature function $T_\\text{id.}(\\tau; \\Lambda_T)$, see \\Eq{eq:TId}) by solving the implicit \nequation\n\\begin{multline}\ne(\\tau_{\\scriptscriptstyle \\text{EKT}})= \n\\nu_{g}~\\frac{\\pi^2}{30}~T_\\text{id.}^4(\\tau_{\\scriptscriptstyle \\text{EKT}};\\Lambda_T) \\\\ \\times \\mathcal{E}\\left[x=\\frac{\\tau_{\\scriptscriptstyle \\text{EKT}}\n T_\\text{id.}(\\tau_{\\scriptscriptstyle \\text{EKT}};\\Lambda_T) }{\\eta\/s}\\right],\\label{eq:implicit}\n\\end{multline}\nwhere $\\mathcal{E}(x)$ corresponds to the universal scaling curve for the \nevolution of the energy density (c.f. \\Sec{subsec:background} and \\app{sec:paramtr}). \n\\Eq{eq:implicit} is the requirement that the initial time and energy density, i.e.\\ $\\tau_{\\scriptscriptstyle \\text{EKT}}$ and\n$e(\\tau_{\\scriptscriptstyle \\text{EKT}})$, lie somewhere on the scaling curve shown in \\Fig{fig:scaledTmunu}.\nOnce the temperature \nfunction $T_\\text{id.}(\\tau; \\Lambda_T)$ is known, the energy density at the hydrodynamic \ninitialization time $\\tau_\\text{hydro}$ can be read off from the same universal curve as\n\\begin{multline}\ne(\\tau_\\text{hydro}) = \n\\nu_{g}~\\frac{\\pi^2}{30}~T_\\text{id.}^4(\\tau_\\text{hydro};\\Lambda_T) \\\\ \\times \\mathcal{E}\\left[x=\\frac{\\tau_\\text{hydro} \nT_\\text{id.}(\\tau_\\text{hydro};\\Lambda_T) }{\\eta\/s}\\right]\\;.\n\\end{multline}\nSimilarly, the background longitudinal and transverse pressure components, $P_L=\\tau^2 T^{\\eta\\eta}$ and $P_T=T^{ii}$, can be determined from the scaling curve as \ndetailed in \\App{sec:paramtr}, and we obtain the background energy-momentum tensor $\\overline{T}^{\\mu\\nu}_{\\mathbf{x}_0}$ at time $\\tau_\\text{hydro}$.\\\\\n\n\\paragraph{ Energy \\& momentum perturbations}\nNext we propagate the initial energy and momentum \nperturbations, $\\delta T_{\\mathbf{x}_0}^{\\tau \\tau}(\\tau_{\\scriptscriptstyle \\text{EKT}},\\mathbf{x})$ and $\\delta T_{\\mathbf{x}_0}^{\\tau i}(\\tau_{\\scriptscriptstyle \\text{EKT}},\\mathbf{x})$, to calculate the contributions to all components of the full energy-momentum tensor at hydrodynamic initialisation time $\\tau_\\text{hydro}$. We use the pre-calculated linear kinetic response functions $G^{\\mu\\nu}_{\\alpha\\beta}$ discussed in \\Sec{sec:response}. First, the tabulated Fourier-space\n functions $\\tilde{G}^{\\mu\\nu}_{\\alpha\\beta}\\big(\\frac{\\tau \\TId}{\\eta\/s}, |\\mathbf{k}|(\\tau-\\tau_0)\\big)$ are \n transformed to the coordinate space \nfor the relevant values of scaled time $\\frac{\\tau \\TId}{\\eta\/s}$ and radius\n$|\\mathbf{x}-\\mathbf{x}_0|\/(\\tau_\\text{hydro}-\\tau_{\\scriptscriptstyle \\text{EKT}})$, see \\app{app:Tmunu_decompose} for details\\footnote{We note that in practice the high $|\\mathbf{k}|$ \ntails are regulated by a small Gaussian damping and free-streaming extensions \nto stabilize numerical calculation of Fourier\/Hankel \ntransforms~\\cite{Keegan:2016cpi}.}. Subsequently, the contributions to the \nenergy-momentum tensor at each point $\\mathbf{x}_0$ at the hydrodynamic initialisation surface \n$\\tau_\\text{hydro}$ are determined by convoluting the coordinate space \nresponse functions with the initial energy and momentum perturbation as in \n\\Eq{eq:dTmunuconvolv}. The coordinate space response functions contributing to $\\delta \nT^{\\mu\\nu}(\\tau_\\text{hydro},\\mathbf{x}_0)$ have only limited support, namely the neigbourhood of points $\\mathbf{x}$ in the causal past of $\\mathbf{x}_0$\n\\begin{equation}\n|\\mathbf{x}-\\mathbf{x}_0|<(\\tau_\\text{hydro}-\\tau_{\\scriptscriptstyle \\text{EKT}}),\n\\end{equation}\nso in practice only a small number of spatial points contribute. \nWe note \nthat, according to the decompositions in Eqs.~\\eqref{eq:G_energy_decomp} and \n\\eqref{eq:G_momentum_decomp}, we explicitly compute the contributions of \nenergy and momentum perturbations to all components of energy-momentum tensor, \nwithout ever enforcing constitutive relations by hand. Adding the perturbations and the background produces the complete energy-momentum tensor at the end of \\kompost{} evolution\n\\begin{equation}\nT^{\\mu\\nu}(\\tau_\\text{hydro},\\mathbf{x}_0)=\\overline{T}^{\\mu\\nu}_{\\mathbf{x}_0}(\\tau_\\text{hydro})+\\delta \nT^{\\mu\\nu}_{\\mathbf{x}_0}(\\tau_\\text{hydro},\\mathbf{x}_0).\n\\end{equation}\n\n\\paragraph{Decomposition of $T^{\\mu\\nu}$ in hydrodynamic variables}\nOnce the full energy-momentum tensor is obtained at hydrodynamic initialization \ntime $\\tau_\\text{hydro}$, we perform a standard tensor decomposition into hydrodynamic \nvariables~\\cite{LandauFluids,Kovtun:2012rj}. Hydrodynamic simulations of heavy ion collisions describe the spacetime evolution of the \nenergy-momentum tensor $T^{\\mu\\nu}$ in terms of the energy density $e$, the \nflow velocity $u^\\mu$, the bulk pressure $\\Pi$ and the shear-stress tensor \n$\\pi^{\\mu\\nu}$. The evolution of these fields is given by the energy-momentum \nconservation equation $\\nabla_\\mu T^{\\mu\\nu}=0$ and, in second-order Israel-Stewart formulations of relativistic viscous hydrodynamics~\\cite{Israel:1976tn}, relaxation-type \nequations for the viscous fields, $\\Pi$ and $\\pi^{\\mu\\nu}$. The bulk pressure $\\Pi$ for conformal systems is exactly zero and the explicit expressions of $T^{\\mu\\nu}$ in terms of hydrodynamics fields is then given by\n\\begin{equation}\nT^{\\mu\\nu}= e u^\\mu u^\\nu + p (e) \\Delta^{\\mu\\nu}+\\pi^{\\mu\\nu}\\;,\n\\end{equation}\nwhere $\\Delta^{\\mu\\nu}\\equiv g^{\\mu\\nu}+u^\\mu u^\\nu$ and the relation between energy and pressure is determined by the equation of state $p=p(e)$\\footnote{In this paper we use mostly-plus metric convention $g^{\\mu\\nu}=\\text{diag}(-1,1,1,\\tfrac{1}{\\tau^2})$.}.\nThe tensor decomposition is done by identifying the local rest-frame velocity $u^{\\mu}$ and local energy density $e$ as the time-like eigenvector and eigenvalue of \n$T^{\\mu}_{\\phantom{\\mu}\\nu}u^\\nu= -e u^{\\mu}$.\nOnce $e$ and $u^{\\mu}$ values are known, $\\pi^{\\mu\\nu}$ is obtained by tensor \nprojection\\footnote{$\\pi^{\\mu\\nu}$ is the symmetric transverse traceless part of $T^{\\mu\\nu}$, i.e.\\ $\\pi^{\\mu\\nu}=\\pi^{\\nu\\mu}$, $\\pi^{\\mu\\nu}u_\\nu=0$ and $\\pi^{\\mu\\nu}g_{\\mu\\nu}=0$.}. \nThe independent hydrodynamic fields $e$, $u^\\mu$ and $\\pi^{\\mu\\nu}$ are then passed to the subsequent hydrodynamic evolution.\n\nNote that the linearized kinetic theory evolution does not guarantees the existence of a local fluid rest frame for arbitrary inputs; for sick cases, the procedure fails to find a meaningful rest frame. Such instances appear for the cases where the initial gradients are \nparticularly steep (e.g.\\ edges or peaks of the medium). \nProblems in extracting the flow velocity $u^{\\mu}$ from $T^{\\mu\\nu}$ in certain \nspatial regions are thus indicative of the linear approximation of the kinetic \ntheory being pushed too far.\n Although these points make only a small fraction of the total points in the transverse extent of the fireball (quantified below), they tend to introduce instabilities in hydrodynamics code.\nRather than attempting to address the problem \nin the hydrodynamic evolution, we developed a selective regulator of the kinetic \ntheory output which we describe at the end of this section.\n\n\n\\paragraph{Hydrodynamic evolution\\label{sec:implem_hydro}}\n\nOnce the hydrodynamic fields are initialized on a constant $\\tau=\\tau_\\text{hydro}$ \nhypersurface, their subsequent spacetime evolution is determined by second \norder relativistic hydrodynamics. In this paper we use the publicly available viscous relativistic hydrodynamic code MUSIC~\\cite{Schenke:2010nt,Schenke:2010rr,Paquet:2015lta} to solve numerically the hydrodynamic equations and the subsequent particlization, which is described below. \nFor the hydrodynamic phase we use a lattice-based QCD equation of state~\\cite{Huovinen:2009yb}, except when comparing to the conformal equation of state $p=e\/3$. As \nfor the first order transport coefficients, a constant shear viscosity over \nentropy density ratio $\\eta\/s=2\/4\\pi\\approx 0.16$ is used, while the bulk viscosity is \nneglected. \nThe second order transport coefficients are determined by relating them to the first order ones in the relaxation-time \napproximation~\\cite{Denicol:2012cn,Molnar:2013lta}. The complete list of second \norder transport coefficients and hydrodynamic equations can be found in Ref.~\\cite{Ryu:2017qzn}.\n \n\n\\paragraph{Hadronization\\label{sec:implem_hadr}}\n\nAt the end of hydrodynamic evolution the hadronic observables are computed from a constant temperature freeze-out surface with $T_\\text{FO}=145\\,\\text{MeV}$ using the standard Cooper-Frye procedure~\\cite{Cooper:1974mv}. We note that for simplicity only thermal hadronic observables, i.e.\\ without hadronic decays, are used in this work, but the viscous corrections to the hadronic momentum distribution are taken into account as described in Refs.~\\cite{Paquet:2015lta,Ryu:2017qzn}.\n\n\\paragraph{Regulators}\\label{par:regulators}\n\nBelow we document the regulator procedure for the limited instances when the energy-momentum tensor $T^{\\mu\\nu}$ obtained at the end of the linear kinetic evolution cannot be inverted to define a local fluid restframe.\nThe regulator we developed was motivated by free streaming, which is a meaningful\nand robust initial stage model.\nThe regulator identifies regions of large gradients or low \ndensities and drives the kinetic theory response towards free streaming in these\nregions by selectively lowering the scaling variable $x=\\tau T_\\text{id.}(\\tau)\/(\\eta\/s)$ \nto compute the kinetic response. We\nemphasize that while the regulator is important for making the hydrodynamic simulations \nrun, it produces only minimal modifications of the hydrodynamic input and does not affect the physical results discussed in \\Sec{sec:results}. Thus, our pragmatic purpose here is to document the computer code.\n\n\nPhysically in heavy ion collisions the low density regions at the edges of the fireball can be never meaningfully described as a hydrodynamic medium. Hence there is no need to assume that \n$\\eta\/s$ is constant throughout. For the purposes of estimating \nthe scaling variable we therefore define $(\\eta\/s)(T)$ which grows large in regions of low density, i.e.\n\\begin{equation}\n (\\eta\/s)(T)\\equiv \\left(\\eta\/s\\right)_0 \\left(1 + \\frac{C_0^2}{T^2} \\right) \\, ,\n\\end{equation}\nwhere $C_0=100\\,{\\rm MeV}$, and $(\\eta\/s)_0$ is a constant physical input parameter, typically of order $1\/4\\pi$\\footnote{$(\\eta\/s)_0$ can easily be replaced with a physical dependence on temperature for the interior of the fireball in future implementations of the code.}. The scaling variable \nis then replaced by the number of \nrelaxation times between times $\\tau_1$ and $\\tau_2$\n \\begin{align}\n\\label{eq:x0}\n x_{0}(\\tau_2, \\tau_1) \\equiv&\n \\frac{3}{2} \\int^{\\tau_2}_{\\tau_1} d\\tau \\frac{T_\\text{id.}(\\tau)}{(\\eta\/s)_0\\left(1 + \\frac{C_0^2}{T_\\text{id.}^2(\\tau)}\\right) } \\, .\n \\end{align}\nTaking $C_0$ and $\\tau_1$ to zero, returns $x_{0}$ to the canonical scaling variable\n\\begin{equation}\n\\label{xcanonical}\n x \\equiv \\frac{\\tau T_\\text{id.}(\\tau)}{(\\eta\/s)_0 } \\, .\n\\end{equation}\nFor obtaining the background energy density from the scaling curve, \\Eq{eq:universalE}, we use the scaling variable value $x_{0}(\\tau_\\text{hydro},0)$, while for propagating\nthe perturbations \nwe use $x_{0}(\\tau_\\text{hydro},\\tau_{\\scriptscriptstyle \\text{EKT}})$, which provides a slightly better scaling parametrization\nof the Green functions\\footnote{Typically $\\tau_\\text{hydro}\\gg\\tau_{\\scriptscriptstyle \\text{EKT}}$ and the difference between the two values of the scaling variable is small.}.\nUsing $x_{0}$ as opposed to the canonical value, \\Eq{xcanonical}, removes a number of instabilities near the edge of the grid, but does not regulate the occasional regions of very high gradients in the central region of the fireball. \n\nThe remaining instabilities arise when non-linearities become important. Examining \nwhen the $T^{\\mu\\nu}$ decomposition fails to find a rest frame, we determined (semi-empirically)\nthat for\n\\begin{equation}\n z \\equiv \\frac{ \\sqrt{\\delta T^{0x} \\delta T^{0x} + \\delta T^{0y} \\delta T^{0y}} }{\\tfrac{4}{3} \\overline{T}^{00} } - \\frac{2}{3} \\frac{\\delta T^{00}}{\\overline{T}^{00}} > 0.5 \\, , \\label{eq:z}\n\\end{equation}\nthe $T^{\\mu\\nu}$ decomposition may fail to find a rest frame. Certainly\nwhen $z$ is of order $0.5$ the linearized kinetic theory has reached its limit of applicability.\nFor large $z$ we regulated $x_{0}$ according to\n\\begin{equation}\n x_{\\rm reg} = x_{0} \\, S(z; z_1, z_2) \\, ,\\label{eq:reg}\n\\end{equation}\nwhere $S(z; z_1, z_2)$ is a monotonic cubic spline interpolating between unity for $z< z_1$, and zero \nfor $z > z_2$. In practice we take $z_1 = 0.4$ and $z_2=0.7$.\n When a regulated value $x_{\\rm reg}$ is used as opposed to $x_{0}$ the dynamics is pushed closer to free streaming limit in localised regions of steep gradients as shown in \\Fig{fig:regulator}.\n\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{Ttautau_vs_x}\n\t\\caption{Profile of the energy density $T^{\\tau\\tau}$ along the $y$-direction for a kinetic theory evolution with and without a regulator, \\Eq{eq:reg}. For points with large gradients, the system response is driven towards free-streaming evolution. Note that only a small fraction of the total number of grid points in the transverse plane are affected, see Table~\\ref{table:reg}. Here IP-Glasma initial conditions are used with pre-equilibrium evolution from $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.2\\,\\text{fm}$ to $\\tau_\\text{hydro}=1.0\\,\\text{fm}$.}\n\t\\label{fig:regulator}\n\\end{figure}\n\n\nTechnically the event is processed in two passes. In the first pass no regulator is used,\nand the scaling variable $x_0$ is calculated according to \\Eq{eq:x0}. From the $\\overline{T}^{\\mu\\nu} + \\delta T^{\\mu\\nu}$\nof the first pass\nwe record the size of the unregulated perturbations as measured by the variable $z$ in \\Eq{eq:z}, but we do\nnot attempt to find a local fluid rest frame yet. In the \nsecond pass we use $z$ from the first pass to determine a regulated value of scaling variable $x_{\\rm reg}$, \\Eq{eq:reg}. Then $x_{\\rm reg}$ is finally used\nto propagate the background and perturbations from $\\tau_{\\scriptscriptstyle \\text{EKT}}$ to $\\tau_\\text{hydro}$. The second pass $T^{\\mu\\nu} = \\overline{T}^{\\mu\\nu} + \\delta T^{\\mu\\nu}$ has a rest frame decomposition which is passed on to the hydrodynamics code. We quantify the effect of the regulator by looking at the relative change in $T^{\\tau\\tau}$ component with and without a regulator\n\\begin{equation}\n\\delta \\equiv \\frac{\\int d^2 x \\left| T^{\\tau\\tau}_{\\textrm{w\/ reg}} - T^{\\tau\\tau}_{\\textrm{w\/o reg}} \\right|}{\\int d^2 x \\left| T^{\\tau\\tau}_{\\textrm{w\/o reg}} \\right| }\\label{eq:delta}.\n\\end{equation}\nThe relative change $\\delta$ for MC-Glauber and IP-Glasma initial conditions for different evolution times is recorded in Table~\\ref{table:reg}. The transverse momentum flow grows with time and, according to criterion in \\Eq{eq:z}, more points need to be regulated. However, from Table~\\ref{table:reg} it is clear that only a small fraction of points in the transverse extend of the fireball are affected and only for longest evolution times the change is at a few percent level.\n\n\\begin{table}\n\t\\centering\n\\begin{tabular}{|c|c|c|}\n\t\\hline \n\t& MC-Glauber & IP-Glasma \\\\ \n$\\tau_\\text{hydro}$ (fm)\t& $\\delta$ & $\\delta$\\\\ \n\t\\hline \n0.4\t & 0.002 & 0.001 \\\\ \n\t\\hline \n0.6\t & 0.004 & 0.005 \\\\ \n\t\\hline \n0.8\t & 0.006 & 0.01 \\\\ \n\t\\hline \n1.0\t & 0.01 & 0.03 \\\\ \n\t\\hline \n1.2\t & 0.02 & 0.05 \\\\ \n\t\\hline \n\\end{tabular} \n\\caption{Effect of the regulator as quantified by relative change in integrated energy density, \\Eq{eq:delta}, for MC-Glauber and IP-Glasma initial conditions, and different kinetic evolution times.\\label{table:reg}}\n\\end{table}\n\n\n\n\n\n\n\n\n\\section{Event-by-event pre-equilibrium dynamics \\& matching to viscous hydrodynamics}\n\\label{sec:results}\nWe will now illustrate the applicability of our framework to perform event-by-event simulations of the pre-equilibrium dynamics of high-energy heavy-ion collisions. Since the kinetic theory equilibration scenario described in the previous sections provides a smooth crossover from the early stage of heavy ion collisions to the viscous hydrodynamics regime, we will demonstrate with the example of two initial state models how initial conditions for hydrodynamic simulations can be obtained within our framework. \n\nWe first consider the Monte Carlo Glauber (MC-Glauber) model~\\cite{Miller:2007ri}, which provides a phenomenological ansatz for the energy deposition in the transverse plane of heavy ion collisions, based on the location of binary nucleon collisions. Since MC-Glauber is a not a dynamical model, most phenomenological studies use an initialization time $\\tau_\\text{hydro}\\sim 0.5-1\\,\\text{fm}$ which is chosen empirically. However, we will show that the framework described in this work greatly reduces the sensitivity of the hydrodynamic evolution to the initialization time $\\tau_\\text{hydro}$. \n\nIn the second part of this section, we will also consider the IP-Glasma model~\\cite{Schenke:2012wb,Schenke:2012fw}, which provides a dynamical description of particle production and energy deposition. In this model color fields in each nucleus are sampled from a saturation model~\\cite{Bartels:2002cj,Kowalski:2003hm} and subsequently evolved with classical Yang-Mills evolution to times $\\tau\\sim 1\/Q_s \\sim 0.1$~fm. Since the early time dynamics of IP-Glasma matches smoothly onto our effective kinetic description, this implementation amounts to a complete dynamical evolution within a weak coupling framework. \n\n\n\nWhile the microscopic IP-Glasma model provides an initialization for the entire energy-momentum tensor of the collision in 2+1D, the MC-Glauber model is typically used as an ansatz only for the transverse energy density\\footnote{It is also common to use Glauber model as an ansatz for the entropy density, which is then related to the energy density through the equation of state. In this work, the Glauber model is used as an ansatz for the energy density directly.}, without specifying the other components of the energy-momentum tensor. In the language of this paper, this means that IP-Glasma initial conditions provide both energy and momentum perturbations\\footnote{We note that IP-Glasma also provides higher order fluctuations, e.g.\\ of the different $T^{ij}$ components. However, as discussed in \\Sec{sec:generalresponse} we limit ourselves to the energy-momentum response to the fluctuations of conserved quantities like energy and momentum.}, \nwhile the Glauber model only contains energy perturbations. \n\nWe note that the hydrodynamic initialization time $\\tau_\\text{hydro}$ is treated as a variable in this section. The values of $\\tau_\\text{hydro}$ used are of the order of the \\emph{background} hydrodynamization time given by Eq.~\\ref{eq:hydrotime}, but not equal to it.\n\n\n\n\n\n\n\n\n\\subsection{Energy perturbations with MC-Glauber initial conditions\\label{sec:glauber}}\n\nWe start our discussion with the Monte Carlo Glauber initial conditions, which provides an ansatz for the energy density distribution $e(\\mathbf{x})$ at each point in the transverse plane of the collision\\footnote{We used the publicly available T\\raisebox{-0.5ex}{R}ENTo code~\\cite{Moreland:2014oya} to generate the event. In our study, the reduced nuclear thickness parameter $p$ of the model was set to unity to obtain a typical participant Glauber model, while the negative binomial parameter for nucleon fluctuations was set to $k=1$, and the nucleon smearing width $w=0.5\\,{\\rm fm}$.}. Energy is deposited at the location of the participant nucleons, and the normalization of the energy distribution (i.e. total deposited energy) is adjusted so as to reproduce experimentally observed charged hadron multiplicities for typical central Pb+Pb events at LHC energies $\\sqrt{s_{NN}}=2.76\\,\\text{TeV}$. The energy-momentum tensor at time $\\tau_{\\scriptscriptstyle \\text{EKT}}$ takes the form\n\\begin{equation}\n\t\\label{eq:GlauberInitial}\n\tT^{\\mu\\nu}(\\tau_{\\scriptscriptstyle \\text{EKT}},\\mathbf{x})=\\begin{pmatrix} e(\\mathbf{x}) &0 &0 &0\\\\\n\t\t0 & \\frac{1}{2} e(\\mathbf{x}) & 0 & 0\\\\\n\t\t0 & 0 & \\frac{1}{2} e(\\mathbf{x}) & 0\\\\\n\t\t0 & 0 & 0 & 0 \n\t\\end{pmatrix\n\\end{equation}\n Even though the Glauber model itself does not provide an intrinsic time scale $\\tau_{\\scriptscriptstyle \\text{EKT}}$ for the dynamic evolution, it is clear that this time scale should be at least on the order of the formation time $\\sim 1\/Q$ required for semi-hard particles to go on-shell. Since we anticipate the typical momenta $Q\\sim 2\\,\\text{GeV}$ for central Pb+Pb events, we will use $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.1\\,\\text{fm}$ in the following if not stated otherwise\\footnote{We checked explicitly that the sensitivity of our results to this choice is relatively small, as long as the initial energy density is rescaled by an appropriate factor, which can be deduced from the relation $e\\tau = e_0\\tau_0+\\int_{\\tau_0}^{\\tau}d\\tau T^{zz}$ for Bjorken expansion. Since at very early times $T^{zz} \\ll e$, such rescaling effectively mimics a free-streaming evolution.}. Similarly, the specific form of the energy-momentum tensor $T^{\\mu\\nu} =e (\\tau_{\\scriptscriptstyle \\text{EKT}},\\mathbf{x}) \\times \\text{diag}\\left( 1,1\/2,1\/2,0 \\right)$ in $(\\tau,x,y,\\eta)$ coordinates, can be motivated from the fact that due to the kinematics of high-energy collisions, the longitudinal momentum of each particle in the local rest frame is negligibly small compared to its transverse energy, such that the longitudinal pressure approximately vanishes at very early times (c.f. \\Sec{sec:ipglasma} for a microscopic description of the early time dynamics). Since the $T^{\\tau i}$ momentum components of the energy-momentum tensor are also initialized as zero in the MC-Glauber model, the effective kinetic theory evolution of the energy-momentum tensor in \\Eq{eq:GlauberInitial} involves only energy perturbations.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{trento3d_tauEKT}\n\t\\caption{Transverse density of $\\tau e^{3\/4}\\sim s\\tau$ for MC-Glauber \n\tevent used in \\Sec{sec:glauber} at initial time $\\tau_\\text{EKT}=0.1\\,\\text{fm}$ corresponding to a central PbPb event at center-of-mass energy $\\sqrt{s_{NN}}=2.76\\,\\text{TeV}$. }\n\t\\label{fig:trento3dtauekt}\n\\end{figure}\n\nWe used a single MC-Glauber event with $b=0.9\\,{\\rm fm}$ impact parameter, $N_{\\rm part} =408$ participants and spatial eccentricity $\\epsilon_2 = 0.064$. The total transverse energy per unit rapidity in the event at $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.1\\,{\\rm fm}$ was set to \n$\n\\int d^2\\mathbf{x}\\, \nT^{\\tau\\tau} = 5.3 \\times 10^4\\, {\\rm GeV\/fm}\n$\nto normalize the energy density distribution.\n The transverse distribution of a proxy quantity $\\tau e^{3\/4}\\sim s\\tau$ for entropy per rapidity is shown in \\Fig{fig:trento3dtauekt} for reference.\n Ultimately, after pre-equilibrium, hydrodynamic evolution, and freeze-out, this event corresponds to a midrapidity charged hadron multiplicity of $dN_{\\textrm{ch}}\/d\\eta=1870$. The event is thus in line with a central LHC at $\\sqrt{s_{NN}}=2.76\\,\\text{TeV}$, which, for reference, have $dN_{\\textrm{ch}}\/d\\eta=1601\\pm 60$ and $\\langle N_{\\textrm{part}} \\rangle=383\\pm 3$ for the 5\\% most central events~\\cite{Aamodt:2010cz}.\n\n\nThe energy momentum tensor in Eq.~(\\ref{eq:GlauberInitial}) features a large pressure anisotropy indicating that the system at $\\tau_{\\scriptscriptstyle \\text{EKT}}$ is still far from local equilibrium, \nand cannot be described properly by ordinary viscous hydrodynamics. However, the use of \\kompost{} to describe the subsequent pre-equilibrium evolution ($\\tau_{\\scriptscriptstyle \\text{EKT}}<\\tau<\\tau_\\text{hydro}$) leads to the onset of hydrodynamic behavior that can be used as proper initial conditions for hydrodynamic evolution at $\\tau_\\text{hydro}$.\nThe overlap in the range of validity of the pre-equilibrium and hydrodynamic phase ensures that the subsequent evolution is essentially independent on the switching time $\\tau_\\text{hydro}$. In practice, the smoothness of the transition from the early stage of heavy ion collisions to hydrodynamics can be quantified in multiple manners. In what follows, we look at averages and profiles of the hydrodynamics fields --- or equivalently the energy-momentum tensor -- as well as hadronic observables, and investigate their dependence on the hydrodynamic initialization time $\\tau_{\\textrm{hydro}}$.\n\n\\subsubsection{Average hydrodynamic fields}\n\\label{sec:Glauber_average}\n\n\\begin{figure*}\n\t\\centering\n(a-c) with lattice QCD equation of state\\\\\n\\subfig{a}{\\includegraphics[width=0.3\\linewidth]{glauber_qcd_energy34_vs_time}}\\quad\n\\subfig{b}{\\includegraphics[width=0.3\\linewidth]{glauber_qcd_radial_v_vs_time}}\\quad\n\\subfig{c}{\\includegraphics[width=0.31\\linewidth]{glauber_qcd_epsilon_prime_p_vs_time_v2}}\n(d-f) with conformal equation of state\\\\\n\\subfig{d}{\\includegraphics[width=0.3\\linewidth]{glauber_conformal_energy34_vs_time_v2}}\\quad\n\\subfig{e}{\\includegraphics[width=0.3\\linewidth]{glauber_conformal_radial_v_vs_time_v2}}\\quad\n\\subfig{f}{\\includegraphics[width=0.31\\linewidth]{glauber_conformal_epsilon_prime_p_vs_time_v2}}\n\t\\caption{(top row) The transverse averages (as defined in \\Eq{eq:energy_average}) of (a) $\\tau \\epsilon^{3\/4}$ and (b) transverse \n\tvelocity $v_\\perp$, and (c) momentum \n\teccentricity (as defined by \\Eq{eq:momentum_aniso}) in the hydrodynamic phase as a function of \n\ttime $\\tau$. The different lines correspond to different \n\thydrodynamic initialization time $\\tau_{\\textrm{hydro}}$, i.e.\\ different duration of kinetic pre-equilibrium evolution. The initial \n\tcondition of the effective kinetic theory at time \n\t$\\tau_{\\scriptscriptstyle \\text{EKT}}=0.1\\,\\text{fm}$ is a central participant MC-Glauber event \n\tnormalized to correspond to a $\\sqrt{s_{NN}}=2.76$~TeV Pb-Pb collisions (see \\Fig{fig:trento3dtauekt}). (bottom row) the same as (a-c), except a conformal equation of state is used in the hydrodynamics evolution instead of a lattice QCD one (see \\Fig{fig:conformality}).}\n\t\\label{fig:average_glauber_conformal}\\label{fig:average_glauber_qcd}\n\\end{figure*}\n\n\n\nSince realistic fluctuating initial conditions are used, hydrodynamic fields have a complicated profile in the transverse $xy$-plane. As a starting \npoint, we look at transversely averaged values of hydrodynamic fields. We define averages $\\left<\\ldots\\right>$ as\n\\begin{equation}\n\\left<\\ldots\\right>\\equiv \\frac{\\int d^2\\mathbf{x} u^\\tau e \\ldots}{\\int d^2\\mathbf{x} u^\\tau e}\n\\label{eq:energy_average}\n\\end{equation}\nwhere $\\int d^2\\mathbf{x}$ denotes an integral over the transverse coordinates. The factor $u^\\tau$ is inserted for covariance, as would be obtained from the (covariant) surface flux $d\\Sigma_\\mu u^\\mu$ with $d\\Sigma_\\mu$ in the (proper) time direction.\n\nIn \\Fig{fig:average_glauber_qcd}(a) we present the evolution of $\\left<\\tau e^{3\/4}\\right>$, which is akin to the entropy per rapidity $\\left$ of the boost-invariant system, as a function of physical time $\\tau$. Initial conditions at $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.1$~fm are evolved with the kinetic theory in \\kompost{} (c.f. Sec.~\\ref{sec:implementation}), and subsequently passed to the hydrodynamic model at five values of $\\tau_{\\textrm{hydro}}$: $0.4, 0.6, 0.8, 1.0\\text{ and }1.2\\,\\text{fm}$. \nWe first note that \\Fig{fig:average_glauber_qcd}(a) shows the expected transition from $\\left<\\tau e^{3\/4}\\right>$ being approximately constant at early time to subsequently dropping when the transverse expansion becomes important.\nMore importantly one observes from \\Fig{fig:average_glauber_qcd}(a) that the evolution of $\\langle\\tau e^{3\/4}\\rangle$ depends very weakly on the \nhydrodynamics initialization time $\\tau_\\text{hydro}$.\n\nIn \\Fig{fig:average_glauber_conformal}(b) we show the rise of radial velocity $\\left< v_\\perp \\right>$=$\\left< \\sqrt{v_x^2+v_y^2} \\right>$ with time $\\tau$. The kinetic theory pre-equilibrium captures well the rapid rise in the radial flow at early times, which levels off to a steady radial increase at later times. It is again remarkable that the spread between the calculations for different values of $\\tau_\\text{hydro}$ between $0.4$~fm and $1.2$~fm is at a percent level. \n\nIn order to follow the evolution of the azimuthal anisotropy, we computed\nthe integrated transverse stress tensor $[T^{ij}]_s = ([T^{xx}]_s, [T^{xy}]_s, [T^{yy}]_s)$, where \n\\begin{equation}\n\\left[\\ldots\\right]_s \\equiv \\int d^2\\mathbf{x} u^\\tau \\ldots \\, .\n\\end{equation}\n denotes an integral over the transverse plane without the energy weight, as $T^{\\mu\\nu}$ is already ``energy weighted''.\nExamining the principal axes of $[T^{ij}]_s$, we define the \nmomentum ellipticity\n\\begin{equation}\n\\varepsilon_p^\\prime(\\tau) \\equiv \\frac{\\sqrt{( \\left[T^{xx}\\right]_s-\\left[T^{yy}\\right]_s)^2+4\\left[T^{xy}\\right]_s^2}}{\\left[T^{xx}\\right]_s+\\left[T^{yy}\\right]_s} \\, ,\n\\label{eq:momentum_aniso}\n\\end{equation}\nwhich provides a measure of the elliptic flow as function of time.\n\nOur results for momentum ellipticity are shown in\n\\Fig{fig:average_glauber_qcd}(c), and indicate that the dependence \nof the averaged hydrodynamic fields on $\\tau_\\text{hydro}$ is modest even if\n$\\tau_\\text{hydro}$ is varied from $0.4$ to $1.2$~fm. Generally, the kinetic theory\nslightly under-predicts the pressure anisotropy\n$\\varepsilon_p^\\prime(\\tau)$ in the hydrodynamic evolution. \n$\\varepsilon_p^\\prime(\\tau)$ is inherently quadratic in the flow velocity, \nsuggesting that including the first nonlinear couplings between\nthe background radial flow and the generated elliptic flow could improve \nthe agreement here.\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=0.95\\linewidth]{conformality_pressure_qcd}\n \\caption{Deviation of the QCD equation of state~\\cite{Huovinen:2009yb} from conformality, as \n quantified by the ratio $e\/(3 p)$.}\n \\label{fig:conformality}\n \\end{figure}\n\n\nThe effective kinetic theory approach used in this work assumes that the system is conformal, i.e. the pressure is $p(e)=\\tfrac{1}{3}e$. \nEven though QCD is nearly conformal at very high temperatures, deviations from conformality are expected for the range of temperatures of order $150{-}600$~MeV encountered in heavy ion collisions at the RHIC and LHC, as can be observed from Fig.~\\ref{fig:conformality}, where the ratio \n$e\/3p$ \nis shown for a QCD equation of state~\\cite{Huovinen:2009yb}. Since hydrodynamic simulations of heavy ion collisions necessarily require a realistic equation of state,\nthis leads to discontinuous matching of the energy-momentum tensor. Specifically, when the energy-momentum tensor of \\kompost{} is decomposed and passed to the hydrodynamics, the energy-momentum tensor of the hydrodynamics becomes:\n\\begin{equation}\nT^{\\mu\\nu}_{\\textrm{hydro}}=e u^\\mu u^\\nu+ p_{\\textrm{QCD}}(\\epsilon) \n\\Delta^{\\mu\\nu}+\\pi^{\\mu\\nu}\n\\label{eq:TmunuIPG}\n\\end{equation} \nwhich is not equal to $T^{\\mu\\nu}$ of the effective kinetic theory because $p_{\\textrm{QCD}}(e) \\neq p_{\\textrm{conformal}}(e)$. Unfortunately, there is no obvious way to improve on this procedure, as a better matching will ultimately require breaking the conformal symmetry in the pre-equilibrium phase which is of higher order in $\\alpha_s$. \nOn the other hand, it is straightforward to study and quantify the effects associated with this break of conformality, by replacing the QCD equation of state by a conformal one and reproducing \\Fig{fig:average_glauber_qcd}(a-c). \nThese results are presented in the bottom row of \\Fig{fig:average_glauber_qcd}, where the different panels (d-f) again show the time evolution of $\\left<\\tau e^{3\/4}\\right>$, $\\left< v_\\perp \\right>$ and $\\varepsilon_p^\\prime$. It is clear from these figures that the $\\tau_\\text{hydro}$ dependence, which was already small for a QCD equation of state, is even smaller with the conformal equation of state.\nIn particular, all of the $\\tau_\\text{hydro}$ dependence for $\\left<\\tau e^{3\/4}\\right>$ --- which amount to approximately $10\\%$ between $\\tau_\\text{hydro}=0.4$ and $1.2$~ --- is explained by the break of conformality between the initial conditions and the hydrodynamic evolution with a QCD equation of state.\nThe flow observables $\\left< v_\\perp \\right>$ and $\\varepsilon_p^\\prime$ also have a smaller dependence on $\\tau_\\text{hydro}$ when a conformal equation of state is used, as can be observed by comparing \\Fig{fig:average_glauber_qcd} (b) with (e), and (c) with (f). The effect is not as significant as for $\\left<\\tau e^{3\/4}\\right>$, however, in part because the $\\tau_\\text{hydro}$ dependence was already small in the first place with the QCD equation of state.\n\n\n\n\n\n\n\n\n\\subsubsection{Transverse plane profiles of hydrodynamic fields}\n\n\\begin{figure*}\n\t\\centering\n\t\\subfig{a}{\\includegraphics[height=0.25\\linewidth]{glauber_e_vs_x_vs_time_t12}}\\quad\n\t\\subfig{b}{\\includegraphics[height=0.25\\linewidth,trim=25 0 0 0, clip]{glauber_e_vs_x_vs_time_t20}}\\quad\n\t\\subfig{c}{\\includegraphics[height=0.25\\linewidth,trim=25 0 0 0, clip]{glauber_e_vs_x_vs_time_t50}}\n\t\\subfig{d}{\\includegraphics[height=0.25\\linewidth]{glauber_vx_vs_x_vs_time_t12}}\\quad\n\t\\subfig{e}{\\includegraphics[height=0.25\\linewidth,trim=28 0 0 0, clip]{glauber_vx_vs_x_vs_time_t20}}\\quad\n\t\\subfig{f}{\\includegraphics[height=0.25\\linewidth,trim=28 0 0 0, clip]{glauber_vx_vs_x_vs_time_t50}}%\n\t\\caption{Single event profiles along $x$-axis ($y=0$) of $\\tau e^{3\/4}$ (top row) and velocity $v^x$ (bottom row) for different hydrodynamics transition times $\\tau_\\text{hydro}$. Different columns correspond to three different times in hydrodynamic evolution: $\\tau=1.2, 2.0$ and $5.0$~fm. The same EKT initialization time $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.1$~fm was used. The equation of state is a realistic QCD one. The transverse velocity is not shown for very low energy densities ($\\tau e^{3\/4}<0.01\\,\\text{GeV}^2$) where numerical errors can generate spurious values of velocity.\t\\label{fig:glauber_profiles}}\n\\end{figure*}\n\nBased on our analysis in the previous section, the average energy, transverse velocity and momentum anisotropy were found to have a weak dependence on the value of $\\tau_\\text{hydro}$. The reason for this insensitivity to $\\tau_\\text{hydro}$ is that after a short evolution, the dynamics of energy-momentum tensor in the effective kinetic theory approaches that of hydrodynamics, and the two descriptions are approximately equivalent. However, the integrated quantities shown in the previous subsection have a significantly reduced sensitivity to any type of fluctuations in the hydrodynamic fields. In this section we show profiles of hydrodynamic fields in the transverse plane of the collisions, to emphasize that the smaller features of the fields do not show any significant sensitivity to the hydrodynamic initialization time $\\tau_\\text{hydro}$ either.\nSince the effect of the transition from a conformal kinetic theory to a non-conformal hydrodynamics evolution was quantified in the previous section, it is not revisited again here and all results that follow were obtained with a QCD equation of state.\n\n\\begin{figure*}\n\t\\centering\n\t\\subfig{a}{\\includegraphics[width=0.4\\linewidth]{pi_vs_x_vs_time_v2}}\\quad\n\t\\subfig{b}{\\includegraphics[width=0.4\\linewidth]{xscaled_vs_x_vs_time_v2}}\n\t\\caption{(a) Comparison of the out-of-equilibrium shear stress tensor (c.f.\\,\\Eq{eq:TmunuIPG}) with the Navier-Stokes estimate at different hydrodynamics initialization times $\\tau_\\text{hydro}=0.8,1.0,1.2$~fm. (b) Scaled evolution time variable $\\frac{\\tau \\TId}{\\eta\/s}$ at different hydro starting times. \n Values of $\\tau T_\\text{id.}\/(4\\pi\\eta\/s)>1$ indicate that the system is close enough to local thermal equilibrium for hydrodynamics to become applicable (see \\Sec{sec:hydtime}).\n }\n\t\\label{fig:glauber_pimunu}\n\\end{figure*}\n\nIn \\Fig{fig:glauber_profiles} we show profile plots of $\\tau \\epsilon^{3\/4}$ (top row) and the flow velocity $v^x$ (bottom row) along the $x$-axis \nat midrapidity. Different panels (a-f) show the profiles at different hydrodynamic evolution times $\\tau=1.2, 2.0$ and $5.0$~fm. The different curves on each panel correspond to a hydrodynamic \\emph{initialization} times $\\tau_\\text{hydro}$ from $0.4$ to $1.2$~fm. One observes that even for differential observables, the sensitivity to the initialization time of the hydrodynamic evolution is very small. Except for a few percent change in the overall normalization of $\\tau \\epsilon^{3\/4}$ between the different curves, which can be attributed to the mismatch between the conformal equation of state in \\kompost{} to the QCD equation of state in the hydrodynamic evolution, the profiles look essentially identical indicating a robust matching of the pre-equilibrium dynamics to viscous hydrodynamics. \n\nOne can further probe the approach of kinetic theory towards a hydrodynamic evolution by comparing the out-of-equilibrium shear-stress tensor $\\pi^{\\mu\\nu}$ \nfrom the kinetic theory evolution with an estimate from the Navier-Stokes value $\\pi^{\\mu\\nu}=-\\eta\\sigma^{\\mu\\nu}$, where $\\sigma^{\\mu\\nu}$ is calculated from \nthe velocity profile, see \\Eq{eq:sigmamunu}. In \\Fig{fig:glauber_pimunu}(a) we plot the value of $\\pi^{xx}+\\pi^{yy}$ for $\\tau_\\text{hydro}=0.8,1.0,1.2\\,\\text{fm}$ and Navier-Stokes \nestimate in dashed lines. We see that in most of the collision area the kinetic theory result approached hydrodynamic constitutive equations. One exception is \nthe sharp edges of the fireball where the small gradient assumption breaks down. However, it is not clear that either hydrodynamics or linearized kinetic theory \\`a la \\kompost{} provide an accurate description of the space-time evolution of the edges. The regions where a good matching between kinetic theory and hydrodynamics is expected can be quantified with the typical momentum relaxation time $\\tau_R(\\tau)$ defined at \\Eq{eq:tauR} in Section~\\ref{subsec:background}, using $\\tau\/\\tau_R(\\tau) \\gtrsim 4 \\pi$, i.e. $\\tau T_\\text{id.}\/(4\\pi\\eta\/s)\\gtrsim 1$. This ratio can be calculated locally\n and indicates how the approach to hydrodynamics varies in the transverse plane. The result is shown in \\Fig{fig:glauber_pimunu}(b). As estimated in \\Sec{sec:hydtime}, we find that for starting times $\\tau_\\text{hydro}>0.8\\,\\text{fm}$, most of the medium is at scaled \ntime $\\tau T_\\text{id.}\/(4\\pi\\eta\/s)>1$ where hydrodynamics becomes applicable. Since the local energy density at the edges is significantly smaller, the edges of the fireball remain at $\\tau T_\\text{id.}\/(4\\pi\\eta\/s)<1$ for a longer time, quantifying the statement that the approach to hydrodynamic behavior does not occur isochronously.\n\n\n\n\n\n\\subsubsection{Hadronic observables}\n\\label{sec:glauber_hadronic}\n\n\\begin{figure*}\n \\centering\n\\subfig{a}{ \\includegraphics[width=0.31\\linewidth]{trento_pion_N_v3}}%\n\\subfig{b}{\\includegraphics[width=0.31\\linewidth]{trento_pion_mean_v3}}%\n\\subfig{c}{\\includegraphics[width=0.31\\linewidth]{trento_pion_v2_v3}}\n \\caption{\n The thermal freeze-out pion\n (a) multiplicity $dN_\\pi\/dy$ (b) radial flow $\\langle p_T^\\pi\\rangle$ and\n (c) elliptic flow $v_2^\\pi$ as a function of hydrodynamic initialization\n time $\\tau_\\text{hydro}=0.4{-}1.2\\,\\text{fm}$, i.e. different duration of\n pre-equilibrium evolution. (Note the suppressed zero in (b) and (c).) The initial MC-Glauber conditions are specified at $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.1\\,\\text{fm}$ as indicated by the grey band. Different\n pre-equilibrium scenarios are: linearized kinetic theory evolution\n \\kompost{}, interaction-less free streaming and simple Bjorken energy\n rescaling with time $\\propto \\tau^{-4\/3}$ with no dynamics. The open symbols\n show the $\\langle p_T^\\pi \\rangle$ and $v_2^\\pi$ for the free streaming and\n Bjorken pre-equilibrium evolution from 0.1\\,fm to 1.2\\,fm with energy\n density scaled to reproduce the same pion multiplicity as \\kompost{} with\n $\\tau_\\text{hydro}=1.2\\,\\text{fm}$.\n\\label{fig:v2WithMom}\\label{fig:multWithMom}\\label{fig:meanptWithMom}}\n\t\t\\label{fig:glauber_hadronic_obs}\n\\end{figure*}\n\nBased on the successful matching of the early time pre-equilibrium stage to the subsequent hydrodynamic regime discussed in the previous sections, we now investigate the impact of a consistent description of the early time dynamics on the final state hadronic observables computed after the freeze-out of the hydrodynamic evolution. We focus on the multiplicity $dN_{\\pi}\/dy$, the average transverse momentum $\\langle p_T^{\\pi} \\rangle$ and the $v_2^{\\pi}$ of thermal pions\\footnote{We reiterate, as noted in \\Sec{sec:implementation}, that hadronic decays are not included.}, which can be thought of as analogues of the integrated hydrodynamic fields shown in \\Fig{fig:average_glauber_qcd}. We note that, although only pion observables are shown in this section, we verified that similar results are found for heavier hadrons such as kaons and protons.\n\n\nStarting with the same Glauber initial conditions at $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.1\\,\\text{fm}$, the effective kinetic theory is used to evolve the energy-momentum tensor up to five different times $\\tau_\\text{hydro}$ from 0.4~fm to 1.2~fm, as in the previous sections. Subsequently, the hydrodynamic evolution is performed up to the isothermal freeze-out where hadronic observables are calculated. In \\Fig{fig:glauber_hadronic_obs}, our results for the pion multiplicity $dN_{\\pi}\/dy$, the mean transverse momentum $\\langle p_T^{\\pi} \\rangle$ and the $v_2^{\\pi}$ are plotted as a function of $\\tau_\\text{hydro}$. In addition to the results obtained from the effective kinetic theory pre-equilibrium evolution, the dependence of hadronic observables on $\\tau_\\text{hydro}$ is also shown for two other types of pre-equilibrium evolution: free-streaming\\footnote{Our results for free-streaming evolution are obtained by replacing the kinetic evolution of the background energy $\\mathcal{E}(x)$ and the corresponding response functions $G^{\\mu\\nu}_{\\alpha\\beta}$ with their free-streaming counterparts (see \\app{sec:freestreaming}).} and simple Bjorken $\\tau^{-4\/3}$ scaling. \nThe use of free-streaming to describe the pre-equilibrium dynamics has been studied previously in~\\cite{Broniowski:2008qk,Liu:2015nwa}. Similarly, the procedure of scaling the energy density of a set initial condition with $\\tau^{-4\/3}$, as would be expected for a system undergoing ideal Bjorken hydrodynamic expansion, is also used regularly in heavy ion physics to rescale the energy density of the initial conditions when changing the initialization time of hydrodynamics.\n\nIn \\Fig{fig:multWithMom}(a) we show pion multiplicity $dN_{\\pi}\/dy$ as a function of hydrodynamic initialization time $\\tau_\\text{hydro}$ for the three different pre-equilibrium evolution scenarios. We find that for all pre-equilibrium scenarios, the multiplicity has an approximately linear dependence on $\\tau_\\text{hydro}$. For the effective kinetic theory, the multiplicity is only approximately $\\sim 5\\%$ smaller if the hydrodynamics is initialized at $\\tau_\\text{hydro}=1.2$~fm rather than $\\tau_\\text{hydro}=0.4$~fm, while in the case of the Bjorken $\\tau^{4\/3}$, this figure is significantly larger, $\\sim 15\\%$. Conversely, for a free-streaming pre-equilibrium dynamics the longitudinal pressure is underestimated during the pre-equilibrium phase, such that the energy decreases much less rapidly than in hydrodynamics, and the multiplicity is $\\sim 8\\%$ \\emph{larger} with $\\tau_\\text{hydro}=1.2$~fm than with $\\tau_\\text{hydro}=0.4\\,\\text{fm}$. We conclude that overall, the pion multiplicity has the smallest dependence on $\\tau_\\text{hydro}$ when \\kompost{} is used to describe the early stage of the medium evolution, although all curves are relatively flat. \n\n\n\n\nIn \\Fig{fig:glauber_hadronic_obs}(b) we look at the radial flow dependence on $\\tau_\\text{hydro}$, again for kinetic theory, free streaming and $\\tau^{4\/3}$ scaling. We find that for \nkinetic theory equilibration, the radial flow build-up is consistent with hydrodynamic evolution and the mean pion $\\left$ is independent of \nswitching time. In contrast, for the free streaming evolution, the overall energy scale grows slightly too rapidly, leading to a $\\sim 3\\%$ change in $\\langle p_T \n\\rangle$ if the hydrodynamics is initialized at $\\tau_\\text{hydro}=1.2$~fm rather than $\\tau_\\text{hydro}=0.4$~fm. Conversely, for Bjorken $\\tau^{4\/3}$ scaling, no pre-flow is built up during the pre-equilibrium stage resulting in a decrease by $\\sim 8\\%$ of $\\left$ for $\\tau_\\text{hydro}=1.2$~fm rather than $\\tau_\\text{hydro}=0.4$~fm. Similar observations can be made for the second order flow harmonic $v_2$, presented in \\Fig{fig:glauber_hadronic_obs}(c), albeit the overall magnitude of the variations is somewhat smaller in this case.\n\nBesides the different $\\tau_\\text{hydro}$-dependence of $\\left$ and $v_2$, it is also clear from \\Fig{fig:glauber_hadronic_obs} that the $\\left$ and $v_2$ obtained after a kinetic pre-equilibrium evolution is different from that obtained through the free-streaming and $\\tau^{4\/3}$ scaling. Of course the same is also true for the multiplicity in \\Fig{fig:glauber_hadronic_obs}(a): given the same initial conditions, $dN_\\pi\/dy$ is larger for a free-streaming evolution than a kinetic one, and smaller for $\\tau^{4\/3}$ scaling.\nIn practice, the normalization of the initial conditions of hydrodynamics simulations of heavy-ion collision is always adjusted so as to reproduce hadronic multiplicities. It is thus relevant to ask how $\\left$ and $v_2$ change for free-streaming and $\\tau^{4\/3}$ scaling if their respective initial condition normalizations are adjusted so that they produced the same pion multiplicity as the kinetic theory evolution. This result is shown with the open symbol on \\Fig{fig:glauber_hadronic_obs}, for a fixed $\\tau_\\text{hydro}=1.2$~fm.\nWe see that even if the pion multiplicity is fixed by hand, different pre-equilibrium dynamics still lead to a difference in $\\left$ and $\\left$, although this difference becomes very small for free-streaming.\nIn the case of Bjorken $\\tau^{4\/3}$ scaling, the discrepancy remains relatively large, which we attribute in part of the lack of pre-equilibrium flow velocity resulting from this simplified scaling.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Energy and momentum perturbations with IP-Glasma initial \nconditions\\label{sec:ipglasma}}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{ipglasma3d_tauEKT}\n\t\\caption{Transverse ``entropy'' density \\ $\\tau e^{3\/4}\\sim s\\tau$ for a single $\\sqrt{s_{NN}}=2.76\\,\\text{TeV}$ central Pb-Pb IP-Glasma \n\tevent at kinetic theory initialization time $\\tau_\\text{EKT}=0.2\\,\\text{fm}$. }\n\t\\label{fig:ipglasma3dtauekt}\n\\end{figure}\n\n\nBesides being applicable to general initial condition ansatzs,\nour framework of kinetic pre-equilibrium evolution can also be applied to microscopically \nmotivated initial states. In this section we use the IP-Glasma model, where the \nevolution at early times of heavy ion collisions is described in terms of classical Yang-Mills (CYM)\ndynamics~\\cite{Schenke:2012wb,Schenke:2012fw}. We note that in contrast to our previous \ndiscussion of the MC-Glauber model, the microscopic IP-Glasma model also \nincludes initial momentum fluctuations $\\delta T^{\\tau i}$, which are propagated by our kinetic theory evolution. Therefore we need the complete set of response functions discussed in \\Sec{sec:generalresponse}.\n\n\n\nSince classical-statistical field theory and effective kinetic theory \nhave an overlapping range of validity, the combination of the two \n allows for a consistent weak coupling description of \nearly time dynamics~\\cite{Mueller:2002gd,Jeon:2004dh, York:2014wja, Kurkela:2015qoa}. In principle, such combined approach describes particle production from the partonic structure of nuclei at high \nenergies to the onset of \nhydrodynamics in the quark-gluon plasma.\n\n\\begin{figure}\n\t\\centering\n\t\\subfig{a}{\\includegraphics[width=0.95\\linewidth]{early_ratio_etaOverS_dep_only_two_curves}}\n\t\\subfig{b}{\\includegraphics[width=0.95\\linewidth]{early_ratio}}%\n \\caption{Time evolution of the longitudinal and transverse \n stress tensor components,\n $\\left\\langle P_L \\right\\rangle$ and $\\left\\langle P_T \\right\\rangle$, \n averaged across the transverse plane, \n relative to the average\n background energy density of a single IP-Glasma event shown in\n \\Fig{fig:ipglasma3dtauekt}. (a) \n $\\left\\langle P_L \\right\\rangle$ and $\\left\\langle P_T \\right\\rangle$ \n in the 2+1D Yang Mills and kinetic theory codes for\n a range of $\\eta\/s$ at early times. (b) \n $\\left\\langle P_L \\right\\rangle$ and $\\left\\langle P_T \\right\\rangle$ \n in the 2+1D Yang Mills, kinetic theory, and viscous hydrodynamics codes with $\\eta\/s{=}2\/4\\pi$ for a wide range of times.\n After $\\sim 2\\,{\\rm fm}$ the 3D hydrodynamic expansion starts, and the (integrated) estimate \n $\\left\\langle P_L \\right\\rangle\/\\left\\langle e \\right\\rangle \\simeq \\tfrac{1}{3}$ does\n not serve as a useful measure of local thermal equilibrium for a flowing fluid. }\n\t\\label{fig:pressure_matching_ipglasma}\n\\end{figure}\n\nIn what follows, we use the original version of the IP-Glasma model, which is effectively 2+1D \nwith boost-invariant fields in the longitudinal direction. \nEven though a qualitative matching between classical-statistical field theory and effective kinetic theory has been \ndemonstrated in previous works~\\cite{Baier:2000sb,Kurkela:2011ub,Berges:2013fga,Kurkela:2015qoa}, no concrete implementation has been \nachieved to date for realistic full 2+1D simulation of heavy ion collisions.\nBecause of the reduction to 2+1D (boost-invariant) fields, the dynamics in IP-Glasma becomes \neffectively free-streaming after $\\tau \\sim 1\/Q_s \\sim \n\\mathcal{O}(0.1~\\textrm{fm})$ of classical Yang-Mills evolution~\\cite{Schenke:2015aqa}. This is different from the first stage of ``bottom-up\" equilibration, which is recovered in a full 3+1D classical-statistical simulation~\\cite{Berges:2013fga,Berges:2013eia}.\nHowever in practice,\nwe find that at weak coupling (larger values of $\\eta\/s$) the matching between classical Yang-Mills and kinetic theory is still rather \nsmooth, as long as the switching to the kinetic description is performed at \nsufficiently early times $(1\/Q_s \\ll \\tau_{\\scriptscriptstyle \\text{EKT}} \\ll (4\\pi \\eta\/s)\/T_\\text{id.})$ as demonstrated below. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\nIn order to illustrate the smooth matching at early times, we analyze the evolution of transversely averaged longitudinal $\\langle P_L\\rangle$ and transverse $\\langle P_T \\rangle$ \npressures defined by \\Eq{eq:defPLPT}, relative to the mean background energy density $\\langle e\\rangle $ in a particular event shown in \\Fig{fig:ipglasma3dtauekt}.\nThe IP-Glasma event used in our analysis is a central event with $N_p=404$ participant nucleons and an eccentricity $\\epsilon_2=0.126$ at time $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.2$~fm.\nThe total energy in the event is\n$\n\\int d^2{\\mathbf{x}} \\tau_{\\scriptscriptstyle \\text{EKT}} T^{\\tau\\tau}\n= 4.4 \\times 10^4\\, \\frac{{\\rm GeV}}{{\\rm fm}} ,\n$\ncorresponding to a charged hadron multiplicity of $dN_\\text{ch}\/d{\\eta}=1419$. Fluctuations smaller than $\\tau_{\\scriptscriptstyle \\text{EKT}}$ have been smeared with a Gaussian.\n\nIn \\Fig{fig:pressure_matching_ipglasma} we present the time dependence of the pressure to energy \nratio\n$\\left\/\\langle e\\rangle$ during various stages of the evolution. \nIn panel a) we see that at \nvery early times ($\\tau \\ll 1\/Q_s$) in the IP-Glasma evolution, the \nlongitudinal \npressure is negative due to the presence of strong longitudinal color fields~\\cite{Lappi:2006fp}.\nHowever, on a time scale $\\tau\\sim1\/Q_s \\sim 0.1\\,\\text{fm}$ the fields decay, and the longitudinal \npressure \napproaches zero, except for a slight overshoot due to the \nresidual pressure in the classical fields. At this time the 2+1D classical Yang-Mills dynamics becomes essentially free streaming, and one should \nthen evolve the\nsystem with QCD kinetics\nin order to describe the subsequent approach towards equilibrium. \nSince at very early times, the expansion-dominated kinetic theory is also effectively free-streaming, the classical Yang-Mills and kinetic theory evolutions\ncan be smoothly matched, provided $\\tau_{\\scriptscriptstyle \\text{EKT}} \\gtrsim 1\/Q_s$\nis small in units of the relaxation time, \n$\\tau_{\\scriptscriptstyle \\text{EKT}} T_\\text{id.}\/(4\\pi\\eta\/s)\\ll 1$. Indeed, \nin \\Fig{fig:pressure_matching_ipglasma}(a) we see that IP-Glasma to kinetic-theory transition becomes increasingly smooth \nas $4\\pi\\eta\/s$ at early times is increased from $2$ to $8$.\nThe energy density is high at such early times,\nand this would naturally lead to larger values of $\\eta\/s$ in a theory where the coupling runs.\nOf course the physics of the running coupling is absent in our leading order\nanalysis where $4\\pi\\eta\/s \\simeq 2$.\n\n\nIn \\Fig{fig:pressure_matching_ipglasma}(b) we see that during \nkinetic phase the pressure anisotropy decreases, and \nafter the system is sufficiently close to local equilibrium $\\tau \nT_\\text{id.}\/(4\\pi\\eta\/s)>1$, the subsequent evolution \ncan be smoothly matched to second order viscous hydrodynamics, as\ndiscussed in \\Sec{subsec:background}.\n\\Fig{fig:pressure_matching_ipglasma} represents the integrated stress and\nenergies across a realistic IP-Glasma event,\nwhich approximately follows a 1D Bjorken expansion\nfor the first $\\tau\\lesssim\n2.5\\,\\text{fm\/c}$. \nAfter this time the 3D hydrodynamic expansion starts, and the (integrated)\nestimate $\\left\\langle P_L \\right\\rangle\/\\left\\langle e \\right\\rangle \\simeq \\tfrac{1}{3}$ does\nnot provide a useful measure of local thermal equilibrium in the flowing fluid.\n\n\nHaving investigated the different stages of the evolution, we will evaluate \nthe transition between kinetic theory and hydrodynamics in \ngreater detail by monitoring the\nstress tensor,\nand by checking that hadronic\nobservables are independent of the crossover time $\\tau_\\text{hydro}$. The analysis parallels the\nMC-Glauber initial conditions described in \\Sec{sec:glauber}, and the discussion will highlight the differences.\n\n\n\n\n\\subsubsection{Average hydrodynamic fields}\n\\begin{figure*}\n\t\\centering\n\t(a-c) with latttice QCD equation of state\\\\\n\t\\subfig{a}{\\includegraphics[width=0.3\\linewidth]{ipglasma_qcd_energy34_vs_time_v2}}\\quad\n\t\\subfig{b}{\\includegraphics[width=0.3\\linewidth]{ipglasma_qcd_radial_v_vs_time_v2}}\\quad\n\t\\subfig{c}{\\includegraphics[width=0.3\\linewidth]{ipglasma_qcd_epsilon_prime_p_vs_time_v2}}\n\t(d-f) with conformal equation of state\\\\\n\t\\subfig{d}{\\includegraphics[width=0.3\\linewidth]{ipglasma_conformal_energy34_vs_time_v2}}\\quad\n\t\\subfig{e}{\\includegraphics[width=0.3\\linewidth]{ipglasma_conformal_radial_v_vs_time_v2}}\\quad\n\t\\subfig{f}{\\includegraphics[width=0.3\\linewidth]{ipglasma_conformal_epsilon_prime_p_vs_time_v2}}\n\t\\caption{(top row) Transverse average of $\\tau e^{3\/4}$ and of the transverse \n\t\tvelocity $v_\\perp$, as defined in \\Eq{eq:energy_average}, and momentum \n\t\teccentricity as defined by \\Eq{eq:momentum_aniso}, as a function of \n\t\ttime $\\tau$. The three average fields are plotted for different \n\t\thydrodynamic initialization time $\\tau_\\text{hydro}$, i.e. different duration of kinetic pre-equilibrium evolution. The initial \n\t\tcondition of the effective kinetic theory at time \n\t\t$\\tau_{\\scriptscriptstyle \\text{EKT}}=0.2$~fm is a central IP-Glasma events normalized to \n\t\tcorrespond to a $\\sqrt{s_{NN}}=2.76$~TeV Pb-Pb collisions (see \\Fig{fig:ipglasma3dtauekt}). (bottom row) the same as (a-c), except a conformal equation of state is used in the hydrodynamics evolution instead of a lattice QCD one. }\n\t\\label{fig:average_ipglasma_conformal}\t\\label{fig:average_ipglasma_qcd}\n\\end{figure*}\n\nAs in \\Sec{sec:Glauber_average}, we consider the transversely averaged hydrodynamic fields of energy $\\langle \\tau e^{3\/4}\\rangle$, velocity $\\langle v_\\perp \\rangle$ and momentum eccentricity $\\epsilon'_p$ (\\Eq{eq:momentum_aniso}) after the linearized kinetic pre-equilibrium evolution \\kompost{} until $\\tau_\\text{hydro}=0.4,0.6,0.8,1.0,1.2\\,\\text{fm}$ starting from IP-glasma initial conditions at $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.2\\,\\text{fm}$.\nThe transversely averaged hydrodynamic fields are shown in Fig~\\ref{fig:average_ipglasma_conformal}(a-c).\nThe transverse average of $\\tau e^{3\/4}$ and the transverse velocity $v_\\perp$ are both showing an overall small dependence on $\\tau_\\text{hydro}$ consistent with what was observed in \\Sec{sec:Glauber_average} for MC-Glauber initial conditions. The momentum eccentricity $\\epsilon_p^\\prime$ (\\Eq{eq:momentum_aniso}), on the other hand, is showing a larger dependence on $\\tau_\\text{hydro}$ than in the Glauber case. We verified that similar dependence is obtained for $\\epsilon_p'$ in peripheral IP-Glasma collisions with appreciable background ellipticity (not shown). \nThe initial momentum perturbations for IP-Glasma initial conditions (which are absent in MC-Glauber initialization) are also propagated by the kinetic theory evolution, but they only make a minor contribution to averaged energy density $\\langle\\tau e^{3\/4}\\rangle$ and radial velocity $\\langle v_\\perp \\rangle$ at later times.\n\nWe note that both IP-Glasma and the kinetic theory are conformal, which means that the breaking of conformality discussed in Section~\\ref{sec:Glauber_average} also occurs and increases the dependence on $\\tau_{\\textrm{hydro}}$ for hydrodynamic evolution with realistic equation of state.\nIn the bottom row of \\Fig{fig:average_ipglasma_qcd} we verified that using a conformal equation of state for the hydrodynamic phase slightly reduces the $\\tau_\\text{hydro}$ dependence of both the transverse average of $\\tau e^{3\/4}$ and of the transverse \nvelocity $v_\\perp$, while leaving $\\epsilon_p^\\prime$ relatively unchanged, as discussed in the previous section.\n\n\n \n \n\n\n\n\nWe also vary the transition time between IP-Glasma and the kinetic theory $\\tau_{\\scriptscriptstyle \\text{EKT}}$, namely we do simulations with $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.1$\\,fm and $0.2$\\,fm, while keeping the crossover time to $\\tau_\\text{hydro}=0.8$\\,fm fixed. As shown in Fig~\\ref{fig:average_ipglasma_qcd_tauEKTdep}, all three averaged fields --- the transverse energy, velocity and momentum eccentricities --- show very little dependence on $\\tau_{\\scriptscriptstyle \\text{EKT}}$. This is expected as at the crossover time the 2+1D Yang Mills evolution and kinetic theory are both close to free streaming. For the rest of this section we will use a fixed transition time of IP-Glasma to kinetic theory of $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.2$\\,fm.\n\n\n\n\n\\begin{figure*}\n\t\\centering\n\\subfig{a}{\\includegraphics[width=0.3\\linewidth]{ipglasma_qcd_energy34_vs_time_tauEKTdep}}%\n\\subfig{b}{\\includegraphics[width=0.3\\linewidth]{ipglasma_qcd_radial_v_vs_time_tauEKTdep}}%\n\\subfig{c}{\\includegraphics[width=0.3\\linewidth]{ipglasma_qcd_epsilon_prime_p_vs_time_tauEKTdep}}\n\t\\caption{Transverse average of $\\tau \\epsilon^{4\/3}$ and of the transverse \n\t\tvelocity $v_\\perp$, as defined in \\Eq{eq:energy_average}, and momentum \n\t\teccentricity as defined by \\Eq{eq:momentum_aniso}, as a function of \n\t\ttime $\\tau$. The three average fields are plotted for a single hydrodynamic \n\t\tinitialization time $\\tau_\\text{hydro}=0.8$~fm, and two kinetic theory \n\t\tinitialization time $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.1$~fm and \n\t\t$\\tau_{\\scriptscriptstyle \\text{EKT}}=0.2$~fm, for IP-Glasma initial conditions. A \n\t\trealistic QCD equation of state is used in the hydrodynamics evolution.}\n\t\\label{fig:average_ipglasma_qcd_tauEKTdep}\n\\end{figure*}\n\\subsubsection{Transverse plane profiles of hydrodynamic fields}\n\n\\begin{figure*}\n\t\\centering\n\t\\subfig{a}{\\includegraphics[height=0.25\\linewidth]{ipglasma_e_vs_x_vs_time_t12}}%\n\t\\subfig{b}{\\includegraphics[height=0.25\\linewidth,trim=25 0 0 0, clip]{ipglasma_e_vs_x_vs_time_t20}}%\n\t\\subfig{c}{\\includegraphics[height=0.25\\linewidth,trim=25 0 0 0, clip]{ipglasma_e_vs_x_vs_time_t50}}\n\t\\subfig{d}{\\includegraphics[height=0.25\\linewidth]{ipglasma_vx_vs_x_vs_time_t12}}%\n\t\\subfig{e}{\\includegraphics[height=0.25\\linewidth,trim=28 0 0 0, clip]{ipglasma_vx_vs_x_vs_time_t20}}%\n\t\\subfig{f}{\\includegraphics[height=0.25\\linewidth,trim=28 0 0 0, clip]{ipglasma_vx_vs_x_vs_time_t50}}\n\t\\caption{Transverse profile of $\\tau e^{3\/4}$ (upper panels) and velocity $v^x$ (lower panels) for different hydrodynamics transition times $\\tau_\\text{hydro}$ at $y=0\\,\\text{fm}$. The different columns correspond to three different hydrodynamic evolution time, $\\tau=1.2, 2.0$ and $5.0$~fm. The same EKT initialization time $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.2$~fm was used. The equation of state is a realistic QCD one. The transverse velocity is not shown for very low energy densities ($\\tau e^{3\/4}<0.01\\,\\text{GeV}^{2}$) where numerical errors can generate spurious values of velocity.}\n\t\\label{fig:ipglasma_profiles}\n\\end{figure*}\n\nNext, we scrutinize the matching between \\kompost{} with IP-Glasma initial conditions and hydrodynamic evolution by looking at the transverse energy and velocity profiles along the $x$-axis ($y=0$) in \\Fig{fig:ipglasma_profiles}.\nThe upper row shows the transverse profile of energy density $\\tau e^{3\/4}$ at different times in hydrodynamic evolution. Different lines represent a varying length of kinetic theory pre-equilibrium evolution with crossover times $\\tau_\\text{hydro}=0.4{-}1.2\\,\\text{fm}$. A good overlap of different curves indicates a smooth matching between kinetic theory and hydrodynamics. The small spread in energy can be attributed to the conformal breaking discussed in previous section, c.f. \\Fig{fig:average_ipglasma_qcd}(a) and (d).\n In the bottom row of \\Fig{fig:ipglasma_profiles}, we show the transverse velocity $v^x$ along $x$-axis. In the central region of the plasma we observe a smooth matching between kinetic theory and a full 2+1D relativistic hydrodynamics. For small energy densities at the edge of the medium ($|x|\\gtrsim 5\\,\\text{fm}$), the velocities\nare not as smoothly matched, but according to the discussion in \\Sec{sec:hydtime}, these regions do not satisfy the criterion of hydrodynamization anyway. Although at later times the pre-equilibrium flow is dominated by the response to initial energy gradients, the IP-Glasma initial conditions have a non-zero initial velocity at $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.2\\,\\text{fm}$, which contribute to the fine details of profiles shown in \\Fig{fig:ipglasma_profiles}.\n\n\n\\begin{figure*}\n\t\\centering\n\\subfig{a}{\\includegraphics[width=0.40\\linewidth]{pi_ip_vs_x_vs_time}}\\quad\\quad\n\\subfig{b}{\\includegraphics[width=0.40\\linewidth]{xscaled_ip_vs_x_vs_time}}\n\t\\caption{(a)\tComparison of $\\pi^{xx}+\\pi^{yy}$ as defined in \n\t\\Eq{eq:TmunuIPG} with its Navier-Stokes counterpart at hydro\n\tinitialization times $\\tau_\\text{hydro}=0.8, 1.0, 1.2\\,\\text{fm}$ with IP-Glasma initial \n\tconditions. (b) Scaled time variable for different locations in the transverse plane and at different crossover times $\\tau_\\text{hydro}$. \n Values of $\\tau T_\\text{id.}\/(4\\pi\\eta\/s)>1$ indicate that the system is close enough to local thermal equilibrium for hydrodynamics to become applicable (see \\Sec{sec:hydtime}).\n }\n\t\\label{fig:ipglasma_pimunu}\n\\end{figure*}\n\nTo test the convergence of viscous components of energy momentum tensor to hydrodynamic expectations, in \\Fig{fig:ipglasma_pimunu}(a) we show the shear stress tensor $\\pi^{\\mu\\nu}$ profile for three values of the hydrodynamic initialization time $\\tau_\\text{hydro}= 0.8\\,\\text{fm}, 1.0\\,\\text{fm}$ and $1.2\\,\\text{fm}$, and compare to the estimated Navier-Stokes value $\\sim \\eta \\sigma ^{\\mu\\nu}$ (obtained from the velocity profile). The agreement with the Navier-Stokes value is not as good as observed for the smoother Glauber initial conditions (\\Fig{fig:glauber_pimunu}), although still reasonable. In panel (b) we show the local scaled evolution time $\\frac{\\tau \\TId}{\\eta\/s}$ for different crossover times, which shows that the central part of the collision is within the hydrodynamic regime $\\tau T_\\text{id.}\/(4\\pi\\eta\/s)>1$ at time $\\tau_\\text{hydro}>0.8\\,\\text{fm}$ as stipulated in \\Sec{sec:hydtime}.\n\n\n\\subsubsection{Hadronic observables}\n\n\\begin{figure*}\n\t\\centering\n\\subfig{a}{\\includegraphics[width=0.31\\textwidth]{ipglasma_pion_N_v3}}%\n\\subfig{b}{\\includegraphics[width=0.31\\textwidth]{ipglasma_pion_mean_v3}}%\n\\subfig{c}{\\includegraphics[width=0.31\\textwidth]{ipglasma_pion_v2_v3}}\n\t\\caption{\nThe thermal freeze-out pion\n (a) multiplicity $dN_\\pi\/dy$ (b) radial flow $\\langle p_T^\\pi\\rangle$ and (c) elliptic flow $v_2^\\pi$ as a function of hydrodynamic initialization time $\\tau_\\text{hydro}=0.4{-}1.2\\,\\text{fm}$, i.e. different duration of pre-equilibrium evolution. The initial IP-Glasma conditions are specified at $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.2\\,\\text{fm}$. The different pre-equilibrium scenarios are linearized kinetic theory (i.e. \\kompost{}) and free streaming.\n \\label{fig:ipglasma_mult}\\label{fig:ipglasma_meanpt}\\label{fig:ipglasma_v2}}\n\t\\label{fig:ipglasma_hadronic_obs}\n\\end{figure*}\n\nAfter checking the smooth matching between individual stages of the evolution, we now test the effect of the hydrodynamic initialization time $\\tau_\\text{hydro}$ on the final state observables. To recap, the IP-Glasma initial conditions are evolved until $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.2\\,\\text{fm}$ at which point the background energy density together with energy and momentum perturbations are passed to \\kompost{}. After linearized kinetic theory pre-evolution the hydrodynamic fields like energy $e$, velocity $u^\\mu$ and shear-stress tensor $\\pi^{\\mu\\nu}$ are passed to viscous hydrodynamic simulation at time $\\tau_\\text{hydro}=0.4{-}1.2\\,\\text{fm}$. Then thermal hadronic observables are computed at constant temperature $T_\\text{FO}=145\\,\\text{MeV}$ freeze-out surface via the standard Cooper-Frye procedure. In \\Fig{fig:ipglasma_hadronic_obs} we show the thermal pion multiplicity $dN_\\pi\/y$, the mean transverse momentum $\\langle p_T^\\pi\\rangle$ and $v_2^\\pi$, as a function of the hydrodynamic initialization time $\\tau_\\text{hydro}$. For comparison, we replace the kinetic theory evolution with free-streaming background evolution and response functions. As observed in the Glauber case (see \\Sec{sec:glauber_hadronic} and \\Fig{fig:glauber_hadronic_obs}), hadronic observables show very little dependence on $\\tau_\\text{hydro}$ when a kinetic theory pre-equilibrium evolution is used. The pion multiplicity changes by less than 4\\%; for free-streaming pre-equilibrium the change is twice as large in the opposite direction. The mean radial $\\langle p_T\\rangle $ is essentially independent of $\\tau_\\text{hydro}$, but is slightly increasing for free streaming pre-equilibrium. Finally, the elliptic flow slightly changes for both types of pre-equilibrium evolutions, but it is still smaller for kinetic theory description. All in all, the dependence on $\\tau_\\text{hydro}$ is small for the \\kompost{} pre-equilibrium evolution and the dependence increase when the kinetic theory is replaced by free-streaming description.\nWe highlight that the $\\tau_\\text{hydro}$ dependence of the pion $v_2$ is still small, despite the somewhat larger dependence of $\\epsilon_p^\\prime$ seen in \\Fig{fig:average_ipglasma_qcd}.\n\n\n\n\n\n\n\\section{Effective descriptions of early time dynamics}\nSo far we have demonstrated the practical performance of our framework to describe early time dynamics of high-energy collisions. We will now investigate in more detail theoretical relations and practical comparison to other approaches previously discussed in the literature~\\cite{Broniowski:2008qk,Liu:2015nwa, Vredevoogd:2008id,vanderSchee:2013pia,Romatschke:2015gxa}.\n\n\\subsection{Long wavelength limit of kinetic theory response\\label{sec:lowk}}\n\n\\begin{figure*}\n\t\\centering\n\\subfig{a}{\\includegraphics[width=0.4\\linewidth]{coef_de}}\t\n\\subfig{b}{\\includegraphics[width=0.4\\linewidth]{coef_g}}\n\t\\caption{\\label{fig:coefde} Linear and quadratic long wavelength response coefficients to\n\t\tinitial energy perturbations (a) and initial momentum perturbations (b) from non-equilibrium kinetic theory evolution. Dashed lines show comparison to viscous hydrodynamic asymptotic derived in \\app{sec:hydrolimit}.}\n\\end{figure*}\n\n\nIn viscous hydrodynamics small scale fluctuations dampen rapidly, and many final state observables are only sensitive to long wavelength perturbations~\\cite{Gardim:2011xv,Teaney:2012ke,Floerchinger:2013rya,Mazeliauskas:2015efa,Mazeliauskas:2015vea,Noronha-Hostler:2015coa,Gardim:2017ruc}. Consequently, one could expect that also the pre-equilibration discussed in previous sections could be well captured by the long wavelength response. We will now discuss how to formalize and test this idea, based on a low $|\\mathbf{k}|$ expansion of our non-equilibrium linear response formalism. Interestingly, it will also be useful to establish the relation of our formalism to previous ideas related to the concept of a ``universal pre-flow\" \\cite{Vredevoogd:2008id}. Details of the derivations presented in this section are worked out in \\App{app:lowk}.\n\n\n\nIn order to study the low $|\\mathbf{k}|$ limit of kinetic theory pre-equilibrium evolution to initial conditions of heavy ion collisions, we first filter out the small wavelength perturbations by performing a Gaussian smearing of the initial energy-momentum tensor $T^{\\mu\\nu}(\\tau_0,\\mathbf{x}_0)$. Specifically, we define a coarse grained energy-momentum tensor\n\\begin{equation}\n\t\\label{eq:smoothTmunu}\n\t\\bar{T}^{\\mu\\nu}(\\tau_0,\\mathbf{x}_0)= \\int \n\td^2\\mathbf{x}'_{0}~S_{\\sigma}(\\mathbf{x}_0-\\mathbf{x}'_0)~T^{\\mu\\nu}(\\tau_0,\\mathbf{x}_0')\\;,\n\\end{equation}\nwith the same Gaussian smearing kernel $S_{\\sigma}(\\mathbf{x}_{0}-\\mathbf{x}'_{0})$ used to define the local background energy $\\bar{T}^{\\mu\\nu}_{\\mathbf{x}}(\\tau_0)$ (c.f. \\Eq{eq:avgBG}).\nConsidering a space-time point $(\\tau,\\mathbf{x})$ on the future hydrodynamic surface, we can then perform our usual decomposition of the energy-momentum tensor into local background $\\bar{T}^{\\mu\\nu}_{\\mathbf{x}}(\\tau)$ and perturbations $\\delta T^{\\mu\\nu}_{\\mathbf{x}}(\\tau_0,\\mathbf{x}_0)$. However, instead of considering fluctuations on all scales, the initial energy-momentum tensor perturbation $\\delta T^{\\mu\\nu}_{\\mathbf{x}}(\\tau_0,\\mathbf{x}_0)$ can now be decomposed further into short and long-wavelength components as \n\\begin{align}\n\t\\delta T^{\\mu\\nu}_{\\mathbf{x}}(\\tau_0,\\mathbf{x}_0)=&\\underbrace{T^{\\mu\\nu}(\\tau_0,\\mathbf{x}_0)-\\bar{T}^{\\mu\\nu}(\\tau_0,\\mathbf{x}_0)}_{\\text{short wavelength}} \\nonumber \\\\\n\t&\\qquad+\\underbrace{\\bar{T}^{\\mu\\nu}(\\tau_0,\\mathbf{x}_0)-\\bar{T}^{\\mu\\nu}_{\\mathbf{x}}(\\tau_0)}_{\\text{long wavelength}}\\;,\n\\end{align}\nwhere, recalling that the definition of the local background $\\bar{T}^{\\mu\\nu}_{\\mathbf{x}}(\\tau_0)$ involves the same coarse graining procedure, the long wave-length components of $\\delta T^{\\mu\\nu}_{\\mathbf{x}}(\\tau_0,\\mathbf{x}_0)$ are then entirely given in terms of smooth fields $\\bar{T}^{\\mu\\nu}$. If the initial profile of the energy-momentum tensor \n$T^{\\mu\\nu}(\\tau,\\mathbf{x})$ is sufficiently smooth on length scales smaller than $\\sigma\\sim |\\tau-\\tau_{0}|$ the short wave-length component is very small and can safely be ignored\\footnote{Note that even if the initial energy-momentum tensor features fluctuations smaller on length scales smaller than $\\sigma$, short wave length components on scales\nless than $\\sigma_{{\\rm visc}} \\sim (\\tau-\\tau_0)\/\\sqrt{\\frac{\\tau \\TId}{\\eta\/s}}$, are subject to strong viscous damping and could still be neglected.}. Neglecting short wavelength components in the following, the energy-momentum tensor at later times is then given by\n\\begin{align}\n\t\\label{eq:EvolutionLowKmain}\n\t\\frac{\\delta \n\t\tT^{\\mu\\nu}(\\tau,\\mathbf{x})}{\\bar{T}^{\\tau\\tau}_{\\mathbf{x}}(\\tau)}&=\\frac{1}{\\bar{T}^{\\tau\\tau}_\\mathbf{x}(\\tau_0)}\\times\\nonumber\\\\\n\t\\int\n\td^2\\mathbf{x}_0&G^{\\mu\\nu}_{\\alpha\\beta}\\Big(\\tau,\\tau_0,\\mathbf{x}-\\mathbf{x}_0\\Big)\n\t~(\\bar{T}^{\\alpha\\beta}(\\tau_0,\\mathbf{x}_0)-\\bar{T}_\\mathbf{x}^{\\alpha\\beta}(\\tau_0))\\;.\n\\end{align}\nBy expressing the convolution in Fourier space, and expanding the response functions in powers of the wave number $|\\mathbf{k}|$, as\n\\begin{align}\n\t\\tilde{G}^{s}_{s}(\\tau,\\tau_0,|\\mathbf{k}|)&=\\tilde{G}_{s}^{0}(\\tau,\\tau_0)~\\Big(\\tilde{s}_s^{(0)}-\\frac{1}{2}\n\t|\\mathbf{k}|^2 (\\tau-\\tau_0)^2~\\tilde{s}^{(2)}_{s}+...\\Big)\\;, \\nonumber \\\\\n\t\\tilde{G}^{v}_{s}(\\tau,\\tau_0,|\\mathbf{k}|)&=\\tilde{G}_{s}^{0}(\\tau,\\tau_0)~\\Big(|\\mathbf{k}|\n\t(\\tau-\\tau_0)~\\tilde{s}^{(1)}_{v}+...\\Big)\\;,\t\\label{eq:lowkcoef}\n\\end{align}\nand similarly for the other components, it is then straightforward to show that the long wave length response in \\Eq{eq:EvolutionLowKmain} is proportional to the gradients of the \n\\emph{smoothened} energy-momentum tensor $\\bar{T}^{\\mu\\nu}(\\tau,\\mathbf{x})$. Saving the details of the derivation for \\app{app:lowk}, the energy-momentum response to initial energy gradients is then given by\n\\begin{subequations}\n\\label{eq:preflowedense}\n\\begin{align}\n\t\\frac{\\delta T^{\\tau\\tau}(\\tau,\\mathbf{x})}{\\bar{T}_\\mathbf{x}^{\\tau\\tau}(\\tau)}&\\approx \n\t\\frac{\\tilde{G}_s^{0}(\\tau,\\tau_0) }{\\bar{T}_\\mathbf{x}^{\\tau\\tau}(\\tau_0)} \n\t\\left[\\frac{1}{2}\\tilde{s}_s^{(2)}(\\tau-\\tau_0)^2 \n\t\\partial_k\\partial^k\\right]\\bar{T}^{\\tau\\tau}(\\tau_0,\\mathbf{x})\\\\\n\t\\frac{\\delta T^{\\tau i}(\\tau,\\mathbf{x})}{\\bar{T}_\\mathbf{x}^{\\tau\\tau}(\\tau)}&\\approx \n\t\\frac{\\tilde{G}_s^{0}(\\tau,\\tau_0) }{\\bar{T}_\\mathbf{x}^{\\tau\\tau}(\\tau_0)} \n\t\\left[- \\tilde{s}_v^{(1)}(\\tau-\\tau_0) \\partial^i \n\t\\right]\\bar{T}^{\\tau\\tau}(\\tau_0,\\mathbf{x}) \n\t\\label{eq:pratt}\n\\end{align}\n\\end{subequations}\nwhich is a generalization of the previous result for the transverse pre-flow derived in~\\cite{Vredevoogd:2008id, Keegan:2016cpi}. \\Eq{eq:preflowedense} says\nthat in addition to the ``pre-flow'' \nwhich develops during \nthe equilibration process due to gradients~\\cite{Vredevoogd:2008id}, \nlocal maxima of the initial energy density, which have negative second derivatives, are depleted during the evolution.\nWe can also obtain the long wavelength response to initial momentum perturbations\n\\begin{align}\n\t\\frac{\\delta T^{\\tau\\tau}(\\tau,\\mathbf{x})}{\\bar{T}^{\\tau\\tau}_{\\mathbf{x}}(\\tau)} &\\approx \\frac{\\tilde{G}_{v}^{0}(\\tau,\\tau_0)}{\\bar{T}^{\\tau\\tau}_\\mathbf{x}(\\tau_0)} \\Big[ - \\tilde{v}^{(1)}_{s} (\\tau-\\tau_0) \\partial_{i} \\Big]~\\bar{T}^{\\tau i}(\\tau_0,\\mathbf{x}) \\;, \\\\\n\t\\frac{\\delta T^{\\tau i}(\\tau,\\mathbf{x})}{\\bar{T}_\\mathbf{x}^{\\tau\\tau}(\\tau)}&\\approx \\frac{\\tilde{G}_{v}^{0}(\\tau,\\tau_0)}{\\bar{T}^{\\tau\\tau}_\\mathbf{x}(\\tau_0)} \\Big[ 1+ \\frac{(\\tau-\\tau_0)^2}{2} \\tilde{v}^{(2)}_{\\delta} \\partial_k\\partial^k\\Big] \\bar{T}^{\\tau i}(\\tau_0,\\mathbf{x}) \\nonumber \\\\\n&+\\frac{\\tilde{G}_{v}^{0}(\\tau,\\tau_0)}{\\bar{T}^{\\tau\\tau}_\\mathbf{x}(\\tau_0)} \\Big[ \\frac{(\\tau-\\tau_0)^2}{2} \\tilde{v}^{(2)}_{k} \\partial^{i}\\partial_{j} \\Big]~\\bar{T}^{\\tau j}(\\tau_0,\\mathbf{x}) \\;. \\nonumber \\\\\n\t\\label{eq:prattmomentum}\n\\end{align}\n\nWe extract the $1$st and $2$nd order coefficients in $|\\mathbf{k}|(\\tau-\\tau_0)$ \nby performing a polynomial \nfit to the first few $|\\mathbf{k}|$ values of kinetic theory response functions, e.g.\\ \n\\Fig{fig:rescaledresponse19de}.\nOur results for the zero momentum response $G^{0}_{s\/v}$ as well as the long wave-length coefficients $s^{(n)}$,$v^{(n)}$ from effective kinetic theory simulations, are compactly summarized in Fig.~\\ref{fig:coefde}, where we plot the various response coefficients as a function of the scaling variables $\\frac{\\tau \\TId}{\\eta\/s}$. We find that the time dependence of the coefficients is rather weak, such that in practice the long wave length response can be approximated rather well by the hydrodynamic asymptotics, see \\app{sec:hydrolimit}.\n\n\n\n\nBefore we compare the long wavelength results to the full treatment in \\kompost, it is important to point out that the above separation into long and short wavelength modes introduces an artificial regulator dependence not present the full treatment. Specifically for the long wave-length filter $\\tilde{S}_{\\sigma}(\\mathbf{k})\\propto e^{-\\frac{\\sigma^2}{2} \\mathbf{k}^2}\\simeq 1-\\frac{\\sigma^2}{2}\\mathbf{k}^2$ in \\Eq{eq:smoothTmunu}, the first difference formally appears at quadratic order $\\mathcal{O}(\\mathbf{k}^2)$. However, the $\\mathcal{O}(\\mathbf{k}^2)$ can always be absorbed into an additive renormalization of the long wavelength coefficients, i.e.\\ by subtracting $\\sigma^2\/(\\tau-\\tau_0)^2$ from the quadratic $\\tilde{s}^{(2)}_s$ coefficient in energy response, as explained in \\app{app:regsigma}.\n Based on this renormalization scheme, the regulator dependence enters only at cubic order $O(|\\mathbf{k}|^3)$ and the residual dependence can be used to quantify the systematic uncertainties of the long wavelength approximation.\n\n\\subsection{Comparison of effective kinetic theory, long wavelength response \\& free streaming}\n\\begin{figure*}\n\t\\centering\n\\subfig{a}{\\includegraphics[width=0.3\\linewidth]{e3_vs_x_vs_time}}%\n\\subfig{b}{\\includegraphics[width=0.3\\linewidth]{vx3_vs_x_vs_time}}%\n\\subfig{c}{\\includegraphics[width=0.3\\linewidth]{pi3_vs_x_vs_time}}\n\\subfig{d}{\\includegraphics[width=0.3\\linewidth]{e6_vs_x_vs_time}}%\n\\subfig{e}{\\includegraphics[width=0.3\\linewidth]{vx6_vs_x_vs_time}}%\n\\subfig{f}{\\includegraphics[width=0.3\\linewidth]{pi6_vs_x_vs_time}}%\n\t\\caption{Comparison of hydrodynamic fields obtained in the long-wavelength (top) and free streaming limits (bottom), with the results of full kinetic theory pre-equilibrium (\\kompost{}). Different columns show\nprofiles of energy density $e$ (left), flow velocity $v^x$ (center) and shear stress $(\\pi^{xx}+\\pi^{yy})$ (right). }\n\t\\label{fig:lowk}\n\\end{figure*}\n\n\nWe now turn to the comparison between the full kinetic theory and the low-$|\\mathbf{k}|$ limit described in previous section. Our results are summarized in the top row of \\Fig{fig:lowk}, where we compare profiles of the energy-momentum tensor at the end of the pre-equilibrium evolution at $\\tau_\\text{hydro}=1.2\\,\\text{fm}$ initialized with the same MC-Glauber initial conditions at $\\tau_{\\scriptscriptstyle \\text{EKT}}=0.1\\,\\text{fm}$. We assess the robustness of the long wavelength results by using two different smearing widths $\\sigma_1=\\sigma_0$ and $\\sigma_2=2\\sigma_0$, where $\\sigma_0=(\\tau_\\text{hydro}-\\tau_{\\scriptscriptstyle \\text{EKT}})\/2$ is the same Gaussian width used to define the local averaged background in the full kinetic theory response. \n\nAs it is visible from \\Fig{fig:lowk}(a) the simplified low-$k$ evolution accurate to quadratic order in small $|\\mathbf{k}|$ reproduces the energy density profile rather well and is largely \ninsensitive to the smearing width. However, most of the energy is evolved as a background according to universal scaling curves, \\Eq{eq:universalE}, which is the the same \nfor both the full kinetic and low $|\\mathbf{k}|$ response.\n\nIn \\Fig{fig:lowk}(b) we compare the transverse velocity profile in kinetic theory and long wavelength limit. Formally, the low-$k$ limit given by \\Eq{eq:pratt} is only accurate to linear order in \n$k$ (c.f.\\ \\cite{Vredevoogd:2008id}). However for the particular choice of the regulator $\\sigma_1=\\sigma_0$, the long wavelength limit approximately reproduces the cubic order \nflow response (see \\App{app:regsigma} for details) and the resulting curve is very close to the full kinetic theory result. Despite this accidental agreement, it is also evident from the comparison with the curve for $\\sigma_2=2\\sigma_0$ that the long wavelength result exhibits a strong regulator dependence, which points to the fact that the actual leading order $(\\sim k)$ velocity profile provides a less accurate approximation of the full kinetic theory result.\n\nSimilar features can be observed in \\Fig{fig:lowk}(c), where we compare the kinetic theory evolution of the shear-stress tensor $\\pi^{\\mu\\nu}$ with the corresponding long wavelength result obtained for simplicity through hydrodynamic constitutive equations. At leading viscous order $\\pi^{\\mu\\nu}\\propto \\eta \\sigma^{\\mu\\nu}$ is mainly dominated by velocity gradients, therefore the agreement is better for the low-$|\\mathbf{k}|$ response with accidental cubic accuracy in velocities, but is also reasonably good for the different choice of coarse-graining.\n\n\nIt is constructive to compare in the same fashion the kinetic theory response with a simple model of the pre-equilibrium evolution prescription based on free streaming~\\cite{Broniowski:2008qk,Liu:2015nwa}. For completeness the free-streaming response functions are summarized in \\app{sec:freestreaming}.\n The comparison is shown in \\Fig{fig:lowk}(d-f). In free streaming evolution the longitudinal pressure is completely neglected and thus the energy density decreases more slowly as a function of time, overshooting the kinetic theory results (see \\Fig{fig:lowk}(d)). For the \nspecial case of initial (scalar) energy perturbations, the free streaming \nresponse functions for transverse \\emph{velocity} have approximately the same \nlow-$|\\mathbf{k}|$ expansion as the full kinetic theory evolution, see \\Eq{eq:cubic}. \nTherefore the transverse velocity profile in \\Fig{fig:lowk}(e) between kinetic \ntheory and free streaming response is almost indistinguishable. However, the physical momentum \nis not correct, because of the incorrect background energy evolution, \n(c.f.\\ \\Fig{fig:lowk}(d)).\n\nFinally in \\Fig{fig:lowk}(f) we compare the shear-stress tensor $\\pi^{\\mu\\nu}$ in free streaming (the dashed-dotted line) to the full kinetic theory result (the solid line). In \nfree streaming the longitudinal pressure is completely neglected, and thus \nthe transverse stress $\\pi^{xx} {+} \\pi^{yy}$ is \ntoo large when the system is passed to hydrodynamics. (We have reduced the free $\\pi^{xx}{+}\\pi^{yy}$ by a factor of two in \\Fig{fig:lowk} for visibility.) If instead of \nthe free streaming result for $\\pi^{xx}{+}\\pi^{yy}$, one just uses the free energy and \nvelocity profiles, \\Fig{fig:lowk}(d-e), together with the hydrodynamic constitutive \nrelations, the shear-stress tensor is closer to the kinetic theory \nresult, as indicated by a Navier-Stokes estimate $\\sim \\eta\n\\sigma^{\\mu\\nu}$ in the figure.\n\nTo summarize, \nthe low-$|\\mathbf{k}|$ limit does a good job at describing the velocity, but it makes no predictions for the background energy density as a function of time. Therefore it must \nbe supplemented by a theory (such as QCD kinetics)\nwhich describes the longitudinal pressure at early times.\nFree streaming \n also correctly describes the velocity response,\nbut in this case the energy density and second order hydrodynamic variables\nmust be continually readjusted in order to have a smooth transition \nto hydrodynamics. On a practical note, using \n{\\kompost } is computationally no more expensive\nthan the other alternatives, and it should become the \npre-hydro engine of choice.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Summary \\& outlook \\label{sec:summary}}\n\nIn this paper we developed a linear response framework to describe the non-equilibrium evolution of the energy-momentum tensor during the early stages of high-energy heavy-ion collisions. Based on microscopic input from effective kinetic theory simulations, we presented a practical implementation of the ``bottom-up`` thermalization scenario with realistic fluctuating heavy-ion initial conditions and demonstrated a consistent matching between the pre-equilibrium and the hydrodynamic evolution on an event-by-event basis. \nOur linear kinetic pre-equilibrium propagator \\kompost{}~\\cite{kompost_github}\nprovides a practical implementation of a systematic procedure to propagate initial out-of-equilibrium perturbations to the hydrodynamic initialization time~\\cite{Keegan:2016cpi}. Crucially, for short evolution times the equilibration dynamics can be described as sum of local background equilibration and linear response to fluctuations of initial conserved energy and momentum density, \\Eq{one}. We developed a concrete realization of the general linear response formalism presented in \\Sec{sec:macro} and \\Sec{sec:generalresponse} using QCD kinetic theory with gluonic degrees of freedom and extrapolation to moderate values of the coupling constant $\\lambda$~\\cite{Arnold:2002zm,Keegan:2016cpi,Kurkela:2015qoa}.\nWithin our framework, the entire kinetic equilibration of $T^{\\mu\\nu}$ is compactly summarized by a single evolution curve of the homogeneous background (c.f.\\ \\Fig{fig:scaledTmunu}) and a handful of response functions (c.f.\\ Figs.~\\ref{fig:plot_grgss} and \\ref{fig:plot_grgvs} ). Interestingly, we find that for the relevant range of coupling constants, corresponding to realistic values of $\\eta\/s$, the kinetic equilibration only depends on the scaling time $\\sim \\tau T\/(\\eta\/s)$ such that pre-tabulated kinetic response functions can be used for event-by-event simulations.\n\nWe found that for typical QGP parameters the leading order kinetic theory predicts a hydrodynamization time around $\\tau_\\text{hydro}\\sim 1\\,\\text{fm}$ (c.f.\\ \\Eq{eq:hydrotime} and Ref.~\\cite{Keegan:2016cpi}). During this time the initial gluon number density per rapidity roughly doubles and, thus, significantly alters the\nrelation between final particle multiplicities and the entropy density per rapidity in the initial state (see \\Fig{fig:entropya}).\n\nWe applied our formalism to two widely used initial state descriptions: a phenomenological MC-Glauber ansatz for transverse energy density deposition in \\Sec{sec:glauber} and the first-principle dynamical IP-Glasma model (which contains both energy and momentum fluctuations) in \\Sec{sec:ipglasma}. In both cases we demonstrated a smooth matching between the kinetic theory and hydrodynamics, both in the hydrodynamics fields, e.g.\\ \\Fig{fig:glauber_profiles}, and in the final hadronic observables, see \\Figs{fig:glauber_hadronic_obs} and \\ref{fig:ipglasma_hadronic_obs}, largely eliminating the dependence on the crossover time $\\tau_\\text{hydro}$. Note that the kinetic response functions reproduce the spatial components of the energy-momentum tensor $\\delta T^{ij}$ without explicitly imposing the constitutive relations, (c.f. \\Figs{fig:glauber_pimunu}(a) and \\ref{fig:ipglasma_pimunu}(a)).\n\nFinally we studied the kinetic response in the ``low wavelength'' (hydrodynamic) and ``fast expansion'' (free-streaming) limits to establish\n connections with the existing literature~\\cite{Broniowski:2008qk,Liu:2015nwa, Vredevoogd:2008id,vanderSchee:2013pia,Romatschke:2015gxa}. We find that the first few terms in small $|\\mathbf{k}|$-expansion are sufficient to capture most of kinetic response (\\Fig{fig:lowk}). Incidentally, the leading order velocity response in free-streaming limit agrees with kinetic theory, but energy density and shear-stress tensor evolution are described incorrectly.\n \nOur publicly available kinetic propagator package \\kompost{}~\\cite{kompost_github} offers a simple, yet non-trivial description of hydrodynamization in heavy ion collisions. The reduced sensitivity on the hydrodynamic initialization time $\\tau_\\text{hydro}$ brings us closer to ab initio initial conditions of heavy ion collisions. Moreover, there are multiple possibilities to build on this work to achieve an even better description of the early stage of heavy ion collisions. The inclusion of quarks as degrees of freedom would lead to a more realistic description of the pre-equilibration stage of collisions, which should provide a better matching with the QCD equation of state. Studying non-trivial background evolution in kinetic theory, e.g.\\ radial gradients, could improve the applicability of presented framework at the edges of the QGP fireball and in small systems. \n\nMore importantly, the formalism derived in this work to propagate linear perturbations on top of a smoother background can be used with response functions computed in limits other than weakly-coupled effective QCD kinetic theory. By using this framework to compare systematically the macroscopic description of equilibration from a weakly-coupled regime and a strongly-coupled one, one can hope to better constrain the real dynamics of the medium produced in heavy ion collisions. %\n\n \n\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\n\n\nThe authors would like to thank Bj\\\"orn Schenke for insightful discussions and for his help adapting the hydrodynamics code MUSIC for this work, and\nLiam Keegan for his contributions at the beginning of this project. Useful discussions with J\\\"urgen Berges, Stefan Fl\\\"orchinger, Yacine Mehtar-Tani, Klaus Reygers, Raju Venugopalan are gratefully acknowledged.\nResults in this paper were obtained using the high-performance computing system \nat the Institute for Advanced Computational Science at Stony Brook University. \nThis work was supported in part by the U.S. Department of Energy, Office of \nScience, Office of Nuclear Physics under Award Numbers \nDE\\nobreakdash-FG02\\nobreakdash-88ER40388 (A.M., J.-F.P., D.T.), DE-FG02-05ER41367 (J.-F.P.) and \nDE-FG02-97ER41014 (S.S.). This work was supported in part by the German Research Foundation (DFG) \nCollaborative Research Centre (SFB) 1225 (ISOQUANT) (A.M.). Finally, A.M.\nwould like to thank CERN Theoretical Physics\nDepartment for the hospitality during the short-term visit.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzasbs b/data_all_eng_slimpj/shuffled/split2/finalzzasbs new file mode 100644 index 0000000000000000000000000000000000000000..8a476faeb7beaafcc1e0d19acc6d569a67c3a6e6 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzasbs @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nIn the recent papers~\\cite{finite, infinite} the fermionic signature operator was introduced\non globally hyperbolic Lorentzian spin manifolds. It is a bounded symmetric operator on the Hilbert space\nof solutions of the Dirac equation which depends on the global geometry of space-time.\nThis raises the question how the geometry of space-time is related to spectral properties of\nthe fermionic signature operator. The first step in developing the resulting ``Lorentzian spectral geometry''\nis the paper~\\cite{drum} where the simplest situation of Lorentzian surfaces is considered.\nIn the present paper, we proceed in a somewhat different direction and \nshow that there is a nontrivial index associated to the fermionic signature operator.\nThis is the first time that an index is defined for a geometric operator on a Lorentzian manifold.\n\nWe make essential use of the decomposition of spinors in even space-time dimension\ninto left- and right-handed components (the ``chiral grading''). The basic idea is to\ndecompose the fermionic signature operator~$\\mathscr{S}$ using the chiral grading as\n\\begin{equation} \\label{Sigprop}\n\\mathscr{S} = \\mathscr{S}_L + \\mathscr{S}_R \\qquad \\text{with} \\qquad \\mathscr{S}_L^* = \\mathscr{S}_R \\:,\n\\end{equation}\nand to define the so-called {\\em{chiral index}} of~$\\mathscr{S}$ as the Noether index of~$\\mathscr{S}_L$.\nAfter providing the necessary preliminaries (Section~\\ref{secprelim}), this definition will be given in\nSection~\\ref{secindex} in space-times of finite lifetime.\nIn order to work out the mathematical essence of our index, in Section~\\ref{seccfs} we \nalso give its definition in the general setting of causal fermion systems\n(for an introduction to causal fermion systems see~\\cite{rrev} or~\\cite{topology}).\nSection~\\ref{secodd} is devoted to a variant of the chiral index which applies in the special case\nof the massless Dirac equation and a Dirac operator which is odd with respect to the chiral grading.\nIn Section~\\ref{secstable} we analyze the invariance properties of the chiral indices when\nspace-time or the Dirac operator are deformed by a homotopy.\nIn Sections~\\ref{secex1}--\\ref{secex3} we construct examples of fermionic signature operators\nwith a non-trivial index and illustrate the homotopy invariance.\nFinally, in Section~\\ref{secoutlook} we discuss our results and and give an outlook on\npotential extensions and applications, like the generalization to space-times of infinite lifetime.\n\nWe point out that the purpose of this paper is to define the chiral index, to study a few basic\nproperties and to show in simple examples that it is in general non-trivial.\nBut we do not work out any physical applications, nor do we make the connection\nto geometric or topological invariants.\nThese shortcomings are mainly due to the fact that we only succeeded in computing\nthe index explicitly in highly symmetric and rather artificial examples.\nMoreover, it does not seem easy to verify the conditions needed for the homotopy invariance.\nFor these reasons, we leave physically interesting examples and geometric stability results\nas a subject of future research.\nAll we can say for the moment is that the chiral index describes the ``chiral asymmetry''\nof the Dirac operator in terms of an integer. This integer seems to depend on the geometry of the\nboundary of space-time and on the singular behavior of the potentials in the\nDirac equation. Smooth potentials in the Dirac equation, however, tend to not affect the\nindex.\n\n\\section{Preliminaries} \\label{secprelim}\nWe recall a few basic constructions from~\\cite{finite}. Let~$(\\mycal M, g)$ be a smooth, globally\nhyperbolic Lorentzian spin manifold of even dimension~$k \\geq 2$.\nFor the signature of the metric we use the convention~$(+ ,-, \\ldots, -)$.\nWe denote the spinor bundle by~$S\\mycal M$. Its fibers~$S_x\\mycal M$ are endowed\nwith an inner product~$\\mathopen{\\prec} .|. \\mathclose{\\succ}_x$ of signature~$(n,n)$\nwith~$n=2^{k\/2-1}$ (for details see~\\cite{baum, lawson+michelsohn}),\nwhich we refer to as the spin scalar product. Clifford multiplication is described by a mapping~$\\gamma$\nwhich satisfies the anti-commutation relations,\n\\[ \\gamma \\::\\: T_x\\mycal M \\rightarrow \\text{\\rm{L}}(S_x\\mycal M) \\qquad\n\\text{with} \\qquad \\gamma(u) \\,\\gamma(v) + \\gamma(v) \\,\\gamma(u) = 2 \\, g(u,v)\\,\\mbox{\\rm 1 \\hspace{-1.05 em} 1}_{S_x(\\mycal M)} \\:. \\]\nWe write Clifford multiplication in components with the Dirac matrices~$\\gamma^j$\nand use the short notation with the Feynman dagger, $\\gamma(u) \\equiv u^j \\gamma_j \\equiv \\slashed{u}$.\nThe metric connections on the tangent bundle and the spinor bundle are denoted by~$\\nabla$.\n\nIn the even-dimensional situation under consideration, the spinor bundle has a decomposition\ninto left- and right-handed components. We describe this chiral grading by an operator~$\\Gamma$\n(the ``pseudoscalar operator,'' in physics usually denoted by~$\\gamma^5$),\n\\[ \\Gamma \\::\\: S_x\\mycal M \\rightarrow S_x\\mycal M \\:, \\]\nhaving for all~$u \\in T_x\\mycal M$ the properties\n\\begin{equation} \\label{gammaprop}\n\\Gamma^* = -\\Gamma\\:,\\qquad \\Gamma^2 = \\mbox{\\rm 1 \\hspace{-1.05 em} 1} \\:,\\qquad \n\\Gamma\\, \\gamma(u) = -\\gamma(u) \\,\\Gamma \\:,\\qquad \\nabla \\Gamma = 0\n\\end{equation}\n(where the star denotes the adjoint with respect to the spin scalar product).\nWe denote the chiral projections to the left- and right-handed components by\n\\begin{equation} \\label{chiLRdef}\n\\chi_L = \\frac{1}{2} \\big( \\mbox{\\rm 1 \\hspace{-1.05 em} 1} - \\Gamma \\big) \\qquad \\text{and} \\qquad \\chi_R = \\frac{1}{2} \\big( \\mbox{\\rm 1 \\hspace{-1.05 em} 1} + \\Gamma \\big) \\:.\n\\end{equation}\n\nThe sections of the spinor bundle are also referred to as wave functions.\nWe denote the smooth sections of the spinor bundle by~$C^\\infty(\\mycal M, S\\mycal M)$.\nSimilarly, $C^\\infty_0(\\mycal M, S\\mycal M)$ denotes the smooth sections with compact support.\nOn the compactly supported wave functions, one can introduce the Lorentz invariant\ninner product\n\\begin{gather}\n\\mathopen{<} .|. \\mathclose{>} \\::\\: C^\\infty_0(\\mycal M, S\\mycal M) \\times C^\\infty_0(\\mycal M, S\\mycal M) \\rightarrow \\mathbb{C} \\:, \\\\\n\\mathopen{<} \\psi|\\phi \\mathclose{>} := \\int_\\mycal M \\mathopen{\\prec} \\psi | \\phi \\mathclose{\\succ}_x\\: d\\mu_\\mycal M\\:. \\label{stip}\n\\end{gather}\nThe Dirac operator~${\\mathcal{D}}$ is defined by\n\\[ {\\mathcal{D}} := i \\gamma^j \\nabla_j + {\\mathscr{B}} \\::\\: C^\\infty(\\mycal M, S\\mycal M) \\rightarrow C^\\infty(\\mycal M, S\\mycal M)\\:, \\]\nwhere~${\\mathscr{B}} \\in \\text{\\rm{L}}(S_x)$ (the ``external potential'') typically is a smooth multiplication operator\nwhich is symmetric with respect to the spin scalar product.\nIn some of our examples, ${\\mathscr{B}}$ will be chosen more generally as a convolution operator\nwhich is symmetric with respect to the inner product~\\eqref{stip}.\nFor a given real parameter~$m \\in \\mathbb{R}$ (the ``mass''), the Dirac equation reads\n\\[ ({\\mathcal{D}} - m) \\,\\psi = 0 \\:. \\]\nWe mainly consider solutions in the class~$C^\\infty_{\\text{sc}}(\\mycal M, S\\mycal M)$ of smooth sections\nwith spatially compact support. On such solutions one has the scalar product\n\\begin{equation} \\label{print}\n(\\psi | \\phi) = 2 \\pi \\int_\\mycal N \\mathopen{\\prec} \\psi | \\slashed{\\nu} \\phi \\mathclose{\\succ}_x\\: d\\mu_\\mycal N(x) \\:,\n\\end{equation}\nwhere~$\\mycal N$ denotes any Cauchy surface and~$\\nu$ its future-directed normal.\nDue to current conservation, the scalar product is\nindependent of the choice of~$\\mycal N$ (for details see~\\cite[Section~2]{finite}).\nForming the completion gives the Hilbert space~$(\\H_m, (.|.))$.\n\nFor the construction of the fermionic signature operator, we need to\nextend the bilinear form~\\eqref{stip} to the solution space of the Dirac equation.\nIn order to ensure that the integral in~\\eqref{stip} exists, we need\nto make the following assumption (for more details see~\\cite[Section~3.2]{finite}).\n\\begin{Def} \\label{defmfinite}\nA globally hyperbolic space-time~$(\\mycal M,g)$ is said to be {\\bf{{\\em{m}}-finite}} if\nthere is a constant~$c>0$ such that for\nall~$\\phi, \\psi \\in \\H_m \\cap C^\\infty_{\\text{sc}}(\\mycal M, S\\mycal M)$, the\nfunction~$\\mathopen{\\prec} \\phi | \\psi \\mathclose{\\succ}_x$ is integrable on~$\\mycal M$ and\n\\[ |\\mathopen{<} \\phi | \\psi \\mathclose{>}| \\leq c \\:\\|\\phi\\|\\: \\|\\psi\\| \\]\n(where~$\\| . \\| = (.|.)^\\frac{1}{2}$ is the norm on~$\\H_m$).\n\\end{Def} \\noindent\nUnder this assumption, the space-time inner product is well-defined as\na bounded bilinear form on~$\\H_m$,\n\\[ \\mathopen{<} .|. \\mathclose{>} \\::\\: \\H_m \\times \\H_m \\rightarrow \\mathbb{C}\\:. \\]\nApplying the Riesz representation theorem, we can\nuniquely represent this bilinear form with a signature operator~$\\mathscr{S}$,\n\\begin{equation} \\label{Sdef}\n\\mathscr{S} \\::\\: \\H_m \\rightarrow \\H_m \\qquad \\text{with} \\qquad\n\\mathopen{<} \\phi | \\psi \\mathclose{>} = ( \\phi \\,|\\, \\mathscr{S}\\, \\psi) \\:.\n\\end{equation}\nWe refer to~$\\mathscr{S}$ as the {\\bf{fermionic signature operator}}.\nIt is obviously a bounded symmetric operator on~$\\H_m$.\nWe note that the construction of the fermionic signature operator is manifestly covariant\nand independent of the choice of a Cauchy surface.\n\n\\section{The Chiral Index} \\label{secindex}\nWe now modify the construction of the fermionic signature operator by inserting the chiral projection\noperators into~\\eqref{stip}. We thus obtain the bilinear forms\n\\begin{equation} \\label{stipLR}\n\\mathopen{<} \\psi|\\phi \\mathclose{>}_{L\\!\/\\!R} = \\int_\\mycal M \\mathopen{\\prec} \\psi \\,|\\, \\chi_{L\\!\/\\!R} \\,\\phi \\mathclose{\\succ}_x\\: d\\mu_\\mycal M\\:.\n\\end{equation}\nFor the space-time integrals to exist, we need the following assumption.\n\\begin{Def} \\label{defGfinite}\nA globally hyperbolic space-time~$(\\mycal M,g)$ is said to be {\\bf{$\\Gamma$-finite}} if\nthere is a constant~$c>0$ such that for\nall~$\\phi, \\psi \\in \\H_m \\cap C^\\infty_{\\text{sc}}(\\mycal M, S\\mycal M)$, the\nfunction~$\\mathopen{\\prec} \\phi | \\Gamma \\psi \\mathclose{\\succ}_x$ is integrable on~$\\mycal M$ and\n\\[ |\\mathopen{<} \\phi | \\Gamma \\psi \\mathclose{>}| \\leq c \\:\\|\\phi\\|\\: \\|\\psi\\| \\:. \\]\n\\end{Def} \\noindent\nThere seems no simple relation between $m$-finiteness and $\\Gamma$-finiteness.\nBut both conditions are satisfied if we assume that the space-time~$(\\mycal M,g)$ has {\\bf{finite lifetime}}\nin the sense that it admits a foliation~$(\\mycal N_t)_{t \\in (t_0, t_1)}$ by Cauchy surfaces with~$t_0, t_1 \\in \\mathbb{R}$\nsuch that the function~$\\langle \\nu, \\partial_t \\rangle$ is bounded on~$\\mycal M$\n(see~\\cite[Definition~3.4]{finite}).\nThe following proposition is an immediate generalization of~\\cite[Proposition~3.5]{finite}.\n\\begin{Prp} Every globally hyperbolic manifold of finite lifetime is $m$-finite\nand $\\Gamma$-finite.\n\\end{Prp}\n\\begin{proof} Let~$\\psi \\in \\H_m \\cap C^\\infty_{\\text{sc}}(\\mycal M, S\\mycal M)$ and~$C(x)$ one of the operators~$\\mbox{\\rm 1 \\hspace{-1.05 em} 1}_{S_x}$ or~$i \\Gamma(x)$.\nApplying Fubini's theorem and decomposing the volume measure, we obtain\n\\[ \\mathopen{<} \\psi | C \\psi \\mathclose{>} = \\int_\\mycal M \\mathopen{\\prec} \\psi | C \\psi \\mathclose{\\succ}(x)\\: d\\mu_\\mycal M(x) \\\\\n=\\int_{t_0}^{t_1} \\int_{\\mycal N_t} \\mathopen{\\prec} \\psi | C \\psi \\mathclose{\\succ}\\, \\langle \\nu, \\partial_t \\rangle \\,dt \\,d\\mu_{\\mycal N_t} \\]\nand thus\n\\[ \\big| \\mathopen{<} \\psi | C \\psi \\mathclose{>} \\big| \\leq \\sup_\\mycal M \\langle \\nu, \\partial_t \\rangle\n\\int_{t_0}^{t_1} dt \\int_{\\mycal N_t} |\\mathopen{\\prec} \\psi | C \\psi \\mathclose{\\succ}|\\,d\\mu_{\\mycal N_t} \\:. \\]\nRewriting the integrand as\n\\[ |\\mathopen{\\prec} \\psi | C \\psi \\mathclose{\\succ}| = |\\mathopen{\\prec} \\psi | \\slashed{\\nu}\\, (\\slashed{\\nu} C) \\psi \\mathclose{\\succ}| \\:, \\]\nthe bilinear form~$\\mathopen{\\prec} .| \\slashed{\\nu} . \\mathclose{\\succ}$ is a scalar product. Moreover, the operator~$\\slashed{\\nu} C$\nis symmetric with respect to this scalar product. Using that\n\\[ (\\slashed{\\nu})^2 = \\mbox{\\rm 1 \\hspace{-1.05 em} 1} = (i \\slashed{\\nu} \\Gamma)^2 \\:, \\]\nwe conclude that the sup-norm corresponding to the\nscalar product~$\\mathopen{\\prec} .| \\slashed{\\nu} . \\mathclose{\\succ}$ of the operator~$\\slashed{\\nu} C$ is equal to one. Hence\n\\[ \\int_{\\mycal N_t} |\\mathopen{\\prec} \\psi | C \\psi \\mathclose{\\succ}|\\,d\\mu_{\\mycal N_t} \\leq\n\\int_{\\mycal N_t} \\mathopen{\\prec} \\psi | \\slashed{\\nu} \\psi \\mathclose{\\succ}\\,d\\mu_{\\mycal N_t} = (\\psi | \\psi) \\:, \\]\nand consequently\n\\[ \\big| \\mathopen{<} \\psi | C \\psi \\mathclose{>} \\big| \\leq (t_1-t_0)\\: \\sup_\\mycal M \\langle \\nu, \\partial_t \\rangle\\; \\|\\psi\\|^2\\:. \\]\nPolarization and a denseness argument give the result.\n\\end{proof} \\noindent\n\nAssuming that our space-time is $m$-finite and $\\Gamma$-finite, the bilinear forms~\\eqref{stipLR}\nare bounded on~$\\H_m \\times \\H_m$. Thus we may represent them with respect to the Hilbert space scalar\nproduct in terms of signature operators~$\\mathscr{S}_{L\\!\/\\!R}$,\n\\begin{equation} \\label{SLRdef2}\n\\mathscr{S}_{L\\!\/\\!R} \\::\\: \\H_m \\rightarrow \\H_m \\qquad \\text{with} \\qquad\n\\mathopen{<} \\phi | \\psi \\mathclose{>}_{L\\!\/\\!R} = ( \\phi \\,|\\, \\mathscr{S}_{L\\!\/\\!R}\\, \\psi) \\:.\n\\end{equation}\nWe refer to~$\\mathscr{S}_{L\\!\/\\!R}$ as the {\\bf{chiral signature operators}}. \nTaking the complex conjugate of the equation in~\\eqref{SLRdef2} and using\nthat~$\\chi_L^*=\\chi_R$, we find that~\\eqref{Sigprop} holds, where\nthe star denotes the adjoint in~$\\text{\\rm{L}}(\\H_m)$.\n\nWe now define the chiral index as the Noether index of~$\\mathscr{S}_L$\n(sometimes called Fredholm index; for basics see for example~\\cite[\\S27.1]{lax}).\n\\begin{Def} \\label{defind}\nThe fermionic signature operator is said to have finite chiral index if the operators of~$\\mathscr{S}_L$\nand~$\\mathscr{S}_R$ both have a finite-dimensional kernel.\nThe {\\bf{chiral index}} of the fermionic signature operator is defined by\n\\begin{equation} \\label{ind}\n\\ind \\mathscr{S} = \\dim \\ker \\mathscr{S}_L - \\dim \\ker \\mathscr{S}_R \\:.\n\\end{equation}\n\\end{Def}\n\n\\section{Generalization to the Setting of Causal Fermion Systems} \\label{seccfs}\nOur starting point is a causal fermion system as introduced in~\\cite{rrev}.\n\\begin{Def} {\\em{\nGiven a complex Hilbert space~$(\\H, \\langle .|. \\rangle_\\H)$ (the {\\em{``particle space''}})\nand a parameter~$n \\in \\mathbb{N}$ (the {\\em{``spin dimension''}}), we let~${\\mathscr{F}} \\subset \\text{\\rm{L}}(\\H)$ be the set of all\nself-adjoint operators on~$\\H$ of finite rank, which (counting with multiplicities) have\nat most~$n$ positive and at most~$n$ negative eigenvalues. On~${\\mathscr{F}}$ we are given\na positive measure~$\\rho$ (defined on a $\\sigma$-algebra of subsets of~${\\mathscr{F}}$), the so-called\n{\\em{universal measure}}. We refer to~$(\\H, {\\mathscr{F}}, \\rho)$ as a {\\em{causal fermion system}}.\n}}\n\\end{Def} \\noindent\nStarting from a Lorentzian spin manifold, one can construct a corresponding causal fermion system by\nchoosing~$\\H$ as a suitable subspace of the solution space of the Dirac equation, \nforming the local correlation operators (possibly introducing an ultraviolet regularization) and defining~$\\rho$\nas the push-forward of the volume measure on~$\\mycal M$\n(see~\\cite[Section~4]{finite} or the examples in~\\cite{topology}).\nThe advantage of working with a causal fermion system\nis that the underlying space-time does not need to be a Lorentzian manifold, but it can be a more\ngeneral ``quantum space-time'' (for more details see~\\cite{lqg}).\n\nWe now recall a few basic notions from~\\cite{rrev}. \nOn~${\\mathscr{F}}$ we consider the topology induced by the\noperator norm~$\\|A\\| := \\sup \\{ \\|A u \\|_\\H \\text{ with } \\| u \\|_\\H = 1 \\}$.\nFor every~$x \\in {\\mathscr{F}}$\nwe define the {\\em{spin space}}~$S_x$ by~$S_x = x(\\H)$; it is a subspace of~$\\H$ of dimension\nat most~$2n$. On~$S_x$ we introduce the {\\em{spin scalar product}} $\\mathopen{\\prec} .|. \\mathclose{\\succ}_x$ by\n\\begin{equation} \\label{ssp}\n\\mathopen{\\prec} u | v \\mathclose{\\succ}_x = -\\langle u | x u \\rangle_\\H \\qquad \\text{(for all $u,v \\in S_x$)}\\:;\n\\end{equation}\nit is an indefinite inner product of signature~$(p,q)$ with~$p,q \\leq n$.\nMoreover, we define {\\em{space-time}}~$M$ as the support of the universal measure, $M = \\text{supp}\\, \\rho$.\nIt is a closed subset of~${\\mathscr{F}}$.\n\nIn order to extend the chiral grading to causal fermion systems, we\nassume for every~$x \\in M$ an operator~$\\Gamma(x) \\in \\text{\\rm{L}}(\\H)$ with the properties\n\\begin{equation} \\label{pseudodef}\n\\Gamma(x)|_{S_x} \\::\\: S_x \\rightarrow S_x \\qquad \\text{and} \\qquad\nx\\, \\Gamma(x) = -\\Gamma(x)^*\\, x \\:.\n\\end{equation}\nWe define the operators~$\\chi_{L\\!\/\\!R}(x) \\in \\text{\\rm{L}}(\\H)$ again by~\\eqref{chiLRdef}.\nIn order to explain the equations~\\eqref{pseudodef}, we first note\nthat the right side of~\\eqref{pseudodef} obviously vanishes on the\northogonal complement of~$S_x$. Using furthermore that, by definition of the spin space,\nthe operator~$x$ is invertible on~$S_x$, we infer that\n\\[ \\Gamma(x)|_{S_x^\\perp} = 0 \\:. \\]\nMoreover, the computation\n\\begin{align*}\n\\mathopen{\\prec} \\psi \\,|\\, \\Gamma(x)\\, \\phi \\mathclose{\\succ}_x &= -\\langle \\psi \\,|\\, x \\,\\Gamma(x)\\, \\phi \\rangle_\\H = \n-\\langle \\Gamma(x)^* x\\, \\psi \\,|\\, \\phi \\rangle_\\H \\\\\n&\\!\\!\\overset{\\eqref{pseudodef}}{=}\n\\langle x \\,\\Gamma(x)\\, \\psi \\,|\\, \\phi \\rangle_\\H = -\\mathopen{\\prec} \\Gamma(x)\\, \\psi \\,|\\, \\phi \\mathclose{\\succ}_x\n\\end{align*}\n(with~$\\psi, \\phi \\in S_x$) shows that~$\\Gamma(x) \\in \\text{\\rm{L}}(S_x)$ is antisymmetric with respect to the\nspin scalar product. Thus the first equation in~\\eqref{gammaprop} again holds.\nThis implies that the adjoint of~$\\chi_L(x)$ with respect to~$\\mathopen{\\prec} .|. \\mathclose{\\succ}_x$ equals~$\\chi_R(x)$.\nHowever, we point out that our assumptions~\\eqref{pseudodef} do not imply that~$\\Gamma(x)$\nis idempotent (in the sense that~$\\Gamma(x)^2|_{S_x}=\\mbox{\\rm 1 \\hspace{-1.05 em} 1}_{S_x}$). Hence the analog\nof the second equation in~\\eqref{gammaprop} does not need to hold on a causal fermion system.\nThis property could be imposed in addition, but will not be needed here.\nThe last two relations in~\\eqref{gammaprop} do not have an obvious correspondence on causal fermion systems,\nand they will also not be needed in what follows.\n\nWe now have all the structures needed for defining the fermionic signature operator\nand its chiral index.\nNamely, replacing the scalar product in~\\eqref{Sdef} by the scalar product on the\nparticle space~$\\langle .|. \\rangle_\\H$, we now demand in analogy to~\\eqref{stip} and~\\eqref{Sdef}\nthat the relation\n\\[ \\langle u | \\mathscr{S} v \\rangle_\\H = \\int_M \\mathopen{\\prec} u | v \\mathclose{\\succ}_x\\: d\\rho(x) \\]\nshould hold for all~$u, v \\in \\H$. Using~\\eqref{ssp}, we find that the fermionic signature operator\nis given by the integral\n\\[ \\mathscr{S} = -\\int_M x \\: d\\rho(x) \\:. \\]\nSimilarly, the left-handed signature operator can be introduced by\n\\begin{equation} \\label{sigLint}\n\\mathscr{S}_L = -\\int_M x \\,\\chi_L\\: d\\rho(x) \\:.\n\\end{equation}\nIn the setting on a globally hyperbolic manifold, \nwe had to assume that the manifold was $m$-finite and $\\Gamma$-finite (see Definitions~\\ref{defmfinite}\nand~\\ref{defGfinite}).\nNow we need to assume correspondingly that the integral~\\eqref{sigLint} converges.\nFor the sake of larger generality we prefer to work with weak convergence.\n\n\\begin{Def} The topological fermion system is {\\bf{$\\mathscr{S}_L$-bounded}} if the integral in~\\eqref{sigLint}\nconverges weakly to a bounded operator, i.e.\\ if there is an operator~$\\mathscr{S}_L \\in \\text{\\rm{L}}(\\H)$\nsuch that for all~$u,v \\in \\H$,\n\\[ -\\int_M \\langle u \\,|\\, x \\,\\chi_L v \\rangle_\\H \\: d\\rho(x) = \\langle u | \\mathscr{S}_L v \\rangle_\\H \\:. \\]\n\\end{Def} \\noindent\nIntroducing the right-handed signature operator by~$\\mathscr{S}_R := \\mathscr{S}_L^*$,\nwe can define the {\\bf{chiral index}} again by~\\eqref{ind}.\n\n\\section{The Chiral Index in the Massless Odd Case} \\label{secodd}\nWe return to the setting of Section~\\ref{secindex}\nand consider the special case that the mass vanishes and that the Dirac operator is odd,\n\\begin{equation} \\label{massless}\nm=0 \\qquad \\text{and} \\qquad \\Gamma \\,{\\mathcal{D}} = - {\\mathcal{D}}\\, \\Gamma \\:.\n\\end{equation}\nIn this case, the solution space of the Dirac equation is obviously invariant under~$\\Gamma$,\n\\[ \\Gamma \\::\\: \\H_0 \\rightarrow \\H_0 \\:. \\]\nTaking the adjoint with respect to the scalar product~\\eqref{print} and noting that~$\\Gamma$\nanti-commutes with~$\\slashed{\\nu}$, one sees that~$\\Gamma$ is symmetric on~$\\H_0$.\nHence~$\\chi_L$ and~$\\chi_R$ are orthogonal projection operators,\ngiving rise to the orthogonal sum decomposition\n\\begin{equation} \\label{HLR}\n\\H_0 = \\H_L \\oplus \\H_R \\qquad \\text{with} \\qquad \\H_{L\\!\/\\!R} := \\chi_{L\\!\/\\!R}\\, \\H_0\\:.\n\\end{equation}\nMoreover, the computation\n\\begin{align*}\n\\mathopen{<} \\chi_L \\psi | \\chi_{c'} \\phi \\mathclose{>}_c &=\n\\int_\\mycal M \\mathopen{\\prec} \\chi_L \\psi \\,|\\, \\chi_c \\,\\chi_{c'} \\phi \\mathclose{\\succ}_x\\: d\\mu_\\mycal M \\\\\n&= \\int_\\mycal M \\mathopen{\\prec} \\psi \\,|\\, \\chi_R \\,\\chi_c \\,\\chi_{c'} \\phi \\mathclose{\\succ}_x\\: d\\mu_\\mycal M\n= \\delta_{Rc} \\:\\delta_{cc'} \\:\\mathopen{<} \\psi | \\phi \\mathclose{>}_c\n\\end{align*}\nwith~$c, c' \\in \\{L, R\\}$ (and similarly for~$L$ replaced by~$R$) shows that~$\\mathscr{S}$\nmaps the right-handed component to the left-handed component and vice versa.\nMoreover, in a block matrix notation corresponding to the decomposition~\\eqref{HLR},\nthe operators~$\\mathscr{S}_L$ and~$\\mathscr{S}_R$ have the simple form\n\\[ \\mathscr{S}_L = \\begin{pmatrix} 0 & 0 \\\\ A & 0 \\end{pmatrix} \\qquad \\text{and} \\qquad\n\\mathscr{S}_R = \\begin{pmatrix} 0 & A^* \\\\ 0 & 0 \\end{pmatrix} \\]\nwith a bounded operator~$A : \\H_L \\rightarrow \\H_R$.\nAs a consequence, both~$\\mathscr{S}_L$ and~$\\mathscr{S}_R$ have an infinite-dimensional kernel,\nso that the index cannot be defined by~\\eqref{ind}. This problem can easily be cured by\nrestricting the operators to the respective subspace~$\\H_L$ and~$\\H_R$.\n\n\\begin{Def} \\label{defind0}\nIn the massless odd case~\\eqref{massless}, the fermionic signature operator is said to have finite\nchiral index if the operators~$\\mathscr{S}_L|_{\\H_L}$ and~$\\mathscr{S}_R|_{\\H_R}$ both have a finite-dimensional kernel.\nWe define the index~$\\ind_0 \\mathscr{S}$ by\n\\[ \\ind_0 \\mathscr{S} = \\dim\\ker (\\mathscr{S}_L)|_{\\H_L} - \\dim \\ker (\\mathscr{S}_R)|_{\\H_R} \\:. \\]\n\\end{Def}\n\n\\section{Homotopy Invariance} \\label{secstable}\nWe first recall Dieudonn{\\'e}'s general theorem on the homotopy invariance of the Noether index\n(see for example~\\cite[Theorem~27.1.5'']{lax}).\n\\begin{Thm} \\label{thmhomotopy}\nLet~$T(t) : U \\rightarrow V$, $0 \\leq t \\leq 1$, be a one-parameter family of bounded linear operators\nbetween Banach spaces~$U$ and~$V$ which is continuous in the norm topology. If\nfor every~$t \\in [0,1]$ the vector spaces\n\\begin{equation} \\label{kerfinite}\n\\text{$\\ker T$ and~$V\/T(\\H)$ are both finite-dimensional}\\:,\n\\end{equation}\nthen\n\\[ \\ind T(0) = \\ind T(1) \\:, \\]\nwhere~$\\ind T := \\dim \\ker(T) - \\dim V\/T(\\H)$.\n\\end{Thm}\n\nIn most applications of this theorem, one knows from general arguments that the index\nof~$T$ remains finite under homotopies (for example, in the prominent example of the\nAtiyah-Singer index, this follows from elliptic estimates on a compact manifold).\nFor our chiral index, however, there is no general reason why the chiral index of~$\\mathscr{S}$ should\nremain finite. Indeed, the fermionic signature operator is bounded and typically has many eigenvalues\nnear zero. It may well happen that for a certain value of~$t$, an infinite number of these\neigenvalues becomes zero (for an explicit example see Example~\\ref{exinstable} below).\n\nAnother complication when applying Theorem~\\ref{thmhomotopy} to the fermionic signature operator is that the\nimage of~$\\mathscr{S}_L$ does not need to be a closed subspace of our Hilbert space.\nTo explain the difficulty, we first consider the chiral index of Definition~\\ref{defind}.\nUsing that~$\\ker \\mathscr{S}_R = \\ker \\mathscr{S}_L^* = \\mathscr{S}_L(\\H_m)^\\perp$,\nthe assumption that the fermionic signature operator has finite chiral index can be restated\nthat the vector spaces~$\\ker \\mathscr{S}_L$ and~$\\mathscr{S}_L(\\H_m)^\\perp$ are finite-dimensional subspaces\nof~$\\H_m$. Since\n\\[ \\dim \\mathscr{S}_L(\\H_m)^\\perp = \\dim \\H_m \/ \\big( \\overline{\\mathscr{S}_L(\\H_m)} \\big) \\:, \\]\nthis implies that the {\\em{closure}} of the image of~$\\mathscr{S}_L$ has finite co-dimension.\nIf the image of~$\\mathscr{S}_L$ were closed in~$\\H_m$, the finiteness of the chiral index\nwould imply that the conditions~\\eqref{kerfinite} hold if we set~$T=\\mathscr{S}_L$ and~$U=V=\\H_m$.\nHowever, the image of~$\\mathscr{S}_L$ will in general {\\em{not}} be a closed subspace of~$\\H_m$,\nand in this case it is possible that the condition~\\eqref{kerfinite} is violated for~$T=\\mathscr{S}_L$\nand~$U=V=\\H_m$, although~$\\mathscr{S}$ has finite chiral index (according to Definition~\\ref{defind}).\nIn the massless odd case, the analogous problem occurs if we choose~$T=\\mathscr{S}_L$,\n$U=\\H_L$ and~$V=\\H_R$ (see Definition~\\ref{defind0}).\n\nOur method for making Theorem~\\ref{thmhomotopy} applicable is to endow a subspace\nof the Hilbert space with a finer topology, such that the image of~$\\mathscr{S}_L$\nlies in this subspace and is closed in this topology.\n\n\\begin{Thm} \\label{thmstable}\nLet~$\\mathscr{S}(t) : \\H_m \\rightarrow \\H_m$, $t \\in [0,1]$, be a family of fermionic signature operators\nwith finite chiral index.\nLet~$E$ be a Banach space together with an embedding~$\\iota : E \\hookrightarrow \\H_m$\nwith the following properties:\n\\begin{itemize}\n\\item[(i)] For every~$t \\in [0,1]$, the image of~$\\mathscr{S}_L(t)$ lies in~$\\iota(E)$, giving rise to the mapping\n\\begin{equation} \\label{SigLE}\n\\mathscr{S}_L(t) \\,:\\, \\H_m \\rightarrow E \\:.\n\\end{equation}\n\\item[(ii)] For every~$t \\in [0,1]$, the image of the operator~$\\mathscr{S}_L$, \\eqref{SigLE}, is a closed subspace of~$E$.\n\\item[(iii)] The family~$\\mathscr{S}_L(t) : \\H_m \\rightarrow E$ is continuous in the norm topology.\n\\end{itemize}\nThen the chiral index is a homotopy invariant,\n\\[ \\ind \\mathscr{S}(0) = \\ind \\mathscr{S}(1)\\:. \\]\n\\end{Thm}\n\nIn the chiral odd case, the analogous result is stated as follows.\n\\begin{Thm} \\label{thmstablem0}\nLet~$\\mathscr{S}(t) : \\H_0 \\rightarrow \\H_0$, $t \\in [0,1]$, be\na family of fermionic signature operators of finite chiral index in the massless odd case (see~\\eqref{massless}).\nMoreover, let~$E$ be a Banach space together with an embedding~$\\iota : E \\hookrightarrow \\H_R$\nsuch that the operator~$\\mathscr{S}_L|_{\\H_L} : \\H_L \\rightarrow \\H_R$ has the following properties:\n\\begin{itemize}\n\\item[(i)] For every~$t \\in [0,1]$, the image of~$\\mathscr{S}_L(t)$ lies in~$\\iota(E)$, giving rise to the mapping\n\\begin{equation} \\label{SigLE2}\n\\mathscr{S}_L(t) \\,:\\, \\H_L \\rightarrow E \\:.\n\\end{equation}\n\\item[(ii)] For every~$t \\in [0,1]$, the image of the operator~$\\mathscr{S}_L$, \\eqref{SigLE2}, is a closed subspace of~$E$.\n\\item[(iii)] The family~$\\mathscr{S}_L(t) : \\H_L \\rightarrow E$ is continuous in the norm topology.\n\\end{itemize}\nThen the chiral index in the massless odd case is a homotopy invariant,\n\\[ \\ind_0 \\mathscr{S}(0) = \\ind_0 \\mathscr{S}(1)\\:. \\]\n\\end{Thm}\nIn Example~\\ref{exstable} below, it will be explained how these theorems can be applied.\n\n\\section{Example: Shift Operators in the Setting of Causal Fermion Systems}\nIn the remainder of this paper we illustrate the previous constructions in several examples.\nThe simplest examples for fermionic signature operators with a non-trivial chiral index\ncan be given in the setting of causal fermion systems.\nWe let~$\\H=\\ell^2(\\mathbb{N})$ be the square-summable sequences with the scalar product\n\\[ \\langle u | v \\rangle_\\H = \\sum_{l=1}^\\infty \\overline{u_l} v_l \\:. \\]\nFor any~$k \\in \\mathbb{N}$ we define the operators~$x_k$ by\n\\[ (x_k \\,u)_k = -u_{k+1} \\:,\\qquad (x_k \\,u)_{k+1} = -u_k \\:, \\]\nand all other components of~$x_k u$ vanish. Thus, writing the series in components,\n\\begin{equation} \\label{xkdef}\nx_k \\,u = (\\underbrace{0,\\ldots, 0}_{\\text{$k-1$ entries}}, -u_{k+1}, -u_k, 0, \\ldots ) \\:.\n\\end{equation}\nEvery operator~$x _k$ obviously has rank two with the non-trivial eigenvalues~$\\pm 1$.\nWe let~$\\mu$ be the counting measure on~$\\mathbb{N}$ and~$\\rho = x_*(\\mu)$ the\npush-forward measure of the mapping~$x : k \\mapsto x_k \\in {\\mathscr{F}} \\subset \\text{\\rm{L}}(\\H)$.\nWe thus obtain a causal fermion system~$(\\H, {\\mathscr{F}}, \\rho)$ of spin dimension one.\n\nNext, we introduce the pseudoscalar operators~$\\Gamma(x_k)$ by\n\\begin{equation} \\label{Gkdef}\n\\Gamma(x_k) \\,u = (\\underbrace{0,\\ldots, 0}_{\\text{$k-1$ entries}}, u_k, -u_{k+1}, 0, \\ldots ) \\:.\n\\end{equation}\nObviously, these operators have the properties~\\eqref{pseudodef}. Moreover,\n\\begin{eqnarray*}\nx\\, \\chi_L(x_k)\\, u = (0,\\ldots, 0, & \\!\\!\\!\\!\\!-u_{k+1}, \\;\\;0, & \\!\\!\\!0, \\ldots ) \\\\\nx\\, \\chi_R(x_k)\\, u = (0,\\ldots, 0, & \\:\\;\\;\\;0, \\;\\;-u_k, & \\!\\!\\!0, \\ldots ) \\:.\n\\end{eqnarray*}\nConsequently, the operators\n\\begin{equation} \\label{shiftsum}\n\\mathscr{S}_{L\\!\/\\!R} = -\\sum_{k=1}^\\infty x\\, \\chi_L(x_k)\n\\end{equation}\ntake the form\n\\[ \\mathscr{S}_L \\,u = (u_2, u_3, u_4, \\ldots) \\:,\\qquad \\mathscr{S}_R \\,u = (0, u_1, u_2, \\ldots) \\]\n(note that the series in~\\eqref{shiftsum} converges weakly; in fact it even converges strongly\nin the sense that the series $\\sum_k (x_k \\chi_L u)$ converges in~$\\H$ for every~$u \\in \\H$).\nThese are the usual shift operators, implying that\n\\[ \\ind \\mathscr{S} = 1 \\:. \\]\n\nWe finally remark that a general index~$p \\in \\mathbb{N}$ can be arranged by modifying~\\eqref{xkdef}\nand~\\eqref{Gkdef} to\n\\begin{align*}\nx_k \\,u &= (\\underbrace{0,\\ldots, 0}_{\\text{$k-1$ entries}}, -u_{k+p}, \\!\\!\n\\underbrace{0, \\ldots, 0}_{\\text{$p-1$ entries}}, \\;-u_k, \\;\\:0, \\ldots ) \\\\\n\\Gamma(x_k) \\,u &= (\\;\\;\\overbrace{0,\\ldots, 0}\\;\\:, \\;\\;\\;\\;u_k,\\;\\;\\; \\overbrace{0, \\ldots, 0}, \\:-u_{k+p}, 0, \\ldots ) \\:.\n\\end{align*}\nMoreover, a negative index can be arranged by exchanging the left- and right-handed components.\n\n\\section{Example: A Dirac Operator with~$\\ind_0 \\mathscr{S} \\neq 0$} \\label{secex1}\nWe now construct a two-dimensional space-time~$(\\mycal M,g)$ together with an odd Dirac operator~${\\mathscr{D}}$\nsuch that the resulting fermionic signature operator in the massless case has a non-trivial chiral\nindex~$\\ind_0$ (see Definition~\\ref{defind0}).\nWe choose~$\\mycal M=(0, 2 \\pi) \\times S^1$ with coordinates~$t \\in (0, 2 \\pi)$ and~$\\varphi \\in [0, 2 \\pi)$.\nWe begin with the flat Lorentzian metric\n\\begin{equation} \\label{2dl}\nds^2 = dt^2 - d\\varphi^2 \\:.\n\\end{equation}\nWe consider two-component complex spinors, with the spin scalar product\n\\begin{equation} \\label{ssprod}\n\\mathopen{\\prec} \\psi | \\phi \\mathclose{\\succ} = \\langle \\psi | \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\phi \\rangle_{\\mathbb{C}^2}\\:.\n\\end{equation}\nWe choose the pseudoscalar matrix as\n\\begin{equation} \\label{pseudoc}\n\\Gamma = \\begin{pmatrix} -1 & 0 \\\\ 0 & 1 \\end{pmatrix} \\:,\n\\end{equation}\nso that\n\\begin{equation} \\label{defchir}\n\\chi_L = \\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix} \\:,\\qquad\n\\chi_R = \\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix} \\:.\n\\end{equation}\nThe space-time inner product~\\eqref{stip} becomes\n\\begin{equation} \\label{stiptorus}\n\\mathopen{<} \\psi|\\phi \\mathclose{>} = \\int_0^{2 \\pi} \\int_0^{2 \\pi} \\mathopen{\\prec} \\psi(t, \\varphi) \\,|\\, \\phi(t, \\varphi) \\mathclose{\\succ}\\:d\\varphi\\, dt \\:.\n\\end{equation}\n\nThe Dirac operator~${\\mathcal{D}}$ should be chosen to be odd (see the right equation in~\\eqref{massless}).\nThis means that~${\\mathcal{D}}$ has the matrix representation\n\\begin{equation} \\label{Direx}\n{\\mathcal{D}} = \\begin{pmatrix} 0 & {\\mathcal{D}}_R \\\\ {\\mathcal{D}}_L & 0 \\end{pmatrix}\n\\end{equation}\nwith suitable operators~${\\mathcal{D}}_L$ and~${\\mathcal{D}}_R$.\nIn order for current conservation to hold, the Dirac operator should be symmetric \nwith respect to the inner product~\\eqref{stiptorus}. This implies that the operators~${\\mathcal{D}}_L$\nand~${\\mathcal{D}}_R$ must both be symmetric,\n\\begin{equation} \\label{DLRsymm}\n{\\mathcal{D}}_L^* = {\\mathcal{D}}_L \\:,\\qquad {\\mathcal{D}}_R^* = {\\mathcal{D}}_R \\:,\n\\end{equation}\nwhere the star denotes the formal adjoint with respect to the scalar product on\nthe Hilbert space~$L^2(\\mycal M, \\mathbb{C})$. We consider the massless Dirac equation\n\\begin{equation} \\label{Dir0}\n{\\mathcal{D}} \\psi = 0 \\:.\n\\end{equation}\nThe scalar product~\\eqref{print} on the solutions takes the form\n\\begin{equation} \\label{printtorus}\n(\\psi | \\phi) = 2 \\pi \\int_0^{2 \\pi} \\langle \\psi(t,\\varphi) | \\phi(t, \\varphi) \\rangle_{\\mathbb{C}^2}\\: d\\varphi \\:,\n\\end{equation}\ngiving rise to the Hilbert space~$(\\H_0, (.|.))$. As a consequence of current conservation,\nthis scalar product is independent of the choice of~$t$.\n\nWe assume that the system is invariant under time translations and is a first order differential\noperator in time. More precisely, we assume that\n\\begin{equation} \\label{DirHam}\n{\\mathcal{D}}_{L\\!\/\\!R} = i \\partial_t - H_{L\\!\/\\!R}\n\\end{equation}\nwith purely spatial operators~$H_{L\\!\/\\!R}$, referred to as the left- and right-handed Hamiltonians.\nMoreover, we assume that these Hamiltonians are homogeneous.\nThis implies that they can be diagonalized by plane waves,\n\\[ {\\mathcal{D}}_c \\:e^{i k \\varphi} = \\omega_{k,c}\\: e^{i k \\varphi} \\qquad \\text{with~$k \\in \\mathbb{Z}$ and~$c \\in \\{L, R\\}$}\\:. \\]\nAs a consequence, the Dirac equation~\\eqref{Dir0} can be solved by the plane waves\n\\begin{equation} \\label{esols}\n{\\mathfrak{e}}_{k, L} = \\frac{1}{2 \\pi}\\: e^{-i \\omega_{k, L} t + i k \\varphi} \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\:,\\qquad\n{\\mathfrak{e}}_{k, R} = \\frac{1}{2 \\pi}\\: e^{-i \\omega_{k, R} t + i k \\varphi} \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} \\:.\n\\end{equation}\nThe vectors~$({\\mathfrak{e}}_{k, c})_{k \\in \\mathbb{Z}, c \\in \\{L, R\\}}$ form an orthonormal basis of the\nHilbert space~$\\H_0$.\nWe remark that the Dirac operator of the Minkowski vacuum is obtained by choosing\n\\[ H_L = i \\partial_\\varphi \\:,\\qquad H_R = -i \\partial_\\varphi \\]\n(see for example~\\cite{drum} or~\\cite[Section~7.2]{topology}).\nIn this case, $\\omega_{k, L\\!\/\\!R} = \\mp k$. More generally, choosing~${\\mathcal{D}}_c$ as a\nhomogeneous differential operator of first order, the eigenvalues~$\\omega_{k,c}$ are linear in~$k$.\nHere we do not want to assume that the operators~${\\mathcal{D}}_c$ are differential operators.\nThen the eigenvalues~$\\omega_{k, L}$ and~$\\omega_{k,R}$\ncan be chosen arbitrarily and independently, except for the constraint coming from the\nsymmetry~\\eqref{DLRsymm} that these eigenvalues must be real.\n\nMore specifically, for a given parameter~$p \\in \\mathbb{N}$ we choose\n\\begin{equation} \\label{omegaex}\n\\omega_{k,L} = -k \\qquad \\text{and} \\qquad\n\\omega_{k, R} = \\left\\{ \\begin{array}{cl} k & \\text{if~$k \\leq 0$} \\\\\nk+p & \\text{if~$k > 0$}\n\\end{array} \\right.\n\\end{equation}\n(see Figure~\\ref{figdisperse}).\n\\begin{figure}\n\\scalebox{1}\n{\n\\begin{pspicture}(0,-2.52)(6.92,2.52)\n\\psdots[dotsize=0.24,fillstyle=solid,dotstyle=o](1.86,0.1)\n\\psdots[dotsize=0.24,fillstyle=solid,dotstyle=o](2.46,-0.5)\n\\usefont{T1}{ptm}{m}{n}\n\\rput(6.61,-0.715){$k$}\n\\psline[linewidth=0.04cm,arrowsize=0.3cm 1.0,arrowlength=1.5,arrowinset=0.5]{->}(0.04,-1.1)(6.7,-1.1)\n\\psline[linewidth=0.04cm,arrowsize=0.3cm 1.0,arrowlength=1.5,arrowinset=0.5]{->}(3.06,-2.5)(3.06,2.5)\n\\usefont{T1}{ptm}{m}{n}\n\\rput(3.45,2.325){$\\omega$}\n\\psline[linewidth=0.02cm,linestyle=dashed,dash=0.16cm 0.16cm](1.66,-2.5)(6.06,1.9)\n\\psline[linewidth=0.02cm,linestyle=dashed,dash=0.16cm 0.16cm](0.06,1.9)(4.46,-2.5)\n\\psdots[dotsize=0.12](3.66,0.7)\n\\psdots[dotsize=0.12](3.06,-1.1)\n\\psdots[dotsize=0.12](2.46,-0.5)\n\\psdots[dotsize=0.12](1.26,0.7)\n\\psdots[dotsize=0.12](0.06,1.9)\n\\psdots[dotsize=0.12](4.26,-2.3)\n\\psdots[dotsize=0.12](2.46,-1.7)\n\\psdots[dotsize=0.12](1.86,-2.3)\n\\psdots[dotsize=0.12](1.86,0.1)\n\\psdots[dotsize=0.12](0.66,1.3)\n\\psdots[dotsize=0.12](4.26,1.3)\n\\psdots[dotsize=0.12](4.86,1.9)\n\\psdots[dotsize=0.12](3.66,-1.7)\n\\psline[linewidth=0.04cm](3.66,-1.0)(3.66,-1.2)\n\\psline[linewidth=0.04cm](4.26,-1.0)(4.26,-1.2)\n\\psline[linewidth=0.04cm](4.86,-1.0)(4.86,-1.2)\n\\psline[linewidth=0.04cm](2.46,-1.0)(2.46,-1.2)\n\\psline[linewidth=0.04cm](1.86,-1.0)(1.86,-1.2)\n\\psline[linewidth=0.04cm](1.26,-1.0)(1.26,-1.2)\n\\psline[linewidth=0.04cm](0.66,-1.0)(0.66,-1.2)\n\\psline[linewidth=0.04cm](6.06,-1.0)(6.06,-1.2)\n\\psline[linewidth=0.04cm](5.46,-1.0)(5.46,-1.2)\n\\psline[linewidth=0.04cm](0.06,-1.0)(0.06,-1.2)\n\\psline[linewidth=0.04cm](3.16,-0.5)(2.96,-0.5)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](0.06,0.7)(6.26,0.7)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](0.06,1.3)(6.26,1.3)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](0.06,1.9)(6.26,1.9)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](0.06,-1.7)(6.26,-1.7)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](0.06,-2.3)(6.26,-2.3)\n\\usefont{T1}{ptm}{m}{n}\n\\rput(3.35,-0.435){$1$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(1.87,-0.715){$\\omega_{-1,L}$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(1.29,-0.135){$\\omega_{-2,L}$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(3.73,-0.815){$1$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(3.72,0.425){$\\omega_{1,R}$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(4.63,-2.055){$\\omega_{2,L}$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(4.28,1.025){$\\omega_{2,R}$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(1.3,-2.055){$\\omega_{-2,R}$}\n\\end{pspicture} \n}\n\\caption{The eigenvalues~$\\omega_{k, L\\!\/\\!R}$ in the case~$p=2$.}\n\\label{figdisperse}\n\\end{figure}\nThen the space-time inner product of the basis vectors~$({\\mathfrak{e}}_{k, c})_{k \\in \\mathbb{Z}, c \\in \\{L, R\\}}$\nis computed by\n\\begin{align*}\n\\mathopen{<} {\\mathfrak{e}}_{k,L} | {\\mathfrak{e}}_{k', L} \\mathclose{>} &= 0 = \\mathopen{<} {\\mathfrak{e}}_{k,R} | {\\mathfrak{e}}_{k', R} \\mathclose{>} \\\\\n\\mathopen{<} {\\mathfrak{e}}_{k,R} | {\\mathfrak{e}}_{k', L} \\mathclose{>} \n&= \\frac{1}{2 \\pi} \\: \\delta_{k,k'} \\int_0^{2 \\pi} e^{i (\\omega_{k,R} - \\omega_{k,L}) t} \\,dt =\n\\delta_{k,k'} \\: \\delta_{\\omega_{k,R}, \\;\\omega_{k,L}} = \\delta_{k,0} \\: \\delta_{k',0} \\:.\n\\end{align*}\nWe conclude~$\\mathscr{S}$ does not have finite chiral index.\n\nIn order to obtain a non-trivial index, we need to modify our example.\nThe idea is to change the space-time inner product in such a way that\nthe inner product between two different plane-wave solutions with\nthe same frequencies becomes non-zero. As a consequence,\nthe corresponding pair of plane-wave solutions will disappear from the kernel.\nThe only vectors which remain in the kernel are those which do not have a partner\nfor pairing, so that\n\\[ \\ker \\mathscr{S}_L|_{\\H_L} = \\text{span} \\big( {\\mathfrak{e}}_{-1,L}, \\ldots, {\\mathfrak{e}}_{-p,L} \\big) \\:,\\qquad\n\\ker \\mathscr{S}_R|_{\\H_R} = \\{0\\}\\:. \\]\n(see again Figure~\\ref{figdisperse}, where the pairs are indicated by horizontal dashed lines,\nwhereas the vectors in the kernel correspond to the circled dots).\nGenerally speaking, the method to modify the space-time inner product for states with\nthe same frequency is to insert a potential into the Dirac equation which\nis time-independent but has a non-trivial spatial dependence.\nIt is most convenient to work with a {\\em{conformal transformation}}.\nThus we go over from the Minkowski metric~\\eqref{2dl} to the conformally flat metric\n\\begin{equation} \\label{conform}\nd\\tilde{s}^2 = f(\\varphi)^2 \\left( dt^2 - d\\varphi^2 \\right) \\:,\n\\end{equation}\nwhere~$f \\in C^\\infty(\\mathbb{R}\/(2 \\pi \\mathbb{Z}))$ is a non-negative, smooth, $2 \\pi$-periodic function.\nThe conformal invariance of the Dirac equation (for details see for example~\\cite[Section~8.1]{topology}\nand the references therein) states in our situation that the Dirac operator transforms as\n\\begin{equation}\n\\tilde{{\\mathcal{D}}} = f^{-\\frac{3}{2}} \\,{\\mathcal{D}}\\, f^{\\frac{1}{2}} \\:, \\label{Dirconf}\n\\end{equation}\nso that\n\\[ \\tilde{{\\mathcal{D}}} = \\begin{pmatrix} 0 & \\tilde{{\\mathcal{D}}}_R \\\\ \\tilde{{\\mathcal{D}}}_L & 0 \\end{pmatrix}\n\\qquad \\text{with} \\qquad\n\\tilde{{\\mathcal{D}}}_{L\\!\/\\!R} = f^{-\\frac{3}{2}} \\,{\\mathcal{D}}_{L\\!\/\\!R}\\, f^{\\frac{1}{2}} \\:. \\]\nThe solutions of the massless Dirac equation are modified simply by a conformal factor,\n\\begin{equation} \\label{psiconf}\n\\tilde{\\psi} = f^{-\\frac{1}{2}}\\: \\psi \\:.\n\\end{equation}\nThe space-time inner product~\\eqref{stiptorus} and the scalar product~\\eqref{printtorus} transform to\n\\begin{align}\n\\mathopen{<} \\tilde{\\psi} | \\tilde{\\phi} \\mathclose{>}\n&= \\int_0^{2 \\pi} \\int_0^{2 \\pi} \\mathopen{\\prec} \\tilde{\\psi}(t, \\varphi) \\,|\\, \\tilde{\\phi}(t, \\varphi) \\mathclose{\\succ}\\:\nf(\\varphi)^2 \\,d\\varphi\\, dt \\nonumber \\\\\n&= \\int_0^{2 \\pi} \\int_0^{2 \\pi} \\mathopen{\\prec} \\psi(t, \\varphi) \\,|\\, \\phi(t, \\varphi) \\mathclose{\\succ}\\:\nf(\\varphi) \\,d\\varphi\\, dt \\label{stiptrans} \\\\\n(\\tilde{\\psi} | \\tilde{\\phi}) &= 2 \\pi \\int_0^{2 \\pi} \\langle \\tilde{\\psi}(t,\\varphi) | \\tilde{\\phi}(t, \\varphi) \\rangle_{\\mathbb{C}^2}\\: \nf(\\varphi) \\:d\\varphi \\nonumber \\\\\n&= 2 \\pi \\int_0^{2 \\pi} \\langle \\psi(t,\\varphi) | \\phi(t, \\varphi) \\rangle_{\\mathbb{C}^2}\\: \nd\\varphi = (\\psi | \\phi) \\:. \\label{printtrans}\n\\end{align}\nTo understand these transformation laws, one should keep in mind that the spin scalar product\nremains unchanged under conformal transformations. The same is true for the \nintegrand~$\\mathopen{\\prec} \\psi | \\slashed{\\nu} \\phi \\mathclose{\\succ}_x$ of the scalar product~\\eqref{print}, because\nthe operator~$\\slashed{\\nu}$ is normalized by~$\\slashed{\\nu}^2=\\mbox{\\rm 1 \\hspace{-1.05 em} 1}$.\n\nFrom~\\eqref{printtrans} we conclude that the scalar product does not change under conformal\ntransformations. In particular, the conformally transformed plane-wave solutions\n\\begin{equation} \\label{vertical}\n\\tilde{{\\mathfrak{e}}}_{k, L\\!\/\\!R} = f(\\varphi)^{-\\frac{1}{2}}\\: {\\mathfrak{e}}_{k, L\\!\/\\!R}\n\\end{equation}\nare an orthonormal basis of~$\\tilde{\\H}_0$. The space-time inner product~\\eqref{stiptrans}, however,\ninvolves a conformal factor~$f(\\varphi)$. As a consequence, the space-time inner product\nof the basis vectors~$(\\tilde{{\\mathfrak{e}}}_{k, c})_{k \\in \\mathbb{Z}, c \\in \\{L, R\\}}$ can be computed by\n\\begin{align*}\n\\mathopen{<} \\tilde{{\\mathfrak{e}}}_{k,L} | \\tilde{{\\mathfrak{e}}}_{k', L} \\mathclose{>} &= 0 = \\mathopen{<} \\tilde{{\\mathfrak{e}}}_{k,R} | \\tilde{{\\mathfrak{e}}}_{k', R} \\mathclose{>} \\\\\n\\mathopen{<} \\tilde{{\\mathfrak{e}}}_{k,R} | \\tilde{{\\mathfrak{e}}}_{k', L} \\mathclose{>} &= \\int_0^{2 \\pi} dt \\int_0^{2 \\pi} f(\\varphi)\\, d\\varphi\\; \n\\mathopen{\\prec} \\tilde{{\\mathfrak{e}}}_{k,R}(t,\\varphi) \\,|\\, \\tilde{{\\mathfrak{e}}}_{k', L}(t,\\varphi) \\mathclose{\\succ} \\\\\n&= \\frac{1}{2 \\pi} \\:\\delta_{\\omega_{k,R}, \\,\\omega_{k',L}}\n\\int_0^{2 \\pi} f(\\varphi)\\: e^{-i (k-k') \\varphi}\\, d\\varphi\n= \\frac{1}{2 \\pi} \\:\\delta_{\\omega_{k,R}, \\omega_{k',L}}\\: \\hat{f}_{k-k'}\\:,\n\\end{align*}\nwhere~$\\hat{f}_k$ is the $k^\\text{th}$ Fourier coefficient of~$f$,\n\\[ f(\\varphi) = \\frac{1}{2 \\pi} \\sum_{k \\in \\mathbb{Z}} \\hat{f}_k\\: e^{i k \\varphi} \\:. \\]\n\nUsing the explicit form of the frequencies~\\eqref{omegaex}, we obtain the following\ninvariant subspaces and corresponding matrix representations of~$\\mathscr{S}$,\n\\begin{align*}\n\\hat{\\mathscr{S}}|_{\\text{span}(\\tilde{{\\mathfrak{e}}}_{-k, L}, \\tilde{{\\mathfrak{e}}}_{k, R})} &=\n\\frac{1}{2 \\pi} \\begin{pmatrix} 0 & \\overline{\\hat{f}_{2k}} \\\\ \\hat{f}_{2k} & 0 \\end{pmatrix} \n&&\\hspace*{-0.8cm} \\text{if~$k \\leq 0$} \\\\\n\\hat{\\mathscr{S}}|_{\\text{span}(\\tilde{{\\mathfrak{e}}}_{-k-p, L}, \\tilde{{\\mathfrak{e}}}_{k, R})} &=\n\\frac{1}{2 \\pi} \\begin{pmatrix} 0 & \\overline{\\hat{f}_{2k+p}} \\\\ \\hat{f}_{2k+p} & 0 \\end{pmatrix} \n&&\\hspace*{-0.8cm} \\text{if~$k>p$} \\\\\n\\hat{\\mathscr{S}}|_{\\text{span}(\\tilde{{\\mathfrak{e}}}_{-1, L}, \\tilde{{\\mathfrak{e}}}_{-p, L})} &= 0 \\:.\n\\end{align*}\nIn particular, we can read off the chiral index:\n\\begin{Prp} \\label{prpindex} Assume that almost all Fourier coefficients~$\\hat{f}_k$ \nof the conformal function in~\\eqref{conform} are non-zero.\nThen the fermionic signature operator in the massless odd case has finite chiral\nindex (see Definition~\\ref{defind0}) and~$\\ind_0 \\mathscr{S} = p$.\n\\end{Prp}\n\nWe finally compute the Dirac operator in position space. The dispersion relations in~\\eqref{omegaex}\nare realized by the operators\n\\begin{align*}\n{\\mathcal{D}}_L &= i (\\partial_t - \\partial_\\varphi) \\\\\n{\\mathcal{D}}_R &= i (\\partial_t + \\partial_\\varphi) + {\\mathscr{B}} \\:,\n\\end{align*}\nwhere~${\\mathscr{B}}$ is the spatial integral operator\n\\[ \\big( {\\mathscr{B}} \\psi \\big)(t,\\varphi) = \\int_0^{2 \\pi} {\\mathscr{B}}(\\varphi, \\varphi')\\: \\psi(t, \\varphi')\\: d\\varphi' \\]\nwith the distributional integral kernel\n\\[ {\\mathscr{B}}(\\varphi, \\varphi') = -\\frac{p}{2 \\pi} \\sum_{k=1}^\\infty e^{i k (\\varphi-\\varphi')} \\nonumber \\\\\n= -\\frac{p}{2}\\: \\delta(\\varphi-\\varphi') -\\frac{p}{2 \\pi}\\:\\frac{\\text{PP}}{e^{-i(\\varphi-\\varphi')}-1} \\:. \\]\nHence, choosing the Dirac matrices as\n\\begin{equation} \\label{gammadef}\n\\gamma^0 = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\:,\\qquad\n\\gamma^1 = \\begin{pmatrix} 0 & 1 \\\\ -1 & 0 \\end{pmatrix}\n\\end{equation}\nand using~\\eqref{defchir}, we obtain\n\\begin{equation} \\label{Diracform}\n{\\mathcal{D}} = i \\gamma^0 \\partial_t + i \\gamma^1 \\partial_\\varphi\n+ \\gamma^1 \\chi_R \\, {\\mathscr{B}} \\:.\n\\end{equation}\nPerforming the conformal transformation~\\eqref{Dirconf}, we finally obtain\n\\begin{align}\n\\big(\\tilde{{\\mathcal{D}}} \\psi)(t, \\varphi) &= \\frac{i}{f(\\varphi)} \\left( \\gamma^0 \\partial_t + \\gamma^1 \\partial_\\varphi \n+ \\frac{f'(\\varphi)}{2 f(\\varphi)} -\\frac{p}{2} \\: \\gamma^1 \\chi_R \\right) \\psi(t, \\varphi) \\label{Dir1} \\\\\n&\\qquad -\\frac{p}{2 \\pi}\\: \\frac{\\gamma^1 \\chi_R}{f(\\varphi)^\\frac{3}{2}}\n\\int_0^{2 \\pi} \\frac{\\text{PP}}{e^{-i(\\varphi-\\varphi')}-1}\\:\n\\psi(t, \\varphi')\\: \\sqrt{f(\\varphi')}\\: d\\varphi' \\:. \\label{Dir2}\n\\end{align}\nThus~\\eqref{Dir1} is the Dirac operator in the Lorentzian metric~\\eqref{conform}\nwith a constant right-handed potential. Moreover, the summand~\\eqref{Dir2} is a nonlocal integral\noperator involving a singular integral kernel.\n\nThis example shows that the index of Proposition~\\ref{prpindex} in general does not encode\nthe topology of space-time, because for a fixed space-time topology the index can take\nany integer value. The way we understand the index is that it gives topological information\non the singular behavior of the potential in the Dirac operator.\n\n\\section{Example: A Dirac Operator with~$\\ind \\mathscr{S} \\neq 0$} \\label{secex2}\nWe now construct an example of a fermionic signature operator for which the\nindex~$\\ind \\mathscr{S}$ of Definition~\\ref{defind} is non-trivial.\nTo this end, we want to modify the example of the previous section.\nThe major difference to the previous setting is that the Hilbert space~$\\H_m$\ndoes not have a decomposition into two subspaces~$\\H_L$ and~$\\H_R$, making\nit necessary to consider the operators~$\\mathscr{S}_L$ and~$\\mathscr{S}_R$ as operators on the whole\nsolution space~$\\H_m$. Our first task is to remove the infinite-dimensional kernels of the operators~$\\mathscr{S}_L$\nand~$\\mathscr{S}_R$. This can typically be achieved by perturbing the Dirac operator, for example by introducing\na rest mass. The second and more substantial modification is to arrange that the\noperators~$\\mathscr{S}_L$ and~$\\mathscr{S}_R$ have {\\em{infinite-dimensional invariant subspaces}}.\nThis is needed for the following reason: In the example of the previous section,\nthe operator~$\\mathscr{S}_L|_{\\H_L} : \\H_L \\rightarrow \\H_R$ mapped one Hilbert\nspace to another Hilbert space. Therefore, we obtained a non-trivial index simply by arranging\nthat the operator~$\\mathscr{S}_L|_{\\H_L}$ gives a non-trivial ``pairing'' of vectors of~$\\H_L$ with\nvectors of~$\\H_R$ (as indicated in Figure~\\ref{figdisperse} by the horizontal dashed lines).\nIn particular, if considered as an operator on~$\\H_0$, the operator~$\\mathscr{S}_L$ had at most two-dimensional\ninvariant subspaces.\nFor the chiral index of Definition~\\ref{defind}, however, we have only one Hilbert space~$\\H_m$\nto our disposal, so that the operator~$\\mathscr{S}_L : \\H_m \\rightarrow \\H_m$ is an endomorphism of~$\\H_m$.\nAs a consequence, the chiral index is trivial whenever~$\\H_m$ splits into a direct sum\nof finite-dimensional subspaces which are invariant under~$\\mathscr{S}_L$\n(because on each invariant subspace, the index is trivial due to the\nrank-nullity theorem of linear algebra).\n\nThe following example is designed with the aim of showing in explicit detail that the\nindex is non-zero. Our starting point are the plane-wave solutions~\\eqref{esols}\nwith the frequencies according to~\\eqref{omegaex} with~$p \\in \\mathbb{N}$.\nIn Figure~\\ref{figsnail} the transformed plane-wave solutions~$\\tilde{{\\mathfrak{e}}}_{k,c}$ \n(where the transformation from~${\\mathfrak{e}}_{k,c}$ to~$\\tilde{{\\mathfrak{e}}}_{k,c}$ will be explained below) are\narranged according to their frequencies and momenta on a lattice.\n\\begin{figure}\n\\scalebox{1}\n{\n\\begin{pspicture}(0,-2.175)(7.06,2.18)\n\\psdots[dotsize=0.24,fillstyle=solid,dotstyle=o](3.22,0.06)\n\\usefont{T1}{ptm}{m}{n}\n\\rput(6.23,-0.175){$k$}\n\\psline[linewidth=0.04cm,arrowsize=0.3cm 1.0,arrowlength=1.5,arrowinset=0.5]{->}(1.62,-0.54)(6.42,-0.54)\n\\psline[linewidth=0.04cm,arrowsize=0.3cm 1.0,arrowlength=1.5,arrowinset=0.5]{->}(3.82,-2.14)(3.82,2.16)\n\\usefont{T1}{ptm}{m}{n}\n\\rput(4.21,1.925){$\\omega$}\n\\psline[linewidth=0.02cm,linestyle=dashed,dash=0.16cm 0.16cm](2.42,-1.94)(6.02,1.66)\n\\psline[linewidth=0.02cm,linestyle=dashed,dash=0.16cm 0.16cm](1.62,1.66)(5.22,-1.94)\n\\psdots[dotsize=0.12](4.42,0.66)\n\\psdots[dotsize=0.12](3.82,-0.54)\n\\psdots[dotsize=0.12](3.22,0.06)\n\\psdots[dotsize=0.12](2.02,1.26)\n\\psdots[dotsize=0.12](5.02,-1.74)\n\\psdots[dotsize=0.12](3.22,-1.14)\n\\psdots[dotsize=0.12](2.62,-1.74)\n\\psdots[dotsize=0.12](2.62,0.66)\n\\psdots[dotsize=0.12](5.02,1.26)\n\\psdots[dotsize=0.12](4.42,-1.14)\n\\psline[linewidth=0.04cm](4.42,-0.44)(4.42,-0.64)\n\\psline[linewidth=0.04cm](5.02,-0.44)(5.02,-0.64)\n\\psline[linewidth=0.04cm](5.62,-0.44)(5.62,-0.64)\n\\psline[linewidth=0.04cm](3.22,-0.44)(3.22,-0.64)\n\\psline[linewidth=0.04cm](2.62,-0.44)(2.62,-0.64)\n\\psline[linewidth=0.04cm](2.02,-0.44)(2.02,-0.64)\n\\psline[linewidth=0.04cm](3.92,0.06)(3.72,0.06)\n\\usefont{T1}{ptm}{m}{n}\n\\rput(4.11,0.125){$1$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(1.44,1.245){$\\tilde{{\\mathfrak{e}}}_{-3,L}$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(4.49,-0.255){$1$}\n\\psline[linewidth=0.03cm,tbarsize=0.07055555cm 5.0]{|*-|*}(2.02,1.46)(2.02,1.06)\n\\psline[linewidth=0.03cm,tbarsize=0.07055555cm 5.0]{|*-|*}(5.02,1.46)(5.02,1.06)\n\\psline[linewidth=0.03cm,tbarsize=0.07055555cm 5.0]{|*-|*}(4.42,0.86)(4.42,0.46)\n\\psline[linewidth=0.03cm,tbarsize=0.07055555cm 5.0]{|*-|*}(2.62,0.86)(2.62,0.46)\n\\psline[linewidth=0.03cm,tbarsize=0.07055555cm 5.0]{|*-|*}(3.22,0.26)(3.22,-0.14)\n\\psline[linewidth=0.03cm,tbarsize=0.07055555cm 5.0]{|*-|*}(4.42,-0.94)(4.42,-1.34)\n\\psline[linewidth=0.03cm,tbarsize=0.07055555cm 5.0]{|*-|*}(3.22,-0.94)(3.22,-1.34)\n\\psline[linewidth=0.03cm,tbarsize=0.07055555cm 5.0]{|*-|*}(2.62,-1.54)(2.62,-1.94)\n\\psline[linewidth=0.03cm,tbarsize=0.07055555cm 5.0]{|*-|*}(5.02,-1.54)(5.02,-1.94)\n\\psbezier[linewidth=0.03,arrowsize=0.08cm 3.0,arrowlength=1.2,arrowinset=0.4]{->}(2.22,1.26)(3.02,1.54)(3.92,1.7)(4.82,1.26)\n\\psbezier[linewidth=0.03,arrowsize=0.08cm 3.0,arrowlength=1.2,arrowinset=0.4]{->}(5.22,1.16)(5.92,0.2)(5.9,-1.0)(5.2,-1.66)\n\\psbezier[linewidth=0.03,arrowsize=0.08cm 3.0,arrowlength=1.2,arrowinset=0.4]{->}(4.84,-1.78)(4.12,-2.1)(3.7,-2.16)(2.8,-1.78)\n\\psbezier[linewidth=0.03,arrowsize=0.08cm 3.0,arrowlength=1.2,arrowinset=0.4]{->}(2.44,-1.7)(1.74,-1.0)(1.72,-0.06)(2.48,0.64)\n\\psbezier[linewidth=0.03,arrowsize=0.08cm 3.0,arrowlength=1.2,arrowinset=0.4]{->}(2.8,0.68)(3.24,0.88)(3.66,1.0)(4.24,0.68)\n\\psbezier[linewidth=0.03,arrowsize=0.08cm 3.0,arrowlength=1.2,arrowinset=0.4]{->}(4.58,0.6)(5.04,-0.02)(4.98,-0.58)(4.58,-1.1)\n\\psbezier[linewidth=0.03,arrowsize=0.08cm 3.0,arrowlength=1.2,arrowinset=0.4]{->}(4.28,-1.18)(4.1,-1.3)(3.68,-1.36)(3.4,-1.18)\n\\psbezier[linewidth=0.03,arrowsize=0.08cm 3.0,arrowlength=1.2,arrowinset=0.4]{->}(3.06,-1.14)(2.72,-0.7)(2.8,-0.32)(3.04,0.04)\n\\usefont{T1}{ptm}{m}{n}\n\\rput(5.64,-1.875){$\\tilde{{\\mathfrak{e}}}_{2,L}$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(5.43,1.645){$\\tilde{{\\mathfrak{e}}}_{2,R}$}\n\\psline[linewidth=0.04cm,tbarsize=0.07055555cm 5.0]{|*-|*}(3.82,-0.34)(3.82,-0.74)\n\\end{pspicture} \n}\n\\caption{The action of~$\\mathscr{S}_L$ on the transformed plane-wave solutions in the case~$p=1$.}\n\\label{figsnail}\n\\end{figure}\nWe shall construct the operator~$\\mathscr{S}_L$ in such a way that\nthese plane-wave solutions are mapped to each other as indicated by the arrows.\nThus similar to a shift operator, $\\mathscr{S}_L$ maps the basis vectors to each other ``spiraling in,''\nimplying that the vector~$\\tilde{{\\mathfrak{e}}}_{-1,L}$ (depicted with the circled dot) is in the kernel of~$\\mathscr{S}_L$.\nLikewise, the operator~$\\mathscr{S}_R$ acts like a ``spiraling out'' shift vector, so that it is injective.\nIn this way, we arrange that~$\\ind \\mathscr{S} = 1$. Similarly, in the case~$p>1$ we shall obtain~$p$\nspirals, so that~$\\ind \\mathscr{S}=p$.\n\nBefore entering the detailed construction, we point out that our method\nis driven by the wish that the example should be explicit and that the kernels of the\nchiral signature operators should be given in closed form.\nThis makes it necessary to introduce a Dirac operator which seems somewhat artificial.\nIn particular, instead of introducing a rest mass,\nwe arrange a mixing of the left- and right-handed components\nusing a time-dependent vectorial gauge transformation.\nMoreover, we again work with a conformal transformation with a carefully adjusted\nspatial and time dependence. We consider these special features merely as a\nrequirement needed in order to make the computations as simple as possible.\nIn view of the stability result of Theorem~\\ref{thmstable}, we expect that the index\nis also non-trivial in more realistic examples involving a rest mass and less fine-tuned potentials.\nBut probably, this goes at the expense of longer computations or less explicit arguments.\n\nWe begin on the cylinder~$\\mycal M=(0, 6 \\pi) \\times S^1$, again with the Minkowski metric~\\eqref{2dl}\nand two-component spinors endowed with the spin scalar product~\\eqref{ssprod}.\nThe space-time inner product~\\eqref{stip} becomes\n\\begin{equation} \\label{stiptoru2}\n\\mathopen{<} \\psi|\\phi \\mathclose{>} = \\int_0^{6 \\pi} \\int_0^{2 \\pi} \\mathopen{\\prec} \\psi(t, \\varphi) \\,|\\, \\phi(t, \\varphi) \\mathclose{\\succ}\\: d\\varphi \\:dt \\:,\n\\end{equation}\nwhereas the scalar product on solutions of the Dirac equation is again given by~\\eqref{printtorus}.\nWe again consider the massless Dirac equation~\\eqref{Dir0}\nwith the Dirac operator~\\eqref{Direx} and the left- and right-handed\noperators according to~\\eqref{DirHam}. Moreover, we again assume that the operators~${\\mathcal{D}}_{L\\!\/\\!R}$\nhave the plane-wave solutions~\\eqref{esols} with frequencies~\\eqref{omegaex}.\nFor a fixed real parameter~$\\nu \\neq 0$, we consider the transformation\n\\begin{equation} \\label{Utdef}\nU(t) = \\frac{\\mbox{\\rm 1 \\hspace{-1.05 em} 1} + i \\nu \\gamma^0 \\cos t\/3}{\\sqrt{1 + \\nu^2 \\cos^2 t\/3}} = \n\\frac{1}{\\sqrt{1 + \\nu^2 \\cos^2 t\/3}}\n\\begin{pmatrix} 1 & i \\nu \\cos t\/3 \\\\ i \\nu \\cos t\/3 & 1 \\end{pmatrix} \\:.\n\\end{equation}\nObviously, $U(t) \\in {\\rm{U}}(2)$ is a unitary matrix. Moreover, it commutes with~$\\gamma^0$,\nimplying that it is also unitary with respect to the spin scalar product.\nAs a consequence, the transformation~$U(t)$ is unitary both on the Hilbert space~$\\H_0$\nand with respect to the inner product~\\eqref{stiptoru2}.\nNext, we again consider a conformal transformation~\\eqref{Dirconf} and~\\eqref{psiconf},\nbut now with a conformal function~$f(t, \\varphi)$ which depends on space and time.\nThus we set\n\\begin{equation} \\label{Dirdef}\n\\tilde{{\\mathcal{D}}} = f^{-\\frac{3}{2}} U \\,{\\mathcal{D}}\\, U^* f^{\\frac{1}{2}} \\qquad \\text{and} \\qquad\n\\tilde{\\psi} = f^{-\\frac{1}{2}}\\: U \\,\\psi \\:.\n\\end{equation}\nSimilar to~\\eqref{stiptrans} and~\\eqref{printtrans}, the inner products transform to\n\\[ \\mathopen{<} \\tilde{\\psi} | \\tilde{\\phi} \\mathclose{>}\n= \\int_0^{6 \\pi} \\int_0^{2 \\pi} \\mathopen{\\prec} \\psi(t, \\varphi) \\,|\\, \\phi(t, \\varphi) \\mathclose{\\succ}\\:\nf(t, \\varphi)\\, d\\varphi\\,dt \\qquad \\text{and} \\qquad\n(\\tilde{\\psi} | \\tilde{\\phi}) = (\\psi | \\phi) \\:. \\]\nIn particular, the transformed plane wave solutions~$\\tilde{e}_{k,c}$ are an orthonormal\nbasis of~$\\H_0$.\nKeeping in mind that the chiral projectors in~\\eqref{stipLR} do {\\em{not}} commute with~$U$,\nwe obtain\n\\[ \\mathopen{<} \\tilde{\\psi} |\\tilde{\\phi} \\mathclose{>}_L\n= \\int_0^{6 \\pi} \\int_0^{2 \\pi} \\mathopen{\\prec} U(t)\\, \\psi(t, \\varphi) \\,|\\, \\chi_L\\, U(t)\\, \\phi(t, \\varphi) \\mathclose{\\succ}\\:\nf(t, \\varphi)\\, d\\varphi\\,dt \\]\nand thus, in view of~\\eqref{SLRdef2},\n\\begin{equation} \\label{SL1}\n(\\tilde{{\\mathfrak{e}}}_{k,c} \\,|\\, \\mathscr{S}_L \\,\\tilde{{\\mathfrak{e}}}_{k',c'}) = \n\\int_0^{6 \\pi} \\int_0^{2 \\pi} \\mathopen{\\prec} U {\\mathfrak{e}}_{k,c} \\,|\\, \\chi_L U {\\mathfrak{e}}_{k',c'} \\mathclose{\\succ}\\:\nf(t, \\varphi)\\, d\\varphi\\,dt \\:.\n\\end{equation}\nIn order to get rid of the square roots in~\\eqref{Utdef}, it is most convenient to set\n\\begin{equation} \\label{lambdadef}\nV(t) = \\begin{pmatrix} 1 & i \\nu \\cos t\/3 \\\\ i \\nu \\cos t\/3 & 1 \\end{pmatrix}\n\\qquad \\text{and} \\qquad \\mu(t, \\varphi) = \\frac{f(t, \\varphi)}{1 + \\nu^2 \\cos^2 t\/3} \\:.\n\\end{equation}\nThen~\\eqref{SL1} simplifies to\n\\begin{equation} \\label{SL2}\n(\\tilde{{\\mathfrak{e}}}_{k,c} \\,|\\, \\mathscr{S}_L \\,\\tilde{{\\mathfrak{e}}}_{k',c'}) = \n\\int_0^{6 \\pi} \\int_0^{2 \\pi} \\mathopen{\\prec} V {\\mathfrak{e}}_{k,c} \\,|\\, \\chi_L V {\\mathfrak{e}}_{k',c'} \\mathclose{\\succ}\\:\n\\mu(t, \\varphi)\\, d\\varphi\\,dt \\:.\n\\end{equation}\n\nLet us first discuss the effect of the transformation~$V$. A left-handed spinor is mapped to\n\\[ V \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} = \n\\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} + \\frac{i}{2}\\: e^{it\/3}\\, \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix}\n+ \\frac{i}{2}\\: e^{-it\/3}\\, \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} \\:. \\]\nThus two right-handed contributions are generated, whose frequency differ from the frequency\nof the left-handed component by~$\\pm1\/3$. Similarly, a right-handed spinor is mapped to\n\\[ V \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} = \n\\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} + \\frac{i}{2}\\: e^{it\/3}\\, \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix}\n+ \\frac{i}{2}\\: e^{-it\/3}\\, \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\:, \\]\ngenerating two left-handed components with frequencies shifted by $\\pm 1\/3$.\nAgain plotting the frequencies vertically, we depict the transformation~$V$ as in Figure~\\ref{figVtrans}.\n\\begin{figure}\n\\scalebox{1}\n{\n\\begin{pspicture}(0,-0.97)(10.51,0.97)\n\\psdots[dotsize=0.2](4.4,-0.15)\n\\psline[linewidth=0.04cm,tbarsize=0.07055555cm 5.0]{|*-|*}(5.6,0.45)(5.6,-0.75)\n\\psdots[dotsize=0.2](5.6,-0.15)\n\\usefont{T1}{ptm}{m}{n}\n\\rput(4.01,-0.125){$V$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(4.98,-0.145){$=$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(4.39,0.215){$L$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(6.19,-0.145){$L \\;\\;,$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(6.0,-0.745){$R$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(6.0,0.455){$R$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(2.01,-0.145){$\\omega$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(1.68,0.455){$\\omega+\\frac{1}{3}$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(1.64,-0.745){$\\omega-\\frac{1}{3}$}\n\\psline[linewidth=0.04cm,arrowsize=0.04cm 4.0,arrowlength=2.0,arrowinset=0.4]{->}(2.4,-0.95)(2.4,0.95)\n\\psline[linewidth=0.04cm](2.3,-0.15)(2.5,-0.15)\n\\psline[linewidth=0.04cm](2.3,0.45)(2.5,0.45)\n\\psline[linewidth=0.04cm](2.3,-0.75)(2.5,-0.75)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](2.4,0.45)(5.5,0.45)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](2.4,-0.15)(3.6,-0.15)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](2.3,-0.75)(5.6,-0.75)\n\\usefont{T1}{ptm}{m}{n}\n\\rput(7.61,-0.125){$V$}\n\\psdots[dotsize=0.2](8.0,-0.15)\n\\usefont{T1}{ptm}{m}{n}\n\\rput(8.58,-0.145){$=$}\n\\psline[linewidth=0.04cm,tbarsize=0.07055555cm 5.0]{|*-|*}(9.2,0.45)(9.2,-0.75)\n\\psdots[dotsize=0.2](9.2,-0.15)\n\\usefont{T1}{ptm}{m}{n}\n\\rput(9.6,-0.145){$R$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(9.59,-0.745){$L$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(9.59,0.455){$L$}\n\\usefont{T1}{ptm}{m}{n}\n\\rput(8.0,0.215){$R$}\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](6.4,0.45)(9.1,0.45)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](6.4,-0.75)(9.1,-0.75)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](6.7,-0.15)(7.3,-0.15)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](10.0,-0.75)(10.5,-0.75)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](10.0,0.45)(10.5,0.45)\n\\psline[linewidth=0.02cm,linestyle=dotted,dotsep=0.16cm](10.0,-0.15)(10.5,-0.15)\n\\end{pspicture} \n}\n\\caption{The transformation~$V$ in momentum space.}\n\\label{figVtrans}\n\\end{figure}\nThe same notation is also used in Figure~\\ref{figsnail} for the transformed\nplane-wave solutions.\n\nThe inner product~$\\mathopen{\\prec} .| \\chi_L . \\mathclose{\\succ}$ in~\\eqref{SL2} only gives a contribution if the\narguments on the left and right have the opposite chirality.\nSince the transformed plane-wave solutions~$V e_{k,c}$ have a fixed chirality\nat every lattice point, one sees in particular that~\\eqref{SL2} vanishes if~$\\mu$ is chosen\nas a constant. By adding to the constant~$\\mu=1$ contributions with different momenta,\nwe can connect the different lattice points in Figure~\\ref{figsnail}. This leads us to the ansatz\n\\begin{equation} \\label{muform}\n\\mu(t, \\varphi) = 1 + \\mu_\\text{hor}(t, \\varphi) + \\mu_\\text{vert}(t, \\varphi) \\:,\n\\end{equation}\nwhere the last two summands should describe the horizontal respectively vertical arrows\nin Figure~\\ref{figsnail}. For the horizontal arrows we can work similar to~\\eqref{vertical} with a\nspatially-dependent conformal transformation. However, in order to make sure that the\nleft-handed component generated by~$V$ (corresponding to the two $L$s at the very right\nof Figure~\\eqref{figsnail}) are not connected horizontally, we include two Fourier modes\nwhich shift the frequency by~$\\pm 2\/3$,\n\\begin{equation} \\label{muhor}\n\\mu_\\text{hor}(t, \\varphi) = a(\\varphi) \\left( 1 - e^{\\frac{2 i t}{3}} - e^{-\\frac{2 i t}{3}} \\right) ,\n\\end{equation}\nwhere~$a$ has the Fourier decomposition\n\\begin{equation} \\label{aser}\na(\\varphi) = \\sum_{k =1}^\\infty \\left( a_k\\, e^{i k \\varphi} + \\overline{a_{-k}}\\, e^{-i k \\varphi} \\right) \\:.\n\\end{equation}\nFor the vertical arrows we must be careful that the left-handed contribution of~$V {\\mathfrak{e}}_{k,L}$\nis not connected to the right-handed component of~$V {\\mathfrak{e}}_{k,R}$, because then the arrow\nwould have the wrong direction. To this end, we avoid integer frequencies.\nInstead, we work with the frequencies in~$\\mathbb{Z} \\pm 1\/3$, because they\nconnect the left-handed component~$V {\\mathfrak{e}}_{k,R}$ to the right-handed component of~$V {\\mathfrak{e}}_{k,L}$.\nThis leads us to the ansatz\n\\begin{equation} \\label{bser}\n\\mu_\\text{vert}(t, \\varphi) = \\mu_\\text{vert}(t) = \\sum_{n \\in \\mathbb{Z}} e^{i n t}\n\\left(b_n \\,e^{\\frac{it}{3}} + \\overline{b_{-n}}\\, e^{\\frac{-it}{3}} \\right) .\n\\end{equation}\nThe ans\\\"atze~\\eqref{aser} and~\\eqref{bser} ensure that~$\\mu$ is real-valued.\nMoreover, by choosing the Fourier coefficients sufficiently small, one can clearly arrange that\nthe first summand in~\\eqref{muform} dominates, so that~$\\mu$ is strictly positive.\nWe thus obtain the following result.\n\n\\begin{Prp} Assume that the Fourier coefficients~$a_k$ and~$b_n$\nin~\\eqref{aser} and~\\eqref{bser} are sufficiently small and\nthat almost all Fourier coefficients are non-zero.\nThen the function~$\\mu$ defined by~\\eqref{muhor} and~\\eqref{muform}, is strictly positive.\nConsider the Dirac operator~\\eqref{Dirdef} with~$U$ and~$f$ according\nto~\\eqref{Utdef} (for some fixed~$\\nu \\in \\mathbb{R} \\setminus \\{0\\}$) and~\\eqref{lambdadef}.\nThen the chiral index of the fermionic signature operator (see Definition~\\ref{defind}) \nis finite and~$\\ind \\mathscr{S} = p$.\n\\end{Prp}\n\nWe finally discuss the form of the Dirac operator in position space.\nSubstituting~\\eqref{Diracform} into~\\eqref{Dirdef} and using\nthe above form of~$U$ and~$f$, the Dirac operator~$\\tilde{{\\mathcal{D}}}$\ncan be computed in closed form. Similar as discussed in the previous section,\nthe Dirac operator contains a nonlocal integral operator with a singular potential.\nMoreover, the transformation~$U$ modifies the Dirac matrix~$\\gamma^1$ to\n\\[ \\gamma^1 \\rightarrow U \\gamma^1 U^* = \n\\frac{1}{1+ \\nu^2 \\cos^2(t\/3)} \\Big( \\big(1-\\nu^2 \\cos^2(t\/3) \\big) \\:\\gamma^1 - 2 \\nu \\cos^2(t\/3)\\: \\gamma^2\n\\Big) \\:, \\]\nwhere\n\\[ \\gamma^2 := \\begin{pmatrix} i & 0 \\\\ 0 & -i \\end{pmatrix}\\:. \\]\nThus the representation of the Dirac matrices becomes time-dependent; this is the main effect\nof the vectorial transformation~$U$. This transformation changes the first order terms in the\nDirac equation. Moreover, the conformal transformation also changes the first order terms\njust as in~\\eqref{Dir1} by a prefactor~$1\/f$.\n\n\\section{Examples Illustrating the Homotopy Invariance} \\label{secex3}\nWe now give two examples to illustrate our considerations on the homotopy invariance of\nthe chiral index.\nWe begin with an example which shows that the dimension of the kernel of~$\\mathscr{S}_L$\ndoes not need to be constant for deformations which are continuous in~$\\text{\\rm{L}}(\\H_0)$.\nIt may even become infinite-dimensional.\n\n\\begin{Example} \\label{exinstable}\n{\\em{ We consider the space-time~$\\mycal M=(0, T) \\times S^1$ with coordinates~$t \\in (0, T)$\nand~$\\varphi \\in [0, 2 \\pi)$ endowed with the Minkowski metric\n\\[ ds^2 = dt^2 - d\\varphi^2 \\:. \\]\nWe again choose two-component complex spinors with the spin scalar product~\\eqref{ssprod}.\nThe Dirac operator is chosen as\n\\[ {\\mathcal{D}} = i \\gamma^0 \\partial_t + i \\gamma^1 \\partial_\\varphi \\:, \\]\nwhere the Dirac matrices are again given by~\\eqref{gammadef}.\nThe pseudoscalar matrix and the chiral projectors are again\nchosen according to~\\eqref{pseudoc} and~\\eqref{defchir}.\n\nWe consider the massless Dirac equation\n\\[ {\\mathcal{D}} \\psi = 0 \\:. \\]\nThis equation~\\eqref{Dir0} can be solved by plane wave solutions, which we write as\n\\begin{equation} \\label{eLR2}\n{\\mathfrak{e}}_{k,L}(\\zeta) = \\frac{1}{2 \\pi}\\: e^{+i k t + i k \\varphi} \\begin{pmatrix} 1 \\\\ 0 \\end{pmatrix} \\:,\\qquad\n{\\mathfrak{e}}_{k,R}(\\zeta) = \\frac{1}{2 \\pi}\\: e^{-i k t + i k \\varphi} \\begin{pmatrix} 0 \\\\ 1 \\end{pmatrix} \\:,\n\\end{equation}\nwhere~$k \\in \\mathbb{Z}$ (the indices~$L$ and~$R$ denote the left- and right-handed components; at the\nsame time they propagate to the left respectively right). By direct computation, one verifies\nthat~$({\\mathfrak{e}}_{k,c})_{k \\in \\mathbb{Z}, c \\in \\{L,R\\}}$ is an orthonormal basis of~$\\H_0$.\n\nWe next compute the space-time inner product~\\eqref{stip},\n\\begin{align*}\n\\mathopen{<} {\\mathfrak{e}}_{k,R} | {\\mathfrak{e}}_{0, L} \\mathclose{>} &= \\int_0^T dt \\int_0^{2 \\pi} d\\varphi\\; \n\\mathopen{\\prec} {\\mathfrak{e}}_{k,R}(t,\\varphi) \\,|\\, {\\mathfrak{e}}_{0, L}(t,\\varphi) \\mathclose{\\succ}\n= \\frac{1}{2 \\pi} \\int_0^T \\delta_{k,0} \\,dt = \\frac{T}{2 \\pi}\\: \\delta_{k,0} \\\\\n\\mathopen{<} {\\mathfrak{e}}_{k,R} | {\\mathfrak{e}}_{k', L} \\mathclose{>} \n&= \\frac{1}{2 \\pi} \\: \\delta_{k,k'} \\int_0^T e^{2 i k t} \\,dt = \\frac{e^{2 i k T} - 1}{4 \\pi i k}\n\\quad \\text{($k' \\neq 0$)} \\\\\n\\mathopen{<} {\\mathfrak{e}}_{k,L} | {\\mathfrak{e}}_{k', L} \\mathclose{>} &= 0 = \\mathopen{<} {\\mathfrak{e}}_{k,R} | {\\mathfrak{e}}_{k', R} \\mathclose{>} \\:.\n\\end{align*}\nThus the fermionic signature operator~$\\mathscr{S}$ is invariant on the subspaces~$\\H^{(k)}_0$\ngenerated by the basis vectors~${\\mathfrak{e}}_{k,L}$ and~${\\mathfrak{e}}_{k,R}$. Moreover, in these bases it has the\nmatrix representations\n\\[ \\mathscr{S}|_{\\H^{(0)}_0} = \\frac{T}{2 \\pi} \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix} \\qquad \\text{and} \\qquad\n\\mathscr{S}|_{\\H^{(k)}_0} = \\frac{1}{4 \\pi i k}\\:\n\\begin{pmatrix} 0 & e^{2 i k T} - 1 \\\\ e^{-2 i k T} - 1 & 0 \\end{pmatrix} \\quad (k \\neq 0)\\:. \\]\n\nIf~$T \\not \\in \\pi \\mathbb{Q}$, the matrix entries~$e^{\\pm 2 i k T} - 1$ are all non-zero. As a consequence,\nthe operators~$\\mathscr{S}_L|_{\\H_L}$ and~$\\mathscr{S}_R|_{\\H_R}$ are both injective.\nThus~$\\mathscr{S}$ has finite chiral chiral index in the massless odd case (see Definition~\\ref{defind0}).\nIf~$T \\in \\pi \\mathbb{Q}$, however, the chiral index vanishes for all~$k$ for which~$2 k T$ is a multiple of~$2 \\pi$.\nAs a consequence, the operators~$\\mathscr{S}_L|_{\\H_L}$ and~$\\mathscr{S}_R|_{\\H_R}$\nboth have an infinite-dimensional kernel, so that~$\\mathscr{S}$ does not have a finite chiral index.\n}} \\ \\hfill $\\Diamond$\n\\end{Example}\nThis example also explains why we need additional assumptions like those in\nTheorems~\\ref{thmstable} and~\\ref{thmstablem0}. In particular, \nwhen considering homotopies of space-time or of the Dirac operator,\none must be careful to ensure that the chiral index remains finite along the chosen path.\n\nWe next want to construct examples of homotopies to which the stability result of\nTheorem~\\ref{thmstablem0} applies. To this end, it is convenient to work similar to~\\eqref{conform}\nwith a conformal transformation.\n\\begin{Example} \\label{exstable}\n{\\em{ As in Example~\\ref{exinstable} we consider the space-time~$(0,T) \\times S^1$,\nbut now with the conformally transformed metric\n\\[ d\\tilde{s}^2 = f(t)^2 \\left( dt^2 - d\\varphi^2 \\right) \\]\nwhere~$f$ is a non-negative $C^2$-function with\n\\[ \\supp f \\subset (-T,T) \\qquad \\text{and} \\qquad f(0) > 0 \\:. \\]\nSimilar to~\\eqref{vertical} and the computation thereafter, \ntransforming the plane-wave solutions~\\eqref{eLR2} conformally\nto~$\\tilde{{\\mathfrak{e}}}_{k, L\\!\/\\!R} = f(t)^{-\\frac{1}{2}}\\: {\\mathfrak{e}}_{k, L\\!\/\\!R}$,\nwe again obtain an orthonormal basis of~$\\H_0$ and\n\\begin{align*}\n\\mathopen{<} \\tilde{{\\mathfrak{e}}}_{k,R} | \\tilde{{\\mathfrak{e}}}_{k', L} \\mathclose{>} \n&= \\frac{1}{2 \\pi} \\: \\delta_{k,k'} \\int_0^T f(t)\\: e^{2 i k t} \\,dt \\\\\n\\mathopen{<} \\tilde{{\\mathfrak{e}}}_{k,L} | \\tilde{{\\mathfrak{e}}}_{k', L} \\mathclose{>} &= 0 = \\mathopen{<} \\tilde{{\\mathfrak{e}}}_{k,R} | \\tilde{{\\mathfrak{e}}}_{k', R} \\mathclose{>}\n\\end{align*}\nfor all~$k, k' \\in \\mathbb{Z}$.\n\nThe integration-by-parts argument\n\\begin{align*}\n\\int_0^T f(t)\\: e^{2 i k t} \\,dt &=\n\\frac{1}{2 i k} \\int_0^T f(t)\\: \\frac{d}{dt} e^{2 i k t} \\,dt = \n-\\frac{f(0)}{2 i k} - \\frac{1}{2 i k} \\int_0^T f'(t)\\: e^{2 i k t} \\,dt \\\\\n&= -\\frac{f(0)}{2 i k} -\\frac{f'(0)}{4 k^2}\n-\\frac{1}{4 k^2} \\int_0^T f''(t)\\: e^{2 i k t} \\,dt \n\\end{align*}\nshows that the space-time inner products have a simple explicit asymptotics for large~$k$ given by\n\\[ \\mathopen{<} \\tilde{{\\mathfrak{e}}}_{k,R} | \\tilde{{\\mathfrak{e}}}_{k', L} \\mathclose{>} = \\frac{f(0)}{4 \\pi i k}\\; \\delta_{k,k'}\n+ \\O \\Big( \\frac{1}{k^2} \\Big) \\:. \\]\nHence the operator~$\\mathscr{S}_L$ has the form\n\\[ \\mathscr{S}_L {\\mathfrak{e}}_{k,L} = c_k\\, {\\mathfrak{e}}_{k,R} \\]\nwith coefficients~$c_k$ having the asymptotics\n\\[ c_k = \\frac{f(0)}{4 \\pi i k} + \\O \\Big( \\frac{1}{k^2} \\Big) \\:. \\]\nFrom this asymptotics we can read off the following facts. First, it is obvious that~$\\mathscr{S}_L|_{\\H_L}$\nhas a finite-dimensional kernel. Exchanging the chirality, the same is true for~$\\mathscr{S}_R|_{\\H_R}$,\nimplying that~$S$ has a finite chiral index (according to Definition~\\ref{defind0}).\nNext, the vectors in the image of~$\\mathscr{S}_L$ are in the Sobolev space $W^{1,2}$,\n\\[ \\mathscr{S}_L|_{\\H_L} \\::\\: \\H_L \\rightarrow \\H_R \\cap W^{1,2}(S^1, \\mathbb{C}^2) \\:. \\]\nMoreover, the image of this operator is closed (in the $W^{1,2}$-norm).\nFinally, our partial integration argument also yields that\n\\[ \\|\\mathscr{S}_L \\psi\\|_{W^{1,2}} \\leq |f|_{C^2}\\: \\|\\psi\\|_{\\H_0} \\:, \\]\nshowing that the family of signature operators is norm continuous\nfor a $C^2$-homotopy of functions~$f$.\n\nHaving verified the assumptions of Theorem~\\ref{thmstablem0},\nwe conclude that the chiral index in the massless odd case is invariant\nunder $C^2$-homotopies of the conformal function~$f$, provided that~$f(0)$\nstays away from zero.\n}} \\ \\hfill $\\Diamond$\n\\end{Example}\n\n\\section{Conclusion and Outlook} \\label{secoutlook}\nOur analysis shows that the chiral index of a fermionic signature operator\nis well-defined and in general non-trivial. Moreover, it is a homotopy invariant provided\nthat the additional conditions stated in Theorems~\\ref{thmstable} and~\\ref{thmstablem0}\nare satisfied. As already mentioned at the end of the introduction,\nthe physical and geometric meaning of this index is yet to be explored.\n\nWe now outline how our definition of the chiral index could be generalized or\nextended other situations. First, our constructions also apply in the {\\em{Riemannian setting}}\nby working instead of causal fermion systems with so-called Riemannian fermion systems\nor general {\\em{topological fermion systems}} as introduced in~\\cite{topology}.\nIn this situation, one again imposes a pseudoscalar operators~$\\Gamma(x) \\in \\text{\\rm{L}}(\\H)$\nwith the properties~\\eqref{pseudodef}. Then all constructions in Section~\\ref{seccfs}\ngo through. Starting on an even-dimensional Riemannian spin manifold, one can proceed as\nexplained in~\\cite{topology} and first construct a corresponding topological fermion system.\nFor this construction, one must choose a particle space, typically of eigensolutions of the Dirac equation.\nOnce the topological fermion system is constructed, one can again work with the index of Section~\\ref{seccfs}.\nIf the Dirac operator anti-commutes with the pseudoscalar operator, one can choose the\nparticle space~$\\H$ to be invariant under the action of~$\\Gamma$. This gives a\ndecomposition of the particle space into two chiral subspaces,\n$\\H = \\H_L \\oplus \\H_R$. Just as explained in Section~\\ref{secodd},\nthis makes it possible to introduce other indices by restricting the chiral signature operators\nto~$\\H_L$ or~$\\H_R$. Moreover, one could compose the operators from the left\nwith the projection operators onto the subspaces~$\\H_{L\\!\/\\!R}$ and consider the Noether indices\nof the resulting operators.\n\nAnother generalization concerns space-times of {\\em{infinite lifetime}}.\nUsing the constructions in~\\cite{infinite}, in such space-times one can still introduce\nthe fermionic signature operator~$\\mathscr{S}_m$ provided that the space-time satisfies\nthe so-called mass oscillation property. By inserting chiral projection operators, one\ncan again define chiral signature operators~$\\mathscr{S}^{L\\!\/\\!R}_m$ and define the\nchiral index as their Noether index. Also the stability results of Theorems~\\ref{thmstable}\nand~\\ref{thmstablem0} again apply. It is unknown whether the resulting indices have a\ngeometric meaning. Since~$\\mathscr{S}_m$ depends essentially on the asymptotic form of\nthe Dirac solutions near infinity, the corresponding chiral indices should encode \ninformation on the metric and the external potential in the asymptotic ends.\n\nWe finally remark that the fermionic signature operator could be {\\em{localized}}\nby restricting the space-time integrals to a measurable subset~$\\Omega \\subset \\mycal M$.\nFor example, one can introduce a chiral signature operator~$\\mathscr{S}_L(\\Omega)$\nsimilar to~\\eqref{stipLR} and~\\eqref{SLRdef2} by\n\\[ ( \\phi \\,|\\, \\mathscr{S}_L(\\Omega)\\, \\psi) = \n\\int_\\Omega \\mathopen{\\prec} \\psi \\,|\\, \\chi_L \\,\\phi \\mathclose{\\succ}_x\\: d\\mu_\\mycal M\\:. \\]\nLikewise, in the setting of causal fermion systems, one can modify~\\eqref{sigLint} to\n\\[ \\mathscr{S}_L(\\Omega) = -\\int_\\Omega x \\,\\chi_L\\: d\\rho(x) \\:. \\]\nThe corresponding indices should encode information on the behavior of the Dirac solutions\nin the space-time region~$\\Omega$.\n\n\\vspace*{.5em} \\noindent \\thanks {{\\em{Acknowledgments:}}\nI would like to thank Niky Kamran and Hermann Schulz-Baldes for helpful discussions.\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{sec:intro}\n\n\nIn the trace reconstruction problem, the goal is to reconstruct an unknown bit string $x \\in \\{0,1\\}^n$ from multiple independent noisy observations of $x$.\nWe focus on the case where the noise is due to\n $x$ going through the deletion channel, where each bit is deleted independently with probability $q$ (and the remaining bits are concatenated, with no space between them, so the observer is uncertain about the original location of a bit in the output).\nThat is, instead of seeing $x$, we see (many independent copies of) $\\wt{X}$, which is obtained as follows:\nstart with an empty string, and for $k = 0, 1, \\dots, n-1$, do the following:\n\\begin{itemize}\n \\item \\textbf{(retention)} with probability $p=1 - q$, copy $x_k$ to the end of $\\wt{X}$ and increase $k$ by $1$;\n \\item \\textbf{(deletion)} with probability $q$, increase $k$ by $1$.\n\\end{itemize}\nVariants, such as including insertions and substitutions, are discussed in Section 5.\n\nGiven $T$ i.i.d.\\ samples (known as {\\em traces\\\/}) $\\wt{X}^{1}, \\dots, \\wt{X}^{T}$, all obtained from passing the same unknown string $x$ through the deletion channel, a trace reconstruction algorithm outputs an estimate\n$\\wh{X}$ which is a function of $ \\wt{X}^{1}, \\dots, \\wt{X}^{T}$. The main question is: Given $\\delta>0$, how many samples are needed so that there is a choice of $\\wh{X}$ that satisfies $\\P_x[\\wh{X} =x] \\ge 1- \\delta$ for every $x \\in \\{0,1\\}^n$ ? \\newline\n(Here $\\P_x$ is the law of $\\wt{X}_{1}, \\dots, \\wt{X}_{T}$ when the original string was $x$.)\nPrior to this work, the best available upper bound, due to \\cite{HMPW08}, was $T=\\exp(\\wt{O}(n^{1\/2}))$. Our main result, proved in Section 2, yields the following improvement.\n\n\\begin{theorem}\\label{thm:main}\nFor any deletion probability $q < 1$ and any $\\delta>0$, there exists a finite constant $C$ such that, for any original string $x \\in \\{0,1\\}^n$,\nit can be reconstructed with probability at least $1-\\delta$ from\n $T = \\exp \\left( C n^{1\/3} \\right)$ i.i.d.\\ samples of the deletion channel applied to $x$.\n\\end{theorem}\n\nOur estimator will only use individual bit statistics from the outputs of the deletion channel.\nThe following Theorem, proved in Section 4, shows that\n$T = \\exp \\left( \\Omega(n^{1\/3}) \\right)$ traces are needed for reconstruction if these are the only data used.\n\n\\begin{theorem}\\label{thm:opt}\nFix a deletion probability $q < 1$. For each $n$ there exist two distinct strings $x,y \\in \\{0,1\\}^{n}$,\nwith the following property: For all $j$, the total variation distance between the laws of $\\Bigl(\\wt{X}_{j}^{t}\\Bigr)_{t=1}^T$\n and $\\Bigl(\\wt{Y}_{j}^{t}\\Bigr)_{t=1}^T$ is at most $T\\exp \\left( -c n^{1\/3} \\right)$, for some $c=c(q)>0$.\n \\end{theorem}\n (Thus, for $c_10$, trace reconstruction algorithms relying on single bit statistics require $\\exp(n^c)$ traces. That proof was not published.\n\n\n\\item\nAfter the results of this paper were obtained, we learned that similar results were obtained independently and simultaneously by Anindya De, Ryan O'Donnell and Rocco Servedio.\n\n\\end{itemize}\n\n\n\n\n\n\\section{Proof of Theorem~\\ref{thm:main}}\n\n\nIn the proof we consider the random power series\n\\begin{equation}\\label{eq:stat}\n\\sum_{j \\geq 0} \\wt{a}_{j} w^{j},\n\\end{equation}\nwhere $ \\wt{{\\mathbf a}}$ is a sample output of the deletion channel and $w \\in \\mathbb{C}$ is chosen appropriately.\nThe first lemma expresses the expectation of such a random series using the original sequence of interest.\n\n\\begin{lemma}\\label{lem:key1}\nLet $w \\in \\mathbb{C}$, let\\\/ ${\\mathbf a} := \\left( a_{0}, a_{1}, \\dots, a_{n-1} \\right) \\in \\mathbb{R}^{n}$, let\\\/ $ \\wt{{\\mathbf a}}$ be the output of the deletion channel with input ${\\mathbf a}$, and pad\\\/ $ \\wt{{\\mathbf a}}$ with zeroes to the right. Write $p=1-q$. Then\n\\begin{equation}\\label{eq:expectation}\n\\E \\left[ \\sum_{j \\geq 0} \\wt{a}_{j} w^{j} \\right] = p \\sum_{k=0}^{n-1} a_{k} \\left( pw+q \\right)^{k}.\n\\end{equation}\n\\end{lemma}\nThe proof is given in the next section. Intuitively speaking, this identity is useful because by averaging samples we can approximate the expectation on the left-hand side of~\\eqref{eq:expectation}, while from the right-hand side of~\\eqref{eq:expectation} we can extract the original sequence ${\\mathbf a} = \\left( a_{0}, a_{1}, \\dots, a_{n-1} \\right)$.\n\nNote that unless $\\left| w \\right| = 1$, either the first or last terms of $ \\wt{{\\mathbf a}}$ will dominate in the left-hand side of~\\eqref{eq:expectation}.\nSimilarly, if we let $z := pw+q$, then unless $\\left| z \\right| = 1$, either the first or the last terms of ${\\mathbf a}$ will dominate in the right-hand side of~\\eqref{eq:expectation}.\nWe wish to give approximately equal weight to all terms, hence we would like $\\left| w \\right|$ and $\\left| z \\right|$ to both be close to 1.\nThis only happens if both $w$ and $z$ are close to $1$; thus we will let $z$ vary along a small arc on the unit circle near $1$.\nThis explains our interest in the following lemma, which is a special case of Theorem 3.2 in~\\cite{BE97}.\n\n\\begin{lemma}[Borwein and Erd{\\'e}lyi~\\cite{BE97}] \\label{lem:key2}\nThere exists a finite constant $c$ such that the following holds.\nLet\n$${\\mathbf a} = \\left( a_{0}, a_{1}, \\dots, a_{n-1} \\right) \\in \\left\\{ -1, 0, 1 \\right\\}^{n}$$\nbe such that ${\\mathbf a} \\neq 0$.\nLet $A \\left( z \\right) := \\sum_{k = 0}^{n-1} a_{k} z^{k}$ and denote by $\\gamma_L$ the arc $\\left\\{ e^{i \\theta} : -\\pi\/L \\leq \\theta \\leq \\pi \/ L \\right\\}$. Then $\\max_{z \\in \\gamma_L} |A(z)| \\geq e^{-cL}$.\n\\end{lemma}\n\nWe will optimize over the length of the arc $\\gamma_L$, and in the end we shall choose $L$ of order $ n^{1\/3}$.\n\nNote that if $z$ is in the arc $\\gamma_L=\\left\\{ e^{i \\theta} : -\\pi \/ L \\leq \\theta \\leq \\pi \/ L \\right\\}$\nand , then\n\\begin{equation} \\label{wbound}\n w = (z-q)\/p \\quad \\mbox{\\rm satisfies} \\; | w | \\le \\exp \\left( C_1 \/ L^{2} \\right) \\,\n\\end{equation}\nfor some constant $C_1=C_1(q)$. This is because writing\n$z = \\cos \\theta + i \\sin \\theta$,\nand using the Taylor expansion of cosine, we get\n\\begin{eqnarray*}\n\\left| w \\right|^{2} &=& \\frac{1+q^2-2q\\cos(\\theta)}{p^2} = \\frac{1+q^2-2q+2q(1-\\cos \\theta)}{p^2} \\\\ &\\le& 1 + \\frac{q}{p^2} \\theta^{2} + O \\left( \\theta^{4} \\right)\n = \\exp \\left( q \\theta^{2}\/p^2 + O \\left( \\theta^{4} \\right) \\right)\\, .\n\\end{eqnarray*}\nThe quadratic term $\\theta^{2}$ is to be expected: when $z$ is on the unit circle, $w=(z-q)\/p$ is on a circle of radius $1\/p$ centered at $-q\/p$; these circles are tangent at 1.\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm:main} using the lemmas}\n\nLet $x,y \\in \\left\\{ 0, 1 \\right\\}^{n}$ be two different bit sequences. Our first goal is to distinguish between $x$ and $y$.\nLet ${\\mathbf a} := x-y$ and let $A(z) := \\sum_{k = 0}^{n-1} a_{k} z^{k}$.\nGiven a large integer $L$ (which we shall choose later), fix $z$ in the arc \n$\\gamma_L=\\left\\{e^{i \\theta} : -\\pi\/L \\leq \\theta \\leq \\pi \/ L \\right\\}$ such that\n$\\left| A (z) \\right| \\geq e^{-cL} $; \nsuch a $z$ exists by Lemma~\\ref{lem:key2}.\nLet $w = (z-q)\/p$. Recall from the previous subsection that $\\left| w \\right| < \\exp \\left( C_1 \/ L^{2} \\right)$ for some $C_1 < \\infty$.\n\nConsidering the random series defined in~\\eqref{eq:stat}, we see via Lemma~\\ref{lem:key1} that\n\\[\n\\E \\Bigl[ \\sum_{j \\geq 0} \\wt{X}_{j} w^{j} \\Bigr] - \\E \\Bigl[ \\sum_{j \\geq 0} \\wt{Y}_{j} w^{j} \\Bigr] = A (z)\\, .\n\\]\nTaking absolute values,\n\\[\n \\sum_{j \\geq 0} \\Bigl|\\E \\Bigl[ \\wt{X}_{j}-\\wt{Y}_{j} \\Bigr] \\Bigr| \\cdot |w|^{j} \\ge |A (z)| \\ge e^{-cL}, \n\\]\nwhence by (\\ref{wbound}),\n\\[\n \\sum_{j \\geq 0} \\Bigl|\\E \\Bigl[ \\wt{X}_{j}-\\wt{Y}_{j} \\Bigr] \\Bigr| \\ge \\exp \\Bigl( -C_1 n \/ L^{2} \\Bigr) \\cdot e^{-cL} \\,\n\\]\nTo approximately maximize the right-hand side, we choose $L$ to be the integer part of $n^{1\/3} $ and obtain that for some constant $C_2$,\n\\[\n \\sum_{j \\geq 0} \\Bigl|\\E \\Bigl[ \\wt{X}_{j}-\\wt{Y}_{j} \\Bigr] \\Bigr| \\ge \\exp \\Bigl(- C_2 n^{1\/3} \\ \\Bigr) \\,.\n\\]\nWe infer that there must exist some smallest $j2C_2$ makes the right-hand side of (\\ref{union}) tend to 0.\n\\hfill $\\Box$\n\n\\section{Proof of the polynomial identity and a simplified inequality}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:key1}]\nFor $j \\leq n-1$, the output bit $\\wt{a}_{j}$ must come from an input bit $a_{k}$ for some $k \\ge j$.\nNow $\\wt{a}_{j}$ comes from $a_{k}$ if and only if exactly $j$ among $a_{0}, a_{1}, \\dots a_{k-1}$ are retained and $a_{k}$ is also retained.\nThere are $\\binom{k}{j}$ ways of choosing which $j$ bits among $a_{0}, a_{1}, \\dots a_{k-1}$ to retain, and the probability of each such choice is $p^j q^{k-j}$.\nThe probability of retaining $a_{k}$ is $p$. Putting everything together, we obtain that\n\\[\n\\E \\Bigl[ \\sum_{j \\geq 0} \\wt{a}_{j} w^{j} \\Bigr] = p \\sum_{j \\geq 0} w^{j} \\sum_{k = j}^{n-1} a_{k} \\binom{k}{j} p^j q^{k-j} \\,.\n\\]\nChanging the order of summation, we infer that\n\\[\n\\E \\Bigl[ \\sum_{j \\geq 0} \\wt{a}_{j} w^{j} \\Bigr] = p \\sum_{k = 0}^{n-1} a_{k} \\sum_{j=0}^{k} \\binom{k}{j} p^j q^{k-j} w^{j}.\n\\]\nFinally, observe that the sum over $j$ on the right-hand side is exactly the binomial expansion of $(pw+q)^{k}$.\n\\end{proof}\n\n\n\nSince the proof of Lemma~\\ref{lem:key2} in \\cite{BE97} is somewhat involved, for expository purposes, we prove here a weaker estimate. This is simpler to prove and it does not result in a much weaker conclusion. Specifically, if we use Lemma~\\ref{lem:key2_weak} below as a black box instead of Lemma~\\ref{lem:key2}, then we obtain that $T = \\exp \\Bigl( c n^{1\/3} \\log n \\Bigr)$ samples suffice for trace reconstruction; comparing this with Theorem~\\ref{thm:main}, we only lose a log factor in the exponent.\n\n\\begin{lemma}\\label{lem:key2_weak}\nLet\n${\\mathbf a} = \\Bigl( a_{0}, a_{1}, \\dots, a_{n-1} \\Bigr) \\in \\Bigl\\{ -1, 0, 1 \\Bigr\\}^{n}$\nbe such that ${\\mathbf a} \\neq 0$.\nLet $A (z) := \\sum_{k = 0}^{n-1} a_{k} z^{k}$.\nIf $ | A (z)| \\leq \\lambda$ on the arc $\\gamma_L:=\\Bigl\\{ z = e^{i \\theta} : -\\pi \/ L \\leq \\theta \\leq \\pi \/ L \\Bigr\\}$,\nthen $\\lambda \\geq n^{- L}$.\n\\end{lemma}\n\n\n\\begin{proof}\nWe may assume w.l.o.g.\\ that $a_{0} = 1$. (Indeed, if $a_{m}$ is the first nonzero entry and $m \\geq 1$, then replace $A(z)$ by $A(z)\/z^{m}$; this does not change the magnitude of the function on the unit circle, and yields a polynomial with $a_{0} \\neq 0$. Multiplying $A(z)$ by an appropriate sign, we can guarantee that $a_{0} = 1$.) In other words, $A(0) = 1$.\n \nConsider the product\n\\begin{equation}\\label{eq:A_rotations}\n{F} (z) := \\prod_{j=0}^{L-1} A \\Bigl( z \\cdot e^{2 \\pi i j \/ L} \\Bigr).\n\\end{equation}\nWe again have that $F(0) = 1$. By the maximum principle, the the maximum absolute value of the polynomial $F(z)$ on the unit disc is attained on the boundary, i.e., on the unit circle.\nThus there exists $z$ such that $ | z | = 1$ and $\\Bigl| F (z) \\Bigr| \\geq 1$.\nOn the other hand, for every $z$ such that $ | z | = 1$, the assumption of the lemma guarantees that there is at least one factor in~\\eqref{eq:A_rotations} whose absolute value is at most $\\lambda$.\nUsing the trivial bound $ |A (z)| \\leq n$ for every other factor,\nwe obtain that $ | F (z) | \\leq \\lambda n^{L-1}$ for every $z$ such that $ | z | = 1$.\nPutting the two inequalities together we obtain that\n$\\lambda \\geq n^{-(L-1)}$.\n\\end{proof}\n\n\\section{Optimality for single bit tests} \n\\begin{proof}[Proof of Theorem~\\ref{thm:opt}] \nLet $L:=n^{1\/3}$. (To keep the notation light, we omit integer parts and use $c_j$ to denote absolute constants and constants that depend only on $q$). By Theorem 3.3 in~\\cite{BEK99}, there exists a polynomial $Q$ of degree $c_2 L^2$, with coefficients in $\\{-1,0,1\\}$,\nsuch that \n$$\\max_{z \\in [0,1]} |Q(z)| \\le \\exp(-c_3 L) \\,.\n$$ \n Write $Q$ in the form $Q=\\varphi -\\psi $\nwhere $\\varphi$ and $\\psi$ are polynomials of degree $c_2 L^2$ with coefficients in $\\{0,1\\}$.\n\n\nLet $\\widehat{E}_L$ denote the ellipse with foci at $1-8\/L$ and $1$ and with major axis $[1-14\/L,1+6\/L]$, i.e.,\n$$\n\\widehat{E}_L=\\{z: |z-(1-8\/L)|+|z-1| \\le 20\/L \\} \\,.\n$$\n As explained on page 11 of~\\cite{BE97},\nCorollary 4.5 of that paper implies that\n$$\n\\max_{z \\in \\widehat{E}_L} |Q(z)| \\le e^{-c_4 L} \\, .\n$$\nRecall that $p=1-q$ and let $\\Gamma$ denote the circle $\\{z: |z-q|=p\\}$. Then $\\Gamma$ intersects the ellipse $\\widehat{E}_L$ in an arc $\\Gamma_L$ of length $c_5\/L$, since $\\widehat{E}_L$ contains the disk of radius $6\/L$ centered at 1.\nThus we may write\n$$\\Gamma_L=\\{pe^{i\\theta}+q : -c_6\/L \\le \\theta \\le c_6\/L \\} \\,.\n$$\n\nLet $m:=(n-c_2L^2)\/2$. Define the string $x \\in \\{0,1\\}^{n}$ where the first $m$ bits are zeros, the next $c_2L^2$ bits are the coefficients of $\\varphi$, and the final $m$ bits are zeros. The string $y \\in \\{0,1\\}^{n}$ is constructed from $\\psi$ in the same way.\n Then \n$$\nA(z):=\\sum_{k=0}^{n-1} (x_j-y_j)z^j=z^mQ(z) \n$$\nsatisfies \n\\begin{equation} \\label{maxA}\n\\max_{z \\in \\Gamma_L} |A(z)| \\le e^{-c_4 L} \\,.\n\\end{equation}\nDefine $b_j:=\\E \\Bigl[ \\wt{X}_{j}-\\wt{Y}_{j}]$ and $B(w):=\\sum_{j=0}^{n-1} b_j w^j$. By Lemma~\\ref{lem:key1}, we have\n$B(w)=pA(pw+q)$. We can extract $b_j$ from $B(\\cdot)$ by integration:\n$$\nb_j=\\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} e^{-ij\\theta} B(e^{i\\theta})\\,d\\theta \\,.\n$$\nTherefore\n\\begin{equation} \\label{eq:b_j}\n|b_j| \\le \\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} |B(e^{i\\theta})|\\, d\\theta \\le \\frac{1}{2\\pi}\\int_{-\\pi}^{\\pi} |A(pe^{i\\theta}+q)|\\, d\\theta \\,.\n\\end{equation}\nFor $\\theta \\in [-c_6\/L,c_6\/L]$, the integrand on the right-hand side is at most $e^{-c_4 L}$ by (\\ref{maxA}).\nTo bound that integrand for larger $\\theta$, observe that \n\\begin{eqnarray} \\label{inside}\n|pe^{i\\theta}+q|^2 &=& p^2\\cos^2\\theta+2pq\\cos\\theta+q^2+p^2\\sin^2\\theta=(p+q)^2+2pq(\\cos\\theta-1) \\\\\n&=& 1-pq\\theta^2+O(\\theta^4) \\le 1-c_7 \\theta^2 \\,.\n\\end{eqnarray}\nSince $|A(z)| \\le |z|^m (1-|z|)^{-1}$ in the unit disk and $m>n\/3$, we infer that if $|\\theta|>c_6\/L$, then\n$$\n|A(pe^{i\\theta}+q)| \\le c_8 L^2 (1-c_9 L^{-2})^{n\/3} \\le \\exp(-c_{10} nL^{-2}) =e^{-c_{10} L} \\,.\n$$\nIn conjunction with (\\ref{maxA}), we conclude that the integrand on the right-hand side of (\\ref{eq:b_j}) is uniformly bounded\nby $e^{-c_{11} L}$, whence\n\\begin{equation} \\label{eq:b_j2}\n|b_j| \\le e^{-c_{11} L} \\quad \\mbox{\\rm for all } \\, j \\,.\n\\end{equation}\n\n\n\nNext, fix $j$. To bound the total variation distance between the laws of $\\bigl(\\wt{X}_{j}^{t}\\bigr)_{t=1}^T$\n and $\\bigl(\\wt{Y}_{j}^{t}\\bigr)_{t=1}^T$, we will use a greedy coupling. More precise estimates can be obtained, e.g., using Hellinger distance, but the improvement will not affect the final result. Let $\\bigl(\\xi_t \\bigr)_{t=1}^T$ be i.i.d. variables, uniform in $[0,1]$.\n Then ${\\bf 1}_{\\xi_t \\le \\E(\\wt{X}_{j})}$ has the law of $\\wt{X}_{j}^t$ and ${\\bf 1}_{\\xi_t \\le \\E(\\wt{Y}_{j})}$ has the law of $\\wt{Y}_{j}^t$.\n These indicators differ with probability $b_j$. Altogether, this coupling implies that the total variation distance between the laws of $\\bigl(\\wt{X}_{j}^{t}\\bigr)_{t=1}^T$ and $\\bigl(\\wt{Y}_{j}^{t}\\bigr)_{t=1}^T$ is at most $Tb_j$.\n Referring to (\\ref{eq:b_j2}) concludes the proof.\n\\end{proof}\n\n\\noindent{\\bf Remark.} Strictly speaking, padding $x$ and $y$ with zeros on the right was not really needed in the above proof.\nThe reason for it is that one can also consider single bit tests on the traces using the $j$th bit from the right in each output;\nthe additional padding and a symmetry argument ensures that these tests will also require $\\exp(\\Omega(n^{1\/3})$ traces for reconstruction.\n\n\n\\section{Substitutions and insertions}\nIf, after $x$ goes through the deletion channel with deletion probability $q$, every bit is flipped with probability $\\lambda<1\/2$,\nthen $\\exp(O(n^{1\/3})$ samples still suffice for reconstruction. Indeed, let $X^\\# $ be the output of this deletion-substitution channel with input $ x$, padded with zeroes to the right. Define $Y^\\#$ from $y$ similarly. Recall that $p=1-q$. Then $\\E(X^\\#_{j}-Y^\\#_j)=(1-2\\lambda)\\E(\\wt{X}_{j}-\\wt{Y}_j)$, so\n(\\ref{eq:expectation}) is replaced by\n\\begin{equation}\\label{eq:expectation2}\n\\E \\left[ \\sum_{j \\geq 0} (X^\\#_{j}-Y^\\#_j) w^{j} \\right] = (1-2\\lambda) p \\sum_{k=0}^{n-1} (x_k-y_k) \\left( pw+q \\right)^{k}.\n\\end{equation}\nThe analysis in Section 2 then proceeds without change, since the pre-factor $1-2\\lambda$ is immaterial.\n\n\\smallskip\n\n{\\bf Insertions} are more interesting. Suppose that before each bit $x_k$ in the input, $G_k-1$ i.i.d.\\ fair bits are inserted, where the variables\n$G_k$ are i.i.d.\\ with a Geometric$(\\alpha)$ distribution, i.e., denoting $\\beta=1-\\alpha$, for all $\\ell \\ge 1$,\n$$\n\\P(G_k=\\ell)=\\alpha\\beta^{\\ell-1} \\,.\n$$\nAfter $x_{n-1}$, at the end of the sequence, $G_{n}-1$ additional fair bits are appended. We call $\\beta=1-\\alpha$ the insertion parameter.\nThus, such an insertion channel will yield an output $X^*$ consisting of $G_0-1$ i.i.d.\\ fair bits, followed by $x_0$, followed by $G_1-1$ i.i.d.\\ fair bits, followed by $x_1$, etc., ending with $x_{n-1}$ and the $G_{n}-1$ bits after it. The next theorem is analogous to Theorem~\\ref{thm:main}.\n\n\\begin{theorem}\\label{thm:insert}\nFor any insertion parameter $\\beta < 1$ and any $\\delta>0$, there exists a finite constant $C$ such that, for any original string $x \\in \\{0,1\\}^n$,\nit can be reconstructed with probability at least $1-\\delta$ from\n $T = \\exp \\left( C n^{1\/3} \\right)$ i.i.d.\\ samples of the insertion channel applied to $x$.\n\\end{theorem}\n\\begin{proof}\nTo prove this theorem, an analog of Lemma~\\ref{lem:key1} is needed:\n\n\\begin{lemma}\\label{lem:keyins}\nGiven strings $x$ and $y$ in $\\{0,1\\}^n$, let $X^*$ and $Y^*$ denote the corresponding outputs of the insertion channel with parameter $\\beta$, where $\\alpha+\\beta=1$. Then for $w \\in \\mathbb{C}$, we have\n\\begin{equation}\\label{eq:expectation3}\n\\E \\left[ \\sum_{j \\geq 0} (X^*_j-Y^*_j) w^{j+1} \\right] = \\sum_{k=0}^{n-1} (x_k-y_k) \\Bigl(\\frac{\\alpha w}{1-\\beta w} \\Bigr)^{k+1} \\,.\n\\end{equation}\n\\end{lemma}\nWe also need an analog of (\\ref{wbound}): if \n$$ \n\\zeta=e^{i\\theta}=\\frac{\\alpha w}{1-\\beta w}\n$$ \nis on the unit circle, then \n\\begin{equation}\\label{analog}\nw=\\zeta\/(\\alpha+\\beta \\zeta) \\quad \\mbox{\\rm satisfies} \\; |w|^2 \\le 1+C\\theta^2 \\,,\n\\end{equation}\n for some constant $C=C(\\alpha)$. This is immediate from (\\ref{inside}), with $\\alpha,\\beta$ replacing $q,p$ there.\n\nWith Lemma~\\ref{lem:keyins} and the inequality (\\ref{analog}) in hand, the rest of the proof of Theorem~\\ref{thm:insert} is identical to the proof of Theorem~\\ref{thm:main}. \n\\end{proof}\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:keyins}]\nIt is convenient to couple $X^*$ and $Y^*$ to use the same geometric variables $G_0,G_1,\\ldots$ and the same inserted bits. Note that the choice of coupling does not affect $\\E(X^*_{j}-Y^*_j)$. Write $a_k=x_k-y_k$ and $D_j:=X^*_{j}-Y^*_j$. Then\n$$\n\\E(D_j)=\\sum_{k=0}^j \\P(G_0+\\dots + G_k=j+1)a_k=\\sum_{k=0}^j {j \\choose k} \\alpha^{k+1} \\beta^{j-k} a_k \\,.\n$$\nTherefore, using the classical expansion\n $$\n\\sum_{j=k}^\\infty {j \\choose k} s^{j-k} =(1-s)^{-k-1}\n$$\nwe obtain that \n\\begin{eqnarray*}\n\\E\\left[ \\sum_{j \\geq 0} D_j w^{j+1} \\right] &=& \\sum_{j \\geq 0} \\sum_{k=0}^j w^{j+1} {j \\choose k} \\alpha^{k+1} \\beta^{j-k} a_k \\\\\n&=& \\sum_{k \\ge 0} a_k (\\alpha w)^{k+1} \\sum_{j \\geq k} {j \\choose k} (\\beta w)^{j-k} \\\\\n&=& \\sum_{k \\ge 0} a_k (\\alpha w)^{k+1} (1-\\beta w)^{-k-1} \\,,\n\\end{eqnarray*}\nwhich is the same as (\\ref{eq:expectation3}).\n\\end{proof} \n\n\\noindent{\\bf Remark.} To combine deletions, insertions and substitutions, simply compose the linear transformation $w \\mapsto pw+q$\nthat appears in (\\ref{eq:expectation2}) with the M\\\"obius transformation $w \\mapsto \\alpha w\/(1-\\beta w)$. Each of these transformations maps the unit circle\nto a smaller circle that is tangent to it at 1, and this also holds for their composition, in any order.\n\n\n \n\n\\section*{Acknowledgements}\nWe first learned of the trace reconstruction problem from Elchanan Mossel and Ben Morris. The second author is grateful to them, as well as to Ronen Eldan, Robin Pemantle and Perla Sousi for many discussions of the problem. The insightful suggestion by Elchanan that one should focus on deciding between\ntwo specific candidates for the original bit string was particularly influential. We are also indebted to Miki Racz and Gireeja Ranade for their help\nwith the exposition.\n\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\nNoise reduction in experiments facilitates reliable extraction of useful information from a smaller amount of data. This allows for more efficient use of experimental and analytical resources as well as enables the study of systems with intrinsically limited measurement time, e.g. cases with sample damage or out-of-equilibrium dynamics. While instrumentation development and optimization of experimental protocols are crucial in noise reduction, there are situations where computational methods can advance the improvements even further.\n\n\\noindent X-ray Photon Correlation Spectroscopy (XPCS) \\cite{Madsen_Fluerasu_Ruta, Shpyrko_2014, Sinha_2014} is a statistics-based technique that extracts information about a sample's dynamics through spatial and temporal analysis of intensity correlations between sequential images (frames) of a speckled pattern collected from coherent X-ray beam scattered from the sample. The two-time intensity-intensity correlation function \\cite{Brown_1997, Madsen_2010} (2TCF) is a matrix calculated as: \n\\begin{equation}\\label{eq:(1)}\nC2(\\pmb{q},t_{1}, t_{2}) = \\frac{\\langle I(\\pmb{q},t_{1})I(\\pmb{q},t_{2})\\rangle}{\\langle I(\\pmb{q},t_{1})\\rangle \\langle I(\\pmb{q},t_{2})\\rangle}\n\\end{equation} \nwhere \\(I(\\pmb{q},t)\\) is the intensity of a detector pixel corresponding to the wave vector \\(\\pmb{q}\\) at time \\(t\\). The average is taken over pixels with equivalent \\(\\pmb{q}\\) values.\nAn example of a 2TCF is shown in Fig.~\\ref{fig:Figure1}. The dimensions of the matrix are \\emph{N}x\\emph{N}, where \\emph{N} is a number of frames in the experimental series. The dynamics can be traced along the lag times \\(\\delta t=|t_{1}-t_{2}|\\). In the case of equilibrium dynamics, information from a 2TCF can be 'condensed' to a single dimension by integrating along the \\emph{(1,1)} diagonal producing a time-averaged one-time photon correlation function (1TCF) \\cite{Luxi_Li_2014}: \n\\begin{equation}\\label{eq:(2)}\nC1(\\pmb{q},\\delta t) = C_{\\infty} + \\beta|f(\\pmb{q},\\delta t)|^2\n\\end{equation}\nwhere \\(f(\\pmb{q},\\delta t)\\) is the intermediate scattering function at lag time \\(\\delta t\\), \\(\\beta\\) is the optical contrast and \\(C_{\\infty}\\) is the baseline that equals to 1 for ergodic samples. While 1TCF can be directly obtained from raw data \\cite{Lumma_2000}, calculating 2TCF as an intermediate step is beneficial even for presumably equilibrium cases. 2TCF contains time-resolved information about both samples' intrinsic dynamics and fluctuations of the experimental conditions, which enables one to determine between stationary and non-stationary dynamics and whether or not the time-averaged 1TCF is a valid representation of the scattering series. Investigation of 2TCF helps to identify single-frame events, such as cosmic rays detection, and beam-induced dynamics, where timescales might vary with the accumulation of X-ray dose absorbed by the sample during the acquisition of the dataset. \n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[]{Figure-1-Konstantinova.jpg}\n\\caption{Data for the model. (A) 2TCF for an experimental series consisting of 400 frames. Red squares show examples of regions selected for the model training. Yellow arrow shows the temporal direction \\emph{t} of the system's dynamics. Yellow solid line shows the 1TCF along \\emph{t}, calculated from the 2TCF. (B) Example of 50$\\times$50 2TCF, passed as an input to the model. (C) Example of the target data for the model, obtained by averaging multiple 50$\\times$50 diagonal sections of the 2TCF. All images have the same intensity scale.}\n\\label{fig:Figure1}\n\\end{figure}\n\n\\noindent XPCS experiments can suffer from various sources of noise and artifacts: probabilistic nature of photon scattering, detector shot noise, and instrumentational instabilities. Significant progress in reduction of the noise involved in photon detection and counting has been made by developing single-photon counting devices \\cite{Grybos_2016, Llopart_2002} and employment of the 'droplet' algorithm \\cite{Livet_2000} or pixel binning \\cite{Falus_2006}. Efforts have been dedicated to integrating feedback loops \\cite{Kongtawong_2020, Strocov_2010} into instrumentational controls to reduce the impact of instabilities. Despite the current advances of experimental setup and methods for data analysis in reduction of noise and instability effects, achieving high signal-to-noise ratio is still a practical challenge in many XPCS experiments. The need to suppress the high-frequency fluctuations leads to extended data collection times \u2013 an approach that itself can introduce additional errors, for instance due to slow changes in experimental conditions. Limited experimental resources may not allow for multiple repeated measurements for systems with very slow dynamics. Besides, a sample's intrinsic properties can limit the time-range, within which the dynamics can be considered \\cite{Madsen_2010} as equilibrium and thus quantitatively evaluated with Eq.~\\ref{eq:(2)}. A tool that helps to accurately extract parameters of the system's equilibrium dynamics from a limited amount of noisy data would be useful, but no generally applicable, out-of-the-box tool exists for XPCS results.\n\n\\noindent Solutions based on artificial neural networks are attractive candidates as they are broadly used for application-specific noise removal. Among such solutions are extensions of autoencoder models \\cite{ Kramer_1991}, which are unsupervised algorithms for learning a condensed representation of an input signal. The principle behind an autoencoder is based on a common fact that the information about significant non-random variations in data is contained in a much smaller number of variables than the dimensionality of the data. An autoencoder model consists of two modules: encoder and decoder. The encoder transforms the input signal to a set of unique variables called latent space. The decoder part then attempts to transform the encoded variables back to the original input. As the number of components in the latent space is generally much smaller than the number of components in the original input, the nonessential information, i.e. random noise, is lost during such transformations. Thus, an autoencoder model on its own can be used as an effective noise reduction tool. However, in the scope of this work we employ a broader idea of noise. We treat all dynamic heterogeneities due to changes in a sample configuration caused by stress or diffusive processes, as well as correlated noise in 2TCF, as an unwanted signal. Such point of view can be preferred when one wants to quantify the average dynamics parameters with Eq.~\\ref{eq:(3)} or to separate the underlying (envelope) dynamics from stochastic heterogeneities. An autoencoder model can be modified to address the removal of a deterministic, application-specific noise by replacing its targets with 'noise-free' versions of the input signals. In the case of an image-like input, such as an XPCS 2TCF, convolutional neural networks (CNN) are the obvious choice for the encoder and decoder modules. CNN-based encoder-decoder (CNN-ED) models have been successfully implemented for noise removal and restoration of impaired signals in audio applications\\cite{Grais_2017, Se_Rim_Park_2017} and images \\cite{Pathak_2016, Xioa-Jiao_Mao_2016}.\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[]{Figure-2-Konstantinova.jpg}\n\\caption{ Architecture of the CNN-ED model. The input and the output images have the same intensity scale.}\n\\label{fig:Figure2}\n\\end{figure}\n\n\\noindent Here, we demonstrate an approach for noise reduction in 2TCFs by means of CNN-ED models. An ensemble of such models, trained on real experimental data, shows noticeable suppression of noise while preserving the functional form of system's equilibrium dynamics and the temporal resolution of the signal. Addressing noise removal from 2TCF instead of the scattering signal at the detector makes the approach agnostic to the type of the registering device, the size of the selected area, the shape of the speckles, the intensity of the scattering signal and the exposure time, enabling the models' application to a wide range of XPCS experiments.\n\n\n\\section*{Results}\n\n\\textbf{Data Processing.}\nThe models are trained using data from the measurements of equilibrium dynamics of nanoparticle filled polymer systems conducted at the Coherent Hard X-ray Scattering (CHX) beamline at NSLS-II. For the nanoparticles' dynamics Eq.~\\ref{eq:(2)} can be approximated by the form \\cite{Madsen_2010}:\n\\begin{equation}\\label{eq:(3)}\nC1(\\pmb{q},t) = C_{\\infty} + \\beta e^{-2(\\Gamma t)^\\alpha}\n\\end{equation}\nwhere \\(\\Gamma\\) is the rate of the dynamics and \\(\\alpha\\) is the compression constant. The baseline \\(C_{\\infty}\\) is nearly 1 in the considered cases. Each experiment contains a series of 200-1000 frames. To augment the training data, additional 2TCFs are constructed using every second frame of the original series, which would be an equivalent to data collection with a twice longer lag period. Multiple regions of interest (ROI) - groups of pixels on the detector with equivalent wavevectors - are analyzed for each series and the 2TCFs are calculated for each ROI.\nFor each model datum, or an \"example\", the input image is obtained by cropping a 50x50 pixels part from a 2TCF with the center on the \\emph{(1,1)} diagonal, starting at the lower left corner, as shown in Fig.~\\ref{fig:Figure1}(A). Each next datum is obtained by shifting the center of the cropped image along the diagonal by 25 frames.\nThe target image for each example is the average of all the cropped inputs extracted from the same 2TCF. Thus, groups of 3 to 39 inputs have the same target. While the target images still contain noise, its level is significantly reduced with respect to the noise of the input images. Here, the size of 50x50 pixels is chosen as for the majority of the examples in the considered dataset the dynamics' parameters can be inferred from the first 50 frames. However, any size can be selected to train a model with little to no modification to its architecture if enough data are available. \n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{|l|l|l|l|}\n\\hline\n & Training & Validation \\\\\n\\hline\n\nUnique Inputs & 12236 & 5449 \\\\\n\\hline\n\nUnique Targets & 722 & 401 \\\\\n\\hline\n\\end{tabular}\n\\caption{\\label{tab:Table1}Distribution of examples between the training and validation sets.}\n\\end{table}\n\n\\noindent The diagonal (lag=0) 2TCF values of the raw data reflect the normalized variance of the photon count. Such values are vastly different between experiments and detector ROIs. They can by far exceed the values of photon correlation between frames (typically on a scale between 1 and 1.5) and are usually excluded from the traditional XPCS analysis. To prevent the influence of the high diagonal 2TCF values on the model cost function, the pixels along the diagonal are replaced with the values randomly drawn from the distribution of 2TCF values at lag=1. In doing so, we avoid artificial discontinuities in the images.\n\n\\noindent For a proper model training process and to ensure its generalizability, we find that all the input data should be introduced to the model on the same scale. However, a commonly applied standard scaling is not suitable for the present case as the level of noise may affect the values of the essential parameters such as the baseline and the contrast. \nTo bring all examples to a similar range, the estimated contrast for each series and each ROI is scaled to unity (see Methods). After processing, the data are split into the training and validation sets as shown in Table~\\ref{tab:Table1}. The splitting is done in a way that no two inputs from different sets have the same target.\n\n\\noindent\\textbf{Model Training.} The ED model architecture used in this work is shown in Fig.~\\ref{fig:Figure2}. The encoder part consists of two convolutional layers with the kernel size 1$\\times$1. Training the model with larger kernel sizes did not improve the performance of the model. While kernels of size 1$\\times$1 are used sometimes in CNN image applications \\cite{Simonyan_2014} for creating non-linear activations, generally, they are not exclusively incorporated across the entire network. The reason for this is that the convolutional kernels are intended to catch distinctive edges, which form characteristic features of an image. To identify an edge, the distribution of intensities among the neighboring pixels is needed. However, the 2TCFs used in this work do not have sharp edges, which can partially explain the lack of improved learning with larger kernels. Besides, an equilibrium 2TCF has a unique structure, with symmetry along the diagonals. An equilibrium 2TCF and its modified copy with pixel values randomly shuffled along the diagonals would produce exactly the same 1TCF. This property is picked up by the model during compression of convolutional outputs to the latent space.\n\n\\noindent Both convolutional layers consist of 10 channels with rectified linear unit (\\emph{ReLU}) activation function applied to the output of each channel. \nWe find that increasing the number of channels does not significantly change the performance of the model and the smaller number of channels gives poorer performance.\nNo pooling layers\\cite{Nagi_2011} were introduced to prevent information loss\\cite{Ronneberger_2015} at the encoding stage. The output of the convolutional layers contains 25,000 features. A linear transformation is performed to convert them to the latent space of a much smaller dimension. \nWhile some ED image applications implement fully convolutional architectures \\cite{Xioa-Jiao_Mao_2016, Se_Rim_Park_2017}, we believe that the introduction of the linear layer for purpose of denoising equilibrium 2TCFs is beneficial. Not only does the bottleneck layer provide the regularization of the model, it also mixes the features derived by convolutional layers from different parts of the input image.\nThe decoder part consists of two transposed convolutional layers, symmetrical to the encoder part, that convert the latent space back to a 50$\\times$50 image.\nThe \\emph{ReLU} function is applied only to the output of the first decoder layer.\n\n\\noindent The mean squared error (MSE) between the denoised output and the target is a natural choice of cost function for many image denoising applications. The MSE is shown to be useful for image denoising even in cases of some noise being present in the target \\cite{Lehtinen_2018}. Moreover, presence of noise in the input data puts a regularization on model weights, enforcing contractive property \\cite{Alain_2014} on the reconstruction function of denoising EDs. The goal of the model presented here is to reduce the noise in 2TCF in such a way that the 1TCF, calculated from the model output, is as close to the target 1TCF as possible. Thus, the model's learning objective is modified by inclusion the MSE between respective 1TCFs into the cost function.\n\n\\noindent We find that the regularization, which is enforced by the noise in both inputs and targets, in conjunction with the early stopping based on the cost function for the validation set, is sufficient for the model to avoid over-fitting. Introducing additional weight regularization reduced the accuracy of the model, especially for the cases of fast dynamics.\n\n\\noindent However, the cost function calculated for the validation set is not the only parameter to consider when selecting the optimal parameters for the model.\nWhen examining models trained for different latent space dimensions, the validation cost function (Fig.~\\ref{fig:Figure3}) does not have a pronounced minimum in the range of dimensions between 1 and 200.\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[]{Figure-3-Konstantinova.jpg}\n\\caption{Selection of the latent space dimension for the model. From top to bottom, as the function of the latent space dimension: the MSE(1TCF) (blue circles) and the cost function (orange triangles) for the validation set; the MRE($\\Gamma$); the MSE of $\\beta$, $C_{\\infty}$ and $\\alpha$, extracted from the fit of the corresponding CNN-ED ensemble's outputs for the examples in the validation set to Eq.~\\ref{eq:(3)}. The vertical line marks the choice of the latent space dimension for the model.}\n\\label{fig:Figure3}\n\\end{figure}\n\nHowever, this metric may not reflect well the systematic errors in reconstructing the\ndynamics parameters, such as $\\beta$, $\\Gamma$, $\\alpha$ and $C_{\\infty}$, which are essential to drawing scientific conclusions. An efficient model would precisely recover these parameters for a broad set of observations. Thus, the optimal dimension is selected based on how well the model output allows to recover those parameters for the validation data.\nHere, the rate of the dynamics, $\\Gamma$, is the most important parameter to consider since the variation of $\\beta$ is taken care of by pre-processing normalization and the variations of $\\alpha$ and $C_{\\infty}$ are naturally very small in the considered examples.\n\n\\noindent To reduce the variance associated with the randomness of the initial weights initialization, ten models with different random initialization are trained for each latent space dimension. For each of the validation examples, the outputs of the ten models are converted to 1TCF, averaged and then fit to Eq.~\\ref{eq:(3)}. The ground truth values, used for comparison, are obtained by fitting the 1TCF calculated from all (100-1000) frames in the same experiment and the same ROI as the input example is taken from.\n\n\\noindent Since values of dynamics rate can be very close to zero, the mean absolute relative error (MRE) is considered for $\\Gamma$. The MSE is calculated for other parameters. The accuracy of $\\Gamma$ keeps improving with increased number of hidden variables. But the rate of improvement slows down considerably above 5-8 variables. The same is observed for $\\alpha$ and $C_{\\infty}$. This is in agreement with the MSE(1TCF) between the model output and the target, as shown in Fig.~\\ref{fig:Figure3}. The accuracy of $\\beta$ is relatively uniform across all the models. Based on the above, we select the models with eight latent variables for further consideration.\n\n\\noindent To address the variance of the selected CNN-ED, we train 100 such models with different random initialization and select among them the 10 best performing models based on the MSE(1TCF) for the validation set. Selecting only a limited number of the best performing models instead of combining all trained models also optimizes the use of storage memory and computational resources. \n\n\n\\noindent\\textbf{Model Testing.} The performance of the ensemble of models is evaluated through several tests. Firstly, we estimate the model applicability range by applying it to experiments similar to the ones used for training.\nAn example of noise removal from a test datum is shown in Fig.~\\ref{fig:Figure4}. Reduction of the noise is especially important for larger lag times, where fewer scattering frames are available for calculating the correlations. \n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[]{Figure-4-Konstantinova.jpg}\n\\caption{Example of 2TCF denoising with the CNN-ED models. (A) From left to right: the raw input 2TCF; the averaged target; the output of the ensemble of CNN-ED models. (B) 1TCF calculated from each 2TCF in (A). The dashed line corresponding to a baseline \\(C_{\\infty} = 1\\) is shown for convenience.}\n\\label{fig:Figure4}\n\\end{figure}\n\n\\noindent As mentioned above, despite the cost function working well for determining the optimal weights for a model, it is not sufficient to assess the reliability of the model output for quantitative analysis of the materials' dynamics. We assess the performance of the ensemble by comparing the fits with Eq.~\\ref{eq:(3)} for the 1TCFs calculated from the cropped 50$\\times$50 pixels raw data (inputs), the corresponding denoised model outputs and the full-range raw data (ground truth target) (see the Supplemental Materials). From the results of the test set, the noise removal from the raw cropped 2TCFs with the CNN-ED ensemble noticeably improves the precision for the dynamics parameters in a wide range of cases with $0.01 frames^{-1}<\\Gamma<0.15 frames^{-1}$ (i.e. the contrast drops by half within approximately the first 3-35 frames) in comparison with fitting the raw cropped 2TCFs. The application of the model enables reasonable estimates even in cases when the low signal-to-noise ratio of the raw cropped data prevents a convergent fit within the parameters' boundaries. In the region $\\Gamma >0.15 frames^{-1}$, the results of the model are no longer more accurate than the raw data in general. \nNote that the precision of the model depends on the accuracy of identifying the optical contrast. Accurate measurements of optical contrast in XPCS experiments with fast dynamics can be challenging as they can involve data collection with reduced exposure or relying on averaged speckle visibility.\nFurthermore, a poor accuracy in identifying dynamics parameters is observed for inputs with very high noise levels (Fig.~S7) and\/or the presence of well pronounced dynamical heterogeneities.\n\n\\noindent If 100 or more frames are available for analysis, 2TCFs with slow dynamics can be reduced by considering every $2^{nd}$, $3^{rd}$, etc. frame, as it is done for augmenting the training data. This to will effectively increase the exposure times and increase the $\\Gamma$ measured in $frames^{-1}$, making the model output more accurate. Alternatively, a model with a larger size of input 2TCF can be trained to handle cases of slow dynamics.\n\n\n\\noindent While it is clear from above how the model performs on average for individual independent 2TCFs, it is useful to see if application of the model can lead to reducing data collection in a typical XPCS experiment. We consider a single 700-frames series of scattering images among those used for creating the test set. The goal is to see if one can extract a sufficient information about the $q$-dependence of the dynamics rate $\\Gamma$ using only the first 50 frames with and without the model application. The target 2TCF for each of the concentric ROIs (shown in Fig.~\\ref{fig:Figure5}(A)) are calculated using all 700 frames. The first $50\\times50$ frames regions of the 2TCFs are considered and the ensemble CNN-ED model is applied to them. The visual comparison between the level of noise in the raw data and in the model output for an ROI with large $q$ is shown in Fig.~\\ref{fig:Figure5}(B). The 1TCFs, calculated from the raw cropped 2TCFs, from the model outputs and from the target 2TCFs for each ROI are fit to Eq.~\\ref{eq:(3)} with $\\alpha = 1$. The results are shown in Fig.~\\ref{fig:Figure5}(C-E). For the parameter $\\Gamma$, at small $q$, where the signal-to-noise ratio is high, all three fits are close. However, as $q$ grows and the noise level increases, the fit for the raw 50-frame 1TCFs starts to deviate from the target values more than the fit for the model outputs. In fact, the outcome of the model remains close to the actual values until the large $q$ values (ROI\\# 16 and above). A similar tendency is observed for parameters $\\beta$ and $C_{\\infty}$. This example demonstrates that application of the model can help to obtain sufficient information about the equilibrium system's dynamics from a smaller amount of data.\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[]{Figure-5-Konstantinova.jpg}\n\\caption{Model application for recovering the $q$-dependence of the dynamics parameters. (A) Scattering image of the sample with the ROI map on top of it. Dark blue corresponds to pixels excluded from the analysis. (B) A 100-frame fragment of 2TCF from ROI \\#14. The first 50-frame part is denoised with the model. Variation of $\\Gamma$ (C), $\\beta$ (D) and $C_{\\infty}$ (E) parameters among ROIs with different $q$.}\n\\label{fig:Figure5}\n\\end{figure}\n\n\\noindent The applicability of the model to non-equilibrium data is also tested. Although the model is trained with the equilibrium examples, it still can be applied to quasi-equilibrium regions of a 2TCF with gradually varying dynamics parameters. Here, the model performance is demonstrated for a sample with ageing dynamics that become slower with time. Since the target values cannot be obtained by averaging many frames for a such case, we calculate two 2TCFs with different noise levels, but carrying the same information, through sub-sampling pixels from the same ROI. The original ROI is a circle of small width with its center at $q$=0. This ROI is used for calculating the target 2TCF. To calculate the test 2TCF with the reduced signal-to-noise ratio we randomly remove 74\\% of pixels from the original ROI. The model is applied along the \\emph{(1,1)} diagonal in a sliding window fashion (see Methods).\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[]{Figure-6-Konstantinova.jpg}\n\\caption{Model application for the case of non-equilibrium dynamics. (A) 2TCF calculated from reduced ROI. (B) The same 2TCF after the model is applied along its diagonal. Temporal evolution of $\\Gamma$ (C), $C_{\\infty}$ (D) and $\\alpha$ (E) parameters extracted from the noisy raw data, denoised data and the target 2TCF.}\n\\label{fig:Figure6}\n\\end{figure}\n\n\\noindent To compare the test 2TCF and the result of the model application, the cuts with width of 1 frame are taken perpendicular to the \\emph{(1,1)} diagonal and the resulting 1TCFs are fit to Eq.~\\ref{eq:(3)} in analogy to other XPCS analyses \\cite{Madsen_2010, Malik_1998}. The target parameters are obtained by taking the cuts of 10 frames with the step of 10 frames from the target 2TCF and fitting the 1TCF, averaged over each cut, to the Eq.~\\ref{eq:(3)}. Averaging is done for improving the accuracy of the target values. \nThe contrast $\\beta$ is estimated as the mean of the respective raw 2TCF(lag=1) at frames 250-300, where the dynamics are fairly slow, and is fixed during the fit.\nThe results for $\\Gamma$, $C_{\\infty}$ and $\\alpha$ are shown in Fig.~\\ref{fig:Figure6}(C-E). While the general trend of $\\Gamma(t)$ could be visually estimated from the raw test data, the output of the model gives much fewer outliers. Moreover, the temporal region, where $\\Gamma$ can be reasonably estimated is wider for the model output. The fit to the raw test data does not allow to estimate $\\Gamma$ in the first 30-40 frames, while the fit to the denoised data is close to the target in that region. The fit of the denoised data only shows a high uncertainty at the corners of the 2TCF, where the corresponding 1TCFs consist of less than 20 points. The variance of parameter $C_{\\infty}$ is also improved for the denoised data, but the most notable improvement in accuracy is observed for the parameter $\\alpha$. The fits to the raw noisy data have high variance, which hides the upward trend of $\\alpha$, in contrast to the fits to the denoised data. \n\n\\noindent In a typical experiment, cuts with width of more than 1 frame are used for estimating the dynamics parameters achieving a better accuracy for the raw data than shown in Fig.~\\ref{fig:Figure6}. However, the selection of regions with quasi-equilibrium dynamics is not trivial. Since the fits to 1-frame-wide cuts from the denoised data have a low variance almost across the entire experiment, application of CNN-EDs makes the data more suitable for automated analysis and for visual inspection of the data when selecting the quasi-equilibrium regions.\n\n\n\\noindent\\textbf{Comparison to Other Techniques.} We compare the performance of our approach to several of-the-shelf solutions for noise reduction in images: linear principle components\u2013based, Gaussian, median and total variation denoising (Chambolles' projection) \\cite{Duran_2013} filters. The comparison of the application of these techniques to the same test example as in Fig.~\\ref{fig:Figure4} is shown in Fig.~\\ref{fig:Figure7}. Principle components filters have the same idea as the ED model \u2013 preserving only the information from a few essential components of the original data. In fact, an autoencoder is a type of non-linear principle component generator. As one would expect, a filter based on linear principle components, trained on the same data as the CNN-EDs, under-performs comparing to the case of non-linear components due to a larger bias of the procedure for the components extraction. Gaussian and median filters are based on smoothing the intensity fluctuations between neighboring pixels and the total variation denoising is a regularized minimization of the additive normally distributed noise. While these approaches help to reduce pixel-to-pixel intensity variations, unlike the demonstrated here CNN-ED models, they do not learn the functional form of the equilibrium 2TCF images and cannot be improved by having a larger training set. By only considering local surrounding of individual pixels in a single image, such algorithms cannot recognize, for example, that the correlation function decays at larger lag times. Consequently, when an isolated high-intensity pixel (noise) is encountered in an image, an application of such filters leads to inflation of intensities in the surrounding pixels, highlighting the noise instead of correcting it. Thus, noise removal with the above filters can introduce false trends in 1TCF, which makes them unsuitable for quantitative XPCS data analysis. On the other hand, a CNN-ED, which is a regression model, learns from numerous examples the characteristic trends in the data and is less likely to introduce artifacts.\n\n\\begin{figure}[!htb]\n\\centering\n\\includegraphics[]{Figure-7-Konstantinova.jpg}\n\\caption{Comparison of various noise removal techniques applied to an example from the test set. Top row: results of applying filters to the raw 2TCF, middle row: 1TCFs calculated from the 2TCFs for the raw input (blue dashed line), the results of the respective filters (orange solid line) and the target (green solid line), bottom row: residuals of the 1TFCs calculated from the example after denoising with the respective filters. }\n\\label{fig:Figure7}\n\\end{figure}\n\n\\section*{Discussion}\nThe CNN-ED approach to noise removal in XPCS shows a reasonable improvement in the quality of the signal, allowing for quantification of a sample's dynamics from a limited amount of data, avoiding extensive data collection, accessing finite regions of reciprocal space and quasi-equilibrium intervals of non-equilibrium dynamics. The CNN-ED models go beyond and are superior to a simple smoothing of intensity between neighboring pixels since these models empirically learn the structural form of the 2TCF. The models are fast to train and do not require an extensive amount of training data. Their accuracy is pretty robust with respect to the choice of hyperparameters such as the number of channels in the hidden layers, the convolutional kernel size and the latent space dimension. The computational resources required for the application of the ensemble of 10 models are smaller than one needs to calculate 2TCFs for a typical number of frames required to achieve the same signal-to-noise ratio. \n\n\\noindent However, there are several limitations to keep in mind when applying CNN-ED models to a 2TCF. The testing results show that the models may not reliably remove the noise for the cases of very fast and very slow dynamics as well as from very noisy data (see the illustration in Supplementary Fig.~\\ref{fig:FigureS7}). Some inaccuracy for the cases of fast dynamics comes from uncertainties in identifying the normalization factors (contrasts) for pre-processing of the inputs, which is also a challenge for traditional analysis. When the speckle visibility drops significantly within a single frame acquisition period, its estimation from the input data can have a high error. As it is seen from the model performance for the validation set and for the non-equilibrium test case in comparison to its performance for the equilibrium test set, a more accurate scaling of the inputs can improve the precision of the model for experiments with faster dynamics.\nBesides, one is advised to consider the context of extracted dynamics for a given material before relying solely on the information extracted from only a single 2TCF regardless of whether a CNN-ED model is applied.\nOne benefits from a series of experiments on a single system, such as a temperature dependence or the demonstrated here $q$-dependence, to lend credibility to extracted dynamics for one particular experiment.\n\n\\noindent In this work, only equilibrium dynamics described by stretched exponents with the baseline close to 1 are used for training. Thus, the model learns to approximate any input with this type of dynamics. This can result in a loss of fine details, such as heterogeneities, oscillations or fast divergence of dynamics parameters in non-equilibrium cases. However, the demonstrated approach to the noise removal can be expanded to other types of dynamics with sufficient amount of data for training. Even in the absence of proper denoised target data, the autoencoder version of the model can significantly reduce the random noise. Furthermore, a CNN-ED model can be trained to correct for specific types of artifacts, such as impact of instrumentational instabilities or outlier frames, leading to a more efficient use of experimental user facilities \\cite{Campbell_2020}. Similarly to other fields\\cite{Baur_2019, Chong_2017}, the autoencoder models can be used for identifying unusual observations in the stream of XPCS data. Additionally, the encoded low-dimensional representation of the 2TCF can be used for classification, regression and clustering tasks, related to samples' dynamics. In the broader scope, the presented here CNN-ED models and their modifications have the potential for application in automated high-rates XPCS data collection\\cite{Zhang_2021} and processing pipelines, reducing the reliance on the human-in-the-loop in decision making during experiments. \n\n\\section*{Methods}\n\n\\noindent\\textbf{Training data.} The data for training and validation set contain experiments for 7 samples from 3 different material groups A(1 sample), B(2 samples) and C(4 samples). The experiments are taken at various exposures, acquisition rates and temperatures. Concentric ROIs with increasing $q$ are used. Depending on the noise level and the dynamics duration, from 2 to 17 ROIs (median 10 ROIs) are considered for each experiment. The diversity of experimental conditions and regions in the reciprocal space allows one to obtain a realistic distribution of dynamics parameters and noise levels. We have not included the cases with very slow dynamics, for which only a small portion is complete within the 50 frames. To cut off the high noise data, we excluded the cases, where the fit to Eq.~\\ref{eq:(3)} did not converge for the full-range 1TCF. \nThe distributions of dynamics parameters for the training and the validation set are shown in Fig.~\\ref{fig:FigureS1}.\nFor the model training purposes, all input data (2TCF$_{noisy}$) are scaled as:\n\\begin{equation}\\label{eq:(4)}\ninput = (2TCF_{noisy} - 1)\/\\beta^{*} + 1\n\\end{equation}\nwhere $\\beta^{*}$ is the estimation of speckle visibility for the integration time of a single frame. It is obtained from fitting the equivalent pixels' intensity fluctuations with a negative binomial distribution \\cite{Luxi_Li_2014}. For this, the speckle visibility, is calculated for each frame and is averaged among all the frames in the series. The target data are reversely scaled accordingly.\n\n\n\\noindent\n\\textbf{Test data.} The test data are collected in a similar fashion to the training\/validation data. Experiments for 5 different samples in the same material group (C) are considered. Experiments are performed for different temperatures, exposure times and acquisition rates. Concentric ROIs with increasing $q$ are used. 10 ROIs with the smallest $q$-s are considered. However, no visual inspection of the data is done prior to model application and the ROIs with slow dynamics are not rejected. ROIs, where the full-range 1TCF fit to Eq.~\\ref{eq:(3)} does not converge, are not considered. Overall, 12060 inputs (679 distinct targets) are considered in the test set. The distribution of the parameters from Eq.~\\ref{eq:(3)} in shown in Fig.~\\ref{fig:FigureS2}.\n\\noindent Unlike training\/validation data, the contrast for normalization of test inputs is estimated from the 1TCF derived from the 2TCF) at lag=1 frame instead of the speckle visibility. This is done to reduce the computation time and to test the model performance for the cases when only the 50$\\times$50 2TCFs, and not the scattered images, are available. No adjustment is done to the baseline as the input data does not provide a good estimate for it. Thus, for each of the noisy 2TCF$_{noisy}$, the model \\emph{input} is calculated as:\n\\begin{equation}\\label{eq:(5)}\ninput = (2TCF_{noisy} - 1)\/1TCF_{noisy}(lag=1) + 1\n\\end{equation}\nThe denoised 2TCF$_{denoised}$ is then obtained from the output as: \n\\begin{equation}\\label{eq:(6)}\n2TCF_{denoised} = (output - 1)*1TCF_{noisy}(lag=1) + 1\n\\end{equation}\n\n\n\\noindent \n\\noindent\\textbf{Non-equilibrium test.}\n\\noindent For the example of ageing dynamics considered in this work, the model is applied to each $50\\times50$ piece of the raw 2TCF along its [1,1] diagonal with the step size 5 frames, starting at the first frame.\nPrior the application of the model, each input is obtained from a raw 2TCF$_{noisy}$ as:\n\\begin{equation}\\label{eq:(7)}\ninput = (2TCF_{noisy} - C_{\\infty}^{*})\/\\beta^{*}(lag = 1) +1 \n\\end{equation}\nwhere $\\beta^{*}(lag = 1)$ is the estimation of contrast at lag=1 as the mean of 2TCF$_{noisy}(lag =1)$ for frames 250-300 and $C_{\\infty}^{*}$ is the estimation of the baseline as the mean of 2TCF$_{noisy}$ at lags 270-300. The reverse transformation is applied to the model output.\nThe overlapping model outputs between the current and the previous steps are averaged. The values outside of the $50\\times50$ diagonal sliding window are remained unchanged. The same procedure is repeated with the model sliding window moving from the last frame towards the first frame. The two results are averaged to reduce the dominating influence of the later dynamics over the earlier dynamics and vice versa. The loss of the temporal resolution due to convolution between the model and the raw signal is not significant for the considered case of slowly-evolving sample dynamics. \n\n\n\\noindent\\textbf{Model Training Details.} The cost function used for training the models is the sum of the Mean Squared Error (MSE) between the target 2TCF and the models' output and the MSE between the respective 1TCFs (without lag=0): \n\\begin{equation}\\label{eq:(8)}\ncost = \\frac{1}{2500\\cdot m}\\sum_{k =1}^{m} ||x^{out}_{k} -x^{target}_{k}||_{2} + \\frac{1}{49 \\cdot m}\\sum_{k =1}^{m} ||1TCF(x^{out}_{k}) -1TCF(x^{target}_{k})||_{2}\n\\end{equation}\nwhere \\(x^{out}_{k}\\) is the model output for the $k$\u2013th training example and \\(x^{target}_{k}\\) is the corresponding target's pixel, \\emph{m} is the number of examples, \\(||\\cdot||_{2}\\) stands for \\emph{2-norm}. \n\n\\noindent At every training epoch, batches of size 8 are processed. Adam \\cite{Kingma_2014} optimizer with initial learning rate from 2.5e-6 to 4e-5 is used. Learning rate is reduced by a factor of 0.9995 at every epoch. Initial weights in the convolutional and linear layers are assigned according to Xavier uniform initialization \\cite{Glorot_2010}. The models are trained with Nvidia GPU accelerator GeForce RTX 2070 Super. For the selected CNN-ED configuration, the average training time is 27 seconds per epoch with 9-82 epochs necessary to train a model. Each input or target takes 30 kB of computer memory.\n\n\\noindent A model application does not require a GPU and, in fact, can be preformed faster without transferring the 2TCF data to a GPU. When using a CUDA accelerator, loading the model from a file, converting the 2TCF from numpy arrays to a CUDA Pytorch tensor, application of the model and converting the result back to a numpy array takes 2.3 ms on average with pure model computation taking 0.48 ms. Without using a CUDA accelerator, the corresponding times are 1.4 ms and 0.57 ms, respectively. \n\n\\section*{Acknowledgements}\n\nThe authors thank A. Fluerasu and M. Fukuto for fruitful discussions. This research used CHX and CSX beamlines and resources of the National Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory(BNL) under Contract No. DE-SC0012704 and under a BNL Laboratory Directed Research and Development (LDRD) project 20-038 \"Machine Learning for Real-Time Data Fidelity, Healing, and Analysis for Coherent X-ray Synchrotron Data\" \n\n\\section*{Author contributions statement}\n\nA.M.B, A.M.D, L.W and T.K. conceived the idea, L.W. performed the beamline experiments, generated the XPCS results, and identified individual scans for model development, T.K. processed the data for the model, A.M.D and T.K. wrote the code, M.R. provided technical consultation, T.K., L.W., A.M.D, A.M.B and M.R. analyzed the model performance, T.K. wrote the manuscript with contribution from all authors.\n\n\\section*{Additional information}\n\n\\noindent\\textbf{Accession codes and Data availability} The code and the data used for the model training can be found at GitHub repository, \\emph{https:\/\/github.com\/bnl\/CNN-Encoder-Decoder} at the time of publication. \n\n\\noindent\\textbf{Competing interests} The authors declare no competing interests. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nA commonly encountered problem in practice is merging databases\ncontaining records collected by different sources, often via\ndissimilar methods. Different variants of this task are known as record linkage, de-duplication, and\nentity resolution. Record linkage is inherently a difficult problem\n\\cite{christen_2011, Herzog_2007,Herzog:2010}. \nThese difficulties are partially\ndue to the noise inherent in the data, which is often hard to\naccurately model \\cite{pasula_2003, steorts_2013b}. A\nmore substantial obstacle, however, is the scalability of the approaches \\cite{WYP:2010}. With $d$ databases of $n$ records each, brute-force approaches,\nusing all-to-all comparisons, require $O(n^d)$ comparisons. This is quickly\nprohibitive for even moderate $n$ or $d$. To avoid this computational\nbottleneck, the number of comparisons made must be drastically reduced, without\ncompromising linkage accuracy. Record linkage is made scalable by ``blocking,'' which involves partitioning datafiles into\n``blocks'' of records and treating records in different blocks as non-co-referent {\\em a\n priori} \\cite{christen_2011, Herzog_2007}. Record linkage methods are only\napplied {\\em within} blocks, reducing the comparisons to\n$O(B n_{\\max}^d)$, with $n_{\\max}$ being the size of the largest of the $B$\nblocks.\n\nThe most basic method for constructing a blocking partition picks certain fields (e.g. geography, or gender and year\nof birth) and places records in the same block if and only if they agree on\nall such fields. This amounts to an {\\em a priori} judgment that these fields\nare error-free. We call this \\emph{traditional blocking} (\\S \\ref{sec:block-naive}).\n\n\nOther data-dependent blocking methods \\cite{christen_2011, WYP:2010}\n are highly application-specific or are based on placing similar records into the\nsame block, using techniques of ``locality-sensitive hashing'' (LSH).\n LSH uses all of\nthe information contained in each record and can be adjusted to ensure that blocks are\nmanageably small, but then does not allow for further record linkage within blocks. For example, \\cite{christen_2014} introduced novel data structures for sorting and fast approximate nearest neighbor look-up within blocks produced by LSH. Their\napproach gave balance between speed and recall, but their technique is\nvery specific to nearest neighbor search with similarity defined by the hash\nfunction. Such\nmethods are fast and have high recall, but suffer from low precision, rather, too\nmany false positives. This approach is called \\emph{private} if, after the blocking is performed, all candidate records pairs are compared and classified into matches\/non-matches using computationally intensive ``private\" comparison and classification techniques \\cite{christen_2009}. \n\nSome blocking schemes involve clustering techniques to partition the records into clusters of similar records. \\cite{mccallum_2000} used canopies, a simple clustering approach to group similar records into overlapping subsets for record linkage. Canopies involves organizing the data into overlapping clusters\/canopies using an inexpensive distance measure. Then a more expensive distance measure is used to link records within each canopy, reducing the number of required comparisons of records.\n \\citep{vatsalan_2013} used a sorted nearest neighborhood clustering approach, combining $k$-anonymous clustering and the use of publicly available reference values to privately link records across multiple files. \n\nSuch clustering-based blocking schemes motivate our variants of LSH methods for blocking. \nThe first, transitive locality sensitive hashing (TLSH), is based upon the community discovery literature such that \\emph{a soft transitivity} (or relaxed transitivity) can be imposed across blocks. The second, $k$-means locality sensitive hashing (KLSH), is based upon the information retrieval literature and clusters similar records into blocks using a vector-space representation and projections.\n (KLSH has been\nused before in information retrieval but never with record linkage\n\\citep{pauleve_2010}.)\n\nThe organization of this paper is as follows. \\S \\ref{sec:block} reviews traditional blocking. We then review other blocking methods in \\S \\ref{sec:block-modern} stemming from the computer science literature. \\S \\ref{sec:lsh} presents two different methods based upon locality sensitive hashing, TLSH and KLSH. We discuss the computational complexity of each approach in \\S \\ref{sec:complex}. We evaluate these methods (\\S \\ref{sec:results}) on simulated data using recall, reduction ratio, and the empirical computational time as our evaluation criteria, comparing to the other methods discussed above. Finally we discuss privacy protection aspects of TLSH and KLSH, given the description of LSH as a ``private\" blocking technique. \n\n\\section{Blocking Methods}\n\\label{sec:block}\nBlocking divides records into mutually\nexclusive and jointly exhaustive ``blocks,'' allowing the linkage\nto be performed within each block. \nThus, only records within the same\nblock can be linked; linkage algorithms may still aggregate information across\nblocks. \nTraditional blocking requires domain knowledge to\npick out highly reliable, if not error-free, fields for blocking. This methodology has at least two drawbacks. The first is that the resulting blocks may still be so large that linkage within them is\ncomputationally impractical. The second is that because blocks {\\em only}\nconsider selected fields, much time may be wasted comparing records that\nhappen to agree on those fields but are otherwise radically different.\n\nWe first review some simple alternatives to traditional blocking on fields, and then introduce\nother blocking approaches that stem from computer science.\n\n\n\\subsection{Simple Alternatives to Blocking}\n\\label{sec:block-naive}\nSince fields can be unreliable for many applications, blocking may miss large proportions of matches. Nevertheless, we can make use of domain-specific knowledge on the types of errors expected for field attributes. To make decisions about matches\/non-matches, we must understand the \\emph{kinds of errors} that are unlikely for a certain field or a combination of them. With this information, we can identify a pair as a non-match when it has strong disagreements in a combination of fields. It is crucial that this calculation be scalable since it must be checked for all pairs of records. Some sequence of these steps reduces the set of pairs to a size such that more computationally expensive comparisons can be made. In \\S \\ref{ss:naive_results}, we apply these concepts.\n\n\\subsection{Cluster-Based Blocking}\n\\label{sec:block-modern}\nOthers have described blocking as a clustering problem, \nsometimes with a special emphasis on privacy, e.g., see \n\\cite{durham_2012,\nkarakasidis_2012, kuzu_2011, vatsalan_2013}. \nThe motivation is\nnatural: the records in a cluster should be similar,\nmaking good candidate pairs for linkage. \n\n\nOne clustering approach proposed for blocking is nearest neighbor clustering.\nThreshold nearest neighbor clustering (TNN) begins with a single record as the base\nof the first cluster, and recursively adds the nearest neighbors of records in\nthe cluster until the distance\\footnote{The distance metric used can vary depending on the\nnature of the records.} to the nearest neighbor exceeds some threshold.\nThen one of the remaining records is picked to be the base for the next\ncluster, and so forth. K-nearest neighbor clustering (KNN) uses a similar procedure, but ensures\nthat each cluster contains at least $k$ records\\footnote{Privacy-preserving versions of these approaches use ``reference\nvalues'' rather than the records themselves to cluster the records \\cite{vatsalan_2013}.}, to help maintain\n``$k$-anonymity'' \\cite{karakasidis_2012}.\n\nA major drawback of nearest neighbor clustering is that it requires\ncomputing a large number of distances between records, $O(n^2)$. \nBlocking a new record means finding its nearest neighbors, an\n $O(n)$ operation. \n\nThe cost of calculating distances between records in\nlarge, high-dimensional datasets led \\cite{mccallum_2000} to propose the\nmethod of \\emph{canopies}. In this approach, a computationally cheap (if inaccurate)\ndistance metric is used to place records into potentially-overlapping sets (canopies). \nAn initial record is picked randomly to be the base of the first\ncanopy; all records within a distance $t_1$ of the base are grouped under that\ncanopy. Those within distance $t_2 \\leq t_1$ of the base are removed from\nlater consideration. A new record is picked to be the base of the next\ncanopy, and the procedure is repeated until the list of candidate records is empty. More accurate but\nexpensive distance measures are computed only between records that fall under\nat least one shared canopy. That is, only record-pairs sharing a canopy are candidates to be linked.\n\n\nCanopies is not strictly a blocking method. They overlap, \nmaking the collection of canopies only a covering of the set\nof records, rather than a partition. We can derive blocks from canopies,\neither set-theoretically or by setting $t_1=t_2$. The complexity of building\nthe canopies is $O(nC_n)$, with $C_n$ being the number of canopies, itself a\ncomplicated and random function of the data, the thresholds, and the order in\nwhich records are chosen as bases. Further, finding fast, rough distance measures for complicated high-dimensional records is non-trivial.\n\n\\subsection{LSH-Based Approaches}\n\\label{sec:lsh}\nWe explore two LSH-based blocking methods. These are based, respectively, on graph\npartitioning or community discovery, and on combining random projections with\nclassical clustering. The main reason for exploring these two methods is that even with comparatively\nefficient algorithms for partitioning the similarity graph, doing that is still\ncomputationally impractical for hundreds of thousands of records\n\n\n\n\n\\subsubsection{Shingling}\nLSH-based blocking schemes ``shingle''\n\\cite{rajaraman_2012} records. That is, each record is treated as a string and\nis replaced by a ``bag'' (or ``multi-set'') of length-$k$ contiguous\nsub-strings that it contains. These are known as ``$k$-grams'', ``shingles'',\nor ``tokens''. For example, the string ``TORONTO'' yields the bag of length-two\nshingles ``TO'', ``OR'', ``RO'', ``ON'', ``NT'', ``TO''. (N.B., ``TO'' appears\ntwice.)\n\nAs alternative to shingling, we might use a bag-of-words representation, or\neven to shingle into consecutive pairs (triples, etc.) of words. \nIn our\nexperiments, shingling at the level of letters worked better than dividing by\nwords.\n\n\\subsubsection{Transitive LSH (TLSH)}\n\\label{subsec:tlsh}\n\nWe create a graph of the similarity between records. \nFor simplicity, assume that all fields are string-valued. Each record is\nshingled with a common $k$, and the bags of shingles for all $n$ records are\nreduced to an $n$-column binary-valued matrix $M$, indicating which\nshingles occur in which records. \n$M$ is large, since the number of length-$k$ shingles typically grows\nexponentially with $k$. As most shingles are absent from most records, $M$ is\n sparse. We reduce its dimension by generating a random\n``minhash'' function and applying it to each column. Such functions map\ncolumns of $M$ to integers, ensuring that the probability of two columns being\nmapped to the same value equals the Jaccard similarity between the columns\n\\cite{rajaraman_2012}. Generating $p$ different minhash functions, we reduce\nthe large, sparse matrix $M$ to a dense $p\\times n$ matrix, $M^{\\prime}$, of integer-valued\n``signatures,'' while preserving information. Each row of\n$M^{\\prime}$ is a random projection of $M$. Finally, we divide\nthe rows of $M^{\\prime}$ into $b$ non-overlapping ``bands,'' apply a hash\nfunction to each band and column, and establish an edge between two records if\ntheir columns of $M^{\\prime}$ are mapped to the same value in any\nband.\\footnote{To be mapped to the same value in a particular band, two columns\n must either be equal, or a low-probability ``collision'' occurred for the\n hash function.}\n\nThese edges define a graph: records are nodes, and edges indicate a certain\ndegree of similarity between them. We form blocks by dividing the graph into its connected components.\nHowever, the largest connected\ncomponents are typically very large, making them unsuitable as blocks.\nThus, we sub-divide the connected components into ``communities'' or ``modules'' \n--- sub-graphs that are densely connected\ninternally, but sparsely connected to the rest of the graph. This \nensures that the blocks produced consist of records that are all highly\nsimilar, while having relatively few ties of similarity to\nrecords in other blocks \\cite{fortunato_2010}. Specifically, we apply the\nalgorithm of \\cite{clauset_2004}\\footnote{We could use other community-discovery algorithms, e.g. \\cite{goldenberg_2010}.}, sub-dividing communities greedily, until even\nthe largest community is smaller than a specified threshold.\\footnote{This maximum\nsize ensures that record linkage is feasible.} \n The end result is a set of blocks that balance false\nnegative errors in linkage (minimized by having a few large blocks) and the\nspeed of linkage (minimized by keeping each block small). We summarize the\nwhole procedure in Algorithm \\ref{subsec:tlsh} (see Appendix \\ref{sec:app}).\n\nTLSH involves many tuning parameters (the length of shingles, the number\nof random permutations, the maximum size of communities, etc.)\nWe chose the shingle such that we have \nthe highest recall possible for each application. We used a random permutation of 100, since the recall was approximately constant for all permutations higher than 100. \nFurthermore, we chose a maximum size of the communities of 500, after tuning this specifically for desired speed.\n\n\n\\subsubsection{K-Means Locality Sensitive Hashing (KLSH)}\n\\label{subsec:klsh}\n\nThe second LSH-based blocking method begins, like TLSH,\nby shingling the records, treated as strings, but then differs in several ways.\nFirst, we do not ignore the number of times each shingle type appears in a\nrecord, but rather keep track of these counts, leading to a bag-of-shingles\nrepresentation for records. Second, we measure similarity between\nrecords using the inner product of bag-of-shingles vectors, with\ninverse-document-frequency (IDF) weighting. Third, we reduce the\ndimensionality of the bag-of-shingles vectors by random projections, followed\nby clustering the low-dimensional projected vectors with the $k$-means\nalgorithm. \nHence, we can control the mean\nnumber of records per cluster to be $n\/c$, where $c$ is the number of block-clusters. In practice, there is a fairly small\ndispersion around this mean, leading to blocks that, by construction, have the roughly the same distribution for all applications.\\footnote{This property is not guaranteed for most LSH methods.} The KLSH algorithm is given in Appendix \\ref{sec:app}.\n\n\n\n\\section{Computational Complexity}\n\\label{sec:complex}\n\n\n\\subsection{Computational Complexity of TLSH}\n\nThe first steps of the algorithm can be done independently across records.\nShingling a single record is $O(1),$ so shingling all the records\nis $O(n)$. Similarly, applying one minhash function to the shingles of one\nrecord is $O(1),$ and there are $p$ minhash functions, so minhashing\ntakes $O(np)$ time. Hashing again, with $b$ bands, takes $O(nb)$ time. We\n assume that $p$ and $b$ are both $O(1)$ as $n$ grows.\n\n\nWe create an edge between every pair of records that get mapped to the same\nvalue by the hash function in some band. Rather than iterating over pairs of\nrecords, it is faster to iterate over values $v$ in the range of the\nhash function. If there are $|v|$ records mapped to the value $v$, creating\ntheir edges takes $O(|v|^2)$ time. On average, $|v| = n V^{-1}$, where $V$ is the\nnumber of points in the range of the hash function, so creating the edge list\ntakes $O(V (n\/V)^2 ) = O(n^2 V^{-1})$ time. \\cite{clauset_2004} shows that creating\nthe communities from the graph is $O(n (\\log{n})^2)$.\n\n\n\nThe total complexity of TLSH is $O(n) + O(np) + O(nb) + O(n^2 V^{-1}) +\nO(n(\\log{n})^2) = O(n^2 V^{-1})$, and is dominated by actually building the graph.\n\n\n\\subsection{Computational Complexity of KLSH}\n\nAs with TLSH, the shingling phase of KLSH takes $O(n)$ time. The time required\nfor the random projections, however, is more complicated. Let $w(n)$ be the\nnumber of distinct words found across the $n$ records. The time needed to do\none random projection of one record is then $O(w(n))$, and the time for the\nwhole random projection phase is $O(npw(n))$. For $k$-means cluster, with a\nconstant number of iterations $I$, the time required to form $b$ clusters of\n$n$ $p$-dimensional vectors is $O(bnpI)$. Hence, the complexity is $O(npw(n))\n+ O(bnpI)$.\n\n\nHeaps's law suggests $w(n) = O(n^\\beta)$, where $0 < \\beta\n< 1$.\\footnote{For English text, $0.4 < \\beta < 0.6$.} Thus, the complexity\nis $O(p n^{1+\\beta}) + O(bnpI)$. \nFor record linkage to run in linear time, it must\nrun in constant time in each block. Thus, the number of records per block must be\nconstant, i.e., $b = O(n)$. Hence, the time-complexity for blocking\nis $O(p n^{1+\\beta}) + O(n^2 pI) = O(n^2 pI),$ a quadratic time algorithm\ndominated by the clustering. Letting $b = O(1)$ yields an over-all time\ncomplexity of $O(p n^{1+\\beta})$, dominated by the projection step. If we\nassume $\\beta = 0.5$ and let $b = O(\\sqrt{n}),$ then both the projection and\nthe clustering steps are $O(pn^{1.5})$. Record linkage in each block is $O(n),$ so record linkage is $O(n^{1.5}),$ \nrather than $O(n^2)$ without blocking.\n\n\\subsection{Computational Complexity of Traditional Blocking Approaches}\nTraditional blocking approaches use attributes of the records to partition records into blocks. As such, calculating the blocks using traditional approaches requires $O(n)$ computations. For example, approaches that block on birth year only require a partition of the records based on these fields. That is, each record is simply mapped to one of the unique birth year values in the dataset, which is an $O(n)$ calculation for a list of size $n$. Some traditional approaches, however, require $O(n^2)$ computations. For example, in Table \\ref{t:naive_results}, we show some effective blocking strategies which require $O(n^2)$ computations, but each operation is so cheap that they can be run in reasonable time for moderately sized files.\n\n\\section{Results}\n\\label{sec:results}\nWe test the previously mentioned approaches on data from the RecordLinkage R package.\\footnote{\\url{http:\/\/www.inside-r.org\/packages\/cran\/RecordLinkage\/docs\/RLdata}}\nThese simulated datasets contain 500 and 10,000 records (denoted \\texttt{RLdata500} and \\texttt{RLdata10000}), with exactly 10\\% duplicates in each list. These datasets contain first and last Germanic name and full date of birth (DOB). Each duplicate contains one error with respect to the original record, and there is maximum of one duplicate per original record. Each record has a unique identifier, allowing us to test the performance of the blocking methods. \n\nWe explore the performance of the previously presented methods under other scenarios of measurement error. \\citep{Christen05, ChristenPudjijono09, ChristenVatsalan13} developed a data generation and corruption tool that creates synthetic datasets containing various field attributes. This tool includes dependencies between fields and permits the generation of different types of errors. We now describe the characteristics of the datafiles used in the simulation. We consider three files having the following field attributes: first and last name, gender, postal code, city, telephone number, credit card number, and age. For each database, we allow either 10, 30, or 50\\% duplicates per file, and each duplicate has five errors with respect to the original record, where these five errors are allocated at random among the fields. Each original record has maximum of five duplicates. We refer to these files as the ``noisy'' files.\n\n\n\\subsection{Traditional Blocking Approaches}\\label{ss:naive_results}\nTables \\ref{t:naive_results} -- \\ref{t:naive_results2} provide results of traditional blocking when applied to the \\texttt{RLdata10000} and ``noisy\" files. While field-specific information \\emph{can} yield favorable blocking solutions, each blocking criteria is application specific. The overall goal of blocking is to reduce the overall set of candidate pairs, while minimizing the false negatives induced. Thus, we find the \\emph{recall} and \\emph{reduction ratio} (RR). This corresponds to the proportion of true matches that the blocking criteria preserves, and the proportion of record-pairs discarded by the blocking, respectively. \n\nCriteria 1 -- 5 (Table \\ref{t:naive_results}) and 1 -- 6 (Table \\ref{t:naive_results2}) show that \\emph{some} blocking approaches are poor, where the recall is never above 90\\%. Criteria requiring exact agreement in a single field or on a combination of them are susceptible to field errors. More reliable criteria are constructed using combinations of fields such that multiple disagreements must be met for a pair to be declared as a non-match. (See Criteria 7--10 and 12 in Table \\ref{t:naive_results}, and 7 -- 8 in Table \\ref{t:naive_results2}.) We obtain high recall and RR using these, but in general their performance is context-dependent. \n\nCriteria 10 (Table \\ref{t:naive_results}) deals with the case when a pair is declared a non-match whenever it disagrees in four or more fields, which is reliable since false-negative pairs are only induced when the datafile contains large amounts of error. For example this criterion does not lead to good results with the noisy files, hence a stronger criteria is needed, such as 7 (Table \\ref{t:naive_results2}). Using Criteria 12 (Table \\ref{t:naive_results}) and 8 (Table \\ref{t:naive_results2}), we further reduce the set of candidate pairs whenever a pair has a strong disagreement in an important field.\\footnote{We use the Levenshtein distance (LD) of first and last names for pairs passing Criterion 10 of Table \\ref{t:naive_results} or Criteria 7 of Table \\ref{t:naive_results2}, and declare pairs as non-matches when LD $\\geq 4$ in either first or last name.} These criteria are robust. In order to induce false negatives, the error in the file must be much higher than expected. \n\n\n\n\n\n\n\n\\begin{table}[htdp]\n\\begin{center}\n\\begin{tabular}{rlrr}\n\t\t\t\t\\hline\\\\[-8pt]\n\t\t\t\t& Declare non-match if disagreement in: & Recall (\\%) & RR (\\%) \\\\\n\t\t\t\t\\hline\\\\[-8pt]\n\t\t\t\t1.&First OR last name & 39.20 & 99.98\\\\\n\t\t\t\t2.&Day OR month OR year of birth & 59.30 & 99.99\\\\\n\t\t\t\t3.&Year of birth & 84.20 & 98.75\\\\\n\t\t\t\t4.&Day of birth & 86.10 & 96.74\\\\\n\t\t\t\t5.&Month of birth & 88.40 & 91.70\\\\\n\t\t\t\t6.&Decade of birth & 93.20 & 87.76\\\\\n\t\t\t\t7.&First AND last name & 99.20 & 97.36 \\\\\n\t\t\t\t8.&\\{First AND last name\\} OR &&\\\\\n\t\t\t\t&\\{day AND month AND year of birth\\} & 99.20 & 99.67 \\\\\n\t\t\t\t9.&Day AND month AND year of birth & 100.00 & 87.61\\\\\n\t\t\t\t10.&More than three fields & 100.00 & 99.26 \\\\\n\t\t\t\t11.&Initial of first OR last name & 100.00 & 99.25\\\\\n\t\t\t\t12.&\\{More than three fields\\} OR &&\\\\\n\t\t\t\t& \\{Levenshtein dist. $\\geq 4$ in first OR last name\\}& 100.00 & 99.97\\\\\n\t\t\t\t\\hline\n\t\t\t\t\t\t\t\\end{tabular}\n\t\t\t\t\t\t\t\n\\end{center}\n \\caption{Criteria for declaring pairs as non-matches, where results correspond to the \\texttt{RLdata10000} datafile. }\n \\label{t:naive_results}\n\n\\end{table}%\n\n\n\n\n\n\\begin{table}[htdp]\n\\begin{center}\n\\begin{tabular}{rlrr}\n \\hline\\\\[-8pt]\n\t& Declare non-match if disagree in: & Recall (\\%) & RR (\\%) \\\\\n \\hline\\\\[-8pt]\n\t1.&Gender &\t 31.96\t&\t 53.39\\\\\n\t2.&City &\t 31.53\t&\t 77.25\\\\\n\t3.&Postal Code &\t 32.65\t&\t 94.20\\\\\n\t4.&First OR last name &\t 1.30\t&\t$>$99.99\\\\\n\t5.&Initial of first OR last name&\t 78.10\t&\t 99.52\\\\\n\t6.&First AND last name &\t 26.97\t&\t 99.02\\\\\n\t7.&All fields &\t 93.28\t&\t 40.63 \\\\\n\t8.&\\{All fields\\} OR \\{Levenshtein dist. &&\\\\\n\t& $\\geq 4$ in first OR last name\\} &\t 92.84\t&\t 99.92\\\\\n\t\\hline\n\\end{tabular}\n\\end{center}\n \\caption{Criteria for declaring pairs as non-matches, where results correspond to the noisy datafile with 10\\% duplicates. Similar results obtained for 30 and 50\\% duplicates. }\n \\label{t:naive_results2}\n\\end{table}%\n\n\n\n\n\\vspace*{-2em}\n\n\\subsection{Clustering Approaches}\n\\label{sec:cluster-results}\n\nOur implementations of \\cite{mccallum_2000}'s canopies approach and \\cite{vatsalan_2013}'s nearest neighbor approach perform poorly on the \\texttt{RLdata10000} and ``noisy\" datasets\\footnote{In our implementations, we use the TF-IDF matrix representation of the records and Euclidean distance to compare pairs of records in TNN and canopies. We tried several other distance measures, each of which gave similar results.}. Figure \\ref{TNN_and_canopies} gives results of these approaches for different threshold parameters ($t$ is the threshold parameter for sorted TNN) for the \\texttt{RLdata10000} dataset. For all thresholds, both TNN and canopies fail to achieve a balance of high recall and a high reduction ratio.\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{tnn-results-rldata10000-Euclidean.pdf} \\includegraphics[width=0.48\\textwidth]{canopies-results-rldata10000.pdf}\n\\caption{Performance of threshold nearest neighbors (left) and canopies (right) on the RLdata10000 datafile.}\n\\label{TNN_and_canopies}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{tnn-results-Euclidean.pdf} \\includegraphics[width=0.48\\textwidth]{canopies-results.pdf}\n\\caption{Performance of TNN (left) and canopies (right) on the ``noisy\" datafile (10\\% duplicates). The other ``noisy\" datafiles exhibited similar behavior as the figures above. }\n\\label{TNN_and_canopies_error}\n\\end{figure}\n\n\nTurning to the ``noisy\" dataset with 10\\% duplicates, we find that TNN fails to achieve a balance of high recall and high reduction ratio, regardless of the threshold $t$ that is used. Similarly, the canopies approach does not yield a balance of high recall while reducing the number of candidate pairs.\n\nClearly, both clustering approaches fail to achieve a balance of high recall and RR for any threshold parameters. The inefficacy of these approaches is likely due the limited number of field attributes (five fields) and the Euclidean distance metric used for these datasets. In particular, only three fields in the ``noisy\" dataset use textual information, which both of these approaches use to identify similar records. Limited field information can make it difficult for clustering approaches to group similar records together, since the resulting term frequency matrices will be very sparse. Thus, we investigate the behavior with the same number of duplicates, but vary the error rate and provide richer information at the field attribute level. Figure \\ref{TNN_and_canopies_error} illustrates that both methods do not have a good balance between recall and RR, which we investigated for various thresholds. As such, further analysis of these approaches on more information-rich datasets is required in order to make sound conclusions about their efficacy for blocking. (We note that the metrics used in TLSH and KLSH, which shingle the records, were chosen so as to not have such problems.) \n\n\n\n\\subsection{LSH Approaches}\n\\label{sec:data500}\nSince the performance of KLSH and TLSH depends on tuning parameters, we tune each application appropriately to these. We empirically measure the scalability of these methods, which are consistent with our derivations in \\S \\ref{sec:complex}.\n\nWe analyze the \\texttt{RLdata10000} database for TLSH and KLSH. As we increase $k$ under TLSH, we see that the recall peaks at $k=5,$ and does very poorly (below 40\\% recall) when $k\\leq 4$. For KLSH, the highest and most consistent recall is when $k=2,$ since it is always above 80\\% and it is about the same no matter the total number of blocks chosen (see Figure \\ref{distort_10000}). In terms of RR, we see that TLSH performs extremely poorly as the total number of blocks increases, whereas KLSH performs extremely well in terms of RR comparatively (Figure \\ref{reduction}). Figure \\ref{comp_time} shows empirically that the running time for both KLSH and TLSH scales quadratically with the $n$, matching our asymptotic derivation. We then analyze the ``noisy\" database for TLSH and KLSH (see Figures \\ref{reduction_new} and \\ref{klsh_recall_new}).\n\n\n\\subsubsection{Comparisons of Methods}\nIn terms of comparing to the methods presented in Table \\ref{t:naive_results}, we find that TLSH is not comparable in terms of recall or RR. However, KLSH easily beats Criteria 1--2 and competes with Criteria 3--4 on both recall and RR. It does not perform as well in terms of recall as the rest of the criteria, however, it \\emph{may} in other applications with more complex information for each record (this is a subject of future work). When comparing the Table \\ref{t:naive_results2} to TLSH and KLSH when run for the noisy datafile, we find that TLSH and KLSH usually do better when tuned properly, however not always. Due to the way these files have been constructed, more investigation need to be done in terms of how naive methods work for real work type applications versus LSH-based methods. \n\nComparing to other blocking methods, both KLSH and TLSH outperform KNN in terms of recall (and RR for the noisy datafiles). We find that for this dataset, canopies do not perform well in terms of recall or RR unless a specific threshold $t_1$ is chosen. However, given this choice of $t_1$, this approach yields either high recall and low RR or vice versa, making canopies undesirable according to our criteria. \n\nFor the \\texttt{RLdata10000} dataset, the simple yet effective traditional blocking methods and KLSH perform best in terms of balancing both high recall and high RR. As already stated, we expect the performance of these to be heavily application-dependent. Additionally, note that each method relies on high-quality labeled record linkage data to measure the recall and RR and the clustering methods require tuning parameters, which can be quite sensitive. Our studies show that TLSH is the least sensitive in general and further explorations should be done here. Future work should explore the characteristics of the underlying datasets for which one method would be preferred over another.\n\n\n\n\n\\subsubsection{Sensitivity Analysis on \\texttt{RLdata500} and \\texttt{RLdata10000}}\nA sensitivity analysis is given for KLSH and TLSH. \nFor TLSH, the \\texttt{RLdata500} dataset is not very sensitive to $b$ since the recall is always above 80\\% whereas the \\texttt{RLdata10000} dataset is quite sensitive to the band, and we recommend the use of a band of 21--22 since the recall for these $b$ is $\\approx$ 96\\%, although this may change for other datasets. We then evaluate TLSH using the ``best'' choice of the band for shingled values from $k=1,\\ldots 5. $ The sensitivity analysis for the ``noisy\" datafiles was quite similar to that described above, where a band of 22 was deemed the most appropriate for TLSH. For KLSH, we found that we needed to increase the number of permutations slightly to improve the recall and recommend $p=150.$\n\n\nFor KLSH, we find that when the number of random permutations $p$ is above 100, the recall does not change considerably. We refer back to Figure \\ref{distort_10000} (right), which illustrates the recall versus number of blocks when $p = 100.$ When $k=4,$ the recall is always above 70\\%. However, we find that when $k=2,$ the recall is always above 80\\%.\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:disc} \n\nWe have explored two LSH methods for blocking, one of which would naturally fit into the privacy preserving record linkage (PPRL) framework, since the method could be made to be private by creating reference values for each individual in the database. This has been done for many blocking methods in the context of PPRL \\citep{durham_2012, vatsalan_2011,karakasidis_2012, kuzu_2011}. KLSH performs just as well or better than commonly used blocking methods, such as some simple traditional blocking methods, nearest neighbor clustering approaches, and canopies \\citep{vatsalan_2013, mccallum_2000}. One drawback is that like LSH-based methods, it must be tuned for each application since it is sensitive to the tuning parameters. Thus, some \\emph{reliable} training data must be available to evaluate the recall and RR (and tune KLSH or clustering type methods). In many situations, a researcher may be better off by using domain-specific knowledge to reduce the set of comparisons, as shown in \\S \\ref{ss:naive_results}.\n\nLSH-methods have been described elsewhere as ``private blocking\" due to the hashing step. However, they do not in fact provide any formal privacy guarantees in our setting. The new variant that we have introduced, KLSH, does satisfy the $k$-anonymity criterion for the de-duplication of a single file. However, the data remain subject to intruder attacks, as the literature on differential privacy makes clear, and the vulnerability is greater the smaller the value of $k$. Our broader goal, however, is to merge and analyze data from multiple files. Privacy protection in that context is far more complicated. Even if one could provide privacy guarantees for each file separately, it would still be possible to identify specific entities or \\emph{sensitive} information regarding entities in the merged database. \n\nThe approach of PPRL reviewed in \\cite{hall12} sets out to deal with this problem. Merging data from multiple files with the same or similar values without releasing their attributes is what PPRL hopes to achieve. Indeed, one of course needs to go further, since performing statistical analyses on the merged database is the real objective of PPRL. Whether the new ``private blocking\" approaches discussed offer any progress on this problem, it is unclear at best. Adequately addressing the PPRL goals remains elusive, as do formal privacy guarantees, be they from differential privacy or other methods. \n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{pics\/complexity_quadratic.pdf}\n\\includegraphics[width=0.45\\textwidth]{pics\/RLdata_bands_tlsh}\n\\caption{\\text{RLdata10000\/RLdata500} datasets. Left: Square Root Elapsed time versus number of records for KLSH and TLSH, illustrating that both methods scale nearly quadratically (matching the computationally complexity findings). We shingle using $k=5$ for both methods. We use a band of 26 for TLSH. Right: Recall versus $b$ for both \\texttt{RLdata500} and \\texttt{RLdata10000} after running TLSH.}\n\\label{comp_time}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{pics\/newpics\/recall_tlsh_rldata10000_5_4_14_k28}\n\\includegraphics[width=0.45\\textwidth]{pics\/newpics\/tlsh_rldata10000_p100_k14.pdf}\n\\caption{\\text{RLdata10000} dataset. Left: Recall versus number of shingles $k$ for KLSH. The highest recall occurs at $k=5$. Right: Recall versus the total number of blocks, where we vary the number of shingles $k$. We find that the highest recall is for $k=2$.}\n\\label{distort_10000}\n\\end{figure}\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{pics\/RR_klsh.pdf}\n\\includegraphics[width=0.45\\textwidth]{pics\/RR_tlsh_new.pdf}\n\\caption{\\text{RLdata10000} dataset. Left: For TLSH, we see the RR versus the number of shingles, where the RR is always very high. We emphasize that TLSH does about as well on the RR as any of the other methods, and certainly does much better than many traditional blocking methods and KNN. (The RR is always above $98\\%$ for all shingles with $b=26$.)\nRight: For KLSH, we illustrate the RR versus the total number of blocks for various $k=1,\\ldots,4$ illustrating that as the number of blocks increases, the RR increases dramatically. When the total block size is at least 25, the RR $\\geq 95\\%.$}\n\\label{reduction}\n\\end{figure}\n\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{pics\/tsh_b22_10000_et_newdata}\n\\includegraphics[width=0.45\\textwidth]{pics\/tsh_b22_et_newdata_2.pdf}\n\\caption{ Left: We run TLSH for 10 percent duplicates, as before, the application is quite sensitive to $b,k.$ Hence, it is quite easy to find values of $b,k$ such that the recall is very low or if tuning is done properly, we can find values of $b,k$ where the recall is acceptable. We note this relies on very good ground truth. The only value of $k$ we recommend is 4 since it is close to 90\\% recall. The computational time is the same as previously. Right: Elapsed time for 10, 30, and 50 percent duplicates on ``noisy\" dataset. }\n\\label{reduction_new}\n\\end{figure}\n\n\n\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{pics\/klsh_newdata_p20_10percent}\n\\includegraphics[width=0.48\\textwidth]{pics\/klsh_p150_10percent}\n\\caption{ We run KLSH at 10 percent duplicates with p=100 (left) and p=150 (right). We see as the number of permutations increases (left figure), the recall increases. The behavior is the same for 30 and 50 percent duplicates. This indicates that KLSH needs to be tuned for each application based on $p.$ \n}\n\\label{klsh_recall_new}\n\\end{figure}\n\n\n\n\n\\section*{Acknowledgements}\nWe thank Peter Christen, Patrick Ball, and Cosma Shalizi for thoughtful conversations that led to to early versions of this manuscript. We also thank the reviewers for their suggestions and comments.\n\n\\clearpage\n\\newpage\n\n\\bibliographystyle{ims}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nPoint processes on the line, generated by transitions of Continuous Time Markov Chains (CTMCs) have been studied intensely by the applied probability community over the past few decades under the umbrella of Matrix Analytic Methods (MAM), see e.g. \\cite{latouche1999introduction}. These have been applied to teletraffic \\cite{akar1998matrix}, business networks \\cite{herbertsson2007pricing}, social operations research \\cite{xing2013operations}, and biological systems \\cite{olsson2015equilibrium}. The typical model referred to as the Markovian Arrival Process (MAP) is comprised of a finite state irreducible CTMC which generates events at selected instances of state change and\/or according to Poisson processes modulated by the CTMC. MAPs have been shown to be dense in the class of point processes so that they can essentially approximate any point process, \\cite{asmussen1993marked}. Yet at the same time, they are analytically tractable and may often be incorporated effectively within more complex stochastic models \\cite{neuts1979versatile}. \n\nIn general, treating point processes as {\\em stationary} often yields a useful mathematical perspective which matches scenarios when there is no known dependence on time. In describing a point process we use $N(t)$ to denote the number of events during $[0,t]$ and further use the sequence $\\{T_n\\}$ to denote the sequence of inter-event times. Two notions of stationarity are useful in this respect. Roughly, a point process is {\\em time-stationary} if the distribution of the number of events within a given interval does not depend on the location of the interval; that is if $N(t_1+s)-N(t_1)$ is distributed as $N(t_2+s)-N(t_2)$ for any non-negative $t_1, t_2$ and $s$. A point process is {\\em event-stationary} if the joint distribution of $T_{k_1},\\ldots,T_{k_n}$ is the same as that of $T_{k_1+\\ell},\\ldots,T_{k_n+\\ell}$ for any integer sequence of indices $k_1, \\ldots,k_n$ and any integer shift $\\ell$. For a given model of a point process, one may often consider either the event-stationary or the time-stationary case. The probability laws of both cases agree in the case of the Poisson process. However, this is not true in general. \nFor MAPs, time-stationarity and event-stationarity are easily characterized by the initial distribution of the background CTMC. Starting it at its stationary distribution yields time-stationarity and starting at the stationary distribution of the embedded Markov chain (jump chain) yields event-stationarity.\n\nA common way to parameterize MAPs is by considering the generator, $Q$, of an irreducible finite state CTMC and setting $Q= C + D$. Roughly speaking, the matrix $C$ determines state transitions without event counts and the matrix $D$ determines event counts. Such parameterization hints at considering two special cases: Markov Modulated Poisson Processes (MMPP) arising from a diagonal matrix $D$, and Markovian Switched Poisson Processes (MSPP) arising from a diagonal matrix $C$. \n\nMMPPs are a widely used class of processes in modelling and are a typical example of a Cox process, also known as a doubly stochastic Poisson process, \\cite{grandell2006doubly} and \\cite{tang2009markov}. For a detailed outline of a variety of classic MMPP results, see~\\cite{fischer1993markov} and references therein. MSPPs were introduced in \\cite{dan1991counter} and to date, have not been as popular for modeling. However, the duality of diagonal $D$ vs. diagonal $C$ motivates us to consider and contrast both these processes. We also note that hyper-exponential renewal processes are special cases of MSPPs as well as Markovian Transition Counting Processes (as introduced in \\cite{asanjarani2016queueing}).\n\nOur focus in this paper is on second order properties of MMPPs and MSPPs and related traits. Consider the squared coefficient of variation and the limiting index of dispersion of counts given by,\n\\begin{equation}\n\\label{eq:3535}\nc^2 = \\frac{\\mathrm{Var}(T_1^{\\boldsymbol{\\alpha}})}{\\mathbb{E}^2\\,[ T_1^{\\boldsymbol{\\alpha}}]},\n\\qquad\n\\mbox{and}\n\\qquad\nd^2 = \\lim_{t \\to \\infty} \\frac{\\mathrm{Var}(N(t))}{\\mathbb{E}[N(t)]},\n\\end{equation} \nwhere $T_1^{\\boldsymbol{\\alpha}}$ is the time of the first event, taken from the event stationary version. Modelling folklore of MMPP sometimes assumes that $c^2 \\ge 1$. This is perhaps due to the fact that $d^2 \\ge 1$ is straightforward to verify and the similarity between these measures (for example for a renewal process, $c^2 = d^2$). However, as we highlight in this paper, establishing such ``burstiness'' properties is not straightforward.\n\nA related property is having $T_1^\\alpha$ exhibit Decreasing Hazard Rate (DHR), where for a random variable with PDF $f(t)$ and CDF $F(t)$ the hazard rate is,\n\\[\nh(t)=\\frac{f(t)}{1-F(t)}.\n\\]\nA further related property is the stochastic order, $T_1^\\pi \\ge_{\\mbox{st}} T_1^\\alpha$ where $T_1^\\pi$ is the first event time in the time-stationary version. We denote the properties as follows:\n\\begin{description}\n\\item (I) $d^2 \\ge 1$.\n\\item (II) $T_1^{\\boldsymbol{\\alpha}}$ exhibits DHR. \n\\item (III) $c^2 \\ge 1$.\n\\item (IV) The stochastic order $T_1^{\\boldsymbol{\\pi}} \\ge_{\\mbox{st}} T_1^{\\boldsymbol{\\alpha}}$.\n\\end{description} \n\nAll these properties are related and in this paper we highlight relationships between (I), (II), (III) and (IV) and establish the following: For MSPPs and MMPPs of order $2$ we show that (I)--(IV) holds. For general MMPPs it is known that (I) holds however, a counter-example of Miklos Telek and Illes Horvath shows that (II) does not hold and we conjecture (and numerically test) that (III) and (IV) holds.\n\nOur interest in this class of problems stemmed from relationships between different types of MAPs as in \\cite{nazarathy2008asymptotic} and \\cite{asanjarani2016queueing}. Once it became evident that $c^2 \\ge 1$ for MMPPs is an open problem even though it is acknowledged as a modeling fact in folklore, we searched for alternative proof avenues. This led to the stochastic order in (IV) as well as to considering DHR properties (the latter via communication with Miklos Telek and Illes Horvaths).\n\nThe remainder of the paper is structured as follows. In Section~\\ref{sec2} we present preliminaries, focusing on the relationships between properties (I) -- (IV) as well as defining MMPPs and MSPPs. In Section~\\ref{sec3} we present our main results and the conjecture. We close in Section~\\ref{sec4}.\n\n\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\n\\label{sec2}\n\nConsider first properties (I)--(IV) and their relationships. With an aim of establishing property (III), $c^2 \\ge 1$, there are several possible avenues based on properties (I), (II) and (IV). We now explain these relationships.\n\n\\vspace{5pt}\n\\noindent\n\\paragraph*{Using property (I):} First, from the theory of simple point processes on the line, note the relationship between $d^2$ and $c^2$: \n\\begin{equation}\\label{eq:dcR}\nd^2 = c^2\\Big(1+2\\sum_{j=1}^\\infty \\frac{\\mathrm{Cov}(T_0^{\\boldsymbol{\\alpha}},T_j^{\\boldsymbol{\\alpha}})}{\\mathrm{Var}(T_0^{\\boldsymbol{\\alpha}})}\\Big).\n\\end{equation}\nHowever, the autocorrelation structure is typically intractable and hence does not yield results. If we were focusing on a renewal process where $T_i$ and $T_j$ are independent for $i \\neq j$ then this immediately shows that $d^2 = c^2$. Our focus is broader and hence property (I) indicating that $d^2 \\ge 1$ does not appear to be of use.\n\n\\vspace{5pt}\n\\noindent\n\\paragraph*{Using property (II):} An alternative way is to consider property (II) and use the fact that for any DHR random variable we have $c^2 \\ge 1$ (see \\cite{stoyan1983comparison}). Hence if property (II) holds then (III) holds.\n\n\n\\vspace{5pt}\n\\noindent\n\\paragraph*{Using property (IV):} We have the following Lemma, implying that (III) is a consequence of the stochastic order (IV).\n\n\n\n\\begin{lemma}\n\\label{lem:soscv}\nConsider a simple non-transient point process on the line, and let $T_1^{\\boldsymbol{\\pi}}$, $T_1^{\\boldsymbol{\\alpha}}$ represent the first inter-event time in the time-stationary case and event-stationary case respectively. Then $c^2 \\ge 1$ if and only if $\\mathbb{E}[T_1^{\\boldsymbol{\\pi}}] \\geq \\mathbb{E}[T_1^{\\boldsymbol{\\alpha}}]$.\n\\end{lemma}\n\\begin{proof}\nFrom point process theory (see for example, Eq. (3.4.17) of \\cite{daley2007introduction}), it holds $$\n\\mathbb{E}[T_1^{\\boldsymbol{\\pi}}]=\\frac{1}{2}\\lambda^* \\mathbb{E}\\big[\\big(T_1^{\\boldsymbol{\\alpha}}\\big)^2\\big],\n$$\nwhere,\n\\[\n\\lambda^* = \\lim_{t \\to \\infty} \\frac{E\\big[N[0,t]\\big]}{t} = \\frac{1}{\\mathbb{E}[T_1^{\\boldsymbol{\\alpha}}]}.\n\\]\nNow,\n\\[\nc^2 =\\frac{\\mathbb{E}[\\big(T_1^{\\boldsymbol{\\alpha}}\\big)^2]-\\big(\\mathbb{E}[T_1^{\\boldsymbol{\\alpha}}]\\big)^2}{\\big(\\mathbb{E}[T_1^{\\boldsymbol{\\alpha}}]\\big)^2} = 2 \\frac{\\mathbb{E}[T_1^{\\boldsymbol{\\pi}}]}{\\mathbb{E}[T_1^{\\boldsymbol{\\alpha}}]} - 1,\n\\]\nand we obtain the result.\n\\end{proof}\n\n\\paragraph*{MAPs:} We now describe Markovian Arrival Process (MAPs). \nA MAP of order $p$ (MAP$_p$) is generated by a two-dimensional Markov process $\\{(N(t), X(t)); t \\geq 0\\}$ on state space $\\{0,1, 2, \\cdots\\}\\times \\{1,2, \\cdots, p\\}$. The counting process $N(\\cdot)$ counts the number of ``events'' in $[0,t]$ with $\\mathbb P(N(0)=0)=1$. The phase process $X(\\cdot)$ is an irreducible CTMC with state space $\\{1, \\ldots, p\\}$, initial distribution $\\boldsymbol{\\eta}$ and generator matrix $Q$. A MAP is characterized by parameters $(\\boldsymbol{\\eta}, C,D)$, where the matrix $C$ has negative diagonal elements and non-negative off-diagonal elements and records the rates of phase transitions which are not associated with an event. The matrix $D$ has non-negative elements and describes the changes of the phase process with an event (increase of $N(t)$ by 1). Moreover, we have $Q=C+D$. More details are in \\cite{asmussen2003applied} (Chapter~XI) and \\cite{he2014fundamentals} (Chapter~2).\n \nMAPs are attractive due to the tractability of many of their properties, including distribution functions, generating functions, and moments of both $N(\\cdot)$ and the sequence of inter-event times $\\{T_n\\}$. \nSince $Q$ is assumed irreducible and finite, it has a unique stationary distribution $\\boldsymbol{\\pi}$ satisfying $\\boldsymbol{\\pi} Q = \\mathbf{0}'$, $\\boldsymbol{\\pi} \\mathbf{1} = 1$. Note that from $Q {\\mathbf 1} = \\mathbf {0}'$ we have $-C {\\mathbf 1}=D {\\mathbf 1}$.\n Of further interest is the embedded discrete-time Markov chain with irreducible stochastic matrix $P = (-C)^{-1}D$ and stationary distribution $\\boldsymbol{\\alpha}$, where $\\boldsymbol{\\alpha} P=\\boldsymbol{\\alpha}$ and $\\boldsymbol{\\alpha} \\mathbf{1}=1$. \n\nObserve the relation between the stationary distributions $\\boldsymbol{\\pi}$ and $\\boldsymbol{\\alpha}$:\n\\begin{equation}\\label{Eq:pi-alpha}\n\\boldsymbol{\\alpha}=\\frac{\\boldsymbol{\\pi} D}{\\boldsymbol{\\pi} D \\mathbf{1}} \\qquad \\text{and} \\qquad \\boldsymbol{\\pi}=\\frac{\\boldsymbol{\\alpha} (-C)^{-1}}{\\boldsymbol{\\alpha} (-C)^{-1}\\mathbf{1}}=\\lambda^*\\boldsymbol{\\alpha} (-C)^{-1},\n\\end{equation}\nwhere $\\lambda^* = \\boldsymbol{\\pi} D {\\mathbf 1} = - \\boldsymbol{\\pi} C {\\mathbf 1}$.\n\n\nThe following known proposition, as distilled from the literature (see for example \\cite{asmussen2003applied}, Chapter XI) provides the key results of MAPs that we use in this paper. It shows that $T_1$ is a Phase Type (PH) random variable with parameters $\\boldsymbol{\\eta}$ for the initial distribution of the phase and $C$ for the sub-generator matrix. It further shows that the initial distribution of the phase process may render the MAP as time stationary or event stationary.\n\n\\begin{proposition}\nConsider a MAP with parameters ($\\boldsymbol{\\eta}$,$C$,$D$), then\n\\begin{equation}\n\\label{eq:T1dist}\n\\mathbb P(T_1 > t) = \\boldsymbol{\\eta} e^{C t} {\\mathbf 1}.\n\\end{equation}\nFurther, if $\\boldsymbol{\\eta} = \\boldsymbol{\\pi}$ then the MAP is time-stationary and if $\\boldsymbol{\\eta}=\\boldsymbol{\\alpha}$ it is event stationary, where $\\boldsymbol{\\pi}$ and $\\boldsymbol{\\alpha}$ are associated stationary distributions.\n\\end{proposition}\n\nNote that for such a $PH(\\boldsymbol{\\eta}, C)$ random variable the density $f(t)$ and the hazard rate $h(t)$, are respectively,\n\\[\nf(t) = \\boldsymbol{\\eta} e^{C t} D {\\mathbf 1},\n\\qquad\nh(t) = \\frac{\\boldsymbol{\\eta}e^{Ct} D \\mathbf{1}}{\\boldsymbol{\\eta}e^{Ct} \\mathbf{1}}.\n\\]\nFurther, as may be used for showing DHR, the derivative of the hazard rate is,\n\\begin{equation}\n\\label{eq:derHazard}\nh^{\\prime}(t)= \\frac{\\boldsymbol{\\eta} C e^{Ct} (-C) \\mathbf{1} \\,\\,\n\\boldsymbol{\\eta}e^{Ct} \\mathbf{1} - \\boldsymbol{\\eta} C e^{Ct} \\mathbf{1}\\,\\, \\boldsymbol{\\eta}e^{Ct} (-C) \\mathbf{1}\n}{(\\boldsymbol{\\eta}e^{Ct} \\mathbf{1})^2}.\n\\end{equation}\n\nWe now describe second-order properties associated with each case. \n\\paragraph*{Event-Stationary Case:} \n The MAP is event-stationary\\footnote{Sometimes an event-stationary MAP is referred to as an interval-stationary MAP, see for instance \\cite{fischer1993markov}.} if \n \n$\\boldsymbol{\\eta}=\\boldsymbol{\\alpha}$. In this case, the (generic) inter-event time is phase-type distributed, $PH(\\boldsymbol{\\alpha}, C)$ and thus has $k$-th moment:\n$$\nM_k=\\mathbb{E}[T_n^k]=k! \\boldsymbol{\\alpha} (-C)^{-k}\\mathbf{1}\n=(-1)^{k+1} \\, k! \\, \\frac{1}{\\lambda^*} \\boldsymbol{\\pi} \\big(C^{-1}\\big)^{k-1} {\\mathbf 1},\n$$\nwith the first and second moments (here represented in terms of $\\boldsymbol{\\pi}$ and $C$):\n\\[\nM_1=\\frac{1}{\\lambda^*} \\boldsymbol{\\pi} {\\mathbf 1} = \\frac{1}{\\lambda^*},\n\\qquad\nM_2=2 \\frac{1}{\\lambda^*} \\boldsymbol{\\pi} (-C)^{-1} {\\mathbf 1}.\n\\]\nThe squared coefficient of variation (SCV) of events (intervals) has a simple formula: \n\\begin{equation}\\label{Eq:SCV-Interval}\nc^2+1 = \\frac{M_2}{M_1^2} = \n\\frac{-2 \\, (1\/\\lambda^*) \\, \\boldsymbol{\\pi} \\, C^{-1} {\\mathbf 1}}{(1\/\\lambda^*)^2}\n= 2 \\boldsymbol{\\pi} C {\\mathbf 1} \\boldsymbol{\\pi} C^{-1} {\\mathbf 1}.\n\\end{equation}\n\n\n\\paragraph*{Time-Stationary Case:} \n A MAP with parameters $(\\boldsymbol{\\eta}, C,D)$ is time-stationary \\index{time-stationary MAP} if $\\boldsymbol{\\eta}=\\boldsymbol{\\pi}$. In the time-stationary case ($\\boldsymbol{\\eta}=\\boldsymbol{\\pi}$), we have (see \\cite{asmussen2003applied}):\n\\begin{align}\n\\label{Eq:Mean}\n\\mathbb{E}[N(t)]&= {\\boldsymbol{\\pi}} D \\mathbf{1}\\,t,\\\\\n\\label{Eq:Var}\n\\mathrm{Var}\\big(N(t)\\big)&=\\{{\\boldsymbol{\\pi}}D \\mathbf{1}+2\\, {\\boldsymbol{\\pi}}D D_Q^{\\sharp} D \\mathbf{1}\\}\\,t- 2 {\\boldsymbol{\\pi}}D D_Q^{\\sharp} D_Q^{\\sharp}(t) D \\mathbf{1},\n\\end{align}\n where $D_Q^{\\sharp}$ is the \\textit{deviation matrix}\\index{deviation matrix} associated with $Q$ defined by the following formula.\n\\begin{equation} \n\\label{Eq:deviation}\nD_Q^{\\sharp}=\\lim_{t\\rightarrow\\infty} D^{\\sharp}_Q(t)=\\int_0^{\\infty}(e^{Qu}-\\mathbf{1}{\\boldsymbol{\\pi}})\\, du.\n\\end{equation}\n Note that in some sources, for instance \\cite{asmussen2003applied} and \\cite{narayana1992first}, the variance formula \\eqref{Eq:Var} is presented in terms of the matrix $Q^{-}:=(\\mathbf{1}{\\boldsymbol{\\pi}} - Q)^{-1}$. The relation between these two matrices is $Q^{-}=D_Q^{\\sharp} +\\mathbf{1}{\\boldsymbol{\\pi}}$, see \\cite{coolen2002deviation}.\n \nApplying \\eqref{Eq:Mean} and \\eqref{Eq:Var}, we can write $d^2$ in terms of a MAP parameters as:\n\\begin{equation}\\label{eq:d}\nd^2= 1+\n\\frac{2}{\\lambda^*}\\, {\\boldsymbol{\\pi}}D D_Q^{\\sharp} D \\mathbf{1}.\n\\end{equation}\n\n\\paragraph*{MMPP:} A MAP with a diagonal matrix $D$ is an MMPP.\nMMPPs correspond to doubly-stochastic Poisson processes (also known as Cox processes) where the modulating process is driven by a CTMC. \nMMPPs have been used extensively in stochastic modelling and analysis, see for example \\cite{fischer1993markov}.\nThe parameters of an MMPP$_p$ are $D= \\text{diag}(\\lambda_i)$, where $\\lambda_i \\geq 0$ for $i=1, \\ldots, p$, and $C=Q-D$. Here, $Q$ is the generator matrix of a CTMC. \nFor MMPPs, \\eqref{Eq:Mean} and \\eqref{Eq:Var} can be simplified by using the following relations:\n$$\n\\boldsymbol{\\pi} D \\mathbf{1}=\\sum_{i=1}^p {\\pi}_i \\lambda_i,\n\\qquad D \\mathbf{1}=\\boldsymbol{\\lambda}=(\\lambda_1, \\cdots, \\lambda_p)^\\prime,\n\\qquad \\boldsymbol{\\pi} D=({\\pi}_1 \\lambda_1, \\cdots, {\\pi}_p\\lambda_p)\\,.\n$$\n\n\\paragraph*{MSPP:} A MAP with a diagonal matrix $C$ is is an MSPP. For MSPP$_p$ events switch between $p$ Poisson processes with rates $\\lambda_1, \\cdots, \\lambda_p$, where each switch also incurs an event. Here as in MMPPs we denote the diagonal elements of $D$ via $\\lambda_1, \\cdots, \\lambda_p$. However, unlike MMPPs, (irreducible) MSPPs don't have a diagonal $D$. We also remark that the modulation in the MSPP is of a discrete nature and it occurs at certain event epochs of the counting process, whereas the modulation of the MMPP is performed at epochs without events. See \\cite{artalejo2010markovian} and \\cite{he2014fundamentals}.\n\nAs our research attempts have shown, analyzing MSPPs is considerably easier than MMPPs, because a diagonal $C$ is much easier to handle than a non-diagonal $C$ and in an (irreducible) MMPP, $C$ must be non-diagonal.\n\n\n\\paragraph*{Properties (I)-(IV) for MAPs:} Using the results above, for any irreducible MAP with matrices $C$ and $D$ we have that the main properties (I)-(IV) of this paper can be formulated as follows:\n\\begin{align}\n \\label{Eq:d2}(I)&\\qquad {\\boldsymbol{\\pi}}D D_Q^{\\sharp} D \\mathbf{1} \\ge 0, \\\\\n \\label{Eq:HDR}\n (II)&\\qquad \\boldsymbol{\\alpha} C e^{Ct} (-C) \\mathbf{1} \\,\\,\n\\boldsymbol{\\alpha}e^{Ct} \\mathbf{1} + (\\boldsymbol{\\alpha} C e^{Ct} \\mathbf{1})^2 \\leq 0 \\qquad \\forall t \\ge 0, \\\\\n \\label{Eq:c2} (III)&\\qquad \\boldsymbol{\\pi} C {\\mathbf 1} \\boldsymbol{\\pi} C^{-1} {\\mathbf 1} \\ge 1, \\\\\n \\label{Eq:SO}(IV)&\\qquad \\boldsymbol{\\pi} e^{C t} \\mathbf{1} \\ge \\boldsymbol{\\alpha} e^{C t} \\mathbf{1},\n\\qquad \\forall t \\ge 0. \n\\end{align} \n\n\n\n\\section{Main Results}\n\\label{sec3}\n\nWe now present results for MSPP and MMPP$_2$ for properties (I)-(IV) as presented in the introduction. Establishing property (I), $d^2 \\ge 1$ is not a difficult task for both MMPPs and MSPPs:\n\n\\begin{proposition}\n\\label{eq:pp22}\nMMPP and MSPP processes have $d^2 \\ge 1$.\n\\end{proposition}\n\n\\begin{proof}\nThis is a well-known result that for all doubly stochastic Poisson processes (Cox processes), $d^2 \\ge 1$. So, we have the proof for an MMPP, for instance see Chapter 6 of \\cite{kingman1993poisson}.\n\nFor an MSPP, using the fact that for a given MMPP, we have $d^2\\geq 1$, results in:\n\\begin{equation}\\label{eq:Ddiag}\n{\\boldsymbol{\\pi}}D D_Q^{\\sharp} D \\mathbf{1} \\geq 0,\\quad \\text{for any diagonal non-negative matrix $D$.}\n\\end{equation}\nOn the other hand, all MAPs satisfy $\n{\\boldsymbol{\\pi}}D D_Q^{\\sharp} D \\mathbf{1}={\\boldsymbol{\\pi}}(-C) D_Q^{\\sharp} (-C) \\mathbf{1}$. Since for an MSPP, $-C$ is a diagonal non-negative matrix, from \\eqref{eq:Ddiag} we have \\eqref{Eq:d2}.\n\\end{proof}\n\n\nIt isn't difficult to show that property (II), DHR holds for MSPP:\n\n\\begin{proposition}\n\\label{eq:pp22}\nFor an MSPP the hazard rate of the stationary inter-event time is non-increasing.\n\\end{proposition}\n\\begin{proof}\nDenote the diagonal matrix $C$ with $C=diag(-c_i)$ and the positive elements of the column vector $e^{Ct} \\mathbf{1}$ with $\\mathbf{u}$.\nSo, Eq.~\\eqref{Eq:HDR} can be written element-wise as:\n\\begin{equation}\\label{eq:dif2}\n-(\\sum _{i=1}^p\\alpha_i c_i^2 u_i)\\,(\\sum_{i=1}^p \\alpha_i u_i)+(\\sum_{i=1}^p \\alpha_i c_i u_i)^2.\n\\end{equation}\nDenoting $v_i=\\alpha_i u_i$ and assuming $p_i=\\frac{v_i}{\\sum_{i=1}^p v_i}$ results in:\n$$\n-(\\sum _{i=1}^p c_i^2 p_i)+(\\sum _{i=1}^p c_i p_i)^2.\n$$\nThe above expression can be viewed as the minus variance of a random variable that takes values $c_i$ with probability $p_i$. Therefore, we have \\eqref{Eq:HDR}.\n\n\\end{proof}\n\nHowever, somewhat surprisingly, MMPPs don't necessarily possess DHR. An exception is MMPP$_2$ as shown in Proposition~\\ref{prop:mmpp2}. However for higher order MMPPs DHR doesn't always hold. The gist of the following example was communicated to us by Milkos Telek and Illes Horvath. Set \n\\begin{equation}\n\\label{Example}\nQ =\\left(\n\\begin{array}{cccc}\n-1 & 1 & 0 & 0 \\\\ \n0 & -1 & 1 & 0\\\\\n0 & 0 & -1 & 1\\\\\n1& 0&0 & -1 \n \\end{array}\n\\right)\n\\qquad\n\\mbox{and}\n\\qquad\nD =\\left(\n\\begin{array}{cccc}\\displaystyle\n0.01 & 0 &\\, \\,0 & \\,\\,\\,\\,0 \\\\ \n0 & 0.01 &\\, \\,0 &\\, \\,\\,\\,0\\\\\n0 & 0 & \\,\\,1 & \\,\\,\\,\\,0\\\\\n0& 0&\\,\\,0 & \\,\\,\\,\\,1\n \\end{array}\n\\right).\n\\end{equation}\nAs shown in Figure~\\ref{Fig:example}, the hazard rate function for an MMPP with the above matrices is not monotone. Hence at least for general MMPPs, trying to show (III), $c^2\\ge 1$, via hazard rates is not a viable avenue.\n\\begin{figure}\n\\center\n\\includegraphics[scale=0.4]{TelekExample.eps}\n\\caption{{{\\small The hazard rate of the MMPP in \\eqref{Example} is not monotone.}}}\n\\label{Fig:example}\n\\end{figure}\n\n\n\n\n\nSince hazards rates don't appear to be a viable paths for establishing (III) for MMPPs, an alternative may be to consider the stochastic order (IV). Starting with MSPPs, we see that this property holds.\n\n\\begin{proposition}\n\\label{eq:msppSO}\nFor an MSPP $T_1^{\\boldsymbol{\\pi}} \\ge_{\\mbox{st}} T_1^{\\boldsymbol{\\alpha}}$.\n\\end{proposition}\n\n\\begin{proof}\n\nUsing \\eqref{Eq:SO}, the claim is,\n\\begin{equation}\n\\label{eq:351}\n(\\boldsymbol{\\pi} - \\boldsymbol{\\alpha})e^{C t} \\mathbf{1} \\ge 0,\n\\qquad \\forall t \\ge 0.\n\\end{equation}\nWithout loss of generality we assume that there is an order $01$ is straightforward. \n\nNow, $\\{\\lambda^*-c_i\\}_{i=1, \\cdots, p}$ is a non-increasing sequence and therefore in the sequence $\\{\\pi_i-\\alpha_i\\}=\\{\\frac{\\pi_i}{\\lambda^*}(\\lambda^*-c_i)\\}$ when an element $\\pi_k-\\alpha_k$ is negative, all the elements $\\pi_i-\\alpha_i$ for $i\\geq k$ are negative. \nMoreover, both $\\boldsymbol{\\pi}$ and $\\boldsymbol{\\alpha}$ are probability vectors, so $(\\boldsymbol{\\pi}-\\boldsymbol{\\alpha})\\mathbf{1}=\\sum_i(\\pi_i-\\alpha_i)=0$. Therefore,\n at least the first element in the sequence $\\{\\pi_i-\\alpha_i\\}=\\{\\frac{\\pi_i}{\\lambda^*}(\\lambda^*-c_i)\\}$ is positive. \nHence, there exists an index $1 < k \\leq p$ such that $\\pi_i-\\alpha_i$ for $i=1, \\cdots, k-1$ is non-negative and for $i=k,\\cdots, p$ is negative. Therefore, we have:\n\n\\begin{align*}\n ({\\pi} - {\\boldsymbol\\alpha}) e^{Ct} \\mathbf{1}\n &=\\underbrace{\\sum_{i=1}^{k-1}({\\pi_i}-{ \\alpha_i})e^{-c_i t}}_{\\text{non-negative}}+\\underbrace{\\sum_{i=k}^p({\\pi_i}-{ \\alpha_i})e^{-c_i t}}_{\\text{negative}}\\\\\n& =\\underbrace{\\sum_{i=1}^{k-1}({\\pi_i}-{ \\alpha_i})e^{-c_i t}}_{\\text{non-negative}}-\\underbrace{\\sum_{i=k}^p({\\alpha_i}-{\\pi_i})e^{-c_i t}}_{\\text{non-negative}}.\n \\end{align*}\n Assume: $({\\pi} - {\\alpha}) e^{Ct} \\mathbf{1} <0$ :\n \\begin{equation}\n \\label{Eq:neg}\n \\sum_{i=1}^{k-1}({\\pi_i}-{\\alpha_i})e^{-c_i t}\n< \n\\sum_{i=k}^p({\\alpha_i}-{\\pi_i})e^{-c_i t}.\n \\end{equation}\n Then, since $01$ and $d^2>1$, $h(t)$ is DHR and the stochastic order $T_1^{\\boldsymbol{\\pi}} \\ge_{\\mbox{st}} T_1^{\\boldsymbol{\\alpha}}$ holds.\n\\end{proposition}\n\\begin{proof}\nConsider an MMPP$_2$ with parameters \n\\[\nD=\\left(\\begin{array}{cc}\n\\lambda_1&0\\\\\n0& \\lambda_2\n\\end{array}\\right)\n\\qquad\n\\mbox{and}\n\\qquad \nC=\\left(\\begin{array}{cc}\n-\\sigma_1-\\lambda_1 & \\sigma_1\\\\\n\\sigma_2 & -\\sigma_2-\\lambda_2\n\\end{array}\\right).\n\\]\nThen, $\\boldsymbol{\\pi}=\\frac{1}{\\sigma_1+\\sigma_2}(\\sigma_2,\\, \\sigma_1)$. As in \\cite{heffes1986markov}, evaluation of the transient deviation matrix through (for e.g.) Laplace transform inversion yields:\n$$\n\\frac{\\mathrm{Var}(N(t))}{\\mathbb{E}[N(t)]}=1+\\frac{2\\sigma_1\\sigma_2(\\lambda_1-\\lambda_2)^2}{(\\sigma_1+\\sigma_2)^2(\\lambda_1\\sigma_2+\\lambda_2\\sigma_1)}-\\frac{2\\sigma_1\\sigma_2(\\lambda_1-\\lambda_2)^2}{(\\sigma_1+\\sigma_2)^3(\\lambda_1\\sigma_2+\\lambda_2\\sigma_1)t}(1-e^{-(\\sigma_1+\\sigma_2)t}).\n$$\nTherefore from~\\eqref{eq:3535}, we have\n$$\nd^2=1+ \\frac{2\\sigma_1\\sigma_2(\\lambda_1-\\lambda_2)^2}{(\\sigma_1+\\sigma_2)^2(\\lambda_1\\sigma_2+\\lambda_2\\sigma_1)}.\n$$ \n\nFurther, explicit computation yields,\n$$\nc^2 = 1 + \\frac{2\\sigma_1\\sigma_2(\\lambda_1-\\lambda_2)^2}{(\\sigma_1+\\sigma_2)^2(\\lambda_2\\sigma_1+\\lambda_1(\\lambda_2+\\sigma_2))}.\n$$\nThus it is evident that the MMPP$_2$ has $d^2 >1\\, , c^2 > 1$\nas long as $\\lambda_1 \\neq \\lambda_2$ and $d^2=c^2=1$ when $\\lambda_1 = \\lambda_2$. \n\nFor DHR and the stochastic order, first we note that for an MMPP$_2$ with the above parameters, $\\boldsymbol{\\alpha}=\\frac{1}{\\sigma_1 \\lambda_1+\\sigma_2 \\lambda_2}(\\sigma_1 \\lambda_1, \\sigma_2 \\lambda_2)$. \nBy setting $B= \\sigma_1 +\\sigma_2 + \\lambda_1+\\lambda_2$ and $A=\\sigma_2 \\lambda_1 +\\lambda_2(\\sigma_1+\\lambda_1)$, after some simplification, Eq.~\\eqref{Eq:HDR} is given by:\n$$\n- \\frac{A e^{-Bt}\\sigma_1 \\sigma_2(\\lambda_1-\\lambda_2)^2}{(\\sigma_2 \\lambda_1+\\sigma_1\\lambda_2)^2},\n$$\nwhich is strictly negative for $\\lambda_1 \\neq \\lambda_2$ and is zero for $\\lambda_1 = \\lambda_2$.\nFor the stochastic order, from Eq.~\\eqref{Eq:SO}, we have:\n$$\n(\\boldsymbol{\\pi} - \\boldsymbol{\\alpha})e^{C t} \\mathbf{1}= \\frac{e^{-\\frac{t}{2} \\big(B+\\sqrt{B^2-4A} \\big)\n}\\big(-1+e^{t\\sqrt{B^2-4A}}\\big)\\sigma_1 \\sigma_2 (\\lambda_1-\\lambda_2)^2 }{(\\sigma_1+\\sigma_2) (\\sigma_1 \\lambda_2+\\sigma_2 \\lambda_1) \\sqrt{B^2-4A}},\n$$\nwhich is strictly positive for $\\lambda_1 \\neq \\lambda_2$ and is zero for $\\lambda_1 = \\lambda_2$.\n\n\n\\vspace{30pt}\n\n\n\\end{proof}\n\n\n\\section{Conjectures for MMPP}\n\\label{sec4}\n\n\nWe embarked on this research due to the folklore assumption that for MMPP, $c^2 \\ge 1$ (III). Initially we believed that it is easy to verify, however to date there isn't a known proof for an arbitrary irreducible MMPP. Still, we conjecture that both (III) and (IV) hold for MMPPs:\n\n\\begin{conjecture}\n\\label{conj:1}\nFor an irreducible MMPP, $c^2~\\ge~1$.\n\\end{conjecture}\n\n\\begin{conjecture}\n\\label{conj:2}\nFor an irreducible MMPP, $T_1^{\\boldsymbol{\\pi}} \\ge_{\\mbox{st}} T_1^{\\boldsymbol{\\alpha}}$.\n\\end{conjecture}\n\nIn an attempt to disprove these conjectures or alternatively gain confidence in their validity, we carried out an extensive numerical experiment. Our experiment works by generating random instances of MMPPs . Each instance is generated by first generating a matrix $Q$ with uniform$(0,1)$ off-diagonal entries and diagonal entries that ensure row sums are $0$. We then generate a matrix $D$ with diagonal elements that are exponentially distributed with rate $1$. Such a $(Q,D)$ pair then implies ${\\boldsymbol{\\pi}}$ and ${\\boldsymbol{\\alpha}}$. For each such MMPP we calculate $\\boldsymbol{\\pi} C {\\mathbf 1} \\boldsymbol{\\pi} C^{-1} {\\mathbf 1} -1 $ as in \\eqref{Eq:c2} and $(\\boldsymbol{\\pi} - \\boldsymbol{\\alpha})e^{C t} \\mathbf{1}$ as in \\eqref{Eq:SO}, where we take $t \\in \\{0,0.2,0.4,\\ldots,9.8,10.0\\}$. We then ensure that both of these quantities are non-negative.\n\nWe repeated this experiment for $10^6$ random MMPP instances of orders $3,4,5$ and $6$. In all cases the calculated quantities were greater that $-10^{-15}$. Note that in certain cases, the quantity associated with (IV) was negative and lying in the range $(-10^{-15},-10^{-16}]$. We attribute this to numerical error stemming from the calculation of the matrix exponential $e^{Ct}$. We ran our experiments with the Julia programming language, V1.0. The calculation time was about 1.5 hours.\n\nThis provides some evidence for the validity of Conjectures 1 and 2, although it is clearly not a proof. Further, we note that it is possible that some extreme cases exist that are not likely to come up by uniformly and randomly generating entries of $Q$. For example the cyclic matrix $Q$ in \\eqref{Example}. For this we have also considered random cyclic $Q$ matrices with non-zero entries similar to \\eqref{Example}. We generated $10^6$ such (order $4$) examples and all agreed with (III) and (IV).\n\n\n\n\\section{Conclusion}\n\\label{sec5}\n\nWe have highlighted various related properties for point processes on the line and MAPs exhibiting diagonal matrices ($C$ or $D$) in particular. Showing that $c^2 \\ge 1$ for MMPP and establishing the stochastic order $T_1^{\\boldsymbol{\\pi}} \\ge_{\\mbox{st}} T_1^{\\boldsymbol{\\alpha}}$ remains an open problem. We have shown this for MMPPs of order $2$ and using a similar technique to our MSPP proof, we can also show it for MMPPs with symmetric $C$ matrices. However for general MMPPs this remains an open problem.\n\nWe note, that stepping outside of the matrix analytic paradigm and considering general Cox processes is also an option. In fact, since any Cox process can be approximated by an MMPP, we believe that versions of conjectures $1$ and $2$ also hold for Cox processes under suitable regularity conditions.\n\nThere is also a related branch of questions dealing with characterizing the Poisson process via $c^2 = 1$ and considering when an MMPP is Poisson. For example, for the general class of MAPs, the authors of \\cite{bean2000map} provide a condition for determining if a given MAP is Poisson. It is not hard to construct a MAP with $c^2 = 1$ that is not Poisson. But, we believe that all MMPPs with $c^2 = 1$ are Poisson. Yet, we don't have a proof. Further, we believe that for an MMPP, if $c^2 = 1$ then all $\\lambda_i$ are equal (the converse is trivially\ntrue). We don't have a proof of this either. Related questions also hold for the more general Cox processes.\n\nWe also note that the MSPP class of process that we considered generalized hyper-exponential renewal processes as well as a class of processes called Markovian Transition Counting Processes (MTCP) as in \\cite{asanjarani2016queueing}.\n\n\n\n\\section*{Acknowledgement}\nAzam Asanjarani's research is supported by the Australian Research Council Centre of Excellence for the Mathematical and Statistical Frontiers (ACEMS). Yoni Nazarathy is supported by Australian Research Council Grant DP180101602. We thank Soren Asmussen, Qi-Ming He, Illes Horvath, Peter Taylor and Miklos Telek for useful discussions and insights related to this problem.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzcdec b/data_all_eng_slimpj/shuffled/split2/finalzzcdec new file mode 100644 index 0000000000000000000000000000000000000000..e8fec0cff8e6d6955bd7b4233fb8a82f3c039bb1 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzcdec @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nAfter Buchberger initiated his celebrated algorithm in his remarkable PhD thesis \\cite{Buc65}, the theory of Gr\\\"obner\\ bases has been established as a standard tool in algebraic geometry and computer algebra, yielding algorithmic solutions to many significant problems in mathematics, science and engineering \\cite{BW98}.\nAs a result, there have been many excellent textbooks on the subject such as \\cite{AL94} \\cite{BW93} \\cite{CLO15} \\cite{KR00} \\cite{DL06} \\cite{GP08} \\cite{EH12} \\cite{GG13}.\n\nNonetheless the computational complexity of Gr\\\"obner\\ bases often demands an enormous amount of computing time and storage space even for problems of moderate sizes, which severely impedes its practicality and dissemination.\nA striking phenomenon is in the computation of Gr\\\"obner\\ bases over the rational field with respect to the \\lex\\ ordering, when the coefficients of the final basis elements swell to extremely complicated rational numerals even though the coefficients of the original ideal generators are quite moderate.\nExample \\ref{Expl:FullModularAlgo} in this paper should be impressive enough to illustrate such a phenomenon.\nAn even more dramatic phenomenon is the ``intermediate expression swell\" referring to a generation of a huge number of intermediate polynomials with much more gigantic coefficients and larger exponents than those of the final basis elements during the implementation of the classical algorithm.\nIn \\cite[P116]{CLO15} and \\cite[P616, 21.7]{GG13} there are some brief reviews on the complexity issues associated with the classical algorithm.\n\nThese challenges have stimulated decades of ardent endeavors in improving the efficiency of the classical algorithm.\nThe methodologies such as the normal selection strategies and signatures effectively diminish the number of intermediate polynomials spawned during the process of algorithmic implementations \\cite{Buc85} \\cite{GMN91} \\cite[P222, 5.5]{BW93} \\cite{Fau02} \\cite{EF17}.\nThe modular and $p$-adic techniques based on ``lucky primes\" and Hensel lifting have been adopted to control the rampant growth of the intermediate coefficients albeit being limited to numeral coefficients only \\cite{Ebe83} \\cite{Win87} \\cite{ST89} \\cite{Tra89} \\cite{Pau92} \\cite{Gra93} \\cite{Arn03}.\nThere are also Gr\\\"obner\\ basis conversion methods such as the FGLM algorithm \\cite{FGL93} and Gr\\\"obner\\ Walk \\cite{CKM97}, a detailed description of which can be found in \\cite[P49, \\S 3; P436, \\S 5]{CLO05} and \\cite{Stu95}.\nThe idea of these methods is to compute another Gr\\\"obner\\ basis with respect to a different but favorable monomial ordering before converting it to the desired Gr\\\"obner\\ basis.\nAlbeit with all these endeavors over the decades, the high-level complexity associated with the Gr\\\"obner\\ basis computations remains a conundrum.\n\nAnother train of thoughts over the past decades is to generalize Gr\\\"obner\\ bases from over fields to over rings.\nAmong the copious and disparate coefficient rings that we shall not enumerate here, the Gr\\\"obner\\ bases over principal ideal rings are pertinent to the new type of bases in this paper.\nThere is an excellent exposition on Gr\\\"obner\\ bases over rings and especially over PIDs in \\cite[Chapter 4]{AL94}.\nHowever the focal point of the exposition is on the strong Gr\\\"obner\\ bases that resemble the Gr\\\"obner\\ bases over fields and hence are still plagued with complexity problems.\n\nIn this paper we take a novel approach by defining a new type of bases over principal quotient rings instead of over numeral fields like Gr\\\"obner\\ bases.\nIt is a natural approach since for a zero-dimensional ideal, the final eliminant is always a univariate polynomial after eliminating all the other variables.\nWith the principal ideals generated by the eliminant factors serving as moduli, we obtain an elegant decomposition of the original ideal into pairwise relatively prime ideals.\nWe also use pseudo-divisions and multipliers to enhance computational efficiency.\nIn the exemplary computations in Section \\ref{Sec:Examples}, it is conspicuous that this new approach scales down both the high-level complexity and gigantic numeral coefficients of the Gr\\\"obner\\ bases over rational fields.\n\nIn practice the Wu's method \\cite{Wu83} is more commonly used than the Gr\\\"obner\\ basis method since it is based on pseudo-divisions and thus more efficient.\nHowever the pseudo-divisions adopted by Wu's method usually lose too much algebraic information of the original ideals.\nIn Section \\ref{Sec:DivisionAlgm} we recall some rudimentary facts on monomial orderings and then define pseudo-divisions over PIDs.\nThe multipliers for the pseudo-divisions in this paper are always univariate polynomials so as to avoid losing too much algebraic information of the original ideals.\nThe pseudo-divisions of this ilk also dispose of the solvability condition for the linear equations of leading coefficients imposed by the classical division algorithm over rings.\nPlease refer to Remark \\ref{Rmk:LinearEqs} for details.\n\nAlgorithm \\ref{Algo:PseudoEliminant} is one of the pivotal algorithms in the paper.\nIt computes the pseudo-eliminant and pseudo-basis as per the elimination ordering in Definition \\ref{Def:EliminationOrdering}.\nThe purpose of Corollary \\ref{Cor:CoprimePair} and Lemma \\ref{Lemma:TriangleIdentity} is to trim down the number of $S$-polynomials\nto be pseudo-reduced.\nThey are highly effective in this respect as illustrated by the exemplary computations in Section \\ref{Sec:Examples}.\n\nThe pseudo-eliminant might contain factors that are not the bona fide ones of the eliminant.\nThe discrimination among these factors for authenticity is based on a crucial methodology, i.e., the pseudo-eliminant should be compared with the multipliers of the pseudo-divisions instead of the leading coefficients of the basis elements.\nExample \\ref{Expl:MultipliersCount} shows that the multipliers of the pseudo-divisions are more reliable than the leading coefficients of the basis elements.\nThe multiplier methodology is incorporated into Theorem \\ref{Thm:CompatiblePart} establishing that the compatible part of the pseudo-eliminant constitutes a bona fide factor of the eliminant.\nThis is one of the primary conclusions of the paper.\nThe multiplier and its property in Lemma \\ref{Lemma:SyzygyTransform} generalize the syzygy theory over fields and PIDs for Gr\\\"obner\\ bases and is another substantiation of the multiplier methodology.\nPlease refer to Remark \\ref{Rmk:MultiplierSyzygy} for the comment.\nThe compatible and incompatible parts of the pseudo-eliminant are defined in Definition \\ref{Def:CompatibleDivisors} and computed via Algorithm \\ref{Algo:CompatiblePartPseudoEliminant}.\nIn particular, we obtain a squarefree decomposition of the incompatible part $\\Ip (\\pel)$ by Algorithm \\ref{Algo:CompatiblePartPseudoEliminant} via a univariate squarefree factorization of the pseudo-eliminant $\\pel$ by Algorithm \\ref{Algo:Squarefree}.\nWe avoid a complete univariate factorization of the pseudo-eliminant $\\pel$ due to the concerns on computational complexity.\n\nWe conduct a complete analysis of the incompatible part $\\Ip (\\pel)$ of the pseudo-eliminant $\\pel$ in Section \\ref{Sec:IncompatibleModular} based on modular algorithms with the composite divisors obtained in Algorithm \\ref{Algo:CompatiblePartPseudoEliminant} as the moduli.\nThe advantages are that we have one less variable than the classical algorithm and the composite divisors are usually small polynomial factors of the pseudo-eliminant $\\pel$.\nHowever the disadvantage is that the computations are over the principal quotient rings (PQR) that might contain zero divisors.\nAs a result, we redefine $S$-polynomials in Definition \\ref{Def:SpolynPQR} carefully in order to obviate the zero multipliers incurred by the least common multiple of leading coefficients.\nAlgorithm \\ref{Algo:ProperEliminant} is pivotal in procuring the proper eliminants and proper bases by proper divisions as in Theorem \\ref{Thm:ProperReduction}.\nWe prove rigorously in Theorem \\ref{Thm:IncompatiblePart} that the nontrivial proper eliminants obtained in Algorithm \\ref{Algo:ProperEliminant} are de facto the bone fide factors of the eliminant of the original ideal.\nThe meticulous arguments in this primary conclusion are to ensure that our arguments are legitimate within the algebra $\\rx$ that contains zero divisors.\nSimilar to Lemma \\ref{Lemma:SyzygyTransform}, we generalize the classical syzygy theory over fields and PIDs for Gr\\\"obner\\ bases to the one in Lemma \\ref{Lemma:nPQRSyzygy}.\nFurther, we also have Corollary \\ref{Cor:PQRCoprime} and Lemma \\ref{Lemma:TriangleIdentityPQR} to trim down the number of $S$-polynomials for proper reductions.\n\nWe render two equivalent characterizations on the pseudo-bases $\\pbs$ obtained in Algorithm \\ref{Algo:PseudoEliminant}.\nThe first characterization is the identity \\eqref{LeadTermMod} in terms of leading terms whereas the second one is Theorem \\ref{Thm:MemberChar} via \\Gd-reductions as defined in Theorem \\ref{Thm:GcdReduction}.\nWe have the same kind of characterizations in Theorem \\ref{Thm:MemberCharModulo} for the proper bases $\\prb$ and $\\Bn$ obtained in Algorithm \\ref{Algo:ProperEliminant}.\nThese bases as in \\eqref{NewBases} correspond to a decomposition of the original ideal in \\eqref{IdealDecomposition} whose modular version is in \\eqref{ChineseModuleTheorem}.\nWe can define a unique normal form of a polynomial in $\\rx$ with respect to the original ideal by the Chinese Remainder Theorem as in Lemma \\ref{Lemma:CRT} since the ideal decomposition in \\eqref{IdealDecomposition} is pairwise relatively prime.\nIn the remaining part of this section we define irredundant, minimal and reduced bases that possess different levels of uniqueness.\n\nIn Section \\ref{Sec:MinorImprovements} we make some further improvements on the algorithms by Principle \\ref{Principle:PseudoRedPrinciple}.\nThe highlight of Section \\ref{Sec:ComplexityComparison} is Lemma \\ref{Lemma:OldGbasis} in which we contrive a special scenario consisting of two basis elements, a detailed analysis of which reveals that the classical algorithm contains the Euclidean algorithm computing the greatest common divisor of the leading coefficients.\nMoreover, the results in \\eqref{CancelLeadTerm} and \\eqref{AlsoCancelLeadTerm} contain the B\\'ezout coefficients\\ $u$ and $v$ that might swell to an enormous size like in Example \\ref{Expl:BezoutCoeffs}.\nBy contrast the computation of our new type of $S$-polynomial as in \\eqref{NewSPolynComput} yields the above results in one step without the B\\'ezout coefficients.\nThis might help to unveil the mystery of intermediate coefficient swell as well as high-level complexity associated with the Gr\\\"obner\\ basis computations.\n\nWe make two exemplary computations in Section \\ref{Sec:Examples} with the second one being more sophisticated than the first one.\nIt contains a paradigmatic computation of proper eliminants and proper bases over principal quotient rings with zero divisors as in Algorithm \\ref{Algo:ProperEliminant}.\nWe provide a detailed explanation for each step of the computation to elucidate the ideas of this new type of bases.\n\nAs usual, we denote the sets of complex, real, rational, integral and natural numbers as $\\mathbb{C}$, $\\mathbb{R}$, $\\bQ$, $\\mathbb{Z}$ and $\\mathbb{N}$ respectively.\nIn this paper, we use the following notations for a ring $R$:\n$R^\\ast:=R\\setminus\\{0\\}$;\n$R^\\times$ denotes the set of units in $R$.\nWith $\\bm{x}=(\\Lst x1n)$ and $\\alpha=(\\Lst\\alpha 1n)$, we denote a \\emph{monomial} $x_1^{\\alpha_1}\\cdots x_n^{\\alpha_n}$ as $\\bm{x}^\\alpha$ and a \\emph{term} as $c\\bm{x}^\\alpha$ with the \\emph{coefficient} $c\\in R^\\ast$.\nWe also use the boldface $\\bm{x}$ to abbreviate the algebra $R[\\Lst x1n]$ over a ring $R$ as $R[\\bm{x}]$.\nThe notation $\\langle A\\rangle$ denotes an ideal generated by a nonempty subset $A\\subset R[\\bm{x}]$.\nFurther, $K$ usually denotes a perfect field that is not necessarily algebraically closed unless specified.\nIn most cases we treat the algebra $K[\\bm{x}]$ as $\\knx$ over the ring $R=\\kxn$ with the variables $\\tilde{\\bx}:=(\\Lst x2n)$.\n\n\\section{A Pseudo-division Algorithm over Principal Ideal Domains}\\label{Sec:DivisionAlgm}\n\nIn this article we adopt a pseudo-divison of polynomials over a principal ideal domain as in Theorem \\ref{Thm:PseudoReduction}.\nWe shall abbreviate a principal ideal domain as a PID and denote it as $R$ henceforth.\n\nLet $R$ be a PID and $R[\\bm{x}]$ a polynomial algebra $R[\\Lst x1n]$ over $R$.\nLet us denote the set of monomials in $\\bm{x}=(\\Lst x1n)$ as $\\bM:=\\{\\bm{x}^\\alpha\\colon\\alpha\\in{\\mathbb{N}^n}\\}$.\nA nonzero ideal $I\\subset R[\\bm{x}]$ is called a \\emph{monomial} ideal if $I$ is generated by monomials in $\\bM$.\nBy Hilbert Basis Theorem we can infer that every monomial ideal in $R[\\bm{x}]$ is finitely generated since a PID $R$ is Noetherian.\n\n\\begin{lemma}\\label{Lemma:MonomialIdealProperty}\nLet $R$ be a PID.\nConsider a monomial ideal $I=\\langle\\bm{x}^\\alpha\\colon\\alpha\\in E\\rangle$ in $R[\\bm{x}]$ with $E\\subset{\\mathbb{N}^n}\\setminus\\{\\bm{0}\\}$.\nWe have the following conclusions:\n\\begin{enumerate}[(i)]\n\\item\\label{item:MonomialIdealDivisionCond} A term $c\\bm{x}^\\beta\\in I$ for $c\\in R^\\ast$ if and only if there exists an $\\alpha\\in E$ such that $\\bm{x}^\\beta$ is divisible by $\\bm{x}^\\alpha$;\n\n\\item\\label{item:MonomialIdealCond} A polynomial $f\\in I$ if and only if every term of $f$ lies in $I$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nIt suffices to prove the necessity of the two conclusions.\nSuppose that $g=\\sum_{j=1}^s\\qr_j\\bm{x}^{\\alpha_j}$ with $g$ representing the term $c\\bm{x}^\\beta\\in I$ as in \\eqref{item:MonomialIdealDivisionCond}, or $f\\in I$ as in \\eqref{item:MonomialIdealCond}.\nHere $\\qr_j\\in R[\\bm{x}]$ and $\\alpha_j\\in E$ for $1\\le j\\le s$.\nWe expand each $\\qr_j$ into individual terms and compare those with the same multi-degrees on both sides of the equality.\nThe conclusion readily follows since every term on the right hand side of the equality is divisible by some $\\bm{x}^{\\alpha_j}$ with $\\alpha_j\\in E$.\n\\end{proof}\n\nA \\emph{total ordering} $\\succ$ on the monomial set $\\bM$ satisfies that for every pair $\\bm{x}^\\alpha,\\bm{x}^\\beta\\in\\bM$, exactly one of the following relations holds: $\\bm{x}^\\alpha\\succ\\bm{x}^\\beta$, $\\bm{x}^\\alpha=\\bm{x}^\\beta$, or $\\bm{x}^\\alpha\\prec\\bm{x}^\\beta$.\nMoreover, $\\bm{x}^\\alpha\\succeq\\bm{x}^\\beta$ means either $\\bm{x}^\\alpha\\succ\\bm{x}^\\beta$ or $\\bm{x}^\\alpha=\\bm{x}^\\beta$.\nA \\emph{well-ordering} $\\succ$ on $\\bM$ satisfies that every nonempty subset $A\\subset\\bM$ has a minimal element.\nThat is, there exists $\\bm{x}^\\alpha\\in A$ such that $\\bm{x}^\\beta\\succeq\\bm{x}^\\alpha$ for every $\\bm{x}^\\beta\\in A$.\nA well-ordered set is always a totally ordered set since every subset consisting of two elements has a minimal element.\n\nIt is evident that under a well-ordering $\\succ$ on $\\bM$, there is no infinite strictly decreasing sequence $\\bm{x}^{\\alpha_1}\\succ\\bm{x}^{\\alpha_2}\\succ\\bm{x}^{\\alpha_3}\\succ\\cdots$ in $\\bM$ (or every strictly decreasing sequence in $\\bM$ terminates).\nNonetheless we have a much easier description as follows under the Noetherian condition.\n\n\\begin{proposition}\nLet $\\succ$ be a total ordering on $\\bM$ such that $\\bm{x}^\\alpha\\cdot\\bm{x}^\\gamma\\succ\\bm{x}^\\beta\\cdot\\bm{x}^\\gamma$ when $\\bm{x}^\\alpha\\succ\\bm{x}^\\beta$ for all $\\bm{x}^\\alpha,\\bm{x}^\\beta,\\bm{x}^\\gamma\\in\\bM$.\nThen $\\succ$ is a well-ordering on $\\bM$ if and only if $\\bm{x}^\\gamma\\succeq 1$ for all $\\gamma\\in{\\mathbb{N}^n}$.\n\\end{proposition}\n\\begin{proof}\nSuppose that $\\succ$ is a well-ordering.\nThen $\\bM$ has the smallest element which we denot as $\\bm{x}^{\\beta_0}$.\nIf $1\\succ\\bm{x}^{\\beta_0}$, then $\\bm{x}^{\\beta_0}\\succ\\bm{x}^{2\\beta_0}$, contradicting the minimality of $\\bm{x}^{\\beta_0}$.\nHence follows the necessity of the conclusion.\n\nSuppose that $A\\subset\\bM$ is nonempty.\nTo prove the sufficiency, it suffices to prove that $A$ has a minimal element in terms of the ordering $\\succ$.\nLet $K$ be a nontrivial field and $\\langle A\\rangle:=\\langle\\{\\bm{x}^\\alpha\\colon\\alpha\\in A\\}\\rangle$ the monomial ideal generated by $A$ in $K[\\bm{x}]$.\nAs per Hilbert Basis Theorem, $\\langle A\\rangle$ has a finite basis as $\\langle A\\rangle=\\langle\\bm{x}^{\\alpha_1},\\bm{x}^{\\alpha_2},\\dotsc,\\bm{x}^{\\alpha_s}\\rangle$.\nSince $\\succ$ is a total ordering, we relabel the subscripts such that $\\bm{x}^{\\alpha_s}\\succ\\cdots\\succ\\bm{x}^{\\alpha_1}$.\nNow $\\bm{x}^{\\alpha_1}$ is the minimal element of $A$.\nIn fact, for every $\\bm{x}^\\alpha\\in\\langle A\\rangle$, according to Lemma \\ref{Lemma:MonomialIdealProperty}, $\\bm{x}^\\alpha$ is divisible by one of $\\{\\bm{x}^{\\alpha_j}\\}$ for $1\\le j\\le s$.\nAssume that $\\bm{x}^\\alpha$ is divisible by $\\bm{x}^{\\alpha_{j_0}}$, i.e., $\\bm{x}^\\alpha=\\bm{x}^{\\alpha_{j_0}}\\cdot\\bm{x}^\\gamma$ for some $\\gamma\\in{\\mathbb{N}^n}$.\nThen $\\bm{x}^\\gamma\\succeq 1$ indicates that $\\bm{x}^\\alpha\\succeq\\bm{x}^{\\alpha_{j_0}}\\succeq\\bm{x}^{\\alpha_1}$.\n\\end{proof}\n\n\\begin{definition}\\label{Def:MonomialOrdering}\nA \\emph{monomial ordering} on $\\bM$ is a well-ordering on $\\bM$ such that $\\bm{x}^\\alpha\\cdot\\bm{x}^\\gamma\\succ\\bm{x}^\\beta\\cdot\\bm{x}^\\gamma$ when $\\bm{x}^\\alpha\\succ\\bm{x}^\\beta$ for all $\\bm{x}^\\alpha,\\bm{x}^\\beta,\\bm{x}^\\gamma\\in\\bM$.\nIn particular, we have $\\bm{x}^\\gamma\\succeq 1$ for all $\\gamma\\in{\\mathbb{N}^n}$.\n\\end{definition}\n\n\\begin{notation}\\label{Notation:LeadingEntities}\nLet $R$ be a PID and $f=\\sum_\\alpha c_\\alpha\\bm{x}^\\alpha$ a polynomial in $R[\\bm{x}]$.\nLet $\\succ$ be a monomial ordering.\nWe denote the \\emph{support} of $f$ as $\\supp f:=\\{\\bm{x}^\\alpha\\in\\bM\\colon c_\\alpha\\ne 0\\}\\subset\\bM$.\nIn particular, we define $\\supp f:=\\{1\\}$ when $f\\in R^\\ast$ and $\\supp f:=\\emptyset$ when $f=0$.\n\nIf $f$ has a term $c_\\beta\\bm{x}^\\beta$ that satisfies $\\bm{x}^\\beta:=\\max_{\\succ}\\{\\bm{x}^\\alpha\\in\\supp f\\}$, then we use the following terminologies hereafter.\nThe \\emph{leading term} of $f$ is denoted as $\\ltc (f):=c_\\beta\\bm{x}^\\beta$;\nThe \\emph{leading monomial}\\footnote{It is also called the ``leading power product\" in the literature.\nHere we adopt the convention that is consistent with the terminology of ``\\emph{monomial} ideals\".} of $f$ is denoted as $\\lmc (f):=\\bm{x}^\\beta$;\nThe \\emph{leading coefficient} of $f$ is denoted as $\\lcc (f):=c_\\beta\\in R^\\ast$.\n\nLet $\\pb=\\{\\ibr_j\\colon 1\\le j\\le s\\}$ be a polynomial set in $R[\\bm{x}]\\setminus\\{0\\}$.\nWe denote the leading monomial set $\\{\\lmc (\\ibr_j)\\colon 1\\le j\\le s\\}$ as $\\lmc (\\pb)$.\nLet us also denote the monomial ideal generated by $\\lmc (\\pb)$ in $R[\\bm{x}]$ as $\\langle\\lmc (\\pb)\\rangle$.\n\nIn what follows we use $\\gcd (a,b)$ and $\\lcm (a,b)$ to denote the greatest common divisor and least common multiple of $a,b\\in R^\\ast$ respectively over a PID $R$.\n\\end{notation}\n\n\\begin{definition}[Term pseudo-reduction in \\text{$R[\\bm{x}]$} over a PID $R$]\\label{Def:TermReduction}\n\\hfill\n\nLet $R$ be a PID and $\\succ$ a monomial ordering on $\\bM$.\nFor $f\\in\\rd$ and $g\\in R[\\bm{x}]\\setminus\\{0\\}$, suppose that $f$ has a term $c_\\alpha\\bm{x}^\\alpha$ such that $\\bm{x}^\\alpha\\in\\supp f\\cap\\langle\\lmc (g)\\rangle$.\nThen we can make a \\emph{pseudo-reduction} of the term $c_\\alpha\\bm{x}^\\alpha$ of $f$ by $g$ as follows.\n\\begin{equation}\\label{TermReduction}\nh=\\iur f-\\frac{\\lmr\\bm{x}^\\alpha}{\\ltc (g)}g\n\\end{equation}\nwith the multipliers $\\lmr:=\\lcm (c_\\alpha,\\lcc (g))$ and $\\iur:=\\lmr\/c_\\alpha\\in R^\\ast$.\nWe call $h$ the \\emph{remainder} of the pseudo-reduction and $\\iur$ the \\emph{interim multiplier} on $f$ with respect to $g$.\n\\end{definition}\n\n\\begin{definition}[Pseudo-reduced polynomial]\\label{Def:PseudoReduced}\n\\hfill\n\nLet $R$ be a PID and $\\succ$ a monomial ordering on $\\bM$.\nA polynomial $\\dr\\in R[\\bm{x}]$ is \\emph{pseudo-reduced} with respect to a polynomial set $\\pb=\\{\\ibr_j\\colon 1\\le j\\le s\\}\\subset\\rd$ if $\\supp\\dr\\cap\\langle\\lmc (\\pb)\\rangle=\\emptyset$.\nIn particular, this includes the special case when $\\dr=0$ and hence $\\supp\\dr=\\emptyset$.\nWe also say that $\\dr$ is \\emph{pseudo-reducible} with respect to $\\pb$ if it is not pseudo-reduced with respect to $\\pb$, i.e., $\\supp\\dr\\cap\\langle\\lmc (\\pb)\\rangle\\ne\\emptyset$.\n\\end{definition}\n\n\\begin{theorem}[Pseudo-division in \\text{$R[\\bm{x}]$} over a PID $R$]\\label{Thm:PseudoReduction\n\\hfil\n\nLet $R$ be a PID and $\\succ$ a monomial ordering on $\\bM$.\nSuppose that $\\pb=\\{\\ibr_j\\colon 1\\le j\\le s\\}\\subset\\rd$ is a polynomial set.\nFor every $f\\in R[\\bm{x}]$, there exist a multiplier $\\mr\\in R^\\ast$ as well as a remainder $\\dr\\in R[\\bm{x}]$ and quotients $\\qr_j\\in R[\\bm{x}]$ for $1\\le j\\le s$ such that\n\\begin{equation}\\label{PseudoDivisionExpression}\n\\mr f=\\sum_{j=1}^s\\qr_j\\ibr_j+\\dr,\n\\end{equation}\nwhere $\\dr$ is pseudo-reduced with respect to $G$.\nMoreover, the polynomials in \\eqref{PseudoDivisionExpression} satisfy the following condition:\n\\begin{equation}\\label{DivisionCond}\n\\lmc (f)=\\max\\bigl\\{\\max_{1\\le j\\le s}\\{\\lmc (\\qr_j\\ibr_j)\\},\\lmc (\\dr)\\bigr\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nIf $f$ is already pseudo-reduced with respect to $\\pb$, we just take $\\dr=f$ and $\\qr_j=0$ for $1\\le j\\le s$.\nOtherwise we define $\\bm{x}^\\alpha:=\\max_{\\succ}\\{\\supp f\\cap\\langle\\lmc (\\pb)\\rangle\\}$.\nThere exists some $j$ such that $\\bm{x}^\\alpha$ is divisible by $\\lmc (\\ibr_j)$ as per Lemma \\ref{Lemma:MonomialIdealProperty} \\eqref{item:MonomialIdealDivisionCond}.\nWe make a pseudo-reduction of the term $c_\\alpha\\bm{x}^\\alpha$ of $f$ by $\\ibr_j$ in the same way as the term pseudo-reduction in \\eqref{TermReduction}.\nWe denote the remainder also as $h$ and $\\bm{x}^\\beta:=\\max_{\\succ}\\{\\supp h\\cap\\langle\\lmc (\\pb)\\rangle\\}$ if $h$ is not pseudo-reduced with respect to $\\pb$.\nIt is easy to see that $\\bm{x}^\\alpha\\succ\\bm{x}^\\beta$ after the above term pseudo-reduction.\nWe repeat such term pseudo-reductions until the remainder $h$ is pseudo-reduced with respect to $\\pb$.\nSince the monomial ordering $\\succ$ is a well-ordering by Definition \\ref{Def:MonomialOrdering}, the term pseudo-reductions terminate in finite steps.\nHence follows the representation \\eqref{PseudoDivisionExpression} in which the multiplier $\\mr\\in R^\\ast$ is a product of such interim multipliers $\\iur$ as in \\eqref{TermReduction}.\n\nTo prove the equality in \\eqref{DivisionCond}, it suffices to prove it for the term pseudo-reduction in \\eqref{TermReduction}.\nIn fact, the pseudo-division in \\eqref{PseudoDivisionExpression} is just a composition of the term pseudo-reductions in \\eqref{TermReduction} and the remainder $h$ in \\eqref{TermReduction} shall eventually become the remainder $\\dr$ in \\eqref{PseudoDivisionExpression}.\nIn \\eqref{TermReduction} the leading monomial of $\\lmr\\bm{x}^\\alpha g\/\\ltc (g)$ is $\\bm{x}^\\alpha$.\nHence either $\\lmc (f)=\\bm{x}^\\alpha$, or $\\lmc (f)\\succ\\bm{x}^\\alpha$ in which case $\\lmc (f)=\\lmc (h)$ in \\eqref{TermReduction}.\nThus follows the equality in \\eqref{DivisionCond}.\n\\end{proof}\n\n\\begin{definition}\\label{Def:Reduction}\nLet $R$ be a PID and $f\\in R[\\bm{x}]$.\nSuppose that $\\pb=\\{\\ibr_j\\colon 1\\le j\\le s\\}\\subset\\rd$ is a polynomial set over $R$.\nWe call the expression in \\eqref{PseudoDivisionExpression} a \\emph{pseudo-division} of $f$ by $\\pb$.\nMore specifically, we name the polynomial $r$ in \\eqref{PseudoDivisionExpression} as a \\emph{remainder} of $f$ on pseudo-division by $\\pb$ and $\\mr\\in R^\\ast$ in \\eqref{PseudoDivisionExpression} a \\emph{multiplier} of the pseudo-division.\nWe say that $f$ \\emph{pseudo-reduces} to the \\emph{remainder} $r$ via the \\emph{multiplier} $\\mr\\in R^\\ast$ \\emph{modulo} $\\pb$.\nWe also call it a \\emph{pseudo-reduction} of $f$ by $\\pb$.\n\\end{definition}\n\nThe proof of Theorem \\ref{Thm:PseudoReduction} shows that the multiplier $\\mr$ in \\eqref{PseudoDivisionExpression} is a finite product of the interim multipliers $\\iur$ as in \\eqref{TermReduction}.\nBased on the proof of Theorem \\ref{Thm:PseudoReduction} we can easily contrive a pseudo-division algorithm.\nWe do not elaborate on it here since it is quite straightforward.\n\n\\begin{remark}\\label{Rmk:LinearEqs}\nThere is a difference between the above pseudo-division algorithm and the traditional division algorithm over a PID.\nIn the traditional one as in \\cite[P207, Algorithm 4.1.1]{AL94}, it is required that the linear equation $\\lcc (f)=\\sum_{j=1}^s\\ibr_j\\cdot\\lcc (f_j)$ as in \\cite[P204, (4.1.1)]{AL94} be solvable for $\\ibr_j$'s over $R$.\nThe pseudo-division algorithm does not have this extra requirement.\nTheir major difference is the multiplier $\\mr\\in R^\\ast$ in \\eqref{PseudoDivisionExpression}.\n\\end{remark}\n\n\\section{Pseudo-eliminants of Zero-dimensional Ideals}\\label{Sec:PseudoGroebner}\n\nLet $K$ be a field and $\\bm{x}$ denote variables $(\\Lst x1n)$ as before.\nIn this section let us consider the case when the PID $R$ in Section \\ref{Sec:DivisionAlgm} bears the particular form $R=\\kxn$ with $x_1$ being the first variable of $\\bm{x}$.\nIn this case the polynomials in the algebra $K[\\bm{x}]$ over $K$ can be viewed as those in $\\knx$ over $\\kxn$ with the variables $\\tilde{\\bx}=(\\Lst x2n)$ and coefficients in $\\kxn$.\nIt is evident that the pseudo-division of polynomials in Theorem \\ref{Thm:PseudoReduction} applies here without any essential change.\n\nUnless specified, in what follows we shall always treat the algebra $K[\\bm{x}]$ over $K$ as the algebra $\\knx$ over $\\kxn$.\nHence for $f\\in\\knx$ in this section, its leading coefficient $\\lcc (f)$ and leading monomial $\\lmc (f)$ in Notation \\ref{Notation:LeadingEntities} now satisfy $\\lcc (f)\\in (\\kxn)^*$ and $\\lmc (f)\\in\\tM$ respectively.\nHere $\\tM$ denotes the set of nonzero monomials in the variables $\\tilde{\\bx}=(\\Lst x2n)$.\nMoreover, we use $\\pid f$ to denote a principal ideal in $\\kxn$ generated by $f\\in\\kxn$.\nRecall that $\\langle f\\rangle$ denotes a principal ideal in $\\knx=K[\\bm{x}]$ generated by either $f\\in\\kxn$ or $f\\in\\knx$.\n\n\\begin{definition}[Elimination ordering\\footnote{Please also refer to \\cite[P69, Definition 2.3.1]{AL94} and \\cite[P33, Definition 3.1]{EH12}.} on $\\knx$]\\label{Def:EliminationOrdering}\n\\hfill\n\nAn \\emph{elimination ordering} on $\\knx$ is a monomial ordering on $\\bM$ such that the $\\tilde{\\bx}$ variables are always larger than the $x_1$ variable.\nThat is, $x_1^\\alpha\\tilde{\\bx}^\\gamma\\succ x_1^\\beta\\tilde{\\bx}^\\delta$ if and only if $\\tilde{\\bx}^\\gamma\\succ\\tilde{\\bx}^\\delta$ or, $\\tilde{\\bx}^\\gamma=\\tilde{\\bx}^\\delta$ and $\\alpha>\\beta$.\n\\end{definition}\n\nIn what follows let us suppose that $I$ is a zero-dimensional ideal\\footnote{Please note that a zero-dimensional ideal $I$ is always a proper ideal such that $I\\ne K[\\bm{x}]$.} of $K[\\bm{x}]=\\knx$.\nWe have the following well-known conclusion.\n\\begin{proposition}\nFor a zero-dimensional ideal $I\\subset\\knx$, we always have $\\Il\\ne\\{0\\}$.\n\\end{proposition}\n\\begin{proof}\nPlease refer to \\cite[P272, Lemma 6.50]{BW93} or \\cite[P243, Proposition 3.7.1(c)]{KR00}.\n\\end{proof}\n\n\\begin{definition}[Eliminant]\\label{Def:Eliminant}\n\\hfill\n\nFor a zero-dimensional ideal $I\\subset\\knx$, the principal ideal $\\Il$ in $\\kxn$ is called the \\emph{elimination ideal} of $I$.\nLet us denote its generator as $\\el$ that satisfies $\\Il=\\pid\\el$ being a principal ideal in $\\kxn$.\nWe call $\\el$ the \\emph{eliminant} of the zero-dimensional ideal $I$ henceforth.\n\\end{definition}\n\nIn what follows let us elaborate on a revised version of Buchberger's algorithm.\nThe purpose is to compute not only a pseudo-basis but also a pseudo-eliminant of the elimination ideal $\\Il$.\nLet us first recall the $S$-polynomial over a PID as in the following definition\\footnote{Please also refer to \\cite[P249, (4.5.1)]{AL94} or \\cite[P457, Definition 10.9]{BW93}.}.\n\n\\begin{definition}[$S$-polynomial]\\label{Def:SPolynomial}\n\\hfill\n\nSuppose that $f,g\\in\\Rd$.\nLet us denote $\\lmr:=\\lcm (\\lcc (f),\\lcc (g))\\in\\kxn$ and $\\clm:=\\lcm (\\lmc (f),\\lmc (g))\\in\\tM$.\nThen the polynomial\n\\begin{equation}\\label{SPolynomialDef}\nS(f,g):=\\frac{\\lmr\\clm}{\\ltc (f)}f-\\frac{\\lmr\\clm}{\\ltc (g)}g\n\\end{equation}\nis called the \\emph{$S$-polynomial} of $f$ and $g$.\n\\end{definition}\n\nIt is easy to verify that the $S$-polynomial satisfies the following inequality due to the cancellation of leading terms in \\eqref{SPolynomialDef}:\n\\begin{equation}\\label{LeadingSMonomials}\n\\lmc (S(f,g))\\prec\\clm=\\lcm (\\lmc (f),\\lmc (g)).\n\\end{equation}\n\nWhen $g\\in (\\kxn)^*$ and $f\\in\\Rd$, we take $\\lmc (g)=1$ and $\\lmr=\\lcm (\\lcc (f),g)$.\nThe $S$-polynomial in \\eqref{SPolynomialDef} becomes:\n\\begin{equation}\\label{SpecialSPoly}\nS(f,g):=\\frac\\lmr{\\lcc (f)}f-\\lmr\\cdot\\lmc (f).\n\\end{equation}\n\n\\begin{lemma}\\label{Lemma:UnnecessaryConst}\nWhen $g\\in (\\kxn)^*$ and $f\\in\\Rd$, the $S$-polynomial in \\eqref{SpecialSPoly} satisfies:\n\\begin{equation}\\label{SpecialSPolyEssence}\n\\idr S(f,g)=(f-\\ltc (f))\\cdot g:=f_1g\n\\end{equation}\nwith $\\idr:=\\gcd (\\lcc (f),g)\\in\\kxn$ and $f_1:=f-\\ltc (f)$.\n\\end{lemma}\n\\begin{proof}\nLet us denote $l_f:=\\lcc (f)$.\nBased on the equality $\\lmr\/l_f=g\/\\idr$, the $S$-polynomial in \\eqref{SpecialSPoly} becomes:\n\\begin{equation*}\nS(f,g)=\\frac{gf}{\\idr}-\\frac{gl_f}\\idr\\cdot\\lmc (f)=\\frac g\\idr (f-\\ltc (f))=\\frac{f_1g}\\idr.\n\\qedhere\n\\end{equation*}\n\\end{proof}\n\nLemma \\ref{Lemma:UnnecessaryConst} can be generalized to the following conclusion:\n\n\\begin{lemma}\\label{Lemma:RelativePrimePairs}\nFor $f,g\\in\\Rd$, suppose that $\\lmc (f)$ and $\\lmc (g)$ are relatively prime.\nLet us denote $\\idr:=\\gcd (\\lcc (f),\\lcc (g))$.\nThen their $S$-polynomial in \\eqref{SPolynomialDef} satisfies:\n\\begin{equation}\\label{CoprimeReduction}\n\\idr S(f,g)=f_1g-g_1f\n\\end{equation}\nwith $f_1:=f-\\ltc (f)$ and $g_1:=g-\\ltc (g)$.\nMoreover, we have:\n\\begin{equation}\\label{LeadTwoMonomial}\n\\lmc (S(f,g))=\\max\\{\\lmc (f_1g),\\lmc (g_1f)\\}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nIf $\\lmc (f)$ and $\\lmc (g)$ are relatively prime, then we have an identity $\\clm=\\lmc (f)\\cdot\\lmc (g)$ in \\eqref{SPolynomialDef}.\nFor convenience, let us denote $l_f:=\\lcc (f)$, $l_g:=\\lcc (g)$ and $\\idr:=\\gcd (l_f,l_g)$.\nThen we have the identities $\\lmr\/l_f=l_g\/\\idr$ and $\\lmr\/l_g=l_f\/\\idr$ with $\\lmr=\\lcm (l_f,l_g)$ as in \\eqref{SPolynomialDef}.\nWe substitute these identities into \\eqref{SPolynomialDef} to obtain:\n\\begin{align}\nS(f,g)&:=\\frac{l_g\\cdot\\lmc (g)}\\idr f-\\frac{l_f\\cdot\\lmc (f)}\\idr g=\\frac 1\\idr (\\ltc (g)\\cdot f-\\ltc (f)\\cdot g)\\notag\\\\\n&=\\frac 1\\idr ((g-g_1)f-(f-f_1)g)=\\frac 1\\idr (f_1g-g_1f)\\label{CoprimeReductProof}\\\\\n&=\\frac 1\\idr (f_1(g_1+\\ltc (g))-g_1(f_1+\\ltc (f)))=\\frac 1\\idr (f_1\\cdot\\ltc (g)-g_1\\cdot\\ltc (f)).\\label{ReductionLeadTerms}\n\\end{align}\nThe identity \\eqref{CoprimeReduction} follows from \\eqref{CoprimeReductProof}.\nNow we show that the conclusion \\eqref{LeadTwoMonomial} is a consequence of the expression \\eqref{ReductionLeadTerms}.\nIn fact, for every term $c_\\alpha\\tilde{\\bx}^\\alpha$ of $f_1$ and every term $c_\\beta\\tilde{\\bx}^\\beta$ of $g_1$, we have $\\tilde{\\bx}^\\alpha\\cdot\\lmc (g)\\ne\\tilde{\\bx}^\\beta\\cdot\\lmc (f)$ since $\\lmc (g)$ and $\\lmc (f)$ are relatively prime and moreover, we have $\\tilde{\\bx}^\\alpha\\prec\\lmc (f)$ and $\\tilde{\\bx}^\\beta\\prec\\lmc (g)$.\nAs a result, no term of $f_1\\cdot\\ltc (g)$ cancels no term of $g_1\\cdot\\ltc (f)$ in \\eqref{ReductionLeadTerms}.\nThus follows the equality \\eqref{LeadTwoMonomial}.\n\\end{proof}\n\n\\begin{corollary}\\label{Cor:CoprimePair}\nSuppose that $\\lmc (f)$ and $\\lmc (g)$ are relatively prime for $f,g\\in\\Rd$.\nThen their $S$-polynomial $S(f,g)$ as in \\eqref{SPolynomialDef} can be pseudo-reduced to $0$ by $f$ and $g$ as in Theorem \\ref{Thm:PseudoReduction} with the multiplier $\\mr=\\idr$ and quotients $f_1$ and $g_1$ as in \\eqref{CoprimeReduction}.\n\nIn particular, for $g\\in (\\kxn)^*$ and $f\\in\\Rd$, their $S$-polynomial $S(f,g)$ in \\eqref{SpecialSPoly} can be pseudo-reduced to $0$ by $g$ with the multiplier $\\mr=\\idr$ and quotient $f_1$ as in \\eqref{SpecialSPolyEssence}.\n\\end{corollary}\n\\begin{proof}\nThe conclusions readily follow from Lemma \\ref{Lemma:RelativePrimePairs} and Lemma \\ref{Lemma:UnnecessaryConst} based on Theorem \\ref{Thm:PseudoReduction}.\n\\end{proof}\n\nFor two terms $c_\\alpha\\tilde{\\bx}^\\alpha,c_\\beta\\tilde{\\bx}^\\beta\\in\\knx$ with $c_\\alpha,c_\\beta\\in\\kxn$, let us denote $\\lcm (c_\\alpha\\tilde{\\bx}^\\alpha,c_\\beta\\tilde{\\bx}^\\beta):=\\lcm (c_\\alpha,c_\\beta)\\cdot\\lcm (\\tilde{\\bx}^\\alpha,\\tilde{\\bx}^\\beta)$.\nThen we have the following notation:\n\\begin{equation*}\n\\lcm (\\ltc (f),\\ltc (g)):=\\lcm (\\lcc (f),\\lcc (g))\\cdot\\lcm (\\lmc (f),\\lmc (g)).\n\\end{equation*}\n\n\\begin{lemma}\\label{Lemma:TriangleIdentity}\nFor $f,g,h\\in\\Rd$, if $\\lcm (\\lmc (f),\\lmc (g))\\in\\langle\\lmc (h)\\rangle$, then we have the following triangular relationship among their $S$-polynomials:\n\\begin{equation}\\label{TriangleIdentity}\n\\mr S(f,g)=\\frac{\\mr\\cdot\\lcm (\\ltc (f),\\ltc (g))}{\\lcm (\\ltc (f),\\ltc (h))}S(f,h)-\\frac{\\mr\\cdot\\lcm (\\ltc (f),\\ltc (g))}{\\lcm (\\ltc (h),\\ltc (g))}S(g,h),\n\\end{equation}\nwhere the multiplier $\\mr:=\\lcc (h)\/\\idr$ with $\\idr:=\\gcd (\\lcm (\\lcc (f),\\lcc (g)),\\lcc (h))\\in\\kxn$.\nHenceforth we also call the identity \\eqref{TriangleIdentity} the \\emph{triangular identity} of $S(f,g)$ with respect to $h$.\n\\end{lemma}\n\\begin{proof}\nFrom $\\lcm (\\lmc (f),\\lmc (g))\\in\\langle\\lmc (h)\\rangle$ we can easily deduce that:\n\\begin{equation*}\n\\lcm (\\lmc (f),\\lmc (g))\\in\\langle\\lcm (\\lmc (f),\\lmc (h))\\rangle\\cap\\langle\\lcm (\\lmc (h),\\lmc (g))\\rangle.\n\\end{equation*}\nIn order to corroborate that the multiplier $\\mr=\\lcc (h)\/\\idr$ suffices to make the two fractions in \\eqref{TriangleIdentity} terms in $\\knx$, we only need to consider the case when $\\mlt_\\ipr (\\lcc (h))>\\gamma:=\\max\\{\\mlt_\\ipr (\\lcc (f)),\\mlt_\\ipr (\\lcc (g))\\}$ for every irreducible factor $\\ipr$ of $\\lcc (h)$.\nIn this case we have $\\mlt_\\ipr (\\lcm (\\lcc (f),\\lcc (h)))=\\mlt_\\ipr (\\lcc (h))=\\mlt_\\ipr (\\lcm (\\lcc (h),\\lcc (g)))$ in the denominators of \\eqref{TriangleIdentity}.\nHence in the numerators of \\eqref{TriangleIdentity} we can take $\\mlt_\\ipr (\\mr)=\\mlt_\\ipr (\\lcc (h))-\\gamma=\\mlt_\\ipr (\\lcc (h))-\\mlt_\\ipr (\\idr)$.\nNow let us write the definition of $S$-polynomial in \\eqref{SPolynomialDef} into the following form:\n\\begin{equation}\\label{OtherFormSPoly}\nS(f,g)=\\frac{\\lcm (\\ltc (f),\\ltc (g))}{\\ltc (f)}f-\\frac{\\lcm (\\ltc (f),\\ltc (g))}{\\ltc (g)}g.\n\\end{equation}\nThen the identity \\eqref{TriangleIdentity} readily follows if we also write $S(f,h)$ and $S(g,h)$ into the form of \\eqref{OtherFormSPoly}.\n\\end{proof}\n\nIt is easy to verify that the identity \\eqref{TriangleIdentity} is consistent with the inequality \\eqref{LeadingSMonomials} for $S$-polynomials since from \\eqref{TriangleIdentity} we can deduce that $\\lmc (S(f,g))$ is dominated by one of the following leading monomials:\n\\begin{equation*}\n\\frac{\\lcm (\\lmc (f),\\lmc (g))}{\\lcm (\\lmc (f),\\lmc (h))}\\lmc (S(f,h)),\\text{~\\,or~\\,}\\frac{\\lcm (\\lmc (f),\\lmc (g))}{\\lcm (\\lmc (h),\\lmc (g))}\\lmc (S(g,h)).\n\\end{equation*}\n\nFor $f\\in\\Rd$ and $g\\in (\\kxn)^*$, we shall use the following relation between \\eqref{SPolynomialDef} and \\eqref{SpecialSPoly}:\n\\begin{equation}\\label{DisappearingMonomial}\nS(f,g\\cdot\\lmc (f))=S(f,g).\n\\end{equation}\n\nMoreover, the $S$-polynomial in \\eqref{SpecialSPoly} coincides with the term pseudo-reduction in \\eqref{TermReduction} for $g\\in R^\\ast=(\\kxn)^*$.\n\n\\begin{algorithm}[Pseudo-eliminant of a zero-dimensional ideal over a PID]\\label{Algo:PseudoEliminant}\n\\hfil\n\nInput: A finite polynomial set $\\tas\\subset\\knx\\setminus K$.\n\nOutput: A pseudo-eliminant $\\pel\\in (\\kxn)^*$, pseudo-basis $\\pbs\\subset\\langle\\tas\\rangle\\setminus\\kxn$ and multiplier set $\\mrs\\subset\\nonk$.\n\nInitialization: A temporary basis set $\\tbs:=\\tas\\setminus\\kxn$; a multiplier set $\\mrs:=\\emptyset$ in $\\kxn$; a temporary set $\\mathfrak{S}:=\\emptyset$ in $\\knx\\setminus K$ for $S$-polynomials.\nIf $\\tas\\cap\\kxn\\ne\\emptyset$, we initialize $\\lel:=\\gcd (\\tas\\cap\\kxn)$; otherwise we initialize $\\lel:=0$.\n\nFor each pair $\\tf,\\tg\\in\\tbs$ with $\\tf\\ne\\tg$, we invoke \\Po Q as follows to compute their $S$-polynomial $S(f,g)$.\n\n\\Po Q:\n\n\\leftskip=5mm\n\\begin{itshape}\nInput: $f,g\\in\\Rd$.\n\nIf $\\lmc (\\tf)$ and $\\lmc (\\tg)$ are relatively prime, we define $\\idr:=\\gcd (\\lcc (f),\\lcc (g))$ as in \\eqref{CoprimeReduction}.\nIf $\\idr\\in\\nonk$, we add $\\idr$ into the multiplier set $\\mrs$.\nThen we disregard the $S$-polynomial $S(\\tf,\\tg)$.\n\nIf there exists an $h\\in\\tbs\\setminus\\{\\tf,\\tg\\}$ such that $\\lcm (\\lmc (f),\\lmc (g))\\in\\langle\\lmc (h)\\rangle$, and the triangular identity \\eqref{TriangleIdentity} has never been applied to the same triplet $\\{f,g,h\\}$, we compute the multiplier $\\mr$ as in \\eqref{TriangleIdentity}.\nIf $\\mr\\in\\nonk$, we add $\\mr$ into the multiplier set $\\mrs$.\nThen we disregard the $S$-polynomial $S(\\tf,\\tg)$.\n\nIf neither of the above two cases is true, we compute their $S$-polynomial $S(\\tf,\\tg)$ as in \\eqref{SPolynomialDef}.\nThen we add $S(\\tf,\\tg)$ into the set $\\mathfrak{S}$.\n\\end{itshape}\n\nEnd of $\\mathcal{Q}$\n\\leftskip=0mm\n\nWe recursively repeat \\Po P as follows for the pseudo-reductions of all the $S$-polynomials in the set $\\mathfrak{S}$.\n\n\\Po P:\n\n\\leftskip=5mm\n\\begin{itshape}\nFor an $S\\in\\mathfrak{S}$, we invoke Theorem \\ref{Thm:PseudoReduction} to make a pseudo-reduction of $S$ by the temporary basis set $\\tbs$.\n\nIf the multiplier $\\mr\\in\\nonk$ in \\eqref{PseudoDivisionExpression}, we add $\\mr$ into the multiplier set $\\mrs$.\n\nIf the remainder $\\dr\\in\\Rd$, we add $\\dr$ into $\\tbs$.\nFor every $\\tf\\in\\tbs\\setminus\\{\\dr\\}$, we invoke \\Po Q to compute the $S$-polynomial $S(\\tf,\\dr)$.\n\nIf the remainder $\\dr\\in\\nonk$, we redefine $\\lel:=\\gcd (\\dr,\\lel)$.\n\nIf the remainder $\\dr\\in K^\\ast$, we halt the algorithm and output $\\tbs=\\{1\\}$.\n\nThen we delete $S$ from the set $\\mathfrak{S}$.\n\\end{itshape}\n\nEnd of $\\mathcal{P}$\n\\leftskip=0mm\n\nFinally we define $\\pel:=\\lel$ and $\\pbs:=\\tbs$ respectively.\n\n\\Po R:\n\n\\leftskip=5mm\n\\begin{itshape}\nFor every $\\tf\\in\\pbs$, if $\\idr:=\\gcd (\\lcc (f),\\pel)\\in\\nonk$, we add $\\idr$ into the multiplier set $\\mrs$.\n\\end{itshape}\n\nEnd of $\\mathcal{R}$\n\\leftskip=0mm\n\nWe output $\\pel$, $\\pbs$ and $\\mrs$.\n\\end{algorithm}\n\n\\begin{remark}\nIn Algorithm \\ref{Algo:PseudoEliminant} we compute both $\\idr:=\\gcd (\\lcc (f),\\lcc (g))$ when $\\lmc (\\tf)$ and $\\lmc (\\tg)$ are relatively prime in \\Po Q and $\\idr:=\\gcd (\\lcc (f),\\pel)$ for every $\\tf\\in\\pbs$ in \\Po R.\nThe purpose of these computations is to procure the multipliers $\\idr$ in \\eqref{CoprimeReduction} of Lemma \\ref{Lemma:RelativePrimePairs} and \\eqref{SpecialSPolyEssence} of Lemma \\ref{Lemma:UnnecessaryConst} respectively for the pseudo-reductions of the $S$-polynomials.\nIt is the reason why we add $\\idr$ into the multiplier set $\\mrs$ when $\\idr\\in\\nonk$.\n\nMoreover, in \\Po Q the condition that there exists an $h\\in\\tbs\\setminus\\{\\tf,\\tg\\}$ such that $\\lcm (\\lmc (f),\\lmc (g))\\in\\langle\\lmc (h)\\rangle$ is a condition for Lemma \\ref{Lemma:TriangleIdentity}.\n\\end{remark}\n\n\\begin{definition}[Pseudo-eliminant; pseudo-basis; multiplier set]\\label{Def:PseudoBasis}\n\\hfill\n\nWe call the univariate polynomial $\\pel$ obtained via Algorithm \\ref{Algo:PseudoEliminant} a \\emph{pseudo-eliminant} of the zero-dimensional ideal $I$.\nWe also call the polynomial set $\\pbs$ a \\emph{pseudo-basis} of $I$ and $\\mrs$ its \\emph{multiplier set}.\n\\end{definition}\n\nPlease note that contrary to the convention, we do not include the pseudo-eliminant $\\pel$ in the pseudo-basis $\\pbs$ since we shall contrive modular algorithms to compute modular bases with the factors of $\\pel$ as moduli in Section \\ref{Sec:IncompatibleModular}.\n\n\\begin{lemma}\\label{Lemma:PseudoEliminantTerminate}\n\\begin{inparaenum}[(i)]\n\\item\\label{item:EliminantDivisibility} A pseudo-eliminant $\\pel$ of a zero-dimensional ideal $I$ is divisible by its eliminant $\\el$.\n\\item\\label{item:AllSPolynRemainder} For each pair $f\\ne g$ in the union set of pseudo-basis and pseudo-eliminant $\\pbs\\cup\\{\\pel\\}$, the pseudo-reduction of their $S$-polynomial $S(f,g)$ by $\\pbs$ yields a remainder $r\\in\\pid\\pel$ in $\\kxn$.\n In particular, this includes the case when $r=0$.\n\\item\\label{item:PseudoEliminantTerminate} Algorithm \\ref{Algo:PseudoEliminant} terminates in a finite number of steps.\n\\end{inparaenum}\n\\end{lemma}\n\\begin{proof}\\begin{inparaenum}[(i)]\n\\item The conclusion readily follows from the fact that $\\pel\\in\\Il=\\pid\\el$.\n\n\\item According to \\Po P in Algorithm \\ref{Algo:PseudoEliminant}, if the remainder $r$ of the pseudo-reduction by an intermediate polynomial set $\\tbs$ satisfies $r\\in\\Rd$, we add it into $\\tbs$.\nThat is, $r$ eventually becomes an element of the pseudo-basis $\\pbs$.\nIt is evident that a pseudo-reduction of $r$ by itself per se leads to the zero remainder.\nOn the other hand, if $r\\in\\nonk$, then as per $\\lel:=\\gcd (\\dr,\\lel)$ in \\Po P of Algorithm \\ref{Algo:PseudoEliminant}, $r$ is divisible by $\\lel$ and hence by the pseudo-eliminant $\\pel$, i.e., $r\\in\\pid\\pel$.\n\n\\item The termination of the algorithm follows from the ring $\\knx=K[\\bm{x}]$ being Noetherian.\nIn fact, whenever the remainder $r\\in\\Rd$ in the \\Po P of the algorithm, we add it to the intermediate polynomial set $\\tbs$.\nAs a result, the monomial ideal $\\langle\\lmc (\\tbs)\\rangle$ is strictly expanded since $r$ is pseudo-reduced with respect to $\\tbs\\setminus\\{r\\}$.\nHence the ascending chain condition imposed on the chain of ideals $\\langle\\lmc (\\tbs)\\rangle$ ensures the termination of the algorithm.\n\\end{inparaenum}\n\\end{proof}\n\nBased on Lemma \\ref{Lemma:PseudoEliminantTerminate} \\eqref{item:EliminantDivisibility}, the following conclusion is immediate:\n\\begin{corollary}\nIf a pseudo-eliminant $\\pel$ of a zero-dimensional ideal $I$ in $K[\\bm{x}]$ satisfies $\\pel\\in K^\\ast$, then the reduced Gr\\\"obner\\ basis\\footnote{Please refer to \\cite[P48, Definition 1.8.5]{AL94} or \\cite[P93, Definition 4]{CLO15} for a definition.} of $I$ is $\\{1\\}$.\n\\end{corollary}\n\nIn what follows we assume that the ideal $I$ is a proper ideal of $K[\\bm{x}]$, that is, $I\\ne K[\\bm{x}]$.\nThus let us exclude the trivial case of $\\pel\\in K^\\ast$ hereafter.\n\n\\section{Pseudo-eliminant Divisors and Compatibility}\\label{Section:PseudoEliminantDivisors}\n\nSuppose that $K$ is a perfect field whose characteristic is denoted as $\\ch K$.\nRecall that finite fields and fields of characteristic zero are perfect fields.\nIn this section we begin to contrive an algorithm retrieving the eliminant $\\el$ of a zero-dimensional ideal $I$ from its pseudo-eliminant $\\pel$ obtained in Algorithm \\ref{Algo:PseudoEliminant}.\nWe first make a factorization of the pseudo-eliminant $\\pel$ into \\emph{compatible} and \\emph{incompatible} divisors.\nWe prove that the compatible divisors of $\\pel$ are the authentic factors of $\\el$.\nThis shows that Algorithm \\ref{Algo:PseudoEliminant} generates the eliminant $\\el$ when the pseudo-eliminant $\\pel$ is compatible.\nWe compute the factors of $\\el$ that correspond to the incompatible divisors of $\\pel$ in Section \\ref{Sec:IncompatibleModular}.\n\n\\begin{definition}[Squarefree factorization of univariate polynomials]\\label{Def:SquarefreePart}\n\\hfill\n\nA univariate polynomial $f\\in\\nok$ is \\emph{squarefree} if it has no quadratic factors in $\\nok$.\nThat is, for every irreducible polynomial $g\\in\\nok$, $f$ is not divisible by $g^2$.\n\nThe \\emph{squarefree factorization} of a univariate polynomial $f\\in\\nok$ refers to a product $f=\\prod_{i=1}^s g_i^i$ with $g_s\\in\\nok$ such that for those $g_i$'s that are not constants, they are both squarefree and pairwise relatively prime.\nMoreover, the \\emph{squarefree part} of $f$ is defined as $\\prod_{i=1}^s g_i$.\n\\end{definition}\n\nThe squarefree factorization is unique up to multiplications by constants in $K^\\ast$.\nIts existence and uniqueness follow from the fact that $K[x]$ is a PID and hence a unique factorization integral domain.\nThere are various algorithms for squarefree factorization depending on the field $K$ being finite or of characteristic zero.\nAlgorithm \\ref{Algo:Squarefree} as follows amalgamates these two cases of field characteristics.\nWe improve the algorithm in \\cite[P539, Algorithm B.1.6]{GP08} over a finite field and then apply it to the squarefree factorization over a filed of characteristic zero.\n\nConsider the integer set $J:=\\Np$ when $\\ch K=0$, and $J:=\\mathbb{N}\\setminus p\\mathbb{N}$ when $\\ch K=p>0$.\nThat is, $J$ stands for the set of positive integers that are not a multiple of $p$ when $\\ch K=p>0$.\nLet us enumerate the positive integers in $J$ by the bijective enumeration map $\\rho\\colon\\Np\\rightarrow J$ such that $\\rho (i)<\\rho (j)$ whenever $i0$.\nWhen $\\ch K=0$, the enumeration map $\\rho$ is the identity map.\nIn the algorithm below, for every $i\\in\\Np$ we simply denote its image $\\rho (i)\\in J$ as $[i]$, i.e., $\\rho (i)=[i]$.\n\n\\begin{algorithm}[Squarefree factorization of a univariate polynomial]\\label{Algo:Squarefree}\n\\hfill\n\nInput: A univariate polynomial $f\\in\\nok$.\n\nOutput: The squarefree decomposition $\\{g_1,\\dotsc,g_s\\}$ of $f$.\n\n\\Po P:\n\n\\leftskip=5mm\n\\begin{itshape}\nIf $f'\\ne 0$, we compute the greatest common divisor $f_\\ro 1:=\\gcd (f,f')$ first.\nThe squarefree part of $f$ is defined as $h_\\ro 1:=f\/f_\\ro 1$.\n\nWe repeat the following procedure\\footnote{If $\\ch K>0$, the exponent $\\ro {i+1}-\\ro i$ of $h_\\ro {i+1}$ in the following definition of $f_\\ro {i+1}$ is the improvement on \\cite[P539, Algorithm B.1.6]{GP08}.} starting with $i=1$ until $i=s$ such that $f'_\\ro s=0$:\n\\begin{equation*}\n\\begin{aligned}\nh_\\ro{i+1}&:=\\gcd (f_\\ro i,h_\\ro i);\\\\\ng_\\ro i&:=h_\\ro i\/h_\\ro {i+1}.\n\\end{aligned}\\quad\n\\biggl\\{\\begin{aligned}\nf_\\ro {i+1}&:=f_\\ro i\/h_\\ro {i+1}^{\\ro {i+1}-\\ro i}\\\\\nf_\\ro {i+1}&:=f_\\ro i\/h_\\ro {i+1}.\n\\end{aligned}\\biggr.~\n\\begin{aligned}\n&\\text{if $\\ch K>0$};\\\\\n&\\text{if $\\ch K=0$}.\n\\end{aligned}\n\\end{equation*}\n\nIf $f_\\ro s\\in K$, we define $g_\\ro s:=h_\\ro s$ to obtain the squarefree factorization $f=\\prod_{i=1}^s g_\\ro i^\\ro i$.\n\nIf $\\ch K=p>0$ and $f_\\ro s\\in\\nok$, we invoke \\Po Q on $f_\\ro s$.\n\\end{itshape}\n\nEnd of $\\mathcal{P}$\n\\leftskip=0mm\n\n\\Po Q:\n\n\\leftskip=5mm\n\\begin{itshape}\nIf $\\ch K=p>0$ and $f\\in\\nok$ satisfies $f'=0$, we repeat the following procedure starting with $x_1:=x$ and $\\psi_1:=f$ until $i=t$ such that $\\psi'_t\\ne 0$:\n\\begin{equation*}\nx_{i+1}:=x_i^p;\\qquad\\psi_{i+1}(x_{i+1}):=\\psi_i(x_i)\n\\end{equation*}\n\nWe treat $\\psi_t$ as $f$ and invoke \\Po P on $\\psi_t$.\n\\end{itshape}\n\nEnd of $\\mathcal{Q}$\n\\leftskip=0mm\n\\end{algorithm}\n\n\\Po Q in Algorithm \\ref{Algo:Squarefree} is a composition of Frobenius automorphism.\n\n\\begin{proposition}\nWe can procure a squarefree factorization of $f$ in finite steps via Algorithm \\ref{Algo:Squarefree}.\n\\end{proposition}\n\\begin{proof}\nThe termination of the algorithm in finite steps readily follows from the fact that $\\deg f_\\ro{i+1}<\\deg f_\\ro i$ in \\Po P as well as $\\deg\\psi_{i+1}<\\deg\\psi_i$ in \\Po Q.\n\nIn the case of $\\ch K=0$, the bijective map $\\rho\\colon\\Np\\rightarrow J$ is an identity map such that $\\rho (i)=[i]=i$.\nTo illustrate the procedure of the algorithm, suppose that $f=\\prod_{i=1}^s g_i^i$ is a squarefree factorization of $f$.\nThen $h_1=\\prod_{i=1}^s g_i$ is the squarefree part of $f$ obtained in the beginning of \\Po P.\nMoreover, the $h_i$ in \\Po P equals $\\prod_{j=i}^s g_j$ for $1\\le i\\le s$.\nHence we have $g_i=h_i\/h_{i+1}$.\nFurther, $f_i$ equals $\\prod_{j=i+1}^s g_j^{j-i}$ for $1\\le i0$, suppose that $f=\\prod_{k\\in p\\mathbb{N}}g_k^k\\cdot\\prod_{i=1}^s g_\\ro i^\\ro i$.\nLet us denote $\\varphi_\\pp p:=\\prod_{k\\in p\\mathbb{N}}g_k^k$ and $\\varphi_q:=\\prod_{i=1}^s g_\\ro i^\\ro i$ such that $f=\\varphi_\\pp p\\varphi_q$.\nThen the $f_\\ro 1$ in \\Po P equals $\\varphi_\\pp p\\cdot\\prod_{i=2}^s g_\\ro i^{\\ro i-1}$.\nHence $h_\\ro 1=\\prod_{i=1}^s g_\\ro i$ is the squarefree part of $\\varphi_q$.\nAnd the $h_\\ro i$ equals $\\prod_{j=i}^s g_\\ro j$ for $1\\le i\\le s$.\nHence we have $g_\\ro i=h_\\ro i\/h_\\ro{i+1}$.\nMoreover, $f_\\ro i$ equals $\\varphi_\\pp p\\cdot\\prod_{j=i+1}^s g_\\ro j^{\\ro j-\\ro i}$ for $1\\le i1$.\n\\end{notation}\n\nWhen $\\qr\\in R^\\ast\\setminus R^\\times$ is irreducible in Definition \\ref{Def:PQR}, i.e., when $s=\\alpha_1=1$ in Notation \\ref{Notation:Factorials}, the PQR $\\bar R$ becomes a field since $\\qr$ is prime in $R$.\nNonetheless when $\\qr\\in R^\\ast\\setminus R^\\times$ is not irreducible in Definition \\ref{Def:PQR}, i.e., in the case of either $s>1$ or $\\alpha_i>1$ for an $i$ satisfying $1\\le i\\le s$, the PQR $\\bar R$ has zero divisors and is not an integral domain.\nIn this case $\\bar R$ is no longer a factorial ring.\nNonetheless $\\bar R$ still has nice properties to which we can resort in our computations.\n\n\\begin{lemma}\\label{Lemma:PQRProperties}\nLet $\\bar R=R\/\\pid\\qr$ be a PQR as in Definition \\ref{Def:PQR} and $\\varphi\\colon R\\rightarrow\\bar R$ the canonical ring homomorphism.\nSuppose that $\\qr\\in R^\\ast\\setminus R^\\times$ has a unique factorization $\\qr=u\\prod_{i=1}^sp_i^{\\alpha_i}$ as in Notation \\ref{Notation:Factorials}.\nIn what follows we also use the notation $\\bar a:=\\varphi (a)$ for every $a\\in R$.\n\\begin{enumerate}[(i)]\n\\item\\label{item:SimpleCoprime} An $\\dr\\in R$ is relatively prime to $\\qr$, i.e., $\\gcd (\\dr,\\qr)=1$, if and only if $\\varphi (\\dr)$ is a unit in $\\bar R$, that is, $\\varphi (\\dr)\\in\\bru R$.\n\n\\item\\label{item:ExpoLimits} For $1\\le i\\le s$ and each $l\\in\\mathbb{N}$ satisfying $l\\ge\\alpha_i$, we have $\\bar p_i^l\\sim\\bar p_i^{\\alpha_i}$.\nHere the notation $\\bar a\\sim\\bar b$ in $\\bar R$ means that $\\bar a$ is an associate of $\\bar b$ in $\\bar R$, i.e., there is a unit $\\bar u\\in\\bru R$ such that $\\bar b=\\bar u\\bar a$.\n\n\\item\\label{item:StandardRep} For every $\\bar a\\in\\bra R$, we have a unique representation $\\bar a\\sim\\prod_{i=1}^s\\bar p_i^{\\beta_i}$ that satisfies $0\\le\\beta_i\\le\\alpha_i$ for $1\\le i\\le s$.\nWe call such kind of representations a \\emph{standard} representation of $\\bar a$ in the PQR $\\bar R$ and denote it as $\\bar a_\\st$.\nIn particular, we define $\\bar a_\\st:=1$ for $\\bar a\\in\\bru R$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nThe conclusion \\eqref{item:SimpleCoprime} readily follows from the fact that $R$ is a PID.\nIn fact, $\\gcd (\\dr,\\qr)=1$ means that there exist $u,v\\in R$ such that $u\\dr+v\\qr=1$, from which we can deduce that $\\varphi (u)\\varphi (\\dr)=1$.\nConversely $\\varphi (u\\dr)=1$ implies that there exists $v\\in R$ such that $u\\dr-1=v\\qr$.\nHence follows the conclusion.\n\nWhen $s=1$ both the conclusions \\eqref{item:ExpoLimits} and \\eqref{item:StandardRep} are evident since $\\bar p_1^l=0$ for $l\\ge\\alpha_1$.\nIn particular, $\\bra R=\\bru R$ when $\\alpha_1=s=1$.\nIn this case every $\\bar a\\in\\bra R$ has a standard representation $\\bar a\\sim 1:=\\bar a_\\st$.\nSo in what follows let us suppose that $s>1$.\n\nWithout loss of generality, let us prove \\eqref{item:ExpoLimits} in the case when $i=s$.\nFor $l>\\alpha_s$, we have $p_s^l\\equiv p_s^l+\\qr\\mod \\qr$.\nMoreover, we have the identity:\n$p_s^l+\\qr=p_s^{\\alpha_s}(p_s^{l-\\alpha_s}+u\\prod_{i=1}^{s-1}p_i^{\\alpha_i})$.\nHere $\\varphi (p_s^{l-\\alpha_s}+u\\prod_{i=1}^{s-1}p_i^{\\alpha_i})$ is a unit in $\\bar R$ by \\eqref{item:SimpleCoprime} since $p_s^{l-\\alpha_s}+u\\prod_{i=1}^{s-1}p_i^{\\alpha_i}$ is relatively prime to $\\qr$ in $R$.\nThus $\\bar p_s^l=\\varphi (p_s^l+\\qr)\\sim\\bar p_s^{\\alpha_s}$.\n\nThe existence of the standard representation in \\eqref{item:StandardRep} readily follows from the fact that $R$ is factorial and the canonical homomorphism $\\varphi$ is an epimorphism.\nIf $\\ipr$ is irreducible and relatively prime to $\\qr$, then $\\bar\\ipr=\\varphi (\\ipr)\\in\\bru R$ is a unit.\nHence for every $\\bar a\\in\\bra R$, its standard representation can only bear the form $\\bar a\\sim\\prod_{i=1}^s\\bar p_i^{\\beta_i}$ with $0\\le\\beta_i\\le\\alpha_i$ for $1\\le i\\le s$.\nNow suppose that $\\bar a$ has another standard representation $\\bar a\\sim\\prod_{i=1}^s\\bar p_i^{\\gamma_i}$ with $0\\le\\gamma_i\\le\\alpha_i$ for $1\\le i\\le s$.\nThen there exists $h\\in R$ such that $\\prod_{i=1}^s\\ipr_i^{\\beta_i}=u\\cdot\\prod_{i=1}^s\\ipr_i^{\\gamma_i}+h\\qr$ with $u\\in R$ being relatively prime to $\\qr$, from which we can easily deduce that $\\beta_i=\\gamma_i$ for $1\\le i\\le s$.\n\\end{proof}\n\nLet $K$ be a perfect field and $\\qr\\in\\nonk$.\nIt is easy to see that $\\kxn\/\\pid\\qr$ is a PQR as defined in Definition \\ref{Def:PQR}.\nHereafter we use $R$ and $\\bar R$ to denote $\\kxn$ and $\\kxn\/\\pid\\qr$ respectively.\nLet us consider the following set:\n\\begin{equation}\\label{ResidueRing}\n\\rr:=\\{r\\in\\kxn\\colon\\deg (r)<\\deg (\\qr)\\}\n\\end{equation}\nwith $\\deg (r)=0$ for every $r\\in K$ including $r=0$.\nLet us deem the canonical ring homomorphism $\\varphi\\colon R\\rightarrow\\bar R$ as a map.\nWe restrict it on $\\rr$ and denote it as $\\varphi_\\qr$.\nIt is evident that $\\varphi_\\qr\\colon\\rr\\rightarrow\\bar R$ is a bijective map with $\\varphi_\\qr (0)=0$.\nWe redefine the two binary operations on $\\rr$, the addition and multiplication, as follows.\n\\begin{equation}\\label{BinaryOperations}\na+b:=\\varphi_\\qr^{-1}(\\bar a+\\bar b);\\quad a\\cdot b:=\\varphi_\\qr^{-1}(\\bar a\\cdot\\bar b).\n\\end{equation}\nIn this way the set $\\rr$ in \\eqref{ResidueRing} becomes a ring, which we still denote as $\\rr$.\nIt is easy to verify that $\\varphi_\\qr$ is a ring isomorphism between $\\rr$ and $\\bar R$.\nAs a result, the conclusions in Lemma \\ref{Lemma:PQRProperties} apply to the ring $\\rr$ as well.\n\n\\begin{definition}[Normal PQR $\\rr$]\\label{Def:nPQR}\n\\hfill\n\nWe call the ring $\\rr$ being constructed as in \\eqref{ResidueRing} and \\eqref{BinaryOperations} a \\emph{normal} PQR henceforth.\n\\end{definition}\n\nLet a normal PQR $\\rr$ be defined as in Definition \\ref{Def:nPQR} for $\\qr\\in\\nonk$.\nFor every $f\\in R=\\kxn$, there exist a quotient $h\\in\\kxn$ and unique remainder $r\\in\\kxn$ satisfying\n\\begin{equation}\\label{DivisionPQR}\nf=h\\qr+r;\\qquad\\deg (r)<\\deg (\\qr).\n\\end{equation}\nHence by \\eqref{ResidueRing} we can define an epimorphism directly as follows.\n\\begin{equation}\\label{ProjectionPQR}\n\\mo\\colon R\\rightarrow\\rr\\colon\\quad\\mo (f):=r.\n\\end{equation}\n\nIt is easy to verify that the epimorphism $\\mo$ is a composition of the canonical ring homomorphism $\\varphi\\colon R\\rightarrow\\bar R=\\kxn\/\\pid\\qr$ and the isomorphism $\\varphi_\\qr^{-1}\\colon\\bar R\\rightarrow\\rr$ in \\eqref{BinaryOperations}.\n\nSince a normal PQR $\\rr$ is a subset of $R=\\kxn$, for every $r\\in\\rr$, we can define an injection as follows.\n\\begin{equation}\\label{PQREmbedding}\n\\io\\colon\\rr\\hookrightarrow R\\colon\\quad\\io (r):=r.\n\\end{equation}\nPlease note that $\\io$ is not a ring homomorphism since the binary operations on the ring $\\rr$ are different from those on $R$.\nNonetheless $\\mo\\circ\\io$ is the identity map on $\\rr$.\nFor each pair $a,b\\in\\rr$, we define the binary operations between $\\io (a)$ and $\\io (b)$ as those defined on $R$.\n\nSuppose that $\\bar a\\in\\bra R$ has a standard representation $\\bar a\\sim\\bar a_\\st=\\prod_{i=1}^s\\bar p_i^{\\beta_i}$ with $0\\le\\beta_i\\le\\alpha_i$ as in Lemma \\ref{Lemma:PQRProperties} \\eqref{item:StandardRep}.\nWe can substitute $p_i=\\varphi_\\qr^{-1}(\\bar p_i)\\in\\rat$ as in \\eqref{BinaryOperations} for $\\bar p_i\\in\\bra R\\setminus\\bru R$ in this representation.\nIn this way we obtain a \\emph{standard} representation of $a:=\\varphi_\\qr^{-1}(\\bar a)\\in\\rr^\\ast$ in the normal PQR $\\rr$ as follows:\n\\begin{equation}\\label{StdRepsPQR}\na\\sim a_\\st:=\\prod_{i=1}^s p_i^{\\beta_i},\\quad 0\\le\\beta_i\\le\\alpha_i;\n\\qquad a=a^\\times\\cdot a_\\st,\n\\end{equation}\nwhere $\\{\\alpha_i\\colon 1\\le i\\le s\\}$ are the exponents for the unique factorization of the moduli $\\qr$ as in Lemma \\ref{Lemma:PQRProperties}.\nFor convenience we use $a^\\times\\in\\rr^\\times$ to denote the unit factor of $a$ with respect to $a_\\st$.\nWe also call $a_\\st$ the \\emph{standard factor} of $a$ henceforth.\nIn particular, we define $a_\\st:=1$ for $a\\in\\rr^\\times$.\nWe can derive the existence and uniqueness of the standard representation $a_\\st$ in \\eqref{StdRepsPQR} from Lemma \\ref{Lemma:PQRProperties} \\eqref{item:StandardRep} since the normal PQR $\\rr$ is isomorphic to the PQR $\\bar R$ under $\\varphi_\\qr$.\n\n\\begin{remark}\nIt is unnecessary to procure a complete factorization of $a_\\st$ as in \\eqref{StdRepsPQR} in our computations.\nIn fact, it suffices to make a factorization $a=a^\\times\\cdot a_\\st$.\nThis can be easily attained by a computation $a_\\st=\\gcd (\\io (a),\\qr)$ with $\\io$ being defined as in \\eqref{PQREmbedding}.\nThe soundness of the computation readily follows from Lemma \\ref{Lemma:PQRProperties} and \\eqref{StdRepsPQR}.\n\\end{remark}\n\nAn apparent difference between the PQR $\\bar R$ and normal PQR $\\rr$ is that the degree function $\\deg$ is well defined on $\\rr$, which is indispensable for polynomial divisions.\nMore specifically, for all $a,b\\in\\rr^\\ast$ with $\\deg (b)>0$, there exist a quotient $h\\in\\rr$ and unique remainder $r\\in\\rr$ satisfying the following equality:\n\\begin{equation}\\label{SimpleDivisionIdentity}\na=hb+r\\text{\\quad such that~}\\deg (r)<\\deg (b).\n\\end{equation}\nSince all polynomials involved here including the product $hb$ have degrees strictly less than $\\deg (\\qr)$ in \\eqref{SimpleDivisionIdentity}, there is no multiplication of zero divisors leading to $0$ for polynomial divisions.\nThis includes the case when $\\deg (a)<\\deg (b)$ and hence $h=0$.\nThat is, we make polynomial divisions on the normal PQR $\\rr$ in the same way as on $R$.\n\nFor $a,b\\in\\rr^\\ast$ and their standard representations $a\\sim a_\\st=\\prod_{i=1}^s p_i^{\\beta_i}$ and $b\\sim b_\\st=\\prod_{i=1}^s p_i^{\\gamma_i}$ as in \\eqref{StdRepsPQR}, let us define:\n\n\\begin{equation}\\label{GcdPQRStd}\n\\gcd\\nolimits_\\st (a,b):=\\gcd (a_\\st,b_\\st)=\\prod_{i=1}^s p_i^{n_i};~\n\\lcm\\nolimits_\\st (a,b):=\\lcm (a_\\st,b_\\st)=\\prod_{i=1}^s p_i^{m_i}\n\\end{equation}\nwith $n_i:=\\min\\{\\beta_i,\\gamma_i\\}$ and $m_i:=\\max\\{\\beta_i,\\gamma_i\\}$.\nIt is evident that we might have $\\lcm_\\st (a,b)=0$ for $a,b\\in\\rr^\\ast$ due to the possible existence of zero divisors in $\\rr^\\ast$.\n\n\\begin{remark}\\label{Rmk:GcdComput}\nThe definition of $\\gcd_\\st (a,b)$ and $\\lcm_\\st (a,b)$ in \\eqref{GcdPQRStd} is based upon a complete factorization of $a$ and $b$ as in \\eqref{StdRepsPQR}.\nIn practice in order to minimize the complexity of our algorithm, we resort to Euclidean algorithm to compute $\\gcd (a,b)$.\nThe normal PQR $\\rr$ might have zero divisors and not be an Euclidean domain.\nHowever from our discussion on polynomial divisions in \\eqref{SimpleDivisionIdentity}, we know that the polynomial division on $\\rr$ is the same as that on $R$.\nMoreover, for the irreducible factor $p_i\\in\\rat$ in Notation \\ref{Notation:Factorials} and $1\\le e\\le\\alpha_i$, if both $a$ and $b$ are divisible by $p_i^e$ in \\eqref{SimpleDivisionIdentity}, then so is the remainder $r$.\nSimilarly if both $b$ and $r$ are divisible by $p_i^e$ in \\eqref{SimpleDivisionIdentity}, then so is $a$.\nThus the computation of $\\gcd (a,b)$ for $a,b\\in\\rr^\\ast$ by Euclidean algorithm on $\\rr$ is sound and feasible.\nIt differs from $\\gcd_\\st (a,b)$ only by a unit factor.\n\\end{remark}\n\nLet $\\lcm (\\io (a),\\io (b))$ be the least common multiple of $\\io (a)$ and $\\io (b)$ on $R$.\nThe same for $\\gcd (\\io (a),\\io (b))$.\nFor each pair $a,b\\in\\rr$, let us define:\n\\begin{equation}\\label{GcdPQRqr}\n\\gcd\\nolimits_\\qr (a,b):=\\mo(\\gcd (\\io (a),\\io (b)));\\quad\n\\lcm\\nolimits_\\qr (a,b):=\\mo(\\lcm (\\io (a),\\io (b)))\n\\end{equation}\nwith the epimorphism $\\mo$ and injection $\\io$ defined as in \\eqref{ProjectionPQR} and \\eqref{PQREmbedding} respectively.\n\nBy Lemma \\ref{Lemma:PQRProperties} and \\eqref{StdRepsPQR}, it is easy to verify the following relationship between the two definitions in \\eqref{GcdPQRStd} and \\eqref{GcdPQRqr}:\n\\begin{equation}\\label{UniqueGcds}\n\\gcd\\nolimits_\\qr (a,b)\\sim\\gcd\\nolimits_\\st (a,b)\\quad\\text{and}\\quad\\lcm\\nolimits_\\qr (a,b)\\sim\\lcm\\nolimits_\\st (a,b).\n\\end{equation}\n\nIn the identity \\eqref{DivisionPQR}, $\\mlt_\\ipr (r)$ is well-defined since $\\kxn$ is a factorial domain and $r\\in\\kxn$.\nTherefore we can deduce that for every irreducible polynomial $\\ipr\\in\\kxn$, if $\\max\\{\\mlt_\\ipr (f),\\mlt_\\ipr (r)\\}\\le\\mlt_\\ipr (q)$, then we have:\n\\begin{equation}\\label{InvariantMultiplicity}\n\\mlt_\\ipr (f)=\\mlt_\\ipr (r).\n\\end{equation}\n\n\\begin{definition}[Elimination ordering on \\text{$\\rx$}]\\label{Def:EliminationOrderingPQR}\n\\hfill\n\nIf the variable $x_1\\in\\rr^\\ast$, the \\emph{elimination ordering} on $\\rx$ is the monomial ordering such that the $\\tilde{\\bx}$ variables are always larger than the variable $x_1\\in\\rr^\\ast$.\nThat is, $x_1^\\alpha\\tilde{\\bx}^\\gamma\\succ x_1^\\beta\\tilde{\\bx}^\\delta$ if and only if $\\tilde{\\bx}^\\gamma\\succ\\tilde{\\bx}^\\delta$ or, $\\tilde{\\bx}^\\gamma=\\tilde{\\bx}^\\delta$ and $\\alpha>\\beta$.\n\nWe also say that the elimination ordering on $\\rx$ is \\emph{induced} from the one on $\\knx$ in Definition \\ref{Def:EliminationOrdering}.\n\\end{definition}\n\n\\begin{definition}[Term reduction in \\text{$\\rx$}]\\label{Def:TermReductionPQR}\n\\hfill\n\nLet $\\rr$ be a normal PQR as in Definition \\ref{Def:nPQR} and $\\succ$ the elimination ordering on $\\rx$ as in Definition \\ref{Def:EliminationOrderingPQR}.\nLet the epimorphism $\\mo\\colon R\\rightarrow\\rr$ and injection $\\io\\colon\\rr\\rightarrow R$ be defined as in \\eqref{ProjectionPQR} and \\eqref{PQREmbedding} respectively.\nFor $f\\in\\rqd$ and $g\\in\\rnd$ with $\\lcc (g)\\in\\rr^\\ast$, suppose that $f$ has a term $\\icr_\\alpha\\tilde{\\bx}^\\alpha$ with $\\tilde{\\bx}^\\alpha\\in\\supp f\\cap\\langle\\lmc (g)\\rangle$.\nWe also define the multipliers $\\iur:=\\mo (\\lcm (l_\\alpha,l_g)\/l_\\alpha)$ and $\\lmr:=\\mo (\\lcm (l_\\alpha,l_g)\/l_g)$ with $l_\\alpha:=\\io (\\icr_\\alpha)$ and $l_g:=\\io (\\lcc (g))$.\nWe can make a \\emph{reduction} of the term $\\icr_\\alpha\\tilde{\\bx}^\\alpha$ of $f$ by $g$ as follows.\n\\begin{equation}\\label{TermReductionPQR}\nh=\\iur f-\\frac{\\lmr\\tilde{\\bx}^\\alpha}{\\lmc (g)}g.\n\\end{equation}\nWe call $h$ the \\emph{remainder} of the reduction and $\\iur$ the \\emph{interim multiplier} on $f$ with respect to $g$.\n\\end{definition}\n\nIn Definition \\ref{Def:TermReductionPQR} we might have $\\lcm_\\st (c_\\alpha,\\lcc (g))=0$ for $c_\\alpha,\\lcc (g)\\in\\rr^\\ast$ due to the possible existence of zero divisors in $\\rr^\\ast$.\nWe postpone to address this issue until Lemma \\ref{Lemma:SPolynRational} \\eqref{item:NonzeroMultipliers} after the definition of $S$-polynomials over a normal PQR because in what follows we only consider a special kind of term reductions whose interim multipliers $\\iur$ in \\eqref{TermReductionPQR} satisfy $\\iur\\in\\rr^\\times$.\n\n\\begin{definition}[Properly reduced polynomial in $\\rx$]\\label{Def:ProperlyReduced}\n\\hfill\n\nLet $\\rr$ be a normal PQR as in Definition \\ref{Def:nPQR} and $\\succ$ the elimination ordering on $\\rx$ as in Definition \\ref{Def:EliminationOrderingPQR}.\nA term $\\icr_\\alpha\\tilde{\\bx}^\\alpha\\in\\rx$ with the coefficient $\\icr_\\alpha\\in\\rr^\\ast$ is said to be \\emph{properly reducible} with respect to $\\tas=\\{\\Lst f1s\\}\\subset\\rqd$ if there exists an $f_j\\in\\tas$ such that $\\tilde{\\bx}^\\alpha\\in\\langle\\lmc (f_j)\\rangle$ and the interim multiplier $\\iur$ with respect to $f_j$ as in \\eqref{TermReductionPQR} satisfies $\\iur\\in\\rr^\\times$.\nWe say that a polynomial $f\\in\\rx$ is \\emph{properly reduced} with respect to $\\tas$ if none of its terms is properly reducible with respect to $\\tas$.\n\\end{definition}\n\nThe condition $\\iur\\in\\rr^\\times$ for the properness in Definition \\ref{Def:ProperlyReduced} indicates that $\\iur=\\mo (\\lcm (l_\\alpha,l_j)\/l_\\alpha)\\in\\rr^\\times$.\nHere $l_\\alpha:=\\io (\\icr_\\alpha)$ and $l_j:=\\io (\\icr_j)$ with $\\icr_j:=\\lcc (f_j)$.\nHence we can deduce that $\\icr_\\alpha\\in\\pid{\\icr_j}\\subset\\rr$.\nCombined with the condition $\\tilde{\\bx}^\\alpha\\in\\langle\\lmc (f_j)\\rangle$, the condition $\\iur\\in\\rr^\\times$ for the properness in Definition \\ref{Def:ProperlyReduced} is equivalent to the following term divisibility condition:\n\\begin{equation}\\label{DivisibleCondProper}\n\\icr_\\alpha\\tilde{\\bx}^\\alpha\\in\\langle\\icr_j\\cdot\\lmc (f_j)\\rangle=\\langle\\ltc (f_j)\\rangle\\subset\\rx.\n\\end{equation}\n\n\\begin{theorem}[Proper division in \\text{$\\rx$}]\\label{Thm:ProperReduction\n\\hfill\n\nLet $\\rr$ be a normal PQR as in Definition \\ref{Def:nPQR} and $\\succ$ the elimination ordering on $\\rx$ as in Definition \\ref{Def:EliminationOrderingPQR}.\nSuppose that $\\tas=\\{\\Lst f1s\\}$ are polynomials in $\\rqd$.\nFor every $f\\in\\rx$, there exist a multiplier $\\mr\\in\\rr^\\times$ as well as a remainder $\\dr\\in\\rx$ and quotients $q_j\\in\\rx$ for $1\\le j\\le s$ such that:\n\\begin{equation}\\label{ProperDivisionExpression}\n\\mr f=\\sum_{j=1}^s q_jf_j+\\dr,\n\\end{equation}\nwhere $\\dr$ is properly reduced with respect to $\\tas$.\nMoreover, the polynomials in \\eqref{ProperDivisionExpression} have to satisfy the following condition:\n\\begin{equation}\\label{ProperDivisionCond}\n\\lmc (f)=\\max\\{\\max_{1\\le j\\le s}\\{\\lmc (q_j)\\cdot\\lmc (f_j)\\},\\lmc (r)\\}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe proof amounts to a verbatim repetition of that for Theorem \\ref{Thm:PseudoReduction} if we substitute the criterion of being properly reduced for that of being pseudo-reduced.\nIn fact, the polynomial division on a normal PQR $\\rr$ as in \\eqref{SimpleDivisionIdentity} is the same as that on $R=\\kxn$.\nMoreover, a normal PQR $\\rr$ as in Definition \\ref{Def:nPQR} is also a Noetherian ring since it is isomorphic to the PQR $\\bar R$ in Definition \\ref{Def:PQR} that is Noetherian.\n\\end{proof}\n\nPlease note that the product $\\lmc (q_j)\\cdot\\lmc (f_j)$ in \\eqref{ProperDivisionCond} is based upon the term divisibility condition \\eqref{DivisibleCondProper}.\n\nWe also call the proper division in Theorem \\ref{Thm:ProperReduction} a \\emph{proper reduction} henceforth.\nWe can easily contrive a proper division algorithm based on Theorem \\ref{Thm:ProperReduction}.\n\nFor $f,g\\in\\rnd$, suppose that $\\lcm_\\st (\\lcc (f),\\lcc (g))=0$ due to the existence of zero divisors in $\\rr^\\ast$.\nIn this case if we employed Definition \\ref{Def:SPolynomial} for $S$-polynomials directly and in particular, the multiplier $\\lmr=\\lcm_\\qr (\\lcc (f),\\lcc (g))$, then their $S$-polynomial $S(f,g)$ would equal $0$ since $\\lmr=\\lcm_\\qr (\\lcc (f),\\lcc (g))=0$ in \\eqref{SPolynomialDef} as per \\eqref{UniqueGcds}.\nHence we need to revise Definition \\ref{Def:SPolynomial} as follows.\n\n\\begin{definition}[$S$-polynomial over a normal PQR $\\rr$]\\label{Def:SpolynPQR}\n\\hfill\n\nLet $\\rr$ be a normal PQR as in Defintion \\ref{Def:nPQR}.\nSuppose that $f\\in\\rqd$ and $g\\in\\rnd$.\nLet us use $c_f$ and $c_g$ to denote $\\lcc (f)$ and $\\lcc (g)$ in $\\rr^\\ast$ respectively.\nWith the epimorphism $\\mo\\colon R\\rightarrow\\rr$ and injection $\\io\\colon\\rr\\rightarrow R$ defined as in \\eqref{ProjectionPQR} and \\eqref{PQREmbedding} respectively, we denote $l_f:=\\io(c_f)$ and $l_g:=\\io(c_g)$.\nWe also define multipliers $\\lmr_f:=\\mo (\\lcm (l_f,l_g)\/l_f)$ and $\\lmr_g:=\\mo (\\lcm (l_f,l_g)\/l_g)$ as well as the monomial $\\clm:=\\lcm (\\lmc (f),\\lmc (g))\\in\\tM$.\nThen the following polynomial:\n\\begin{equation}\\label{SPolyPQR}\nS(f,g):=\\frac{\\lmr_f\\clm}{\\lmc (f)}f-\\frac{\\lmr_g\\clm}{\\lmc (g)}g\n\\end{equation}\nis called the \\emph{$S$-polynomial} of $f$ and $g$ in $\\rx$.\n\\end{definition}\n\nIn particular, when $f\\in\\rqd$ and $\\tg\\in\\rat$, we can take $\\lmc (\\tg)=1$ and $c_g=\\lcc (\\tg)=g$ in Definition \\ref{Def:SpolynPQR}.\nIf we define $l_\\tg:=\\io (\\tg)$, the definitions for $\\lmr_f$ and $\\lmr_\\tg$ in \\eqref{SPolyPQR} are unaltered.\nNow $\\clm=\\lmc (f)$ and the $S$-polynomial in \\eqref{SPolyPQR} becomes:\n\\begin{equation}\\label{SpecialSPolyPQR}\nS(f,\\tg):=\\lmr_ff-\\lmr_\\tg\\tg\\cdot\\lmc (f).\n\\end{equation}\n\n\\begin{lemma}\\label{Lemma:IdentitySpecialSPoly}\nFor $f\\in\\rqd$ and $\\tg\\in\\rat$, and with the same notations as in \\eqref{SpecialSPolyPQR}, let us further denote $\\idr:=\\gcd (l_\\tf,l_\\tg)$.\nThen the $S$-polynomial $S(\\tf,\\tg)$ in \\eqref{SpecialSPolyPQR} satisfies the following identity:\n\\begin{equation}\\label{IdentitySpecialSPoly}\nS(\\tf,\\tg)=\\mo\\Bigl(\\frac {l_\\tg}\\idr\\Bigr)(f-\\ltc (f))=\\lmr_\\tf (f-\\ltc (f))\n\\end{equation}\nwith $\\lmr_f=\\mo (\\lcm (l_f,l_g)\/l_f)$ being defined as in \\eqref{SPolyPQR}.\n\\end{lemma}\n\\begin{proof}\nIt is evident that $\\lmr_\\tf=\\mo (l_g\/\\idr)$ and $\\lmr_\\tg=\\mo (l_\\tf\/\\idr)$.\nHence from \\eqref{SpecialSPolyPQR} follows directly:\n\\begin{equation*}\nS(f,\\tg)=\\mo\\Bigl(\\frac {l_\\tg}\\idr\\Bigr)f-\\mo\\Bigl(\\frac {l_\\tf}\\idr\\Bigr)\\mo (l_\\tg)\\cdot\\lmc (f)=\\mo\\Bigl(\\frac {l_\\tg}\\idr\\Bigr)(f-\\ltc (f))\n\\end{equation*}\nsince $\\mo\\circ\\io$ is the identity map on $\\rr$.\n\\end{proof}\nThere is a special kind of $S$-polynomials for $f\\in\\rqd$ when $c_f=\\lcc (f)\\in\\rat$.\n\\begin{equation}\\label{BeheadSPolyPQR}\nS(f,\\qr):=\\lnr_ff=\\lnr_f (f-\\ltc (f))\n\\end{equation}\nwith $\\lnr_f:=\\mo (\\lcm (l_f,\\qr)\/l_f)=\\mo (\\qr\/\\gcd (l_f,\\qr))$.\nHere $l_f:=\\io (c_f)$ as in \\eqref{SPolyPQR}.\n\n\\begin{lemma}\\label{Lemma:RelativePrimePQR}\nFor $f,g\\in\\rqd$, suppose that $\\lmc (f)$ and $\\lmc (g)$ are relatively prime.\nWith the same notations as in Definition \\ref{Def:SpolynPQR}, let us also denote $\\idr:=\\gcd (l_f,l_g)$.\nThen their $S$-polynomial in \\eqref{SPolyPQR} satisfies:\n\\begin{equation}\\label{CoprimeReductionPQR}\nS(f,g)=\\frac{f_1\\cdot\\ltc (g)-g_1\\cdot\\ltc (f)}{\\mo (\\idr)}=\\frac{f_1g-g_1f}{\\gcd_\\qr (\\lcc (f),\\lcc (g))}\n\\end{equation}\nwith $f_1:=f-\\ltc (f)$ and $g_1:=g-\\ltc (g)$.\nMoreover, we have:\n\\begin{equation}\\label{LeadTwoMonomialPQR}\n\\max\\{\\lmc (f_1)\\cdot\\lmc (g),\\lmc (g_1)\\cdot\\lmc (f)\\}\\prec\\lmc (f)\\cdot\\lmc (g).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nIn the definition \\eqref{SPolyPQR} we have $\\clm=\\lmc (f)\\cdot\\lmc (g)$.\nFurthermore, $\\lmr_f=\\mo (l_g\/\\idr)$ and $\\lmr_g=\\mo (l_f\/\\idr)$.\nThus the first equality in \\eqref{CoprimeReductionPQR} follows from \\eqref{SPolyPQR} as follows.\n\\begin{equation*}\nS(f,g)=\\mo\\Bigl(\\frac{l_g}{\\idr}\\Bigr)\\cdot\\lmc (g)(f_1+\\ltc (f))-\n\\mo\\Bigl(\\frac{l_f}{\\idr}\\Bigr)\\cdot\\lmc (f)(g_1+\\ltc (g)),\n\\end{equation*}\nwhere we can write $\\mo (l_g\/\\idr)$ as $\\mo (l_g)\/\\mo (\\idr)=c_g\/\\mo (\\idr)$ and same for $\\mo (l_f\/\\idr)$.\n\nThe second equality in \\eqref{CoprimeReductionPQR} follows from the first one by substituting $g-g_1$ and $f-f_1$ for $\\ltc (g)$ and $\\ltc (f)$ respectively.\nAnd the denominator $\\mo (\\idr)=\\gcd_\\qr (\\lcc (f),\\lcc (g))$ is defined as in \\eqref{GcdPQRqr}.\n\nThe inequality \\eqref{LeadTwoMonomialPQR} is evident since $\\lmc (f_1)\\prec\\lmc (f)$ and $\\lmc (g_1)\\prec\\lmc (g)$.\n\\end{proof}\n\nFrom Lemma \\ref{Lemma:IdentitySpecialSPoly} and Lemma \\ref{Lemma:RelativePrimePQR} and based on Theorem \\ref{Thm:ProperReduction}, we can easily deduce the following corollary for algorithmic simplifications.\n\n\\begin{corollary}\\label{Cor:PQRCoprime}\nFor $f,g\\in\\rqd$, suppose that $\\lmc (f)$ and $\\lmc (g)$ are relatively prime.\nWith the same notations as in Lemma \\ref{Lemma:RelativePrimePQR}, if $\\mo (\\idr)\\in\\rr^\\times$, then their $S$-polynomial $S(f,g)$ as in \\eqref{CoprimeReductionPQR} can be properly reduced to $0$ by $f$ and $g$ as in Theorem \\ref{Thm:ProperReduction} with the multiplier $\\mo (\\idr)$ and quotients $f_1$ and $-g_1$.\n\nIn particular, for $f\\in\\rqd$ and $\\tg\\in\\rat$, with the same notations as in Lemma \\ref{Lemma:IdentitySpecialSPoly}, if $\\mo (\\idr)=\\gcd_\\qr (\\lcc (f),\\tg)\\in\\rr^\\times$, then their $S$-polynomial $S(f,g)$ as in \\eqref{IdentitySpecialSPoly} can be properly reduced to $0$ by $g$ with the multiplier $\\mo (\\idr)\\in\\rr^\\times$ and quotient $f-\\ltc (f)$.\n\\end{corollary}\n\n\\begin{definition}[\\Lcm\\ representation\\footnote{Please refer to \\cite[P107, Definition 5]{CLO15} for its definition over a field instead.}]\\label{Def:LcmRep}\n\\hfill\n\nFor $\\tas=\\{\\Lst f1s\\}\\subset\\rnd$, we say that an $S$-polynomial $S(f,g)$ has an \\emph{\\Lcm\\ representation} with respect to $\\tas$ if there exist $\\{\\Lst h1s\\}\\subset\\rx$ satisfying:\n\\begin{equation*}\nS(f,g)=\\sum_{j=1}^sh_jf_j\n\\end{equation*}\nsuch that the following condition holds:\n\\begin{equation}\\label{LcmRepCond}\n\\max_{1\\le j\\le s}\\{\\lmc (h_j)\\cdot\\lmc (f_j)\\}\\prec\\lcm (\\lmc (f),\\lmc (g)).\n\\end{equation}\n\\end{definition}\n\n\\begin{remark}\\label{Rmk:LCMRep}\nThe \\Lcm\\ representation is especially suitable for the representation of $S$-polynomials over such rings with zero divisors as the PQR $\\rr$.\nIn particular, the condition \\eqref{LeadTwoMonomialPQR} in Lemma \\ref{Lemma:RelativePrimePQR} amounts to the condition \\eqref{LcmRepCond} for the \\Lcm\\ representation with respect to $\\{\\tf,\\tg\\}$ when the multiplier $\\mo (\\idr)\\in\\rr^\\times$ in \\eqref{CoprimeReductionPQR} as in Corollary \\ref{Cor:PQRCoprime}.\nSimilarly the identity \\eqref{IdentitySpecialSPoly} is also an \\Lcm\\ representation of $S(f,g)$ with respect to $\\tg\\in\\rat$ when the multiplier $\\mo (\\idr)\\in\\rr^\\times$.\n\\end{remark}\n\nFor $g\\in\\rat$ and $f\\in\\rqd$, we shall also use the following relation between \\eqref{SPolyPQR} and \\eqref{SpecialSPolyPQR}:\n\\begin{equation}\\label{DisappearingMonomialPQR}\nS(f,g\\cdot\\lmc (f))=S(f,g).\n\\end{equation}\n\n\\begin{lemma}\\label{Lemma:SPolynRational}\n\\begin{inparaenum}[(i)]\n\\item\\label{item:SPolyRational} The $S$-polynomial in \\eqref{SPolyPQR} satisfies $\\lmc (S(f,g))\\prec\\clm$.\nThe $S$-polynomials in \\eqref{SpecialSPolyPQR} and \\eqref{BeheadSPolyPQR} satisfy $\\lmc (S(f,g))\\prec\\lmc (f)$ and $\\lmc (S(f,\\qr))\\prec\\lmc (f)$ respectively.\n\\item\\label{item:NonzeroMultipliers} The two multipliers $\\lmr_f$ and $\\lmr_g$ in \\eqref{SPolyPQR} and \\eqref{SpecialSPolyPQR} are not zero, that is, we always have $\\lmr_f,\\lmr_g\\in\\rr^\\ast$ even if $\\lcm_\\qr (\\lcc (f),\\lcc (g))=0$ with $\\lcm_\\qr$ defined as in \\eqref{GcdPQRqr}.\n\\end{inparaenum}\n\\end{lemma}\n\\begin{proof}\nTo prove the conclusion \\eqref{item:SPolyRational}, it suffices to prove that $\\lmr_fc_f=\\lmr_gc_g$.\nIn particular, we have $c_g=g$ in the case of \\eqref{SpecialSPolyPQR}.\nThe composition $\\mo\\circ\\io$ of the epimorphism $\\mo$ in \\eqref{ProjectionPQR} and injection $\\io$ in \\eqref{PQREmbedding} is the identity map on $\\rr$.\nHence $c_f=\\mo (\\io (c_f))=\\mo (l_f)$ and we have the following verification:\n\\begin{equation*}\n\\lmr_fc_f=\\mo\\Bigl(\\frac{\\lcm (l_f,l_g)}{l_f}\\Bigr)\\cdot\\mo (l_f)=\\mo (\\lcm (l_f,l_g)).\n\\end{equation*}\nSimilarly we have $\\lmr_gc_g=\\mo (\\lcm (l_f,l_g))$ and thus the conclusion follows.\nThe conclusion for \\eqref{BeheadSPolyPQR} is easy to corroborate.\n\nThe conclusion \\eqref{item:NonzeroMultipliers} follows from the identity:\n\\begin{equation}\\label{NonzeroMultipliers}\n\\lmr_f=\\mo\\Bigl(\\frac{\\lcm (l_f,l_g)}{l_f}\\Bigr)=\\mo\\Bigl(\\frac{l_g}{\\gcd (l_f,l_g)}\\Bigr).\n\\end{equation}\nIn fact, according to the definition of leading coefficients in Notation \\ref{Notation:LeadingEntities}, $c_g=\\lcc (g)\\in\\rr^\\ast$.\nHence $l_g=\\io (c_g)\\in (\\kxn)^*$ and $\\deg (l_g)<\\deg (\\qr)$ as in \\eqref{ResidueRing}.\nThus the multiplier $\\lmr_f\\in\\rr^\\ast$ by \\eqref{NonzeroMultipliers}.\nThe same holds for $\\lmr_g$.\n\\end{proof}\n\nA conspicuous difference between the $S$-polynomials in Definition \\ref{Def:SpolynPQR} over a normal PQR and those in Definition \\ref{Def:SPolynomial} over a PID is that the leading coefficients $\\lmr_f\\cdot\\lcc (f)=\\lmr_fc_f$ and $\\lmr_g\\cdot\\lcc (g)=\\lmr_gc_g$ in \\eqref{SPolyPQR} and \\eqref{SpecialSPolyPQR} might be zero due to the possible existence of zero divisors in $\\rr^\\ast$.\nWe shall prove that this imposes no hindrance to the viability of our computations.\nFor $S$-polynomials over a normal PQR $\\rr$, Lemma \\ref{Lemma:SPolynRational} \\eqref{item:SPolyRational} is equivalent to the inequality \\eqref{LeadingSMonomials}.\n\nFor $f,g\\in\\rnd$, let us define:\n\\begin{equation}\\label{LCMstTerm}\n\\lcm\\nolimits_\\qr (\\ltc (f),\\ltc (g)):=\\lcm\\nolimits_\\qr (\\lcc (f),\\lcc (g))\\cdot\\lcm (\\lmc (f),\\lmc (g))\n\\end{equation}\nwith $\\lcm_\\qr$ being defined as in \\eqref{GcdPQRqr}.\n\nLet us use the same notations as in Definition \\ref{Def:SpolynPQR} for $S$-polynomials.\nFor $f,g\\in\\rnd$ without both of them in $\\rat$, we define the \\emph{\\Lcm\\ multiplier} of $f$ and $g$ as:\n\\begin{equation}\\label{CMR}\n\\cmr (g\\vert f):=\\mo\\Bigl(\\frac{\\lcm (l_f,l_g)}{l_f}\\Bigr)\\frac{\\lcm (\\lmc (f),\\lmc (g))}{\\lmc (f)}=\\frac{\\lmr_f\\clm}{\\lmc (f)}.\n\\end{equation}\nThen the definition for the $S$-polynomial $S(f,g)$ in \\eqref{SPolyPQR} can be written as:\n\\begin{equation}\\label{LMRRepSPoly}\nS(f,g)=\\cmr (g\\vert f)\\cdot f-\\cmr (f\\vert g)\\cdot g.\n\\end{equation}\n\n\\begin{lemma}\\label{Lemma:TriangleIdentityPQR}\nFor $f,g,h\\in\\rnd$ with at most one of them in $\\rat$, if $\\lcm (\\lmc (f),\\lmc (g))\\in\\langle\\lmc (h)\\rangle$, then we have the following relationship between their $S$-polynomials:\n\\begin{equation}\\label{TriangleIdentityPQR}\n\\mr S(f,g)=\\frac{\\mr\\cdot\\cmr (g\\vert f)}{\\cmr (h\\vert f)}S(f,h)-\\frac{\\mr\\cdot\\cmr (f\\vert g)}{\\cmr (h\\vert g)}S(g,h)\n\\end{equation}\nwith the \\Lcm\\ multiplier $\\cmr$ being defined as in \\eqref{CMR}.\nHere the multiplier $\\mr:=\\mo (l_h\/\\idr)\\in\\rr^\\ast$ with $l_h:=\\io (\\lcc (h))$ and $\\idr:=\\gcd (\\lcm (l_f,l_g),l_h)$.\n\\end{lemma}\n\\begin{proof}\nSame as the conclusion in Lemma \\ref{Lemma:SPolynRational} \\eqref{item:NonzeroMultipliers}, the denominators $\\cmr (h\\vert f)$ and $\\cmr (h\\vert g)$ in \\eqref{TriangleIdentityPQR} are nonzero, which is the reason why we use the \\Lcm\\ multiplier $\\cmr$ as in \\eqref{CMR} instead of the $\\lcm_\\qr$ as in \\eqref{LCMstTerm}.\n\nThe multiplier $\\mr:=\\mo (l_h\/\\idr)\\in\\rr^\\ast$ can indeed render the two fractions in \\eqref{TriangleIdentityPQR} terms in $\\rx$.\nThe proof is totally similar to that for the multiplier $\\mr$ in the identity \\eqref{TriangleIdentity} in Lemma \\ref{Lemma:TriangleIdentity}.\n\nWe can corroborate the identity in \\eqref{TriangleIdentityPQR} directly by the form of $S$-polynomials in \\eqref{LMRRepSPoly} as well as the definition of \\Lcm\\ multipliers in \\eqref{CMR}.\n\\end{proof}\n\nBased on the above discussions we now analyze the incompatible part $\\Ip (\\pel)$ of the pseudo-eliminant $\\pel$.\nOur goal is to determine the corresponding factors of the eliminant $\\el$ of the original ideal $I$.\n\nLet $\\{\\ids_i\\colon 1\\le i\\le s\\}$ be the composite divisor sets of the incompatible part $\\Ip (\\pel)$ as in Definition \\ref{Def:CompositeDivisor}.\nFor a multiplicity $i$ satisfying $1\\le i\\le s$ and composite divisor $\\sfr^i$ with $\\sfr\\in\\ids_i\\subset\\kxn$, let us denote $\\sfr^i$ as $\\qr$ and consider the normal PQR $\\rr$ that is isomorphic to the PQR $\\bar R=\\kxn\/\\pid\\qr=\\kxn\/\\pid{\\sfr^i}$ as in Definition \\ref{Def:nPQR}.\nIn short, from now on our discussions and computations are over the normal PQR $\\rr$ as follows.\n\\begin{equation}\\label{kxnPQR}\n\\rr\\cong\\kxn\/\\pid\\qr,\\quad\\qr=\\sfr^i,\\quad\\sfr\\in\\ids_i\\subset\\kxn.\n\\end{equation}\n\nWe shall follow the pseudo-eliminant computation in Algorithm \\ref{Algo:PseudoEliminant} to compute the eliminant of the ideal $I+\\pid\\qr=I+\\pid{\\sfr^i}$ except that we shall compute it over the normal PQR $\\rr$.\n\nIf we extend the ring epimorphism $\\mo$ in \\eqref{ProjectionPQR} such that it is the identity map on the variables $\\tilde{\\bx}$, then $\\mo$ induces a ring epimorphism from $\\knx$ to $\\rx$ which we still denote as $\\mo$ as follows.\n\\begin{equation}\\label{ExtendedProjectionPQR}\n\\mo\\colon\\knx\\rightarrow\\rx\\colon\\quad\\mo\\Bigl(\\sum_{j=1}^s c_j\\tilde{\\bx}^{\\alpha_j}\\Bigr):=\\sum_{j=1}^s\\mo(c_j)\\tilde{\\bx}^{\\alpha_j}.\n\\end{equation}\n\nPlease note that when the composite divisor $\\qr$ bears the form $x_1-a$ with $a\\in K$, the epimorphism $\\mo$ in \\eqref{ProjectionPQR} becomes $\\mo (f)=f(a)\\in K$ for $f\\in\\kxn$.\nIn this case the coefficients $\\mo(c_j)$ in \\eqref{ExtendedProjectionPQR} become $c_j(a)\\in K$.\nWe call the induced epimorphism $\\mo$ in \\eqref{ExtendedProjectionPQR} a \\emph{specialization} associated with $a\\in K$.\n\nSimilarly we can extend the injection $\\io$ in \\eqref{PQREmbedding} to an injection of $\\rx$ into $\\knx$ in the way that it is the identity map on the variables $\\tilde{\\bx}$ as follows.\n\\begin{equation}\\label{ExtendedEmbedding}\n\\io\\colon\\rx\\rightarrow\\knx\\colon\\quad\\io\\Bigl(\\sum_{j=1}^s c_j\\tilde{\\bx}^{\\alpha_j}\\Bigr):=\\sum_{j=1}^s\\io(c_j)\\tilde{\\bx}^{\\alpha_j}.\n\\end{equation}\n\nFurther, it is evident that $\\mo\\circ\\io$ is the identity map on $\\rx$.\n\n\\begin{lemma}\\label{Lemma:InitialBasisPQR}\nLet $\\succ$ be an elimination ordering on $\\bM$ as in Definition \\ref{Def:EliminationOrdering} and $\\tas:=\\{f_j\\colon 0\\le j\\le s\\}\\subset\\knx\\setminus K$ a basis of a zero-dimensional ideal $I$.\nSuppose that $\\qr=\\sfr^i$ is a composite divisor of the incompatible part $\\Ip (\\pel)$ as in Definition \\ref{Def:CompositeDivisor} and $\\rr$ a normal PQR as in Definition \\ref{Def:nPQR}.\nThen for $\\tas\\cap\\kxn$ and $\\tbs:=\\tas\\setminus\\kxn$ as in the Initialization of Algorithm \\ref{Algo:PseudoEliminant}, we have $\\mo (\\tas\\cap\\kxn)=\\{0\\}$\nand $\\mo (\\tbs)$ is a basis of $\\Iq:=\\mo (I)$ under the epimorphism $\\mo$ in \\eqref{ExtendedProjectionPQR}.\n\\end{lemma}\n\\begin{proof}\nThe construction of the composite divisor set $\\ids_i$ in Algorithm \\ref{Algo:CompatiblePartPseudoEliminant} indicates that the pseudo-eliminant $\\pel$ is divisible by the composite divisor $\\qr=\\sfr^i$.\nThe computation of $\\pel$ in Algorithm \\ref{Algo:PseudoEliminant} shows that every element of $\\tas\\cap\\kxn$ is divisible by $\\lel$ in the Initialization of Algorithm \\ref{Algo:PseudoEliminant} and thus by $\\pel$.\nHence $\\tas\\cap\\kxn\\subset\\pid{\\sfr^i}=\\pid\\qr\\subset\\kxn$, which yields $\\mo (\\tas\\cap\\kxn)=\\{0\\}$.\nThen readily follows the conclusion $\\Iq=\\langle\\mo (\\tbs)\\rangle$ with $\\mo (\\tbs)\\subset\\rqd$.\n\\end{proof}\n\nIn the following Algorithm \\ref{Algo:ProperEliminant} that is parallel to Algorithm \\ref{Algo:PseudoEliminant}, please note that all the binary operations over the ring $\\rr$ in \\eqref{kxnPQR}, i.e., the additions and multiplications over the ring $\\rr$ in \\eqref{kxnPQR}, are performed according to those defined in \\eqref{BinaryOperations}.\n\nBased on Lemma \\ref{Lemma:InitialBasisPQR}, in what follows let us abuse the notations a bit and simply denote $\\mo (\\tbs)$ as $\\tas=\\{f_j\\colon 1\\le j\\le s\\}\\subset\\rqd$.\n\n\\begin{algorithm}[Proper eliminant and proper basis over a normal PQR $\\rr$]\\label{Algo:ProperEliminant}\n\\hfil\n\nInput: A finite polynomial set $\\tas\\subset\\rqd$.\n\nOutput: A proper eliminant $\\ee\\in\\rr$ and proper basis $\\prb\\subset\\rqd$.\n\nInitialization: A temporary set $\\mathfrak{S}:=\\emptyset$ in $\\rx\\setminus\\rr$ for $S$-polynomials; a temporary $e\\in\\rr$ as $e:=0$.\n\nFor each pair $\\tf,\\tg\\in\\tas$ with $\\tf\\ne\\tg$, we invoke \\Po R as follows to compute their $S$-polynomial $S(f,g)$.\n\n\\Po R:\n\n\\leftskip=5mm\n\\begin{itshape}\nIf $\\lmc (\\tf)$ and $\\lmc (\\tg)$ are relatively prime, we compute the multiplier $\\mo (\\idr):=\\gcd_\\qr (\\lcc (f),\\lcc (g))$ as in \\eqref{CoprimeReductionPQR}.\nIf $\\mo (\\idr)\\in\\rat$, we compute and then add the $S$-polynomial $S(\\tf,\\tg)$ into the set $\\mathfrak{S}$.\nIf $\\mo (\\idr)\\in\\rr^\\times$ as in Corollary \\ref{Cor:PQRCoprime}, we disregard $S(\\tf,\\tg)$.\n\nIf there exists an $h\\in\\tas\\setminus\\{\\tf,\\tg\\}$ such that $\\lcm (\\lmc (f),\\lmc (g))\\in\\langle\\lmc (h)\\rangle$, and the triangular identity \\eqref{TriangleIdentityPQR} has not been applied to the same triplet $\\{f,g,h\\}$ before, we compute the multiplier $\\mr$ as in \\eqref{TriangleIdentityPQR}.\nIf $\\mr\\in\\rat$, we compute and then add the $S$-polynomial $S(\\tf,\\tg)$ into the set $\\mathfrak{S}$.\nIf $\\mr\\in\\rr^\\times$, we disregard $S(\\tf,\\tg)$.\n\nIf neither of the above two cases is true, we compute their $S$-polynomial $S(\\tf,\\tg)$ as in \\eqref{SPolyPQR}.\nThen we add $S(\\tf,\\tg)$ into the set $\\mathfrak{S}$.\n\\end{itshape}\n\nEnd of $\\mathcal{R}$\n\\leftskip=0mm\n\nWe recursively repeat \\Po P as follows for proper reductions of all the $S$-polynomials in $\\mathfrak{S}$.\n\n\\Po P:\n\n\\leftskip=5mm\n\\begin{itshape}\nFor an $S\\in\\mathfrak{S}$, we invoke Theorem \\ref{Thm:ProperReduction} to make a proper reduction of $S$ by $\\tas$.\n\nIf the remainder $\\dr\\in\\rqd$, we add $\\dr$ into $\\tas$.\nFor every $\\tf\\in\\tas\\setminus\\{\\dr\\}$, we invoke \\Po R to compute the $S$-polynomial $S(\\tf,\\dr)$.\n\nIf the remainder $\\dr\\in\\rat$ and $e=0$, we redefine $\\et:=\\mo(\\gcd (\\io (\\dr),q))$ with $\\mo$ and $\\io$ as in \\eqref{ProjectionPQR} and \\eqref{PQREmbedding} respectively.\n\nIf the remainder $\\dr\\in\\rat$ and $\\et\\in\\rr^\\ast$, we compute $\\idr=\\gcd_\\qr (\\dr,\\et)$ as in \\eqref{GcdPQRqr}.\nIf $\\idr$ is not an associate of $\\et$, we redefine $\\et:=\\idr$.\n\nIf the remainder $\\dr\\in\\rr^\\times$, we halt the algorithm and output $\\ee=1$.\n\nThen we delete $S$ from $\\mathfrak{S}$.\n\\end{itshape}\n\nEnd of $\\mathcal{P}$\n\\leftskip=0mm\n\nNext we recursively repeat \\Po Q as follows for proper reductions of the special kinds of $S$-polynomials in \\eqref{SpecialSPolyPQR} and \\eqref{BeheadSPolyPQR}.\n\n\\Po Q:\n\n\\leftskip=5mm\n\\begin{itshape}\nIf $\\mathfrak{S}=\\emptyset$ and $\\et=0$, then for every $\\tf\\in\\tas$ with $\\lcc (\\tf)\\in\\rat$, we compute the $S$-polynomial $S(\\tf,\\qr)$ as in \\eqref{BeheadSPolyPQR} and add it into $\\mathfrak{S}$ if this has not been done for $\\tf$ in a previous step.\n\nThen we recursively repeat \\Po P.\n\nIf $\\mathfrak{S}=\\emptyset$ and $\\et\\in\\rr^\\ast$, then for every $\\tf\\in\\tas$ with $\\lcc (\\tf)\\in\\rat$, if $\\mo (\\idr):=\\gcd_\\qr (\\lcc (\\tf),\\et)\\in\\rat$ as in Corollary \\ref{Cor:PQRCoprime}, we compute the $S$-polynomial $S(\\tf,\\et)$ as in \\eqref{IdentitySpecialSPoly} and add it into $\\mathfrak{S}$ unless an $S$-polynomial equal to $uS(\\tf,\\et)$ with $u\\in\\rr^\\times$ had been added into $\\mathfrak{S}$ in a previous step.\n\nThen we recursively repeat \\Po P.\n\\end{itshape}\n\nEnd of $\\mathcal{Q}$\n\\leftskip=0mm\n\nFinally we define $\\ee:=\\et$ and $\\prb:=\\tas$ respectively.\n\nWe output $\\ee$ and $\\prb$.\n\\end{algorithm}\n\n\\begin{remark}\nIn \\Po Q of Algorithm \\ref{Algo:ProperEliminant}, the condition $\\lcc (f)\\in\\rat$ in the case of $\\et\\in\\rr^\\ast$ is necessary because if $\\lcc (f)\\in\\rr^\\times$, we would have $\\idr:=\\gcd_\\qr (\\lcc (\\tf),\\et)\\in\\rr^\\times$ as in Corollary \\ref{Cor:PQRCoprime}.\n\\end{remark}\n\n\\begin{definition}[Proper eliminant $\\ee$; proper basis $\\prb$; modular eliminant $\\mel$]\\label{Def:ProperEliminant}\n\\hfill\n\nWe call the standard representation $\\ee^\\st$ as in \\eqref{StdRepsPQR} of the univariate polynomial $\\ee\\in\\Iq\\cap\\rr$ obtained in Algorithm \\ref{Algo:ProperEliminant}, whether it is zero or not, a \\emph{proper} eliminant of the ideal $\\Iq$.\nIn what follows we shall simply denote $\\ee:=\\ee^\\st$ except for a necessary discrimination in the context.\nWe also call the final polynomial set $\\prb$ obtained in Algorithm \\ref{Algo:ProperEliminant} a \\emph{proper} basis of $\\Iq$.\n\nLet $\\el$ be the eliminant of a zero-dimensional ideal $I$ and $\\qr$ a composite divisor as in \\eqref{kxnPQR}.\nSuppose that $\\mo$ is the epimorphism as in \\eqref{ExtendedProjectionPQR}.\nThen we define $\\mel:=\\mo (\\el)$ as the \\emph{modular} eliminant of $\\Iq=\\mo (I)$.\n\\end{definition}\n\n\\begin{lemma}\\label{Lemma:ProperEliminant}\nLet $\\mel$ and $\\ee$ be the modular and proper eliminants of the ideal $\\Iq$ respectively as in Definition \\ref{Def:ProperEliminant}.\nThen the following conclusions hold.\n\\begin{enumerate}[(a)]\n\\item\\label{item:EliminantAndItsPart} If the modular eliminant $\\mel\\in\\rr^\\ast$, then the eliminant $\\el$ is divisible by $\\io (\\mel^\\st)$ with $\\io$ being the injection as in \\eqref{PQREmbedding} and $\\mel^\\st$ the standard representation of $\\mel$ in $\\rr$ as in \\eqref{StdRepsPQR}.\nAnd we have $\\mlt_\\ipr (\\el)=\\mlt_\\ipr (\\mel^\\st)$ for every irreducible factor $\\ipr$ of the composite divisor $\\qr=\\sfr^i$.\nIf $\\mel=0$, then $\\el$ is divisible by $\\qr$ and we have $\\mlt_\\ipr (\\el)=\\mlt_\\ipr (\\qr)$ for every irreducible factor $\\ipr$ of $\\qr$.\n\n\\item\\label{item:ProperEliminant} The epimorphism $\\mo$ as in \\eqref{ExtendedProjectionPQR} is also an epimorphism from $I\\cap\\kxn$ to $\\Iq\\cap\\rr$.\nMoreover, the proper and modular eliminants $\\ee$ and $\\mel$ of $\\Iq$ satisfy $\\ee\\in\\pid\\mel=\\Iq\\cap\\rr$.\nIn particular, we have $\\ee=0$ if $\\mel=0$.\n\n\\item\\label{item:SPolynReductionPQR} For each pair $f\\ne g$ in the polynomial set $\\prb\\cup\\{\\ee\\}$ with $\\ee\\in\\rr^\\ast$ and $\\prb$ being the proper basis of $\\Iq$, the proper reduction of their $S$-polynomial $S(f,g)$ by $\\prb$ yields a remainder $\\dr\\in\\pid\\ee\\subset\\pid\\mel\\subset\\rr$ including the special case of $\\dr=0$.\n The same holds for the polynomial set $\\prb\\cup\\{\\qr\\}$ when $\\ee=0$.\n\n\\item\\label{item:ProperEliminantTermination} Algorithm \\ref{Algo:ProperEliminant} terminates in finite steps.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nLet us first prove \\eqref{item:EliminantAndItsPart}.\nWhen $\\mel\\in\\rr^\\ast$, by Lemma \\ref{Lemma:PQRProperties} as well as the definition of the standard representation $\\mel^\\st$ of $\\mel$ as in \\eqref{StdRepsPQR}, we know that $\\mel=u\\mel^\\st$ with $u\\in\\rr^\\times$ being a unit.\nMoreover, for every irreducible factor $\\ipr$ of $\\qr$, we have $0\\le\\mlt_\\ipr (\\mel^\\st)\\le\\mlt_\\ipr (\\qr)$ as per Lemma \\ref{Lemma:PQRProperties} \\eqref{item:StandardRep} and \\eqref{StdRepsPQR}.\nAs per Lemma \\ref{Lemma:PseudoEliminantTerminate} \\eqref{item:EliminantDivisibility}, the pseudo-eliminant $\\pel$ is divisible by $\\el$.\nBy Definition \\ref{Def:CompositeDivisor}, the composite divisor $\\qr=\\sfr^i$ satisfies $\\mlt_\\ipr (\\qr)=i=\\mlt_\\ipr (\\pel)$ for every irreducible factor $\\ipr$ of $\\qr$.\nHence follows the following inequality:\n\\begin{equation}\\label{MinimumEliMultiplicity}\n0<\\mlt_\\ipr (\\el)\\le\\mlt_\\ipr (\\qr)=i\n\\end{equation}\nfor every irreducible factor $\\ipr$ of $\\qr$.\nBased on the division identity $\\el=h\\qr+\\io (\\mel)=h\\qr+\\io (u\\mel^\\st)$ that is parallel to \\eqref{DivisionPQR}, we can deduce that $\\mlt_\\ipr (\\mel^\\st)=\\mlt_\\ipr (\\io (\\mel^\\st))=\\mlt_\\ipr (\\el)$ for every irreducible factor $\\ipr$ of $\\qr$, which is similar to the deduction of \\eqref{InvariantMultiplicity}.\nThus the eliminant $\\el$ is divisible by $\\io (\\mel^\\st)$ as in the conclusion \\eqref{item:EliminantAndItsPart} due to the arbitrariness of the irreducible factor $\\ipr$ of $\\qr$.\n\nWhen the modular eliminant $\\mel=0$, the divisibility of $\\el$ by $\\qr$ can be readily deduced from the definition of the epimorphism $\\mo$ in \\eqref{ProjectionPQR}.\nThen the equality $\\mlt_\\ipr (\\el)=\\mlt_\\ipr (\\qr)$ for every irreducible factor $\\ipr$ of $\\qr$ can be deduced from \\eqref{MinimumEliMultiplicity}.\n\nNext let us prove \\eqref{item:ProperEliminant}.\nFor every $r\\in\\Iq\\cap\\rr$, assume that there exists $f\\in I\\setminus\\kxn$ such that $\\mo (f)=r$.\nThen $f$ can be written into $f=g\\qr+\\io (r)$ with $\\io (r)\\in\\kxn$.\nLet us denote $\\idr:=\\gcd (\\el,\\qr)\\in\\kxn$.\nIt is evident that we have $g\\qr\\el\/\\idr\\in\\langle\\el\\rangle\\subset I$ and hence $(f-g\\qr)\\el\/\\idr=\\el\\cdot\\io (\\dr)\/\\idr\n\\in\\Il$.\nMoreover, $\\mo (\\el\\cdot\\io (\\dr)\/\\idr)=\\dr\\mo (\\el\/\\idr)$ such that $\\mo (\\el\/\\idr)\\in\\rr^\\times$ by Lemma \\ref{Lemma:PQRProperties} \\eqref{item:SimpleCoprime} since $\\el\/\\idr$ is relatively prime to $\\qr$.\nThus $\\mo\\colon\\Il\\longrightarrow\\Iq\\cap\\rr$ is an epimorphism.\nAs a result, we have $\\Iq\\cap\\rr=\\pid\\mel$ based on $\\Il=\\pid\\el$ as per Definition \\ref{Def:Eliminant}.\nThen the conclusion \\eqref{item:ProperEliminant} readily follows from the fact that $\\ee\\in\\Iq\\cap\\rr$.\n\nThe proofs for the conclusions \\eqref{item:SPolynReductionPQR} and \\eqref{item:ProperEliminantTermination} are almost verbatim repetitions of those for Lemma \\ref{Lemma:PseudoEliminantTerminate} \\eqref{item:AllSPolynRemainder} and \\eqref{item:PseudoEliminantTerminate}.\nIn particular, the argument for \\eqref{item:ProperEliminantTermination} is based on the fact that $\\rx$ is also a Noetherian ring.\nIn fact, the normal PQR $\\rr$ in Definition \\ref{Def:nPQR} is isomorphic to the Noetherian PQR $\\bar R$ in Definition \\ref{Def:PQR}.\n\\end{proof}\n\n\\begin{corollary}\nIf the proper eliminant $\\ee\\in\\rr^\\ast$, then the eliminant $\\el$ is not divisible by the composite divisor $\\qr=\\sfr^i$ of the incompatible part $\\Ip (\\pel)$.\nMoreover, if the proper eliminant $\\ee\\in\\rr^\\times$, then the eliminant $\\el$ is relatively prime to the composite divisor $\\qr=\\sfr^i$.\n\\end{corollary}\n\\begin{proof}\nIf the eliminant $\\el$ is divisible by the composite divisor $\\qr$, then the modular eliminant $\\mel=\\mo (\\el)=0$.\nBy Lemma \\ref{Lemma:ProperEliminant} \\eqref{item:ProperEliminant} we can deduce that $\\ee=0$, contradicting $\\ee\\in\\rr^\\ast$.\n\nIf the proper eliminant $\\ee\\in\\rr^\\times$, there exists $b\\in\\rr^\\times$ such that $1=b\\ee\\in\\pid\\mel$ since we have $\\ee\\in\\pid\\mel$ by Lemma \\ref{Lemma:ProperEliminant} \\eqref{item:ProperEliminant}.\nHence the modular eliminant $\\mel=\\mo (\\el)\\in\\rr^\\times$, from which we can deduce that the eliminant $\\el$ is relatively prime to the composite divisor $\\qr=\\sfr^i$ by Lemma \\ref{Lemma:PQRProperties} \\eqref{item:SimpleCoprime}.\n\\end{proof}\n\nIn what follows let us exclude the trivial case when the proper eliminant $\\ee\\in\\rr^\\times$.\nThat is, let us assume that the eliminant $\\el$ is not relatively prime to the composite divisor $\\qr=\\sfr^i$.\n\n\\begin{lemma}\\label{Lemma:nPQRSyzygy}\nLet $\\tas=\\{f_j\\colon 1\\le j\\le s\\}\\subset\\rqd$ be a polynomial set over a normal PQR $\\rr$ as in \\eqref{kxnPQR}.\nSuppose that for $1\\le j\\le s$, each $f_j$ has the same leading monomial $\\lmc (f_j)=\\tilde{\\bx}^\\alpha\\in\\tM$.\n\n\\begin{inparaenum}[(a)]\n\\item\\label{item:SPolynomialExpansionPQR} If $f=\\sum_{j=1}^s f_j$ satisfies $\\lmc (f)\\prec\\tilde{\\bx}^\\alpha$, then\nthere exist multipliers $\\ibr,\\ibr_j\\in\\rr^\\ast$ for $1\\le j1.7\\times 10^{11}\\times g_{ds}^V \\hspace{1cm}\\mathrm{GeV}.\n\\end{equation} \nThus for Model I, $v_a>3.7\\times 10^{10}$ GeV. The limits for all models,\nanalagously derived, are listed in Table \\ref{t:DevAxLims}, taken \nfrom \\cite{HinMou98a}.\n\\begin{table}[hbt]\n\\setlength{\\tabcolsep}{1.5pc}\n\\catcode`?=\\active \\def?{\\kern\\digitwidth}\n\\caption{\\label{t:DevAxLims}Limits on the axion scale $v_a$ from \nthe flavour-changing process $K^+\\to \\pi^+ a$.}\n\\begin{tabular}{lll}\n\\hline\nModel & Charged & Limit\\\\\n & doublets & (GeV)\\\\[1pt]\n\\hline\nI & $ud$ & $3.7\\cdot 10^{10}$\\\\\nII & $tb$ & $6.1\\cdot 10^7 $\\\\\nIII & $ud,tb$ & $3.7\\cdot 10^{10}$\\\\\nIV & $cs$ & $3.6\\cdot 10^{10} $ \\\\\nV & $cs,ud$ & $7.3\\cdot 10^{10}$\\\\\nVI & $cs,tb$ & $3.6\\cdot 10^{10}$\\\\\n\\hline\n\\end{tabular}\n\\end{table}\nWith the exception of Model III, these bounds are all stronger than the \ncorresponding supernova bound \\cite{HinMou98a}. The bound on Model III is\nweak because of the smallness of $V_{td}$ and $V_{ts}$.\n\nA theory with four Higgs doublets and a Higgs singlet has an additional three \nmassive pseudoscalars. All of them can in principle mediate neutral meson \nmixing. $B\\bar B$ is a particularly strong constraint, which forces \nthe masses of the pseudoscalars to be greater than about $10^6$ GeV, \n\\cite{HinMou98a} unless they are for some reason very weakly coupled to the $b$. \nMaking a massive pseudoscalar is not a problem in principle, as there are in \ngeneral terms such as $\\lambda\\phi^2\\phi^T_1 \\phi_2$, which contribute \npieces of order $\\lambda v^2$ to the masses. However, this leads \nto some kind of fine tuning in the Higgs potential, \nan ugly feature of these models which seems unavoidable.\n\nOne of the lessons we learn from these studies is that there is\nplenty of freedom in assigning the Peccei-Quinn symmetry properties \nof the quarks, particularly in the right-handed sector. If we wish \nto assign different charges in the left-handed sector, we must \npay the price in inducing flavour-changing neutral currents mediated \nby the extra pseudoscalar bosons that are a result of the extra \nHiggs fields in such models. To keep these at an acceptable level \nwe must tune the Higgs potential such that these pseudoscalars have\nmasses greater than about $10^6$ GeV, while keeping the electroweak \nHiggs vacuum expectation value at $246$ GeV.\n\n\nM.H.\\ is supported by PPARC Advanced Fellowship \nB\/93\/AF\/1642 and by PPARC grants GR\/L56305 and GR\/L55759. \n \n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nSemiconductor quantum dots (QDs) are unique non-classical light emitters. In addition to their now well-known single photon emission properties,\ntheir potential as sources of entangled photons on demand \\cite{Akopian, Shields, Shields2} was recently demonstrated. The main obstacle to\npolarization entanglement of photons emitted in the biexciton-exciton radiative cascade is the lifted of degeneracy of the two optically active\nexciton states due to the anisotropic electron-hole exchange interaction. It is revealed in experiment by a splitting of the exciton and\nbiexciton lines into doublets with orthogonal linear polarizations. The origin of the symmetry breakdown governing this fine structure splitting\n(FSS) is not fully established. FSS can result from any combination of in-plane shape anisotropy of a dot (elongation of the dot due to\npreferential growth direction), piezoelectric potential in the dot vicinity (due to the vertically asymmetric strain field), and local symmetry\nbreakdown due to chemical bond alignment at the dot interfaces~\\cite{Zunger, Sequin}. As controlling FSS is of utmost importance for quantum\noptics applications, different strategies for restoring higher symmetry were tested, either by influencing material properties of\nheterostructures (annealing or strain engineering~\\cite{Young, Tartakovskii}) or by applying external perturbations compensating the native\nasymmetry : in-plane electric field \\cite{Kowalik-APL}, uniaxial strain~\\cite{Seidl}, and in-plane magnetic field~\\cite{Stevenson, Shields} were\ntried. The last-mentioned method has given the most satisfactory results so far in GaAs-based self-assembled QDs. II-VI systems have promising\nfeatures in this context, based on their more robust excitonic states allowing to study non-classical light emission at higher\ntemperatures~\\cite{Tinjod-temp}. However, they generally exhibit stronger anisotropy splittings~\\cite{Puls}, so it is essential to devise\nefficient methods of symmetry control suited to II-VI QDs. In this paper we report a study of exciton FSS in CdTe\/ZnTe QDs in the presence\nof an in-plane magnetic field and show that it can be increased or decreased, depending on the in-plane field direction.\\\\\n\\section{Sample and Experiment}\n\\indent The studied sample was grown by MBE on a $($001$)$-oriented GaAs substrate. It consists of following layers: a thick (4.3$\\mu$m) CdTe\nbuffer, followed by a 0.35$\\mu$m ZnTe barrier, a CdTe quantum dot layer, and a 114nm ZnTe cap. QD formation was induced by desorption of\namorphous Tellurium deposited on six monolayers of CdTe \\cite{Tinjod}. Before the cap deposition the dot layer was annealed in-situ at\n$480^{o}$C for 25 minutes. The sample emission is characterized by a broad micro-luminescence ($\\mu$PL) spectrum, with well resolved individual\nQD lines appearing on the low energy tail. For the experiment the sample was mounted directly on a specially designed Cassegrain-type microscope\nobjective immersed in liquid helium \\cite{Jasny}, in a cryostat with superconducting coil allowing to apply magnetic field up to 7T. The field\nwas applied parallel to one of the $\\langle110\\rangle$ crystallographic axes, in Voigt configuration. For the excitation, a CW doubled YAG laser\nat 532nm was focused on a $1-2\\mu m^{2}$ area spot on the sample surface. The photoluminescence signal was collected by the same objective,\nanalyzed using a linear polarizer, filtered by a monochromator, and recorded by a CCD camera. Excellent mechanical stability of this set-up\nenabled PL measurement of a single QD for many hours. For well isolated dots, lineshape fits allowed us to determine the line position with a\nprecision of 30$\\mu$eV. A distinctive characteristic of CdTe quantum dots is that their optical axes are randomly oriented\n\\cite{Kudelski-ICPS,Marsal}, contrarily to InAs QDs where they are clamped to the $\\langle110\\rangle$ direction. This allows simultaneous\nmeasurements of different relative orientations of dot and applied field in a fixed geometry, by selecting different dots and rotating the\nanalyzer accordingly. We define the dot orientation as the polarization direction of\nthe lower component of the excitonic doublet and denote $\\theta$ the angle between the field and this direction.\\\\\n\\section{Results}\n\\begin{figure}[h]\n\\includegraphics[width=0.45 \\textwidth,keepaspectratio]{fig1b.eps}%\n\\caption{(Color online)(a) Linearly polarized excitonic doublet (solid and dashed lines) for magnetic fields $B=$0,~2, and 4~T in a dot\ncharacterized by $\\theta =20^{\\circ}$. Inset shows the direction of the magnetic field with respect to the QD polarization eigenaxes.(b) Mean\nenergy of the \"bright\" excitonic doublet versus magnetic field and (c) splitting of bright to dark states vs magnetic field. For (b) and (c)\nLines are theoretical curves calculated with the model described in the text.}\\label{fig1}\n\\end{figure}\n\\indent The polarization-resolved $\\mu$-PL spectra of a QD at various fields are shown in Fig.~\\ref{fig1} for the peculiar case of a dot nearly\noriented along $\\bm{B}$ ($\\theta=20^{\\circ}$). The most salient feature is a line appearing $\\sim$1~meV below the excitonic doublet, and\ndeveloping when the field is increased. This line is attributed to the dark exciton that becomes optically active due to field-induced mixing of\nbright and dark states~\\cite{Bayer-PRB65}. Detailed analysis shows in general a blueshift of the excitonic doublet (105~$\\mu$eV at 7~T), a\nsimilar increase of the bright-dark exciton splitting, and in this case, an increase of FSS from 87~$\\mu$eV at $B=$ 0 to 185~$\\mu$eV at $B=$7~T.\nIn fact, the magnetic field influence on the PL fine structure depends not only on the field magnitude, but also on some intrinsic properties\nand orientation of the quantum dot. In particular an increase of FSS is observed when the field is applied parallel to the orientation of the\nlower energy component of excitonic doublet ($\\theta\\approx0$, see Fig.~\\ref{fig2}(a)), while a decrease is produced for perpendicular direction\nof the field ($\\theta\\approx\\pi\/2$, see Fig.\\ref{fig2}(b)).\n\\begin{figure}[h]\n\\includegraphics[width=0.45 \\textwidth,keepaspectratio]{fig2b.eps}%\n\\caption{(Color online) Upper panel: $\\mu$-PL spectra at $B=$0~T for two dots ((a) and (b)) with perpendicular anisotropy orientations,\n$\\theta$=20$^{\\circ}$ and $\\theta$=110$^{\\circ}$, (open symbols and dashed line: measured at $20^{\\circ}$, closed symbols and solid line:\nmeasured at $110^{\\circ}$). Lower panel: Fine structure splitting (FSS) vs in-plane magnetic field $B$ for the dots from upper panel. Solid\nlines are theoretical curves according to the model discussed in the last section.}\\label{fig2}\n\\end{figure} \\\\\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45 \\textwidth,keepaspectratio]{fig3b.eps}%\n\\end{center}\n\\caption{(a) Fine structure splitting (FSS), and (b) PL polarization orientation $\\theta$ against in-plane magnetic field of four different QDs\n(different symbols). Lines are guide to the eyes.}\\label{fig3}\n\\end{figure}\\\\\n\\indent Finally, we have also investigated some dots with a strongly tilted field configuration (i.e. $\\theta\\approx\\pi\/4$ or $\\theta\\approx\n3\\pi\/4$). In such cases, in addition to the FSS change, the dot orientation shows a clear rotation when increasing the field. Figure~\\ref{fig3}\nillustrates this effect for only a few selected dots, but observations were the same for all the QDs that we have studied (around 20). As shown,\nthe reference QD eigenaxis rotates systematically towards the direction of the applied field. This indicates that the in-plane magnetic field\n$\\bm{B}$ contributes to the FSS by an effective spin splitting of the bright exciton states characterized by the low (high) energy component\npolarized parallel (orthogonal) to the field. This conclusion is also supported by the corresponding FSS modification which depends both\nqualitatively and quantitatively on the initial angle $\\theta$ between the field direction and QD orientation. The initial angle has to be\nclose to $\\pi\/2$ in order to get a reduction of FSS. If not, there is first a rotation of the optical orientation followed by an increase of FSS\ndue to the field. The theoretical discussion presented below sheds some light on this qualitative description.\\\\\n\\section{Discussion}\n\\indent The electron-hole exchange Hamiltonian responsible for the ground state exciton fine structure in an anisotropic quantum dot can be\nrepresented by~\\cite{Ivchenko} :\n\\begin{equation}\n\\hat{H}_{ex}=\\frac{\\delta_{0}}{2}\\hat{\\sigma}^{e}_{z}\\hat{\\sigma}^{h}_{z}+\n\\frac{\\delta_{1}}{4}(\\hat{\\sigma}^{e}_{x}\\hat{\\sigma}^{h}_{x}-\\hat{\\sigma}^{e}_{y}\\hat{\\sigma}^{h}_{y})+\n\\frac{\\delta_{2}}{4}(\\hat{\\sigma}^{e}_{x}\\hat{\\sigma}^{h}_{x}+\\hat{\\sigma}^{e}_{y}\\hat{\\sigma}^{h}_{y})\n\\label{Exchange}\n\\end{equation}\nwhere the Pauli matrices $\\sigma_{i}^{e,h}$ act on the spin components of the electron (e) or hole (h) respectively. Here, we used a $\\pm 1\/2$\npseudo-spin to describe the QD hole ground states with angular momentum $J_{z}=\\mp 3\/2$ along $z$. The quantities $\\delta_{0}$, $\\delta_{1}$,\nand $\\delta_{2}$ describe the exciton quartet fine structure as follows : $\\delta_{0}$ \\-- between states of angular momentum $|M|=1 $ and\n$|M|=2$ (or $\\sigma_z^{e}+\\sigma_z^{h}$=0), $|\\delta_{1}|$ (i.e. FSS) \\-- between the components of the optically active doublet ($ M=\\pm 1$),\nand $|\\delta_{2}|$ \\-- between the dark states ($ M=\\pm 2$). These parameters are determined by the quantum dot properties (size, shape,\ncomposition, strain field, etc). In this formalism, the arbitrary $x$,~$y$ directions of the Pauli matrices correspond to the eigenaxes of the\nQD. In the following we assume that the parameters $\\delta_{0}$, $\\delta_{1}$, and $\\delta_{2}$ are not directly modified by the in-plane\nmagnetic field, although for high field values the magnetic confinement likely affects the electron-hole exchange. Therefore, to the first order\nwe only consider the Zeeman Hamiltonian to describe the effect of the in-plane field $\\bm{B}_{\\perp}$ as recently done for self-assembled InAs\nQDs~\\cite{Stevenson}. To derive properly the $g$~factor for hole ground states, we start from the general expression available for bulk excitons\nand given by~\\cite{van-Kesteren}:\n\\begin{equation}\n \\hat{H}_{Z}^{bulk}(\\bm{B})=\\mu_{B}\\sum\n_{i=x,y,z}(\\frac{1}{2}g^{e}\\hat{\\sigma}_{i}^{e}B_{i}-2\\kappa\\hat{J}_{i}B_{i}-2q\\hat{J}^{3}_{i}B_{i}) \\label{Bulk_HZ}\n\\end{equation}\nwhere $\\mu_{B}$ is the Bohr magneton, $g^{e}$ is the electron Land\\'{e} factor, $q, \\kappa$ are Luttinger coefficients and the $\\hat{J}_{i}$'s\nare the angular momentum projections of the Bloch states in the $\\Gamma_{8}$ hole band along the crystallographic axes $\\langle100\\rangle$.\nUsually, the main term driving the hole Zeeman splitting is the linear term $-2 \\mu_{B}\\kappa\\hat{\\bm{J}}\\cdot\\bm{B}$ while the cubic term in\nEq.~\\ref{Bulk_HZ} is considered as negligible. Yet, for a transverse magnetic field, only the hole states which differ by $|\\Delta J|$=1 are\ncoupled by $\\hat{J}_{x}$ or $\\hat{J}_{y}$. As a result, in the quantum dots investigated here, the hole ground states which are essentially pure\nheavy-holes with $J_{z}=\\pm3\/2$ are not directly split by this term~\\cite{AngularMomentum}. To obtain a non-zero transverse $g$~factor, which is\nrequired to modify the exciton FSS~\\cite{Stevenson}, it seems thus necessary either to take into account the cubic term in Eq.~\\ref{Bulk_HZ}, or\nto include in the model a light-hole doublet state ($J_{z}=\\pm1\/2$) split by an energy $\\Delta_{h-l}$ of a few tens meV's from the hole ground\nstate doublet. Actually, the sole Zeeman coupling to the light-hole states produces only a weak third-order contribution to the heavy hole\nsplitting in a transverse magnetic field. We could conclude that only the cubic term contributes to the effective $g$~factor. However, including\nthe light-holes also enables us to take into account the QD symmetry reduction to $C_{2v}$ or even $C_{2}$ (responsible for the FSS) which\nimplies a direct coupling between the heavy and light hole ground states. The latter is proportional to the symmetrized product of the in-plane\nangular momentum components $\\{\\hat{J}_{x'}\\hat{J}_{y'}\\}$~\\cite{Ivchenko,PRB63-Toropov}. Here, the indexes $x',\\, y'$ denote axes which are\nrotated by $\\pi\/4$ with respect to the QD eigenaxes. After a $-\\pi\/4$ rotation to use the same referential axes as Eq.~(\\ref{Exchange}), we\nobtain the following Hamiltonian for the heavy-hole to light-hole coupling:\n\\begin{equation}\n \\hat{H}_{h\\!-\\!l}=\\beta \\left( \\hat{J}_{x}^{2}-\\hat{J}_{y}^{2}\\right)\n\\label{C2v_term}\n\\end{equation}\nwhere $\\beta$ represents the strength of the coupling. In the above formalism, based essentially on symmetry considerations, the respective\nsigns of $\\delta_{1}$ and $\\beta$ are not \\textsl{a priori} correlated although they are necessarily determined by the features of a given QD.\nThis unknown sign correlation could however reveal of importance for the control of the FSS as discussed below~\\cite{HexcC2v} and somehow\nenlightens the concept of \\textquotedblleft inverted\\textquotedblright FSS in Ref.~\\onlinecite{Stevenson}. Experimentally the perturbation\n$\\hat{H}_{h\\!-\\!l}$ leads also to dichroism of the ground state excitonic transition (i.e. a difference in oscillator strength of the\nlinearly-polarized doublet components) as reported in the past for the quantum wells of $C_{2v}$ symmetry~\\cite{PRB63-Toropov} and more recently\nfor trions in CdSe QDs~\\cite{Koudinov}. It is worth mentioning that in InAs QD's similar dichroism has been reported and that no correlation was\nfound between the sign of the linear polarization degree (related to $\\beta$) and the sign of $\\delta_{1}$~\\cite{APL-Favero}. For taking into\naccount both $\\hat{H}_{Z}^{bulk}$ and $\\hat{H}_{h\\!-\\!l}$ we have to introduce the angle $\\phi$ between the QD eigenaxis $x$ and the\ncrystallographic direction [100] in order to rotate the cubic term of $\\hat{H}_{Z}^{bulk}$ in the QD coordinate frame (see Fig.~\\ref{schema}).\nIn this way, we obtain the effective Zeeman Hamiltonian to the first order in $\\beta\/\\Delta_{h\\!-\\!l}$ and in the basis of electron spin and\nhole pseudo-spin :\n\\begin{equation}\n \\hat{H}_{Z}(\\bm{B}_{\\perp})=\\frac{1}{2}\\mu_{B}\\left(\\sum_{i}g^{e}\\hat{\\sigma}^{e}_{i}B_{i}+\\sum_{i,j}\n\\hat{\\sigma}^{h}_{i}g^{h}_{ij}B_{j}\\right)\\label{QD_HZ}\n\\end{equation}\nwith the hole $g$~factor tensor :\n\\begin{eqnarray}\ng^{h}&=&3q\\left(\n \\begin{array}{cc}\n \\cos4\\phi +\\rho_{g} & \\sin4\\phi \\\\\n -\\sin4\\phi & \\cos4\\phi -\\rho_{g}\\\\\n \\end{array}\n\\right)\\label{g_factor}\\\\\n\\nonumber\\mbox{where}\\quad \\rho_{g}&=&\\frac{(4\\kappa+7q)\\beta}{q\\Delta_{h\\!-\\! l}}\n\\end{eqnarray}\n As it could be expected, the transverse hole $g$~factor gets anisotropic due to the term proportional to $\\beta$ as already emphasized in\nRef.~\\onlinecite{Koudinov}. Of course, the parameters $g^{e}$,~$\\kappa$ and $q$ in Eqs.~\\ref{QD_HZ},~\\ref{g_factor} are not the bulk parameters\nof a quantum dot (or a barrier) material: in QDs the strong confinement of eigenstates considerably affects the values of these parameters as\npredicted~\\cite{PhysRevB.58.16353,PRL96-Pryor} and experimentally observed~\\cite{Bayer-PRB65}. In particular the mixing allowed in $D_{2d}$\nsymmetry between heavy-hole and light-hole with \\emph{envelope} wave functions of different angular momentum may explain the significative\nvalue often reported for the transverse hole $g$~factor~\\cite{AngularMomentum}. In the following we assume first a symmetric transverse\n$g$~factor for the hole ($\\beta$=0) and then discuss the effect produced by an antisymmetric term ($\\beta\\neq$0).\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45 \\textwidth,keepaspectratio]{fig4b.eps}%\n\\end{center}\n\\caption{(Color online) Schematics of the crystallographic axis configuration with respect to the QD principal axes $x,\\,y$. Thick red dashes\nrepresent the exciton polarization orientation produced by the field-induced splitting only for different directions of the field. (a) Effect\nof the cubic term (see text), (b) effect of the linear term, including heavy-hole to light-hole coupling. The grey shaded area represents a QD\nelongated along the polarization eigenaxis $x$.} \\label{schema}\n\\end{figure}\\\\\n\\indent The total Hamiltonian $\\hat{H}_{ex}+\\hat{H}_{Z}$ enables us to predict the evolution of the bright state FSS as a function of\n$\\bm{B}_{\\perp}$. The results are displayed in Fig.~\\ref{fig_q-term} which shows the absolute splitting of the bright excitons (panel (a)) and\ntheir polarization rotation angle $\\Delta\\theta$ (panel (b)). Both are plotted as a function of field magnitude and orientation represented here\nby $\\theta-2\\phi$. Before commenting further these figures, let us consider the effect of the cubic term in $\\hat{H}_{Z}$ for a symmetrical QD\nwith $\\delta_{1}=0$ (and $\\beta=0$). In this case, the choice of the QD $x,\\, y$ axes to define the angles $\\theta$ and $\\phi$ is arbitrary.\nChoosing $\\phi=0$ shows that the hole $g$~factor tensor reduces to the scalar value $3q$ which leads to an isotropic FSS induced by the field\nwhen its orientation is varied. On the other hand, if we fix $\\theta=0$ we observe that in the referential attached to the field, the\npolarization of the upper excitonic doublet split by the field rotates faster by an angle $2\\phi$ than the field from the [100] axis. This is\nillustrated in the left-hand part of Fig.~\\ref{schema}. In the general case, it is thus clear that depending on the field direction $\\theta$,\nthe field-induced splitting will add to or subtract from an initial finite splitting $\\delta_{1}$. Calculations presented in\nFig.~\\ref{fig_q-term}~(a) show that cancelation can be achieved for $\\theta=\\pi\/2 + 2\\phi$ (changing the sign of $q$ with respect to $g^{e}$\nshifts this angle by $\\pm\\pi\/2$), i.e. for a field oriented symmetrically to the low energy component of the bright doublet ($y$ axis) with\nrespect to the crystallographic direction [100]. A small discrepancy of the field orientation leads to a continuous rotation of the eigenaxes\ndirections when passing near the critical point $(B_{crit.},\\pi\/2+ 2\\phi)$ of exact cancelation as shown in Fig.~\\ref{fig_q-term}(b).\nNevertheless, choosing correctly the field direction with respect to the QD orientation allows in principle to reduce the FSS of any QDs. This\npoint was not clearly established in the analytical treatment presented by Stevenson~\\textit{et al.}~\\cite{Stevenson} where however only the\nisotropic contribution of Eq.~(\\ref{QD_HZ}) was considered~\\cite{InAsQDs}.\\\\\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45 \\textwidth,keepaspectratio]{fig5b.eps}%\n\\end{center}\n\\caption{(a) FSS absolute value vs in-plane field direction $\\theta-2 \\phi$ and magnitude encoded on a color scale. Cross-sections for three\ndirections of the field are displayed on the right-hand side. (b)Rotation angle $\\Delta\\theta$ of the PL polarization orientation vs in-plane\nmagnetic field direction $\\theta$ and magnitude. Three cross-sections for $\\theta=0,\\pi\/4$~and~$\\pi\/2$ are also shown. Calculations are made\nwith an isotropic $g$~factor ($\\beta$=0), $\\delta_{0}$=~80~$\\mu$eV and a FSS $\\delta_{1}$=~80~$\\mu$eV in zero field.}\\label{fig_q-term}\n\\end{figure}\n\n\\indent Taking now into account the $g$~factor anisotropy ($\\rho_{g}\\propto\\beta\\neq$0) may considerably change the above phenomenology. As\nshown in Fig.~\\ref{fig_Beta-term} the key feature turns out to be the sign of $\\rho_{g}$ (determined by that of $\\kappa\\beta$) with respect to\n$\\delta_{1}$. If opposite, we find that it is still possible to reduce to zero the FSS for $\\theta=\\pi\/2+2\\phi$, and actually when the\nantisymmetric part really dominates ($|\\rho_{g}|\\gg$1) this can be achieved for any field direction. On the contrary, for $\\rho_{g}$ and\n$\\delta_{1}$ of same sign the critical field $B_{crit.}$ for which cancelation could be achieved, diverges when $|\\rho_{g}|$ approaches 1, a\nsituation which indeed corresponds to $g^{h}_{yy}$=0. For larger values of the anisotropy the magnetic field produces an increase of the FSS\nwhatever its in-plane orientation $\\theta$ is, as illustrated in Fig.~\\ref{fig_Beta-term}(b). In both cases, when the anisotropy dominates\n($|\\rho_{g}|\\gg1$) the principal axes of the PL polarization remain essentially parallel to their initial orientation (see\nFig.~\\ref{schema}(b)), in contrast to the case $\\beta=0$ as obvious in Fig.~\\ref{fig_q-term}(b) for fields above $\\sim$4T. Our analysis\nreveals thus that in the case of a strong anisotropy of the hole $g$~factor the possibility of tuning the QD FSS by a magnetic field depends\nessentially on the sign of $\\beta$ with respect to $\\delta_1$.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.45 \\textwidth,keepaspectratio]{fig6b.eps}%\n\\end{center}\n\\caption{FSS absolute value on a color scale vs in-plane field direction $\\theta-2 \\phi$ and magnitude for a strong $g$~factor anisotropy\n$\\rho_{g}$. Depending on its sign with respect to the initial splitting (here $\\delta_{1}$=~80$\\mu$eV) it produces very different behaviors :\n(a) $\\rho_{g}$=-1.3, (b) $\\rho_{g}$=+1.3. We used the same average $g$~factor $\\bar{g}_{h}$=0.78, $g^{e}$=1.75 and $\\delta_{0}$=~0.8~meV as in\nFig.~\\ref{fig_q-term}.}\\label{fig_Beta-term}\n\\end{figure}\n\n \\indent In our experiments we did not vary the in-plane field orientation $\\theta$ for a given QD. Therefore it turns out to be rather delicate\nto determine the strength of the in-plane $g$~factor anisotropy. Yet, the fact that in most cases we observed a rotation of the QD principal\naxes towards those defined by the field indicates that the average value ($3q$) of the $g$~factor likely dominates over its anisotropic\ncontribution $\\propto \\beta$. Besides, we also did not observe significative degree of linear polarization between the PL intensities of both\nexciton components, which means that $\\beta\/\\Delta_{h\\!-\\!l}\\ll 1$. This situation clearly differs from that reported for CdSe\nquantum dots~\\cite{Koudinov}.\\\\\n\n\\indent For the quantum dot QD1 shown in Figs.~\\ref{fig1},~\\ref{fig2}, we could produce a good fit of the FSS evolution by taking the values of\n$\\delta_{1}$ ($\\delta_{0}$) observed (extrapolated) in zero magnetic field, and $\\delta_{2}$ assumed to be around 1~$\\mu$eV. The only fitting\nparameters were thus the electron and hole $g$~factors considered as isotropic. We included in the model the field orientation $\\theta$ with\nrespect to the dot main axis, which also defines $\\phi$, as in our experiment the field was parallel to one of the cleaved edge of the sample\ncorresponding to $|\\theta\\!-\\!\\phi|=\\pi\/4$. We obtained the following $g$~factor values $|g^{e}|=1.75\\pm0.1$ and $|g^{h}|=0.78\\pm0.1$ for the\nfit shown in Fig.~\\ref{fig1}. The decrease of FSS observed for QD2 (characterized by $\\theta\\approx\\pi\/2$) could also be reproduced by the model\nwith values $|g^{e}|=1.75\\pm0.2$ and $|g^{h}|=0.9\\pm0.2$. The dark states being absent from the spectra, $\\delta_{0}$=1~meV was arbitrary chosen\nfor the latter fitting. In the general case of tilted QD orientation the almost systematic rotation of the PL polarization towards the field\ndirection ($\\theta+\\Delta\\theta\\rightarrow 0$ in Fig.~\\ref{fig3}(b)) agrees well with our model with a negligible $g$~factor anisotropy. But\nclearly, further experimental investigations consisting in a full mapping of the influence of the magnetic field on FSS as a function of its\nin-plane orientation and magnitude should be performed to determine more quantitatively the $g$~factor anisotropy in these quantum dots.\n\n\\indent In all studied QDs, the observed dark states were fully linearly polarized (see for example Fig.~\\ref{fig1}(a)) and we noticed only one\ncomponent of dark excitonic doublet. The lack of one \"dark\" state in the spectrum can be explained within our interpretation. For the used\nexperimental configuration $\\phi=\\pi\/4$, and for quantum dots with eigenaxes parallel to the crystallographic direction $[110]$ or $[-110]$\n($\\theta=0$ or $=\\pi\/2$) we obtain the Hamiltonian in a particularly simple form. It may be used to describe the case of QD1, for which $\\theta$\nis close to zero. After writing down the discussed Hamiltonian in a basis of linearly polarized states~\\cite{LinearBasis} it has a following\nform: \\small\n\\begin{align}\n&\\hat{H}_{QD1}=\\nonumber\\\\\\frac{1}{2}&\\left(\n\\begin{array}{cccc}\n \\delta_{0}+\\delta_{1} & 0 & \\mu_{B}B (g^{e}+g^{h}) & 0 \\\\\n 0 & \\delta_{0}-\\delta_{1} & 0 & \\mu_{B}B (g^{e}-g^{h}) \\\\\n \\mu_{B}B (g^{e}+g^{h}) & 0 & -\\delta_{0}+\\delta_{2} & 0 \\\\\n 0 & \\mu_{B}B (g^{e}-g^{h})& 0 & -\\delta_{0}-\\delta_{2} \\\\\n\\end{array}\n\\right)\n\\end{align}\n\\normalsize\n\nThe diagonal elements refer to energy levels of states in absence of magnetic field, the non-diagonal ones show the mixing induced by the field.\nIn this representation it is clearly visible that mixing of dark and bright states occurs independently for each linear polarization. The mixing\nmatrix elements for $x$ and $y$ polarizations are proportional either to $(g^{e}+g^{h})$ or to $(g^{e}-g^{h})$, respectively. The corresponding\nintensities of the transitions are proportional to the squares of corresponding matrix elements. For the obtained values of g-factors the\ntheoretically predicted intensity ratio for dark transitions is $7.84$. It explains qualitatively the strong asymmetry in the intensities of the\ntwo components of the dark excitonic doublet.\n\n\n\\section{Conclusion}\n\\indent In summary, we have shown that an in-plane magnetic field modifies the fine structure splitting of the excitonic emission of CdTe\/ZnTe\nquantum dots. This effect depends on the field direction. If applied along one of the main axes of the dot, the field can either increase or\ndecrease the splitting. If not, a rotation of the bright exciton eigenaxes towards the axis defined by the field direction is generally\nobserved together with a change of the splitting. These effects are in qualitative (polarization rotation) and rough quantitative (splitting\nvariation) agreement with a simple model based on a Zeeman spin Hamiltonian. We find that in the QDs investigated here the anisotropy of the\n$g$~factor is likely negligible, in contrast to results reported for other types of self-assembled QDs. This could be an advantage as in this\ncase the possibility to cancel the fine structure splitting does not seem to be hindered by light---heavy hole mixing.\n\\\\\n\\begin{acknowledgments}\nThis work has been partially supported by Polish Ministry of Science and Higher Education (Grants 1PO3B-114-30 and 2PO3B-015-25). One of us\n(K.K.) is supported by the European network of excellence SANDIE. We would like to thank Pr. E. Ivchenko for enlightening discussions.\\\\\n\\end{acknowledgments}\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction.}\nQuantum computer science has come to know numerous examples in which a quantum computer can outperform its classical counterpart \\cite{chuang}. Among the celebrated algorithms is that a quantum computer can search for a marked element from a list of $N$ items using the oracle only $\\bigO{\\sqrt{N}}$ number of times rather than the classical $\\Omega(N)$ \\cite{grover}. The quantum algorithm, which gives the picture of a state initially being an equal superposition of all candidates rotating onto the target, is entirely dissimilar to the simple process of elimination of classical physics. Indeed even a classical computer operates according to quantum laws, yet we do not understand what happens differently that gives a quantum computer its speed-up. Numerous elements including entanglement, interference, and quantum correlation have been suspected as the key resource responsible for the quantum enhancement \\cite{josza, bruss,lloyd,meyer, cast}. On a related note cosmologists have invented the theory of decoherent histories as a way to derive classical physics from fundamental quantum laws\\cite{hartle, grif2}. In this paradigm there is no collapse of the wave-function or classical measuring apparatus. The Schrodinger equation is the only dynamical law, yet deterministic and classical laws governing coarse-grained observables can arise. The theory of decoherent histories has been applied to quantum information processing to identify information that can be made classical without losing substantially quantum computing power \\cite{poulin}. Unfortunately only a few dynamical models have been solved exactly to demonstrate classical emergence \\cite{brun, brun2, hartle-path}. \n\nIn the current paper we apply the theory of decoherent histories to Grover's algorithm as a particular example illustrating how classical computational power can arise within a closed quantum system. We are able to compute the main figure of merit in closed form and recover the well known quadratic speed-up. Furthermore, changing the Grover dynamics to a generic unitary about which we need not know anything except that it makes records about how the target is found available, we recover classical search time. Although we make a number of simplifying assumptions in the derivation, our results show how computing power depends on the decoherence of histories. In the next section we will set up the search problem within the theory of decoherent histories and derive the famous quadratic speed-up. Then we present the modified Grover dynamics under a generic environmental influence and derive the search time as a function of decoherence. \n\\section{Quantum search.}\nConsider the $N$ dimensional Hilbert space of $N$ independent spins spanned by $\\{\\ket{m}\\}_{m=1}^N$ where $\\ket{m}$ denotes spin $m$ being up while the rest of them down. The marked element $w$ which is to be found is distinguished in the following manner in the Hamiltonian\n\\begin{equation}\nH_w = -\\sum_{m=1}^N (-1)^{\\delta_{mw}} \\sigma_m -(N-2)\n\\end{equation}\nwhere $\\sigma_m$, in the ordered basis $\\{\\uparrow, \\downarrow\\}$, is the tensor product of $\\begin{pmatrix}\n1 & 0\\\\\n0 &-1\n\\end{pmatrix}$\non the $m$-th spin and identity on the rest. This Hamiltonian $H_w$ has the desired spectrum of all the ground states having energy $0$ except for the marked state having energy $2$. To this Hamiltonian we add the driving Hamiltonian $2 \\proj{s}$, where \n\\begin{equation}\n\\ket{s} = \\frac{1}{\\sqrt{N}}\\sum_{m=1}^N \\ket{m}\n\\end{equation}\nand consider henceforth the total Hamiltonian\n\\begin{equation}\nH = H_w + 2 \\proj{s}. \n\\end{equation}\nThe unitary evolution driven by $H$ will rotate the initial state $\\ket{s}$ onto the target $\\ket{w}$ in a time proportional to $O(\\sqrt{N})$ \\cite{farhi}. We will make $t$ short enough so that $U(t)$ is equivalent to $1$ application of the oracle in discrete time and apply the oracle for a total of $K$ times. With the unity partition of projectors\n\\begin{align}\nP_1 &= \\proj{w} \\\\\nP_0 &= 1 - P_1,\n\\end{align}\ngenerally\n\\begin{align}\nU(tK)\\ket{s} & = \\label{eq:Utk}\n((P_1+P_0)U(t))^{K}\\ket{s} \\\\\n& = \\sum_{\\alpha \\in \\{0,1\\}^K} C_\\alpha \\ket{s}. \\label{eq:sumCa}\n\\end{align}\nwhere\n\\begin{equation}\\label{def:Ca}\nC_{\\alpha} \\coloneqq P_{\\alpha_K} U(t) \\cdots P_{\\alpha_i} U(t) \\cdots P_{\\alpha_1} U(t).\n\\end{equation}\n While the well-known perspective shows Grover search to be a rotation, the theory of decoherent histories considers the unitary evolution as a sum of history branches $C_\\alpha \\ket{s}$ as in Equation~\\eqref{eq:sumCa}. Each query is an attempt to ask the question ``Is the target found,'' and a history is a string of answers ``yes'' or ``no'' to the question, as represented by the projectors $P_1$ and $P_0$, respectively. For brevity, when successive components of $\\alpha$ are alike, we can collect them into a streak. For example, if a history consists of $0$'s for the first $s_1$ events, $1$'s for the next $s_2$ events, and so forth up to $1$'s for the last $s_n$ events, then we would write it as $\\alpha = 1^{s_n}\\dots 1^{s_2}0^{s_1}$, where $n$ is the number of streaks in the history with $1 \\leq n \\leq K$ and $s_j$ the length of streak $j$. The projectors at each time slice provide for an algebra of events, and each $\\alpha$ has the same meaning as a path in a random walk, yet we may not be able to assign probabilities to them. Let us define the following quantities in terms\nof $N$ and $w$:\n\\begin{align}\n\\ket{\\xi} &\\coloneqq \\frac{1}{\\sqrt{N-1}}\\sum_{m = \\{1,\\dots, N\\} \\setminus \\{w\\}} \\ket{m},\\\\\nx &\\coloneqq \\frac{1}{\\sqrt{N}},\\\\\nf &\\coloneqq \\cos xt - \\i x \\sin xt = \\abs{f}{\\mathrm{e}^{\\i \\phi}}\\\\\nb &\\coloneqq -i \\sqrt{\\frac{N-1}{N}} \\sin xt\n\\end{align}\nStarting from the definition of operators $C_\\alpha$ of Equation~\\eqref{def:Ca}, one can show that\n\\begin{widetext}\n\\begin{equation}\\label{main}\nC_\\alpha \\ket{s} = \\abs{f}^{K-n}b^{n-1}\n\\begin{cases}\n\\displaystyle\\exp\\Bigl( \\i \\phi \\Bigl( \\sum_{k=\\textrm{even}}\\alpha_k - \\sum_{k=\\textrm{odd}}\\alpha_k\\Bigr) \\Bigr) \n\\Bigl(f^*\\sqrt{\\frac{N-1}{N}}+bx\\Bigr)\\ket{w}, &\\mbox{if } \\alpha = 1^{s_n},\\dots,0^{s_1}, n = \\textrm{even}\\\\\n\\displaystyle\n\\exp\\Bigl( \\i \\phi \\Bigl( \\sum_{k=\\textrm{odd}}\\alpha_k - \\sum_{k=\\textrm{even}}\\alpha_k\\Bigr) \\Bigr)\\Bigl(b\\mathrm{e}^{-\\i \\phi}\\sqrt{\\frac{N-1}{N}} + \\abs{f} x\\Bigr)\\ket{w}, &\\mbox{if } \\alpha =1^{s_n},\\dots, 1^{s_1}, n = \\textrm{odd}\\\\\n\\displaystyle\n\\exp\\Bigl( \\i \\phi \\Bigl( \\sum_{k=\\textrm{odd}}\\alpha_k - \\sum_{k=\\textrm{even}}\\alpha_k\\Bigr)\\Bigr)\n\\Bigl(b\\sqrt{\\frac{N-1}{N}} + fx\\Bigr)\\ket{\\xi}, &\\mbox{if } \\alpha =0^{s_n},\\dots, 1^{s_1}, n = \\textrm{even}\\\\\n\\displaystyle\n\\exp\\Bigl(\\i \\phi \\Bigl( \\sum_{k=\\textrm{even}}\\alpha_k - \\sum_{k=\\textrm{odd}}\\alpha_k\\Bigr)\\Bigr)\n\\Bigl(\\abs{f}\\sqrt{\\frac{N-1}{N}} + \\mathrm{e}^{\\i \\phi} bx\\Bigr)\\ket{\\xi}, &\\mbox{if } \\alpha =0^{s_n},\\dots, 0^{s_1}, n = \\textrm{odd}\n\\end{cases}\n\\end{equation}\n\\end{widetext}\nClearly there exist $\\alpha \\neq \\alpha'$ such that the decoherence functional\n\\begin{equation}\nD(\\alpha, \\alpha') \\coloneqq \\bra{s}C^{\\dagger}_{\\alpha'}C_{\\alpha}\\ket{s} \\label{def:med}\n\\end{equation}\ndoes not vanish. If the decoherence functional vanishes $\\forall \\alpha \\neq \\alpha'$, we would have medium decoherence, which is equivalent to existence of records \\cite{grif,halliwell}.\nMoreover, probabilities would be assigned to each $\\alpha$, thus making the Grover search a classical stochastic process.\n\\subsection{Time of Grover's Search Algorithm}\nWe will now derive the time required to find the target. In the regime $N \\gg 1$ we have $xt \\in \\bigO{x} = \\bigO{1\/\\sqrt{N}} = o(1), \\abs{b} \\approx xt, \\abs{f} \\approx 1$, and $\\abs{\\phi} \\approx 1\/N$. Moreover, the total time for which we study the process is proportional to $K$, so long as $K \\lesssim \\sqrt{N}$, the phases are of order $\\phi K \\approx 1\/\\sqrt{N}$ and therefore negligible. As a result, Equation~\\eqref{main} is simplified to\n\\begin{equation}\\label{c_approx}\nC_\\alpha \\ket{s} \\approx \\abs{f}^{K-n}b^{n-1} \n\\begin{cases}\n\\displaystyle\nS(n)\\ket{w}, & \\mbox{if }\\alpha = 1^{s_n}\\ldots \\\\\n\\displaystyle\nF(n) \\ket{\\xi}, & \\mbox{if }\\alpha =0^{s_n}\\ldots\n\\end{cases}\n\\end{equation}\nwhere \n\\begin{equation}\\label{eq:defS}\nS(n) = \\begin{cases}\nf^*, &n = \\textrm{even}\\\\\nb + \\abs{f} x, &n = \\textrm{odd},\n\\end{cases} \n\\end{equation}\nand\n\\begin{equation}\\label{eq:defF}\nF(n) = \\begin{cases}\nb+fx, &n=\\textrm{even}\\\\ \n\\abs{f}+ bx, &n=\\textrm{odd}.\n\\end{cases}\n\\end{equation}\nFrom Equations~\\eqref{eq:sumCa} and \\eqref{c_approx}, the probability of success after time $tK$ is the amplitude square of the following branch\n\\begin{align}\n& P_1 U(tK) \\ket{s} = \\sum_{\\alpha \\in \\{1\\}\\times \\{0,1\\}^{K-1}} C_\\alpha \\ket{s}\\\\\n&\\approx \\sum_{n=1}^{K} \\binom{K -1}{n-1}\\abs{f}^{K-n}b^{n-1}S(n)\\ket{w},\n\\end{align}\nwhere $\\binom{K -1}{n-1}$ is the number of histories with $n$ streaks for a given $K$. Moreover, since $\\abs{f}^2 + \\abs{b}^2 = 1$, there exists a $\\theta$ such that\n\\begin{align}\n\\sin \\theta &= \\abs{b} \\, \\textrm{and}\\\\\n\\cos \\theta &= \\abs{f}\n\\end{align}\nUsing the identity found in \\cite{table} to compute the sum above, we find\n\\begin{equation}\\label{suc}\nP_1 U(tK) \\ket{s} \\approx -\\i S(\\textrm{even})\\sin (K-1)\\theta + S(\\textrm{odd})\\cos (K-1)\\theta\n\\end{equation} \nSimilarly the sum of the branches that ends in failure is\n\\begin{equation}\\label{fail}\nP_0 U(tK) \\ket{s} \\approx -\\i F(\\textrm{even})\\sin (K-1)\\theta + F(\\textrm{odd})\\cos (K-1)\\theta\n\\end{equation}\nNote that $\\tan \\theta = \\bigO{1\/\\sqrt{N}}$ and is small. Thus the number of oracle queries sufficient to find the target is when $(K-1) \\theta = \\pi\/2$, yielding $K = \\bigO{1\/x} = \\bigO{\\sqrt{N}}$ consistent with Grover's well-known result. \n\\section{Classical search.}\nWe now expand the treatment so as to include classical as well as quantum results in one framework. Consider the tensor product space of the system of $N$ spins that executes Grover algorithm and its environment. Let the initial state be $\\ket{s} \\otimes \\ket{\\Psi_e}$, where $\\ket{\\Psi_e}$ is the state of the environment, and $\\tilde{U}(t)$ be the unitary evolution on the composite. Between applications of the oracle, we still ask the question ``Is the target found,'' and the answers ``yes'' and ``no'' are represented respectively as\n\\begin{align}\n\\Pi_1 &= \\proj{w} \\otimes 1\\\\\n\\Pi_0 &= (1-\\proj{w}) \\otimes 1\n\\end{align}\nAgain denoting histories by $\\alpha \\in \\{0,1\\}^K$, we define the history branch operators similarly to Equation~\\eqref{def:Ca} as\n\\begin{equation}\\label{def:Ga}\nG_{\\alpha} \\coloneqq \\Pi_{\\alpha_K} \\tilde{U}(t) \\cdots \\Pi_{\\alpha_i} \\tilde{U}(t) \\cdots \\Pi_{\\alpha_1} \\tilde{U}(t).\n\\end{equation}\nRather than specifying the interaction Hamiltonian, we make the following assumptions:\n\\begin{enumerate}\n\\item $G_\\alpha \\ket{s}\\otimes \\ket{\\Psi_e} = \\frac{1}{A} C_\\alpha \\ket{s} \\otimes \\ket{e_\\alpha},$ where the environment states $\\ket{e_\\alpha}$ are assumed to be normalized, and $A$ is a normalizing factor independent of $\\alpha$ such that\n\\begin{equation}\n\\abs{\\sum_\\alpha G_\\alpha \\ket{s}\\otimes \\ket{\\Psi_e}}^2 = 1. \n\\end{equation}\n\\item \\label{eq:va}\n$\\braket{e_{\\alpha'}}{e_\\alpha} = 0$\nfor histories $\\alpha$ and $\\alpha'$ having different numbers of streaks.\n\\item \\label{eq:defn-d}\n\\begin{equation}\\delta \\coloneqq \\frac{1}{\\binom{K-1}{n-1}(\\binom{K-1}{n-1}-1)}\\sum_{\\substack{\\alpha \\neq \\alpha'\\\\ \\textrm{histories of }n\\textrm{ streaks}}}\\braket{e_{\\alpha'}}{e_\\alpha}\n\\end{equation} to be independent of $n$.\n\\end{enumerate}\nUp to this point we have left the interaction with the environment completely unspecified, and it could be so destructive that the system of $N$ spins no longer executes the search. Therefore, the first assumption is necessary since it restricts the interaction in a way that Grover search is still being executed with the additional complication that each history branch is now correlated with a state in the environment. Records of which branch are kept by the environment in the states $e_{\\alpha}$, but they may not be orthogonal to each other. Therefore, the information of which branch is not necessarily available, but the second assumption makes available the information about a branch's number of streaks. Since $\\delta$ is defined to be proportional to the average cosine of the angle between two branches having a given number of streaks, if the branches are orthogonal or have randomized phases relative to each other, then $\\delta = 0$. Either case is a signature of classical behavior, and we take $\\delta$ to be a measure of decoherence with $0$ corresponding to classical and $1$ to possibly quantum. Finally, assuming $\\delta$ to be independent of $n$ is for convenience yet is general enough to reveal the quantum-ness responsible for the square root speed-up. Using the last two assumptions and the series representation of Legendre polynomials found in \\cite{nist}, we obtain\n\\begin{equation}\n\\abs{A}^2 \\approx \\abs{f}^{2K} \\left[ \\delta \\left(1- \\tan^2\\theta\\right)^{K-1}P_{K-1}\\left(\\frac{1+\\tan^2 \\theta}{1-\\tan^2 \\theta}\\right)\n+ (1-\\delta) \\left(1 + \\tan^2 \\theta\\right)^{K-1}\\right],\n\\end{equation}\nwhere $P_{K-1}$ is a Legendre polynomial. After $K$ iterations of the oracle, similar to the Grover analysis, the probability of success is\n\\begin{widetext}\n\\begin{align}\n\\mathrm{Pr}(\\textrm{Success}) &= \\abs{\\sum_{\\alpha \\in \\{1\\}\\times\\{0,1\\}^{K-1} } G_\\alpha\\ket{s}\\otimes \\ket{\\Psi_e}}^2 \\\\\n&\\approx \\frac{\\abs{f}^{2K}}{\\abs{A}^2}(1-\\delta) \\left[ \\frac{(1+\\tan^2 \\theta)^K - (1-\\tan^2 \\theta)^K}{2} \\right] + \\frac{\\abs{f}^{2K}}{\\abs{A}^2} \\delta\\times \\nonumber \\\\\n&\\left[ (1-\\tan^2 \\theta)^{K-1} \\left(\\frac{1+\\tan^2 \\theta}{2}\\right) P_{K-1}\\left(\\frac{1+\\tan^2 \\theta}{1-\\tan^2 \\theta}\\right) -(1+\\tan^2 \\theta)^{K-1}\\left(\\frac{1-\\tan^2 \\theta}{2}\\right)P_{K-1}\\left(\\frac{1-\\tan^2 \\theta}{1+\\tan^2 \\theta}\\right) \\right] \\label{eq:qc}\n\\end{align}\n\\end{widetext}\nWhen $\\delta = 0$, the above equation simplifies\n\\begin{equation}\n\\mathrm{Pr}(\\textrm{Success}) = \\frac{1+\\tan^2 \\theta}{2}\\left(1- \\left(\\frac{1-\\tan^2 \\theta}{1+\\tan^2 \\theta}\\right)^K \\right)\n\\end{equation}\nHence, the probability of finding the target is of order unity when $K = \\bigO{N}$, which is the classical search time. When $\\delta = 1$, we have\n\\begin{equation}\n\\mathrm{Pr}(\\textrm{Success}) = \\frac{1+\\tan^2 \\theta}{2}- \\frac{1-\\tan^2 \\theta}{2}\n\\left( \\frac{1+\\tan^2 \\theta}{1-\\tan^2 \\theta}\\right)^{K-1}\\frac{P_{K-1}\\left(\\frac{1-\\tan^2 \\theta}{1+\\tan^2 \\theta}\\right)}{P_{K-1}\\left(\\frac{1+\\tan^2 \\theta}{1-\\tan^2 \\theta}\\right)}.\n\\end{equation}\nSince the expansion of the Legendre polynomial is $P_{K-1}\\left(\\frac{1-\\tan^2 \\theta}{1+\\tan^2 \\theta}\\right) \\approx 1 - K(K-1)\\tan^2 \\theta$, the probability of success is of order unity when $K = \\bigO{\\sqrt{N}}$, which is quantum time. As shown in Equation~\\eqref{eq:qc} the probability of success in general is a convex combination between classical and quantum search times. \n\\section{Conclusion.}\nWe find studying the quantum search in the framework of decoherent histories to be beneficial in many ways. Firstly, the Grover system makes an instructive toy application for the theory of decoherent histories, which allows for an alternative derivation of the quantum search time. Secondly, we see an exactly solvable demonstration of how classical computing can arise out of quantum systems. After each application of the oracle, we ask the question ``Is the target found'', and a search history, which is a series of answers ``yes'' or ``no'' to the question, is a vector, whose orientation relative to each other determines the search time and is affected by the environment. In the pure Grover dynamics case, all the history branches that end in success are phase coherent and aligned in one direction as evident in Equation~\\eqref{main}. Thus, no information about the search is made available. The role of the environmental interaction is that it can disrupt the history branches' orientations to affect the search time. In contrast to the pure Grover dynamics, our model shows that a little bit of information, namely the number of streaks in the search history, can be leaked out to the environment without sacrificing the quadratic speed-up. However, if the history branches have randomized phases or are orthogonal to each other, which is equivalent to the environment keeping complete records of the search, then the search time will be classical scaling. For a general level of decoherence the search time is a convex combination between classical and quantum as given in Equation~\\eqref{eq:qc}. Thus, applying the theory of decoherent histories to the search problem allows us to see how the same quantum laws at the fundamental level can give rise to both quantum and classical computing powers depending on the amount of information obtained by the environment. \n\n\\paragraph*{Acknowledgements:}\nThis material is based upon work supported by the National Science Foundation under\nGrant No.~0747526.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{intro}\n\nThe affine Toda field theories associated to an affine Kac-Moody\nalgebra ${\\bf \\hat g}$ have received a great deal of attention in recent years.\nThe classical equation of motion which describes them is\n\n\\begin{equation} \\Box^2\n\\phi+{2\\mu^2\\over\\beta}\\sum_{i=0}^r m_i H_i\ne^{\\beta\\alpha_i(\\phi)}=0,\\label{affext2}\\end{equation}\nwhere $\\phi$ is the dynamical field, taking values in the Cartan\nsubalgebra of ${\\bf \\hat g}$ (for whom Chevalley generators are the $H_i$ and\nthe Coxeter number is $h$), $\\mu$\nand $\\beta$ are real parameters, and $\\alpha_i$ are the simple roots\nof ${\\bf \\hat g}$. Taking $\\beta$ to be real we obtain a model with a particle\nspectrum whose properties are algebraic in origin (see, for example,\nrefs.\n\\cite{FLO91,Do92}), and whose S-matrix is exactly known\n\\cite{Do91,Do92,BCDS89a,BCDS89b,BCDS90,FO92,KM90,CD91,MF91}.\n\nIf the coupling constant $\\beta$ is pure imaginary then the model\npossesses a series of vacuum solutions. We expect soliton\nsolutions to be those of minimal energy which interpolate these vacua,\nthe difference between the final and the initial vacua being known as\nthe topological charge of the soliton. The seminal calculation was\nmade by Hollowood \\cite{Ho92} for the $\\hat A_r$ models using\nHirota's method. This technique was later developed to include the\nrest of the theories \\cite{ACFGZ92,CZ93}. A new method,\ninvolving the use of the general solution discovered by Leznov and\nSaveliev (see e.g. \\cite{LS92}), has recently proved to be very useful in\nextracting particular properties of the solitons \\cite{OTU93}. This method can\nbe\nmotivated by consideration of the form of the canonical\nenergy-momentum tensor\n\\begin{equation}\nT_{\\mu\\nu}= \\left(\\partial_\\mu\\phi ,\\partial_\\nu \\phi\\right) -\n{g_{\\mu\\nu}\\over 2}\n\\left(\\partial_\\alpha \\phi,\\partial^\\alpha\n\\phi\\right)\n-\\frac{2g_{\\mu\\nu}}{\\beta^2}\n\\sum_{k=0}^{r}n_k\\left(e^{\\beta{\\alpha}_k(\\phi)}-1\\right).\\label{em}\\end{equation}\nUsing arguments given in ref. \\cite{OTU93} it can be shown that\n$T_{\\mu\\nu}$ can be split into a sum of two parts,\n$T_{\\mu\\nu}=C_{\\mu\\nu}+\\Theta_{\\mu\\nu}$, where $\\Theta_{\\mu\\nu}$ is traceless\nand $C_{\\mu\\nu}$ is a total derivative.\nDue to the topological nature of solitons, their properties, in particular\ntheir energies and momenta,\ndepend only on their behaviour at\ninfinity, and hence we might expect $\\Theta_{\\mu\\nu}$ to vanish for soliton\nsolutions.\nFurther work \\cite{OTU93,Un93} shows that $\\Theta_{\\mu\\nu}$ can be written\npurely in terms of the chiral fields which are the parameters of the\nLeznov-Saveliev solution, so it is plausible that the soliton solutions\narise if these fields vanish.\n\nIn order to find the N-soliton solutions a further ansatz for the form\nof a constant of integration appearing in the general solution must be\nmade \\cite{OU93}; the N-soliton specialised solution turns out to be\n\\begin{equation} e^{-\\beta\\phi_k} =\\langle\\Lambda_k\n|\\exp\\left(W_1\\hat F^{i_1}(z_1)\\right)\\ldots\n\\exp\\left(W_N\\hat F^{i_N}(z_N)\\right) |\\Lambda_k\\rangle. \\label{fulsol} \\end{equation}\nFor a detailed explanation of the quantities appearing in the formula\nthe reader should consult ref. \\cite{OU93}; we will need only the following\ndetails. The parameters $i_j$ are integers lying between 1 and $r$\nwhich describe the species of the N solitons. $z_j$ are complex\nparameters related to the soliton rapidities $\\rho_j$ in the following way:\n\\begin{equation} z_j=\\theta_j\ne^{-\\rho_j}\\frac{|q^+_{i_j}|}{q^+_{i_j}},\\label{onereal}\\end{equation}\nwhere $\\theta_j=\\pm\n1$, and the $q_{i_j}$ are structure constants of ${\\bf \\hat g}$ which can be calculated\nfrom the following formula of ref. \\cite{FLO91}\n\\begin{equation} q_p^+\\equiv \\gamma_p\\cdot\nq(1)=2\\imath x_p(1)e^{-\\delta_{pB}\\imath\\nu\\pi\/h} \\label{gq} \\end{equation}\n($\\i \\equiv \\sqrt{-1}$). The functions $W_j$\nencode the space-time dependence of the solution,\nand are given by the formula\n\\begin{equation} W_j= Q_j\\exp \\left\\lbrace \\sqrt 2\\mu\\theta_j\\mid\nq_{i_j}^+\\mid\n\\left(t\\sinh \\rho_j- x\\cosh\\rho_j\\right)\\right\\rbrace.\\label{wndef}\\end{equation}\n\nThe first major use to which this formalism was put was a calculation\nof the masses of single solitons of species $p$. These turned out to\nbe \\cite{OTU93}:\n\\begin{equation} M_p=-{4\\sqrt\n2h\\over\\beta^2\\gamma_p^2}\\mu |q^+_{p}|.\\label{mass}\\end{equation}\nAn interesting fact which is not yet fully understood is that these\nmasses are in a certain sense dual to those of the particles.\n\nIn this letter we will use further features of the solution\n(\\ref{fulsol}) to calculate the Poisson brackets on the N-soliton phase\nspace and will perform a canonical\nquantisation to extract the S-matrix.\n\n\n\n\n\\section{Poisson Brackets}\\label{pobr}\n\nThe classical phase space of the affine Toda theories has been discussed\nrecently in ref. \\cite{PS94}.\n The\nsymplectic form $\\Omega$ is calculated by integrating the symplectic current\nover all space at some time,\nand is simply\n\\begin{equation}\n\\Omega=\\int dx \\left(\\delta\\phi,\\delta\\partial_t\\phi\\right)\\label{sform}\\end{equation}\nwhere $(\\ ,\\ )$ denotes the usual Killing form on the algebra ${\\bf \\hat g}$.\n\nWe wish to investigate the phase space of the soliton solutions. To do this,\nwe insert the soliton solutions\ninto the symplectic form $\\Omega$.\nFirst let us calculate the symplectic form for a one-soliton solution.\nWe will need to use two important properties of this solution, but our\nmethod will mean that we do not need the explicit form of the solution\nitself. This is a typical feature of calculations concerning\nsolitons in affine Toda theory. The first feature we need is that the soliton\nis a relativistic object, and so the Poincare algebra must be\nrealised on the phase space. What this means for the solution is that\nthe field $\\phi$ must appear only as a function of $u$, where\n\\begin{equation} u=t\\sinh\\rho\n-(x-x_0)\\cosh\\rho.\\label{udef}\\end{equation}\nThe rapidity $\\rho$ and centre of mass $x_0$ are the real parameters of the\nsoliton solution. The relationship of these parameters to those\nin the algebraic ansatz for the soliton solution, as well as the explicit\ndependence of $\\phi$ upon $u$, can be determined from the formul\\ae\\ of\n\\cite{OTU93}, \\cite{OU93}, but we shall not need these yet. Using the\nantisymmetry properties of the wedge product we obtain\n\\begin{eqnarray} \\Omega &=& \\int dx\n\\left(\\frac{d\\phi}{du},\\frac{d\\phi}{du}\\right)\\; \\cosh\\rho\\, \\delta x_0\\wedge\n\\delta\n(\\sinh \\rho), \\nonumber \\\\ &=& -\\int dx\n\\left(\\partial_t\\phi,\\partial_x\\phi\\right)\\; \\frac{\\delta x_0\\wedge\n\\delta\n(\\sinh \\rho)}{\\sinh\\rho},\\nonumber \\\\&=& -\\int\nT_{tx}\\; \\frac{\\delta x_0\\wedge \\delta (\\sinh \\rho)}{\\sinh\\rho},\n\\label{step}\\end{eqnarray}\nwhere the last step is performed using the\nexpression for the energy-momentum tensor (\\ref{em}).\nFinally,\nsubstituting $P=-\\int dx T_{01}= M\\sinh\\rho$ into (\\ref{step}), we obtain\n\\begin{equation} \\Omega=\\delta \\xi\\wedge\\delta \\rho,\\label{onesym}\\end{equation}\nwhere $\\xi=Mx_0\n\\cosh\\rho$ is the canonical variable conjugate to $\\rho$. This is of\ncourse the result we expect.\n\nIn order to evaluate the symplectic form in the more general case of\n$N$ solitons we consider the form of the solution as $t\\rightarrow\\pm \\infty$.\nIn\nthe generic situation the solitons will be\nwell separated and since $\\Omega$ is a\nlocal expression we can just add up the contributions from each of the\nsolitons in turn. Thus we find\n\\begin{equation} \\Omega= \\sum_{i=1}^N \\delta \\xi_i^{\\stackrel{\\rm in}{\\rm out}}\\wedge\n\\delta \\rho_i^{\\stackrel{\\rm in}{\\rm out}}.\\label{inout}\\end{equation}\nWe now need\nto relate these variables to those which parameterise the solution.\n\nLet us consider the form of the solution (\\ref{fulsol}) as $x$ increases\nfrom $-\\infty$.\nThe first significant departure from the vacuum will\noccur with the soliton of greatest rapidity, $\\rho_N$. We know from\nref. \\cite{OU93} that the greatest non-vanishing power of $\\hat F^{i_N}$\nwithin a representation of level $x$ is $x$.\\footnote{We are only\nconsidering the theories where ${\\bf \\hat g}$ is simply-laced from now on.}\nThis in turn means that the solution for the component field $\\phi_k$\nwill be the logarithm of a polynomial of degree $m_k$ in $W_N$ defined\nby equation (\\ref{wndef}).\nAs we move through this soliton $W_N$ becomes much greater than 1 and\nso we can ignore all but the highest power in the polynomial as far as\ncalculating the form of the solution for greater $x$ is concerned.\nThis term of course multiplies an algebraic factor $\\hat\nF^{i_N}(z_N)$. Normal ordering the vertex operator expression\n\\cite{KO93}, \\cite{OU93} for this yields\n\\begin{eqnarray} \\lefteqn{e^{-\\beta\\phi_k} =\\langle\\Lambda_k\n|F^{i_N}(z_N) |\\Lambda_k\\rangle W_N^{m_k}\\langle\\Lambda_k\n|\\exp\\left(X_{i_1,i_N}(z_1,z_N)W_1\\hat F^{i_1}(z_1)\\right)\n\\ldots}\n\\label{nextsol}\n\\\\ &&\\hspace{2cm}\\ldots\n\\exp\\left(X_{N-1,N}(z_{N-1},z_N) W_{N-1}\\hat\nF^{i_{N-1}}(z_{N-1})\\right) |\\Lambda_k\\rangle, \\nonumber \\end{eqnarray} where\n\\begin{equation} X_{i,j}(z_1,z_2)=\\prod_{n=0}^{h-1} \\left(1-e^{-2\\pi\nin\/h}\\frac{z_2}{z_1}\n\\right)^{w^n(\\gamma_i)\\cdot\\gamma_j}.\\label{psmat}\\end{equation}\nThe roots\n$\\gamma_i$ are the simple roots $\\alpha_i$ up to a sign \\cite{OU93},\nand $w$ is the Coxeter element of the Weyl group.\nThus, aside from a constant shift in the field $\\phi$, the only effect\nis to change each of the $Q_j$ for $jj}X_{j,p}(z_j,z_p).\\label{qin}\\end{equation}\nThe rapidities remain unchanged under the transformation to `in'\nvariables. Now all we need to do is relate the variables $Q_j$ to the\n$\\xi_j$. Comparing expressions (\\ref{wndef}) and (\\ref{udef}) we find that\nup to an irrelevant constant\n\\begin{equation} x_0= \\frac{\\theta\\ln Q}{\\sqrt 2\\mu|q^+_i|\\cosh\\rho}.\\label{bored}\\end{equation}\nUsing\nthe mass formula equation (\\ref{mass}) we obtain\n\\begin{equation}\\xi=\\frac{2h\\theta\\ln Q}{|\\beta|^2}\\label{xicq},\\end{equation}\nand so\n\\begin{eqnarray} \\xi_j^{\\rm in} &=& \\xi_j + \\frac{2h}{|\\beta|^2} \\sum_{p>j}\\ln\nX_{j,p}(z_j,z_p),\\nonumber \\\\ \\rho^{\\rm\nin}_j&=&\\rho_j.\\label{inpar}\\end{eqnarray}\nSimilar arguments yield the following relationships\nfor the out variables\n\\begin{eqnarray} \\xi_j^{\\rm out} &=& \\xi_j + \\frac{2h}{|\\beta|^2} \\sum_{p j}\\ln\nX_{j,p}(z_j,z_p)\\right)\\nonumber \\\\ \\rho^{\\rm out}_j&=&\\rho^{\\rm in}_j\n=\\rho_j. \\label{inout2}\\end{eqnarray}\nWe can see from these equations that the S matrix\n$S$ is purely a function of the rapidity differences, and satisfies the\nequation\n\\begin{equation} {\\partial ({\\rm log} S)\\over \\partial\\rho_j} = {-2h{\\it\\i}\\over\\vert\n\\beta\\vert^2\\hbar}\n \\left(\\sum_{pj} {\\rm ln} X_{j,p}(\\rho_j,\\rho_p) \\right).\n\\label{anotherone}\\end{equation}\nThe solution of this is\n\\begin{equation} S=\\exp\\left\\lbrace\n\\frac{2h\\imath}{|\\beta|^2\\hbar}\\sum_{b> a}\n\\sum_{n=0}^{h-1}w^n(\\gamma_a)\n\\cdot(\\gamma_b)\\; Li_2\\left[\\exp\\left(\n\\rho_a-\\rho_b+\\frac{\\imath\\pi}{h}\\left(\\delta_{i_b,B}-\\delta_{i_a,B}-2n\n\\right)\\right) \\right]\n\\right\\rbrace\n.\\label{shakeitallabout}\\end{equation}\nWe have used the definition of the classical Euler dilogarithm\n\\begin{equation} Li_2(v)=\\sum_{s=1}^{\\infty}\\frac{v^s}{s^2}=-\\int_0^v\n{\\ln\\left(\n1-y\\right)\\over y} dy.\\label{whatacorker}\\end{equation}\nMany of the interesting properties\nof this function were studied by one William Spence\nin the early nineteenth century \\cite{S26}. A more recent discussion\nin the mathematical literature can be found in ref. \\cite{Le81}.\nRecent developments relating to conformal field theory are reviewed\nin ref. \\cite{K94}.\nThe matrix (\\ref{shakeitallabout}) satisfies various conditions, which\nare essentially related to properties of the time-delay functions $X_{ij}$.\nThis function was studied recently in ref. \\cite{FJKO94}, where it was noted\nthat it has properties\nreminiscent of those of $S$ matrices. Indeed, one can think\nof these properties as being consequences of the fact that the $S$ matrix\n(\\ref{shakeitallabout}) \nis periodic, symmetric, etc. Let us define\n\\begin{equation} T_{ab}(\\rho) =\n\\sum_{n=0}^{h-1}w^n(\\gamma_a)\n\\cdot(\\gamma_b)\\; Li_2\\left[\\exp\\left(\n\\rho + \\frac{\\imath\\pi}{h}\\left({c(a) - c(b) \\over 2} - 2n\n\\right)\\right) \\right].\n\\label{why bother} \\end{equation}\nThen the matrix function $T_{ab}(\\rho)$ satisfies the following relations:\n\\begin{description}\n\\item[{\\rm(i)}] $T_{ab}(\\rho+2\\i\\pi) = T_{ab}(\\rho)$\n\\item[{\\rm(ii)}] $T_{ab}(\\rho) = T_{ba}(\\rho)$\n\\item[{\\rm(iii)}] $T_{ab}(\\rho+i\\pi) = -T_{\\bar a b}(\\rho)$\n\\item[{\\rm (iv)}] $(T_{ab}(\\rho^*))^* = T_{ab}(\\rho)$\n\\item[{\\rm (v)}] $T_{ab}(\\rho) = -T_{ab}(-\\rho) + 2T_{ab}(0)$\n\\item[{\\rm (vi)}] $\\sum_{t=i,j,k} T_{lt}(\\rho+\\i\\eta_t) = 0$, where $l$ is\na free label and $i,j,k$ satisfy a fusing rule; $\\eta_t=-2\\xi_t +\n{c(t)-1\\over2}$,\nwhere $c(t)$ is $1(-1)$ if the root $\\alpha_t$ is black(white), and $\\xi_t$\nare the integers in the fusing relation\n$\\sum_{t=i,j,k}\\omega^{-\\xi_t}\\gamma_t=0$.\n\\end{description}\nThese relations can be shown directly:\nProperty (i) is obvious; properties (ii), (iii) and (iv)\nfollow using arguments analogous to those\nused for the time-delay functions\n$X_{ab}(\\rho)$ in ref. \\cite{FJKO94}.\nProperty (vi) similarly follows from an argument analogous to that given in\nref. \\cite{FO92} when discussing the affine Toda particle $S$-matrix.\nProperty (v) is most easily proved by noting firstly that the\n$\\rho$ derivative of this equation is true - this\nfollows using eqn. (\\ref{whatacorker})\nand the fact that $X_{ab}(\\rho)=X_{ab}(-\\rho)$\n(see ref. \\cite{FJKO94}). Thus the left-hand side of (v) equals the\nright-hand side up to an additive constant, and\nputting $\\rho=0$ one sees that this constant must\nvanish. A direct proof of (v) seems more involved - for example, for the $A_1$\ncase\nthis property reduces to the equation\n\\begin{equation} Li_2(-e^\\rho) - Li_2(e^\\rho) = -Li_2(-e^{-\\rho}) + Li_2(e^{-\\rho})\n +2Li_2(-1) - 2Li_2(1). \\label{brmmm}\\end{equation}\nThe Euler dilogarithm satisfies the following \\lq inversion' relation\n\\cite{K94}\n\\begin{equation} Li_2(-y) + Li_2(-1\/y) = -{\\pi^2\\over6} - {1\\over2}({\\rm log}\\, y)^2.\n\\end{equation}\nThe function $Li_2(y)$ is divergent for\nreal $y$, $y>1$; however, one can define a continuous function on the real line\nby setting \\cite{K94}\n\\begin{equation}\n Li_2(y) = {\\pi^2\\over3} - Li_2(1\/y) - {1\\over2}({\\rm log}\\,y)^2, \\qquad\n {\\rm for}\\;\\; y>1.\n\\end{equation}\nThen, using the above two equations and the facts that $Li_2(1)=\\pi^2\/6$ and\n$Li_2(-1)=-\\pi^2\/12$, equation (\\ref{brmmm}) may be proved.\n\nThe constant term in the relation (v) means that the S matrix\n(\\ref{shakeitallabout})\nwill be invariant under $\\rho\\rightarrow-\\rho$ only if it is normalised so that\n$S(\\rho=0)=1$. Physically this requirement is obvious - that there be no\nscattering\nwhen the two solitons have the same rapidity.\nWe note that the properties (i)-(vi) above reflect identities satisfied by\nthe Euler dilogarithm.\n\nThus the expression (\\ref{shakeitallabout}) satisfies relations expected for\nan $S$-matrix. Note, however, that the Euler dilogarithm can be continued\nto a multi-valued function on the complex plane, minus the segment\n$(1,\\infty)$ of the real axis \\cite{Le81,K94}.\nHence our proposed $S$-matrix does not have the expected pole structure.\nWe will comment upon this in the following section.\n\n\n\n\n\n\\section{Conclusions and Developments}\\label{theend}\n\nIt is important to be clear about what this S-matrix is, and what it\nis not. We have only made a semi-classical approximation to the\nquantum theory of affine Toda solitons, and so do not expect to see\nall of the behaviour typical of a quantum field theory. In particular\nas noted above, there are no poles in formula (\\ref{shakeitallabout}), and we\nhave made no mention of the renormalisation of the soliton masses. The\ninterested reader should consult the recent papers \\cite{MW94} and\n\\cite{DG94} for a discussion of this latter point. Very little is\nknown about the expected behaviour of the poles on account of\ndifficulties concerning the unitarity of the affine Toda theories in\nthe imaginary coupling r\\'egime. Another feature expected of the\nquantum theory which this kind of approximation will not possess is\nthat there will be no processes which change the topological charges\nof the scattering solitons. This is because such these are absent\nclassically. An obvious conjecture worthy of exploration is that\na full quantum version of (\\ref{shakeitallabout}) involves the\n{\\it quantum} dilogarithm\nof ref. \\cite{FK93}.\n\nWhat we believe we have is the S-matrix of an quantum integrable\nparticle theory, generalising that of Ruijsenaars \\cite{Ru,Ru2}. In\nthe case of $\\hat A$-series solitons of equal mass this particle model\nalready reproduces the appropriate scattering shifts. In the more\ngeneral case however, no such model is known. Starting from the formula for\nthe shifts which follows from Ruijsenaars model, and fixing the\nrapidities in such a way that the soliton fusing rule \\cite{OU93} is\nsatisfied, we have obtained correct expressions for the shifts\nwhen two solitons of different masses scatter (it is reasonably easy\nto show that all of the $\\hat A$-series solitons can be obtained by\nrepeated fusing from the lightest in this manner). The question is\nwhether this procedure can be implemented at the level of the\nRuijsenaars Hamiltonian. This is a non-trivial problem since the\nfusings correspond to imposing imaginary constraints on the particle\nrapidities, and it is unclear what effect this will have on the\nsymplectic structure of the phase space. We remark that the change\n$\\rho\\rightarrow \\rho+ 2in\\pi\/h$ is a symplectic transformation,\nalbeit a complex one, which gives us some hope that the above\nprocedure can be made to work.\n\nA natural generalisation of this work would seem to be to try and\ninclude the breathers. In the sine-Gordon theory at least these\nbreathers are related to the particles, and possess a number of\ndiscrete energy levels according to the value of the coupling constant\n$\\beta$. It is a long-standing problem in affine Toda soliton theory\nto try and characterise the breather spectrum for more general\ntheories, and in particular to see if the particle-soliton\ncorrespondence remains. We have obtained preliminary results in\nthis direction, including a description of the energy levels of the\n$\\hat A$-series breathers of remarkable and rather mysterious\nsimplicity. We plan to discuss these issues in a forthcoming paper \\cite{SU95}.\n\\hfill\\break\n\n\n\\noindent{\\bf Acknowledgements}\\hfill\\break\n\\noindent Part of this work was done under the auspices of a Royal Society\nVisiting Fellowship award to one of us (JWRU). JWRU thanks the\nlate UK Science and Engineering Research Council, and\nBS acknowledges\n support from the Australian Research Council and\nthe UK Engineering and Physical Sciences Research Council.\n\\bibliographystyle{unsrt}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzcpoz b/data_all_eng_slimpj/shuffled/split2/finalzzcpoz new file mode 100644 index 0000000000000000000000000000000000000000..cfc7e694b7c6ba10373d48d7dbc5ce47944bf861 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzcpoz @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThis project is part of a broader ongoing investigation into the use\nof methods from data analysis to identify the presence of \nstructures and relations between the syntactic\nparameters of the world languages, considered either globally\nacross all languages, or within specific language families and\nin comparative analysis between different families.\n\n\\smallskip\n\nWe analyze the SSWL database of syntactic structures of world languages, \nusing methods from {\\em topological data analysis}. After performing principal \ncomponent analysis to reduce the dimensionality of the data set, we compute\nthe persistent homology. The generators behave erratically when computed\nover the entire set of languages in the database. However, if restricted to\nspecific language families, non-trivial persistent homology appears, which\nbehaves differently for different families. We focus our analysis on the two\nlargest language families covered by the SSWL database: the Niger-Congo\nfamily and the Indo-European family. We show that the Indo-European family\nhas a non-trivial persistent generator in the first homology. By performing\ncluster analysis, we show that the four major language families in the database\n(Indo-European, Niger-Congo, Austronesian, Afro-Asiatic) exhibit different\ncluster structures in their syntactic parameters. \nThis allows us to focus on specific cluster filtering values,\nwhere other non-trivial persistent homology can be found, in both the Indo-European\nand the Niger-Congo cases. \n\n\\smallskip\n\nThis analysis shows that the Indo-European family has a non-trivial\npersistent generator of the first homology, and two persistent generators\nof the zeroth homology (persistent connected components), with substructures\nemerging at specific cluster filtering values. The Niger-Congo family, on the\nother hand, does not show presence of persistent first homology, and\nhas one persistent connected component. \n\n\\smallskip\n\nWe discuss the possible linguistic significance of persistent connected\ncomponents and persistent generators of the first homology. We propose\nan interpretation of persistent components in terms of subfamilies, and\nwe analyze different possible historical linguistic mechanisms that may give\nrise to non-trivial persistent first homology. \n\n\\smallskip\n\nWe focus on the non-trivial persistent first homology generator in the\nIndo-European family and we try to trace its origin in the structure \nof the phylogenetic network of Indo-European languages. The first\nhypothesis we consider is the possibility that the non-trivial loop in the \nspace of syntactic parameters may be a reflection of the presence \nof a non-trivial loop in the phylogenetic network, due to the historical\n``Anglo-Norman bridge\" connecting French to Middle English, hence\ncreating a non-trivial loop between the Latin and the Germanic subtrees.\nHowever, we show by analyzing the syntactic parameters of these\ntwo subtrees alone that the persistent first homology is not coming\nfrom this part of the Indo-European family. We show that \nit is also not coming from the Indo-Iranian branch. Moreover, we\nshow that adding or removing the Hellenic branch from the remaining\ngroup of Indo-European languages causes a change in both\nthe persistent first homology and the number of persistent component.\n\n\n\\medskip\n\\subsection*{Acknowledgment} This work was performed within the activities of the last author's \nMathematical and Computational Linguistics lab and CS101\/Ma191 class at Caltech. The last author \nis partially supported by NSF grants DMS-1007207, DMS-1201512, and PHY-1205440. \n\n\n\\section{Syntactic parameters and Data Analysis}\n\nThe idea of codifying different syntactic structures through\n{\\em parameters} is central to the Principles and Parameters\nmodel of syntax, \\cite{Chomsky}, \\cite{ChoLa}, within Generative Linguistics. \nIn this approach,\none associates to a language a string of binary ($\\pm$ or $0\/1$ valued) variables, \nthe syntactic parameters, that encode many features of its \nsyntactic structures. Examples of such parameters include \n{\\em Subject Verb}, which has the value $+$ when in a clause \nwith intransitive verb the order Subject Verb can be used;\n{\\em Noun Possessor}, which has value $+$ when a possessor \ncan follow the noun it modifies; {\\em Initial Polar Q Marker}, which\nhas value $+$ when a direct yes\/no question is marked by a clause \ninitial question-marker; etc.\\footnote{See {\\tt http:\/\/sswl.railsplayground.net\/browse\/properties}\nfor a list and description of all the syntactic parameters covered by the SSWL database.}\nThe ``Syntactic Structures of the World's Languages\" (SSWL) database, which \nwe used in this investigation, includes a set of $115$ different parameters,\n(partially) mapped for a set of $252$ of the known world languages.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.62]{AllLing1.jpg}\n\\includegraphics[scale=0.62]{AllLing2.jpg}\n\\includegraphics[scale=0.55]{RandomLing.jpg}\n\\caption{Barcode graph over languages with $60\\%$ \nof parameters known and $60\\%$ of variance preserved; and with $80\\%$ of\nparameters known and $60\\%$ of variance preserved. Third graph: barcode\nfor a random subset of 15 languages, $100\\%$ of variance preserved.\n\\label{AllLing1}}\n\\end{center}\n\\end{figure}\n\n\n\\smallskip\n\nThe comparative study of syntactic structures across different world\nlanguages plays an important role in Linguistics, see \\cite{Sholpen} \nfor a recent extensive treatment. In particular, in this study, we focus\non data of syntactic parameters for two of the major families of\nworld languages: the Indo-European family and the Niger-Congo\nfamily. These are the two families that are best represented in the\nSSWL database, which includes 79 Indo-European languages\nand 49 Niger-Congo languages. The Niger-Congo family is the\nlargest language family in the world (by number of languages it\ncomprises). General studies of syntactic structures of Niger-Congo \nlanguages are available, see for instance \\cite{BeSa}, \\cite{MaRe}, \nthough many of the languages within this family\nare still not very well mapped when it comes to their syntactic parameters in the SSWL database.\nThe Indo-European family, on the other hand, is very extensively\nstudied, and more of the syntactic parameters are mapped. Despite\nthis difference, the data available in the SSWL database provide\nenough material for a comparative data analysis between these\ntwo families.\n\n\\smallskip\n\nThe point of view based on syntactic\nparameters has also come to play a role in the study of Historical\nLinguistics and language change, see for instance \\cite{Galv}.\nAn excellent expository account of the parametric approach to syntax\nis given in \\cite{Baker}.\n\n\\smallskip\n\nOne of the sources of criticism to the Principles and Parameters model\nis the lack of a good understanding of the space of syntactic\nparameters, \\cite{Hasp2}. In particular, the theory does not clearly identify a\nset of independent binary variables that can be thought of as \na ``universal set of parameters\", and relations between syntactic\nparameters are not sufficiently well understood. \n\n\\smallskip\n\nIt is only in recent years, however, that accessible online databases of \nsyntactic structures have become available, such as the WALS \ndatabase of \\cite{Hasp} or the SSWL database \\cite{SSWL}.\nThe existence of databases that record syntactic parameters \nacross different world languages for the first time makes them\naccessible to techniques of modern {\\em data analysis}. Our hope\nis that a computational approach, based on various data analysis \ntechniques applied to the \nsyntactic parameters of world languages, may help the\ninvestigation of possible dependence relations between \ndifferent parameters and a better understanding of their overall structure. In the present\nstudy, we focused on the data collected in the SSWL database,\nand on {\\em topological data analysis} based on {\\em persistent homology}.\n\n\\smallskip\n\nThe structures we observe do not, at present, have a clear explanation\nin terms of Linguistic theory and of the Principles and Parameters model\nof syntax. The presence of persistent homology in the syntactic\nparameter data, and its different behavior for different language\nfamilies begs for a better understanding of the formation and\npersistence of topological structures from the Historical Linguistics\nviewpoint, and from the viewpoint of Syntactic Theory. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.41]{clusters_indoeuropean.jpg}\n\\includegraphics[scale=0.41]{clusters_niger-congo.jpg}\n\\includegraphics[scale=0.41]{clusters_austronesian.jpg}\n\\includegraphics[scale=0.41]{clusters_afro-asiatic.jpg}\n\\caption{Cluster structure of syntactic parameters for the Indo-European, the Niger-Congo, the Austronesian,\nand the Afro-Asiatic language families. \\label{ClusterFig}}\n\\end{center}\n\\end{figure}\n\n\n\\section{Persistent homology}\n\nAn important and fast developing area of data analysis, in recent years,\nhas been the study of high dimensional structures in large\nsets of data points, via topological methods, see \\cite{Carlsson},\n\\cite{EdHar}, \\cite{Ghrist}. These methods of {\\em topological\ndata analysis} allow one to infer global features from discrete \nsubsets of data as well as find commonalities of discrete \nsub-objects from a given continuous object. The techniques\ndeveloped within this framework have found applications in\nfields such as pure mathematics (geometric group theory,\nanalysis, coarse geometry), as well as in other sciences (biology,\ncomputer science), where one has to deal with large sets of data. Topology\nis very well-suited in tackling these problems, being qualitative in nature. \nSpecifically, topological data analysis achieves its goal by transforming \nthe data set under study into a family of simplicial complexes, indexed by a proximity parameter. One analyzes said complexes by computing their {\\em persistent homology}, \nand then encoding the persistent homology of the data set in the form of a parametrized \nversion of a Betti number called a {\\em barcode graph}. Such graphs exhibit\nexplicitly the number of connected components and of higher-dimensional holes in the data. \nWe refer the reader to \\cite{Carlsson}, \\cite{EdHar}, \\cite{Ghrist} for a general overview\nand a detailed treatment of topological data analysis and persistent homology.\nAs an example, persistent homology was used recently to study\nthe topology of a space of 3D images \\cite{3D}, where the authors\ndetermined that the barcode representation from persistent homology \nmatched the homology of a Klein bottle.\n\n\n\\medskip\n\\subsection{The Vietoris-Rips complex}\n\nSuppose given a set $X=\\{ x_\\alpha \\}$ of points in some Euclidean\nspace ${\\mathbb E}^N$. Let $d(x,y)=\\| x-y \\|=(\\sum_{j=1}^N (x_j-y_j)^2)^{1\/2}$ \ndenote the Euclidean distance function in ${\\mathbb E}^N$. \nThe Vietoris-Rips complex $R(X,\\epsilon)$ of scale $\\epsilon$, over a field ${\\mathbb K}$, \nis defined as the chain complex whose space $R_n(X,\\epsilon)$ of $n$-simplices corresponds \nto the ${\\mathbb K}$-vector space spanned by all the unordered\n$(n+1)$-tuples of points $\\{ x_{\\alpha_0}, x_{\\alpha_1}, \\ldots, x_{\\alpha_n} \\}$\nwhere each pair $x_{\\alpha_i}, x_{\\alpha_j}$ has distance\n$d(x_{\\alpha_i}, x_{\\alpha_j})\\leq \\epsilon$. The boundary maps \n$\\partial_n: R_n(X,\\epsilon) \\to R_{n-1}(X,\\epsilon)$, with $\\partial_n \\circ \\partial_{n+1}=0$, are the\nusual ones determined by the incidence relations of $(n+1)$ and $n$-dimensional\nsimplices. For $n\\geq 0$, one denotes by \n$$ H_n(X,\\epsilon) :=H_n(R(X,\\epsilon),\\partial) $$ $$ \n= {\\rm Ker}\\{ \\partial_n: R_n(X,\\epsilon) \\to R_{n-1}(X,\\epsilon)\\} \/\n{\\rm Range}\\{ \\partial_{n+1}: R_{n+1}(X,\\epsilon) \\to R_n(X,\\epsilon)\\} $$\nthe $n$-th homology with coefficients in ${\\mathbb K}$ of\nthe Vietoris-Rips complex. When the scale $\\epsilon$ varies, one obtains\na system of inclusion maps between the Vietoris-Rips complexes,\n$R(X,\\epsilon_1) \\hookrightarrow R(X,\\epsilon_2)$, for $\\epsilon_1 < \\epsilon_2$.\nBy functoriality of homology, these maps induce corresponding morphisms\nbetween the homologies, $H_n(X,\\epsilon_1) \\to H_n(X,\\epsilon_2)$.\nA homology class in $H_n(X,\\epsilon_2)$ that is not in the image of $H_n(X,\\epsilon_1)$\nis a birth; a nontrivial homology class in $H_n(X,\\epsilon_1)$ that maps to the zero element of\n$H_n(X,\\epsilon_2)$ is a death, and a nontrivial homology class in \n$H_n(X,\\epsilon_1)$ that maps to a nontrivial homology class in $H_n(X,\\epsilon_2)$\nis said to persist. Mapping the deaths, births, and persistence of a set of generators\nof the homology, as the radius $\\epsilon$ grows gives rise to a barcode graph for the\nBetti numbers of these homology groups. \nThose homology generators that survive only over short intervals of $\\epsilon$ radii\nare attributed to noise, while those that persist for longer intervals are considered to\nrepresent actual structure in the data set. \n\n\n\\medskip\n\\subsection{Linguistic significance of persistent homology} \n\nWhen we analyze the persistent topology of different linguistic families \n(see the detailed discussion of results in \\S \\ref{TopLingSec}), we find\ndifferent behaviors, in the number of persistent generators in both $H_0$\nand $H_1$. As typically happens in many data sets, the generators for\n$H_n$ with $n\\geq 2$ behave too erratically to identify any meaningful\nstructure beyond topological noise. \n\n\\smallskip\n\nIn general, the rank of the $n$-th homology group $H_n$ of a \ncomplex counts the ``number of $(n+1)$-dimensional holes\" that\ncannot be filled by an $(n+1)$-dimensional patch. In the \ntopological analysis of a point cloud data set, the presence of\na non-trivial generator of the $H_n$ at a given scale of the \nVietoris-Rips complex implies the existence of a set of data\npoints that is well described by an $n$-dimensional set of\nparameters, whose shape in the ambient space encloses\nan $(n+1)$-dimensional hole, which is not filled by other \ndata in the same set. In this sense, the presence of generators\nof persistent homology reveal the presence of structure in the\ndata set. \n\n\\smallskip\n\nIn our case, the database provides a data point for each recorded \nworld language (or for each language within a given family), and\nthe data points live in the space of syntactic parameters, or in a\nspace of a more manageable lower dimension after performing \nprincipal component analysis. In this setting, the presence of\nan ``$(n+1)$-dimensional hole\" in the data (a generator of the\npersistent $H_n$) shows that (part of) the data cluster around an\n$n$-dimensional manifold that is not ``filled in\" by other data points.\nPossible coordinates on such $n$-dimensional structures represent\nrelations among the syntactic parameters, over certain linguistic\n(sub)families. \n\n\\smallskip\n\nSince the only persistent generators we encountered\nare in the $H_0$ and $H_1$, we discuss more in detail\ntheir respective meanings.\n\n\n\\medskip\n\\subsection{Linguistic significance of persistent $H_0$}\n\nThe rank of the persistent $H_0$ counts the number of connected\ncomponents of the Vietoris-Rips complex. It reveals the presence of\nclusters of data, within which data are closer to each other (in\nthe range of scales considered for the Vietoris-Rips complex) than to\nany point in any other component. Thus, a language family exhibiting\nmore than one persistent generator of $H_0$ has linguistic parameters \nthat naturally group together into different subfamilies. It is not known,\nat this stage of the analysis, whether in such cases the subsets of\nlanguages that belong to the same connected component correspond\nto historical linguistic subfamilies or whether they cut across them.\nWe will give some evidence, in the case of the Indo-European family,\nin favor of matching persistent generators of the $H_0$ to\nmajor historical linguistic sub-families within the same family.\nCertainly, in all cases, the connected components identified by different generators\nof the persistent $H_0$ can be used to define a grouping into subfamilies, \nwhose relation to historical linguistics remains to be investigated.\n\n\\medskip\n\\subsection{Linguistic significance of persistent $H_1$}\n\nThe presence of an $H_1$-generators also means that\npart of the data (corresponding to one of the components\nof the Vietoris-Rips complex) clusters around a one-dimensional\nclosed curve. More precisely, one can identify the\nfirst homology group $H_1(X)$ of a space with the group\nof homotopy classes $[X,S^1]$ of (basepoint preserving)\nmaps $f: X \\to S^1$ to the circle. This means that, if there\nis a non-trivial generator of the persistent $H_1$, then there\nis a choice of a circle coordinate that best describes that part of\nthe data. The freedom to change the map up to homotopy\nmakes it possible to look for a smoothing of the circle coordinate. \nIt is not obvious how to interpret these circles from the Linguistic\npoint of view. The fact that a generator of the $H_1$ represents\na $2$-dimensional hole means that, given the data that cluster along this\ncircle, no further data point determine a $2$-dimensional surface\ninterpolating across the circle. As the topological structures we\nare investigating stem from a Vietoris-Rips complex that measures\nproximity between syntactic parameters of different languages,\nwe can propose a heuristic interpretation for the presence of such circles\nas the case of a (sub)family of languages where each language\nin the subfamily has other ``neighboring\" languages with sufficiently \nsimilar syntactic parameters, so that one can go around the whole\nsubfamily via changes of syntactic parameters described by a single\ncircle coordinate, while parameter changes that move along \ntwo-dimensional manifolds and interpolate between data points on\nthe circle cannot be performed while remain within the same (sub)family. \n\nTwo different possible models of how a non-trivial generator of the persistent first\nhomology can arise point to different possible explanations in historical-linguistic terms.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.25]{hopfbif.jpg} \\ \\ \\ \\ \\ \\ \\ \n\\caption{Two models of the development of a non-trivial loop in the space of parameters. \\label{bifFig}}\n\\end{center}\n\\end{figure}\nAs shown in Figure \\ref{bifFig}, the first model is a typical Hopf bifurcation \npicture, where a circle arises from a point\n(with the horizontal direction as time coordinate). This model would be compatible\nwith a phylogenetic network of the corresponding language family that is a tree,\nwhere one of the nodes generates a set of daughter nodes whose points in the\nparameter space contain a nontrivial loop. The second possibility is of a\nline closing up into a circle. This may arise in the case of a language\nfamily whose phylogenetic network is not a tree, but it contains itself a loop that\ncloses off two previously distant branches. There are well known cases where\nthe phylogenetic network of a language family is not necessarily best described\nby a tree. The most famous case is probably the Anglo-Norman bridge in the phylogenetic\n``tree\" of the Indo-European languages, see Figure \\ref{IEtreeFig}. However, it is\nimportant to point out that the presence of a loop in the phylogenetic network\nof a language family does not imply that this loop will leave a trace in the syntactic\nparameters, in the form of a non-trivial first persistent homology. Conversely, the\npresence of persistent first homology, by itself, is no guarantee that loops may be \npresent in the phylogenetic network, for example due\nto possibilities like the Hopf bifurcation picture described above.\nThus, one cannot infer directly from the presence or absence of a persistent $H_1$\nconclusions about the topology of the historical phylogenetic network.\nThe only conclusion of this sort that can be drawn is that a persistent $H_1$\nsuggests a phylogenetic loop as one of the possible causes. Conversely, one can \nread the absence of non-trivial persistent first homology as a suggestion\n(but not an implication) of the fact that the phylogenetic network may be a tree\nand that phenomena like the Anglo-Norman bridge did not occur\nin the historical development of that family.\n\n\\smallskip\n\nWe will discuss this point more in detail in the case of the Indo-European\nlanguage family. This is a very good example, which shows how \nthe possible correlation between loops in the space of\nsyntactic parameters and in the phylogenetic network is by no means an implication.\nIndeed, the Indo-European language family contains both a known loop\nin the phylogenetic network, due to the Anglo-Norman bridge (see Figure \\ref{IEtreeFig}),\nand a non-trivial generator of the persistent $H_1$. However, we will show using\nour topological data analysis method that these two loops are in fact unrelated,\ncontrary to what intuition might have suggested. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.6]{IEtree.jpg}\n\\caption{The family ``tree\" of the Indo-European languages (by Jack Lynch) showing the\nloop between the Latin and the Germanic subtrees. \\label{IEtreeFig}}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.41]{indoeuropean-7-5-96.jpg}\n\\includegraphics[scale=0.41]{indoeuropean-10-0-96.jpg}\n\\includegraphics[scale=0.6]{indoeuropean-10-5-98.jpg}\n\\caption{Barcode diagram for the Indo-European language family, at indices $(7,5,96)$,\n$(10,0,96)$, and $(10,5,98)$. \\label{IE1fig}}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{Data Analysis Procedure}\\label{DataSec}\n\nThe SSWL database \\cite{SSWL} was first imported into a pivot table in Excel. \nThe on-off parameters are represented \nin binary, in order to compute the distances between languages. \nHowever, the parameter values are not known for many of the languages in the\ndatabase: over one hundred of the languages have, at present, less than half of\ntheir parameters known. Thus, we decided to replace empty language parameters \nwith a value of 0.5. All together, we ended up with 252 languages, each \nwith 115 different parameters.\n\n\\smallskip\n\nWe then proceeded to our analysis based on the results from Perseus\nhomology software \\cite{Perseus}. This is achieved through a series of Matlab\nscripts\\footnote{A repository of the code used for this project is available at \\newline\n{\\tt https:\/\/github.com\/cosmicomic\/cs101-project5}}. The script named {\\tt data\\_select\\_full.m} allows for selection\nof subsets of the raw data. It performs Principal Component Analysis\non the raw parameter data and saves it to a text file for use in Perseus.\nThe format of the data is that of a Vietoris-Rips complex. This script \nhas two important parameters: a completeness threshold, and a percent\nvariance to preserve. The completeness threshold removes the languages\nbelow a threshold of known parameters. The percent variance allows\nus to reduce the dimensionality of our data.\n\n\\smallskip\n\nThe next script, named {\\tt barcode.m}, was used to create barcode graphs\nfor data visualization. Perseus outputs the birth and death times\nfor each persistent homology generator, which are then used to construct\nthe barcode graph of the persistence intervals to visualize the structure\nand determine the generators. The radii in our complexes are incremented \nby $1\\%$ of the mean distance between languages.\n\n\\smallskip \n\nData analysis was initially set up as a three step process: select the data\nwith the script {\\tt data\\_select\\_full.m}, analyze it with Perseus, and use {\\tt barcode.m}\nto visualize the results. The final script, named {\\tt run\\_all}, streamlines\nthis process under a single input command.\n\n\\smallskip\n\nFinally, our analysis includes examining how many data points belong to clusters\nof points at any given radius. Clusters are constructed by creating\n$n$-spheres of uniform radius centered at each data point. If the $n$-spheres\nof two data points overlap, then those data points are in the same\ncluster. A non-trivial cluster is a cluster with at least two data\npoints contained within. The scripts {\\tt group\\_select.m} and {\\tt graph\\_clusters.m}\nmake it possible to visualize the number of clusters and non-trivial clusters\nas radius increases.\n\n\\smallskip\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.41]{indoeuropean_cluster165-7-5-95.jpg}\n\\includegraphics[scale=0.41]{indoeuropean_cluster165-10-5-95.jpg}\n\\caption{Barcode diagram for the Indo-European language family, for \ncluster filtering value $165$ at indices $(7,5,95)$\nand $(10,5,95)$. \\label{IE3fig}}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{The persistent topology of linguistic families}\\label{TopLingSec}\n\nA preliminary analysis performed over the entire set of languages in the SSWL\ndatabase shows that the non-trivial homology generators of $H_1$ and $H_2$ \nbehave erratically. Moreover, there are too many generators of $H_2$ and $H_3$\nto draw any meaningful conclusion about the structure of the underlying topological\nspace. One can see the typical behavior represented in Figure \\ref{AllLing1}. In \nthe first graph of Figure \\ref{AllLing1} we included the languages with more than $60\\%$ \nof the parameters known, while in the second we removed all languages with \nmore than $20\\%$ of the parameters unaccounted for. Here percentage of parameters is\nwith respect to the largest number of syntactic parameters considered in the SSWL\ndatabase. One can compare this with the case of a randomly generated subset of\nlanguages, presented in the third graph of Figure \\ref{AllLing1}. Notice that, while in the cases\nrepresented in the first two graphs of \nFigure \\ref{AllLing1} there is ``noise\" in the $H_1$ and $H_2$ region,\nthat prevents a clear identification of persistent generators, the homology of random \nsubsets of the data, as displayed in the third graph of Figure \\ref{AllLing1}, \nis relatively sparse, \ncontaining only topologically trivial information. This observation lead us to the hypothesis\nthat the behavior seen in Figure \\ref{AllLing1} stems from a superposition of some\nmore precise, but non-uniform, topological information associated to the various different\nlinguistic families. In order to test this hypothesis,\nwe decided to examine specific \nlanguage families as an additional method of data filtering. We chose\nthe four largest families represented in the original database: Indo-European\nwith 79 languages, Niger-Congo with 49, Austronesian with 18, and\nAfro-Asiatic with 14. Although some of the languages in the database\nincluded latitude and longitude coordinates, these were ignored when\ndetermining language family.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.41]{niger-congo-7-5-107.jpg}\n\\includegraphics[scale=0.41]{niger-congo-10-0-100.jpg}\n\\includegraphics[scale=0.6]{niger-congo-10-5-105.jpg}\n\\caption{Barcode diagram for the Niger-Congo language family, \nat indices $(7,5,107)$, $(10,0,100)$, and $(10,5,105)$. \\label{NC1fig}}\n\\end{center}\n\\end{figure}\n\n\\medskip\n\\subsection{Cluster structures in major language families}\\label{clustersec}\n\nA first observation, when comparing syntactic parameters of different linguistic families, \nis that they exhibit different cluster structure of the syntactic parameters. \nThis is illustrated in Figure \\ref{ClusterFig},\nin the case of the our largest families in the SSWL database. \n\n\\smallskip\n\nBased on this \ncluster analysis, we then focused on the cases of the Indo-European and\nthe Niger-Congo language family and we searched for nontrivial generators\nof the first homology $H_1$ in appropriate ranges of cluster filtering. \n\n\\smallskip\n\nThe cluster analysis of Figure~\\ref{ClusterFig} suggests that\ncluster filter values between $150$ and $200$ should provide \ninteresting information. We computed additional barcode\ndiagrams corresponding to cluster filtering values $165$ and $190$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.41]{niger-congo_cluster165-7-3-100.jpg}\n\\includegraphics[scale=0.41]{niger-congo_cluster165-10-0-100.jpg}\n\\includegraphics[scale=0.41]{niger-congo_cluster190-7-5-104.jpg}\n\\includegraphics[scale=0.41]{niger-congo_cluster190-10-0-104.jpg}\n\\caption{Barcode diagram for the Niger-Congo language family, for \ncluster filtering value $165$ and \nindices $(7,3,100)$ and $(10,0,100)$, and for \ncluster filtering value $190$ and indices $(7,5,104)$ and\n$(10,0,104)$. \\label{NC2fig}}\n\\end{center}\n\\end{figure}\n\n\n\\smallskip\n\\subsection{Indexing in barcode graphs}\nIn the graphs presented in the following subsection, the barcode graphs\nare labeled by a set of three indices. The first two indices refer to the\nPrincipal Component Analysis and the third index to the runs of the Perseus program\ncomputing births and deaths of homology generators of the Vietoris-Rips complex.\nMore precisely, the first index (7 or 10) refers\nto the percent variance divided by 10, while the second index (0, 3 or 5)\nrefers to the percent complete divided by 10. They are discussed above in \\S \\ref{DataSec}.\nThe third parameter\nis the number of steps in Perseus. If present, the additional parameter given by\nthe number after ``cluster\" is one hundred times the radius used for cluster filtering.\n\n\\medskip\n\\subsection{Persistent topology of the Indo-European family}\n\nWe analyzed the persistent homology of the syntactic parameters for the\nIndo-European language family. As shown in Figure \\ref{IE1fig},\nat values $(7,5,96)$ and $(10,0,96)$ one sees persistent generators of $H_0$\nand intervals in the varying $n$-sphere radius, \nfor which nontrivial $H_1$ generators exist. At values $(10,5,98)$, as shown in\nFigure \\ref{IE1fig}, one sees one persistent generator of $H_1$ and two persistent\ngenerators of $H_0$. The existence of a persistent generator for the $H_1$ suggests\nthat there should be a ``circle coordinate\" description for at part of the syntactic\nparameters of the Indo-European languages. The fact that there are two persistent\ngenerators of $H_0$ in the same diagram indicates two connected components, only\none of which is a circle: this component determines which subset of syntactic parameters \nadmits a parameterization by values of a circle coordinate. \n\n\\smallskip\n\nBased on the cluster analysis described in \\S \\ref{clustersec} above, we then focused on \nspecific regions of cluster filtering values that were more likely to exhibit interesting topology. \nFor example, for cluster filtering value $165$, the results show, respectively, \none generator of $H_0$ and one generator of $H_1$, for indices $(7,5,95)$, and\none generator of $H_0$ and a possibility of two persistent \ngenerators of the $H_1$, for indices $(10,5,95)$, see\nFigure~\\ref{IE3fig}. The appearance of persistent generators of the $H_1$ as\nspecific cluster filtering values identifies other groups of syntactic parameters that may\nadmit circle variable parameterizations. What these topological structures in the\nspace of syntactic parameters, and these subsets admitting circle variables description, \nmean in terms of linguistic theory remains to be fully understood. We analyze some\nhistorical-linguistic hypotheses in the following subsection.\n\n\\medskip\n\\subsection{Indo-European persistent topology and historical linguistics}\n\nIt is often argued that the phylogenetic ``tree\" of the family of Indo-European languages\nshould not really be a tree, because of the historical influence of French on Middle English,\nsee Figure~\\ref{IE3fig}, which can be viewed as creating a bridge (sometimes referred to\nas the Anglo-Norman bridge) connecting the Latin and the Germanic subtrees and\nintroducing non-trivial topology in the Indo-European phylogenetic network.\nIt is well known that the influx of French was extensive at the lexical\nlevel, but it is not clear whether one should expect to see a trace of this historical phenomenon\nwhen analyzing languages at the level of syntactic structures. It is, however, a natural question\nto ask whether the non-trivial loop one sees in the persistent topology of syntactic\nparameters of the Indo-European family may perhaps be a syntactic remnant of the \nAnglo-Norman bridge.\n\n\\smallskip\n\nHowever, a further analysis of the SSWL dataset of syntactic parameters appears to\nexclude this possibility. Indeed, we computed the persistent homology using only the\nIndo-European languages in the Latin and Germanic groups. If the persistent generator\nof $H_1$ were due to the Anglo-Norman bridge one would still find this non-trivial\ngenerator when using only this group of languages, while what we find is that\nthe group of Latin and Germanic languages alone carry no non-trivial persistent first homology,\nsee Figure~\\ref{noANfig}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.41]{group-1-7-5-129.jpg}\n\\includegraphics[scale=0.41]{group-1-10-5-130.jpg}\n\\caption{Barcode diagram for the Latin+Germanic languages, \nat indices $(7,5,129)$\nand $(10,5,130)$. \\label{noANfig}}\n\\end{center}\n\\end{figure}\n\n\\smallskip\n\nIn order to understand the nature of the two persistent generators of $H_0$,\nwe separated out the Indo-Iranian subfamily of the Indo-European family, to\ntest whether the two persistent connected components would be related to\nthe natural subdivision into the two main branches of the Indo-European family.\nEven though the Indo-Iranian branch is the largest subfamily of Indo-European\nlanguages, it is much less extensively mapped in SSWL \nthan the rest of the Indo-European family, with only 9 languages recorded in the database.\nThus, a topological data analysis performed directly on the Indo-Iranian subfamily is less reliable,\nbut one can gain sufficient information by analyzing the remaining set of\nIndo-European languages, after removing the Indo-Iranian subfamily. The result\nis illustrated in Figure~\\ref{group3aFig}. We see that indeed the number of\npersistent connected component is now just one, which supports the proposal\nof relating persistent generators of $H_0$ to major subdivisions into historical\nlinguistic subfamilies. Moreover, the persistent generator of the $H_1$ is still\npresent, which shows that the non-trivial first homology is not located in the \nIndo-Iranian subfamily.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.6]{group-3a-7-5-97.jpg}\n\\caption{Barcode diagram for the Indo-European family\nwith the Indo-Iranian subfamily removed, \nat indices $(7,5,97)$. \\label{group3aFig}}\n\\end{center}\n\\end{figure}\n\n\n\\smallskip\n\nIn order to understand more precisely where the non-trivial persistent\nfirst homology is located in the Indo-European family, we performed\nthe analysis again, after removing the Indo-Iranian languages and\nalso removing the Hellenic branch, including both Ancient and Modern\nGreek. The resulting persistent topology is illustrated in Figure~\\ref{group3bFig}.\nBy comparing Figures~\\ref{group3aFig} and \\ref{group3bFig} one sees that\nthe position of the Hellenic branch of the Indo-European family has a direct\nrole in determining the persistent topology. When this subfamily is removed,\nthe number of persistent connected components (generators of $H_0$) \njumps from one to three, while the non-trivial single generator of $H_1$\ndisappears. Although this observation by itself does not provide an explanation\nof the persistent topology in terms of historical linguistics of the Indo-European\nfamily, it points to the fact that, if historical linguistic phenomena are involved\nin determining the topology, they appear to be related to the role that \nAncient Greek and the Hellenic branch played in the historical development of \nthe Indo-European languages.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.6]{group-3b-7-5-96.jpg}\n\\caption{Barcode diagram for the Indo-European family\nwith the Indo-Iranian and the Hellenic subfamilies removed, \nat indices $(7,5,96)$. \\label{group3bFig}}\n\\end{center}\n\\end{figure}\n\n\n\n\\smallskip\n\nWhen performing a more detailed cluster analysis on the Indo-European\nfamily, one finds sub-structures in the persistent topology. For instance, as\nshown in Figure~\\ref{IE3fig}, one sees a possible second generator of\nthe persistent $H_1$ for cluster filtering value 165, with indices $(10,5,95)$.\nThese substructures may also be possible traces of other historical linguistic\nphenomena.\n\n\\medskip\n\\subsection{Persistent topology of the Niger-Congo family}\n\nWe performed the same type of analysis on the syntactic parameters of the \nNiger-Congo language family. The interesting result we observed is that the behavior of\npersistent homology seems to be quite different for different language families.\nFigure \\ref{NC1fig} shows the barcode diagrams for persistent\nhomology at index values $(7,5,107)$, $(10,0,100)$, and $(10,5,105)$, which can\nbe compared with the diagrams of Figure \\ref{IE1fig} for the\nIndo-European family. In the Niger-Congo family, we now see persistent $H_0$ \nhomology, respectively, of ranks $1$, $3$, and $1$ (compare with ranks $2$, $3$, $2$\nin the Indo-European case). A lower rank in the $H_0$ means fewer connected\ncomponents in the Vietoris-Rips complex, which seems to indicate that the syntactic\nparameters are more concentrated and homogeneously distributed \nacross the linguistic family, and less ``spread out\" into different sub-clusters. \n\n\n\\smallskip\n\nFollowing the cluster analysis of \\S \\ref{clustersec}, we also considered the\npersistent homology for the Niger-Congo family at specific cluster filtering values.\nWhile for cluster filtering value $165$ and indices $(7,3,100)$ one sees one\npersistent generator of $H_0$ and a possibility of a persistent generator in the\n$H_1$, cluster filtering value $165$ with indices $(10,0,100)$, as well as\ncluster filtering value $190$ with indices $(7,5,104)$ and\n$(10,0,104)$ only show one persistent generator in the $H_0$.\n\n\\smallskip\n\nThis persistent homology viewpoint seems to suggest that syntactic\nparameters within the Niger-Congo language family may be spread out more\nevenly across the family than they are in the Indo-European case, with\na single persistent connected component, whereas the Indo-European \nones have two different persistent connected component, one of which has\ncircle topology. \n\n\n\\section{Further questions}\n\nWe showed that methods from topological data analysis, in particular persistent\nhomology, can be used to analyze how syntactic parameters are distributed\nover different language families. In particular we compared the cases of\nIndo-European and Niger-Congo languages. \n\n\\smallskip\n\nWe list here some questions\nthat naturally arise from this perspective, which we believe are worthy of\nfurther investigation.\n\\begin{enumerate}\n\\item To what extent do persistent generators of the $H_0$ (that is, the \npersistent connected components) of the data space\nof syntactic parameter correspond to different (sub)families of languages\nin the historical linguistic sense? For example, are the three $H_0$ generators\nvisible at scale $(10,0,100)$ in the Congo-Niger family a remnant of the historical \nsubdivision into the Mande, Atlantic-Congo, and Kordofanian subfamilies? \n\\item What is the meaning, in historical linguistic terms, of the circle components\n(persistent generators of $H_1$) in the data space\nof syntactic parameters of language families? \nIs there a historical-linguistic interpretation for the second $H_1$ generator\none sees at cluster filtering value 165 and scale $(10,5,95)$ in the Indo-European\nfamily? Or for the $H_1$ generator one sees with the same cluster filtering, at scale\n$(7,3,100)$ in the Niger-Congo case?\n\\item To what extent does persistent topology describe different\ndistribution of syntactic parameters across languages for different\nlinguistic families?\n\\end{enumerate}\n\n\\bigskip\n\\bigskip\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nRecent analyses of data from \\mbox{$p$+$p$}\\xspace and \\mbox{$p$+Pb}\\xspace collisions at the LHC, as well as from \\mbox{$d$+Au}\\xspace and \\mbox{$^3\\text{He}$+Au}\\xspace at RHIC have revealed the existence of azimuthal particle correlations reminiscent of those observed in $A+A$ collisions~\\cite{adare_measurement_2014,Adare:2015ita,atlas_observation_2012,alice_long_2013,cms_observation_2012,khachatryan_observation_2010}. In the latter, this signal has been interpreted as evidence for the liquid-like nature of the Quark-Gluon Plasma (QGP)---a locally equilibrated, strongly coupled medium undergoing hydrodynamic expansion with viscosity near the quantum lower bound. However, these results were largely unexpected in $p(d)+$A, long considered control systems \nto understand initial-state effects in heavier systems. \n\nThe success of nearly inviscid hydrodynamics in describing $A+A$ bulk observables makes it natural to ask if droplets of QGP are being formed in small systems, and if the created medium is sufficiently long lived to equilibrate locally and translate initial spatial anisotropies into final-state particle momentum correlations. However, this is not the only possibility, with potentially different physics being able to account for these measurements~\\cite{bzdak_elliptic_2014,ma_long-range_2014,dusling_azimuthal_2012}. Further insight into this matter will come from the confrontation of different model calculations with the full data sets available.\n\nIt has been found that azimuthal two-particle correlations from high multiplicity \\mbox{$p$+$p$}\\xspace, \\mbox{$p$+Pb}\\xspace, \\mbox{$d$+Au}\\xspace, and \\mbox{$^3\\text{He}$+Au}\\xspace events exhibit an\nenhancement around $\\Delta \\phi \\approx 0$ (i.e. near-side), even when the particles have a large separation in \npseudo-rapidity ($\\Delta \\eta > 2$), where\njet contributions are expected to be minimal. There is an additional enhancement from \\mbox{$p$+$p$}\\xspace and peripheral \\mbox{$p$+Pb}\\xspace(\\mbox{$d$+Au}\\xspace)\nto central \\mbox{$p$+Pb}\\xspace(\\mbox{$d$+Au}\\xspace) around $\\Delta \\phi \\approx \\pi$ (i.e. away-side) that has been interpreted as the full \nazimuthal continuation of elliptical and triangular flow coefficients, $v_2$ and $v_3$. Alternative readings \nof the away-side pattern include modification of the dijet correlations for the most central \\mbox{$d$+Au}\\xspace \ncollisions~\\cite{Adamczyk:2014fcx}.\n\nPredictions for LHC-energy \\mbox{$p$+Pb}\\xspace anisotropies using nearly inviscid hydrodynamics~\\cite{bozek_collective_2012} provide a reasonable description\nof the flow coefficients measured at the LHC. However, as expected, an exact quantitative description depends on \non the shear viscosity to entropy density ratio ($\\eta\/s$) and the details\nof how initial geometry is modeled, for which there are quite different possibilities in $p+A$ collisions---see\nfor example Ref.~\\cite{Schlichting:2014ip,Bzdak:2013zma}. \nThere are also competing calculations where final-state QGP or flow effects\nare deemed negligible, and it is initial-state glasma diagrams that give rise to the correlations~\\cite{dusling_azimuthal_2012}. In\n\\mbox{$d$+Au}\\xspace and \\mbox{$^3\\text{He}$+Au}\\xspace collisions, the initial geometry is dominated by the spatial separation of the two nucleons in the deuteron, reducing\ndifferences between models of geometry. For this case, nearly inviscid hydrodynamic calculations give a reasonable description of the \nexperimentally extracted flow coefficients~\\cite{nagle_exploiting_2013,Bzdak:2013zma,Bozek:2014xsa}. \n\nHowever, questions regarding the validity of the near-inviscid hydrodynamic calculations have been raised in terms of the expansion around steep energy density gradients in these small systems~\\cite{Romatschke:2015dha,Romatschke:2015gxa,Niemi:2014wta,Denicol:2015bpa}. It is thus quite interesting that incoherent elastic parton-parton scattering---as implemented in A-Multi-Phase-Transport-Model (\\textsc{AMPT})~\\cite{lin_multiphase_2005}---can effectively reproduce the long-range azimuthal correlations~\\cite{ma_long-range_2014} and $v_2$ coefficients~\\cite{bzdak_elliptic_2014} observed in high multiplicity \\mbox{$p$+$p$}\\xspace and \\mbox{$p$+Pb}\\xspace events at the LHC. Notice, however, that in the case of $v_2$ in \\mbox{$p$+Pb}\\xspace, a good reproduction of the measured values is only achieved for \\mbox{$p_T$}\\xspace $\\lesssim$ 2 GeV\/c, above which the calculations underestimate the data. These \\textsc{AMPT} results were obtained using a parton scattering cross section of $\\sigma=1.5-3.0$ mb, and incorporating the so-called \\textit{string melting} mechanism in the model (thus including a time stage dominated by parton-parton scattering). \n\nThese results raise the question of whether a similar description can be achieved for different collision geometries\nat the RHIC energy scale. In particular, we use \\textsc{AMPT} to simulate \\mbox{$p$+Au}\\xspace, \\mbox{$d$+Au}\\xspace and \\mbox{$^3\\text{He}$+Au}\\xspace at $\\mbox{$\\sqrt{s_{_{NN}}}$}\\xspace = 200 \\text{ GeV}$ since they have been proposed as an excellent testing ground to disentangle the properties of the medium created in small collision systems~\\cite{nagle_exploiting_2013}. In this paper, we begin by describing the \\textsc{AMPT} model and the methodology used to compute azimuthal anisotropies of final state hadrons with respect to the participant plane. We then compare our $v_2$ and $v_3$ results in \\mbox{$d$+Au}\\xspace and \\mbox{$^3\\text{He}$+Au}\\xspace to available data, and present predictions for $v_2$ and $v_3$ in \\mbox{$p$+Au}\\xspace. Finally, we discuss our results and provide some conclusions.\n\n\\section{Methods}\n\nThe \\textsc{AMPT} event generator~\\cite{lin_multiphase_2005} is a useful tool for the study of heavy-ion collision dynamics. The \\textsc{AMPT} model\nuses the HIJING model~\\cite{wang_hijing_1994} just to generate initial conditions via Monte Carlo Glauber, Zhang's Parton Cascade (ZPC) to model partonic scattering, and A-Relativistic-Transport (ART) to model late stage hadronic scattering. We utilize the \\textsc{AMPT} model with string melting turned on, such that a stage with parton-parton scattering is included and subsequent hadronization is described via a coalescence model. In this coalescence model, quark-antiquark pairs and sets of three (anti) quarks in close spatial proximity are grouped to form mesons and baryons, respectively.\n\nIn order to better understand how \\textsc{AMPT} translates anisotropies in the initial geometry to anisotropies in final-state momentum, we modified the internal Monte Carlo Glauber to more closely resemble the standard approach described in~\\cite{Miller:2007ri,Loizides:2014vua}, and used in~Ref.\\cite{nagle_exploiting_2013}. The position of nucleons in each colliding nucleus is sampled from the appropriate wave function on an event-by-event basis, after which a nucleon-nucleon inelastic cross section of 42 mb is used to geometrically determine which nucleons were wounded in the collision. In the case of the deuteron, coordinates are sampled from the two-nucleon Hulth\\'en wavefunction; in the case of $^3$He, coordinates are obtained with Green's function Monte Carlo calculations using the AV18+UIX model of three-body interactions~\\citep{carlson_structure_1998}. \n\nWe have run approximately 10 million central (i.e. impact parameter $b < 2$ fm) \\textsc{AMPT} events with a parton-parton scattering cross section $\\sigma = 1.5$ mb for each system at \\mbox{$\\sqrt{s_{_{NN}}}$}\\xspace = 200 GeV. There are numerous publications utilizing the \\textsc{AMPT} model to describe $A+A$ collisions at RHIC, quoting a full range of input cross section values ranging from $\\sigma = 1.5$ mb up to $\\sigma = 10$ mb~\\cite{PhysRevC.86.054908,JPhysG30.2004.S263-S270,ChinesePhysC37.014104}. For this study, we have chosen the smallest value from this range to understand how a minimum parton scattering stage contributes to collective motion in these small systems. Table \\ref{tab_partproduction} summarizes the mean number of nucleon participants for each system in \\textsc{AMPT}, as well as the corresponding yield of partons at the end of the parton scattering stage and the yield of final-state hadrons at the end\nof the hadronic scattering stage. Note that all partons reported by \\textsc{AMPT} at the end of the parton scattering stage are\nquarks and anti-quarks, and no gluons, which are then input to the particular coalescence calculation for hadronization.\n\n\\begin{table}[tbh]\n\\caption{\\label{tab_partproduction} Particle production and eccentricity in central \\textsc{AMPT} small-system collisions. For each collision system, we show the mean number of participant nucleons per event, the mean number of partons (i.e. quarks and antiquarks) at freeze out, the mean number of hadrons after the hadron cascade, and the mean elliptical and triangular initial state eccentricities.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\n System & $\\langle N_{part}\\rangle$ & $\\langle N_{partons} \\rangle$ & $\\langle N_{hadrons}\\rangle$ & $\\langle\\varepsilon_2 \\rangle$ & $\\langle \\varepsilon_3 \\rangle$ \\\\\\hline\n \\mbox{$p$+Au}\\xspace & 10.45 & 182 & 131 & 0.24 & 0.16 \\\\\n \\mbox{$d$+Au}\\xspace & 18.3 & 336 & 233 & 0.57 & 0.17 \\\\\n \\mbox{$^3\\text{He}$+Au}\\xspace & 22.3 & 446 & 326 & 0.48 & 0.23 \\\\\n \\end{tabular}\n \\end{ruledtabular}\n\\end{table}\n\nFor each collision system, we compute the eccentricity $\\varepsilon_n$, and participant plane angle $\\Psi_n$ on an event-by-event basis using the initial-state coordinates $(r_i,\\phi_i)$ of participant nucleons with the spatial distribution of a Gaussian of width $\\sigma=0.4$ fm, as follows for $n=2,3$\n\n\\begin{equation}\n\\varepsilon_n = \\frac{\\sqrt{\\langle r^2\\cos(n\\phi) \\rangle^2 + \\langle r^2\\sin (n\\phi) \\rangle^2}}{\\langle r^2\\rangle},\n\\end{equation}\n\n\\begin{equation}\n\\Psi_n = \\frac{\\text{atan2}(\\langle r^2 \\sin (n\\phi) \\rangle,\\langle r^2 \\cos (n\\phi) \\rangle)}{n} +\\frac{\\pi}{n}.\n\\end{equation}\nAverage values of $\\varepsilon_{2}$ and $\\varepsilon_{3}$ for central \\mbox{$p$+Au}\\xspace, \\mbox{$d$+Au}\\xspace, and \\mbox{$^3\\text{He}$+Au}\\xspace are shown in Table~\\ref{tab_partproduction}.\n\nHaving measured $\\Psi_n$ from the initial-state geometry, we compute the second and third order azimuthal anisotropy moments $v_2$ and $v_3$ of final-state unidentified charged hadrons within $|\\eta| < 2$, with respect to the participant planes, as follows\n\\begin{equation}\nv_n = \\langle \\cos[n(\\varphi-\\Psi_n)] \\rangle.\n\\end{equation} \n\nIn addition to extracting anisotropy moments with respect to the participant plane, we are also interested in qualitatively examining the long-range azimuthal correlations of these hadrons. To that end, we follow an analysis procedure similar to that put forth by the ATLAS experiment for \\mbox{$p$+Pb}\\xspace collisions in Ref.~\\cite{atlas_observation_2012}. We take all final-state charged particles and consider all pairs separated by $2.0 < |\\Delta \\eta| < 3.0$ within a common \\mbox{$p_T$}\\xspace bin. We then form a long-range two-particle azimuthal correlation function for each \\mbox{$p_T$}\\xspace bin:\n\n\\begin{equation}\n\\label{C}\nC(\\Delta\\phi,\\mbox{$p_T$}\\xspace) = \\frac{1}{N_{\\text{trig}}}\\frac{dN(\\mbox{$p_T$}\\xspace)}{d\\Delta\\phi}.\n\\end{equation} \n \nShown in the top panel of Figure~\\ref{fig_jet_contr} are the two-particle correlations from \\mbox{$p$+$p$}\\xspace and central \\mbox{$^3\\text{He}$+Au}\\xspace collisions for particles within $0.9<\\mbox{$p_T$}\\xspace<1.04$ GeV\/c. As expected, a flat near-side (around $|\\Delta\\phi | \\approx 0$) distribution is observed in \\mbox{$p$+$p$}\\xspace since two particles from the same jet fragmentation or resonance decay are very\nunlikely to be separated by more than two units of pseudo-rapidity. There is also a significant \\mbox{$p$+$p$}\\xspace away-side enhancement\n(around $|\\Delta\\phi | \\approx \\pi$) from jets back-to-back at leading order in azimuth and with a significant pseudo-rapidity\nswing between them from an imbalance in the initial parton momentum fractions $x_{1}$ and $x_{2}$. \n\nIn contrast, in the \\mbox{$^3\\text{He}$+Au}\\xspace distribution we observe a prominent near-side peak and a nearly two-fold enhancement of the away-side yield relative to \\mbox{$p$+$p$}\\xspace. Assuming the jet contribution to be the same in \\mbox{$p$+$p$}\\xspace and \\mbox{$^3\\text{He}$+Au}\\xspace, we subtract the two distributions, thus obtaining the result shown in the bottom panel of Figure~\\ref{fig_jet_contr}. We choose to use \\mbox{$p$+$p$}\\xspace data for the jet subtraction, analogously to what has been done in experimental data with peripheral \\mbox{$p$+Pb}\\xspace(\\mbox{$d$+Au}\\xspace) events~\\cite{atlas_observation_2012,alice_long_2013,PhysRevLett.111.212301}.\nIt is important to note that we do not use long-range two-particle correlations to extract anisotropy moments, but rather to provide a qualitative comparison with experimental results where similar correlations are presented. Hence, all results presented in the remainder of this article are computed with respect to the participant plane using initial state geometry, as previously described.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.55]{figure_jetfrag_subtraction_singlepanel.pdf}\n\\caption{(a) Two-particle correlation for charged particles within $0.901$ GeV\/c and by roughly 100\\% for \\mbox{$p_T$}\\xspace$<$ 0.5 GeV\/c. This effect is even more pronounced for $v_3$. Additionally, the green curves show that even when the partonic phase is turned off, a sizable $v_2$ and $v_3$ still develop by virtue of final-state hadronic interactions, with the effect being more pronounced for $v_3$. Finally, as a consistency check, disabling both the parton and the hadron cascades results in the collapse of $v_n$, as shown by the orange curves.\n\nThe same analysis is carried out for \\mbox{$p$+Au}\\xspace in the bottom panels of Figure \\ref{fig_cascade}. We observe that including a hadron cascade after the partonic scattering phase has a much smaller effect on the measured $v_2$ or $v_3$. Furthermore, the $v_n$ that develops from the hadron cascade alone when turning off partonic scattering is much less substantial than in the case of \\mbox{$^3\\text{He}$+Au}\\xspace, as evidenced by the green curves, showing that in \\mbox{$p$+Au}\\xspace the bulk of the $v_n$ originates in the parton cascade.\n\nWe now examine the role of hadronization in the development of $v_n$. Figure \\ref{fig_freezeout} shows $v_2$ and $v_3$ calculated for \\mbox{$^3\\text{He}$+Au}\\xspace and \\mbox{$p$+Au}\\xspace collisions with a partonic scattering phase using $(i)$ partons at freeze out and $(ii)$ hadrons immediately after coalescence. For high \\mbox{$p_T$}\\xspace, we observe that hadronization increases $v_2$ and $v_3$ in both collision systems. However, for \\mbox{$p_T$}\\xspace$<$ 0.5 GeV\/c, the effect of hadronization is to reduce $v_n$, as evidenced by the crossing of the curves in the figure. This can be understood in terms of quark coalescence dynamics. Since hadrons are produced by aggregating partons in spatial proximity and with collimated momenta, the coalescence yields hadrons with transverse momentum greater than that of their constituent quarks, hence increasing $v_n$ at higher \\mbox{$p_T$}\\xspace. It is notable that this effect is greater in \\mbox{$p$+Au}\\xspace than in \\mbox{$^3\\text{He}$+Au}\\xspace.\n\nThe detailed mechanism and its relation to the number of parton scatterings in the early \\textsc{AMPT} stage still require further\nelucidation. That said, it is clear that the \\textsc{AMPT} coalescence prescription and following hadronic cascade substantially\nmodify and amplify this early stage effect. Therefore, the deviations from geometric scaling shown in Figure~\\ref{fig_vn_ratio} \nappear to arise from the relative dominance of these stages as a function of \\mbox{$p_T$}\\xspace and collision system.\n\n\\section{Summary}\nRecent intriguing experimental observations at RHIC and the LHC have raised the question of whether small droplets of QGP \ncan be formed in small collision systems. From among several competing models, nearly inviscid hydrodynamic calculations, both at RHIC and the LHC, give reasonable account of the measured anisotropy coefficients. \nHowever, parton scattering in transport models---\\textsc{AMPT} with string melting, in particular---has also been shown to provide an adequate description of the long-range azimuthal correlations and momentum anisotropy coefficients measured in \\mbox{$p$+$p$}\\xspace and \\mbox{$p$+Pb}\\xspace at LHC energies. In this paper, we extend these calculations to RHIC energies, focusing on the insight that can be gained by varying the initial geometry of the projectile nucleus. \n\nWe find that \\textsc{AMPT} is capable of reasonably reproducing the measured elliptic and triangular flow coefficients for central \\mbox{$d$+Au}\\xspace and \\mbox{$^3\\text{He}$+Au}\\xspace collisions at \\mbox{$\\sqrt{s_{_{NN}}}$}\\xspace = 200 GeV for $\\mbox{$p_T$}\\xspace <1 \\text{ GeV\/c}$. However, \\textsc{AMPT} underestimates the measured values for higher \\mbox{$p_T$}\\xspace. With this observation, we ascertain the validity of the model for rendering initial geometric anisotropy into final-state particle momentum correlations for small systems at both the RHIC and LHC energy scales. We also make predictions for elliptic and triangular anisotropy coefficients in \\mbox{$p$+Au}\\xspace collisions at \\mbox{$\\sqrt{s_{_{NN}}}$}\\xspace = 200 GeV and qualitatively relate these results to calculated initial-state geometric anisotropy. \n\nHowever, we also find that partonic scattering is not the only source of the substantial elliptic and triangular momentum anisotropies in the \\textsc{AMPT} model. Hadronization and the subsequent hadron cascade exert important modifications on the $v_n$ from partonic scattering, with strong dependences both on \\mbox{$p_T$}\\xspace and the intrinsic initial geometry of the system.\nDirect comparisons with experimental data in these new systems, with both \\textsc{AMPT} and various hydrodynamic models, is anticipated to shed light on the physical dynamics involved in these collision systems. To finalize, we highlight the need to identify additional observables that provide a more stringent discrimination between initial geometry and its translation to final-state correlations. \n\n\\begin{acknowledgments}\n\nWe acknowledge funding from the Division of Nuclear\nPhysics of the U.S. Department of Energy under Grant\nNo. DE-FG02-00ER41152. We also thank Paul Romatschke, Paul Stankus, and Shengli Huang for useful discussions.\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:intro}\n\n{\\Large I}{\\large N \\textsc{the}} \\textsc{penultimate paragraph} of the first section of his 1931 celebrated paper on the incompleteness theorems, {\\sc G\\\"odel} wrote\n\n\\begin{quote}\n\\vspace{-1em}\n\\begin{itemize}\n\\item[]\nThe method of proof just explained can clearly be applied to any formal system that, first, [...] and in which, second, every provable formula is true in the interpretation considered. The purpose of carrying out the above proof with full precision in what follows is, among other things, to replace the second of the assumptions just mentioned by a purely formal and much weaker one.\n\\hfill \\cite[p.~151]{Godel}\n\\end{itemize}\n\\vspace{-1em}\n\\end{quote}\n\nHe began the next section with the sentence, ``We now proceed to carry out with full precision the proof sketched above''. It is clear then that {\\sc G\\\"odel} sketched his proof of the first incompleteness theorem in Section~1 of \\cite{Godel} for the system of {\\em Principia Mathematica}, and then noted that his method of proof works for any formal system that, first, is {\\em sufficiently strong} (in today's terminology) and, second, is {\\em sound} (with respect to the standard model of natural numbers). He then said that in the rest of the article the proof would be carried out with full precision, while the second assumption (that of soundness) was replaced by a ``purely formal and much weaker one''. This assumption was called $\\omega$-consistency by him (see Definition~\\ref{def:oc} below). A question pursued in this paper is the following:\n\n\\qquad \\qquad {\\em Why is the purely formal notion of $\\omega$-consistency much weaker than soundness?}\\,\\footnote{\\!See also \\cite[p.~141, the paragraph after the proof of Proposition 19]{Isaacson}.}\n\nOne possible answer could be the pure formality of $\\omega$-consistency itself!\n {\\sc G\\\"odel} knew that soundness (or truth) is not purely formal (what we know today from {\\sc Tarski}'s Undefinability Theorem); see e.g.\\ \\cite{Murawski}. And, $\\omega$-consistency is purely formal (arithmetically definable, see Definition~\\ref{def:oic} below). Since soundness implies $\\omega$-consistency, and the latter is definable while the former is not, then $\\omega$-consistency should be (much) weaker than soundness. Could {\\sc G\\\"odel} have meant in the penultimate paragraph ``to replace\nthe\nsecond of the assumptions just mentioned by a purely formal and (thus) much weaker one''? In the other words, could his reason for the weakness of $\\omega$-consistency in front of soundness be the pure formality (arithmetical definability) of the former (and undefinability of the latter)?\n\nOn the other hand, the independence of the G\\\"odelian sentences can be guaranteed by much weaker assumptions (much weaker than $\\omega$-consistncy!); $1$-consistency is more than enough. Even {\\sc G\\\"odel} mentioned in the last page of \\cite{Godel} that the consistency of the theory with its (standard) Consistency Statement is sufficient (and also necessary, see e.g.\\ \\cite[Theorem~35]{Isaacson}).\nThese stronger (than simple consistency) assumptions were all removed by Rosser \\cite{Rosser36} who showed in 1936 the independence of other sentences from consistent theories (which are recursively enumerable and contain sufficient arithmetic).\nBefore going to technical details, let us quote {\\sc Smory\\'nski} about $\\omega$-consistency:\n\n\\begin{quote}\n\\vspace{-1em}\n\\begin{itemize}\n\\item[]\nOne weakness of {\\sc G\\\"odel}'s original work was his introduction of the semantic notion of $\\omega$-consistency. I find this notion to be pointless, but I admit many proof theorists take it seriously. \\hfill \\cite[p.~158, Remark]{Smorynski}\n\\end{itemize}\n\\vspace{-1em}\n\\end{quote}\n\nIt is notable that some prominent logicians, of the caliber of {\\sc Henkin} \\cite{Henkin}, studied, and even generalized, the concept of $\\omega$-consistency. Which is a {\\em syntactic} (purely formal, proof-theoretic) notion; not ``semantic''! (see Remark~\\ref{remark} below). However, we can agree with {\\sc Smory\\'nski} that $\\omega$-consistency could be ``pointless'', and may be dismissed with.\n\n\n\\section{$\\boldsymbol\\omega$-Consistency and Some of Its Semantic Properties}\\label{sec:sem}\nLet us fix a sufficiently strong theory $\\mathbb{P}$ over an arithmetical language (which contains, $+,\\times$, and probably some other constant, relation, or function symbols). This could be {\\sc Peano}'s Arithmetic, which is more than enough, or some of its weaker fragments, such as ${\\rm I\\Sigma_1}$.\n\\newline\\centerline{\\em All our theories are \\textup{(usually {\\sc re})} sets of arithmetical sentences that contain $\\mathbb{P}$.} Let us be given a fixed {\\sc G\\\"odel} coding and arithmetization by which we have the provability predicate ${\\tt Pr}_T(x)$, for a fixed coding of the theory $T$, saying that ``(the sentence with code) $x$ is $T$-provable''. We assume familiarity with the notions of $\\Sigma_m$ and $\\Pi_m$ formulas.\n\n\n\\begin{definition}[{\\rm $\\omega$-Consistency}]\n\\label{def:oc}\n\\noindent\n\n\\noindent\nThe theory $T$ is called $\\omega$-consistent, when there is {\\em no} formula $\\varphi(x)$, with the only free variable $x$, such that $T\\vdash\\neg\\forall x\\varphi(x)$ and $T\\vdash\\varphi(\\overline{n})$ for each $n\\!\\in\\!\\mathbb{N}$. Here, $\\overline{n}$ denotes the standard term that represents $n$.\n\\hfill\\ding{71}\\end{definition}\n\n\n\\begin{exampl}[{\\rm of an $\\omega$-consistent and an $\\omega$-inconsistent theory}]\\label{example}\n\\noindent\n\n\\noindent\nEvery sound theory is $\\omega$-consistent. To see a natural $\\omega$-{\\em in}consistent theory, let us consider the negation of the (formal) Induction Principle. For a formula $\\varphi(x)$, the formal Induction Principle of $\\varphi$ is\n\\newline\\centerline{${\\rm IND}_\\varphi:\\quad \\varphi(\\overline{0})\\wedge\\forall x\\,[\\varphi(x)\\!\\rightarrow\\!\\varphi(x\\!+\\!\\overline{1})]\n\\longrightarrow\\forall x\\,\\varphi(x).$}\nIt is known that the IND of formulas with smaller complexity do not imply the IND of formulas with higher complexity. So, $\\neg{\\rm IND}_\\varphi$ could be consistent with some weak arithmetical theories, when $\\varphi$ is a sufficiently complex formula. We show that $\\neg{\\rm IND}_\\varphi$ entails an $\\omega$-inconsistency.\nFirst, note that $\\neg{\\rm IND}_\\varphi\\vdash\\neg\\forall x\\varphi(x)$.\nSecond, we have\n$\\neg{\\rm IND}_\\varphi\\vdash\n\\varphi(\\overline{n})\\!\\rightarrow\\!\n\\varphi(\\overline{n\\!+\\!1})$ for every $n\\!\\in\\!\\mathbb{N}$, and so by (meta-)induction on $n$ one can show $\\neg{\\rm IND}_\\varphi\\vdash\n\\varphi(\\overline{n})$ for every $n\\!\\in\\!\\mathbb{N}$, noting that\n $\\neg{\\rm IND}_\\varphi\\vdash\\varphi(\\overline{0})$. Therefore, $\\neg{\\rm IND}_\\varphi$ is $\\omega$-inconsistent.\n\\hfill\\ding{71}\\end{exampl}\n\n\nWe will use the following result of {\\sc Isaacson} \\cite[Theorem 21]{Isaacson}, which is the $\\omega$-version of {\\sc Lindenbaum}'s Lemma:\n\n\n\n\\begin{fact}[$\\omega\\textrm{-}{\\tt Con}_{T}\\Longrightarrow\\forall\\psi\\!\\!: \\omega\\textrm{-}{\\tt Con}_{T+\\psi}\\vee\\omega\\textrm{-}{\\tt Con}_{T+\\neg\\psi}$]\\label{fact:1}\n\\noindent\n\n\\noindent\nIf $T$ is $\\omega$-consistent, then for every sentence $\\psi$ either $T\\!+\\!\\psi$ or $T\\!+\\!\\neg\\psi$ is $\\omega$-consistent.\n\\hfill \\ding{113}\n\\end{fact}\n\nFor a proof, assume (for the sake of a contradiction) that both $T\\!+\\!\\psi$ and $T\\!+\\!\\neg\\psi$ are $\\omega$-inconsistent. Then for some formulas $\\alpha(x)$ and $\\beta(x)$ we have\n $T\\!+\\!\\psi\\vdash\\neg\\forall x\\,\\alpha(x)$ and $T\\!+\\!\\psi\\vdash\\alpha(\\overline{n})$ for each $n\\!\\in\\!\\mathbb{N}$, also $T\\!+\\!\\neg\\psi\\vdash\\neg\\forall x\\,\\beta(x)$ and $T\\!+\\!\\neg\\psi\\vdash\\beta(\\overline{n})$ for each $n\\!\\in\\!\\mathbb{N}$.\nBy Deduction Theorem we have $T\\vdash\\neg\\forall x\\,[\\psi\\!\\rightarrow\\!\\alpha(x)]$ and $T\\vdash\\neg\\forall x\\,[\\neg\\psi\\!\\rightarrow\\!\\beta(x)]$, and so by Classical Logic we have\n(I) $T\\vdash\\neg\\forall x\\,\\big([\\psi\\!\\rightarrow\\!\\alpha(x)]\n\\wedge[\\neg\\psi\\!\\rightarrow\\!\\beta(x)]\\big)$.\nAgain, by Deduction Theorem, for every $n\\!\\in\\!\\mathbb{N}$ we have (II) $T\\vdash[\\psi\\!\\rightarrow\\!\\alpha(\\overline{n})]\n\\wedge[\\neg\\psi\\!\\rightarrow\\!\\beta(\\overline{n})]$. Now, (I) and (II) imply that $T$ is not $\\omega$-consistent, which contradicts the assumption.\n\\hfill {\\scriptsize\\texttt{\\em QED}}\n\nAn interesting consequence of Fact~\\ref{fact:1}, in the light of Example~\\ref{example}, is that if $T$ is $\\omega$-consistent then $T\\!+\\!{\\rm IND}_\\varphi$, for any formula $\\varphi(x)$, is $\\omega$-consistent too. So is the theory $T\\!+\\!\\{{\\rm IND}_{\\varphi_1},\\cdots,{\\rm IND}_{\\varphi_n}\\}$, for any finite set of formulas $\\{\\varphi_1(x),\\cdots,\\varphi_n(x)\\}$. Thus, any $\\omega$-consistent theory is consistent with {\\sc Peano}'s Arithmetic.\n\n\nOur next observation is that $\\omega$-consistency is preserved by adding true $\\Sigma_3$-sentences; this was first proved for (adding true) $\\Pi_1$-sentences in \\cite[Theorem~22]{Isaacson} with a proof attributed\nto {\\sc Kreisel} 2005. Let us recall that our base theory $\\mathbb{P}$ is $\\Sigma_1$-complete, i.e., can prove every true $\\Sigma_1$-sentence.\n\n\\begin{theorem}[$\\omega\\textrm{-}{\\tt Con}_T\\wedge\\sigma\\!\\in\\!\\Sigma_3\\textrm{-}{\\rm Th}(\\mathbb{N})\\Longrightarrow\\omega\\textrm{-}{\\tt Con}_{T+\\sigma}\\wedge\\neg\\omega\\textrm{-}{\\tt Con}_{T+\\neg\\sigma}$]\\label{thm:sigma3}\n\\noindent\n\n\\noindent\nIf $T$ is an $\\omega$-consistent theory and $\\sigma$ is a true $\\Sigma_3$-sentence, then the theory $T\\!+\\!\\sigma$ is $\\omega$-consistent and the theory $T\\!+\\!\\neg\\sigma$ is $\\omega$-inconsistent.\n\\end{theorem}\n\\begin{proof}\n\n\\noindent\nWrite $\\sigma=\\exists x\\,\\pi(x)$ for a $\\Pi_2$-formula $\\pi$. Since $\\sigma$ is true, then there exists some $k\\!\\in\\!\\mathbb{N}$ such that $\\mathbb{N}\\vDash\\pi(\\overline{k})$. Write $\\pi(\\overline{k})=\\forall y\\,\\theta(y)$ for some $\\Sigma_1$-formula $\\theta$. Then for every $n\\!\\in\\!\\mathbb{N}$ we have $\\mathbb{N}\\vDash\\theta(\\overline{n})$. So, by the $\\Sigma_1$-completeness of $\\mathbb{P}$ we have $\\mathbb{P}\\vdash\\theta(\\overline{n})$ for each $n\\!\\in\\!\\mathbb{N}$. Now, from $\\neg\\sigma\\vdash\\neg\\pi(\\overline{k})$ we have $\\neg\\sigma\\vdash\\neg\\forall x\\,\\theta(x)$. Thus, $\\mathbb{P}\\!+\\!\\neg\\sigma$ is $\\omega$-inconsistent; so is $T\\!+\\!\\neg\\sigma$. Since $T$ is $\\omega$-consistent, then by Fact~\\ref{fact:1}, $T\\!+\\!\\sigma$ must be $\\omega$-consistent.\n\\end{proof}\n\nLater, we will see that this result is optimal: adding a true $\\Pi_3$-sentence to an $\\omega$-consistent theory does not necessarily result in an $\\omega$-consistent theory (see Corollary~\\ref{cor:api3} below). Let us now note that $\\omega$-consistency implies $\\Pi_3$-soundness.\n\n\\begin{corollary}[$\\omega\\textrm{-}{\\tt Con}_T\\Longrightarrow\\Pi_3\\textrm{-}{\\tt Sound}_T$]\\label{cor:pi3}\n\\noindent\n\n\\noindent\nEvery $\\omega$-consistent theory is $\\Pi_3$-sound, i.e., every provable $\\Pi_3$-sentence of it is true.\n\\end{corollary}\n\\begin{proof}\n\n\\noindent\nIf $T$ is $\\omega$-consistent, and $\\pi$ is a $\\Pi_3$-sentence such that $T\\vdash\\pi$, then $\\pi$ must be true, since otherwise $\\neg\\pi$ would be a true $\\Sigma_3$-sentence, and so by Theorem~\\ref{thm:sigma3} the theory $T\\!+\\!\\neg\\pi$ would be $\\omega$-consistent, but this is a contradiction since $T\\!+\\!\\neg\\pi$ is inconsistent by the assumption $T\\vdash\\pi$.\n\\end{proof}\n\nWe now note that the notion of $\\omega$-consistency is arithmetically definable.\n\n\\begin{definition}[$\\mho_T(\\varphi)$: {\\rm the formula $\\varphi$ is a witness for the $\\omega$-inconsistency of $T$}]\n\\label{def:oic}\n\\noindent\n\n\\noindent\n Let $\\mho_T(\\varphi)$ be the following formula: ${\\tt Pr}_T(\\ulcorner\\neg\\forall v\\varphi(v)\\urcorner)\\wedge\\forall w\\,{\\tt Pr}_T(\\ulcorner\\varphi(\\overline{w})\\urcorner)$.\n Here $\\ulcorner\\alpha\\urcorner$ (which is a term in the language of $\\mathbb{P}$) denotes the {\\sc G\\\"odel} code of the expression $\\alpha$.\n Let $\\omega\\textrm{-}{\\tt Con}_T$ be the sentence $\\neg\\exists\\chi\\,\\mho_T(\\chi)$.\n\\hfill\\ding{71}\\end{definition}\n\n\nWe note that when $T$ is an {\\sc re} theory, then ${\\tt Pr}_T(x)$ is a $\\Sigma_1$-formula, so $\\mho_T(x)$ is a $\\Pi_2$-formula, thus $\\omega\\textrm{-}{\\tt Con}_T$ is a $\\Pi_3$-sentence.\n\n\nAs far as we know, the first proof of the weakness of $\\omega$-consistency with respect to soundness appeared in print at \\cite{Kreisel}, what is referred to as ({\\sc Kreisel} 1955) in \\cite[Proposition~19]{Isaacson}.\n\n\n\n\n\n\\begin{definition}[{\\rm Kreiselian $\\Sigma_3$-Sentences of $T$, $\\kappa$: I am $\\omega$-inconsistent with $T$}]\n\\label{def:kappa}\n\\noindent\n\n\\noindent\nFor a theory $T$, any $\\Sigma_3$-sentence $\\kappa$ that satisfies $\\mathbb{P}\\vdash\\kappa\\!\\leftrightarrow\\!\\neg\n\\omega\\textrm{-}{\\tt Con}_{T+\\kappa}$ is called a Kreiselian sentence of $T$.\n\\hfill\\ding{71}\\end{definition}\n\nIf we think for a moment that $\\omega$-consistency equals to soundness, then Kreiselian sentences correspond to the Liar sentences. Now, {\\sc Kreisel}'s proof of the non-equality of $\\omega$-consistency with soundness corresponds to the classical proof of {\\sc Tarski}'s Undefinability Theorem. Let us note that by Diagonal Lemma there exist some Kreiselian sentences for any {\\sc re} theory, be it $\\omega$-consistent or not.\n\n\n\n\n\n\\begin{theorem}[$\\omega\\textrm{-}{\\tt Con}_T\\Longrightarrow\\omega\\textrm{-}{\\tt Con}_{T+\\kappa}\\;\\&\\;\n\\mathbb{N}\\nvDash\\kappa$]\\label{thm:kappa}\n\\noindent\n\n\\noindent\nIf $T$ is {\\sc re} and $\\omega$-consistent, and $\\kappa$ is a Kreiselian sentence of $T$, then $\\kappa$ is false and $T\\!+\\!\\kappa$ is $\\omega$-consistent.\n\\end{theorem}\n\\begin{proof}\n\n\\noindent\nIf $\\kappa$ were true, then by Definition~\\ref{def:kappa} and the soundness of $\\mathbb{P}$, the theory $T\\!+\\!\\kappa$ would be $\\omega$-inconsistent. But by Theorem~\\ref{thm:sigma3}, and the assumed truth of the $\\Sigma_3$-sentence $\\kappa$, the theory $T\\!+\\!\\kappa$ should be $\\omega$-consistent; a contradiction. Thus, $\\kappa$ is false; and so, by the soundness of $\\mathbb{P}$, the theory $T\\!+\\!\\kappa$ is $\\omega$-consistent.\n\\end{proof}\n\n\nSo, for an {\\sc re} $\\omega$-consistent theory $T$ and a Kreiselian sentence $\\kappa$ of $T$, the theory $T\\!+\\!\\kappa$ is $\\omega$-consistent but not sound (since $\\kappa$ is false); $T\\!+\\!\\kappa$ is not even $\\Sigma_3$-sound.\n\n\n\\begin{corollary}[$\\omega\\textrm{-}{\\tt Con}_T\\,\\not\\!\\!\\Longrightarrow\\!\\Sigma_3\\textrm{-}{\\tt Sound}_T,\\; \\Sigma_m\\textrm{-}{\\tt Sound}_T\\,\\not\\!\\!\\Longrightarrow\\!\\omega\\textrm{-}{\\tt Con}_T$]\\label{cor:s3n}\n\\noindent\n\n\\noindent\n$\\omega$-consistency does not imply $\\Sigma_3$-soundness, and for any $m\\!\\in\\!\\mathbb{N}$, $\\Sigma_m$-soundness does not imply $\\omega$-consistency.\n\\end{corollary}\n\n\\begin{proof}\n\n\\noindent\nIt is rather easy to see that $\\Sigma_m$-soundness (the truth of provable $\\Sigma_m$-sentences) is equivalent to consistency with $\\Pi_m\\textrm{-}{\\rm Th}(\\mathbb{N})$, the set of true $\\Pi_m$-sentences. By \\cite[Theorem~2.5]{SalSer} there exists a true $\\Pi_{m+1}$-sentence $\\gamma$ such that $\\mathbb{P}\\!+\\!\\Pi_m\\textrm{-}{\\rm Th}(\\mathbb{N})\\nvdash\\gamma$. So, the theory $U=\\mathbb{P}\\!+\\!\\Pi_m\\textrm{-}{\\rm Th}(\\mathbb{N})\\!+\\!\\neg\\gamma$ is consistent. We show that $U$ is not $\\omega$-consistent. Write $\\gamma=\\forall x\\,\\sigma(x)$ for some $\\Sigma_m$-formula $\\sigma$. By the truth of $\\gamma$ we have $\\Pi_m\\textrm{-}{\\rm Th}(\\mathbb{N})\\vdash\\sigma(\\overline{n})$ for each $n\\!\\in\\!\\mathbb{N}$. So, we have $U\\vdash\\neg\\forall x\\,\\sigma(x)$ and $U\\vdash\\sigma(\\overline{n})$ for each $n\\!\\in\\!\\mathbb{N}$. Thus, $U$ is not $\\omega$-consistent, but it is $\\Sigma_m$-sound (being consistent with $\\Pi_m\\textrm{-}{\\rm Th}(\\mathbb{N})$). However, $U$ is not {\\sc re}; let us consider its sub-theory $T=\\mathbb{P}\\!+\\!\\neg\\forall x\\,\\sigma(x)\\!+\\!\n\\{\\sigma(\\overline{n})\\}_{n\\in\\mathbb{N}}$. The theory $T$ is {\\sc re} and $\\Sigma_m$-sound, but not $\\omega$-consistent.\n\\end{proof}\n\nIt is worth noting that while soundness implies $\\omega$-consistency, $\\Sigma_m$-soundness, even for large $m$'s, does not imply $\\omega$-consistency.\nWe can now show the optimality of Theorem~\\ref{thm:sigma3}.\n\n\\begin{corollary}[$\\omega\\textrm{-}{\\tt Con}_T\\wedge\\pi\\!\\in\\!\\Pi_3\\textrm{-}{\\rm Th}(\\mathbb{N})\\,\\not\\!\\!\\Longrightarrow\\!\\omega\\textrm{-}{\\tt Con}_{T+\\pi}$]\\label{cor:api3}\n\\noindent\n\n\\noindent\nAdding a true $\\Pi_3$-sentence to an $\\omega$-consistent theory does not necessarily result in an $\\omega$-consistent theory.\n\\end{corollary}\n\\begin{proof}\n\n\\noindent\nLet $\\kappa_0$ be a Kreiselian sentence of $\\mathbb{P}$. Then, by Theorem~\\ref{thm:kappa}, the theory $T_0=\\mathbb{P}\\!+\\!\\kappa_0$ is $\\omega$-consistent and $\\neg\\kappa_0$ is a true $\\Pi_3$-sentence. But $T_0\\!+\\!\\neg\\kappa_0$ is not even consistent.\n\\end{proof}\n\nFinally, we can show that adding a Kreiselian sentence or its negation to a sound theory results, in both cases, in an $\\omega$-consistent theory (cf.\\ Theorem~\\ref{thm:rosser} below).\n\n\\begin{corollary}[$\\mathbb{N}\\vDash T\\Longrightarrow\\omega\\textrm{-}{\\tt Con}_{T+\\kappa}\\wedge\\omega\\textrm{-}{\\tt Con}_{T+\\neg\\kappa}$]\\label{cor:k}\n\\noindent\n\n\\noindent\nIf $T$ is a sound {\\sc re} theory and $\\kappa$ is a Kreiselian sentence of $T$, then both $T\\!+\\!\\kappa$ and $T\\!+\\!\\neg\\kappa$ are $\\omega$-consistent.\n\\end{corollary}\n\\begin{proof}\n\n\\noindent\nThe theory $T\\!+\\!\\neg\\kappa$ is sound and the theory $T\\!+\\!\\kappa$ is $\\omega$-consistent by Theorem~\\ref{thm:kappa}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Some Syntactic Properties of $\\boldsymbol\\omega$-Consistency}\\label{sec:syn}\n\nLet us begin this section, like the previous one, with another interesting result of {\\sc Isaacson} \\cite[Theorem 20]{Isaacson}; see \\cite[Proposition 3.2]{SalSer} for a generalization.\n\n\n\n\\begin{fact}[$\\omega\\textrm{-}{\\tt Con}_{T}\\wedge{\\tt Complete}_T\\Longrightarrow T\\!=\\!{\\rm Th}(\\mathbb{N})$]\\label{fact:2}\n\\noindent\n\n\\noindent\nTrue Arithmetic, ${\\rm Th}(\\mathbb{N})$, is the only $\\omega$-consistent theory which is complete.\n\\hfill \\ding{113}\n\\end{fact}\n\n\n\n\n\n\nFor a proof, note that a complete and $\\omega$-consistent theory $T$ is $\\Pi_3$-sound by Corollary~\\ref{cor:pi3}. So, we have $(\\mathcal{C}_2)$\n$T\\vdash\\Sigma_2\\textrm{-}{\\rm Th}(\\mathbb{N})\\!\\cup\\!\\Pi_2\\textrm{-}{\\rm Th}(\\mathbb{N})$; since if $\\eta\\!\\in\\!\\Sigma_2\\textrm{-}{\\rm Th}(\\mathbb{N})\\!\\cup\\!\\Pi_2\\textrm{-}{\\rm Th}(\\mathbb{N})$ and $T\\nvdash\\eta$, then $T\\vdash\\neg\\eta$, by the completeness of $T$, which would contradict the $\\Pi_3$-soundness of $T$.\nWe now show, by induction on $m$, that $(\\mathcal{C}_m)$ $T\\vdash\\Sigma_m\\textrm{-}{\\rm Th}(\\mathbb{N})\\!\\cup\\!\\Pi_m\\textrm{-}{\\rm Th}(\\mathbb{N})$. For proving $(\\mathcal{C}_m\\!\\Rightarrow\\!\\mathcal{C}_{m+1})$ suppose that $(\\mathcal{C}_m)$ holds.\nIt is easy to see that $\\Pi_m\\textrm{-}{\\rm Th}(\\mathbb{N})\\vdash\\Sigma_{m+1}\\textrm{-}{\\rm Th}(\\mathbb{N})$; so by $(\\mathcal{C}_m)$ we already have $T\\vdash\\Sigma_{m+1}\\textrm{-}{\\rm Th}(\\mathbb{N})$. We now show $T\\vdash\\Pi_{m+1}\\textrm{-}{\\rm Th}(\\mathbb{N})$. Let $\\pi$ be a true $\\Pi_{m+1}$-sentence; write $\\pi\\!=\\!\\forall x\\,\\sigma(x)$ for some $\\Sigma_m$-formula $\\sigma$. For every $n\\!\\in\\!\\mathbb{N}$ we have $\\mathbb{N}\\vDash\\sigma(\\overline{n})$, so $\\Sigma_m\\textrm{-}{\\rm Th}(\\mathbb{N})\\vdash\\sigma(\\overline{n})$, thus by $(\\mathcal{C}_m)$ we have $T\\vdash\\sigma(\\overline{n})$. Now, the $\\omega$-consistency of $T$ implies $T\\nvdash\\neg\\forall x\\,\\sigma(x)$; so from the completeness of $T$ we have $T\\vdash\\forall x\\,\\sigma(x)$, thus $T\\vdash\\pi$.\n\\hfill {\\scriptsize\\texttt{\\em QED}}\n\n\\begin{corollary}[$\\lim_\\subseteq\\omega\\textrm{-}{\\tt Con}\\neq\\omega\\textrm{-}{\\tt Con}$]\\label{cor:lindenb}\n\\noindent\n\n\\noindent\nThe limit (union) of a chain of $\\omega$-consistent theories is not necessarily $\\omega$-consistent.\n\\end{corollary}\n\\begin{proof}\n\n\\noindent\nLet $\\kappa_0$ be a Kreiselian sentence of $\\mathbb{P}$ and put $T_0=\\mathbb{P}\\!+\\!\\kappa_0$. Then $T_0$ is $\\omega$-consistent by Theorem~\\ref{thm:kappa}. Now, by Fact~\\ref{fact:1} one can expand $T_0$ in stages $T_0\\!\\subseteq\\!T_1\\!\\subseteq\\!T_2\\!\\subseteq\\!\\cdots$ in a way that each $T_m$ is $\\omega$-consistent and their union $T^\\ast=\\bigcup_mT_m$ is complete. But by Fact~\\ref{fact:2}, $T^\\ast$ cannot be $\\omega$-consistent since $\\mathbb{N}\\nvDash\\kappa_0$ by Theorem~\\ref{thm:kappa} and so $T^\\ast\\neq{\\rm Th}(\\mathbb{N})$. Thus, the limit $T^\\ast$ of the chain $\\{T_m\\}_m$ of $\\omega$-consistent theories is not $\\omega$-consistent.\n\\end{proof}\n\n\n\n\n\\begin{remark}[{\\rm Non-Semanticity of $\\omega$-Consistency}]\\label{remark}\n\\noindent\n\n\\noindent\nFact~\\ref{fact:2} enables us to show that $\\omega$-consistency is not a semantic (model-theoretic) notion.\nAssume, for the sake of a contradiction, that for a class $\\mathscr{C}$ of structures (over the language of $\\mathbb{P}$) and for every theory $T$,\n\n\\qquad \\qquad (\\ding{75}) \\quad $T$ is $\\omega$-consistent if and only if $\\mathcal{M}\\!\\vDash\\!T$ for some $\\mathcal{M}\\!\\in\\!\\mathscr{C}$.\n\nA candidate for such a $\\mathscr{C}$ that comes to mind naturally is the class of $\\omega$-type structures: a model $\\mathfrak{A}$ is called $\\omega$-type, when there is no formula $\\varphi(x)$ such that $\\mathfrak{A}\\!\\vDash\\!\\neg\\forall x\\,\\varphi(x)$ and at the same time $\\mathfrak{A}\\!\\vDash\\!\\varphi(\\overline{n})$ for every $n\\!\\in\\!\\mathbb{N}$. It is clear that if a theory has an $\\omega$-type model, then it is an $\\omega$-consistent theory.\n\nFor showing the impossibility of (\\ding{75}), take $T_0$ to be an unsound $\\omega$-consistent theory (as in e.g.\\ the proof of Corollary~\\ref{cor:lindenb}) and assume that for $\\mathcal{M}_0\\!\\in\\!\\mathscr{C}$ we have $\\mathcal{M}_0\\!\\vDash\\!T_0$. Then the full first-order theory ${\\rm Th}(\\mathcal{M}_0)$ of $\\mathcal{M}_0$, the set of all sentences that are true in $\\mathcal{M}_0$, is a complete $\\omega$-consistent theory. So, by Fact~\\ref{fact:2}, ${\\rm Th}(\\mathcal{M}_0)$ should be equal to ${\\rm Th}(\\mathbb{N})$; thus $\\mathcal{M}_0\\equiv\\mathbb{N}$ whence $\\mathbb{N}\\!\\vDash\\!T_0$, a contradiction.\n\\hfill\\ding{71}\\end{remark}\n\n\nWe now show that Theorem~\\ref{thm:sigma3} can be formalized in $\\mathbb{P}$.\n\n\\begin{theorem}[$\\sigma\\!\\in\\!\\Sigma_3\\Longrightarrow\n\\mathbb{P}\\vdash\\sigma\\!\\wedge\\!\\omega\\textrm{-}{\\tt Con}_T\\!\\rightarrow\\!\\omega\\textrm{-}{\\tt Con}_{T+\\sigma}$]\\label{thm:formal}\n\\noindent\n\n\\noindent\nFor every $\\Sigma_3$-sentence $\\sigma$ and any theory $T$ we have $\\mathbb{P}\\vdash\\sigma\\!\\wedge\\!\\omega\\textrm{-}{\\tt Con}_T\\!\\rightarrow\\!\\omega\\textrm{-}{\\tt Con}_{T+\\sigma}$.\n\\end{theorem}\n\\begin{proof}\n\n\\noindent\nThe proofs of Theorem~\\ref{thm:sigma3} and Fact~\\ref{fact:1} can be formalized in $\\mathbb{P}$ with some hard work. We now present a more direct proof for Theorem~\\ref{thm:sigma3} whose formalizability in $\\mathbb{P}$ is straightforward. Suppose that $\\omega\\textrm{-}{\\tt Con}_T$ and that $\\sigma$ is a true $\\Sigma_3$-sentence. If $\\neg\\omega\\textrm{-}{\\tt Con}_{T+\\sigma}$ then for some formula $\\varphi(x)$ we have $T\\!+\\!\\sigma\\vdash\\neg\\forall x\\,\\varphi(x)$ and $T\\!+\\!\\sigma\\vdash\\varphi(\\overline{n})$ for every $n\\!\\in\\!\\mathbb{N}$. Write $\\sigma=\\exists x\\,\\pi(x)$ for a $\\Pi_2$-formula $\\pi$. Since $\\sigma$ is true, then there exists some $u$($\\in\\!\\mathbb{N}$) such that $\\pi(u)$ is true. Write $\\pi(u)=\\forall y\\,\\theta(y)$ for some $\\Sigma_1$-formula $\\theta$. Then $\\theta(z)$ is true for every $z$. So, by the $\\Sigma_1$-completeness of $T$ we have (\\ding{92}) $T\\vdash\\theta(\\overline{n})$ for each $n\\!\\in\\!\\mathbb{N}$.\nFor reaching to a contradiction, we show that $T$ is $\\omega$-inconsistent and the formula $\\theta(x)\\wedge[\\pi(u)\\!\\rightarrow\\!\\varphi(x)]$ is a witness for that.\nBy Deduction Theorem we have $T\\vdash\\sigma\\!\\rightarrow\\!\\neg\\forall x\\,\\varphi(x)$ and so $T\\vdash\\pi(u)\\!\\rightarrow\\!\\neg\\forall x\\,[\\pi(u)\\!\\rightarrow\\!\\varphi(x)]$ therefore $T\\vdash\\neg\\forall y\\,\\theta(y)\\vee\\neg\\forall x\\,[\\pi(u)\\!\\rightarrow\\!\\varphi(x)]$, thus (i) $T\\vdash\\neg\\forall x\\,\\big(\\theta(x)\\wedge[\\pi(u)\\!\\rightarrow\\!\\varphi(x)]\\big)$.\nOn the other hand, for every $n\\!\\in\\!\\mathbb{N}$ we have $T\\vdash\\pi(u)\\!\\rightarrow\\!\\varphi(\\overline{n})$, which by (\\ding{92}) implies that (ii) $T\\vdash\\theta(\\overline{n})\\wedge[\\pi(u)\n\\!\\rightarrow\\!\\varphi(\\overline{n})]$ for each $n\\!\\in\\!\\mathbb{N}$. Thus, by (i) and (ii) the theory $T$ is $\\omega$-inconsistent, a contradiction. So, $T\\!+\\!\\sigma$ must be $\\omega$-consistent.\n\\end{proof}\n\n\nAs a corollary, we show that all the Kreiselian sentences are $\\mathbb{P}$-provably equivalent to one another.\n\n\n\\begin{corollary}[$\\mathbb{P}\\vdash\\kappa\\!\\equiv\\!\\neg\n\\omega\\textrm{-}{\\tt Con}_{T}$]\\label{cor:kk}\n\\noindent\n\n\\noindent\nIf $\\kappa$ is a Kreiselian sentence of the {\\sc re} theory $T$, then $\\mathbb{P}\\vdash\\kappa\\!\\leftrightarrow\\!\\neg\n\\omega\\textrm{-}{\\tt Con}_{T}$.\n\\end{corollary}\n\\begin{proof}\n\n\\noindent\nArgue inside $\\mathbb{P}$: If\n$\\kappa$ then\n$\\neg\\omega\\textrm{-}{\\tt Con}_{T+\\kappa}$ by Definition~\\ref{def:kappa}, which implies $\\neg\\omega\\textrm{-}{\\tt Con}_{T}$\nby Theorem~\\ref{thm:formal} (noting that $\\kappa\\!\\in\\!\\Sigma_3$);\n therefore, $\\kappa$ implies $\\neg\\omega\\textrm{-}{\\tt Con}_{T}$.\nConversely, if\n$\\neg\n\\omega\\textrm{-}{\\tt Con}_{T}$ then $\\neg\n\\omega\\textrm{-}{\\tt Con}_{T+\\kappa}$ and so $\\kappa$ by Definition~\\ref{def:kappa}; therefore, $\\neg\n\\omega\\textrm{-}{\\tt Con}_{T}$ implies $\\kappa$.\n\\end{proof}\n\n\nBy the $\\mathbb{P}$-provable equivalence of $\\kappa$ with $\\neg\\omega\\textrm{-}{\\tt Con}_{T}$, we have the following corollary which is the $\\omega$-version of {\\sc G\\\"odel}'s Second Incompleteness Theorem, that was first proved by {\\sc Rosser} \\cite{Rosser37} (see also \\cite[p.~{\\sf xxxi}]{Boolos}).\n\n\\begin{corollary}[$\\omega\\textrm{-}{\\tt Con}_T\\Longrightarrow\\omega\\textrm{-}{\\tt Con}_{T+\\neg\\omega\\textrm{-}{\\tt Con}_T}$]\\label{cor:g2}\n\\noindent\n\n\\noindent\nIf the {\\sc re} theory $T$ is $\\omega$-consistent, then so is $T\\!+\\!\\neg\\omega\\textrm{-}{\\tt Con}_T$.\n\\hfill \\ding{113}\n\\end{corollary}\n\n\n\nThus far, we have seen some $\\omega$-versions of {\\sc Lindenbaum}'s lemma and also {\\sc G\\\"odel}'s first and second incompleteness theorems. We do not claim novelty for any of these results;\\footnote{\\!See e.g.\\ the \\texttt{\\scriptsize \\url{https:\/\/t.ly\/GmquO}} link of MathOverFlow whose Proposition 1 (due to {\\sc Je\\v{r}\\'{a}bek}) is half of our Theorem~\\ref{thm:sigma3}.} nevertheless, the following $\\omega$-version of {\\sc Rosser}'s incompleteness theorem seems to be new.\n\n\\begin{theorem}[$\\omega\\textrm{-}{\\tt Con}_T\\Longrightarrow\\exists\\rho\\!\\in\\!\\Pi_3\\textrm{-}\n{\\rm Th}(\\mathbb{N})\\!\\!:\\omega\\textrm{-}{\\tt Con}_{T+\\rho}\\wedge\\omega\\textrm{-}{\\tt Con}_{T+\\neg\\rho}$]\\label{thm:rosser}\n\\noindent\n\n\\noindent\nIf $T$ is an $\\omega$-consistent {\\sc re} theory, then there exists some true $\\Pi_3$-sentence $\\rho$ such that both $T\\!+\\!\\rho$ and $T\\!+\\!\\neg\\rho$ are $\\omega$-consistent.\n\\end{theorem}\n\\begin{proof}\n\n\\noindent\nBy Diagonal Lemma there exists a $\\Pi_3$-sentence $\\rho$ such that (see Definition~\\ref{def:oic})\n\n\\qquad \\qquad \\qquad (\\ding{99}) \\quad {$\\mathbb{P}\\vdash\\rho\\leftrightarrow\n\\forall\\chi\\,[\\mho_{T+\\neg\\rho}(\\chi)\\!\\rightarrow\\!\n\\exists\\xi\\!<\\!\\chi\\,\\mho_{T+\\rho}(\\xi)]$.}\n\n\\begin{enumerate}\n \\item[{\\large \\textgoth{a}}.] We first show that $T\\!+\\!\\rho$ is $\\omega$-consistent.\n\n Assume not; then for some (fixed, standard) formula $\\boldsymbol\\varphi(x)$ the $\\Pi_2$-sentence $\\mho_{T+\\rho}(\\boldsymbol\\varphi)$ is true, so we have $\\Pi_2\\textrm{-}{\\rm Th}(\\mathbb{N})\\vdash\\mho_{T+\\rho}(\\boldsymbol\\varphi)$. Also, by Fact~\\ref{fact:1}, $T\\!+\\!\\neg\\rho$ is $\\omega$-consistent. Thus, $U=T\\!+\\!\\neg\\rho\\!+\\!\\Pi_2\\textrm{-}{\\rm Th}(\\mathbb{N})$ is consistent by Corollary~\\ref{cor:pi3}. Now, by (\\ding{99}) we have\n{$U\\vdash\n\\exists\\chi[\\mho_{T+\\neg\\rho}(\\chi)\\!\\wedge\\!\n\\forall\\xi\\!<\\!\\chi\\neg\\mho_{T+\\rho}(\\xi)]$.}\nSince $U\\vdash\\mho_{T+\\rho}(\\boldsymbol\\varphi)$ then we have $U\\vdash\n\\exists\\chi\\!\\leqslant\\!\\boldsymbol\\varphi\\,\\mho_{T+\\neg\\rho}(\\chi)$.\nBut by the $\\omega$-consistency of the theory $T+\\neg\\rho$, the $\\Sigma_2$-sentence $\\forall\\chi\\!\\leqslant\\!\\boldsymbol\\varphi\n\\neg\\mho_{T+\\neg\\rho}(\\chi)$ is true, and so should be $\\Pi_2\\textrm{-}{\\rm Th}(\\mathbb{N})$-provable. Thus, $U$ is inconsistent; a contradiction. Therefore, the theory $T\\!+\\!\\rho$ must be $\\omega$-consistent.\n\n \\item[{\\large \\textgoth{b}}.] We now show that $T\\!+\\!\\neg\\rho$ is $\\omega$-consistent.\n\n\nIf not, then by Fact~\\ref{fact:1}, $T\\!+\\!\\rho$ should be $\\omega$-consistent, and so $U=T\\!+\\!\\rho\\!+\\!\\Pi_2\\textrm{-}{\\rm Th}(\\mathbb{N})$ should be consistent by Corollary~\\ref{cor:pi3}. Also, for some formula $\\boldsymbol\\varphi(x)$ we should have $\\Pi_2\\textrm{-}{\\rm Th}(\\mathbb{N})\\vdash\\mho_{T+\\neg\\rho}(\\boldsymbol\\varphi)$.\nNow, by (\\ding{99}) we have $U\\vdash\\exists\\xi\\!<\\!\\boldsymbol\\varphi\\,\\mho_{T+\\rho}(\\xi)$.\nBut $\\forall\\xi\\!<\\!\\boldsymbol\\varphi\\,\\neg\\mho_{T+\\rho}(\\xi)$ is a true $\\Sigma_2$-sentence by the $\\omega$-consistency of the theory $T\\!+\\!\\rho$. So, $\\forall\\xi\\!<\\!\\boldsymbol\\varphi\\,\\neg\\mho_{T+\\rho}(\\xi)$ is $\\Pi_2\\textrm{-}{\\rm Th}(\\mathbb{N})$-provable, which implies that $U$ is inconsistent; a contradiction. Therefore, the theory $T\\!+\\!\\neg\\rho$ must be $\\omega$-consistent too.\n\\end{enumerate}\nSo, both $T\\!+\\!\\rho$ and $T\\!+\\!\\neg\\rho$ are $\\omega$-consistent, whence the $\\Pi_3$-sentence $\\rho$ is true (by the soundness of $\\mathbb{P}$).\n\\end{proof}\n\n\n\n\n\n\n\nNote that Theorem~\\ref{thm:rosser} is optimal in a sense, since by Theorem~\\ref{thm:sigma3} for no true $\\Sigma_3$-sentence $\\sigma$ can the theory $T\\!+\\!\\neg\\sigma$ be $\\omega$-consistent.\nWe end the paper with the observation that Theorem~\\ref{thm:rosser} can be formalized in $\\mathbb{P}$.\n\n\n\n\\begin{corollary}[$\\omega\\textrm{-}{\\tt Con}_T\\rightarrow\\rho\\not\\rightarrow\\omega\\textrm{-}{\\tt Con}_T$]\\label{cor:formalr}\n\\noindent\n\n\\noindent\nIf $\\rho$ is a $\\Pi_3$-sentence that was constructed for the {\\sc re} theory $T$ in the Proof of Theorem~\\ref{thm:rosser} (\\ding{99}), then\n\n(1) $\\mathbb{P}\\vdash\\omega\\textrm{-}{\\tt Con}_T \\longrightarrow \\rho \\wedge \\omega\\textrm{-}{\\tt Con}_{T+\\rho} \\wedge \\omega\\textrm{-}{\\tt Con}_{T+\\neg\\rho}$, and\n\n(2) if $T$ is $\\omega$-consistent, then $T\\nvdash\\rho\\rightarrow\\omega\\textrm{-}{\\tt Con}_T$; moreover, $T\\!+\\!\\rho\\!+\\!\\neg\\omega\\textrm{-}{\\tt Con}_T$ is $\\omega$-consistent.\n\\end{corollary}\n\\begin{proof}\n\n\\noindent\n For part (2) it suffices to note that since $T\\!+\\!\\rho\\!+\\!\\neg\\omega\\textrm{-}{\\tt Con}_{T+\\rho}$ is $\\omega$-consistent by Theorem~\\ref{thm:rosser} and Corollary~\\ref{cor:g2}, and $\\mathbb{P}\\vdash\\neg\\omega\\textrm{-}{\\tt Con}_{T+\\rho}\\leftrightarrow\\neg\\omega\\textrm{-}{\\tt Con}_T$ by part (1), then $T\\!+\\!\\rho\\!+\\!\\neg\\omega\\textrm{-}{\\tt Con}_T$\n is $\\omega$-consistent as well.\n\\end{proof}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\textcolor{black}{Multicarrier index keying - orthogonal frequency division multiplexing\n(MCIK-OFDM) or the so-called OFDM with index modulation (OFDM-IM)\nis an emerging multicarrier scheme \\cite{HassIM2009,Tsonev2011,basar3013},\nwhich can offer higher energy efficiency and reliability over conventional\nOFDM. In MCIK-OFDM, a subset of subcarriers are active to carry\ndata bits through both the conventional $M$-ary symbols and the indices of active subcarriers. Hence, MCIK-OFDM provides a promising\ntrade-off between spectral efficiency (SE) and reliability compared\nto OFDM just by varying the number of active sub-carriers.}\n\nRecently, various MCIK or IM concepts have been proposed for OFDM,\nwhich can be found in the overview \\cite{SurveyIM}. \\textcolor{black}{Particularly, the IM concept was first applied to OFDM-based multicarrier modulation in \\cite{HassIM2009}, and its enhanced version was proposed in \\cite{Tsonev2011}, while its generalized version which independently applies the IM to different subcarrier groups was developed in \\cite{basar3013}.} For the performance analysis,\nin \\cite{tightBound2014}, a tight bound on the bit error rate (BER)\nof OFDM-IM using the maximum likelihood (ML) detection was derived.\nThe MCIK concept was applied to multiple input multiple output (MIMO)\nsystems in \\cite{mimoIMbasar2016}. In \\cite{GeneralizedIM}, the\ngeneralized MCIK scheme with a variable number of active subcarriers\nwas proposed. In \\cite{CIbasar2015}, coordinate interleaving OFDM-IM\nwas proposed to improve the diversity order. Also inspired by the MCIK\nconcept, code index modulation (CIM) as well as its generalized version were\nstudied in \\cite{Kaddoum2015,Kaddoum2016}. Aiming to enhance the\nerror performance of MCIK-OFDM, several transmit diversity schemes\nare reported in \\cite{ThienTWC2018,codedIM2017choi,Le2020repeated,ThienTVT2018},\nin which the repetition code for either the index or $M$-ary symbol\nwas used in \\cite{ThienTWC2018,codedIM2017choi,Le2020repeated}, while\nthe spreading code was used in \\cite{ThienTVT2018}. Meanwhile, there\nare a number of studies in \\cite{inPhaseQ,dualmode,multimodeIM2017}\nthat focus on improving the SE of MCIK-OFDM, where the IM-based transmitters\nare designed to increase the number of either index or $M$-ary bits. \\textcolor{black}{Recently, deep neural networks (DNNs) have been\napplied to the MCIK signal detection in \\cite{DeepIM2019,Wang2020Conv},\nwhich can provide a near-optimal performance at low runtime complexity. Additionally, the use of a DNN structure called autoencoder for jointly optimizing both the transmitter and receiver of multicarrier systems was reported in \\cite{Luong2020engery, Luong2021mcae,Chao2022tubro, Luong2022optical}, where the resulting learning-based systems can even achieve better error performance than IM-based multicarrier systems. Finally, the IM technique was applied to visible light communications for improving the BER performance in \\cite{Khalid2021}.}\n\n\\begin{table*}[ht]\n\\centering \\caption{Contribution Comparison of MCIK-OFDM Performance Analysis}\n\\label{tab:contribution} \\linespread{1.0} %\n\\begin{tabular}{|l|c|c|c|c|c|c|}\n\\hline \nContribution & \\cite{basar3013} & \\cite{tightBound2014} & \\cite{ThienTVT2017} & \\cite{thienBERGD} & \\cite{JamesTVT} & This work\\tabularnewline\n\\hline \n\\hline \nBER analysis & \\checkmark & \\checkmark & & \\checkmark & & \\checkmark\\tabularnewline\n\\hline \nSEP analysis & & & \\checkmark & & \\checkmark & \\tabularnewline\n\\hline \nImperfect CSI & \\checkmark & & \\checkmark & \\checkmark & & \\checkmark\\tabularnewline\n\\hline \nSC-based multiple-antenna receivers & & & & & \\checkmark & \\checkmark\\tabularnewline\n\\hline \nGreedy detector & & & \\checkmark & \\checkmark & \\checkmark & \\checkmark\\tabularnewline\n\\hline \nML detector & \\checkmark & \\checkmark & \\checkmark & & & \\checkmark\\tabularnewline\n\\hline \nAsysmptotic analysis & & & \\checkmark & \\checkmark & \\checkmark & \\checkmark\\tabularnewline\n\\hline \nTheoretical guideline for detector selection & & & & & & \\checkmark\\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table*}\n\n\nMost of the aforementioned papers consider the ML or log-likelihood\nratio (LLR) detector for MCIK-OFDM, which still has a significantly\nhigher complexity than the classical OFDM. In \\cite{GDjamesPIMRC2015},\na low-complexity greedy detector (GD) was developed, which utilizes the\nenergy detection method to estimate the active indices before decoding the\n$M$-ary symbols conveyed on these active sub-carriers. The outage\nprobabilities and the pair-wise error probability of the GD under generalized\nfading were analyzed in \\cite{Pout2017} and \\cite{Lefteris2016},\nrespectively. The symbol error probability (SEP) and BER of the\nGD in the presence of channel state information (CSI) imperfection\nwere investigated in \\cite{ThienTVT2017,thienBERGD}, which reveal\nthat the GD detector is less sensitive to imperfect CSI than its ML counterpart. \\textcolor{black}{In order to further\nimprove the diversity gain of GD, MCIK-OFDM\nwith hybrid GD and diversity receptions, namely selection combining (SC) and maximal ratio combining\n(MRC), was proposed in \\cite{JamesTVT} to examine the SEP,\nhowever only for the perfect CSI case. Moreover, \\cite{JamesTVT}\nfails to provide an analytical comparison between the MRC\/SC-based\nGD and ML detectors, and its theoretical results are not tight, even at\nhigh signal-to-noise ratios (SNRs). Hence, this work is unable to provide a theoretical guideline of selecting a suitable detection method, particularly under different CSI uncertainties. Meanwhile, the GD shown in \n\\cite{ThienTVT2017,thienBERGD} is more effective in practical\nsystems with imperfect CSI. Therefore, it is worth investigating the\nperformance of MCIK-OFDM with such low-complexity MRC\/SC-based GD\nreceivers under practical CSI uncertainty, and compare with its\nML counterpart. In addition, the performance analysis of MCIK-OFDM\nusing both MRC\/SC and ML detection has been overlooked in the literature.}\n\n\n\\textcolor{black}{To address the aforementioned issues, in this paper, we first\nanalyze and compare the BERs of MCIK-OFDM with the SC-based multiple-antenna\nreceivers called MCIK-OFDM-SC, employing both ML and GD detectors, over\nuncertain CSI. In particular, the main contributions of this work compared with the existing works are listed in Table~\\ref{tab:contribution}, and are summarized as follows:}\n\\begin{itemize}\n\\item We propose a generalized framework for deriving the BERs of both the\nGD and the ML receivers for MCIK-OFDM, where the BER is represented\nas a linear combination of the\nSEP and index error probability (IEP) of the classical $M$-ary data symbols. \n\\item Utilizing this proposed framework, tight, closed-form expressions for the BERs\nof MCIK-OFDM-SC employing both the GD and ML detectors are derived in\npresence of various CSI conditions, namely perfect CSI, and fixed\nor variable CSI uncertainties. \n\\item Based on the derived expressions, asymptotic results are demonstrated\nto further investigate effects of different CSI uncertainties on the BERs\nof the two detectors. More importantly, we asymptotically develop\nconditions under which using the GD instead of the ML is desired for\nMCIK-OFDM-SC under each CSI condition, particularly when the number\nof antennas at the receiver increases. \n\\item Simulation results are provided to validate the derived expressions, as well as theoretical guidelines for selecting detection type for each CSI condition. Unlike \\cite{JamesTVT}, our theoretical results are\ntight in a wide range of SNRs.\n\\end{itemize}\nThe rest of our paper is as follows. Section~II describes\n MCIK-OFDM-SC and its signal detection under uncertain CSI.\nIn Section~III analyzes the BERs of both ML and GD, followed\nby asymptotic analysis in Section~IV. Simulation results are performed\nin Section~V. Section~VI concludes our paper.\n\n\\textit{Notation:} Lower-case bold and Upper-case bold letters and\nare used for vectors and matrices, respectively. $C\\left(,\\right)$ and $(.)^{T}$ denotes the binomial coefficient and transpose operation, respectively.\nThe floor function is represented by $\\left\\lfloor .\\right\\rfloor $. \n$\\mathcal{CN}\\left(0,\\sigma^{2}\\right)$ stands for the complex Gaussian\ndistribution with zero mean and variance $\\sigma^{2}$. $\\mathbb{E}\\left\\{ .\\right\\} $\nand $\\mathcal{M}\\left(.\\right)$ present the expectation operator\nand the moment generating function (MGF), respectively.\n\n\\section{System Model}\n\n\\subsection{MCIK-OFDM-SC}\n\nConsider a uplink single-input multi-output (SIMO) MCIK-OFDM scheme\nwith $N_{c}=NG$ sub-carriers that are divided into $G$ clusters\nwith $N$ sub-carriers per cluster. The transmitter employs a single antennas\nwhile the receiver uses $L$ antennas. At the receiver, the SC technique\nis employed to combine signals received from $L$ branches. Then,\nthe output of the SC is used to estimate transmitted data bits using either the ML or\nthe GD \\cite{JamesTVT}. The resulting scheme is called as MCIK-OFDM-SC. \\textcolor{black}{Since each cluster independently operates the MCIK-OFDM technique,\nfor simplicity and without loss of generality, hereinafter we address\nthe problem of only one cluster, whose block diagram is illustrated in Fig.~\\ref{fig:mcik-sc}. Here, the role of OFDM framework is to make sub-carriers orthogonal to each other so that we can independently apply the MCIK concept to each cluster, reducing the transceiver complexity.}\n\n\\begin{figure*}[t]\n\\begin{centering}\n\\includegraphics[width=1.6\\columnwidth]{MICK-SC.pdf}\n\\par\\end{centering}\n\\caption{\\textcolor{black}{The block diagram of MCIK-OFDM-SC.}\\label{fig:mcik-sc}}\n\\end{figure*}\n\nIn every MCIK-OFDM transmission per cluster, only $K$ out of $N$\nsub-carriers are activated to carry information bits with $K$ complex $M$-ary\nsymbols, while additional data bits are delivered by the indices of\nactive sub-carriers. More specifically, $p$ incoming bits are partitioned\ninto two streams ($p=p_{1}+p_{2}$) at the transmitter. Utilizing\ncombinatorial method or look-up table (LUT) \\cite{basar3013}, the\nfirst $p_{1}$ bits are mapped to a pattern of $K$ active sub-carriers.\nDenote by $\\theta=\\left\\{ \\alpha_{1},...,\\alpha_{K}\\right\\} $ the\nset of $K$ active sub-carrier indices, where $\\alpha_{k}\\in\\left\\{ 1,...,N\\right\\} $\nfor $k=1,...,K$. Note that $\\theta$ can be referred to as an index\nsymbol, which is identified by $p_{1}$ index bits. The remaining\n$p_{2}$ bits are mapped to $K$ $M$-ary symbols. For given\n$N,K$ and $M$, the number of index bits and symbol bits are given\nby $p_{1}=\\left\\lfloor \\log_{2}C\\left(N,K\\right)\\right\\rfloor $ and\n$p_{2}=K\\log_{2}M$, respectively. Denote by $\\mathcal{S}$ the $M$-ary\nconstellation. Using $\\theta$ and $K$ non-zero symbols (determined\nby $p$ incoming bits), the transmitted signal for each cluster is\ngiven as $\\mathbf{x}=\\left[x\\left(1\\right),...,x\\left(N\\right)\\right]^{T},$\nwhere $x\\left(\\alpha\\right)=0$ for $\\alpha\\notin\\theta$ and $x\\left(\\alpha\\right)\\in\\mathcal{S}$\nfor $\\alpha\\in\\theta$. \\textcolor{black}{Here, note that the $K$ non-zero data symbols conveyed on active sub-carriers are denoted as vector $\\mathbf{s}$ in Fig.~\\ref{fig:mcik-sc}. The frequency domain signal $\\mathbf{x}$ is then processed by the inverse fast Fourier transform (IFFT) before being transmitted to the receiver.}\n\n\\textcolor{black}{The received signal at the $l$-th antenna in the frequency domain, i.e., the signal obtained after the FFT,\nis given by} \n\\begin{equation}\n\\mathbf{y}_{l}=\\mathbf{H}_{l}\\mathbf{x}+\\mathbf{n}_{l},\n\\end{equation}\nwhere $\\mathbf{H}_{l}=\\text{diag}\\left\\{ h_{l}\\left(1\\right),...,h_{l}\\left(N\\right)\\right\\} $\nis the channel matrix between the transmitter and the $l$-th receiver\nantenna, while $\\mathbf{n}_{l}=\\left[n_{l}\\left(1\\right),...,n_{l}\\left(N\\right)\\right]^{T}$\nis the noise vector with $n_{l}\\left(\\alpha\\right)\\sim\\mathcal{CN}\\left(0,N_{0}\\right)$,\nfor $\\alpha=1,...,N$ and $l=1,...,L$. \\textcolor{black}{Particularly, $h_{l}\\left(\\alpha\\right)$ represents the Rayleigh fading\nchannel, which is identical and independent to each other, where $h_{l}\\left(\\alpha\\right)\\sim\\mathcal{CN}\\left(0,1\\right)$. Here, we assume that the cyclic prefix inserted to each OFDM symbol in the time domain is large enough to completely combat the inter-symbol interference \\cite{basar3013}.} As such, the average SNR per active\nsub-carrier is given by $\\bar{\\gamma}=\\varphi E_{s}\/N_{0},$ where\n$E_{s}$ denotes the average power per non-zero $M$-ary symbol and\n$\\varphi=N\/K$ is the power allocation ratio.\n\n\n\\subsection{Post-Combining Detection under CSI Uncertainty}\n\nWe consider a practical MCIK-OFDM-SC system where the receiver imperfectly\nknows the CSI. Particularly, denote by $\\hat{h}_{l}\\left(\\alpha\\right)$\nthe estimate of the true channel $h_{l}\\left(\\alpha\\right)$, and\nwe have \n\\begin{equation}\n\\hat{h}_{l}\\left(\\alpha\\right)=h_{l}\\left(\\alpha\\right)-e_{l}\\left(\\alpha\\right),\\label{eq:h_hat}\n\\end{equation}\nwhere $e_{l}\\left(\\alpha\\right)$ represents the channel estimation\nerror as being independent of $\\hat{h}_{l}\\left(\\alpha\\right)$ and\n$e_{l}\\left(\\alpha\\right)\\sim\\mathcal{CN}\\left(0,\\epsilon^{2}\\right)$,\nand $\\hat{h}_{l}\\left(\\alpha\\right)\\sim\\mathcal{CN}\\left(0,1-\\epsilon^{2}\\right),$\nwhere $\\epsilon^{2}\\in\\left[0,1\\right)$ denotes the error variance.\n\nFor sub-carrier $\\alpha$, the $l^{*}$-th branch is selected as the\noutput of the SC such that $l^{*}=\\arg\\max_{l}\\left|\\hat{h}_{l}\\left(\\alpha\\right)\\right|^{2}$.\nHence, the output signal of the SC can be given by \n\\begin{equation}\n\\mathbf{y}=\\mathbf{H}\\mathbf{x}+\\mathbf{n},\\label{eq:y_SC}\n\\end{equation}\nwhere $\\mathbf{H}=\\text{diag}\\left\\{ h\\left(1\\right),...,h\\left(N\\right)\\right\\} $\ndenotes the channel matrix of the SC and the corresponding noise vector\nis $\\mathbf{n}=\\left[n\\left(1\\right),...,n\\left(N\\right)\\right]^{T}$,\nwhere $h\\left(\\alpha\\right)=h_{l^{*}}\\left(\\alpha\\right)$ and $n\\left(\\alpha\\right)=n_{l^{*}}\\left(\\alpha\\right)$\nfor $\\alpha=1,...,N$. Notice in \\eqref{eq:y_SC} that $\\mathbf{y}=\\left[y\\left(1\\right),...,y\\left(N\\right)\\right]^{T},$\nwith $y\\left(\\alpha\\right)=h\\left(\\alpha\\right)x\\left(\\alpha\\right)+n\\left(\\alpha\\right)$.\n\nLet $\\hat{h}\\left(\\alpha\\right)$ be the estimate of $h\\left(\\alpha\\right)$,\ni.e., $\\hat{h}\\left(\\alpha\\right)=\\hat{h}_{l^{*}}\\left(\\alpha\\right)$.\nBased on $\\mathbf{y}$ and $\\hat{h}\\left(\\alpha\\right),$ either the\nML or the GD can be employed for the signal detection as follows.\n\n\\subsubsection{Post-Combining ML }\n\nUnder imperfect CSI, the estimated signal $\\hat{\\mathbf{x}}$ is calculated\nby the ML criterion as \n\\[\n\\hat{\\mathbf{x}}=\\arg\\min_{\\mathbf{x}}\\left\\Vert \\mathbf{y}-\\mathbf{\\hat{H}x}\\right\\Vert ^{2},\n\\]\nwhere $\\hat{\\mathbf{H}}=\\text{diag}\\left\\{ \\hat{h}\\left(1\\right),...,\\hat{h}\\left(N\\right)\\right\\} $\ndenotes the estimate of the channel matrix after the SC. Utilizing\n$\\hat{\\mathbf{x}}$, the index symbol $\\hat{\\theta}$ and $K$ non-zero\nsymbols $x\\left(\\alpha\\right)$ with $\\alpha\\in\\hat{\\theta}$ are\nrecovered.\n\n\\subsubsection{Post-Combining GD }\n\nPost-combining GD makes best antenna selections per sub-carrier before\nGD processing. For given $\\mathbf{H}$, the GD detects signals through\ntwo following steps. Firstly, the active indices are estimated by\n$K$ sub-carriers that have the largest SC-output energies, i.e.,\n$\\left|y\\left(\\alpha\\right)\\right|^{2}.$ Secondly, the non-zero $M$-ary\nsymbols are detected by applying the ML decision to activated sub-carrier\n$\\alpha$ as\n\n\\begin{equation}\nx\\left(\\alpha\\right)=\\arg\\min_{x\\left(\\alpha\\right)\\in\\mathcal{S}}\\left|y\\left(\\alpha\\right)-\\hat{h}\\left(\\alpha\\right)x\\left(\\alpha\\right)\\right|^{2}.\\label{eq:step2_GD}\n\\end{equation}\n\nNote that the GD detector has not only lower complexity, but also less sensitivity\nto CSI imperfection, than the ML detector \\cite{thienBERGD}. However, when\nthe number of antennas is limited to one, the ML still\nperform much better than GD under certain CSI conditions, especially when $M$\nis small (e.g., $M=2,4$) \\cite{ThienTVT2017}.\n\nAs a result, we are prompted to examine the BER performance of both\nthe GD and the ML in MCIK-OFDM-SC, in order to understand if the post-combining\nGD receiver benefits from diversity gain. For this, we intend to derive\nthe closed-form expressions for the BERs of the two detectors, taking CSI uncertainty into consideration in the next section.\n\n\\section{BER Analysis With CSI Uncertainty}\n\nWe note that the ML performs the same performance as the log-likelihood\nratio (LLR) detector \\cite{codedIM2017choi} which also has two separate\nsteps as the GD. Thus, we now introduce a generalized framework to\nderive the BERs of both the ML and the GD. Particularly, we consider\nbit error event consisting of two parts: the index bit error ($p_{1}$\nbits) and the symbol bit error ($p_{2}$ bits). Let $P_{1}$ be the\nindex BER (IBER) and $P_{2}$ be the symbol BER (SBER). Then, the\nBER of either the ML or the GD is given by \n\\begin{equation}\nP_{b}=\\frac{p_{1}P_{1}+p_{2}P_{2}}{p_{1}+p_{2}}.\\label{eq:Pb_start}\n\\end{equation}\nThe IBER and the SBER are obtained by \\cite{thienBERGD} \n\\begin{equation}\nP_{1}\\approx\\eta\\overline{P}_{I}\/2,\\label{eq:P1}\n\\end{equation}\n\\begin{equation}\nP_{2}\\le\\frac{\\overline{P}_{I}}{2K}+\\frac{\\overline{P}_{M}}{\\log_{2}M},\\label{eq:P2}\n\\end{equation}\nwhere $\\overline{P}_{I}$ denotes the average index error probability\n(IEP), $\\eta=1$ for $N>2$ and $\\eta=2$ for $N=2$, and $\\overline{P}_{M}$\nis the average SEP of the $M$-ary symbol detection as long as\nthe activated indices are correctly detected. Plugging \\eqref{eq:P1}\nand \\eqref{eq:P2} into \\eqref{eq:Pb_start}, the generalized BER\nexpression for both the ML and the GD is given by\n\\begin{equation}\nP_{b}\\approx\\frac{\\left(\\eta p_{1}+m\\right)\\overline{P}_{I}\/2+K\\overline{P}_{M}}{p},\\label{eq:Pb_general}\n\\end{equation}\nwhere $m=\\log_{2}M$ and $p=p_{1}+p_{2}$.\n\n\\textit{Remark 1:} As seen from \\eqref{eq:Pb_general}, when $K$\nincreases to $N$, the BER of either the ML or the GD approaches that\nof classical OFDM, which is $\\overline{P}_{M}\/m$. As a result, the\nperformance gap between these two detectors gets smaller when $K$\ngets larger.\n\n\\textit{Remark 2:} $\\overline{P}_{M}$ in \\eqref{eq:Pb_general} is\nthe same for both the ML and the GD, while $\\overline{P}_{I}$ depends\non the detection type employed. Thus, to find out the BER expressions for\nthe GD and the ML in MCIK-OFDM-SC, we need to derive $\\overline{P}_{I}$\nfor them, considering CSI uncertainty. Meanwhile, $\\overline{P}_{M}$\nis provided in the following lemma when the $M$-ary PSK modulation is\nemployed.\n\\begin{lem}\nUnder CSI uncertainty with the error variance $\\epsilon^{2}$, the\naverage SEP of the conventional $M$-ary PSK symbol detection in MCIK-OFDM-SC\nis approximated by \n\\begin{equation}\n\\overline{P}_{M}\\approx\\frac{\\xi}{12}\\left\\{ \\frac{L!}{\\prod_{l=1}^{L}\\left[l+\\frac{\\left(1-\\epsilon^{2}\\right)\\rho\\bar{\\gamma}}{1+\\epsilon^{2}\\bar{\\gamma}}\\right]}+\\frac{3L!}{\\prod_{l=1}^{L}\\left[l+\\frac{4\\left(1-\\epsilon^{2}\\right)\\rho\\bar{\\gamma}}{3\\left(1+\\epsilon^{2}\\bar{\\gamma}\\right)}\\right]}\\right\\} ,\\label{eq:PM}\n\\end{equation}\nwhere $\\rho=\\sin^{2}\\left(\\pi\/M\\right)$, $\\xi=1$ for $M=2$ and\n$\\xi=2$ for $M>2$. \n\\end{lem}\n\\begin{IEEEproof}\nSee Appendix A. \n\\end{IEEEproof}\n\n\\subsection{BER for ML with SC Reception and CSI Uncertainty}\n\nWe first consider the IEP of the ML in MCIK-OFDM with the SC and imperfect\nCSI. Denote by $P_{I_{1}}$ the instantaneous IEP of the ML, which\nis approximated by \\cite{ThienTVT2017} \n\\begin{equation}\nP_{I_{1}}\\approx\\frac{K}{N}\\sum_{\\alpha=1}^{N}\\sum_{\\tilde{\\alpha}\\ne\\alpha=1}^{N-K}\\left[\\frac{1}{12}e^{-\\frac{\\bar{\\gamma}\\left(\\hat{\\nu}_{\\alpha}+\\hat{\\nu}_{\\tilde{\\alpha}}\\right)}{4+2\\bar{\\gamma}\\epsilon^{2}}}+\\frac{1}{4}e^{-\\frac{2\\bar{\\gamma}\\left(\\hat{\\nu}_{\\alpha}+\\hat{\\nu}_{\\tilde{\\alpha}}\\right)}{6+3\\bar{\\gamma}\\epsilon^{2}}}\\right],\\label{eq:PI_1}\n\\end{equation}\nwhere $\\hat{\\nu}_{\\alpha}=\\left|\\hat{h}\\left(\\alpha\\right)\\right|^{2}$,\n$\\hat{\\nu}_{\\tilde{\\alpha}}=\\left|\\hat{h}\\left(\\tilde{\\alpha}\\right)\\right|^{2}$.\n\nDenote $\\hat{\\nu}_{\\varSigma}=\\hat{\\nu}_{\\alpha}+\\hat{\\nu}_{\\tilde{\\alpha}}$.\nThe moment generating function (MGF) of $\\hat{\\nu}_{\\varSigma}$ can\nbe attained by $\\mathcal{M}_{\\hat{\\nu}_{\\varSigma}}\\left(s\\right)=\\mathcal{M}_{\\hat{\\nu}}^{2}\\left(s\\right),$\nwhere $\\mathcal{M}_{\\hat{\\nu}}\\left(s\\right)$ is the MGF of $\\hat{\\nu}_{\\alpha}$\nwhich is given in \\eqref{eq:MGF_v_hat}. Here, applying the MGF approach\nto \\eqref{eq:PI_1}, we obtain the average IEP of the ML with the\nSC and uncertain CSI as follows \n\\begin{equation}\n\\overline{P}_{I_{1}}\\approx\\frac{\\Psi_{1}}{12}\\left\\{ \\frac{\\left(L!\\right)^{2}}{\\prod_{l=1}^{L}\\left[l+\\frac{\\left(1-\\epsilon^{2}\\right)\\bar{\\gamma}}{4+2\\bar{\\gamma}\\epsilon^{2}}\\right]^{2}}+\\frac{3\\left(L!\\right)^{2}}{\\prod_{l=1}^{L}\\left[l+\\frac{2\\left(1-\\epsilon^{2}\\right)\\bar{\\gamma}}{6+3\\bar{\\gamma}\\epsilon^{2}}\\right]^{2}}\\right\\} ,\\label{eq:PI_1_ave}\n\\end{equation}\nwhere $\\Psi_{1}=K\\left(N-K\\right)$.\n\nAs observed from \\eqref{eq:PI_1_ave}, note that as $L=1$, the\naverage IEP of the ML in \\eqref{eq:PI_1_ave} reduces to \\cite[Eq. (16)]{ThienTVT2017}.\nIn addition, $\\overline{P}_{I_{1}}$ mainly relies on $N,K$ and $L$,\nwhile being less influenced by the modulation size $M$.\n\nFinally, the BER of the ML (denoted by $P_{b_{1}}$) can be obtained\nby inserting \\eqref{eq:PM} and \\eqref{eq:PI_1_ave} to \\eqref{eq:Pb_general}\nas \n\\begin{align}\nP_{b_{1}} & \\approx\\frac{\\widetilde{\\Psi}_{1}}{24p}\\left\\{ \\frac{\\left(L!\\right)^{2}}{\\prod_{l=1}^{L}\\left[l+\\frac{\\left(1-\\epsilon^{2}\\right)\\bar{\\gamma}}{4+2\\bar{\\gamma}\\epsilon^{2}}\\right]^{2}}+\\frac{3\\left(L!\\right)^{2}}{\\prod_{l=1}^{L}\\left[l+\\frac{2\\left(1-\\epsilon^{2}\\right)\\bar{\\gamma}}{6+3\\bar{\\gamma}\\epsilon^{2}}\\right]^{2}}\\right\\} \\nonumber \\\\\n & +\\frac{K\\xi}{12p}\\left\\{ \\frac{L!}{\\prod_{l=1}^{L}\\left[l+\\frac{\\left(1-\\epsilon^{2}\\right)\\rho\\bar{\\gamma}}{1+\\epsilon^{2}\\bar{\\gamma}}\\right]}+\\frac{3L!}{\\prod_{l=1}^{L}\\left[l+\\frac{4\\left(1-\\epsilon^{2}\\right)\\rho\\bar{\\gamma}}{3\\left(1+\\epsilon^{2}\\bar{\\gamma}\\right)}\\right]}\\right\\} ,\\label{eq:Pb_1}\n\\end{align}\nwhere $\\widetilde{\\Psi}_{1}=\\Psi_{1}\\left(\\eta p_{1}+m\\right)=K\\left(N-K\\right)\\left(\\eta p_{1}+m\\right).$\n\nIt is shown from \\eqref{eq:Pb_1} that increasing $L$ improves the\nBER of the ML. Moreover, for given $N,$ $L$ and $\\bar{\\gamma}$,\nthe BER $P_{b_{1}}$ depends on both $K$ and $\\epsilon^{2}$. For\nexample, when $K$ gets larger, the second term, which is related to the $M$-ary\nsymbol detection, will dominate over $P_{b_{1}}$. Especially, as $K=N$,\n\\eqref{eq:Pb_1} reduces to the BER of the classical OFDM.\n\n\\subsection{BER for GD with SC Reception and CSI Uncertainty}\n\nIn MCIK-OFDM with the single antenna used at both the transmitter\nand the receiver, the IEP of the GD is independent of CSI conditions\n\\cite{thienBERGD}. However, this is no longer true when employing\nthe SC for MCIK-OFDM. Particularly, the instantaneous IEP of the GD\nis given by \\cite{thienBERGD,JamesTVT} \n\\begin{equation}\nP_{I_{2}}=\\frac{K}{N}\\sum_{\\alpha=1}^{N}\\sum_{i=1}^{N-K}\\frac{\\left(-1\\right)^{i+1}C\\left(N-K,i\\right)}{i+1}e^{-\\frac{i\\bar{\\gamma}\\nu_{\\alpha}}{i+1}},\\label{eq:PI_start}\n\\end{equation}\nwhere $\\nu_{\\alpha}=\\left|h\\left(\\alpha\\right)\\right|^{2}$ which\nis obviously affected by the estimate $\\hat{h}_{l}\\left(\\alpha\\right)$\ndue to $h\\left(\\alpha\\right)=h_{l^{*}}\\left(\\alpha\\right)$ with $l^{*}=\\max_{l}\\left|\\hat{h}_{l}\\left(\\alpha\\right)\\right|^{2}$. \\textcolor{black}{The detailed derivation of \\eqref{eq:PI_start} over Rayleigh fading channels was presented in \\cite{GDjamesPIMRC2015}, which is not included here for the sake of brevity.} \nThus, the IEP of the GD in our system depends on the channel estimation\nerrors. This makes the derivation of the average IEP for this detector\nnon-trivial as follows.\n\nFirst, it is needed to figure out the MGF of $\\nu_{\\alpha}$. Using\n\\eqref{eq:h_hat}, $h\\left(\\alpha\\right)$ can be represented as $h\\left(\\alpha\\right)=e^{j\\phi}\\left|\\hat{h}\\left(\\alpha\\right)\\right|+e\\left(\\alpha\\right)=e^{j\\phi}\\left(\\left|\\hat{h}\\left(\\alpha\\right)\\right|+\\tilde{e}\\left(\\alpha\\right)\\right)$,\nwhere $\\tilde{e}\\left(\\alpha\\right)=e^{-j\\phi}e\\left(\\alpha\\right)\\sim\\mathcal{CN}\\left(0,\\epsilon^{2}\\right)$\nand $\\phi$ denotes the argument of $\\hat{h}\\left(\\alpha\\right)$.\nThis results in \n\\begin{equation}\n\\left|h\\left(\\alpha\\right)\\right|^{2}=\\left|\\left|\\hat{h}\\left(\\alpha\\right)\\right|+\\tilde{e}\\left(\\alpha\\right)\\right|^{2}.\\label{eq:v_alp}\n\\end{equation}\nFrom \\eqref{eq:v_alp}, the MGF of $\\nu_{\\alpha}$ can be computed\nas \n\\begin{align}\n\\mathcal{M}_{\\nu}\\left(t\\right) & =\\mathbb{E}_{\\left|h\\left(\\alpha\\right)\\right|^{2}}\\left\\{ e^{\\left|h\\left(\\alpha\\right)\\right|^{2}t}\\right\\} \\nonumber \\\\\n & =\\mathbb{E}_{\\left|\\hat{h}\\left(\\alpha\\right)\\right|^{2}}\\left\\{ \\mathbb{E}_{\\left|\\left|\\hat{h}\\left(\\alpha\\right)\\right|+\\tilde{e}\\left(\\alpha\\right)\\right|^{2}}\\left\\{ e^{\\left|\\left|\\hat{h}\\left(\\alpha\\right)\\right|+\\tilde{e}\\left(\\alpha\\right)\\right|^{2}t}\\right\\} \\right\\} \\nonumber \\\\\n & =\\int_{0}^{\\infty}f_{\\left|\\hat{h}\\left(\\alpha\\right)\\right|^{2}}\\left(x\\right)\\mathcal{M}_{\\left|\\left|\\hat{h}\\left(\\alpha\\right)\\right|+\\tilde{e}\\left(\\alpha\\right)\\right|^{2}}\\left(t\\right)dx,\\label{eq:MGF_v_start}\n\\end{align}\nwhich motivates us to propose the following lemma. \n\\begin{lem}\nLet $\\tilde{e}\\left(\\alpha\\right)\\sim\\mathcal{CN}\\left(0,\\epsilon^{2}\\right),$\nthen for given $\\left|\\hat{h}\\left(\\alpha\\right)\\right|^{2},$ the\nMGF of $\\left|\\left|\\hat{h}\\left(\\alpha\\right)\\right|+\\tilde{e}\\left(\\alpha\\right)\\right|^{2}$\nis given by \n\\begin{equation}\n\\mathcal{M}_{\\left|\\left|\\hat{h}\\left(\\alpha\\right)\\right|+\\tilde{e}\\left(\\alpha\\right)\\right|^{2}}\\left(t\\right)=\\frac{e^{\\frac{\\left|\\hat{h}\\left(\\alpha\\right)\\right|^{2}t}{1-\\epsilon^{2}t}}}{1-\\epsilon^{2}t}.\\label{eq:MGF_non_central}\n\\end{equation}\n\\end{lem}\n\\begin{IEEEproof}\nSee Appendix B. \n\\end{IEEEproof}\nInserting \\eqref{eq:PDF_v_hat} and \\eqref{eq:MGF_non_central} into\n\\eqref{eq:MGF_v_start}, through simple manipulations, we obtain \n\\begin{equation}\n\\mathcal{M}_{\\nu}\\left(t\\right)=\\frac{L!}{\\left(1-\\epsilon^{2}t\\right)\\prod_{l=1}^{L}\\left[l-\\frac{\\left(1-\\epsilon^{2}\\right)t}{1-\\epsilon^{2}t}\\right]}.\\label{eq:MGF_v_final}\n\\end{equation}\n\nNote that to the best of our knowledge, the approach to derive the\nMGF of $\\nu_{\\alpha}$ in closed-form \\eqref{eq:MGF_v_final} is novel.\nThis interestingly results in a simple, exact closed-form expression\nfor the average IEP of the GD with the SC and uncertain CSI, by applying\nthe MGF approach to \\eqref{eq:PI_start} and using \\eqref{eq:MGF_v_final},\nas \n\\begin{equation}\n\\overline{P}_{I_{1}}=K\\sum_{i=1}^{N-K}\\frac{\\left(-1\\right)^{i+1}C\\left(N-K,i\\right)L!}{\\left(i+1+i\\epsilon^{2}\\bar{\\gamma}\\right)\\prod_{l=1}^{L}\\left[l+\\frac{\\left(1-\\epsilon^{2}\\right)i\\bar{\\gamma}}{i+1+i\\epsilon^{2}\\bar{\\gamma}}\\right]}.\\label{eq:PI_ave}\n\\end{equation}\n\nAs observed from \\eqref{eq:PI_ave}, when $L=1$, the expression for\n$\\overline{P}_{I}$ becomes \\cite[Eq. (8)]{thienBERGD} which no longer\ndepends on $\\epsilon^{2}.$ In addition, as $L>1$, the IEP performance\nsuffers from a degradation caused by CSI uncertainty, i.e., $\\epsilon^{2}.$\nNote that for any $\\epsilon^{2}\\in\\left[0,1\\right)$, $\\overline{P}_{I_{1}}$\nalways tends to 0 as $\\bar{\\gamma}$ increases to infinity, even for\nthe worst case of $\\epsilon^{2}=1$.\n\nFinally, the BER of the GD with the SC and uncertain CSI can be attained \nby substituting \\eqref{eq:PM} and \\eqref{eq:PI_ave} to \\eqref{eq:Pb_general}\nas follows:\n\\begin{align}\nP_{b_{2}} & \\approx\\frac{K\\left(\\eta p_{1}+m\\right)}{2p}\\sum_{i=1}^{N-K}\\frac{\\left(-1\\right)^{i+1}C\\left(N-K,i\\right)L!}{\\left(i+1+i\\epsilon^{2}\\bar{\\gamma}\\right)\\prod_{l=1}^{L}\\left[l+\\frac{\\left(1-\\epsilon^{2}\\right)i\\bar{\\gamma}}{i+1+i\\epsilon^{2}\\bar{\\gamma}}\\right]}\\nonumber \\\\\n & +\\frac{K\\xi}{12p}\\left\\{ \\frac{L!}{\\prod_{l=1}^{L}\\left[l+\\frac{\\left(1-\\epsilon^{2}\\right)\\rho\\bar{\\gamma}}{1+\\epsilon^{2}\\bar{\\gamma}}\\right]}+\\frac{3L!}{\\prod_{l=1}^{L}\\left[l+\\frac{4\\left(1-\\epsilon^{2}\\right)\\rho\\bar{\\gamma}}{3\\left(1+\\epsilon^{2}\\bar{\\gamma}\\right)}\\right]}\\right\\} .\\label{eq:Pb_2}\n\\end{align}\n\nObserve from \\eqref{eq:Pb_2} that different from MCIK-OFDM with the\nsingle antenna \\cite{thienBERGD}, where $\\epsilon^{2}$ affects only\nthe term related to the $M$-ary symbol detection, in MCIK-OFDM-SC\nhaving multiple anttenas, $\\epsilon^{2}$ influences on both the index\ndetection error and the $M$-ary symbol detection error. As $L=1$,\n\\eqref{eq:Pb_2} reduces to \\cite[Eq. (15)]{thienBERGD}, which confirms\nthe accuracy of our derivation for the BER expression of MCIK-OFDM-SC.\n\n\\section{Asymptotic Analysis}\n\nWe now carry out the asymptotic analysis for the BERs of both ML\nand GD detectors at high SNRs and in a large number of antennas. In particular,\nwe investigate the impact of various CSI uncertainties, namely perfect CSI,\nfixed CSI uncertainty, and minimum mean square (MMSE) based variable CSI uncertainty. In addition,\nthe performance comparison between the two detectors is provided.\nThis allows to recommend that when the GD should be used under each\nCSI condition as the number of antennas increases.\n\nNote that existing studies \\cite{JamesTVT,ThienTVT2017,thienBERGD}\nhave not provided any analytical comparisons between the ML and the\nGD such as the behavior of the coding gain gap between them when the\nnumber of antennas changes. Moreover, \\cite{JamesTVT} even has not\nincluded any asymptotic analysis for the GD with the SC. \n\n\\subsection{Perfect CSI $(\\epsilon^{2}=0)$}\n\nAs $\\epsilon^{2}=0$ and $\\bar{\\gamma}$ tends to infinity, the BERs\nin \\eqref{eq:Pb_1} and \\eqref{eq:Pb_2} can be asymptotically approximated\nby \n\\begin{equation}\nP_{b_{1}}\\approx\\Upsilon\\left(\\frac{\\xi\\Omega}{6\\rho^{L}}\\right)\\frac{1}{\\gamma_{0}^{L}},\\label{eq:Pb_1_perfect}\n\\end{equation}\n\\begin{equation}\nP_{b_{2}}\\approx\\Upsilon\\left[\\left(\\eta p_{1}+m\\right)\\omega+\\frac{\\xi\\Omega}{6\\rho^{L}}\\right]\\frac{1}{\\gamma_{0}^{L}},\\label{eq:Pb_2_perfect}\n\\end{equation}\nwhere $\\Upsilon=K^{L+1}L!\/2pN^{L}$, $\\Omega=1+3^{L+1}\/4^{L}$, $\\omega=\\sum_{i=1}^{N-K}\\left(-1\\right)^{i+1}C\\left(N-K,i\\right)\\left(1+i\\right)^{L-1}\/i^{L}$,\nand $\\gamma_{0}=E_{s}\/N_{0}$ is the average SNR per sub-carrier.\n\nAs observed from \\eqref{eq:Pb_1_perfect} and \\eqref{eq:Pb_2_perfect},\nboth the ML and the GD attain a diversity order of $L$ under perfect\nCSI. Moreover, for given $N$ and $L$, a smaller $K$ provides lower\nBERs.\n\nRegarding the comparison between the GD and the ML, we consider the\ncoding gain attained by the ML over the GD under perfect CSI (denoted\nby $\\Delta_{1}$), which can be denoted by $\\Delta_{1}=10\\log_{10}\\left(P_{b_{2}}\/P_{b_{1}}\\right)^{1\/L}$.\nUsing \\eqref{eq:Pb_1_perfect} and \\eqref{eq:Pb_2_perfect}, we have\n\\begin{equation}\n\\Delta_{1}=\\frac{10}{L}\\log_{10}\\left(1+\\eta_{1}\\right)\\text{\\,(dB)},\\label{eq:Delta_1}\n\\end{equation}\nwhere $\\eta_{1}=6\\left(\\eta p_{1}+m\\right)\\omega\\rho^{L}\/\\xi\\Omega$.\nBased on this result, we introduce the following theorem.\n\n\\textbf{Theorem 1.} \\textit{Consider MCIK-OFDM with the SC and perfect\nCSI. For $M=2$, the ML performs better than the GD in terms of the BER by\n3 dB, at large $L$, i.e, $\\lim_{L\\rightarrow\\infty}\\Delta_{1}\\approx3$\n(dB). For $M\\ge4$, the BER of GD approaches to that of ML\nwhen increasing $L$, i.e., $\\lim_{L\\rightarrow\\infty}\\Delta_{1}=0$\n(dB). Especially, when $M\\ge8$, the BERs of the two detectors rapidly\nconverge to each other as $L$ increases, i.e., $\\lim_{L\\rightarrow\\infty}\\eta_{1}=0$.} \n\\begin{IEEEproof}\nSince $\\omega$ in \\eqref{eq:Pb_2_perfect} can be approximated by\n$\\omega\\approx\\left(N-K\\right)2^{L-1}$ at large $L$, we approximate\n$\\eta_{1}$ at large $L$ as \n\\begin{equation}\n\\eta_{1}\\approx\\beta_{1}\\lambda_{1}^{L},\\label{eq:eta_1}\n\\end{equation}\nwhere $\\lambda_{1}=2\\rho,$ recalling $\\rho=\\sin^{2}\\left(\\pi\/M\\right),$\nand $\\beta_{1}=3\\left(\\eta p_{1}+m\\right)\\left(N-K\\right)\/\\xi\\Omega$\nwhich decreases when increasing $L$ due to $\\Omega=1+3^{L+1}\/4^{L}.$\n\nFor $M=2$, we obtain $\\lambda_{1}=2$, thus $\\eta_{1}\\approx\\beta_{1}2^{L}$.\nUsing \\eqref{eq:Delta_1}, $\\lim_{L\\rightarrow\\infty}\\Delta_{1}=\\lim_{L\\rightarrow\\infty}\\left(10\/L\\right)\\log_{10}\\left(1+\\beta_{1}2^{L}\\right)=10\\log_{10}2\\approx3$\n(dB).\n\nFor $M\\ge4,$ we obtain $\\lambda_{1}\\le1$, thus $1<\\eta_{1}\\le1+\\beta_{1}$.\nThis leads to $\\lim_{L\\rightarrow\\infty}\\Delta_{1}=0$ (dB).\n\nFor $M\\ge8,$ we attain $\\lambda_{1}\\le2\\sin^{2}\\left(\\pi\/8\\right)<0.3$,\nwhich results in $\\lim_{L\\rightarrow\\infty}\\eta_{1}=\\lim_{L\\rightarrow\\infty}\\beta_{1}\\lambda_{1}^{L}=0$. \n\\end{IEEEproof}\n\\textit{Remark 3.} From Theorem~1, it is recommended that the GD\nshould be used rather than the ML under perfect CSI as $M\\ge8$, especially\nwhen $L$ gets larger. This is because the GD can achieve a nearly\noptimal BER at a significantly lower complexity than the ML detector for large $M$\nand $L$. Note that the complexities of the ML and GD in MCIK-OFDM\nwith the SC are $\\mathcal{C}_{ML-SC}=N+2CM^{K}$ and $\\mathcal{C}_{GD-SC}=2N+2KM,$\nrespectively, where $C=2^{p_{1}}$ \\cite{JamesTVT}. Obviously, when\n$K$ and $M$ become larger, we attain $\\mathcal{C}_{ML-SC}\\gg\\mathcal{C}_{GD-SC}$.\n\n\\subsection{Fixed CSI Uncertainty $(\\epsilon^{2}>0)$}\n\nAs $\\epsilon^{2}>0$ is fixed, the BERs in \\eqref{eq:PI_ave} and\n\\eqref{eq:Pb_2} can be rewritten at high SNRs, respectively, as follows:\n\\begin{align}\nP_{b_{1}} & \\approx\\underbrace{\\frac{\\widetilde{\\Psi}_{1}}{24p}\\left\\{ \\frac{1}{\\prod_{l=1}^{L}\\left[1+\\frac{\\left(1-\\epsilon^{2}\\right)}{2l\\epsilon^{2}}\\right]^{2}}+\\frac{3}{\\prod_{l=1}^{L}\\left[1+\\frac{2\\left(1-\\epsilon^{2}\\right)}{3l\\epsilon^{2}}\\right]^{2}}\\right\\} }_{A_{1}}\\nonumber \\\\\n & +\\frac{K\\xi}{12p}\\left\\{ \\frac{1}{\\prod_{l=1}^{L}\\left[1+\\frac{\\left(1-\\epsilon^{2}\\right)\\rho}{l\\epsilon^{2}}\\right]}+\\frac{3}{\\prod_{l=1}^{L}\\left[1+\\frac{4\\left(1-\\epsilon^{2}\\right)\\rho}{3l\\epsilon^{2}}\\right]}\\right\\} ,\\label{eq:Pb_1_fixed}\n\\end{align}\n\\begin{equation}\nP_{b_{2}}\\approx\\frac{K\\xi}{12p}\\left\\{ \\frac{1}{\\prod_{l=1}^{L}\\left[1+\\frac{\\left(1-\\epsilon^{2}\\right)\\rho}{l\\epsilon^{2}}\\right]}+\\frac{3}{\\prod_{l=1}^{L}\\left[1+\\frac{4\\left(1-\\epsilon^{2}\\right)\\rho}{3l\\epsilon^{2}}\\right]}\\right\\} ,\\label{eq:Pb_2_fixed}\n\\end{equation}\nwhere we recall that $\\widetilde{\\Psi}_{1}=K\\left(N-K\\right)\\left(\\eta p_{1}+m\\right).$\n\nAs seen from \\eqref{eq:Pb_1_fixed} and \\eqref{eq:Pb_2_fixed}, for\nfixed $\\epsilon^{2}$, there exists error floors on the BERs of both\nthe ML and the GD, or equivalently, increasing the SNR does not improve\nthe BER. Thus, these two detectors in this case achieve a zero diversity\ngain for any $L$. Furthermore, when $L$ gets larger or $\\epsilon^{2}$\ngets smaller, the error floors in \\eqref{eq:Pb_1_fixed} and \\eqref{eq:Pb_2_fixed}\nbecome lower.\n\nThe following theorem compares the BER between the ML and the GD in\nMCIK-OFDM with the SC and fixed $\\epsilon^{2}$.\n\n\\textbf{Theorem 2.}\\textit{ In MCIK-OFDM using the SC under fixed CSI\nuncertainty, the GD achieves a better BER than the ML detector at high SNRs,}\ni.e., $P_{b_{1}}>P_{b_{2}}$. \n\\begin{IEEEproof}\nIt is shown from \\eqref{eq:Pb_1_fixed} and \\eqref{eq:Pb_2_fixed}\nthat at high SNRs, $P_{b_{1}}=P_{b_{2}}+A_{1}>P_{b_{2}},$ where the\nterm $A_{1}$ is related to the index detection error of the ML. This\nconcludes the proof. \n\\end{IEEEproof}\n\\textit{Remark 4.} As a result of Theorem~2, under fixed CSI imperfection,\nthe GD is able to outperform the ML in terms of both the BER and \ncomputational complexity, even for any $M$. This is obviously contrary\nto the perfect CSI case, where the BER of ML is always lower\nthan that of GD.\n\n\\subsection{MMSE-Based Variable CSI Uncertainty}\n\nNote that the error variance provided by the MMSE channel estimator\nis given by \\cite{thienBERGD} \n\\begin{equation}\n\\epsilon^{2}=\\frac{1}{1+\\gamma_{0}},\\label{eq:mmse}\n\\end{equation}\nwhich varies as a decreasing function of the SNR$.$\n\nInserting \\eqref{eq:mmse} to \\eqref{eq:Pb_1} and \\eqref{eq:Pb_2},\nwe obtain the asymptotic BERs for the ML and the GD in this case as\n\\begin{equation}\nP_{b_{1}}\\approx\\Upsilon\\left[\\frac{\\xi\\Omega\\left(1+N\/K\\right)^{L}}{6\\rho^{L}}\\right]\\frac{1}{\\gamma_{0}^{L}},\\label{eq:Pb_1_MMSE}\n\\end{equation}\n\\begin{equation}\nP_{b_{2}}\\approx\\Upsilon\\left[\\psi\\left(\\eta p_{1}+m\\right)+\\frac{\\xi\\Omega\\left(1+N\/K\\right)^{L}}{6\\rho^{L}}\\right]\\frac{1}{\\gamma_{0}^{L}},\\label{eq:Pb_2_MMSE}\n\\end{equation}\nwhere $\\Upsilon$ and $\\Omega$ are defined in \\eqref{eq:Pb_2_perfect},\nand $\\psi=\\sum_{i=1}^{N-K}\\left(-1\\right)^{i+1}C\\left(N-K,i\\right)\\left(i+1+iN\/K\\right)^{L-1}\/i^{L}$\n\nAs seen from \\eqref{eq:Pb_1_MMSE} and \\eqref{eq:Pb_2_MMSE},\nboth the GD and the ML of MCIK-OFDM with the SC achieves the same\ndiversity order of $L$ in this case. However, due to the impact of\nMMSE channel estimation errors, the BERs in \\eqref{eq:Pb_1_MMSE}\nand \\eqref{eq:Pb_2_MMSE} are obviously greater than that of the perfect\nCSI case. For example, we can see from \\eqref{eq:Pb_1_perfect} and\n\\eqref{eq:Pb_1_MMSE} that under the MMSE CSI imperfection, the ML\nendures a coding gain loss of $10\\log_{10}\\left(1+N\/K\\right)$\n(dB) compared with the perfect CSI case.\n\nAs for the comparison in the BER between the ML and the GD, denote\nby $\\Delta_{2}$ the coding gain attained by the ML over GD detector under\nMMSE variable CSI uncertainty, which can be obtained from \\eqref{eq:Pb_1_MMSE}\nand \\eqref{eq:Pb_2_MMSE} as \n\\begin{equation}\n\\Delta_{2}=\\frac{10}{L}\\log_{10}\\left(1+\\eta_{2}\\right)\\text{\\,(dB)},\\label{eq:Delta_2}\n\\end{equation}\nwhere $\\eta_{2}=6\\psi\\left(\\eta p_{1}+m\\right)\\rho^{L}\/\\xi\\Omega\\left(1+N\/K\\right)^{L}.$\nSimilar to Theorem~1, utilizing \\eqref{eq:Delta_2} we propose the\nfollowing theorem.\n\n\\textbf{Theorem 3.} \\textit{Consider MCIK-OFDM using the SC and the\nMMSE-based variable CSI imperfection. For $M\\ge4$, the BERs of the\nML and the GD rapidly converge to each other as increasing }$L$,\ni.e., $\\lim_{L\\rightarrow\\infty}\\eta_{2}=0$. \\textit{When $M=2$,\nthe ML performs better than the GD in terms of the BER by a coding gain of\n$10\\log_{10}\\left[1+K\/\\left(N+K\\right)\\right](dB)$, at large $L$,\ni.e., $\\lim_{L\\rightarrow\\infty}\\Delta_{2}=10\\log_{10}\\left[1+K\/\\left(N+K\\right)\\right]$\n(dB), moreover $\\lim_{L\\rightarrow\\infty}\\Delta_{2}<\\lim_{L\\rightarrow\\infty}\\Delta_{1}.$} \n\\begin{IEEEproof}\nAkin to Theorem~1, at large $L$, $\\psi$ in \\eqref{eq:Pb_2_MMSE}\ncan be approximated as $\\psi\\approx\\left(N-K\\right)\\left(2+N\/K\\right)^{L-1}.$\nThus, \n\\begin{equation}\n\\eta_{2}\\approx\\beta_{2}\\lambda_{2}^{L},\n\\end{equation}\nwhere $\\beta_{2}=6\\left(\\eta p_{1}+m\\right)\\left(N-K\\right)\/\\xi\\Omega\\left(2+N\/K\\right)$\nwhich is a decreasing function of $L$ and $\\lambda_{2}=\\rho\\left[1+K\/\\left(N+K\\right)\\right].$\n\nFor $M\\ge4$, we have $\\lambda_{2}\\le\\left[1+K\/\\left(N+K\\right)\\right]\/2<1$\nfor any $K1$. Thus,\n$\\lim_{L\\rightarrow\\infty}\\Delta_{2}=\\left(10\/L\\right)\\log_{10}\\left[1+K\/\\left(N+K\\right)\\right]^{L}=10\\log_{10}\\left[1+K\/\\left(N+K\\right)\\right]$\n(dB). Moreover, due to $1+K\/\\left(N+K\\right)<1.5$, $\\lim_{L\\rightarrow\\infty}\\Delta_{2}<\\left(10\/L\\right)\\log_{10}\\left(1.5^{L}\\right)\\approx1.76<\\lim_{L\\rightarrow\\infty}\\Delta_{1}\\approx3$\n(dB). \n\\end{IEEEproof}\n\\textit{Remark 5.} Compared to the perfect CSI case (Theorem~1),\nTheorem~3 indicates that for given $M$, the performance gap between\nthe two detectors under uncertain CSI gets smaller than that under perfect\nCSI. Therefore, the GD becomes more attractive than the ML under the\nMMSE CSI condition, particularly when the receiver has more antennas.\n\n\n\\section{Simulation Results}\n\nWe provide simulation results for MCIK-OFDM-SC having $N_{c}=128$ total\nsub-carriers, which are divided into $G$ clusters, each having $N$\nsub-channels. For illustrations, we consider $N\\in\\left\\{ 2,4\\right\\} $,\n$K<4$, $M\\in\\left\\{ 2,4,8\\right\\} $, and $L\\in\\{1,2,4,8,12\\}$.\nThe BER simulation results for the GD are compared to the ML under\nvarious MCIK parameters and CSI conditions.\n\n\\subsection{Accuracy of Theoretical and Asymptotic Expressions}\n\n\\begin{figure}[tb]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{D01}\n\\par\\end{centering}\n\\caption{BER of the GD detector in MCIK-OFDM-SC under various CSI conditions, with $(N,K,M,L)=(4,1,4,2)$.\n\\label{fig:D01}}\n\\end{figure}\n\\begin{figure}[tb]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{D03}\n\\par\\end{centering}\n\\caption{BER of the ML detector in MCIK-OFDM-SC under various CSI conditions, with $(N,K,M,L)=(4,2,4,3)$.\n\\label{fig:D03}}\n\\end{figure}\n\\begin{figure}[tb]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{A01}\n\\par\\end{centering}\n\\caption{BER comparison between the ML and the GD in MCIK-OFDM-SC under perfect\nCSI, with $(N,K,M)=(2,1,2)$ and $L=1,2,4,8$. \\label{fig:A01}}\n\\end{figure}\nFig.~\\ref{fig:D01} depicts the simulation results of MCIK-OFDM-SC\nusing the GD, along with the theoretical and asymptotic BER expressions\nwhen $(N,K,M,L)=(4,1,4,2)$, under various CSI conditions. \\textcolor{black}{As observed \nfrom Fig.~\\ref{fig:D01}, the theoretical BER expressions derived for the GD \n are very tight, i.e., very close to simulation results in a broad range of SNRs, while the asymptotic results\nare accurate in high SNR regions. This observation clearly confirms the accuracy of our theoretical analysis provided in Section~III and Section~IV.} In addition, under fixed or variable\n$\\epsilon^{2}$, the GD suffers from a considerable loss in the BER\ncompared to the perfect CSI case ($\\epsilon^{2}=0$). For example,\nat BER of $10^{-3}$ in Fig.~\\ref{fig:D01}, the loss of SNR gain\ncaused by fixed or variable CSI uncertainty is more than 4 dB. Note \nthat a similar observation can be seen in Fig.~\\ref{fig:D03} for\nthe ML detector.\n\n\\subsection{BER under Perfect CSI}\n\n\\begin{figure}[tb]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{A02}\n\\par\\end{centering}\n\\caption{BER comparison between the ML and the GD in MCIK-OFDM-SC under perfect\nCSI, with $(N,K,M)=(4,3,4)$ and $L=1,2,8$. \\label{fig:A02}}\n\\end{figure}\nFig.~\\ref{fig:A01} depicts the BERs for the ML and the GD in MCIK-OFDM-SC\nunder perfect CSI, with $(N,K,M)=(2,1,2)$ and $L=1,2,4,8$. As observed\nfrom Fig.~\\ref{fig:A01}, the ML always outperforms the GD even as\n$L$ increases. For instance, as $L=8$, at BER of $10^{-4}$, the\nML achieves the SNR gain of 3 dB over the GD. This confirms Theorem~1\nas $M=2$.\n\nIn Fig.~\\ref{fig:A02}, the BER comparison between the two detectors\nunder perfect CSI is illustrated for MCIK-OFDM-SC with $(N,K,M)=(4,3,4)$\nand $L=1,2,8.$ It is shown from Fig.~\\ref{fig:A02} that when $M=4$,\nthe BER of the GD approaches to that of the ML as $L$ gets larger.\nIn particular, at BER of $10^{-3}$, the coding gain attained by the\nML over the GD is about 5 dB when $L=1$, while this gain reduces\nto only 1 dB when $L=8$. This validates Theorem~1 for the case of\n$M=4.$\n\nFig.~\\ref{fig:A03} illustrates the BERs for the ML and the GD when\n$(N,K,M)=(4,2,8)$ and $L=1,2,4,8$. It is clear from this figure\nthat the BER of GD rapidly tends to that of the ML as $L$ increases.\nSpecifically, as $L=4$, the performance gap between these two detectors\nbecomes negligible. This confirms Theorem~1 for the case of $M\\ge8.$\n\n\\begin{figure}[tb]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{A03} \n\\par\\end{centering}\n\\caption{BER comparison between the ML and the GD in MCIK-OFDM-SC under perfect\nCSI, with $(N,K,M)=(4,2,8)$ and $L=1,2,4,8$. \\label{fig:A03}}\n\\end{figure}\n\n\n\\subsection{BER under Fixed CSI Uncertainty}\n\nFig.~\\ref{fig:E03} depicts the BER comparison between the ML and\nthe GD under fixed CSI uncertainty, with $(N,K,M)=(4,2,2)$, $L=2,4,8,12$\nand $\\epsilon^{2}=0.2$. \\textcolor{black}{Interestingly, it can be seen from this figure that at high SNRs, the GD outperforms the ML in terms of the BER.\nFor example, as $L=4,$ the GD achieves the BER lower than the ML\nwhen $E_{s}\/N_{0}\\ge15$ dB. This is due to the fact that under the fixed CSI uncertainty, using the energy detection, the GD achieves better index detection performance than its ML counterpart, leading to better BER performance, as theoretically proved in Subsection~IV-B.} Moreover, due to the fixed error variance,\ni.e., $\\epsilon^{2}=0.2$, there exists error floors on the BERs of\nthe two detectors. These floors get lower as $L$ increases. This\nobservations validate Theorem~2.\n\n\\begin{figure}[tb]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{E03} \n\\par\\end{centering}\n\\caption{BER comparison between the ML and the GD in MCIK-OFDM-SC under fixed\nCSI, with $(N,K,M)=(4,2,2)$, $L=2,4,8,12$ and $\\epsilon^{2}=0.2$.\n\\label{fig:E03}}\n\\end{figure}\n\n\n\\subsection{BER under MMSE Variable CSI Uncertainty}\n\nFig.~\\ref{fig:C01} depicts the BER comparison between the GD and\nML detectors under MMSE-based variable CSI uncertainty, with $(N,K,M)=(2,1,2)$\nand $L=1,2,4,8$. As seen via Fig.~\\ref{fig:C01}, when $L$ gets\nlarger, the BERs of the two detectors become closer. However, the\nML always outperforms the GD. In addition, the performance gap between\nthem under variable CSI uncertainty gets smaller than that under perfect CSI in\nFig.~\\ref{fig:A01}. These observations validate Theorem~3 for $M=2$. \n\n\\begin{figure}[tb]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{C01} \n\\par\\end{centering}\n\\caption{BER comparison between the ML and the GD in MCIK-OFDM-SC under MMSE\nvariable CSI uncertainty, with $(N,K,M)=(2,1,2)$ and $L=1,2,4,8$.\n\\label{fig:C01}}\n\\end{figure}\n\nFig.~\\ref{fig:C02} compares the BER between the two detectors\nunder MMSE variable CSI, when $(N,K,M)=(4,1,4)$ and $L=1,2,4,8$.\nUnlike the perfect CSI case, the BERs of the ML and\nthe GD under this CSI condition quickly converge to each other as $L$ increases even when\n$M=4$. Similar to Fig.~\\ref{fig:A03}, as $L\\ge2$ there is a marginal\ngap in the BER between the two detectors. Hence, Theorem~3\nwith $M\\ge4$ is clearly validated.\n\n\\begin{figure}[tb]\n\\begin{centering}\n\\includegraphics[width=0.9\\columnwidth]{C02} \n\\par\\end{centering}\n\\caption{BER comparison between the ML and the GD in MCIK-OFDM-SC under MMSE\nvariable CSI uncertainty, with $(N,K,M)=(4,1,4)$ and $L=1,2,4,8$.\n\\label{fig:C02}}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nWe proposed a generalized framework for the BER analysis of MCIK-OFDM\nusing either the GD or ML detector. Based on this, we derived tight, closed-form\nexpressions for the BERs of MCIK-OFDM with the selection combining\nML (or GD) receiver, taking effects of CSI uncertainty into account.\nWe provided the asymptotic analysis to investigate impacts of imperfect\nCSI on their BERs. Furthermore, the BER comparison between\nthe GD and ML detectors under various CSI conditions was presented, which\nallows to provide a theoretical guideline on the signal detection\nof MCIK-OFDM-SC under each specific CSI condition. For example, under\nMMSE-based variable CSI, the SC-based GD was shown to approach the\nSC-based ML in terms of the BER as the number of antennas increases\nand $M\\ge4$. More interestingly, under fixed CSI uncertainty and\nat high SNRs, the SC-based GD always outperforms the SC-based ML in\nterms of the BER for any value of $M$. Finally, the derived BER expressions\nand theoretical guideline are validated via simulation results. It\nis noteworthy that the derived expressions and proposed guideline\nfor using the GD would be useful for various designs of the practical\nimplementation of MCIK-OFDM. \\textcolor{black}{In our future work, we plan to investigate the performance of MCIK-OFDM-SC in combination with a number of diversity enhancement techniques, such as coordinate interleaving \\cite{CIbasar2015}, repetition codes \\cite{ThienTWC2018}, and spreading matrix \\cite{ThienTVT2018}.}\n\n\\appendices{}\n\n\\section{Proof of Lemma 1}\n\nThe instantaneous SEP of the classical PSK symbol detection per sub-carrier\n$\\alpha$ (denoted by $P_{M}\\left(\\alpha\\right)$) is given by \\cite{thienBERGD}\n\\begin{equation}\nP_{M}\\left(\\alpha\\right)\\approx\\frac{\\xi}{12}\\left[e^{-\\frac{\\rho\\bar{\\gamma}\\hat{\\nu}_{\\alpha}}{1+\\epsilon^{2}\\bar{\\gamma}}}+3e^{-\\frac{4\\rho\\bar{\\gamma}\\hat{\\nu}_{\\alpha}}{3\\left(1+\\epsilon^{2}\\bar{\\gamma}\\right)}}\\right],\\label{eq:ins_PM}\n\\end{equation}\nwhere $\\xi=1$ for $M=2$ and $\\xi=2$ for $M>2$, and $\\hat{\\nu}_{\\alpha}=\\left|\\hat{h}\\left(\\alpha\\right)\\right|^{2}$\nwhich is chi-square distributed with degrees of freedom of two, .i.e,\n$\\hat{\\nu}_{\\alpha}\\sim\\mathcal{X}_{2}^{2}.$ Note that $\\left|\\hat{h}\\left(\\alpha\\right)\\right|^{2}=\\max_{l}\\left|\\hat{h}_{l}\\left(\\alpha\\right)\\right|^{2}$\nand using the order statistics theory, the probability density function\n(PDF) of $\\hat{\\nu}_{\\alpha}$ is given as \n\\begin{equation}\nf_{\\hat{\\nu}}\\left(x\\right)=\\frac{L}{a}e^{-\\frac{x}{a}}\\left(1-e^{-\\frac{x}{a}}\\right)^{L-1},\\label{eq:PDF_v_hat}\n\\end{equation}\nwhere $a=1-\\epsilon^{2}$. Using \\eqref{eq:PDF_v_hat}, the MGF of\n$\\hat{\\nu}_{\\alpha}$ can be obtained, after simple manipulations,\nas \n\\begin{equation}\n\\mathcal{M}_{\\hat{\\nu}}\\left(t\\right)=\\frac{L!}{\\prod_{l=1}^{L}\\left(l-at\\right)}.\\label{eq:MGF_v_hat}\n\\end{equation}\n\nFinally, applying the MGF approach to \\eqref{eq:ins_PM} and using\n\\eqref{eq:MGF_v_hat}, the average SEP of \\eqref{eq:ins_PM} is attained\nas \\eqref{eq:PM}.\n\n\\section{Proof of Lemma 2 }\n\nLet $b=\\left|\\hat{h}\\left(\\alpha\\right)\\right|$ and $Z=\\left|\\left|\\hat{h}\\left(\\alpha\\right)\\right|+\\tilde{e}\\left(\\alpha\\right)\\right|^{2}$.\nAssume that $\\tilde{e}\\left(\\alpha\\right)=c+jd$, where $c,d\\sim\\mathcal{N}\\left(0,\\epsilon^{2}\/2\\right)$,\nwe obtain\n\n\\begin{equation}\nZ=(b+c)^{2}+d^{2}.\n\\end{equation}\nLet $Z'=2Z\/\\epsilon^{2}=\\left[\\sqrt{2}(b+c)\/\\epsilon\\right]^{2}+\\left(\\sqrt{2}d\/\\epsilon\\right)^{2}.$\nDue to $\\sqrt{2}(b+c)\/\\epsilon\\sim\\mathcal{N}\\left(\\sqrt{2}b\/\\epsilon,1\\right)$\nand $\\sqrt{2}d\/\\epsilon\\sim\\mathcal{N}\\left(0,1\\right)$, $Z'$ is\ndistributed according to the noncentral chi-squared distribution with\ntwo degrees of freedom, i.e., $\\mathcal{X}_{2}^{2}\\left(\\lambda\\right)$,\nwhere $\\lambda=2b^{2}\/\\epsilon^{2}$ is the non-centrality parameter\n\\cite{johnson1995continuous}. Thus, the MGF of $Z'$ is given by\n\\cite{johnson1995continuous} \n\\begin{equation}\n\\mathcal{M}_{Z'}\\left(t\\right)=\\frac{e^{\\frac{2b^{2}t\/\\epsilon^{2}}{1-2t}}}{1-2t}.\\label{eq:MGF_Z_comma}\n\\end{equation}\n\nFinally, the MGF of $Z$ can be computed, using $\\mathcal{M}_{Z'}\\left(t\\right)$\nin \\eqref{eq:MGF_Z_comma} as $\\mathcal{M}_{Z}\\left(t\\right)=\\mathcal{M}_{Z'}\\left(\\epsilon^{2}t\/2\\right)$,\nwhich leads to \\eqref{eq:MGF_non_central}.\n\n \\bibliographystyle{IEEEtran}\n\\phantomsection\\addcontentsline{toc}{section}{\\refname}","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n Ethanol has been considered for fuel cells given its low toxicity and abundance \\cite{Wang2004}. The electro-oxidation of ethanol on the surface of catalysts can follow multiple reaction pathways leading to several different products, which are strongly affected by, e.g., concentration, presence of impurities, and temperature. In this work, we introduce a model, inspired on the random sequential adsorption model (RSA) of dimers, to analyze properties of the oxidation process such as their dependence on the binding configuration, binding rates, and reaction pathway probabilities.\n\n Ethanol electro-oxidation has recently been studied through density function theory \\cite{Wang2008a,Wang2007}, providing possible reaction pathways for the adsorption and catalysis of ethanol.\n However, the time-dependence of the coverage based on the various reaction pathways would become computationally prohibitive using density functional theory.\n Nonetheless, in the limit of low mobility of bound reaction products, large systems sizes, and long timescales a study based on the adoption of a square lattice to represent the (100) substrate for the various reaction products becomes appropriate.\n In this limit, ethanol electro-oxidation can be described as adsorption of a dimer on the substrate, thus occupying two adjacent lattice sites, as it cleaves \\cite{Wang2008a,Wang2007} or, in the presence of neighboring pre-adsorbed species, adsorb as a monomer. A key feature of the model is to provide a configuration dependent rule for the landing site and study the influence on the adsorption rates of the oxidation process. We also study the influence of immobile impurities and understand their role in achieving selectivity of adsorbed species. Our model can also be extended to analyze other cleavage mechanisms like, e.g., the one of sugars \\cite{Parpot2006}.\n\n The random sequential adsorption model (RSA) has been utilized to describe adsorption in the limit of low surface mobility and of negligible desorption rate \\cite{Evans1993a, Cadilhe2007, Araujo2006, Araujo2010b,Bonnier2001,Bonnier1994,Privman1994,Brosilow1991,Feder1980a,Araujo2010}. Adsorption attempts occur sequentially at randomly selected sites, where particles solely interact through excluded volume. Generalized versions have been proposed where the rates of adsorption are dependent on the local configuration \\cite{Evans1993a,Evans1985}, which might, e.g., explain the selectivity of adsorbed species \\cite{Lopez2008}. Further extensions have been considered to study a wide range of problems, such as chemical reactions \\cite{Gonz1974, Flory1939,McLeod1999}, adsorption on membranes \\cite{Finegold1979}, as well as protein and colloid adsorption \\cite{Feder1980,Onoda1986} with and without pre-adsorbed impurities \\cite{RamirezPastor2000,Zuppa1999,Stacchiola1998,Kondrat2006, Lee1996, Bennaim1994}.\n\n In this paper, we study the model above delineated both in one and two dimensions. In the one-dimensional study, we were able to establish a closed hierarchy of rate equations for which we could obtain exact, closed form solutions. To complement and extend the insight provided by this approach, we also performed a Monte Carlo based study for the relevant two-dimensional case.\n\n The paper is organized as follows: in the next section we introduce the model, while in Section~\\ref{sec:analytical} an analytical derivation is exactly solved in three specific limits. Monte Carlo simulations extend the one-dimensional results to the more realistic case of a substrate as described in section~\\ref{sec:MC_1d} with results provided in section~\\ref{sec:2d_results} for substrates with and without impurities. Final remarks are presented in section~\\ref{sec:conclusion}.\n\n\\section{Model}\\label{sec:model}\n\n Ethanol oxidation is of great relevance to the society, since each molecule releases twice as much energy as one methanol molecule \\cite{Farias2007}, posing it as a candidate to replace several sources of energy \\cite{Lynd1996}. Recently, Wang and Liu \\cite{Wang2008a} proposed a mechanism for ethanol electro-oxidation on Pt(100) and Pt(111) substrates, which can be summarized in three pathways \\cite{Wang2007}: \n \\begin{enumerate}\n \\item The \\textit{OH path}, where the cleavage of the hydroxyl group leads to the formation of acetaldehyde which is then adsorbed; \n \\item The \\textit{CH path} where $CH_3CHOH$ is an intermediate product that degrades into $CH_3COH$; \n \\item The \\textit{concerted path} where the ethanol molecule looses two hydrogens followed by the desorption of acetaldehyde.\n \\end{enumerate}\n\n Wang and Liu have shown that for the Pt(100) surface (which can be mapped onto a square lattice) the relevant pathway is mainly the \\textit{CH path} \\cite{Wang2008a}, where ethanol adsorption leads to the formation of acetyl ($CH_3CO$). The work of Wang and Liu \\cite{Wang2008a} discloses that the adsorption mechanism is strongly influenced by the actual surface coverage. At low surface coverages, the acetyl dehydrogenates into $CH_2CO$ or $CHCO$, which leads to a $C-C$ bond cleavage, yielding $CH$ and $CO$ fragments. At oxidative conditions, both fragments react with the oxygen, $O$, present on the surface and desorb as carbon dioxide $CO_2$. Desorption of cleaved products can be neglected for non-oxidative conditions, which leads to a jammed state, whereas, when the surface coverage is high, the $C-C$ bond cleavage is blocked, and the surface becomes poisoned by acetyl.\n \n Based on the proposed mechanism, we introduce a model which can be summarized by the following rules,\n \\begin{align}\n A + 2v \\stackrel{k_d}{\\longrightarrow} 2B \\nonumber \\\\\n A + v \\stackrel{k_m}{\\longrightarrow} C,\n \\label{eq.adsorption_mechanism}\n \\end{align}\nwhere $A$ represents ethanol, $v$ an empty site, $2B$ represents cleaved products, $C$ is acetyl, and $k_d$ ($k_m$) stands for dimer (monomer) production rates. Note that, as discussed below, adsorption as a monomer (with product $C$) can only occur in the neighborhood of an occupied site.\n\n \\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=8cm]{model_rules.pdf}\\\\\n \\caption{Schematic representation of the adsorption rules (1). The\nA-species (red) can deposit either as dimers, on two empty sites,\nyielding B-products (blue), or as monomers, on one empty site with at\nleast one occupied neighbor, yielding C-products (green). In the ethanol\noxidation, $A$ is the ethanol, $B$ stands for the\ntwo products ($CH$ and $CO$), and $C$ the acetyl.}\n \\label{fig.model_rules}\n \\end{center}\n \\end{figure}\n\n As cartooned in Fig.~\\ref{fig.model_rules}, dimers are uniformly formed on the substrate (lattice). Successful adsorption of dimers requires two neighboring empty sites. If only one is available, the species adsorbs as a monomer. When both sites are occupied, due to the excluded volume interaction, the adsorption attempt fails and the particle attempting adsorption is no longer considered. The model differs from traditional cooperative sequential adsorption models \\cite{Evans1993a} since, for the latter, the rates of adsorption depend on the state of the nearest-neighbors but not, as in the present model, on the occupation of the local configuration provided by neighboring adsorbed sites. Despite the focus on the electro-oxidation of ethanol, the model could be utilized for the study of any other process dependent on the local configuration rather than on constant, configuration independent, deposition rates.\n\n\\section{Analytical Study}\\label{sec:analytical}\n\n The time dependence of the coverage and distribution of empty sites can be analytically obtained by establishing a closed hierarchy of rate equations as explained below. To account for different rates for monomer and dimer adsorption (see Eq.~\\ref{eq.adsorption_mechanism}), we consider a competitive deposition of monomers and dimers, with different deposition rates ($k_m$ and $k_d$, respectively), under the constraint that monomers can only deposit in the neighborhood of occupied sites. Results are divided into three limiting cases: equal rates for dimers and monomers (Section \\ref{sec:eq_abs}), preferential dimer site adsorption (Section \\ref{sec:pref_carb}), and different deposition rates for monomers and dimers (Section \\ref{sec:diff_rates}).\n\n Let us start by considering a segment of empty sites with size $n$ which is part of a larger (or equal) one with size $\\ell\\geq n$. Since the neighboring sites of this segment are not necessarily occupied, the possible adsorption events depend on the configuration of the neighbors. A segment can be reduced in size by adsorption of a monomer on the left-hand side of the segment (Fig.~\\ref{fig.anal_rules}b), the right-hand side (Fig.~\\ref{fig.anal_rules}c), or on both sides (Fig.~\\ref{fig.anal_rules}e). It can also be split (or reduced in size) by the adsorption of a dimer on any site of the segment (Fig.~\\ref{fig.anal_rules}a), the right-hand side (Fig.~\\ref{fig.anal_rules}b), the left-hand side (Fig.~\\ref{fig.anal_rules}c), or on both sides (Fig.~\\ref{fig.anal_rules}d). \n\n \\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=8cm]{anal_rules.pdf}\\\\\n \\caption{Catalog of possible adsorption attempts of a dimer on a segment of empty sites with length $n=5$, which is part of a larger segment with length $\\ell$ ($\\ell\\geq n$). For adsorption events with both landing sites in the segment $n$, a dimer is adsorbed (a). For adsorption events with a single landing site in the $n$ segment either a dimer or a monomer is adsorbed, depending on the occupation state of the neighboring site (b, c, d, and e).\n \\label{fig.anal_rules}}\n \\end{center}\n \\end{figure}\n\n We define $P_n$ as the probability that a randomly selected site belongs to any segment $n$ defined before. Each configuration of the catalog in Fig.~\\ref{fig.anal_rules} is obtained with a probability $P[\\cdots]$ given by,\n\n \\begin{align}\n P[\\circ\\circ\\circ\\circ\\circ]&=P_n \\nonumber \\\\\n P[\\bullet\\circ\\circ\\circ\\circ\\circ\\circ]&=P[\\circ\\circ\\circ\\circ\\circ\\circ\\bullet]=P_{n+1}-P_{n+2}\n \\label{eq.anal_mechanism} \\\\\n P[\\circ\\circ\\circ\\circ\\circ\\circ\\circ]&=P_{n+2} \\nonumber \\\\\n P[\\bullet\\circ\\circ\\circ\\circ\\circ\\bullet]&=2P_n-2P\\mbox[\\bullet\\circ\\circ\\circ\\circ\\circ\\circ]-P[\\circ\\circ\\circ\\circ\\circ\\circ\\circ] \\nonumber \\\\\n &=P_n-2P_{n+1}+P_{n+2}. \\nonumber\n \\end{align}\nThe proper set of rate equations depends on the case. Below, we describe this set for equal rates of dimers and monomers, adsorption of a preferential dimer site, and different rates for dimers and monomers.\n\n\\subsection{Equal deposition rate of dimers and monomers}\\label{sec:eq_abs}\n\n For equal deposition rates of monomers and dimers, the rate of both species is considered as $k$. Since $P_n$ refers to the probability that a randomly selected site belongs to any segment of $n$ empty sites, which can be part of a larger one, this probability can never increase with time. The rate of change of $P_n$ due to the adsorption of dimers is given by,\n \\begin{align}\n \\left(\\dot{P}_n\\right)_d=&-k(n-1)P_n-kP[\\bullet\\circ\\circ\\circ\\circ\\circ\\circ] \\nonumber \\\\\n &-kP[\\circ\\circ\\circ\\circ\\circ\\circ\\bullet]-2kP[\\circ\\circ\\circ\\circ\\circ\\circ\\circ] \\label{eq.Pn_dimers_eqrates} \\\\\n =&-(n-1)kP_n-2kP_{n+1}, \\nonumber\n \\end{align}\nwhere $(n-1)$ corresponds to the destruction rate of a segment with, at least, $n$ empty sites, which is zero for $n=1$, see Fig.~\\ref{fig.anal_rules}(a). The rate of change due to monomers adsorption is,\n \\begin{align}\n \\left(\\dot{P}_n\\right)_m=&-2kP[\\bullet\\circ\\circ\\circ\\circ\\circ\\bullet]-kP[\\bullet\\circ\\circ\\circ\\circ\\circ\\circ] \\nonumber \\\\\n &-kP[\\circ\\circ\\circ\\circ\\circ\\circ\\bullet]\\label{eq.Pn_monomers_eqrates} \\\\\n =&-2k(P_n-P_{n+1}). \\nonumber\n \\end{align}\n From Eqs.~(\\ref{eq.Pn_dimers_eqrates})~and~(\\ref{eq.Pn_monomers_eqrates}), the total rate of change is given by,\n \\begin{align}\n \\left(\\dot{P}_n\\right)_T=\\left(\\dot{P}_n\\right)_d+\\left(\\dot{P}_n\\right)_m=-(n+1)kP_n. \\label{eq.Pn_total_eqrates}\n \\end{align}\n This result is equivalent to consider that, regardless the type of adsorption, a segment of $n$, or more, empty sites can be destroyed by adsorption on $(n+1)$ different places. This equation gives $P_n(t)=\\exp{\\left[-(n+1)kt\\right]}$. The coverage $\\theta$ can then be obtained from the probability that a certain site is part of a segment of size $n\\geq1$, i.e.,\n\t \\begin{align}\n \\theta(t)=1-P_1(t)=1-\\exp{(-2kt)}. \\label{eq.coverage_eqrates}\n\t \\end{align}\n Defining $s_m(n)$ as the rate of monomers adsorption on a $n$-segment, i.e., $s_m(n)=-\\left(\\dot{P}_n\\right)_m$, we obtain,\n \\begin{align}\n s_m(n)&=2k(P_n-P_{n+1}) \\label{eq.sm_eq1} \\\\\n &=2k\\left[\\exp{(-\\left[n+1\\right]kt)}-\\exp{(-\\left[n+2\\right]kt)}\\right].\\nonumber\n \\end{align}\n The adsorption of monomers on a segment of size $n\\geq1$, is given by,\n\t \\begin{equation}\n s_m(n=1)=2k\\left[\\exp{(-2kt)}-\\exp{(-3kt)}\\right].\n \\label{eq.sm_n1}\n\t \\end{equation}\n From the rate of adsorption, the coverage of monomers can be obtained from $\\dot{\\theta}_m=s_m$ for $n=1$ giving,\n \\begin{equation}\n \\theta_m(t)=\\left[1-\\exp{(-2kt)}\\right]+\\frac{2}{3}\\left[\\exp{(-3kt)}-1\\right].\n \\label{eq.thetam_eqrates2}\n \\end{equation}\n Defining now $s_d(n)$ as the rate of dimers adsorption on a $n$-segment, i.e., $s_d(n)=-\\left(\\dot{P}_n\\right)_d$, we obtain,\n \\begin{align}\n s_d(n)=&k[(n-1)\\exp{(-\\left[n+1\\right]kt)} \\label{eq.sd_eq1} \\\\\n &+2\\exp{(-\\left[n+2\\right]kt)}]. \\nonumber\n \\end{align}\n The adsorption of dimers on a segment of size $n\\geq1$, is then given by,\n\t \\begin{equation}\n s_d(n=1)=2k\\exp{(-3kt)}.\n \\label{eq.sd_n1}\n\t \\end{equation}\n Knowing the rate of adsorption, the coverage of dimers can be obtained from the rate equation, $\\dot{\\theta}_d=s_d(n=1)$,\n \\begin{equation}\n \\theta_d(t)=\\frac{2}{3}\\left[1-\\exp{(-3kt)}\\right].\n \\label{eq.thetad_eqrates2}\n \\end{equation}\n\n\\subsection{Adsorption of a preferential dimer site}\\label{sec:pref_carb}\n\n Considering a preferential dimer site means that the symmetry is broken and the first adsorption on the substrate occurs through a specific compound of the dimer. The cleavage only takes place if there is a neighboring empty site, in any direction, to adsorb the other compound. In 1D, the left site of the dimer is considered as the preferred one. By symmetry, results are independent on the considered one. For this special case, the change on the $P_n$ by dimers is,\n \\begin{align}\n \\left(\\dot{P}_n\\right)_d=&-k(n-1)P_n-kP[\\bullet\\circ\\circ\\circ\\circ\\circ\\circ] \\nonumber\\\\\n &-kP[\\circ\\circ\\circ\\circ\\circ\\circ\\bullet]-2kP[\\circ\\circ\\circ\\circ\\circ\\circ\\circ]\n \\label{eq.Pn_dimers_leftcarb} \\\\\n =&-(n-1)kP_n-2kP_{n+1}, \\nonumber\n \\end{align}\nwhile by monomers is,\n \\begin{align}\n \\left(\\dot{P}_n\\right)_m&=-kP[\\bullet\\circ\\circ\\circ\\circ\\circ\\bullet]-kP[\\circ\\circ\\circ\\circ\\circ\\circ\\bullet]\\nonumber\\\\\n &=-k(P_n-P_{n+1}). \\label{eq.Pn_monomers_leftcarb}\n \\end{align}\n For the total change on $P_n$ we obtain,\n \\begin{align}\n \\left(\\dot{P}_n\\right)_T=\\left(\\dot{P}_n\\right)_d+\\left(\\dot{P}_n\\right)_m=-nkP_n-kP_{n+1}. \\label{eq.Pn_total_leftcarb}\n \\end{align}\n Considering the relation between $P_n$ and $P_{n+1}$ as $P_{n+1}=Q_nP_n$ \\cite{Evans1997}, then\n \\begin{align}\n \\dot{P}_{n+1}=-(n+1)kQ_nP_n-kQ_{n+1}Q_nP_n, \\label{eq.Pn_total_leftcarb2}\n \\end{align}\nand plugging it back into Eq.~(\\ref{eq.Pn_total_leftcarb}) we obtain,\n \\begin{align}\n \\frac{dQ_n}{dt}\\frac{1}{Q_n}=-(n+1)k+nk-k(Q_{n+1}-Q_n). \\label{eq.Pn_total_leftcarb3}\n \\end{align}\nIf we assume $Q_{n+1}=Q_n$, then $\\frac{dQ_n}{Q_n}=-kdt$. Therefore, $Q_n(t)=\\exp{(-kt)}$, which when replaced in Eq.~(\\ref{eq.Pn_total_leftcarb3}) gives $\\dot{P}_n=-\\left[nk+k\\exp{(-kt)}\\right]P_n$ and so,\n \\begin{equation}\n P_n(t)=\\exp{\\left[-nkt+\\left(\\exp{\\left[-kt\\right]}-1\\right)\\right]}.\n \\label{eq.Pn_total_leftcarb7}\n \\end{equation}\n Since the coverage $\\theta$ is dependent on the evolution of the probability of finding a segment of size $n\\geq1$,\n\t \\begin{align}\n \\theta(t)&=1-P_1(t) \\label{eq.coverage_leftcarb} \\\\\n &=1-\\exp{\\left[-kt+\\left(\\exp{\\left[-kt\\right]}-1\\right)\\right]}. \\nonumber\n\t \\end{align}\nThe independent rates of adsorption for dimers and monomers and the subsequent calculation of the coverage for each species are obtained as before.\n\n\\subsection{Different deposition rates for dimers and monomers}\\label{sec:diff_rates}\n\n To attempt a generic solution for the rules given by Eq.~(\\ref{eq.adsorption_mechanism}), it is necessary to consider different deposition rates for monomers ($k_m$) and dimers ($k_d$). Accounting for the rate of change of $P_n$ by dimers given by Eq.~(\\ref{eq.Pn_dimers_eqrates}),\n \\begin{equation}\n \\left(\\dot{P}_n\\right)_d=-(n-1)k_dP_n-2k_dP_{n+1},\n \\label{eq.Pn_dimers_difrates}\n \\end{equation}\nwhile by monomers,\n \\begin{equation}\n \\left(\\dot{P}_n\\right)_m=-2k_m(P_n-P_{n+1}).\n \\label{eq.Pn_monomers_difrates}\n \\end{equation}\n The change on the the total $P_n$ over time is then given by,\n \\begin{align}\n \\left(\\dot{P}_n\\right)_T&=\\left(\\dot{P}_n\\right)_d+\\left(\\dot{P}_n\\right)_m\n \\label{eq.Pn_total_difrates} \\\\\n &=-\\left[k_d(n-1)+2k_m\\right]P_n-2(k_d-k_m)P_{n+1}. \\nonumber\n \\end{align}\n For the sake of simplicity, we define $\\alpha_n=(n-1)k_d+2k_m$ and $\\beta=k_d-k_m$. In the same way as before, applying the relation $P_{n+1}=Q_nP_n$, the rate equation for $P_{n+1}$ is,\n \\begin{align}\n \\dot{P}_{n+1}&=\\dot{Q}_nP_n+Q_n\\dot{P}_n \\label{eq.Pn_total_difrates2} \\\\\n &=-\\alpha_{n+1}Q_nP_n-2\\beta Q_{n+1}Q_nP_n, \\nonumber\n \\end{align}\nand replacing $\\dot{P_n}$ by Eq.~(\\ref{eq.Pn_monomers_difrates}),\n \\begin{align}\n &\\dot{Q}_nP_n+Q_n\\left[-(\\alpha_n+2\\beta Q_n)P_n\\right]= \\nonumber \\\\ \n &-\\left(\\alpha_{n+1}Q_n+2\\beta Q_{n+1}Q_n\\right)P_n\n \\label{eq.Pn_total_difrates3} \\\\\n &\\dot{Q}_n=-k_dQ_n-2\\beta(Q_{n+1}-Q_n)Q_n. \\nonumber\n \\end{align}\nConsidering $Q_{n+1}=Q_n$ then $Q_n(t)=\\exp{(-k_dt)}$. From the above result, Eq.~(\\ref{eq.Pn_total_difrates}) simplifies as $\\dot{P}_n=-\\left(\\alpha_n-2\\beta Q_n\\right)P_n$, which gives,\n \\begin{equation}\n\\label{eq.Pn_total_difrates6}\n \\begin{split}\n P_n(t)=&\\exp\\biggl[-(\\left[n-1\\right]k_d+2k_m)t\\biggr. \\\\\n &\\left.-\\frac{2(k_d-k_m)}{k_d}\\left(1-\\exp{\\left[-k_dt\\right]}\\right)\\right].\n \\end{split}\n \\end{equation}\nFrom Eq.~(\\ref{eq.coverage_eqrates}),\n\t \\begin{align}\n \\label{eq.coverage_difrates}\n \\begin{split}\n \\theta(t)=&1-P_1(t) \\\\\n\t =&1-\\exp\\biggl[-2k_mt\\biggr.\\\\\n &\\biggl.-\\frac{2(k_d-k_m)}{k_d}\\left(1-\\exp{\\left[-k_dt\\right]}\\right)\\biggr],\n \\end{split}\n\t \\end{align}\nwhich for $k_m=k_d=k$ boils down to Eq.~(\\ref{eq.coverage_eqrates}). If $s_m(n)$ is defined as the rate of monomers adsorption on a $n$-segment $s_m(n)=-\\left(\\dot{P}_n\\right)_m$ and so\n \\begin{align}\n s_m(n)=2k_m(1-Q_n)P_n. \\label{eq.sm_eq1_difrates}\n \\end{align}\n The relations $y=\\exp{(-k_dt)}$ and $\\gamma=\\frac{k_m}{k_d}$ are considered and the adsorption of monomers on a segment of size $n\\geq1$ is given by,\n\t \\begin{equation}\n s_m(1)=2k_m\\left(1-y\\right)y^{2\\gamma}\\exp{\\left[-2(1-\\gamma)(1-y)\\right]}.\n \\label{eq.sm_n1_difrates}\n\t \\end{equation}\n From the rate of adsorption, the coverage of monomers is given by $\\dot{\\theta}_m=s_m$ for $n=1$, and so,\n \\begin{equation}\n \\theta_m=-\\int^{\\exp{(-k_dt)}}_1s_m(1)\\left(yk_d\\right)^{-1}dy.\n \\label{eq.thetam_difrates3}\n \\end{equation}\n The dimers rate of adsorption is then,\n \\begin{align}\n s_d(n)=k_d(n-1)P_n+2k_dQ_nP_n. \\label{eq.sd_eq1_difrates}\n \\end{align}\n For the sake of simplicity, the relation $y=\\exp{(-t)}$ is used, and the adsorption of dimers on a segment of size $n\\geq1$ is given by,\n\t \\begin{equation}\n s_d(1)=2k_dy^{k_d+2k_m}\\exp{\\left[-\\frac{2\\left(k_d-k_m\\right)}{k_d}\\left(1-y^{k_d}\\right)\\right]}.\n \\label{eq.sd_n1_difrates}\n\t \\end{equation}\n For the rate of adsorption, the coverage of monomers can be given by, $\\dot{\\theta}_d=s_d$ for $n=1$, which by integrating over $y$ gives,\n \\begin{equation}\n \\theta_d=-\\int^{\\exp{(-t)}}_1s_d(1)y^{-1}dy. \\label{eq.thetad_difrates3}\n \\end{equation}\n\n\\section{1D Monte Carlo Simulations}\\label{sec:MC_1d}\n We numerically studied the proposed model through Monte Carlo simulations, performed on a lattice with $10^4$ sites, where periodic boundary conditions have been applied and results have been averaged over $10^2$ samples.\n\n \\begin{figure}[t]\n \\begin{center}\n \\begin{tabular}{cc}\n \\includegraphics[width=4cm]{coverage_1d_3plots.pdf} & \\includegraphics[width=4cm]{rates_1d_3plots.pdf}\\\\\n \\end{tabular}\n \\caption{Coverage $\\theta(t)$ and rates of adsorption $s(t)$ as a function of time obtained through Monte Carlo simulations (symbols) and analytically (solid line) for equal rates of adsorption (a) and (d), preferential carbon adsorption (b) and (e), and different rates of adsorption ($k_d=0.5$ and $k_m=1$) (c) and (f). Total coverage and rates of adsorption (open squares), dimers (full squares), and monomers (open circles).}\n \\label{fig.coverage_eqrates}\n \\end{center}\n \\end{figure}\n\n The coverage as a function of time is plotted on Figs.~\\ref{fig.coverage_eqrates}(a),~(b),~and~(c) for the three previously described cases. The total coverage $\\theta_T$, the coverage of dimers $\\theta_d$, and the coverage of monomers $\\theta_m$ are computed over 10 Monte Carlo (MC) sweeps, where one MC sweep corresponds to one adsorption attempt per lattice site. Figure~\\ref{fig.coverage_eqrates}(a) shows the case of equal deposition rates for dimers and monomers, where $k_d=k_m=1$. The solid lines in the plot represent the analytical solution given by Eqs.~(\\ref{eq.coverage_eqrates}),~(\\ref{eq.thetam_eqrates2}),~and~(\\ref{eq.thetad_eqrates2}). The preferential dimer site rule of adsorption is in Fig.~\\ref{fig.coverage_eqrates}(b), where despite the values of the deposition rates being given by $k_d=k_m=1$, the results are equivalent to the ones for $k_d=1$ and $k_m=0.5$, since the deposition of a monomer by the non-favored dimer site is not allowed. For these rules of deposition, the exact results are given by Eqs.~(\\ref{eq.coverage_leftcarb}),~(\\ref{eq.Pn_dimers_leftcarb}),~and~(\\ref{eq.Pn_monomers_leftcarb}). The final case, where a different deposition rate for dimers and monomers is considered, is plotted in Fig.~\\ref{fig.coverage_eqrates}(c), with deposition rates of $k_d=0.5$ and $k_m=1$. The exact solution is given by Eqs.~(\\ref{eq.coverage_difrates}),~(\\ref{eq.thetam_difrates3}),~and~(\\ref{eq.thetad_difrates3}). For all cases, data points from Monte Carlo lay on the line given by the exact solution.\n\n Figures~\\ref{fig.coverage_eqrates}~(d),~(e),~and~(f) show the rates of adsorption as a function of time (only five MC sweeps are shown). The plots (d), (e), and (f) correspond, respectively, to equal deposition rates, preferential dimer site rule, and different deposition rates. Under the same conditions as for the coverage study, some particular aspects can be observed. The dimers rate of adsorption, for instance, starts at a value of two for $k_d=1$ since dimers occupy two sites at each deposition. The monomers rate of adsorption starts at zero, and increases as it requires previously adsorption of, at least, one particle. The rate of monomers adsorption increases due to the large influence of the substrate coverage and reaches a maximum when the number of isolated empty sites start to decrease. Exact results are shown for each plot, consistent with the ones obtained with Monte Carlo simulations.\n\n Since desorption is neglected, a jamming limit is obtained where no further particles can be adsorbed. In Fig.~\\ref{fig.coverage_varrates}, we see the coverage in the jamming limit $\\theta_{\\infty}$ as a function of the ratio between dimers and monomers deposition rates, $R=k_d\/k_m$, where the solid line is the analytical solution, open circles are dimers, and full squares monomers. It can be observed that in the limit of $R\\ll1$, a complete coverage of monomers is found, except for one dimer that always need to be adsorbed to start the monomers deposition. With the decrease in the coverage of monomers an increase in the coverage of dimers is observed, where equal coverage is reached for a ratio $R=0.207\\pm0.006$. In the limit of $R\\gg1$, a maximum coverage of dimers is obtained in agreement with the classical adsorption of dimers in a one-dimensional lattice \\cite{Evans1993a}. Monte Carlo results are also in agreement with the analytical solution.\n\n \\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=8cm]{coverage_total_varrates.pdf}\\\\\n \\caption{Coverage of dimers (open circles) and monomers (full squares) for Monte Carlo simulations and obtained analytically (solid line), with the analytical solution (solid line), at the jamming limit as a function of the ratio between the rate of deposition of dimers and monomers $R$ for the one-dimensional case (two dimensions Monte Carlo results on the inset).}\n \\label{fig.coverage_varrates}\n \\end{center}\n \\end{figure}\n\n\\section{2D Monte Carlo Simulations}\\label{sec:2d_results}\n\n \\begin{figure*}[t]\n \\begin{center}\n \\begin{tabular}{cc}\n \\includegraphics[width=8cm]{percolation_clean.pdf} & \\includegraphics[width=8cm]{percolation_threshold_D0.pdf} \\\\\n \\end{tabular}\n \\caption{a) Dependency on the ratio $R$ of the spanning probability $R_L$ for dimers (open) and monomers (full). Square lattices have been considered with $128^2$ (squares), $256^2$ (circles), and $512^2$ (triangles) lattice sites, for the spanning probability $R_L$ and fraction of sites belonging to the largest cluster $P_\\infty$ (inset). b) Percolation threshold ($R_c$) as a function of the system sizes, for linear sizes of $L=\\{32,64,128,256,512\\}$, for dimers (full circles) and monomers (full squares).}\n \\label{fig.percolation_clean}\n \\end{center}\n \\end{figure*}\n\n \\begin{figure*}[t]\n \\begin{center}\n \\begin{tabular}{cc}\n \\includegraphics[width=8cm]{percolation_pinf_D0.pdf} & \\includegraphics[width=8cm]{L1024_correlation.pdf}\\\\\n \\end{tabular}\n \\caption{a) Largest cluster $s_{max}$ and (inset) second moment of the cluster size distribution function $M_2$ at $R_c$ as a function of the system sizes, for linear sizes of $L=\\{32, 64, 128, 256, 512\\}$, for dimers (full circles), and monomers (full squares). Fractal dimension for monomers and dimers of $D_m=1.898\\pm 0.008$ and $D_m=1.890\\pm 0.009$ respectively. b) correlation function $g(r)$ for dimers (monomers on the inset) for a system of linear size $L=1024$ and averaged over 100 samples with power-law exponent of $\\eta=0.2101\\pm 0.0002$ ($\\eta=0.1361\\pm 0.0002$).}\n \\label{fig.percolation_pinf}\n \\end{center}\n \\end{figure*}\n\n In two dimensions, even for the simplest case of dimer adsorption, no analytical solution have been found. However, it is a case of interest, specially the regular square lattice which reproduces, for example, the topology of the Pt(100) surface. In this section, we study the proposed model on a square lattice through Monte Carlo simulations. We devote special attention to the percolation properties of aggregates of monomers and dimers \\cite{Rampf2002}.\n\n Monte Carlo simulations have been performed on square lattices of linear sizes $L=\\{128,256,512\\}$ in units of lattice sites, with periodic boundary conditions in both directions. Results have been averaged over $10^4$ samples. To decrease the computational effort, a rejection free algorithm was implemented, where the next adsorption trial takes place on an empty site randomly selected from a list of available sites, where the weight of each configuration is properly taken into account. To accurately follow the time evolution, the entire population of events is considered as well as the rate of monomers and dimers adsorption.\n\n Simulations have been performed on both clean and impurities-covered substrates. Impurities are considered quenched and to solely influence the adsorption process by purely geometrical restrictions.\n\n \\subsection{Clean Substrate}\\label{sec:clean}\n On a clean substrate, the coverage of dimers is larger than the coverage of monomers and the rate of adsorption of monomers as a function of time has a maximum, as also observed in the one-dimensional case. However, the coverage of monomers is favored in two dimensions since each deposited dimer has a greater influence on monomer deposition than in one dimension, mainly due to a larger fraction of configurations with occupied and empty neighboring sites. This can be observed on the inset of Fig.~\\ref{fig.coverage_varrates}, where we plot the jamming limit $\\theta_{\\infty}$ as a function of the ratio $R$. In this case, the point of equal coverage for dimers and monomers occurs for a ratio $R\\approx0.6$, which is larger than in one dimension, corroborating that monomers are favored.\n\n \\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=8cm]{coverage_diff_varrates_2d.pdf}\\\\\n \\caption{Difference in the jamming coverage of monomers and dimers as a function of the impurities coverage, in 2D, for $R=0.1$ (open circles), $R=1$ (full squares), and $R=10$ (full circles).}\n \\label{fig.coverage_diff_varrates_2d}\n \\end{center}\n \\end{figure}\n\n The percolation properties are analyzed by identifying clusters with the Hoshen-Kopelman algorithm \\cite{Hoshen1976}. While for lower values of R the system is dominated by monomers, as discussed for 1D, for higher values of R dimers dominate. Percolation of monomers or dimers is then observed with R as a control parameter. In Fig.~\\ref{fig.percolation_clean}(a) we plot the spanning probability $R_L$ defined as the probability of having a percolation cluster touching opposite borders of the lattice. At the percolation transition of both monomers and dimers, the fraction of sites occupied by the specie under study is compatible with the percolation threshold for site percolation in the considered topology \\cite{Stauffer1994}. In the inset of Fig.~\\ref{fig.percolation_clean}(a) is the fraction of sites belonging to the largest cluster $P_{\\infty}$ (the order parameter of the percolation transition).\n\n The percolation threshold $R_c$ can be estimated analyzing the maximum value on the standard deviation of the spanning probability. It can be observed in Fig.~\\ref{fig.percolation_clean}(b), that the percolation threshold scales linearly with the inverse of the lattice lateral size, L. Obtaining for dimers $R_c (L_{\\infty})=0.98\\pm0.01$, and for monomers $R_c(L_{\\infty})=0.41\\pm0.01$.\n\n The scaling of the average size of the largest cluster $\\langle s_{max}\\rangle $ at $R_c$ is in Fig.~\\ref{fig.percolation_pinf}(a) and scales as $\\sim L^{D_f}$, where $D_f$ is the fractal dimension. For both monomers and dimers the obtained scaling for the mass of the largest cluster is consistent with the fractal dimension $D_f=91\/48=1.8958$ of the classical percolation universality class \\cite{Stauffer1994}. \n\n \\begin{figure}[t]\n \\begin{center}\n \\includegraphics[width=8cm]{percolation_imp_both.pdf} \\\\\n \\caption{Wrapping probability as a function of the ratio $R$, in 2D, for a) monomers with impurities coverage from right to left of 0\\%, 10\\%, 20\\%, 30\\% and, 40\\%, and b) dimers with impurities coverage from left to right of 0\\%, 10\\%, 20\\%, and 30\\%. Simulation of a system size of $512^2$. Fraction of sites belonging to the largest cluster $P_\\infty$ on the inset.}\n \\label{fig.percolation_imp}\n \\end{center}\n \\end{figure}\n\n Another important parameter to be taken into account is the second momentum of the cluster-size distribution given by,\n \\begin{equation}\n M_2=\\frac{\\sum_{i\\neq max} s_i^2}{N},\n \\label{eq.m2}\n \\end{equation}\nwhere $s$ is the cluster size, $N$ is the total number of lattice sites, and the sum runs over all clusters excluding the largest one. The variable $M_2$ at $R_c$ as a function of the system size is in the inset of Fig.~\\ref{fig.percolation_pinf}(a), where another scaling behavior is observed consistent with $M_2\\sim L^{\\frac{\\gamma}{\\nu}}$, with $\\frac{\\gamma}{\\nu}=1.80\\pm 0.02$, in agreement with the scaling relation $\\frac{\\gamma}{\\nu}=2D_f-d$.\n\n We measured the correlation function, also known as connectivity correlation function, defined as\n \\begin{equation}\n g(r)=\\langle \\delta_{ij}\\rangle -\\frac{s_{max}^2}{N^2},\n \\label{eq.correlation}\n \\end{equation}\nwhere $\\delta_{ij}$ is 1 if both sites $i$ and $j$ are occupied by the same cluster and zero otherwise and $s_{max}$ is the size of the largest cluster. Figure~\\ref{fig.percolation_pinf}(b) shows that at $R_c$ both correlation functions are power laws with an exponent consistent with the one for random percolation \\cite{Stauffer1994}.\n\n \n\n \\subsection{Substrate with impurities}\\label{sec:impurity}\n\n To account for the presence of impurities (e.g., Pb atoms \\cite{Spencer1983}), quenched impurities are randomly distributed on the substrate. These impurities do not react and remain immobile, influencing only the adsorption, as an occupied site, which promotes the adsorption of monomers.\n \n Since impurities geometrically favor the coverage of monomers, the value of $\\theta_m-\\theta_d$ as a function of impurities coverage $\\theta_{imp}$ is plotted in Fig.~\\ref{fig.coverage_diff_varrates_2d}. A maximum is observed at a specific value of the coverage by impurities. The position at which the maximum occurs depends on the ratio $R$; low values of $R$ favor monomers leading to an earlier maximum while a high ratio disfavors monomers, thus, shifting the position of the maximum to larger values. These results open up the possibility of tuning the production of monomers by controlling the fraction of impurities.\n\n Additionally, impurities also shift the percolation threshold. Figures~\\ref{fig.percolation_imp}(a)~and~(b), show the spanning probability $R_L$ as a function of the ratio R for different values of impurities coverage. The monomers percolation transition is mainly affected at larger impurities coverage and is shifted to lower values of $R$. The dimer percolation transition, on the other hand, is shifted to higher values of $R$. At higher values of impurities coverage, the clusters of impurities predominate on the surface and neither monomers nor dimers percolate. The insets of Figs.~\\ref{fig.percolation_imp}(a)~and~(b) show the fraction of sites belonging to the largest cluster $P_{\\infty}$ as a function of $R$, with $P_{\\infty}=\\frac{\\langle s_{max}\\rangle }{N\\left(1-\\theta_{imp}\\right)}$, where $N$ is the total number of lattice sites, and $\\theta_{imp}$ is the coverage of impurities. In the case of monomers, as disclosed by the behavior of the spanning probability, the size of the largest cluster is only significantly affected by impurities for values of impurities coverage above 30\\%. In the case of dimers, impurities have a larger effect on $R_L$ and $P_{\\infty}$.\n\n\\section{Final remarks}\\label{sec:conclusion}\n\n We introduced a model based on random sequential adsorption of monomers and dimers, representing, acetyl and cleaved products, respectively, in the low desorption limit.\n The kinetic rules based on recent results by Wang and Liu are dependent on the local configuration provided by the neighboring adsorbed sites instead of configuration-independent rates.\n\n The properties of the model were studied in the 1D lattice and also extended to a 2D square lattice. In the latter case, the model describes the mechanisms of ethanol electro-oxidation on a Pt(100) surface, suggested by Wang and Liu \\cite{Wang2008a}. \n In 1D, we have analytically solved the model in three different cases: same deposition rate for dimer and monomer adsorption, preferential dimer site adsorption, and different deposition rates. Monte Carlo simulations are in agreement with the analytical solution. \n In 2D we have studied the jammed state and percolation transition through Monte Carlo simulations. The percolation properties of the adsorbed species reveal that, while monomers percolate at low ratios of dimers\/monomers deposition rate, dimers percolate at high ratios. The influence of impurities has also been monitored, disclosing that the coverage of monomers is significantly improved by their presence.\n\n In the present work, we restricted ourselves to study a system as simplified as proposed in the Introduction. One can clearly devise extensions to the basic model to include desorption and reaction pathways not included in the present paper.\n Molecular dynamics studies could, in principle, provide a more complete picture of the particles arrangements on the surface, but at a shorter timescale.\n It would certainly be interesting to study the atomistic mechanisms based on cooperative thermal effects that could affect some of the reaction pathways.\n \n \\acknowledgments{\n The authors are thankful to Fundac\\~ao para a Ci\\^encia e a Tecnologia fellowships (CD SFRH\/BD\/31833\/2006, AC SFRH\/BPD\/3475\/2007) and the warm hospitality of the T-1 group at the Los Alamos National Laboratory during the tenure of the fellowships. Work at Los Alamos National Laboratory (LANL) was supported by United States Department of Energy (U.S. DOE). LANL is operated by Los Alamos National Security, LLC for National Nuclear Security Administration of U.S. DOE under Contract No. DE-AC52-06NA25396.\n AC acknowledges useful comments by Neil Henson on the early stages of this work.}\n\n \\bibliographystyle{apsrev}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzeavm b/data_all_eng_slimpj/shuffled/split2/finalzzeavm new file mode 100644 index 0000000000000000000000000000000000000000..f15239184147c46134a66d6fe41e2488b0bde535 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzeavm @@ -0,0 +1,5 @@ +{"text":"\\section*{Abstract}\nDetermining how synaptic coupling within and between regions is modulated during sensory processing is an important topic in neuroscience. Electrophysiological recordings provide detailed information about neural spiking but have traditionally been confined to a particular region or layer of cortex. \nHere we develop new theoretical methods to study interactions between and within two brain regions, based on experimental measurements of spiking activity simultaneously recorded from the two regions. By \nsystematically comparing experimentally-obtained spiking statistics to (efficiently computed) model spike rate statistics, we identify regions in model parameter space that are consistent with the experimental \ndata. We apply our new technique to dual micro-electrode array \\textit{in vivo} recordings from two distinct regions: olfactory bulb ({\\bf OB}) and anterior piriform cortex ({\\bf PC}). \nOur analysis predicts that: i) inhibition within the afferent region (OB) has to be weaker than the inhibition within PC, ii) excitation from PC to OB is generally stronger than excitation from OB to PC, \niii) excitation from PC to OB and inhibition \nwithin PC have to both be relatively strong compared to presynaptic inputs from OB. \nThese predictions are validated in a spiking \nneural network model of the OB--PC pathway that satisfies the many constraints from our experimental data. \nWe find when the derived relationships are violated, the spiking statistics no longer satisfy the constraints from the data. In principle this modeling framework can be adapted to other systems and be used to investigate \nrelationships between other neural attributes besides network connection strengths. \nThus, this work can serve as a guide to further investigations into the relationships of various neural attributes within and across different regions during sensory processing.\n \n\n\\section*{Author Summary}\n\n\nSensory processing is known to span multiple regions of the nervous system. \nHowever, electrophysiological recordings during sensory processing have traditionally been limited to a single region or brain layer. \nWith recent advances in experimental techniques, recorded spiking activity from multiple regions simultaneously is feasible. However, other important quantities--- such as inter-region connection strengths --- \ncannot yet be measured. \nHere, we develop new theoretical tools to leverage data obtained by recording from two different brain regions simultaneously. We address the following questions: \nwhat are the crucial neural network attributes that enable sensory processing across different regions, and how are these attributes related to one another? \n\nWith a novel theoretical framework to efficiently calculate spiking statistics, we can characterize a high dimensional parameter space that satisfies \ndata constraints. We apply our results to the olfactory system to make specific predictions about effective network connectivity. \nOur framework relies on incorporating relatively easy-to-measure quantities to predict hard-to-measure interactions across multiple brain regions. \nBecause this work is adaptable to other systems, we anticipate it will be a valuable tool for analysis of other larger scale brain recordings. \n\n\n\\section*{Introduction}\n\nAs experimental tools advance, measuring whole-brain dynamics with single-neuron resolution becomes closer to reality~\\cite{prevedel14,ahrens13,lemon15,brainInit_13}. \nHowever, a task that remains technically elusive is to measure the interactions within and across brain regions that govern such system-wide dynamics. \nHere we develop a theoretical approach to elucidate such interactions based on easily-recorded properties such as mean and (co-)variance of firing rates, when they can be measured in multiple regions and in multiple activity states. \nAlthough previous theoretical studies have addressed how spiking statistics depend on various mechanisms~\\cite{brunel,brunelhakim,doiron16,BarreiroLy_RecrCorr_17}, these studies have typically been limited to a single region, \nleaving open the challenge of how inter-regional interactions impact the system dynamics, and ultimately the coding of sensory signals ~\\cite{zohary94,bair01,ecker11,moreno14,kohn16}.\n\nAs a test case for our new theoretical tools, we studied interactions in the olfactory system. We used two micro-electrode arrays to simultaneously record from olfactory bulb ({\\bf OB}) and anterior piriform cortex ({\\bf PC}). \nConstrained by these experimental data, we developed computational models and theory to investigate interactions within and between OB and PC. \nThe modeling framework includes two distinct regions: a \nnetwork that receives direct sensory stimuli (here, the OB), and a second neural network (PC) that is reciprocally coupled to the afferent region. Each region contains multiple individual populations, each of which is modeled with a firing rate model~\\cite{wilsoncowan1}; thus even this minimal model involves several coupled stochastic differential equations (here, six) and has a large-dimensional parameter space.\nAnalysis of this system would be unwieldy in general; we address this by developing a novel method to compute firing statistics that is computationally efficient, captures the results of Monte Carlo simulations, and can provide analytic insight. \n\nThorough analysis of experimental data in both the spontaneous and stimulus-evoked states leads to a number of constraints on first- and second-order spiking statistics--- many of which could not be observed using data \nfrom just one micro-electrode array. In particular, we find twelve (12) constraints that are consistent across different odorant stimuli. We use our theory and modeling to study an important subset of neural attributes (synaptic strengths) \nand investigate what relationships, if any, must be satisfied in order to robustly capture the many constraints observed in the data. We find that: i) inhibition within OB has to be weaker than the inhibition in PC, \nii) excitation from PC to OB is generally stronger than excitation from OB to PC, iii) excitation from PC to OB and inhibition within PC have to both be relatively strong compared to inputs originating in OB (inhibition within OB and excitation from OB to PC). \nWe validate these guiding principles in a large spiking neural network (leaky integrate-and-fire, or {\\bf LIF}) model,\nby showing that the many constraints from the experimental data are {\\it all} satisfied. \nFinally, we demonstrate that violating these relationships in the LIF model results in spiking statistics that do not satisfy all of the data constraints.\n\n\nOur predictions provide insights into interactions in the olfactory system that are difficult to directly measure experimentally. Importantly, these predictions were inferred from spike rates and variability, which are relatively easy to measure. \nWe believe that the general approach we have developed -- using easy-to-measure quantities to predict hard-to-measure interactions -- will be valuable in diverse future investigations of how whole-brain function \nemerges from interactions among its constituent components.\n\n\n\\section*{Results}\n\nOur main result is the development of a theoretical framework to infer hard-to-measure connection strengths in a minimal firing rate model, constrained by spike count statistics from \nsimultaneous array recordings. \n\nWe performed simultaneous dual micro-electrode recordings in the olfactory bulb ({\\bf OB}) and the anterior piriform cortex ({\\bf PC})\n\n(see {\\bf Materials and Methods}). First, we use the experimental data to \ncompute population-averaged (across cells or cell pairs) first and second order spike count statistics, \ncomparing across regions (OB or PC) and activity states (spontaneous or stimulus-evoked). \nWe use these statistics to \nconstrain a minimal firing rate model of the coupled OB-PC system, aided by a quick and efficient method for calculating firing statistics without Monte Carlo simulations. \n\nAs a test case for our methods, we investigate the structure of four important parameters: within-region inhibitory connection strengths and between-region excitatory connection strengths. We find several relationships that must hold, in\norder to satisfy all constraints from the experimental data. These results are then validated with a large \nspiking network of leaky integrate-and-fire ({\\bf LIF}) model neurons. \n\n\\subsection*{Consistent Trends in the Experimental Data}\n\n\\begin{figure}[!h]\n\\centering\n\t\\includegraphics[width=5.25in]{Fig1-eps-converted-to.pdf}\n\\caption{{\\bf Population firing rates in anterior piriform cortex (PC) and olfactory bulb (OB) from simultaneous dual array recordings.}\n(A) Trial-averaged population firing rate in time from 73 PC cells (38 and 35 cells from two recordings). The inset shows a closeup view, to highlight the distinction between spontaneous and evoked states. \n(B) Trial-averaged population firing rate in time from 41 OB cells (23 and 18 cells from two recordings). Inset as in (A); both (A) and (B) use 5\\,ms time bins. \n(C) The PC firing rate (averaged in time and over trials) of individual cells in the spontaneous (black) and \nevoked states (red). The arrows indicate the mean across 73 cells; the mean$\\pm$std. dev. in the spontaneous state is: $0.75\\pm 0.93$\\,Hz, in the evoked state is: $1.5\\pm1.6$\\,Hz. \n(D) Similar to (C), but for the OB cells described in (B). The mean$\\pm$std. dev. in the spontaneous state is: $2\\pm3.3$\\,Hz, in the evoked state is: $4.7\\pm7.1$\\,Hz.}\n\\label{fig1}\n\\end{figure}\n\nWe first present our data from simultaneous dual micro-electrode array recordings in anesthetized rats. \nDuring each 30-second trial an odor was presented for roughly one second; recordings continued for a total of 30 seconds. \nThis sequence was repeated for 10 trials with 2-3 minutes in between trials; the protocol was repeated for another odor. \nRecordings were processed to extract single-unit activity; the number of units identified was: 23 in OB and 38 in PC (first recording, two odors), 18 in OB and 35 in PC (second recording, another two odors). In total, there were four\ndifferent odors presented.\n\nIn this paper, we focus on the spike count statistics rather than the detailed temporal structure of the neural activity (Fig~\\ref{fig1}A--B). We divided each 30\\,s trial into two segments, representing the odor-{\\bf evoked} state (first 2 seconds) and the {\\bf spontaneous} state (remaining 28 seconds). We computed first- and second-order statistics for identified units; i.e., firing rate \n$\\nu_k$, spike count variance, and spike count covariance (we also computed two derived statistics, Fano Factor and Pearson's correlation coefficient, for each cell or cell pair). Spike count variances, covariances and correlations were computed \nusing time windows $T_{win}$ ranging between 5\\,ms and 2\\,s. In computing population statistics we distinguished between different odors (four total), different regions (OB vs. PC), and different activity states (spontaneous vs. evoked); \notherwise, we assumed statistics were stationary over time.\n\nWe then sought to identify relationships among these standard measures of spiking activity. For example, we found that mean firing rate of OB cells in the evoked state was higher than the mean firing rate in the spontaneous state, or $\\nu_{OB}^{Ev} > \\nu_{OB}^{Sp}$ (although there is significant variability across the population, we focus on population-averaged statistics here).\nWe found twelve (12) robust relationships \nthat held across all odors. \nTable~\\ref{table:constr} summarizes the consistent \nrelationships we found in our data, and Fig~\\ref{fig1}C--D, Fig~\\ref{fig2}, Fig~\\ref{fig3} show the data exhibiting these relationships when combining all odorant stimuli (see \\nameref{S1_file} for statistics plotted by distinct odors). \nThroughout the paper, when comparing activity states the spontaneous state is in black and the evoked state in red; when comparing regions the OB cells are in blue and PC cells in green. \n\n\n\\begin{figure}[!h]\n\\centering\n\t\\includegraphics[width=5.25in]{Fig2-eps-converted-to.pdf}\n\\caption{{\\bf A subset of the important relationships between the spiking statistics in spontaneous and evoked states.}\nConsistent trends that hold for {\\it all} 4 odorant stimuli in the experimental data. Each panel shows two spike count statistics, as a function of the time window. \nThe shaded error bars show the \\textit{standard error of the mean} above and below the mean statistic.\n(A) Stimulus-induced decorrelation of PC cell pairs (red) compared to the spontaneous state (black).\n(B) The variability in PC (measured by Fano Factor) is lower in the evoked state (red) than in the spontaneous state (black). \n(C) In the spontaneous state, the average correlation of PC pairs (green) is \\textit{higher} than that of OB pairs (blue).\n(D) In the evoked state, the average correlation of PC pairs (green) is \\textit{lower} than that of OB pairs (blue), for long time windows. There were 406 total OB pairs and 1298 total PC pairs. (Although the trends reverse in (A) and (D) for smaller time windows, our focus is on the larger time windows because stimuli were held for 1\\,s; smaller time windows are shown for completeness.)}\n\\label{fig2}\n\\end{figure}\n\n\n\\begin{table}[!ht]\n\\centering\n\\caption{ {\\bf The 12 relationships (constraints) that hold in the experimental data across all odors.}}\n\\label{table:constr}\n\\begin{tabular}{l|l|l|l|}\n\\cline{2-4}\n & \\textbf{Spont.} & \\textbf{Evoked} & \\textbf{Spon. to Evoked} \\\\ \\thickhline\n\\multicolumn{1}{|l|}{} & & & $\\nu_{PC}^{Sp}<\\nu_{PC}^{Ev}$ \\\\ \\cline{4-4} \n\\multicolumn{1}{|l|}{\\multirow{-2}{*}{\\textbf{Firing Rate}}} & \\multirow{-2}{*}{$\\nu_{PC}<\\nu_{OB}$} & \\multirow{-2}{*}{$\\nu_{PC}<\\nu_{OB}$} & $\\nu_{OB}^{Sp}<\\nu_{OB}^{Ev}$ \\\\ \\hline\n\\multicolumn{1}{|l|}{} & \\cellcolor[HTML]{C0C0C0} & $\\text{Var}_{PC}<\\text{Var}_{OB}$ & $\\text{Var}_{OB}^{Sp}<\\text{Var}_{OB}^{Ev} $ \\\\ \\cline{2-4} \n\\multicolumn{1}{|l|}{\\multirow{-2}{*}{\\textbf{Variability}}} & $FF_{PC}>FF_{OB}$ & \\cellcolor[HTML]{C0C0C0} & $FF_{PC}^{Sp}>FF_{PC}^{Ev}$ \\\\ \\hline\n\\multicolumn{1}{|l|}{} & \\cellcolor[HTML]{C0C0C0} & $\\text{Cov}_{PC}<\\text{Cov}_{OB}$ & \\cellcolor[HTML]{C0C0C0} \\\\ \\cline{2-4} \n\\multicolumn{1}{|l|}{\\multirow{-2}{*}{\\textbf{Co-variability}}} & $\\rho_{PC}>\\rho_{OB}$ & $\\rho_{PC}<\\rho_{OB}$ & $\\rho_{PC}^{Sp}>\\rho_{PC}^{Ev}$ \\\\ \\hline\n\\end{tabular}\n\\begin{flushleft} Relationships between population-averaged statistics (averages are across all cells or cell pairs) that were consistent across all odors. Other possible relationships were left out because they were \nambiguous and\/or odor dependent.\n\\end{flushleft}\n\\end{table}\n\nA common observation across different animals and sensory systems, is that firing rates increase in the evoked state (see, for example, Figure 3 in~\\cite{churchland10}). \nIndeed, we observed that \naverage firing rates in both the OB and PC were higher in the evoked state than in the spontaneous state (Fig~\\ref{fig1}C--D). \nFurthermore, the firing rate in the OB was larger than the firing rate in the PC, in both spontaneous and evoked states (see mean values in Fig~\\ref{fig1}C--D). \n\n\\textit{Stimulus-induced decorrelation} appears to be a widespread phenomena in \nmany sensory systems and in many animals~\\cite{doiron16}; stimulus-induced decorrelation was previously reported in PC cells under different experimental conditions~\\cite{miura12}.\nHere, we found that in the PC, the average spike count correlation is lower in the evoked state (red) than in the spontaneous state (black), at least for time windows of 0.5\\,s and above (Fig~\\ref{fig2}A). \nAlthough we show a range of time windows for completeness, \nwe focus on the larger time windows because in our experiments the odors are held for 1\\,s; furthermore, our theoretical methods only address long time-averaged spiking statistics. \nNote that stimulus-induced decorrelation in the OB cells was not consistently observed across odors. \n\nAnother common observation in cortex, is for variability to decrease at the onset of stimulus~\\cite{churchland10}: \nin Fig~\\ref{fig2}B we see that the Fano Factor of spike counts in PC cells decreases in the evoked state (red) compared to the spontaneous state (black); note that other experimental labs have also\nobserved this decrease in the Fano factor of PC cells (see supplemental figure S6D in~\\cite{miura12}). \nFig~\\ref{fig2}C--D shows a comparison of PC and OB spike count correlation in the spontaneous state and evoked state, respectively. Spike count correlation in PC (green) \nis larger than correlation in OB (blue) in the spontaneous state, but in the evoked state the relationship switches, at least for time windows larger than 0.5\\,sec. \n\n\\begin{figure}[!h]\n\\centering\n \\includegraphics[width=5.25in]{Fig3-eps-converted-to.pdf}\n\\caption{\\label{fig3} Showing the other trends from the experimental data that are consistent with all odors and for all time windows. The shaded error bars show the \\textit{standard error of the mean} above and below the mean statistic.\n(A) Fano Factor of spontaneous activity is larger in PC (green) than in OB (blue). (B) The spike count variance in the evoked state is smaller in PC (green) than in OB (blue). \n(C) Spike count covariance in the evoked state is smaller in PC (green) than in OB (blue). (D) In OB cells, the evoked spike count variance (red) is larger than the spontaneous (black). \nThe number of cells and number of pairs are the same as in Fig~\\ref{fig2}. Throughout we scale spike count variance and covariance by time window $T$ for aesthetic reasons. \n} \n\\end{figure}\n\nFig~\\ref{fig3} shows the four remaining constraints that are consistent for all odors and for all time windows. The Fano Factor in PC (green) is larger than in OB (blue), in the spontaneous state (Fig~\\ref{fig3}A); spike count variance in PC (green) \nis smaller than in OB (blue) in the evoked state (Fig~\\ref{fig3}B); spike count covariance in PC (green) is smaller than in OB (blue) in the evoked state (Fig~\\ref{fig3}C); and in OB the spike count variance in the evoked state (red) is larger than spontaneous (black, Fig~\\ref{fig3}D). \nThroughout the paper, we scale the spike count variance and covariance by time window for aesthetic reasons; \nthis does not affect the relative relationships.\n\n\\subsection*{A Minimal Firing Rate Model to Capture Data Constraints}\n\nWe model two distinct regions (OB and PC) with a system of six (6) stochastic differential equations, each representing the averaged activity of a neural \npopulation~\\cite{wilsoncowan1} or representative cell (see Fig~\\ref{fig4} for a schematic of the network). \nFor simplicity, in this section we use the word ``cell\" to refer to one of these populations. Each region has two excitatory ({\\bf E}) and one inhibitory ({\\bf I}) cell to account for a variety of spiking correlations. \n\nWe chose to include two E cells for two reasons: first, excitatory cells are the dominant source of projections between regions; we need at least two E cells to compute an E-to-E correlation. Moreover, in our experimental data, we are most likely \nrecording from excitatory mitral and tufted cells (we do not distinguish between mitral vs tufted here, and therefore refer to them as M\/T cells); therefore, the experimental measurements of correlations are likely to have many E-to-E correlations. \nThe arrays likely record from I cell spiking activity as well, and the inclusion of the I cell is also important for capturing the stimulus-induced decreases in correlation and Fano factor~\\cite{churchland10,doiron16} \n(also see~\\cite{LMD_whisker_12} who similarly used these same cell types to analyze spiking correlations in larger spiking network models).\n\nWe use $j\\in\\{1,2,3\\}$ to denote three OB ``cells\" and $j\\in\\{4,5,6\\}$ for three PC cells, with $j=1$ as the inhibitory OB granule cell and $j=4$ as the inhibitory PC cell.\nThe equations are:\n\\begin{eqnarray}\\label{eqn:gen_WC_pop}\n\t\\tau \\frac{d x_j}{dt} & = & -x_j + \\mu_j + \\sigma_j \\eta_j + \\sum_k g_{jk} F(x_k) \\label{eqn:dxdt_SDE}\n\\end{eqnarray}\nwhere $F(x_k)$ is a transfer function mapping activity to firing rate. Thus, the firing rate is:\n\\begin{equation}\n\t\\nu_j = F(x_j). \n\\end{equation}\nWe set the transfer function to $F(X)=\\frac{1}{2}\\left(1+\\tanh((X-0.5)\/0.1) \\right)$, a commonly used sigmoidal function~\\cite{wilsoncowan1} for all cells; experimental recordings of this function demonstrate it can be sigmoidal~\\cite{fellous03,prescott03,cardin08}. \nAll cells receive noise $\\eta_j$, the increment of a Weiner process, uncorrelated in time but correlated within a region: i.e. $\\langle \\eta_j(t) \\rangle = 0$, $\\langle \\eta_j(t) \\eta_j(t+s) \\rangle = \\delta(s)$, \nand $\\langle \\eta_j(t) \\eta_k(t+s) \\rangle = c_{jk} \\delta(s)$. We set $c_{jk}$ to:\n\\begin{eqnarray} c_{jk} = \\left\\{\n\\begin{array}{ll}\n 0, & \\hbox{if }j\\in\\{1,2,3\\}; k\\in\\{4,5,6\\} \\\\\n 1, & \\hbox{if } j=k \\\\\n c_{OB} & \\hbox{if }j\\neq k; j,k\\in\\{1,2,3\\} \\\\\n c_{PC} & \\hbox{if }j\\neq k; j,k\\in\\{4,5,6\\} \\\\\n\\end{array} \\right.\n\\end{eqnarray}\nThe parameters $\\mu_j$ and $\\sigma_j$ are constants that give the input mean and input standard deviation, respectively.\nWithin a particular region (OB or PC), all three cells receive correlated background noisy input, but there is {\\bf no} correlated background input provided to both PC and OB cells. This is justified \nby the experimental data (see Fig S9 in \\nameref{S2_file}); average pairwise OB-to-PC correlations are all relatively small, and in particular, less than pairwise correlations \\textit{within} the OB and PC. Furthermore, \nanatomically there are no known common inputs to both regions that are active at the same time. \n\nWe also set the background correlations to be higher in PC than in OB: i.e., \n$$ c_{PC} > c_{OB} . $$\nThis is justified in part by our array recordings, where correlated local field potential fluctuations are larger in PC than in OB. \nFurthermore, one source of background correlation is global synchronous activity; Murakami et al.~\\cite{murakami05} has demonstrated that state changes \n(i.e., slow or fast waves as measured by EEG) strongly affect odorant responses in piriform cortex but only minimally effect olfactory bulb cells. \nFinally, PC has more recurrent activity than the olfactory bulb; this\ncould lead to more recurrent common input, if not cancelled by inhibition~\\cite{renart10}.\n\n\\begin{figure}[!h]\n\\centering\n \\includegraphics[width=4.25in]{Fig4-eps-converted-to.pdf}\n\\caption{{\\bf Minimal firing rate model to analyze important synaptic conductance strengths.}\nA firing rate model (Wilson-Cowan) with background correlated noisy inputs is analyzed to derive principles relating these network attributes (see Eq~\\ref{eqn:gen_WC_pop} and {\\bf Materials and Methods} section). \nThis model only incorporates some of the anatomical connections that are known to exist {\\it and} are important for modulation of statistics of firing (see main text for further discussion). \nEach neuron within a region (OB or PC) receives correlated background noisy input with $c_{OB} gEO$, once the covariance constraints are omitted.\n\n\\subsection*{Generality of Firing Rate Model Predictions}\n\nIn general, we should expect that if we change the wiring diagram of our simple firing rate model (Fig~\\ref{fig4}), then the same experimental constraints might result in different predictions. \nThis could be a concern since our simple firing rate model is lacking many connections and cell types that exist in the real olfactory system~\\cite{oswald12}. \nHowever, we tested one alternative wiring diagram with different neurons receiving stimulus input, no E-to-I connections within OB, and no E-to-I connections within PC. \nOur predictions were robust to these changes. Second and most importantly, we tested whether our predictions held in a larger network of leaky integrate-and-fire neurons. \nThis spiking network model also had more realistic network connectivity, more closely mimicking known anatomy of real olfactory systems. \n\nThe following highlight the differences between the spiking model and the firing rate model:\n\\begin{itemize}\n\t\\item Include E-to-E connections from OB to PC (lateral olfactory tract). Also include strong E-to-I drive within PC because input from OB results in balanced \n\texcitation and inhibition in PC~\\cite{large16};\n\t\\item Remove the E-to-I connections from OB to PC ($gEO$ in the firing rate model) so that the recurrent activity in PC is driven by E inputs along the lateral olfactory tract;\n\t\\item Remove the direct sensory input to I cells in OB since granule cells do not receive direct sensory input~\\cite{oswald12};\n\t\\item Include substantial recurrent E-to-E connections within PC (see Table~\\ref{table:lif_parms} for strength relative to other connections). \n\\end{itemize}\n\nThe parameter $gEO$ will now refer to the strength of E-to-E connections, rather than E-to-I connections, from OB to PC. The next two sections demonstrate that our predictions hold for this LIF network model (also see \\nameref{S3_file}).\n\n\n\\subsection*{Results are Validated in a Spiking LIF Network}\n\nHere we show that a general leaky integrate-and-fire ({\\bf LIF}) spiking neuron model of the coupled OB-PC system can satisfy all 12 data constraints. \nRather than try to model the exact underlying physiological details of the olfactory bulb or anterior piriform cortex, \nour goal is to demonstrate that the results from the minimal firing rate model can be used as a guiding principle in a more realistic coupled \nspiking model with conductance-based synaptic input. The LIF model does not contain all of the attributes and cell types of the olfactory system, but is a plausible model that contains: \ni) more granule than M\/T cells in OB (a 4-to-1 ratio, comparable to the 3-to-1 ratio used in ~\\cite{grabska17}); \nii) E-to-E connections from OB to PC that drive the entire network within PC; iii) E-to-I (granule cell) feedback from PC to OB; iv) lack of \nsensory input to granule I cells in OB. \n\nWe also show that the minimal firing rate model results can be applied to a generic cortical-cortical coupled population (see ~\\nameref{S3_file}).\n\nWe set the four conductance strength values to:\n\\begin{eqnarray}\n\tgIO &=& 7\t\t\\nonumber \\\\\n\tgEO &=& \t10\t\\nonumber \\\\\n\tgIP &=& \t20\t\\nonumber \\\\\n\tgEP &=& \t15\t; \\label{def_gs_lif}\n\\end{eqnarray}\nSee Fig~\\ref{fig6} or Eq~\\ref{ob_lif}--\\ref{pc_lif} for exact definitions of $gXY$; these conductance strength values are dimensionless scale factors. \nThese values were selected to satisfy the relationships derived from the analysis of the rate model (see Fig~\\ref{fig4}). \nIn contrast to the minimal firing rate model, here the conductance values are all necessarily positive; an inhibitory reversal potential is used to capture the hyperpolarization that occurs upon receiving synaptic input. \n\nWith the conductance strengths in Eq~\\ref{def_gs_lif}, and other standard parameter values (see Table~\\ref{table:lif_parms}) in a typical LIF model, we were able to easily satisfy all 12 constraints: see \nTable~\\ref{table:frate_lif} and Fig~\\ref{fig6}. \n\n\\begin{figure}[!h]\n \\includegraphics[width=\\columnwidth]{Fig6-eps-converted-to.pdf}\n\\caption{{\\bf Detailed spiking LIF model confirms the results from analytic rate model.}\nSchematic of the LIF model with 2 sets of recurrently coupled E and I cells. There are 12 types of synaptic connections. \n(A) Pairwise correlations in PC, spontaneous vs. evoked: $\\rho_{PC}^{Sp}>\\rho_{PC}^{Ev}$. \n(B) Variability (Fano factor) in PC, spontaneous vs evoked: $FF_{PC}^{Sp}>FF_{PC}^{Ev}$. \n(C) Correlations in the spontaneous state, PC vs. OB: $\\rho_{PC}^{Sp}>\\rho_{OB}^{Sp}$.\n(D) Correlations in the evoked state, PC vs. OB: $\\rho_{PC}^{Ev}<\\rho_{OB}^{Ev}$. \n(E) Variability (Fano factor) in the spontaneous state, PC vs. OB: $FF_{PC}^{Sp}>FF_{OB}^{Sp}$. \n(F) Variability (Fano factor) in the evoked state, PC vs. OB: $\\text{Var}_{PC}^{Ev}<\\text{Var}_{OB}^{Ev}$ in evoked state. \n(G) Covariances in the evoked state, PC vs. OB: $\\text{Cov}_{PC}^{Ev}<\\text{Cov}_{OB}^{Ev}$.\n(H) Variability (spike count variance) in OB, spontaneous vs. evoked: $\\text{Var}_{OB}^{Sp} < \\text{Var}_{OB}^{Ev}$. \nThe curves show the average statistics over all $N_{OB\/PC}$ cells or over a large random sample of all possible pairs. \nSee {\\bf Materials and Methods} for model details, and Table~\\ref{table:lif_parms} and Eq~\\ref{def_gs_lif} for parameter values.\n}\n\\label{fig6}\n\\end{figure}\n\n\\begin{table}[!ht]\n\\centering\n\\caption{ {\\bf Population firing rate statistics from an LIF model of the OB--PC pathway. }}\n\\label{table:frate_lif}\n\\begin{tabular}{|c+c|c|}\n\\hline\n\\multicolumn{1}{|l|}{} & Mean Firing Rate (Hz) & Std. Dev. (Hz) \\\\ \\thickhline\n$\\nu_{OB}^{Sp}$ & 5.5 & 4.6 \\\\\n$\\nu_{OB}^{Ev}$ & 6.2 & 4.8 \\\\ \\hline\n$\\nu_{PC}^{Sp}$ & 2.1 & 2.6 \\\\\n$\\nu_{PC}^{Ev}$ & 4.1 & 5.8 \\\\ \\hline\n\\end{tabular}\n\\begin{flushleft} See {\\bf Materials and Methods} for model details, and Table~\\ref{table:lif_parms} and Eq~\\ref{def_gs_lif} for parameter values. The mean and standard deviations are across the heterogeneous population.\n\\end{flushleft}\n\\end{table}\n\nWhile the firing rates in the LIF network (Table~\\ref{table:frate_lif}) do not \\textit{quantitatively} match with the firing rates from the experimental data, a few \\textit{qualitative} trends are apparent: (i) the ratio of mean \nspontaneous to evoked firing rates are similar to that observed in experimental data, for both OB and PC, (ii) the same is true of the standard deviation, (iii) the ratio of the mean OB firing rate to PC firing rate is \nsimilar to what is observed in the experimental data, in both spontaneous and evoked states. Therefore, the LIF network captures the mean firing rates reasonably well. \n\n\nOne difference between the LIF spiking network and the minimal firing rate model is that in the evoked state, mean background input to \\textit{both} the OB and PC cells is increased, compared to the \nspontaneous state (recall that in the minimal firing rate model, only the mean input to the OB cells increased in the evoked state; this ensured that stimulus-induced changes in \nPC were due to network activity). When the mean input to the PC cells is the same in the spontaneous and evoked states, \n10 of the 12 constraints were satisfied -- the exception was the correlation of PC in the evoked state, which decreased but is still larger than the spontaneous correlation \n(see Fig. S13 in \\nameref{S2_file}). The reason is that as firing rates increase, the OB spiking is more variable and the synaptic input from OB to PC is noisier, so the input to PC activity is diffused. \n\nTo capture the final two constraints, we allowed mean input drive to PC to increase in the evoked state. \nThis has also been used in previous theoretical studies to achieve stimulus-induced decreases in spiking variability and co-variability~\\cite{litwin_nn_12}. \nChurchland et al.~\\cite{churchland10} used an extra source of variability in the spike generating mechanism, a doubly stochastic model, which was \nsimply removed with stimulus onset. \nThus, the mechanism we employ (increased mean input with lower input variability) is consistent with other studies that analyzed stimulus-induced changes in variability~\\cite{churchland10,litwin_nn_12}.\n\n\\subsubsection*{Results of Violating Derived Relationships Between Conductance Strengths}\n\nWhat happens in the full LIF spiking network when the derived relationships between the conductance strengths are violated? Since the minimal firing rate model is different than the \ndetailed spiking model in many ways, we do not expect the relationships between the conductance strengths to hold precisely. However, \nthe minimal firing rate model is still useful in providing intuition for what would otherwise be a complicated network \nwith a high-dimensional parameter space.\nWe now demonstrate that when the relationships derived in firing rate model are violated, \na subset of the constraints in the experimental data (Table~\\ref{table:constr}) will no longer be satisfied in the large spiking network.\n\nBecause our network is heterogeneous, our ability to subsample cell pairs is limited, relative to a homogeneous network of the same size. Also, computation for even a single \nparameter set in the spiking network require enormous computing resources. \nThus, we cannot exhaustively explore the parameter space; indeed, the purpose of the reduction method of the firing rate model is to probe large dimensions quickly. Instead, we perform three tests that violate the firing rate model results:\n\\begin{enumerate}\n\t\\item Make $gIO > g IP$ by setting $gIO=20$ and $gIP=7$.\n\t\\item Make $gEO > gEP$ by setting $gEO=15$ and $gEP=1$\n\t\\item Make $gEP$ and $gIP$ relatively smaller by setting $gEP=10$ and $gIP=10$\n\\end{enumerate}\nThe original values (used in Fig~\\ref{fig6}) for these parameters were given in Eq~\\ref{def_gs_lif}.\n\nThe result of Test 1 is that 2 of the 12 constraints are violated (see Fig S14 in \\nameref{S2_file}); most importantly stimulus-induced decorrelation of the PC cells, which is particularly important in the context of coding, was not present. \nIn addition,\nthe evoked PC correlation is larger than evoked OB correlation,\nviolating another constraint.\n\nThe result of Test 2 is that 3 of the 12 constraints are violated (see Fig S15 in \\nameref{S2_file}). \nThe evoked PC correlation is larger than evoked OB correlation, and \nboth the variance and covariance in PC are larger than the corresponding quantities in OB in the evoked state, which is not consistent with our data. \n\nThe result of Test 3 is that 3 of the 12 constraints are violated: they are the same constraints that are violated in Test 2, despite quantitative differences in the statistics (see Fig S16 in \\nameref{S2_file}). \nThe stimulus-induced decorrelation of the PC cells does not hold for small windows, but this is also observed in our data (Fig~\\ref{fig2}A), so we do not formally count this as a clear violation of data constraints. \nHowever, Test 1 and Test 3 show that strong PC inhibition is key for stimulus-induced decorrelation~\\cite{doiron16,diesmann12,middleton12,LMD_whisker_12,litwin12,litwin11,renart10}.\n \n\\section*{Discussion}\n\nAs electrophysiological recording technology advances, there will be more datasets with simultaneous recordings of neurons, spanning larger regions of the nervous system. \nSuch networks are inherently high-dimensional, making mechanistic analyses generally intractable without fast and reasonably accurate approximation methods. \nWe have developed a computational reduction method for a multi-population firing rate model~\\cite{wilsoncowan1} \n that enables analysis of the spiking statistics. Our work specifically enables theoretical characterizations of an important, yet hard-to-measure quantity -- synaptic connection strength -- using easy-to-measure \n spiking statistics. The method is computationally efficient, is validated with Monte Carlo simulations of spiking neural networks, and can provide insight into network structure. \n\nWe applied our computational methods to \nsimultaneous dual-array recordings in two distinct regions of the olfactory system: the olfactory bulb (OB) and anterior piriform cortex (PC). \nOur unique experimental dataset enables a detailed analysis of the first- and second-order spike count statistics in two activity states,\nand a comparison of how these \nstatistics are related between OB and PC cells. We found twelve (12) consistent trends that held across four odors in the dataset (Table~\\ref{table:constr}), and sought to identify \nwhat neural network attributes would account for these trends.\nWe focused on four important network attributes, specifically the conductance strengths in the following connections: feedforward inhibition within OB and within PC, excitatory projections from OB to PC neurons, and finally \nexcitatory projections from PC to OB. Our reduced firing rate model predicts several relationships that are then verified with a more detailed spiking network model, specifically: \ni) inhibition within the OB has to be weaker than the inhibition in PC, ii) excitation from PC to OB is generally stronger than excitation from OB to PC, \niii) connections that originate within PC have to relatively strong compared to connections that originate within OB.\nThese results make a strong prediction that to the best of our knowledge is new and might be testable with simultaneous patch-clamp recordings. \n\t\nIn principle our theory could be used to study the structure of other network features such as background correlation, noise level, transfer function, etc.. \nIt is straightforward mathematically to incorporate other desired neural attributes (with the caveat of perhaps increasing the overall number of equations and terms in the approximations) without changing the basic structure of the framework. \nHere we have focused on the role of the strength of synaptic coupling; \nof course, other neural attributes can affect spike statistics (in particular, spike count correlation~\\cite{cohen11,doiron16}), some of which can conceivably change with stimuli. \nSpike count correlations can depend on intrinsic neural properties~\\cite{hong12,marella08,abouzeid09,barreiro10,barreiro12,ocker14}, network architecture~\\cite{rosenbaum10,litwin_nn_12,rosenbaum17} \nand synaptic inputs~\\cite{renart10,diesmann12,litwin11,litwin12,middleton12,LMD_whisker_12} \n(or combinations of these~\\cite{ostojic09,ly_ermentrout_09,trousdale12,BarreiroLy_RecrCorr_17}), plasticity~\\cite{rosenbaum13}, as well as top-down mechanisms~\\cite{mitchell09,cohen09,ruff14}. Thus, correlation \nmodulation is a rich and deep field of study, and we do not presume our result is the only plausible explanation for spike statistics modulation.\n\nAlthough the minimal firing rate model did not include certain anatomical connections that are known to exist (e.g., recurrent excitation in the PC), \nthe model is meant for deriving qualitative principles rather than precise quantitative modeling of the \npathway. We based our simplifications on insights from recent experimental work: \nrecent slice physiology work \nhas shown that within PC, recurrent activity is dominated by inhibition~\\cite{large16}; previous \nwork has also shown that inhibitory synaptic events are much more common (than excitatory synaptic events) in PC and are much easier to elicit~\\cite{poo09}. Thus, the connection from excitatory OB cells to inhibitory PC cells ($gEO$ in Fig~\\ref{fig4}) \nshould be thought of as the net effect of these connections along the lateral olfactory tract. \nOther theoretical analyses of effective feedforward inhibitory networks have also neglected anatomical E-to-E connections~\\cite{middleton12,LMD_whisker_12}. Furthermore, this minimal model \nwas validated with a more realistic, recurrently coupled spiking network, which did include within-region excitatory connections (see Fig~\\ref{fig6} and Fig S14--S16 in \\nameref{S2_file}, as well as \\nameref{S3_file}).\n\n\nWe have only focused on first- and second-order firing statistics, even though in principle other, higher-order statistics may be important~\\cite{ohiorhenuan10,trousdale13,jovanovic16}. \nIf downstream neurons use a linear decoding scheme,\n then first- and second-order spiking statistics are sufficient in quantitative measures of neural coding~\\cite{kaybook,dayan2001theoretical}. \nIt is currently unknown whether downstream neurons decode olfactory signals with a nonlinear decoder, but there is evidence in other sensory systems that second-order statistics are sufficient~\\cite{kohn16}. \nRecent work has shown conflicting results for coding in olfactory bulb; one study found that decoding an odor in the presence of other odors might be more efficient using nonlinear decoding~\\cite{grabska17}, but another has shown that linear decoding is still plausible~\\cite{mathis16}. \n\nA second reason to neglect higher-order statistics is suggested by Fig. 5, where we show how the various data constraints narrow the scope of plausible models. Here, we saw that even with first and second- order statistics, only 1\\% of the parameter sets satisfy the data constraints; including more constraints would limit the space further. In order to usefully include higher-order constraints, we would need to use a more detailed model and\/or larger \nparameter spaces.\n\nAs a test case for our method, we used recordings from anesthetized animals. The absence of breathing in tracheotomized rats \nin these experiments is only an approximation to olfactory processing in awake animals. \nHowever, there is a benefit to tracheotomized animals: the \ncomplex temporal firing patterns are removed, \nso that firing statistics are closer to stationarity. \nIn principle, we can incorporate breathing dynamics into our framework by including an oscillatory forcing term in Eq~\\ref{eqn:gen_WC_pop}; this will be the subject of future work. \nIn support of this simplification, we note that there is evidence that in the anterior piriform cortex, spike count --- rather than the timing --- is most consequential for odor discrimination~\\cite{miura12}. \nHowever, other studies have reported that timing of the stimuli in the olfactory bulb is important:~\\cite{cury10,gschwend12,grabska17} \nshowed decoding performance is best at the onset of odors in mammals and worsens as time \nproceeds, whereas~\\cite{friedrich01} found that decoding performance improved with time in zebrafish. These important issues are beyond the scope of this current study.\n\n\\subsection*{Relationship to Other Reduction Methods}\n\nIn computing statistics for the minimal firing rate model, we only considered equilibrium firing statistics, in which a set of stationary statistics can be solved self-consistently. \nMore sophisticated methods might be used to address oscillatory firing statistics (see~\\cite{nlc_15} where the adaptive quadratic integrate-and-fire model was successfully analyzed with a reduced method); capturing the firing statistics in these other regimes is a potentially interesting direction of research. The limitation to steady-state statistics is not unique, but is shared by other approximation methods. \nSome methods are known to have issues when the system bifurcates~\\cite{buice07,buice10} because truncation methods can fail~\\cite{ly_tranchina_07}. \n\nSeveral authors have proposed procedures to derive population-averaged first- and second-order \nspiking statistics from the dynamics of single neurons. The microscopic dynamics in question may be given by a master equation ~\\cite{buice07,bressloff09,buice10,touboul11,bressloff15}, a generalized linear model \\cite{toyoizumi09,ocker2017}, or the theta model~\\cite{BC_JSM_2013,BC_PLOSCB_2013}. \n(Other authors have derived rate equations at the single-neuron level, by starting with a spike response model \\cite{aviel06} or by taking the limit of slow synapses~\\cite{ermentrout94}.) While we would ideally use a similar procedure to derive our rate equations, none of the approaches we note here is yet adapted to deal with our setting, a heterogeneous network of leaky integrate-and-fire neurons. Instead, we focused here on perturbing from a background state in which several populations \n(each population modeled by a single equation) \nreceive correlated background input but are otherwise uncoupled. \nThis allows us to narrow our focus to how spike count co-variability from common input is modulated by recurrent connections. \n\nWe also note that other recent works have used firing rate models to explain observed patterns of correlated spiking activity in response to stimuli. \nRosenbaum et al.~\\cite{rosenbaum17} have studied the spatial structure of correlation in primate visual cortex with balanced networks~\\cite{van96}; \nKeane \\& Gong~\\cite{keaneGong15} studied wave propagation in balanced network models. \n \n\n\n\n\n\\section*{Conclusion}\n\nDesigning a spiking neural network model of two different regions that satisfies the many experimental data constraints we have outlined is a difficult problem that would often be addressed \nvia undirected simulations. We have shown that \nsystematic analysis of a minimal firing rate model can yield valuable insights into the relative strength of unmeasured network connections. Furthermore, these insights are transferable to a more complex, physiologically realistic spiking model of the OB--PC pathway. \nIndeed, incorporating the relative relationships of the four conductance strengths resulted in spiking network models that satisfied the constraints. \nStrongly violating the relative relationships of these conductance strengths led to multiple violations of the data constraints. Because our approach can be extended to other network features, we are hopeful that\nthe general approach we have developed -- using easy-to-measure quantities to predict hard-to-measure interactions -- will be valuable in future investigations into how whole-brain function emerges from interactions among its constituent components.\n\n\n\n\\section*{Materials and Methods}\n\n\\subsection*{Electrophysiological Recordings}\n\n \n\n{\\bf Subjects.} All procedures were carried out in accordance with the recommendations in the Guide for the Care and Use of Laboratory Animals of the National Institutes of Health and approved by \nUniversity of Arkansas Institutional Animal Care and Use Committee (protocol \\#14049). Experimental data was obtained from one adult male rat (289\\,g ; \\textit{Rattus Norvegicus}, Sprague-Dawley \noutbred, Harlan Laboratories, TX, USA) housed in an environment of controlled humidity (60\\%) and temperature (23$^{\\circ}$C) with 12\\,h light-dark cycles. The experiments were performed in the light phase. \n\n\\vspace{.1in}\n\\hspace{-.25in} {\\bf Anesthesia.} Anesthesia was induced with isoflurane inhalation and maintained with urethane (1.5\\,g\/kg body weight ({\\bf bw}) dissolved in saline, intraperitoneal injection ({\\bf ip})). \nDexamethasone (2\\,mg\/kg bw, ip) and atropine sulphate (0.4\\,mg\/kg bw, ip) were administered before performing surgical procedures.\n\n\\vspace{.1in}\n\\hspace{-.25in} {\\bf Double tracheotomy surgery.} To facilitate ortho- and retronasal delivery of the odorants a double tracheotomy surgery was performed as described previously \\cite{gautam12}. This \nallowed for the rat to sniff artificially while breathing naturally through the trachea bypassing the nose. A Teflon tube (OD 2.1\\,mm, upper tracheotomy tube) was inserted 10\\,mm into the nasopharynx through the rostral \nend of the tracheal cut. Another Teflon tube (OD 2.3\\,mm, lower tracheotomy tube) was inserted in to the caudal end of the tracheal cut to allow breathing. Both tubes were fixed and sealed to the tissues using surgical \nthread. Local anesthetic (2\\% Lidocaine) was applied at all pressure points and incisions. \nThroughout the surgery and electrophysiological recordings rats' core body temperature was maintained at 37$^{\\circ}$C with a thermostatically controlled heating pad.\n\n\\vspace{.1in}\n\\hspace{-.25in} {\\bf Craniotomy surgery.} Subsequently, a craniotomy surgery was performed on the dorsal surface of the skull at two locations, one over the right Olfactory Bulb \n(2\\,mm $\\times$ 2\\,mm, centered 8.5\\,mm rostral to bregma and 1.5\\,mm lateral from midline) and the other over the right anterior Pyriform Cortex (2\\,mm $\\times$ 2\\,mm, centered 1.5\\.mm caudal to bregma and \n5.5\\,mm lateral from midline).\n\n\\vspace{.1in}\n\\hspace{-.25in} {\\bf Presentation of ortho- and retronasal odorants.} The bidirectional artificial sniffing paradigm previously used for the presentation of ortho- and retronasal odorants \\cite{gautam12} \nwere slightly modified such that instead of a nose mask a Teflon tube was inserted into the right nostril and the left nostril was sealed by suturing. The upper tracheotomy tube inserted into the nasopharynx was used \nto deliver odor stimuli retronasally (Fig~\\ref{fig1}. We used two different odorants, Hexanal ({\\bf Hexa}) and Ethyl Butyrate ({\\bf EB}) by both ortho- and retronasal routes, there by constituting 4 \ndifferent odor stimuli. Each trial consisted of 10 one-second pulse presentations of an odor with 30 second interval in between two pulses, and 2-3 min in between two trials.\n\n\\vspace{.1in}\n\\hspace{-.25in} {\\bf Electrophysiology.} Extracellular voltage was recorded simultaneously from OB and aPC using two different sets of 32-channel microelectrode arrays ({\\bf MEAs}).\n (OB: A4x2tet, 4 shanks x 2 iridium tetrodes per shank, inserted 400 $\\mu$m deep from dorsal surface; aPC: Buzsaki 32L, 4 shanks x 8 iridium electrode sites per shank, \n 6.5\\,mm deep from dorsal surface; NeuroNexus, MI, USA). Voltages were measured with respect to an AgCl ground pellet placed in the saline-soaked gel foams covering the exposed brain surface around the inserted \n MEAs. Voltages were digitized with 30\\,kHz sample rate as described previously \\cite{gautam15} using Cereplex + Cerebus, Blackrock Microsystems (UT, USA). \n\nRecordings were filtered between 300 and 3000\\,Hz and semiautomatic spike sorting was performed using Klustakwik software, which is optimized for the types of electode arrays used here \\cite{rossant16}. \nAfter automatic sorting, each unit was visually inspected to ensure quality of sorting. \n\n\\subsection*{Data processing}\n\nAfter the array recordings were spike sorted to identify activity from distinct cells, we further processed the data as follows:\n\\begin{itemize}\n\\item We computed average firing rate for each cell, where the average was taken over all trials and over the entire trial length (i.e., not distinguishing between spontaneous and evoked periods); units with firing rates below 0.008\\,Hz and above 49\\,Hz were excluded. \n\t\\item When spike times from the same unit were within 0.1\\,ms of each other, only the first (smaller) of the spike time was used and the subsequent spike times were discarded\n\\end{itemize}\n\nWe divided each 30\\,s trial into two segments, representing the odor-{\\bf evoked} state (first 2 seconds) and the {\\bf spontaneous} state (remaining 28 seconds). \nIn each state, we are interested in the random spike counts of the population in a particular window of size $T_{win}$. For a particular time window, \nthe $j^{th}$ neuron has a spike count instance $N_j$ in the time interval $[t,t+T_{win})$:\n\\begin{equation}\\label{n_cnt}\n\tN_j = \\sum_k \\int_{t}^{t+T_{win}} \\delta(t-t_k)\\,dt\n\\end{equation}\n\nThe spike count correlation between cells $j$ and $k$ is given by:\n\\begin{equation}\\label{rho_defn}\n\t\\rho_{T} = \\frac{ \\text{Cov}(N_j,N_k) }{ \\sqrt{ \\text{Var}(N_j) \\text{Var}(N_k) } },\n\\end{equation}\nwhere the {\\it covariance} of spike counts is: \n\\begin{equation}\\label{cov_defn}\n\t\\text{Cov}(N_j,N_k) = \\frac{1}{n-1} \\sum \\left( N_j - \\mu(N_j) \\right) \\left( N_k - \\mu(N_k) \\right).\n\\end{equation}\n Here $n$ is the total number of observations of $N_{j\/k}$, and $\\mu(N_j):=\\frac{1}{n}\\sum N_j$ is the mean spike count across $T_{win}$-windows and trials. \n The correlation $\\rho_{T}$ is a normalized measure of the the trial-to-trial variability (i.e., noise correlation), satisfying $\\rho_{T}\\in[-1,1]$; it is also referred to as the {\\it Pearson's correlation coefficient}. \nFor each cell pair, the covariance $\\text{Cov}(N_j,N_k)$ and variance $\\text{Var}(N_j)$ are empirically calculated by averaging across different time windows within a trial {\\it and} different trials. \n\nA standard measure of \nvariability is the Fano Factor of spike counts, which is the variance scaled by the mean:\n\\begin{equation}\\label{FF_defn}\n\tFF_k = \\frac{\\text{Var}(N_k)}{\\mu(N_k)}.\n\\end{equation}\n\nIn principle, any of the statistics defined here might depend on the time $t$ as well as time window size $T_{win}$; here, we assume that $\\text{Var}$, $\\text{Cov}$, $FF$, and $\\rho_T$ are stationary in time, and thus separate time windows based only on whether they occur in the evoked (first 2 seconds) \nor spontaneous (last 28 seconds) state. \n\nEach trial of experimental \ndata has many time windows\\footnote{an exception is when $T_{win}=2\\,$s; in the evoked state, there is only 1 window per trial}; the exact number depends on the state, the value of $T_{win}$, and whether \ndisjoint or overlapping windows are used. In this paper we use overlapping windows by half the length of $T_{win}$\\footnote{e.g. if the trial length is 2\\,s and $T_{win}=1\\,$s, then there are 3 total windows per trial: [0\\,s, 1\\,s], [0.5\\,s, 1.5\\,s], and [1\\,s, 2\\,s]} \nto calculate the spiking statistics. The results are qualitatively similar for disjoint windows and importantly the relationships\/constraints are the same with disjoint windows. We limit the size of $T_{win}\\leq 2\\,$s \nbecause this is the maximum duration of the evoked state, within each trial.\n\n\nThe average spike count $\\mu(N_j)$ of the $j^{th}$ neuron with a particular time window $T_{win}$ is related to the average firing rate $\\nu_j$ of that neuron:\n\\begin{equation}\\label{frate_defn}\n\t\\nu_j := \\frac{\\mu(N_j)}{T_{win}}\n\\end{equation}\n\n\n\\subsection*{Firing Rate Model}\n\nRecall that the activity in each representative cell is modeled by:\n\\begin{equation}\\label{frate_firsteqn}\n\t\\tau \\frac{d x_j}{dt} = -x_j + \\mu_j + \\sigma_j \\eta_j + \\sum_k g_{jk} F(x_k) \n\\end{equation}\nwhere $F(x_k)$ is a transfer function mapping activity to firing rate. Thus, the firing rate is:\n\\begin{equation}\n\t\\nu_j = F(x_j). \n\\end{equation}\n\nThe index of each region is denoted as follows: $j\\in\\{1,2,3\\}$ for the 3 OB cells, and $j\\in\\{4,5,6\\}$ for the 3 PC cells, with $j=1$ as the inhibitory granule OB cell and $j=4$ as the inhibitory PC cell (see Fig~\\ref{fig4}). In \nthis paper, we set $\\sigma_1=\\sigma_2=\\sigma_3=\\sigma_{OB}$ and $\\sigma_4=\\sigma_5=\\sigma_6=\\sigma_{PC}$ (see Table~\\ref{table:rateMod_parms}). \n\n\\begin{table}[!ht]\n\\caption{ {\\bf Parameters of the rate model (Eq~\\ref{eqn:gen_WC_pop}). The only difference between the spontaneous and evoked states, is that the mean input to OB increased in the evoked state. \nWe set $\\tau=1$ throughout.}}\n\\label{table:rateMod_parms}\n\\begin{tabular}{l|c|l|cc|}\n\\cline{2-5}\n & \\multicolumn{1}{l|}{\\textbf{Parameter}} & \\multicolumn{1}{c|}{\\textbf{Definition}} & \\textbf{Spontaneous Value} & \\multicolumn{1}{l|}{\\textbf{Evoked Value}} \\\\ \\hline\n\\multicolumn{1}{|l|}{\\multirow{5}{*}{Olfactory Bulb}} & $\\mu_1$ & \\multicolumn{1}{c|}{Mean Input} & 13\/60 & \\textbf{26\/60} \\\\\n\\multicolumn{1}{|l|}{} & $\\mu_2$ & \\multicolumn{1}{c|}{} & 9\/60 & \\textbf{18\/60} \\\\\n\\multicolumn{1}{|l|}{} & $\\mu_3$ & & 7\/60 & \\textbf{14\/60} \\\\\n\\multicolumn{1}{|l|}{} & $\\sigma_{OB}$ & Background Noise Level & 1.4 & 1.4 \\\\\n\\multicolumn{1}{|l|}{} & $c_{OB}$ & \\multicolumn{1}{c|}{OB Background Correlation} & 0.3 & 0.3 \\\\ \\hline\n\\multicolumn{1}{|l|}{\\multirow{5}{*}{Piriform Cortex}} & $\\mu_4$ & \\multicolumn{1}{c|}{Mean Input} & 9\/60 & 9\/60 \\\\\n\\multicolumn{1}{|l|}{} & $\\mu_5$ & & 5\/60 & 5\/60 \\\\\n\\multicolumn{1}{|l|}{} & $\\mu_6$ & & 3\/60 & 3\/60 \\\\\n\\multicolumn{1}{|l|}{} & $\\sigma_{PC}$ & Background Noise Level & 2 & 2 \\\\\n\\multicolumn{1}{|l|}{} & $c_{PC}$ & PC Background Correlation & 0.35 & 0.35 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nIn the absence of coupling (i.e. $g_{jk} = 0$), any pair of activity variables, $(x_j,x_k)$, are bivariate normally distributed because the equations:\n\\begin{eqnarray}\n\t\\tau \\frac{d x_j}{dt} & = & -x_j + \\mu_j + \\sigma_j \\left( \\sqrt{1-c_{jk}}\\xi_j(t) + \\sqrt{c_{jk}} \\xi_c(t) \\right) \\\\\n\t\\tau \\frac{d x_k}{dt} & = & -x_k + \\mu_k + \\sigma_k \\left( \\sqrt{1-c_{jk}}\\xi_k(t) + \\sqrt{c_{jk}} \\xi_c(t) \\right) \n\\end{eqnarray}\ndescribe a multi-dimensional Ornstein-Uhlenbeck process~\\cite{gardiner}. Note that we have re-written $\\eta_{j\/k}(t)$ as sums of independent white noise processes $\\xi(t)$, which is always possible for Gaussian white noise. \nSince $x_j(t) = \\frac{1}{\\tau}\\int_0^t e^{-(t-u)\/\\tau} \\Big[ \\mu_j + \\sigma_j\\eta_j(u) \\Big]\\,du$, we calculate marginal statistics as follows: \n\\begin{equation}\n\t\\mu(j) \\equiv \\langle x_j \\rangle = \\mu_j + 0 \\label{eqn:mu_uncoupled}\n\\end{equation}\n\n\\begin{eqnarray*}\n\t\\sigma^2(j) & \\equiv & \\langle (x_j - \\mu(j) )^2 \\rangle \\\\\n\t & = & \\left\\langle \\frac{\\sigma^2_j}{\\tau^2} \\int_0^t \\int_0^t e^{-(t-u)\/\\tau} \\eta_j(u) e^{-(t-v)\/\\tau} \\eta_j(v) \\,du\\,dv \\right\\rangle \\\\\n\t & = &\\frac{\\sigma^2_j}{\\tau^2} \\lim_{t\\to\\infty} \\int^t_0 e^{-2(t-u)\/\\tau} \\,du=\\frac{\\sigma^2_j}{2\\tau}\n\\end{eqnarray*}\n\nA similar calculation shows that in general we have:\n\\begin{equation}\n\t\\text{Cov}(j,k) = \\frac{c_{jk}}{2\\tau} \\sigma_j \\sigma_k \\label{eqn:cov_uncoupled}\n\\end{equation}\n\nThus, $(x_j,x_k)\\sim \\mathcal{N}\\left( \\left(\\begin{smallmatrix}\\mu_j \\\\ \\mu_k \\end{smallmatrix}\\right) , \\frac{1}{2\\tau} \\left(\\begin{smallmatrix} \\sigma^2_j & \\sigma_j\\sigma_k c_{jk} \\\\ \\sigma_j\\sigma_k c_{jk} & \\sigma^2_k \\end{smallmatrix}\\right) \\right)$.\n\nTo simplify notation, we define:\n\\begin{eqnarray}\n\t\\rho_{SN}(y) &:=& \\frac{1}{\\sqrt{2\\pi}} e^{-y^2\/2}, \\hbox{ the standard normal PDF} \\\\\n\t\\rho_{2D}(y_1,y_2) &:=& \\frac{1}{2\\pi\\sqrt{1-c_{jk}^2}} \\exp\\Big( -\\frac{1}{2}\\vec{y}^T \\left(\\begin{smallmatrix} 1 & c_{jk} \\\\ c_{jk} & 1 \\end{smallmatrix}\\right)^{-1} \\vec{y} \\Big), \\hbox{ bivariate standard normal} \\nonumber \\\\\n\t\t\t\t\t & &\n\\end{eqnarray}\nWith coupling, an exact expression for a joint distribution for $(x_1, x_2, x_3, x_4, x_5, x_6)$ is not explicitly known. However, we can estimate this distribution (and any derived statistics, such as means and variances) using Monte Carlo simulations. All \nMonte Carlo simulations of the six (6) coupled SDEs were performed using a time step of 0.01 with a standard Euler-Maruyama method, for a time of 500 units \n(arbitrary, but relative to the characteristic time scale $\\tau=1$) for each of the 3000 realizations. \nThe activity $x_j$ was sampled at each time step after an equilibration period. \n\nFurthermore, we can approximate moments of the joint distribution under the assumption of weak coupling, as described in the next section.\n\n\\subsection*{Approximation of Firing Statistics in the Firing Rate Model}\nWe will now show how to compute approximate first and second order statistics for the firing rate model \n\\textit{with coupling}; i.e., we aim to compute the mean activity $\\langle x_j \\rangle$, mean firing rate $\\langle F(x_j) \\rangle$, variance and covariances of both: $\\langle x_j x_k \\rangle$ and $\\langle F(x_j) F(x_k) \\rangle$.\nFor a simpler exposition, we have only included twelve synaptic connections; we have excluded self (autaptic) connections and E$\\to$E connections. \n\nAn equation for each statistic can be derived by first writing Eq~\\ref{frate_firsteqn} as a low-pass filter of the right-hand-side: \n\\begin{eqnarray}\nx_j(t) &= & \\frac{1}{\\tau}\\int_0^t e^{-(t-u)\/\\tau} \\Big[ \\mu_j + \\sigma_j\\eta_j(u) + \\sum_k g_{jk} F(x_k) \\Big]\\,du\n\\end{eqnarray}\nWe then take expectations, letting $t \\rightarrow \\infty$, we have:\n\\begin{eqnarray}\n\\mu(j):=\\langle x_j \\rangle & = & \\mu_j + \\langle \\sum_k g_{jk} F(x_k) \\rangle = \\mu_j + \\sum_{k} g_{jk} \\langle F(x_k) \\rangle \n\\end{eqnarray}\nWe assume the stochastic processes are ergodic, which is generally true for these types of stochastic differential equations, so that averaging over time is equivalent to averaging over the \ninvariant measure. \n\nWe will make several assumptions for computational efficiency. \nFirst, we only account for direct connections in the formulas for the first and second order statistics, assuming the terms from the indirect connections \nare either small or already accounted for in the direct connections. We further make the following assumptions to simplify the calculations:\n\\begin{align}\n\t& \\left\\langle \\int_0^t F(x_k(u))e^{-(t-u)\/\\tau}\\,du \\int_0^t F(x_k(v))e^{-(t-v)\/\\tau}\\,dv \\right\\rangle \\approx \\frac{\\tau}{2} \\mathbb{E}\\left[ F^2(x_k) \\right] \\label{ass_Fvar} \\\\\n\t& \\hbox{where } \\mathbb{E}\\left[ F^2(x_k) \\right] := \\int F^2(\\sigma(k)y+\\mu(k))\\,\\rho_{SN}(y)\\,dy \\\\\t\n\t& \\left\\langle \\int_0^t \\sigma_j\\eta_j(u)e^{-(t-u)\/\\tau}\\,du \\int_0^t F(x_k(v))e^{-(t-v)\/\\tau}\\,dv \\right\\rangle \\approx \\frac{\\tau}{2} \\mathbb{E}\\left[ N_j F(x_k) \\right], \\hbox{ if }j\\neq k \\label{ass_nzFa} \\\\\n\t& \\hbox{where } \\mathbb{E}\\left[ N_j F(x_k) \\right] := \\frac{\\sigma_j}{\\sqrt{2}} \\iint y_1 F(\\sigma(k)y_2+\\mu(k))\\,\\rho_{2D}(y_1,y_2)\\,dy_1 dy_2 \\\\\n\t& \\left\\langle \\int_0^t \\sigma_j\\eta_j(u)e^{-(t-u)\/\\tau}\\,du \\int_0^t F(x_k(v))e^{-(t-v)\/\\tau}\\,dv \\right\\rangle \\nonumber \\\\\n\t& \\approx \\frac{\\tau}{2} \\frac{\\sigma_k}{\\sqrt{2}} \\int y F(\\sigma(k)y +\\mu(k))\\,\\rho_{SN}(y)\\,dy, \\hbox{ if } j = k \\label{ass_nzFb} \\\\\n\t& \\left\\langle \\int_0^t F(x_j(u))e^{-(t-u)\/\\tau}\\,du \\int_0^t F(x_k(v))e^{-(t-v)\/\\tau}\\,dv \\right\\rangle \\approx \\frac{\\tau}{2} \\mathbb{E}\\left[ F(x_j)F(x_k) \\right] \t\t\\label{ass_Fcov} \\\\ \n\t& \\hbox{where } \\mathbb{E}\\left[ F(x_j) F(x_k) \\right] := \\nonumber \\\\\n\t & \\iint F(\\sigma(j)y_1+\\mu(j)) F(\\sigma(k)y_2+\\mu(k))\\,\\rho_{2D}(y_1,y_2)\\,dy_1 dy_2 \\label{ass_end}\n\\end{align}\nand $N_j$ denotes the random variable $\\int_0^t \\sigma_j\\eta_j(u) e^{-(t-u)\/\\tau}\\,du$, which is by itself normally distributed with mean 0 and variance $\\sigma_j^2\\tau\/2$. \n\nThe first assumption, Eq~\\ref{ass_Fvar}, states that time-average of $F(x_j(t))$ multiplied by an exponential function (low-pass filter) is equal to the expected value scaled by $\\tau\/2$; \nthe second and third, Eq~\\ref{ass_nzFa} and Eq~\\ref{ass_nzFb}, address $N_j$ and $F(x_k(t))$, for $j \\not= k$ and $j=k$ respectively (similarly for Eq~\\ref{ass_Fcov}). \n\nIn all of the definitions for the expected values with $\\rho_{2D}$, note that the underlying correlation correlation $c_{jk}$ depend on the pair of interest $(j,k)$. \nFinally, we assume that the activity variables $(x_j,x_k)$ are pairwise normally distributed with the subsequent statistics; this is sufficient to ``close\" our model and solve for the statistical quantities self-consistently. \nThis is implicitly a weak coupling assumption because with no coupling, $(x_j,x_k)$ are bivariate normal random variables. \n\nThe resulting approximations for the mean activity are:\n\\begin{align}\n\t& \\mu(1) = \\mu_1 + \\sum_{k=2,3,5,6} g_{1k}\\int F(\\sigma(k)y+\\mu(k))\\,\\rho_{SN}(y)\\,dy\t\\label{mn_x1} \\\\\n\t& \\mu(2) = \\mu_2 + g_{21}\\int F(\\sigma(1)y+\\mu(1))\\,\\rho_{SN}(y)\\,dy \t\t\\label{mn_x2} \\\\\n\t& \\mu(3) = \\mu_3 + g_{31}\\int F(\\sigma(1)y+\\mu(1))\\,\\rho_{SN}(y)\\,dy \t\t\\label{mn_x3} \\\\ \n\t& \\mu(4) = \\mu_4 + \\sum_{k=2,3,5,6} g_{4k}\\int F(\\sigma(k)y+\\mu(k))\\,\\rho_{SN}(y)\\,dy \\label{mn_x4} \\\\\t\n\t& \\mu(5) = \\mu_5 + g_{54}\\int F(\\sigma(4)y+\\mu(4))\\,\\rho_{SN}(y)\\,dy \t\t\\label{mn_x5} \\\\\n\t& \\mu(6) = \\mu_6 + g_{64}\\int F(\\sigma(4)y+\\mu(4))\\,\\rho_{SN}(y)\\,dy. \t\t\\label{mn_x6}\n\\end{align}\nThe resulting approximation to the variances of the mean activity are:\n\\begin{eqnarray}\n\t\\tau\\sigma^2(1) &=& \\frac{\\sigma_1^2}{2} + \\sum_{k=2,3,5,6} \\frac{g^2_{1k}}{2} \\text{Var}\\big(F(\\sigma(k)Y+\\mu(k))\\big) \\nonumber \\\\\n\t\t\t\t& &+ \\sum_{(j,k)\\in\\{(2,3);(5,6)\\}} g_{1j}g_{1k}\\text{Cov}\\big(F(\\sigma(j)Y_1+\\mu(j)),F(\\sigma(k)Y_2+\\mu(k))\\big) \\label{var_x1} \\\\\n\t\\tau\\sigma^2(2) &=& \\frac{\\sigma_2^2}{2} + \\frac{g^2_{21}}{2} \\text{Var}\\big(F(\\sigma(1)Y+\\mu(1))\\big) \\nonumber \\\\\t\n\t\t\t\t& & + \\sigma_2 g_{21} \\iint \\frac{y_1}{\\sqrt{2}} F(\\sigma(1)y_2+\\mu(1))\\,\\rho_{2D}(y_1,y_2)\\,dy_1 dy_2 \\label{var_x2} \\\\\n\t\\tau\\sigma^2(3) &=& \\frac{\\sigma_3^2}{2} + \\frac{g^2_{31}}{2} \\text{Var}\\big(F(\\sigma(1)Y+\\mu(1))\\big) \\nonumber \\\\\t\n\t\t\t\t& & + \\sigma_3 g_{31} \\iint \\frac{y_1}{\\sqrt{2}} F(\\sigma(1)y_2+\\mu(1))\\,\\rho_{2D}(y_1,y_2)\\,dy_1 dy_2 \\label{var_x3} \\\\\n\t\\tau\\sigma^2(4) &=& \\frac{\\sigma_4^2}{2} + \\sum_{k=2,3,5,6} \\frac{g^2_{4k}}{2} \\text{Var}\\big(F(\\sigma(k)Y+\\mu(k))\\big) \\nonumber \\\\\n\t\t\t\t& &+ \\sum_{(j,k)\\in\\{(2,3);(5,6)\\}} g_{4j}g_{4k}\\text{Cov}\\big(F(\\sigma(j)Y_1+\\mu(j)),F(\\sigma(k)Y_2+\\mu(k))\\big) \\\\\n\t\\tau\\sigma^2(5) &=& \\frac{\\sigma_5^2}{2} + \\frac{g^2_{54}}{2} \\text{Var}\\big(F(\\sigma(4)Y+\\mu(4))\\big) \\nonumber \\\\\t\n\t\t\t\t& & + \\sigma_5 g_{54} \\iint \\frac{y_1}{\\sqrt{2}} F(\\sigma(4)y_2+\\mu(4))\\,\\rho_{2D}(y_1,y_2)\\,dy_1 dy_2 \\label{var_x5} \\\\\n\t\\tau\\sigma^2(6) &=& \\frac{\\sigma_6^2}{2} + \\frac{g^2_{64}}{2} \\text{Var}\\big(F(\\sigma(4)Y+\\mu(4))\\big) \\nonumber \\\\\t\n\t\t\t\t& & + \\sigma_6 g_{64} \\iint \\frac{y_1}{\\sqrt{2}} F(\\sigma(4)y_2+\\mu(4))\\,\\rho_{2D}(y_1,y_2)\\,dy_1 dy_2 \\label{var_x6} \t\t\t\t\n\\end{eqnarray}\n\nIn Eq~\\ref{mn_x1}--\\ref{var_x6}, all of the $\\text{Var}$ and $\\text{Cov}$ are with respect to $Y\\sim \\mathcal{N}(0,1)$ (for $\\text{Var}$) and \n $(Y_1,Y_2) \\sim \\mathcal{N}\\left( \\left(\\begin{smallmatrix} 0 \\\\ 0 \\end{smallmatrix}\\right) , \\frac{1}{2} \\left(\\begin{smallmatrix} 1 & c_{jk} \\\\ c_{jk} & 1 \\end{smallmatrix}\\right) \\right)$ (for $\\text{Cov}$);\nboth are easy to calculate. The value $c_{jk}$ depends on the pairs; for example in Eq~\\ref{var_x2}, the $\\rho_{2D}$ has $c_{jk}=c_{OB}$, the background correlation value in the olfactory bulb but \nin Eq~\\ref{var_x1}, the $\\text{Cov}$ term is with respect to $\\rho_{2D}$ with $c_{jk}=c_{PC}$, the background correlation value in the piriform cortex. \n\nLastly, we state the formulas for the approximations to the covariances. Although there are 15 total covariance values, we are only concerned with 6 covariance values (3 within OB and 3 within PC); we neglect all covariances \\textit{between} regions.\nFirst, our experimental data set shows that these covariance (and correlation) values are small (see Fig S9 in S2 Text). \nSecond, because there is no background correlation (i.e., common input) between PC and OB in our model, \nany nonzero covariance\/correlation arises strictly via direct coupling. Thus, we cannot view OB-PC covariance from coupling as a small perturbation of the background (uncoupled) state; we do not expect our model to yield qualitatively accurate predictions for these statistics. The formulas for the Cov of interest are:\n\\begin{eqnarray}\n\t\\tau \\text{Cov}(1,2) &=& \\frac{1}{2}c_{OB}\\sigma_1\\sigma_2 +\\sigma_1 \\frac{g_{21}}{2} \\int \\frac{y}{\\sqrt{2}} F(\\sigma(1)y+\\mu(1))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\n\t\t\t& & +\\sigma_2\\frac{g_{12}}{2}\\int \\frac{y}{\\sqrt{2}} F(\\sigma(2)y+\\mu(2))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\t \t\n\t\t\t& & +\\sigma_2\\frac{g_{13}}{2}\\int \\frac{y}{\\sqrt{2}} F(\\sigma(3)y+\\mu(3))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\t\n\t\t\t& & +\\frac{1}{2}\\sum_{(j,k)} g_{1j} g_{2k} \\mathcal{C}(j,k) \\label{cov_12} \\\\\n\t\\tau \\text{Cov}(1,3) &=& \\frac{1}{2}c_{OB}\\sigma_1\\sigma_3 +\\sigma_1 \\frac{g_{31}}{2} \\int \\frac{y}{\\sqrt{2}} F(\\sigma(1)y+\\mu(1))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\n\t& & +\\sigma_3\\frac{g_{12}}{2}\\int \\frac{y}{\\sqrt{2}} F(\\sigma(2)y+\\mu(2))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\t \t\n\t\t\t& & +\\sigma_3\\frac{g_{13}}{2}\\int \\frac{y}{\\sqrt{2}} F(\\sigma(3)y+\\mu(3))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\t\n\t\t\t& & +\\frac{1}{2}\\sum_{(j,k)} g_{1j} g_{3k} \\mathcal{C}(j,k) \\label{cov_13} \\\\\n\t\\tau \\text{Cov}(2,3) &=& \\frac{1}{2}c_{OB}\\sigma_2\\sigma_3 +\\frac{g_{21}g_{31}}{2} \\text{Var}\\big(F(\\sigma(1)Y+\\mu(1))\\big) \t\\nonumber \\\\\t\n\t\t\t & & + \\frac{\\sigma_3 g_{21}+\\sigma_2 g_{31}}{2}\\iint \\frac{y_1}{\\sqrt{2}} F(\\sigma(1)y_2+\\mu(1))\\,\\rho_{2D}(y_1,y_2)\\,dy_1 dy_2 \\label{cov_23} \\\\\n\t\\tau \\text{Cov}(4,5) &=& \\frac{1}{2}c_{PC}\\sigma_4\\sigma_5 +\\sigma_4 \\frac{g_{54}}{2} \\int \\frac{y}{\\sqrt{2}} F(\\sigma(4)y+\\mu(4))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\n\t\t\t\t& & +\\sigma_5\\frac{g_{45}}{2}\\int \\frac{y}{\\sqrt{2}} F(\\sigma(5)y+\\mu(5))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\t \t\n\t\t\t& & +\\sigma_5\\frac{g_{46}}{2}\\int \\frac{y}{\\sqrt{2}} F(\\sigma(6)y+\\mu(6))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\t\n\t\t\t& & +\\frac{1}{2}\\sum_{(j,k)} g_{4j} g_{5k} \\mathcal{C}(j,k) \\label{cov_45} \\\\\n\t\\tau \\text{Cov}(4,6) &=& \\frac{1}{2}c_{PC}\\sigma_4\\sigma_6 +\\sigma_4 \\frac{g_{64}}{2} \\int \\frac{y}{\\sqrt{2}} F(\\sigma(4)y+\\mu(4))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\n\t\t\t\t\t& & +\\sigma_6\\frac{g_{45}}{2}\\int \\frac{y}{\\sqrt{2}} F(\\sigma(5)y+\\mu(5))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\t \t\n\t\t\t& & +\\sigma_6\\frac{g_{46}}{2}\\int \\frac{y}{\\sqrt{2}} F(\\sigma(6)y+\\mu(6))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\t\n\t\t\t& & +\\frac{1}{2}\\sum_{(j,k)} g_{4j} g_{6k} \\mathcal{C}(j,k) \\label{cov_46} \\\\\n\t\\tau \\text{Cov}(5,6) &=& \\frac{1}{2}c_{PC}\\sigma_5\\sigma_6 +\\frac{g_{54}g_{64}}{2} \\text{Var}\\big(F(\\sigma(4)Y+\\mu(4))\\big) \t\\nonumber \\\\\t\n\t\t\t & & + \\frac{\\sigma_6 g_{54}+\\sigma_5 g_{64}}{2}\\iint \\frac{y_1}{\\sqrt{2}} F(\\sigma(4)y_2+\\mu(4))\\,\\rho_{2D}(y_1,y_2)\\,dy_1 dy_2 \\label{cov_56} \n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\mathcal{C}(j,k) =& \\iint F(\\sigma(j)y_1+\\mu(j))F(\\sigma(k)y_2+\\mu(k))\\rho_{2D}(y_1,y_2)\\,dy_1dy_2 \\nonumber \\\\\n\t\t\t&- \\left( \\int F(\\sigma(j)y+\\mu(j))\\rho_{SN}(y)\\,dy \\right) \\left( \\int F(\\sigma(k)y+\\mu(k))\\rho_{SN}(y)\\,dy \\right) \\label{mathcal_defn}\n\\end{eqnarray}\n\n\\subsubsection*{Iteration procedure to solve for the approximate statistics self-consistently}\nBased on the approximations and resulting equations described in the previous section, our objective is to solve for the statistics of $x_j$ self-consistently. Once these are determined, the statistics of the firing rates $F(x_j)$ are approximated with the \nsame pairwise normal assumption on $(x_j,x_k)$; we are {\\bf not} assuming that $(F(x_j),F(x_k))$ are bivariate normal random variables. \n\nWe use a simple iterative procedure to solve the system of coupled algebraic expression for the statistics of $x_j$. \nWe first solve the system in the absence of coupling (i.e. Eq~\\ref{eqn:mu_uncoupled}, \\ref{eqn:cov_uncoupled}), and use these values to start the iteration;\nat each step, the formulas for the means (Eq~\\ref{mn_x1}--\\ref{mn_x6}), variances (Eq~\\ref{var_x1}--\\ref{var_x6}), and covariances (Eq~\\ref{cov_12}--\\ref{cov_56}) are \nrecalculated numerically, using the results of the previous step. The iteration stops once \\underline{all 18} statistical quantities of the activity match up to a relative tolerance of $10^{-6}$ (convergence), or after 50 total iterations (non-convergence). The \nresult with a given parameter set can either be: i) convergence, ii) non-convergence, iii) a pair of statistics with invalid covariance (non-positive definite covariance matrix), which is checked after i) and ii). We only consider parameter sets \nwhere the iteration has converged and all of the covariances are valid, after which we determine whether the constraints are satisfied.\n \nOne subtle point is that we did not use any of the numerically calculated $\\text{Cov}$ values in the bivariate normal distributions $\\rho_{2D}$; rather, the correlation value is always $c_{jk}$ which is either 0, $c_{OB}$, \nor $c_{PC}$ depending on the pair. In principle, one can use a fully iterative procedure where the formulas for the $\\text{Cov}$ (Eq~\\ref{cov_12}--\\ref{cov_56}) are used in $\\rho_{2D}$; however, we found that the resulting covariance matrices (for \n$\\rho_{2D}$) can fail to be positive semi-definite. \nHandling this case requires additional code in the program and slower calculations for each parameter set, which \ndetracts from \nthe purpose of our method. We checked some parameter sets comparing the results of the two procedures, \nand the results are quantitatively similar.\n\nThe standard normal $\\rho_{SN}$ and bivariate $\\rho_{2D}$ PDFs have state variable(s) $y_{1,2}$ discretized from -3 to 3 with a mesh size of 0.01; integrals in Eq~\\ref{mn_x1}--\\ref{cov_56} are computed using the trapezoidal rule.\n\n\\subsubsection*{Simplified network with four coupling parameters}\n\nTo further simplify the network, we:\n\\begin{itemize}\n\t\\item set $\\tau=1$,\n\t\\item assume feedforward inhibitory connections within a region have the same strength: $g_{21} = g_{31} =: gIO$ and $g_{54} = g_{64} =: gIP$,\n\t\\item assume cross-region excitatory connections are equal from the presynaptic cell, i.e., $g_{15} = g_{16} =: gEP$ and $g_{42} = g_{43} =: gEO$.\n\t\\item assume $\\sigma_1=\\sigma_2=\\sigma_3=:\\sigma_{OB}$ and $\\sigma_4=\\sigma_5=\\sigma_6=:\\sigma_{PC}$\n\t\\item assume $g_{12}=g_{13}=g_{45}=g_{46}=:g_\\epsilon=0.1$\n\\end{itemize}\nNow there are only 4 variable coupling parameters: $gIO$, $gEO$, $gIP$, $gEP$.\n\n\n\nThe above formulas for the statistics of $x_j$ reduce to:\n\\begin{eqnarray}\n\t\\mu(1) &=& \\mu_1 + gEP \\int \\Big( F(\\sigma(5)y+\\mu(5)) + F(\\sigma(6)y+\\mu(6)) \\Big)\\,\\rho_{SN}(y)\\,dy\t\\nonumber \\\\\n\t\t & & \t + g_\\epsilon \\int \\Big( F(\\sigma(2)y+\\mu(2)) + F(\\sigma(3)y+\\mu(3)) \\Big)\\,\\rho_{SN}(y)\\,dy \\label{mnX1} \\\\\n\t\\mu(2) &=& \\mu_2 + gIO\\int F(\\sigma(1)y+\\mu(1))\\,\\rho_{SN}(y)\\,dy \t\t\\label{mnX2} \\\\\n\t\\mu(3) &=& \\mu_3 + gIO\\int F(\\sigma(1)y+\\mu(1))\\,\\rho_{SN}(y)\\,dy \t\t\\label{mnX3} \\\\ \n\t\\mu(4) &=& \\mu_4 + gEO \\int \\Big( F(\\sigma(2)y+\\mu(2)) + F(\\sigma(3)y+\\mu(3)) \\Big)\\,\\rho_{SN}(y)\\,dy \t \\nonumber \\\\\n\t\t & & + g_\\epsilon \\int \\Big( F(\\sigma(5)y+\\mu(5)) + F(\\sigma(6)y+\\mu(6)) \\Big)\\,\\rho_{SN}(y)\\,dy\t\\label{mnX4} \\\\\t\n\t\\mu(5) &=& \\mu_5 + gIP\\int F(\\sigma(4)y+\\mu(4))\\,\\rho_{SN}(y)\\,dy \t\t\\label{mnX5} \\\\\n\t\\mu(6) &=& \\mu_6 + gIP \\int F(\\sigma(4)y+\\mu(4))\\,\\rho_{SN}(y)\\,dy; \t\t\\label{mnX6}\n\\end{eqnarray}\nthe variances are: \n\\begin{eqnarray}\n\t\\sigma^2(1) &=& \\frac{\\sigma^2_{OB}}{2} + \\frac{(gEP)^2}{2} \\text{Var}\\Big(F(\\sigma(5)Y_1+\\mu(5)) + F(\\sigma(6)Y_2+\\mu(6))\\Big) \\nonumber \\\\\t\n\t\t\t& & +\\frac{g^2_\\epsilon}{2} \\text{Var}\\Big(F(\\sigma(2)Y_1+\\mu(2)) + F(\\sigma(3)Y_2+\\mu(3))\\Big) \\label{varX1} \\\\\n\t\\sigma^2(2) &=& \\frac{\\sigma^2_{OB}}{2} + \\frac{(gIO)^2}{2} \\text{Var}\\big(F(\\sigma(1)Y+\\mu(1))\\big) \\nonumber \\\\\t\n\t\t\t\t& & + \\sigma_{OB} gIO \\iint \\frac{y_1}{\\sqrt{2}} F(\\sigma(1)y_2+\\mu(1))\\,\\rho_{2D}(y_1,y_2)\\,dy_1 dy_2 \\label{varX2} \\\\\n\t\\sigma^2(3) &=& \\sigma^2(2)\t \\label{varX3} \\\\\n\t\\sigma^2(4) &=& \\frac{\\sigma^2_{PC}}{2} + \\frac{(gEO)^2}{2} \\text{Var}\\Big(F(\\sigma(2)Y_1+\\mu(2)) + F(\\sigma(3)Y_2+\\mu(3))\\Big) \\nonumber \\\\\n\t\t& & +\\frac{g^2_\\epsilon}{2} \\text{Var}\\Big(F(\\sigma(5)Y_1+\\mu(5)) + F(\\sigma(6)Y_2+\\mu(6))\\Big) \\label{varX4}\t\\\\\n\t\\sigma^2(5) &=& \\frac{\\sigma^2_{PC}}{2} + \\frac{(gIP)^2}{2} \\text{Var}\\big(F(\\sigma(4)Y+\\mu(4))\\big) \\nonumber \\\\\t\n\t\t\t\t& & + \\sigma_{PC} gIP \\iint \\frac{y_1}{\\sqrt{2}} F(\\sigma(4)y_2+\\mu(4))\\,\\rho_{2D}(y_1,y_2)\\,dy_1 dy_2 \\label{varX5} \\\\\n\t\\sigma^2(6) &=& \\sigma^2(5); \\label{varX6} \t\t\t\t\n\\end{eqnarray}\nthe covariances are:\n\\begin{eqnarray}\n\t\\text{Cov}(1,2) &=& \\frac{1}{2}c_{OB}\\sigma^2_{OB} +\\sigma_{OB} \\frac{gIO}{2} \\int \\frac{y}{\\sqrt{2}} F(\\sigma(1)y+\\mu(1))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\n\t\t\t\t& & \\sigma_{OB} \\frac{g_\\epsilon}{2} \\int \\frac{y}{\\sqrt{2}} F(\\sigma(2)y+\\mu(2))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\n\t\t\t\t& & g_\\epsilon gIO * \\mathcal{C}(1,2) \\label{covX12} \\\\\n\t\\text{Cov}(1,3) &=& \\text{Cov}(1,2) \\label{covX13} \\\\\n\t\\text{Cov}(2,3) &=& \\frac{1}{2}c_{OB}\\sigma^2_{OB}+\\frac{g^2_{IO}}{2} \\text{Var}\\big(F(\\sigma(1)Y+\\mu(1))\\big) \t\\nonumber \\\\\t\n\t\t\t & & + \\sigma_{OB} gIO \\iint \\frac{y_1}{\\sqrt{2}} F(\\sigma(1)y_2+\\mu(1))\\,\\rho_{2D}(y_1,y_2)\\,dy_1 dy_2 \\label{covX23} \\\\\n\t\\text{Cov}(4,5) &=& \\frac{1}{2}c_{PC}\\sigma^2_{PC} +\\sigma_{PC} \\frac{gIP}{2} \\int \\frac{y}{\\sqrt{2}} F(\\sigma(4)y+\\mu(4))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\n\t\t\t& & \\sigma_{PC} \\frac{g_\\epsilon}{2} \\int \\frac{y}{\\sqrt{2}} F(\\sigma(5)y+\\mu(5))\\,\\rho_{SN}(y)\\,dy \\nonumber \\\\\n\t\t\t\t& & g_\\epsilon gIP * \\mathcal{C}(4,5)\t\\label{covX45} \\\\\n\t\\text{Cov}(4,6) &=& \\text{Cov}(4,5) \\label{covX46} \\\\\n\t\\text{Cov}(5,6) &=& \\frac{1}{2}c_{PC}\\sigma^2_{PC} +\\frac{g^2_{IP}}{2} \\text{Var}\\big(F(\\sigma(4)Y+\\mu(4))\\big) \t\\nonumber \\\\\t\n\t\t\t & & + \\sigma_{PC} gIP \\iint \\frac{y_1}{\\sqrt{2}} F(\\sigma(4)y_2+\\mu(4))\\,\\rho_{2D}(y_1,y_2)\\,dy_1 dy_2 \\label{covX56} \n\\end{eqnarray}\nSee Eq~\\ref{mathcal_defn} for the definition of $\\mathcal{C}$.\n\n\\subsection*{Leaky Integrate-and-Fire Model of the OB--PC Circuit}\n\nWe use a generic spiking neural network model of leaky integrate-and-fire neurons to test the results of the theory. \nThere were $N_{OB}=100$ total OB cells, of which we set 80\\% (80) to be granule (I-)cells and 20\\% (20) to be mitral\/tufted ({\\bf M\/T}) E-cells. There are known to be many \nmore granule cells than M\/T cells in the OB; this ratio of 4-to-1 is similar to other models of OB (see~\\cite{grabska17} who used 3-to-1). \nThe equations for the OB cells are, indexed by $k\\in\\{1,2,\\dots,N_{OB}\\}$:\n\\begin{eqnarray}\\label{ob_lif}\n\t\\tau_m \\frac{d v_k}{dt} & = & \\mu_{OB}-v_k-g_{k, XI}(t)(v_k-\\mathcal{E}_I)-g_{k, XE}(t)(v_k-\\mathcal{E}_E) \\nonumber \\\\\t\n\t & & - g_{k,XPC}(t - \\tau_{\\Delta,PC})(v_k-\\mathcal{E}_I) +\\sigma_{OB}\\left(\\sqrt{1-\\tilde{c}_{OB}}\\eta_k(t) + \\sqrt{\\tilde{c}_{OB}}\\xi_o(t) \\right) \\nonumber \\\\\n\tv_k(t^*) & \\geq & \\theta_k \\Rightarrow v_k(t^*+\\tau_{ref})=0 \\nonumber \\\\\n\tg_{k,XE}(t) &=& \\frac{\\gamma_{XE}}{p_{XE} \\left(0.2 N_{OB} \\right) }\\sum_{k'\\in\\{\\hbox{ presyn OB E-cells}\\} } G_{k'}(t) \\nonumber \\\\\n\tg_{k,XI}(t) &=& \\frac{\\gamma_{XI}}{p_{XI} \\left(0.8 N_{OB} \\right)}\\sum_{k'\\in\\{\\hbox{presyn OB I-cells}\\}} G_{k'}(t) \\nonumber \\\\\n\tg_{k,XPC}(t) &=& \\frac{\\gamma_{X,PC}}{p_{X,PC} \\left(0.8 N_{PC} \\right)} \\sum_{j'\\in\\{\\hbox{presyn PC E-cells}\\}} G_{j'}(t) \\nonumber \\\\\n\t\\tau_{d,X}\\frac{d G_k}{dt} &=& -G_k + A_k \\nonumber \\\\\n\t\\tau_{r,X} \\frac{d A_k}{dt} &=& -A_k + \\tau_{r,X} \\alpha_X \\sum_{l} \\delta(t-t_{k,l}). \\label{eqn:OB_LIF_all}\n\\end{eqnarray}\nThe conductance values in the first equation $g_{k,XI}$, $g_{k,XE}$, and $g_{k,XPC}$ depend on the type of neuron $v_k$ ($X\\in\\{ E, I\\}$). The last conductance, \n$g_{X,PC}(t - \\tau_{\\Delta,PC})(v_k-\\mathcal{E}_E)$, models the excitatory presynaptic input (feedback) from the PC cells with a time delay of $\\tau_{\\Delta,PC}$. The conductance variables $g_{k,XY}(t)$ are dimensionless because this model was \nderived from scaling the original (raw) conductance variables by the leak conductance with the same dimension. \nThe leak, inhibitory and excitatory reversal potentials are 0, $\\mathcal{E}_I$, and $\\mathcal{E}_E$, respectively with $\\mathcal{E}_I<0<\\mathcal{E}_E$ \n(the voltage is scaled to be dimensionless, see Table~\\ref{table:lif_parms}). \n$\\xi_k(t)$ are uncorrelated white noise processes and $\\xi_o(t)$ is the common noise term to all $N_{OB}$ cells.\n\nThe second equation describes the refractory period at spike time $t^*$: when the neuron's voltage crosses \nthreshold $\\theta_j$ (see below for distribution of thresholds), \nthe neuron goes into a refractory period for $\\tau_{ref}$, after which we set the neuron's voltage to 0. \n\nThe parameter $\\gamma_{XY}$ gives the relative weight of a connection from neuron type $Y$ to neuron type $X$; the parameter $p_{XY}$ is the probability that any such connection exists ($X,Y\\in\\{E,I\\}$). $G_k$ is the synaptic variable associated with each cell, and dependent only on that cell's spike times; its dynamics are given by the final two equations in Eq~\\ref{eqn:OB_LIF_all} and depend on whether $k \\in \\{E,I\\}$.\n\nFinally, two of the parameters above can be equated with coupling parameters in the reduced model:\n\\begin{equation}\ngEP = \\gamma_{E,PC}; \\quad gIO = \\gamma_{EI}\n\\end{equation}\nwhich are dimensionless scale factors for the synaptic conductances.\n\n\n\\begin{table}[!ht]\n\\centering\n\\caption{{\\bf Fixed parameters for the LIF OB--PC model, see Eqs~\\ref{ob_lif}--\\ref{pc_lif}. }}\n\\label{table:lif_parms}\n\\begin{tabular}{|lcccccccccc|}\n\\hline\n\\multicolumn{11}{|c|}{\\textbf{Same for both OB and PC}} \\\\ \\hline\n\\textbf{Parameter} & $\\tau_m$ & $\\tau_{ref}$ & $\\mathcal{E}_I$ & $\\mathcal{E}_E$ & $\\tau_{d,I}$ & $\\tau_{r,I}$ & $\\tau_{d,E}$ & $\\tau_{r,E}$ & $\\alpha_I$ & $\\alpha_E$ \\\\ \\hline\n & 20\\,ms & 2\\,ms & -2.5 & 6.5 & 10\\,ms & 2\\,ms & 5\\,ms & 1\\,ms \t\t\t\t\t\t\t\t& 2\\,Hz & 1\\,Hz \\\\ \\thickhline\n\\textbf{Parameter} & $N$ \t & Spont. $\\mu$ & Evoked $\\mu$ & $\\sigma$ & $\\tilde{c}$ & $\\gamma_{EE}$ & $\\gamma_{IE}$ & $\\gamma_{II}$ & $\\tau_{\\Delta,PC\/OB}$ & \\\\ \\hline\n\\textbf{OB} & 100 \t\t & 0.6 & 0.9$^*$ \t\t\t\t& 0.05 & 0.5 & 2 & 4 & 2 \t\t & \t10\\,ms & $ $ \\\\\n\\textbf{PC} & 100 \t\t & 0 & 0.4 \t\t\t\t\t& 0.1 & 0.8 & 5 & 8 & 6 \t\t & \t5\\,ms & $ $ \\\\ \\hline\n\\end{tabular}\n\\begin{flushleft} All 12 probabilities of connections are set to $p_{XY}=0.30$; otherwise connections were chosen randomly and independently (Erd\\H{o}s-R\\'enyi graphs). \nThe synaptic time delay from OB to PC is $\\tau_{\\Delta,OB}=10\\,$ms, and from PC to OB is $\\tau_{\\Delta,PC}=5\\,$ms. \nThe scaled voltages from mV is: (V+Vreset)\/(Vth+Vreset), corresponding for \nexample to Vreset=Vleak=-65\\,mV, Vth=-55\\,mV (on average), excitatory reversal potential of 0\\,mV and inhibitory reversal potential of -90\\,mV. {\\bf *}Note: in the evoked state, \nonly the {\\bf M\/T} (E-cells) in OB receive a larger $\\mu$ input from 0.6 to 0.9; the granule cells in OB have $\\mu=0.6$ even in the evoked state.\n\\end{flushleft}\n\\end{table}\n\nThe PC cells had similar functional form but with different parameters (see Table~\\ref{table:lif_parms} for parameter values). We modeled $N_{PC}=100$ total PC cells, of which 80\\% were excitatory and 20\\% inhibitory. \nThe equations, indexed by $j\\in\\{1,2,\\dots,N_{PC}\\}$ are:\n\\begin{eqnarray}\\label{pc_lif}\n\t\\tau_m \\frac{d v_j}{dt} & = & \\mu_{PC}-v_j-g_{j,XI}(t)(v_j-\\mathcal{E}_I)-g_{j,XE}(t)(v_j-\\mathcal{E}_E) \\nonumber \\\\\t\n\t & & - g_{j,XOB}(t - \\tau_{\\Delta,OB})(v_j-\\mathcal{E}_E) +\\sigma_{PC}\\left(\\sqrt{1-\\tilde{c}_{PC}}\\eta_j(t) + \\sqrt{\\tilde{c}_{PC}}\\xi_p(t) \\right) \\nonumber \\\\\n\tv_j(t^*) & \\geq & \\theta_j \\Rightarrow v_j(t^*+\\tau_{ref})=0 \\nonumber \\\\\n\tg_{j,XE}(t) &=& \\frac{\\gamma_{XE}}{p_{XE} \\left(0.8 N_{PC} \\right)}\\sum_{j'\\in\\{\\hbox{presyn PC E-cells}\\}} G_{j'}(t) \\nonumber \\\\\n\tg_{j,XI}(t) &=& \\frac{\\gamma_{XI}}{p_{XI} \\left(0.2 N_{PC} \\right)}\\sum_{j'\\in\\{\\hbox{presyn PC I-cells}\\}} G_{j'}(t) \\nonumber \\\\\n\t\tg_{j,XOB}(t) &=& \\frac{\\gamma_{X,OB}}{p_{X,OB} \\left(0.2 N_{OB} \\right)} \\sum_{k'\\in\\{\\hbox{presyn OB E-cells}\\}} G_{k'}(t) \\nonumber \\\\\n\t\\tau_{d,X}\\frac{d G_j}{dt} &=& -G_j + A_j \\nonumber \\\\\n\t\\tau_{r,X} \\frac{d A_j}{dt} &=& -A_j + \\tau_{r,X} \\alpha_X \\sum_{l} \\delta(t-t_{j,l}).\n\\end{eqnarray}\nExcitatory synaptic input from the OB cells along the lateral olfactory tract is modeled by: $g_{X,OB}(t - \\tau_{\\Delta,OB})(v_j-\\mathcal{E}_E)$. The common noise term for the \nPC cells $\\xi_p(t)$ is independent of the common noise term for the OB cells $\\xi_o(t)$. \nTwo of the parameters above can be equated with coupling parameters in the reduced model:\n\\begin{equation}\ngEO = \\gamma_{E,OB}; \\quad gIP = \\gamma_{EI}\n\\end{equation}\n\n\nThe values of the parameters \nthat were not stated in Table~\\ref{table:lif_parms} were varied in the paper: \n$$ gIO, \\hspace{.5in} gEO, \\hspace{.5in} gIP, \\hspace{.5in} gEP. $$\n\nTo model two activity states, we allowed mean inputs to vary (see Table~\\ref{table:lif_parms}). In contrast to the reduced model, we increased both inputs to PC cells (from $\\mu_{PC}=0$ in the spontaneous state to \n$\\mu_{PC}=0.4$ in the evoked state) as well as to OB cells; $\\mu_{OB}=0.6$ in the spontaneous state to $\\mu_{OB}=0.9$ in the evoked state only for {\\bf M\/T} cells (OB granule cell \ninput is the same for spontaneous and evoked). \n\n\nFinally, we model heterogeneity by setting the threshold values $\\theta_j$ in the following way. Both OB and PC cells had the following distributions for $\\theta_j$:\n\\begin{eqnarray}\\label{thres_distr}\n\t\\theta_j &\\sim& e^{\\mathcal{N}} \n\\end{eqnarray}\nwhere $\\mathcal{N}$ is normal distribution with mean $-\\sigma^2_\\theta\/2$ and standard deviation $\\sigma_\\theta$, so that $\\{\\theta_j\\}$ has a \nlog-normal distribution with mean 1 and variance: $e^{\\sigma_\\theta^2}-1$. We set $\\sigma_\\theta=0.1$, which results in firing rates ranges seen in the experimental data. \nSince the number of cells are modest with regards to sampling ($N_{OB}=100$, $N_{PC}=100$), we evenly sampled the log-normal distribution from the 5$^{th}$ to 95$^{th}$ percentiles (inclusive). \n\nWe remark that the synaptic delays of $\\tau_{\\Delta,PC}$ and $\\tau_{\\Delta,OB}$ were set to modest values to capture the appreciable distances between OB and PC. This is a reasonable choice \nbased on evidence that stimulation in PC elicit a response in OB 5-10\\,ms later~\\cite{neville03}.\n\nIn all Monte Carlo simulations of the coupled LIF network, we used a time step of 0.1\\,ms, with 2\\,s of biology time for each of the 50,000 realizations (i.e., over 27.7 hours of biology time), enough simulated statistics to effectively have convergence.\n\n\\section*{Supporting Information}\n\n\\paragraph*{S1 Text.}\n\\label{S1_file}\n{\\bf Experimental Data Statistics by Odor.} This file shows the trial-averaged spiking statistics of the experimental data dissected by a specific odor.\nContains Figs. S1-S8, and Tables S1-S2.\n\n\\paragraph*{S2 Text.}\n\\label{S2_file}\n{\\bf Supplementary Figures for the Main Modeling.} This file contains supplemental figures from modeling and analysis. Contains Figs. S9-S16.\n\n\\paragraph*{S3 Text.}\n\\label{S3_file}\n{\\bf Supplementary Material: Cortical-Cortical Network.} This file contains supplemental modeling results on a generic cortical-cortical coupled network. Contains Figs. S17-S21, and Table S3.\n\n\\paragraph*{S1 Table.} \n{\\bf Average population firing rate by odor and activity state.}\n\\paragraph*{S2 Table.} \n{\\bf Standard deviation of population firing rate by odor and activity state.}\n\\paragraph*{S3 Table.} \n{\\bf Fixed parameters for the LIF Cortical-cortical model.}\n\n\n\\paragraph*{S1 Figure.} \n{\\bf Experimental statistics by odor and activity state: Fano Factor.}\n\\paragraph*{S2 Figure.}\n{\\bf Experimental statistics by odor and activity state: spike count variance.}\n\\paragraph*{S3 Figure.}\n{\\bf Experimental statistics by odor and region: Fano Factor.}\n\\paragraph*{S4 Figure.}\n{\\bf Experimental statistics by odor and region: spike count variance.}\n\\paragraph*{S5 Figure.}\n{\\bf Experimental statistics by odor and activity state: spike count correlation.}\n\\paragraph*{S6 Figure.}\n{\\bf Experimental statistics by odor and activity state: spike count covariance.}\n\\paragraph*{S7 Figure.}\n{\\bf Experimental statistics by odor and region: spike count correlation.}\n\\paragraph*{S8 Figure.}\n{\\bf Experimental statistics by odor and region: spike count covariance.}\n\\paragraph*{S9 Figure.}\n{\\bf Cross-region correlations are smaller than within-region correlations.}\n\\paragraph*{S10 Figure.}\n{\\bf Fast analytic approximation accurately captures statistics of a multi-population firing rate model.}\n\\paragraph*{S11 Figure.}\n{\\bf Experimental observations constrain conductance parameters in analytic model.}\n\\paragraph*{S12 Figure.}\n{\\bf Analytic approximation results are robust to choice of transfer function.}\n\\paragraph*{S13 Figure.}\n{\\bf Mean input to PC must increase in the evoked state.}\n\\paragraph*{S14 Figure.}\n{\\bf Violating derived relationship $gIO < gIP$ results in statistics that are inconsistent with experimental observations.}\n\\paragraph*{S15 Figure.}\n{\\bf Violating derived relationship $gEP > gEO$ results in statistics that are inconsistent with experimental observations.}\n\\paragraph*{S16 Figure.}\n{\\bf Violating derived relationship $gEP, gIP \\gg gEO, gIO$ results in statistics that are inconsistent with experimental observations.}\n\\paragraph*{S17 Figure.}\n{\\bf Minimal firing rate model to analyze synaptic conductance strengths.}\n\\paragraph*{S18 Figure.}\n{\\bf Detailed spiking LIF model confirms the results from analytic rate model.}\n\\paragraph*{S19 Figure.}\n{\\bf Violating derived relationship $\\vert gI1\\vert < \\vert gI2\\vert$ results in statistics that are inconsistent with experimental observations.}\n\\paragraph*{S20 Figure.}\n{\\bf Violating derived relationship $gE2, gI2 \\gg gE1, gI1$ results in statistics that are inconsistent with experimental observations.}\n\\paragraph*{S21 Figure.}\n{\\bf iolating derived relationship $gE2 > gE1$ results in statistics that are inconsistent with experimental observations.}\n\n\n\n\\section*{Author Contributions}\nConceived and designed research: AKB CL. Derived expressions in theoretical methods: CL. Analyzed the data: AKB SHG WLS CL. Conceived and designed electrophysiological experiments: SHG WLS. \nWrote the paper: AKB SHG WLS CL. \n\n\\nolinenumbers\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nMotion planning and task planning have gained an enormous thrust in the robotics community in the past decade or so. Though, motion (task) planning has attracted a great deal of research in the past few decades, however recently, researchers have come up with new metrics and methodology to represent motion and task specifications. Initially, motion planning for a mobile robot started with the aim of moving a point mass from an initial position to a final position in some optimal fashion. With course of time, people started to consider planning in cluttered domains (i.e. in presence of obstacles) and also accounted for the dimensionality and the physical constraints of the robot. \n\nThough we have efficient approaches for general motion planning, very few are available or scalable to plan in dynamic environments or under finite time constraints. Temporal logics have been used greatly \nto address complex motion specifications, motion sequencing and timing behaviors etc. Historically temporal logic was originated for model checking, validation and verification in software community \\cite{baier2008}\nand later on researchers found it very helpful to use Linear Temporal logic (LTL), Computational Tree logic (CTL), Signal Temporal logic (STL) etc. for representing complex motion (or task) specifications. \nThe developments of tools such as SPIN \\cite{SPIN}, NuSMV \\cite{NuSMV} made it easier to check if a given specifications can be met by creating a suitable automaton and looking \nfor a feasible path on that automaton. However, the construction of the automaton from a given temporal logic formula is based on the implicit assumption that there is no time constraints associated with the specification.\n\nCurrently motion planning for robots is in such a stage where it is very crucial to incorporate time constraints since these constraints can arise from different aspects of the problem: dynamic environment, sequential \nprocessing, time optimality etc. Planning with time bounded objectives is inherently hard due to the fact that every transition from one state to another in the associated automaton has to be carried out, \nby some controller, exactly in time from an initial configuration to the final configuration. Time bounded motion planning has been done in heuristic ways \\cite{kant, Erdmann} and also by using \nmixed integer linear programming (MILP) framework \\cite{richards, zhou}. In this paper, we are interested in extending the idea of using LTL for time-unconstrained planning to use MITL for time-constrained \nmotion planning. In \\cite{maity}, the authors proposed a method to represent time constrained planning task as an LTL formula rather than MITL formula. This formulation reduced the complexity of \n\\textsc{Exp-space}-complete for MITL to \\textsc{Pspace}-complete for LTL. However, the number of states in the generated B\\\"{u}chi automata increases with time steps. \n\nIn this paper, we mainly focus on motion planning based on the construction of an efficient timed automaton from a given MITL specification. \nA dedicated controller to navigate the robot can be constructed for the general planning problem\nonce the discrete path is obtained from the automaton. The earlier results on construction of algorithms to verify timing properties of real \ntime systems can be found in \\cite{Alur1996}. The complexity of satisfiability and model checking problems for MTL formulas has been already studied in \\cite{SurveyOuaknine08} and it has been shown that commonly \nused real-time properties such as bounded response and invariance can be decided in polynomial time or exponential space. More works on the decidability on MTL can be found in \\cite{MTLOuaknine05} \nand the references there in. The concept of alternating timed automata for bounded time model checking can be \nfound in \\cite{Jenkins2010}. \\cite{Nickovic2010} talks about constructing deterministic timed automata from MTL specifications and this provides a unified framework to include all the future operators\nof MTL. The key to the approach of \\cite{Nickovic2010} was in separating the continuous time monitoring from the discrete time predictions of the future. We restrict our attention to generate timed automata \nfrom MITL based on the work done in \\cite{Maler2006a}. It is done by constructing a timed automaton to generate a sequence of states and another to check whether the sequence generated is actually a valid one in the sense that it satisfies the given MITL specification.\n\nThe rest of the paper is organized as follows, section \\ref{sec:pre} provides a background on MITL and the timed automata based approach for MITL. Section \\ref{sec:motion_planning} illustrates how the timed automata can be used to motion synthesis and we also provide UPPAAL \\cite{uppaal} implementation of the same. Section \\ref{sec:case} gives some examples on different time bounded tasks and shows the implementation results. Section \\ref{sec:continuous} provides a brief overview of how a continuous trajectory can be generated from the discrete plan. Finally, we conclude in section \\ref{sec:conclusion}.\n\n\\section{Preliminaries} \\label{sec:pre}\n\nIn this paper, we consider a surveying task in an area by a robot whose motion is abstracted to a graph. In particular for our particular setup, the robot motion is captured as a timed automaton (Fig. \\ref{fig:map}). Every edge is a timed transition that represents navigation of the robot from one location to other in space and every vertex of the graph represents a partition of the space. Our objective is to find an optimal time path that satisfies the specification given by timed temporal logic.\n\n\\begin{figure}%\n\\centering\n\\begin{tikzpicture}[->,>=stealth']\n\n\n\n\n \\node[normal] (S0) \n {\\begin{tabular}{l}\n pos0\\\\\n\t$z\\leq 1$\\\\\n \\end{tabular}};\n \n \\node[normal, \n right of=S0, \n node distance=4cm, \n anchor=center] (S1) \n {\\begin{tabular}{l}\n pos1:$B$\\\\\n\t$z\\leq 1$\\\\\n \\end{tabular}\n };\n \n \\node[normal,\n below of=S0,\n yshift=-1cm,\n anchor=center] (S3) \n {\\begin{tabular}{l}\n pos3:$A$\\\\\n\t$z\\leq 1$\\\\\n \\end{tabular}\n };\n\n\n \\node[normal,\n right of=S3,\n node distance=4cm,\n anchor=center] (S2) \n {\\begin{tabular}{l}\n pos2\\\\\n\t$z\\leq 1$\\\\\n \\end{tabular}\n };\n\n\n \\path \n\t(S0) \tedge node[above]{$z\\geq 1 | z:=0$} (S1)\n\t(S1)\tedge (S0)\n (S1) edge node[left,align=center]{$z \\geq 1$\\\\ $z:=0$} (S2)\n\t(S2) \tedge (S1)\n\t(S2)\tedge\tnode[below]{$z\\geq 1 | z=0$} (S3)\n\t(S3) edge (S2)\n\t(S0) \tedge node[left=0cm,align=center]{$z>0$\\\\$z:=0$} (S3)\n\t(S3)\tedge\t (S0);\n \n\\end{tikzpicture}\n\\caption{Timed Automata based on cell decomposition and robot dynamics}\n\\label{fig:map}\n\\end{figure}\n\n \\subsection{Metric Interval Temporal Logic (MITL)}\n\nMetric interval temporal logic is a specification that includes timed temporal specification for model checking. It differs from Linear Temporal Logic on the part that it has constraints on the temporal operators.\n\nThe formulas for LTL are build on atomic propositions by obeying the following grammar.\n\\begin{df} \\label{defLTL}\n \\textit{The syntax of LTL formulas are defined according to the following grammar rules:}\n \\begin{center}\n $\\phi ::= \\top ~| ~\\pi~ |~\\neg \\phi~ | ~\\phi \\vee \\phi ~| ~\\mathbf{X} \\phi | ~\\phi \\mathbf{U} \\phi ~ $\n \\end{center}\n \\end{df}\n$\\pi \\in \\Pi$ the set of propositions, $\\top$ and $\\bot(=\\neg\\top)$ are the Boolean constants $true$ and $false$ respectively. $\\vee$ denotes the disjunction operator and $\\neg$ denotes the negation operator. $\\mathbf{U}$ represents the Until operator. MITL extends the Until operator to incorporate timing constraints.\n\n\\begin{df} \\label{def1}\n \\textit{The syntax of MITL formulas are defined according to the following grammar rules:}\n \\begin{center}\n $\\phi ::= \\top ~| ~\\pi~ |~\\neg \\phi~ | ~\\phi \\vee \\phi ~|~\\phi \\mathbf{U}_I \\phi ~ $\n \\end{center} \n \\end{df}\n where $I\\subseteq [0, \\infty]$ is an interval with end points in $\\mathbb{N} \\cup \\{\\infty\\}$. $\\mathbf{U}_I$ symbolizes the timed Until operator. Sometimes we will represent $\\mathbf{U}_{[0, \\infty]}$ by $\\mathbf{U}$. \nOther Boolean and temporal operators such as conjunction ($\\wedge$), eventually within $I$ ($\\Diamond_I$), always on $I$ ($\\Box_I$) etc. can be represented using the grammar desired in definition \\ref{def1}. For example, we can express time constrained eventually operator $\\Diamond_I\\phi \\equiv \\top \\mathbf{U}_I\\phi$ and so on. In this paper all the untimed temporal logic is transformed into until operator and all the timed operator is transformed to eventually within $I$, to make it easier to generate a timed automaton.\n\nMITL is interpreted over $n$-dimensional Boolean $\\omega$-sequences of the form $\\xi: \\mathbb{N} \\rightarrow \\mathbb{B}^n$, where $n$ is the number of propositions.\n\\begin{df}\\label{ltlsym}\n \\textit{The semantics of any MTL formula $\\phi$ is recursively defined over a trajectory $(\\xi, t)$ as:\\\\\n $(\\xi, t) \\models \\pi$ iff $(\\xi, t)$ satisfies $\\pi$ at time $t$\\\\\n $(\\xi, t) \\models \\neg \\pi$ iff $(\\xi, t)$ does not satisfy $\\pi$ at time $t$\\\\\n $(\\xi, t) \\models \\phi_1\\vee \\phi_2$ iff $(\\xi, t) \\models \\phi_1$ or $(\\xi, t) \\models \\phi_2$\\\\\n $(\\xi, t) \\models \\phi_1\\wedge \\phi_2$ iff $(\\xi, t) \\models \\phi_1$ and $(\\xi, t) \\models \\phi_2$\\\\\n $(\\xi, t) \\models \\bigcirc \\phi$ iff $(\\xi, t+1) \\models \\phi$ \\\\\n $(\\xi, t) \\models \\phi_1\\mathbf{U}_I \\phi_2$ iff $\\exists s \\in I$ s.t. $(\\xi, t+s) \\models \\phi_2$ and $\\forall$ $s' \\leq s, ~ (\\xi, t+s') \\models \\phi_1$.}\n\n\\end{df}\nThus, the expression $\\phi_1 \\mathbf{U}_I \\phi_2$ means that $\\phi_2$ will be true within time interval $I$ and until $\\phi_2$ becomes true, $\\phi_1$ must be true. \nThe MITL operator $\\bigcirc \\phi$ means that the specification $\\phi$ is true at next time instance, $\\Box_I \\phi$ means that $\\phi$ is always true for the time duration $I$, \n$\\Diamond_I \\phi$ means that $\\phi$ will eventually become true within the time interval $I$. Composition of two or more MITL operators can express very sophisticated \nspecifications; for example $\\Diamond_{I_1} \\Box_{I_2} \\phi$ means that within time interval $I_1$, $\\phi$ will be true and from that instance it will hold true always for a duration\nof $I_2$. Other Boolean operators such as implication ($\\Rightarrow$) and equivalence ($\\Leftrightarrow$) can be expressed using the grammar rules and semantics given in definitions \n\\ref{def1} and \\ref{ltlsym}. More details on MITL grammar and semantics can be found in \\cite{MTL}, \\cite{Alur1996}. \n\n\\subsection{MITL and Timed Automata Based Approach}\nAn LTL formula can be transformed into a B\\\"{u}chi automaton which can be used in optimal path synthesis \\cite{Smith2010} and automata based guidance \\cite{wolff_automatonguided_2013}. Similarly, in this paper, we focus on developing a timed automata based approach for MITL based motion planning. MITL, a modification of Metric Temporal Logics (MTL), disallows the punctuation in the temporal interval, so that the left boundary and the right boundary have to be different. \nIn general the complexity of model checking for MTL related logic is higher than that of LTL. The theoretical model checking complexity for LTL is \\textsc{Pspace}-complete \\cite{Sistla1985}. The algorithm that has been implemented is exponential to the size of the formula. MTL by itself is undecidable. \nThe model checking process of MITL includes transforming it into a timed automaton \\cite{Alur1996}\\cite{Maler2006a}. CoFlatMTL and BoundedMTL defined in \\cite{CoFlatMTLBouyer08} are more expressive fragments of MTL than MITL, which can be translated to LTL-Past but with exponential increase in size. SafetyMTL \\cite{MTLOuaknine05} and MTL, evaluated over finite and discrete timed word, can be translated into alternative timed automata. Although theoretically, the results suggest many fragments of MTL are usable, many algorithms developed for model checking are based on language emptiness check, which are very different from the control synthesis i.e. finding a feasible path. From best of our knowledge, the algorithm that is close to implementation for motion planning is that of \\cite{Maler2006a}.\n\nThis paper uses the MITL and timed automaton generation based on \\cite{Maler2006a}. In the following section, the summary of the transformation and our implementation for control synthesis are discussed.\n\n\\section{MITL for Motion Planning}\\label{sec:motion_planning}\n\n\\subsection{MITL to Timed Automata Transformation}\nConsider the following requirements: a robot has to eventually visit an area $A$ and another area $B$ in time interval $[l,r]$, and the area $A$ has to be visited first. This can be captured in the following MITL,\n\\[\n\\phi = (\\neg B \\mathbf{U} A)\\wedge (\\Diamond_{[l,r]} B)\n\\] \n\nIt can be represented by a logic tree structure, where every node that has children is a temporal logic operator and every leaf node is an atomic proposition, as shown in Fig. \\ref{fig:tree}. Every link represents an input output relationship.\n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[\nlevel 1\/.style={sibling distance=30mm},level 2\/.style={sibling distance=20mm}, level distance=30pt,\nedge from parent path={\n(\\tikzparentnode) |- \n($(\\tikzparentnode)!0.5!(\\tikzchildnode)$) -|\n(\\tikzchildnode)}] \n\n \\node[normal, text width=1cm] {$\\wedge$}\n child {node[normal,text width=1cm] {$\\mathbf{U}$}\n\t child {node[normal,text width=1cm] {$\\neg$}\n\t\t\tchild {node[normal,text width=1cm]{$p(B)$}}\n\t\t}\n child {node[normal,text width=1cm] {$p(A)$}}\n\t\t}\n child {node[normal,text width=1cm] {$\\Diamond_{[l,r]}$}\n child {node[normal,text width=1cm] {$p(B)$}}\n };\n\\end{tikzpicture}\n\\caption{Logic tree representation of $\\phi$.}\n\\label{fig:tree}\n\\end{figure}\n\nThe authors in \\cite{Maler2006a} propose to change every temporal logic operator into a timed signal transducer, which is a temporal automaton that accepts input and generates output. Based on their definition the Input Output Timed Automaton (IOTA) used in this paper is defined as the following to fit the control synthesis problem,\n\n\\begin{df}[Input Output Timed Automaton]\\label{df:iota}\n\\textit{An input output timed automaton is a tuple $\\mathcal{A} =(\\Sigma, Q, \\Gamma, \\mathcal{C}, \\lambda, \\gamma, I, \\Delta, q_0, F)$, where \\\\\n$\\Sigma$ is the input alphabet, $Q$ is the finite set of discrete states, \\\\\n$\\Gamma$ is the output alphabet, $\\mathcal{C}$ is the set of clock variables, and \\\\\n$I$ is the invariant condition defined by conjunction of inequalities of clock variables. The clock variables can be disabled and activated by setting the rate of the clock $0$ or $1$ in the invariant $I$. \\\\\n$\\lambda : Q \\rightarrow \\Sigma$ is the input function, which labels every state to an input, while \\\\\n$\\gamma : Q \\rightarrow \\Gamma$ is the output function, which labels every state to an output. \\\\\n$\\Delta$ is the transition relationship between states which is defined by $(p,q,r,g)$, where $p$ is the start state, $q$ is the end state, $r$ is the clock resets, and $g$ is the guard condition on the clock variables.\\\\\n $q_0$ is the initial state of the timed automaton. \\\\\n$F$ is the set of B\\\"{u}chi states that have to be visited infinitely often.}\n\\end{df}\n\nThe transformation of Until operator and timed Eventually operator is summarized in Figs. \\ref{fig:pUq}, \\ref{fig:EI_GEN} and \\ref{fig:EI_CHK}. This is based on \\cite{Maler2006a} with minor changes to match with our definition of IOTA. In Fig. \\ref{fig:pUq}, the timed automaton for $p \\mathcal{U} q$ is shown. The inputs outputs of the states are specified in the second line within the box of each state. $p\\bar{q}$ means the inputs are $[1,0]$ and $\\bar{p}$ means the inputs can be $[0,1]$ or $[0,0]$, and $\\gamma=1$ means the output is 1. Transitions are specified in the format of $g|r$. In this case, all the transitions have guard $z>0$ and reset clock $z$. All states in this automaton are B\\\"{u}chi accepting states except $s_{p\\bar{q}}$. The B\\\"{u}chi accepting states are highlighted.\n\n\\begin{figure}%\n\\centering\n\\begin{tikzpicture}[->,>=stealth']\n\n\n\n\n \\node[state,\n\ttext width=2cm] (S0) \n {\\begin{tabular}{l}\n $s_{\\bar{p}}$\\\\\n\t\\quad $\\bar{p}\/\\gamma=0$\\\\\n \\end{tabular}};\n \n \\node[state, \n right of=S0, \n node distance=5cm, \n anchor=center,\n\ttext width=2cm] (S1) \n {%\n \\begin{tabular}{l} \n $\\bar{s}_{p\\bar{q}}$\\\\\n\t\\quad $p\\bar{q}\/\\gamma=0$\\\\\n \\end{tabular}\n };\n \n \\node[normal,\n below of=S0,\n yshift=-2cm,\n anchor=center,\n text width=2cm] (S2) \n {%\n \\begin{tabular}{l}\n $s_{p\\bar{q}}$\\\\\n \\quad $p\\bar{q}\/\\gamma=1$\\\\\n \\end{tabular}\n };\n\n\n \\node[state,\n right of=S2,\n node distance=5cm,\n anchor=center] (S3) \n {%\n \\begin{tabular}{l}\n $s_{pq}$\\\\\n \\quad $pq\/\\gamma=1$\\\\\n \\end{tabular}\n };\n\n\n \\path \n\t(S0.5) \tedge node[above]{$z>0 | z:=0$} (S1.175)\n\t(S1.185)\tedge\tnode[below]{$z>0 | z:=0$} (S0.355)\n (S0.270) \tedge node[left,align=center]{$z>0$\\\\ $z:=0$} (S2.90)\n\t(S2.5) \tedge node[above]{$z>0 | z:=0$} (S3.175)\n\t(S3.185)\tedge\tnode[below]{$z>0 | z:=0$} (S2.355)\n\t(S3.90) \tedge node[right,align=center]{$z>0$\\\\$z:=0$} (S1.270)\n\t(S0.320) \tedge node[left=0cm,align=center]{$z>0$\\\\\\quad\\quad$z:=0$} (S3.160)\n\t(S3.150)\tedge\tnode[right=-0.2cm,align=center]{$z>0$\\\\\\quad\\quad$z:=0$} (S0.332);\n\n\\end{tikzpicture}\n\n\\caption{The timed automaton for $p \\mathcal{U} q$. The inputs and outputs of the states are specified in the second line of each state. $p\\bar{q}$ means the inputs are $[1,0]$ and $\\bar{p}$ means the inputs can be $[0,1]$ or $[0,0]$, and $\\gamma=1$ means the output is 1. Transitions are specified in the format of guard$|$reset. In this case all the transitions have guard $z>0$ and reset clock $z$. All states in this automaton are B\\\"{u}chi accepting states except $s_{p\\bar{q}}$. The B\\\"{u}chi accepting states are highlighted.}%\n\\label{fig:pUq}%\n\\end{figure}\n\n\\begin{figure}%\n\\centering\n\\begin{tikzpicture}[->,>=stealth']\n\n\n\n\n \\node[normal,\n\ttext width=2cm] (S0) \n {\\begin{tabular}{l}\n $\\text{Gen}_1$\\\\\n\t\\quad $x_1'==1$\\\\\n\t\\quad $*\/\\gamma=0$\\\\\n \\end{tabular}};\n\n \\node[normal,\n\ttext width=6cm,\n\tright of=S0,\n\tnode distance=2cm,\n\tyshift=1.5cm] (Init) \n {\\begin{tabular}{l}\n $\\text{Gen}_0$\\\\\n\t\\quad $x_i'==0$, $y_i'==0$, $\\forall i=1,\\ldots, m$\\\\\n \\end{tabular}};\n \n \\node[normal, \n right of=S0, \n node distance=4cm, \n anchor=center,\n\ttext width=2cm] (S1) \n {%\n \\begin{tabular}{l} \n $\\text{Gen}_2$\\\\\n\t\\quad $y_1'==1$\\\\\n\t\\quad $*\/\\gamma=1$\\\\\n \\end{tabular}\n };\n \n \\node[normal,\n below of=S0,\n yshift=-0.5cm,\n anchor=center,\n text width=2cm] (S2) \n {%\n \\begin{tabular}{l}\n $\\text{Gen}_3$\\\\\n\t\\quad $x_2'==1$\\\\\n\t\\quad $*\/\\gamma=0$\\\\\n \\end{tabular}\n };\n\n \\node[normal,\n right of=S2,\n node distance=4cm,\n anchor=center] (S3) \n {%\n \\begin{tabular}{l}\n $\\text{Gen}_4$\\\\\n\t\\quad $y_2'==1$\\\\\n\t\\quad $*\/\\gamma=1$\\\\\n \\end{tabular}\n };\n\n\\node[state] (Sdots) [below of=S2,draw=none,node distance=1cm] {$\\ldots$};\n\\node[state] (Sdots2) [below of=S3,draw=none,node distance=1cm] {$\\ldots$};\n\n \\node[normal,\n below of=Sdots,\n yshift=0.1cm,\n anchor=center,\n text width=2cm,\n\tdraw=black, thin] (S4) \n {%\n \\begin{tabular}{l}\n $\\text{Gen}_{2m-1}$\\\\\n\t\\quad $x_m'==1$\\\\\n\t\\quad $*\/\\gamma=0$\\\\\n \\end{tabular}\n };\n\n\n \\node[normal,\n right of=S4,\n node distance=4cm,\n anchor=center] (S5) \n {%\n \\begin{tabular}{l}\n $\\text{Gen}_{2m}$\\\\\n\t\\quad $y_m'==1$\\\\\n\t\\quad $*\/\\gamma=1$\\\\\n \\end{tabular}\n };\n\n\n \\path \t(S0) \tedge node[above]{$*| y_1:=0$} (S1);\n\t\\path \t(Init.190) \tedge node[right]{$*| x_1:=0$} (S0);\n\t\\path \t(Init.350) \tedge node[right]{$*| y_1:=0$} (S1);\n\t\n \\path \t(S1)\tedge\tnode[right]{$* | x_2:=0$} (S2);\n \\path \t(S2)\tedge\tnode[above]{$* | y_2:=0$} (S3);\n \\path (S3)\tedge\tnode[right=0.2cm,align=left]{$* | x_3:=0$ \\\\ $\\ldots$} (S4);\n\n\t\\path (S4)\tedge\tnode[above]{$* | y_m:=0$} (S5);\n\t\\draw [->] (S5) -- ++(0,-0.7cm) -- node[below]{$* | x_1:=0$} ++(-5.5cm,0cm) |- (S0.west);\n\\end{tikzpicture}\n\n\\caption{The timed automaton for the generator part of $\\Diamond_{I} a$ for motion planning. $2m$ is the number of clocks required to store the states of the timed eventually ($\\Diamond_I$) operator. It is computed based on the interval $I$. Detailed computation and derivation can be found in \\cite{Maler2006a}. $x_i'$ represents the rate of the clock $x_i$. By setting the rate to be 0, we essentially deactivate the clock. The \\lq{$*$}\\rq ~ symbol means that there is no value for that particular input, output or guard for that state. There are no B\\\"{u}chi states since the time is bounded}%\n\\label{fig:EI_GEN}%\n\\end{figure}\n\n\\begin{figure}%\n\\centering\n\\begin{tikzpicture}[->,>=stealth']\n\n\n\n\n \\node[normal,\n\ttext width=1.5cm] (S0) \n {\\begin{tabular}{l}\n $\\text{Chk}_1$\\\\\n\t$y_1\\leq b$\\\\\n\t$\\bar{p}\/*$\\\\\n \\end{tabular}};\n\n \\node[normal,\n\ttext width=1.5cm,\n\tyshift=1.5cm] (Init0) \n {\\begin{tabular}{l}\n $\\text{Chk}_{00}$\\\\\n\t$x_1 \\leq a$\\\\\n \\end{tabular}};\n\n \\node[normal,\n\ttext width=1.5cm,\n\tright of=Init0,\n\tnode distance=4.5cm] (Init1) \n {\\begin{tabular}{l}\n $\\text{Chk}_{01}$\\\\\n\t$y_1 \\leq a$\\\\\n \\end{tabular}};\n \n \\node[normal, \n right of=S0, \n node distance=3cm, \n anchor=center,\n\ttext width=1.5cm] (S1) \n {%\n \\begin{tabular}{l} \n $\\text{Chk}_2$\\\\\n\t$x_2\\leq a$\\\\\n\t$p\/*$\\\\\n \\end{tabular}\n };\n\n \\node[normal, \n right of=S1, \n node distance=3cm, \n anchor=center,\n\ttext width=1.75cm] (S6) \n {%\n \\begin{tabular}{l} \n $\\text{Chk}_3$\\\\\n\t$z] \t(Init1.west) \t-| node[left,yshift=-0.5cm]{$ y_1\\geq a | *$} (S1);\n\t\\draw [->] \t(Init1.east) \t-| node[right,yshift=-0.5cm,align=center]{$y_1\\geq a$\\\\$z:=0$} (S6);\n\t\n \\path \t(S1)\tedge\tnode[right]{$x_2 \\geq a |*$}node[left]{ch!} (S2);\n \\path \t(S2)\tedge\tnode[above]{$ y_2\\geq b|*$} (S3);\n\t\\path \t(S3.5) \tedge node[above]{$ *|z:=0$} (S7.175);\n\t\\path \t(S7.184) \tedge node[below]{} (S3.355);\n\n\n \\path (S3)\tedge\tnode[right=0.2cm,align=left]{$x_3\\geq a | *$ \\\\ $\\ldots$} (S4);\n\n\t\\path (S4)\tedge\tnode[above]{$y_m\\geq b | *$} (S5);\n\t\n\t\\path \t(S5.5) \tedge node[above]{$ *|z:=0$} (S8.175);\n\t\\path \t(S8.184) \tedge node[below]{} (S5.355);\n\t\n\t\\draw [->] (S5) -- ++(0,-0.7cm) -- node[below]{$ x_1 \\geq a | *$} ++(-4.1cm,0cm) |- (S0.west);\n\\end{tikzpicture}\n\n\\caption{The timed automaton for the checker part of $\\Diamond_{I} a$ for motion planning. $2m$ is the number of clocks required for the timed eventually ($\\Diamond_I$) operator. There are no B\\\"{u}chi states since the time is bounded}%\n\\label{fig:EI_CHK}%\n\\end{figure}\n\nThe IOTA for timed eventually ($\\Diamond_{I} a$) is decomposed into two automata, the generator generates predictions of the future outputs of the system, while the checker verifies that the generated outputs actually fit the inputs. Detailed derivations and verifications of the models can be found in \\cite{Maler2006a}. The composition between them is achieved through the shared clock variables. Additional synchronization (`ch!') is added in our case to determine the final satisfaction condition for the control synthesis. A finite time trajectory satisfies the MITL, when the output signal of the generator automaton (Fig. \\ref{fig:EI_GEN}) includes a pair of raising edge and falling edge verified by the checker automaton. The transition from $\\text{Chk}_2$ to $\\text{Chk}_4$ (Fig. \\ref{fig:EI_CHK}) marks the exact time when such falling edge is verified. This guarantees that the time trajectory before the synchronization is a finite time trajectory that satisfies the MITL. \n\nThe composition of IOTA based on logic trees such as that of Fig. \\ref{fig:tree} is defined similar to \\cite{Maler2006a} with some modifications to handle cases when logic nodes have two children, for example the until and conjunction operators.\n\n\\begin{df}[I\/O Composition] \\hfill \\\\\n\\textit{\nLet $\\mathcal{A}^1_1 =(\\Sigma^1_1, Q^1_1, \\Gamma^1_1, \\mathcal{C}^1_1, \\lambda^1_1, \\gamma^1_1, I^1_1, \\Delta^1_1, {q^1_1}_0, F^1_1)$, $\\mathcal{A}^1_2 =(\\Sigma^1_2, Q^1_2, \\Gamma^1_2, \\mathcal{C}^1_2, \\lambda^1_2, \\gamma^1_2, I^1_2, \\Delta^1_2, {q^1_2}_0, F^1_2)$ be the input sides of the automaton. If there is only one, then $\\mathcal{A}^1_1$ is used. Let $\\mathcal{A}^2 =(\\Sigma^2, Q^2, \\Gamma^2, \\mathcal{C}^2, \\lambda^2, \\gamma^2, I^2, \\Delta^2, q_0^2, F^2)$ be the output side of the automaton. Because of the input output relationship between them, they should satisfies the condition that $[\\Gamma^1_1, \\Gamma^1_2] = \\Sigma^2$. The composition is an new IOTA such that,\n\\[\n\\mathcal{A} =(\\mathcal{A}^1_1, \\mathcal{A}^1_2) \\otimes (\\mathcal{A}^2) = ([\\Sigma_1^1,\\Sigma_2^1], Q, \\Gamma^2, \\mathcal{C}, \\lambda, \\gamma, I, \\Delta, q_0, F)\n\\]\nwhere\n\\begin{align*}\nQ=&\\{(q^1_1,q^1_2,q^2) \\in Q^1_1 \\times Q^1_2 \\times Q^2, \\\\\n& s.t. (\\gamma_1^1(q_1^1),\\gamma_2^1(q_2^1)) = \\lambda^2(q^2)\\}\n\\end{align*}\n$\\mathcal{C} = (\\mathcal{C}^1_1 \\cup \\mathcal{C}^1_2 \\cup \\mathcal{C}^2)$, $\\lambda (q^1_1,q^1_2,q^2) = [\\lambda^1_1(q^1_1),\\lambda^1_2(q^1_2)]$, $I_{(q^1_1,q^1_2,q^2)} = I^1_{(q^1_1,q^1_2)} \\cap I^2_{q^2}$, $q_0 = ({q^1_1}_0, {q^1_2}_0, q_0^2)$ and $F = F^1_1 \\cap F^1_2 \\cup F^2$.\n}\n\\end{df}\n\n\\begin{figure*}%\n\\centering\n\\includegraphics[width=6.6in]{until}%\n\\caption{The Resulting timed automaton in UPPAAL of $\\phi_1$. The purple colored texts under the state names represent $I$. The green colored texts along the edges represent guard conditions, while the blue ones represent clock resets. The B\\\"{u}chi accepting states are represented by a subscript b in state names. }%\n\\label{fig:until}%\n\\end{figure*}\n\n\\subsection{Path Synthesis using UPPAAL}\nThe overall path synthesis framework is summarized as following,\n\\begin{itemize}\n\t\\item First, the robot and the environments are abstracted to a timed automaton (TA) $\\mathcal{T}_\\text{map}$ using cell decomposition, and the time to navigate from one cell to another is estimated based on the robot's dynamics. For example Fig. \\ref{fig:map}.\n\t\\item Second, MITL formula is translated to IOTA $\\mathcal{A}$ using method described in previous section.\n\t\\item IOTA $\\mathcal{A}$ is then taken product with the TA $\\mathcal{T}_\\text{map}$ using the location label. For instance $pos1:B$ in Fig. \\ref{fig:map} will be taken product with all states in IOTA that do not satisfy the predicate $p(a)$ but satisfies $p(b)$.\n\t\\item The resulting timed automata are then automatically transformed to an UPPAAL \\cite{uppaal} model with additional satisfaction condition verifier. An initial state is chosen so that the output at that state is 1. Any finite trajectory which initiated from that state and satisfying the following conditions will satisfy the MITL specification. Firstly, it has to visit at least one of the B\\\"{u}chi accepting states, and secondly, it has to meet the acceptance condition for the timed eventually operator. To perform such a search in UPPAAL, a final state is added to allow transitions from any B\\\"{u}chi accepting state to itself. A verification automaton is created to check the finite acceptance conditions for every timed eventually operator.\n\t\\item An optimal timed path is then synthesized using the UPPAAL verification tool.\n\\end{itemize}\nThe implementation of the first and the second step is based on parsing and simplification functions of ltl2ba tool \\cite{gastin_fast_2001} with additional capabilities to generate IOTA. We then use the generated IOTA to autogenerate a python script which constructs the UPPAAL model automatically through PyUPPAAL, a python interface to create and layout models for UPPAAL. The complete set of tools\\footnote{The tool is available on \\url{https:\/\/github.com\/yzh89\/MITL2Timed}} is implemented in C to optimize speed.\n\n\\section{Case Study and Discussion} \\label{sec:case}\nWe demonstrate our framework for a simple environment and for some typical temporal logic formulas. Although our tool is not limited by the complexity of the environment, we use a simple environment to make the resulting timed automaton easy to visualize. Let us consider the timed automaton from the abstraction in Fig. \\ref{fig:map} and the LTL formula is given as the following,\n\\[\\phi_1 = (\\neg A \\mathbf{U} B) \\wedge (\\Diamond A).\n\\]\nThis specification requires the robot to visit the area $B$ first and eventually visit $A$ also. The resulting automaton based on the methods in the previous section is as shown in Fig. \\ref{fig:until}. Each state corresponds to a product state between a state in $\\mathcal{T}_\\text{map}$ and a state in IOTA $\\mathcal{A}$. The B\\\"{u}chi accepting states are indicated by an additional \\textit{b} in their state names. We obtained the optimal path by first adding a final state and linking every accepting states to it, and then using UPPAAL to find one of the shortest path that satisfies condition ``$E<> final$''. UPPAAL will then compute one fastest path in the timed automaton that goes to final state, if one such exists. If such exists, this feasible path is a finite trajectory that satisfies the specification. \nIn this paper, we are more interested in planning a path that satisfies MITL, so finite time trajectory is a valid solution.\nThe initial states of the automaton is loc0 which is the only state at pos0 that outputs 1. The optimal trajectory is $loc0 \\rightarrow loc2 \\rightarrow loc7 \\rightarrow loc6_b$, in the product automaton. This trajectory means that the optimal way for a robot to satisfy the LTL is to traverse the map in the following order, $pos0 \\rightarrow pos1:B \\rightarrow pos0 \\rightarrow pos3:A$.\n\n\\begin{figure*}%\n\\includegraphics[width=7in]{always_eventually_I}%\n\\caption{This shows one of the resulting timed automata in UPPAAL of $\\phi_2$ corresponding to the checker of timed eventually operator and untimed always. Some of the edges are further annotated by synchronization signal (ch!).}%\n\\label{fig:always_event_I}%\n\\end{figure*}\n\n\\begin{figure*}[!tb]\n \\centering{\\subfloat[]{\\includegraphics[width=3.5in]{always_eventually_I_0}\n\t\t\\label{fig:always_event_I1}}\n\t\t\\hfil\n\t\t\\subfloat[]{\\includegraphics[width=1.5in]{always_eventually_I_1}\n\t\t\\label{fig:always_event_I2}}}\n\t\t\\caption{Fig. (a) shows the other timed automata of $\\phi_2$ corresponding to the generator of timed eventually. Fig. (b) shows the verification timed automaton, that checks if the falling edge of generator is ever detected, i.e. if a synchronization signal (ch!) has happened. This signal marks the end of a full eventually cycle. Similar to the LTL case, we ask UPPAAL to check for us the following property, if there is a trajectory that leads to the final states in (a) and (b). The optimal path in this case is $(loc19,loc28)\\rightarrow (loc3,loc28) \\rightarrow (loc3,loc22_b)$. The states are products of states of Fig. \\ref{fig:always_event_I} and Fig. (a). This path corresponds to $(pos0,t\\in[0,1])\\rightarrow (pos3:A,t\\in[1,2]) \\rightarrow (pos0,t\\in[2,3])$ in physical space. Repeating this path will satisfy $\\phi_2$. }\n\\label{fig:always_event_I_2}\n\\end{figure*}\n\nIn the second test case, the environment stays the same and the requirement is captured in a MITL formula $\\phi_2$\n\\[\n\\phi_2 = \\Box \\Diamond_{[0,2]} A\n\\]\nThis requires the robot to perform periodic survey of area \\textit{A} every 2s. The resulting timed automata are shown in Fig. \\ref{fig:always_event_I} and Fig. \\ref{fig:always_event_I_2}. As we discussed earlier, if a synchronization signal (ch!) is sent, the falling edge for output of generator automaton is detected and verified. This marks the end of a finite trajectory that satisfies the MITL constraints. We used the automaton in Fig. \\ref{fig:always_event_I_2} (b) to receive such signal. Similar to the LTL case, we ask UPPAAL to find a fastest path that leads to the final states in Fig. \\ref{fig:always_event_I_2}(a) and \\ref{fig:always_event_I_2}(b) if such exists.\n\nThe optimal trajectory in this case is $(loc19,loc28)\\rightarrow (loc3,loc28) \\rightarrow (loc3,loc22_b)$, which corresponds to $(pos0,t\\in[0,1])\\rightarrow (pos3:A,t\\in[1,2]) \\rightarrow (pos0,t\\in[2,3])$. Then this trajectory repeats itself.\n\nAll the computations are done on a computer with 3.4GHz processor and 8GB memory. Both of the previous examples require very small amount of time $(<0.03s)$. We also tested our implementation against various other complex environments and MITL formulas. The Table \\ref{MITLTime} summarizes our results for complex systems and formulas. The map we demonstrated earlier is a 2x2 map (Fig. \\ref{fig:map}), we also examine the cases for 4x4 and 8x8 grid maps. The used temporal logic formulas are listed below. The time intervals in the formula is scaled accordingly to the map size.\n\\[\n\\phi_3 = \\Diamond_{[0,4]} A \\wedge \\Diamond_{[0,4]} B\n\\]\n\\[\\phi_4 = \\Diamond_{[2,4]} A \\wedge \\Diamond_{[0,2]} B\\]\n\n\\begin{table}\n\\begin{center}\n\\caption{Computation Time for typical MITL formula}\n\\label{MITLTime}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nMITL& Map & Transformation & Num of Timed & Synthesis\\\\\nFormula& Grid &Time & Automata Transitions & Time \\\\\\hline\n$\\phi_1$ &2x2 & $<0.001s$ & 22 & 0.016s\\\\ \\hline\n$\\phi_2$ &2x2 & $0.004s$ & 69 & 0.018s\\\\ \\hline\n$\\phi_3$ &2x2 & $0.40s$ & 532 & 0.10s\\\\ \\hline\n$\\phi_4$ &2x2 & $0.46s$ & 681 & 0.12s\\\\ \\hline\n$\\phi_1$ &4x4 & $0.004s$ & 181 & 0.062s\\\\ \\hline\n$\\phi_1$ &8x8 & $0.015s$ & 886 & 0.21s\\\\ \\hline\n$\\phi_2$ &8x8 & $0.015s$ & 1795 & 0.32s\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIt can be seen from the Table \\ref{MITLTime} that our algorithm works very well with common MITL formulas and scales satisfactorily with the dimensions of the map.\n\n\\section{Continuous Trajectory generation}\\label{sec:continuous} \n\nIn this section, we briefly talk about generating a continuous trajectory from the discrete motion plan obtained from the timed automaton. Let us consider the nonholonomic dynamics of a unicycle car as given in (\\ref{eq:dynamics}).\n\n\\begin{equation} \\label{eq:dynamics}\n \\dot{\\begin{bmatrix}\nx\\\\y\\\\ \\theta\n\\end{bmatrix}} =u\\begin{bmatrix}\n\\cos\\theta \\\\ \\sin\\theta \\\\0\n\\end{bmatrix}+ \\omega \\begin{bmatrix}\n0\\\\0\\\\1\n\\end{bmatrix}\n\\end{equation}\nwhere $\\omega$ and $u$ are the control inputs. It should be noted that the above nonholonomic dynamics is controllable and we assume no constraints on the control inputs at this point. \nThe above sections provide the sequence of cells to be visited in the grid like environment (Fig. \\ref{fig:traj}). \n\\begin{figure}\n\\centering\n\\includegraphics[width=3in]{Continuous_trajectory.png} \n\\caption{Workspace and the continuous trajectory for the specification $\\phi_1$. The initial location is the top-left corner cell (I).}\n\\label{fig:traj}\n\\end{figure}\n\nThe output of the timed automaton are treated as the time-stamped way points for the robot to move. We have to assure that the robot moves from one way point to the next with the given initial and final time and at the same time, the trajectory should remain within the associated cells. \n\nSince our environment is decomposed in rectangular cells, the robot will only move forward, turn right, turn left and make a U-turn. We synthesize a controller that can make the robot to perform these elementary motion segments within the given time.\n\n For moving forward the input $\\omega$ is chosen to be $0$ and the velocity $u$ is tuned so that the robot reaches the final position in time. For turning left and turning right $\\omega$ is chosen to take positive and negative values respectively so that a circular arc is traversed. Similarly the U-turn is also implemented so that the robot performs the U-turn within a single cell. \n \n Let us denote the state of the system at time $t$ by the pair $(q,t)$ i.e. $x(t)=q_1,~y(t)=q_2$ and $\\theta(t)=q_3$ where $q=[q_1,~q_2,~q_3]$. Then we have the following lemma on the optimality of the \n control inputs.\n \n \\begin{lm} \\label{lem:opt}\n \\textit{ If $\\bar u(t)$ and $\\bar \\omega (t)$, $t \\in [0,1]$ is a pair of control inputs s.t. the dynamics moves from the state $(q_0,0)$ to $(q_1,1)$, then $u(t_0+t)=\\frac1\\lambda \\bar u(\n \\frac t\\lambda)$ and $\\omega(t_0+t)=\\frac1\\lambda \\bar \\omega( \\frac t\\lambda)$ move the system from $(q_0,t_0)$ to $(q_1,t_0+\\lambda)$ for any $\\lambda >0$. \\\\\n Moreover, if $\\bar u$ and $\\bar \\omega$ move the system optimally, i.e.\n \\begin{equation}\n J(\\bar u,\\bar \\omega)=\\min_{u(\\cdot),w(\\cdot)} \\int_{0}^{1} [r_1 u^2(t)+r_2 w^2(t)]dt\n \\end{equation}\n then $u$ and $\\omega$ given above are also optimal for moving the system from $(q_0,t_0)$ to $(q_1, t_0+\\lambda)$, i.e.\n \\begin{equation}\n J_1(u,\\omega)=\\min_{u_1(\\cdot),w_1(\\cdot)} \\int_{t_0}^{t_0+\\lambda}[r_1 u^2_1(t)+r_2 w^2_1(t)]dt. \n \\end{equation}\n}\n \\end{lm}\n\n\\begin{proof}\nLet us first denote \n\\[\nG(q)=\\begin{bmatrix}\n \\cos(\\theta(t)) && 0\\\\ \\sin(\\theta(t)) && 0\\\\0 && 1\n \\end{bmatrix}\n\t\t\t\t\t\t\t\t\t\t\t\t\t\\]\n where $q=[x(t), y(t),\\theta(t)]$. Therefore, dynamics (\\ref{eq:dynamics}) can be written as $\\dot q=G(q) \\begin{bmatrix}\n u \\\\\\omega\n \\end{bmatrix}.\n$\nLet us now consider $\\bar q(t)=[x(t_0+\\lambda t),~y(t_0+\\lambda t),~\\theta(t_0+\\lambda t)]$. \nTherefore, $\\dot{\\bar q} =\\lambda G(\\bar q)\\begin{bmatrix} u(t_0+\\lambda t) \\\\ \\omega (t_0+\\lambda t) \\end{bmatrix}$. \nUsing the definition of $u$ and $\\omega$ in the lemma, we get $\\dot{\\bar q}=G(\\bar q)\n\\begin{bmatrix}\n\\bar u \\\\\\bar \\omega\n\\end{bmatrix}$\nBy the hypothesis of the lemma, $\\bar{u}$ and $\\bar \\omega$ move the system from $(q_0,0)$ to $(q_1,1)$ i.e. from $[x(t_0),y(t_0),\\theta(t_0)]=q_0$ to $q_1=\\bar q(1)=[x(t_0+\\lambda),y(t_0+\\lambda),\\theta(t_0+\\lambda)]$.\\\\\n\nFor optimality, let the proposed $u,~\\omega$ be not optimal and $u^*$ and $\\omega^*$ are optimal ones i.e. \n\\begin{equation}\n\\int_{t_0}^{t_0+\\lambda}[r_1{u^*}^2(t)+r_2{\\omega^*(t)}^2]dt \\le \\int_{t_0}^{t_0+\\lambda}[r_1u^2(t)+r_2\\omega^2(t)]dt\n\\label{eq:opt1}\n\\end{equation}\nNow let us construct $\\bar u^*(t)=\\lambda u^*(t_0+\\lambda t)$ and $\\bar \\omega^*(t)=\\lambda \\omega^*(t_0+\\lambda t)$.\n\nTherefore from (\\ref{eq:opt1}),\n\\begin{multline*}\n\\int_{0}^{1}[r_1{u^*}^2(t_0+\\lambda s)+r_2{\\omega^*}^2(t_0 +\\lambda s)]ds \\\\\n\\leq \\int_{0}^{1}[r_1u^2(t_0+\\lambda s)+r_2\\omega^2(t_0+\\lambda s)]ds\n\\end{multline*}\n\n\\begin{equation} \\int_{0}^{1}[r_1 {\\bar {u^*}}^2(s)+r_2{\\bar {\\omega^*}}^2(s)]ds \\le \\int_{0}^{1}[r_1{\\bar u}^2(s)+r_2\\bar \\omega^2(s)]ds \\label{Eqn:ineq1} \\end{equation}\n\nBut, by the hypothesis, $\\bar u$ and $\\bar \\omega$ are optimal and hence \n\\begin{equation} \\label{Eqn:ineq2}\n \\int_{0}^{1}[r_1 {\\bar {u^*}}^2(s)+r_2{\\bar {\\omega^*}}^2(s)]ds \\ge \\int_{0}^{1}[r_1{\\bar u}^2(s)+r_2\\bar \\omega^2(s)]ds\n\\end{equation}\nCombining (\\ref{Eqn:ineq1}) and (\\ref{Eqn:ineq2}) we get,\n\\begin{equation}\n \\int_{0}^{1}[r_1 {\\bar {u^*}}^2(s)+r_2{\\bar {\\omega^*}}^2(s)]ds = \\int_{0}^{1}[r_1{\\bar u}^2(s)+r_2\\bar \\omega^2(s)]ds\n\\end{equation}\n\nAfter changing the dummy variables inside integration again, one can obtain \n\n\\begin{equation}\n \\int_{t_0}^{t_0+\\lambda}[r_1 { {u^*}}^2(s)+r_2{{\\omega^*}}^2(s)]ds = \\int_{t_0}^{t_0+\\lambda}[r_1{u}^2(s)+r_2\\omega^2(s)]ds\n\\end{equation}\n\nHence the proposed $u$ and $\\omega$ are optimal whenever $\\bar u$ and $\\bar \\omega$ are optimal.\n \\end{proof}\n \\begin{rem}\n Lemma \\ref{lem:opt} states that if the controls for elementary motions from initial time $0$ to final time $1$ are synthesized, then by properly scaling and shifting in the time and scaling the magnitude, controls for any movement from any initial time to any final time can be synthesized without further solving any optimization problem.\n \\end{rem}\n\n\n\\section{Conclusion} \\label{sec:conclusion}\nIn this paper, we have presented a timed automaton based approach to generate a discrete plan for the robot to perform temporal tasks with finite time constraints. We implemented the algorithm in an efficient and generic way so that it can translate the time constraints and temporal specifications to timed automaton models in UPPAAL and synthesize the path accordingly. We then demonstrated our algorithm in grid type environments with different MITL formulas. We have considered grid type of environment for our case studies, but it can be generalized to most of the motion planning problems when the environment can be decomposed into cells. We also provide a brief overview of how an optimal continuous trajectory can be generated from the discrete plan. For future works, we are considering to extend the work to include dynamic obstacles as well as for multiagent system. \n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Swift detection}\n\nA bright long-soft GRB\\,060117 was detected by \\swift\\\/ satellite on\nJanuary 17, 2006, at 6:50:01.6\\,UT.\nIt showed a multi-peak structure with T$_\\mathrm{90}$=$16\\pm1$\\,s with\nmaximum peak flux $48.9\\pm1.6\\,{\\rm ph\\,cm}^{\\rm -2} {\\rm s}^{\\rm -1}$. Thus, \nGRB060117 was --- in terms of peak flux --- the most intense GRB detected so far by Swift\nCoordinates computed by \\swift\\\/ were available within 19\\,s and\nimmediately distributed by GCN \\cite{gcn4538}.\n\n\\section{FRAM and optical transient observation}\n\nFRAM is part of the Pierre Auger cosmic-ray observatory \\cite{auger},\nand its main purpose is to immediately monitor the atmospheric\ntransmission. \nFRAM works as an independent, RTS2-driven, fully robotic\nsystem, and it performs a photometric calibration of the sky on various\nUV-to-optical wavelengths using a 0.2\\,m telescope and a photoelectric\nphotomultiplier. \n\nFRAM received the notice at 06:50:20.8\\,UT, 19.2\\,s after the trigger\nand immediately started the slew. \nThe first exposure started at 06:52:05.4, 123.8\\,s after the GRB. \nEight images with different exposures were taken before the observation\nwas terminated. \nA bright, rapidly decaying object was found, and its presence was\nreported by \\cite{gcn4535} soon after the discovery.\nThe FRAM lightcurve for this optical transient is in Figure 1.\n\n\\begin{figure}[t!] \n \\begin{center} \\label{fig4}\n \\resizebox{0.4\\hsize}{!}{\n\t\\includegraphics{jelinekm_fig1.eps}\n\t} \n \\caption{\\small The R-band afterglow lightcurve of GRB\\,060117. \n The lightcurve is fitted as a superposition of reverse shock\n (dotted line) and forward shock (dashed line).\n\t}\n \\end{center} \n\\end{figure}\n\n\\section{Interpretation}\n\nOur preffered interpretation (based on the work of \\cite{shao05}) is to fit the \ndata as a \ntransition between the reverse and the forward shock with the passage of\nthe typical frequency break $\\nu_m$ through the observed passband at\ntime $t_{m,f}$. \nCorresponding decay indices are $\\alpha_{\\mathrm Reverse}$=2.49$\\pm$0.05 and\n$\\alpha_{\\mathrm Forward}$=1.47$\\pm$0.03 (see Fig\\,3).\n\nOther possible interpretations and more details about FRAM telescope, data processing and other\nfollow-up attempts can be found in \\cite{aal}.\n\n\\acknowledgments\n{\\small The telescope FRAM was built and is operated under the support of the\nCzech Ministry of Education, Youth, and Sports through its grant\nprograms LA134 and LC527.\nMJ would like to thank to the Spanish Ministry of Education and Science\nfor the support via grants AP2003-1407, ESP2002-04124-C03-01, and\nAYA2004-01515 (+ FEDER funds), MP was supported by the Grant Agency of\nthe Academy of Sciences of the Czech Republic grant B300100502.}\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nA wide range of phenomena in biology and social sciences can be described by the combination of classical (local) - linear or nonlinear - \\emph{diffusion} with some \\emph{nonlocal transport} effects. Examples can be found in bacterial chemotaxis \\cite{keller_segel,painter_hillen}, animal swarming phenomena \\cite{okubo,capasso}, pedestrian movements in a dense crowd \\cite{hughes}, and more in general in socio-economical sciences \\cite{sznajd,aletti}. In a fairly general setting, a set of $N$ individuals $x_1,\\ldots,x_N$ located in a sub-region of the Euclidean space $\\mathbb R^d$ are subject to a drift which is affected by the status of each other individual. In most of the above-mentioned applications, such a ``biased drift'' can be expressed through a set of first order ordinary differential equations\n\\begin{equation}\\label{eq:intro_discrete}\n\\dot{x}_i(t)=v[(x_1(t),\\ldots,x_N(t)],\\qquad i=1,\\ldots,N,\n\\end{equation}\nin which the velocity law $v$ is known. Having in mind a particle system obeying the laws of classical mechanics or electromagnetism, the set of equations \\eqref{eq:intro_discrete} is quite unconventional due to the absence of inertia. On the other hand, this choice is very common in the modelling of socio-biological systems, mainly due to the following three reasons. \n\\begin{itemize}\n\\item Inertial effects are negligible in many socio-biological aggregation phenomena. Even in cases in which the system is appropriate for a fluid-dynamical description, a `thinking fluid' model, with a velocity field already adjusted to equilibrium conditions, is often preferable compared to a second order approach. The typical examples are in traffic flow and pedestrian flow modelling. Moreover, it is well known in the context of cells aggregation modelling that the time of response to the chemoattractant signal is, most of the times, negligible. Finally, inertia is almost irrelevant in many contexts of socio-economical sciences, such as opinion formation dynamics.\n\\item First-order modelling turns out to simulate real patterns in concrete relevant situations arising in traffic flow, pedestrian motion, and cell-aggregation, and such an achievement is satisfactory in many situations, in applied fields often lacking a unified rigorous modelling approach.\n\\item In several practical problem such as the behaviour of a crowd in a panic situation, the model can be seen as the outcome of an optimization process performed externally, in which the \"best strategy\" needed to solve the problem under study (reaching the exit in the shortest possible time, in the crowd example) is transmitted to the individuals in real time (e.g. a set of ``dynamic\" evacuation signals in a smart building).\n\\end{itemize}\n\nFurther to the `discrete' approach \\eqref{eq:intro_discrete}, these models are often posed in terms of a ``continuum\" PDE approach via a continuity equation\n\\begin{equation}\\label{eq:intro_continuum}\n\\partial_t \\rho + \\mathrm{div}(\\rho v[\\rho]) = 0,\n\\end{equation}\nin which $\\rho(\\cdot,t)$ is a time dependent probability measure expressing the distribution of individuals on a given region at a given time, and in which the continuum velocity map $v=v[\\rho]$ is detected as a reasonable ``cross-grained\" version of its discrete counterpart in \\eqref{eq:intro_discrete}. The modelling of biological movements and socio-economical dynamics are often simulated at the continuum level as the PDE approach is more easy-to-handle in order to analyse the qualitative behaviour of the whole system, in the form e.g. of the emergence of a specific pattern, or the occurrence of concentration phenomena, or the formation of shock waves or travelling waves. In this regard, the descriptive power of the qualitative properties of the solutions in the continuum setting is an argument in favour of the PDE approach \\eqref{eq:intro_continuum}. On the other hand, the intrinsic discrete nature of the applied target situations under study would rather suggest an `individual based' description as the most natural one. For this reason, the justification of continuum models \\eqref{eq:intro_continuum} as a \\emph{many particle limits} of \\eqref{eq:intro_discrete} in this context is an essential requirement to validate the use of PDE models.\n\nAs briefly mentioned above, the velocity law $v=v[\\rho]$ in the PDE approach \\eqref{eq:intro_continuum} may include several effects ranging from diffusion effect to external force fields, from nonlinear convection effects to nonlocal interaction terms. We produce here a non-exhaustive list of results available in the literature in which the continuum PDE \\eqref{eq:intro_continuum} is obtained as a limit of a system of interacting particles, with a special focus on \\emph{deterministic} particle limits, i.e. in which particles move according to a system of ordinary differential equations (i.e. without any stochastic term). The presence of a diffusion operator has several possible counterparts at the discrete level. The literature on this subject involving probabilistic methods is extremely rich and, by now, well established, see e.g. \\cite{varadhan1,varadhan2,presutti} only to mention a few. A first attempt (mainly numerical) to a fully deterministic approach to diffusion equations is due to \\cite{russo}, see \\cite{gosse} for the case of nonlinear diffusion. \n\nWithout diffusion and with only a local dependency $v=v(\\rho)$, an extensive literature has been produced based on probabilistic methods (exclusion processes), see e.g. \\cite{Ferrari,Ferrari-Nejjar}. A first rigorous result based on fully deterministic ODEs at the microscopic level for a nonlinear conservation law was recently obtained in \\cite{DiFra-Rosini}. Nonlocal velocities $v=W\\ast \\rho$ have been considered as a special case of the theory developed in \\cite{carrillo_choi_hauray}, with $W$ a given kernel (possibly singular) using techniques coming from kinetic equations, see \\cite{hauray_jabin}. In all the above mentioned results, the particle system is obtained as a discretised version of the Lagrangian formulation of the system.\n\nA slightly more difficult class of problems is the one in which the velocity $v=v[\\rho]$ depends \\emph{both locally and non-locally} from $\\rho$.\nSeveral results about the mathematical well-posedness of such models are available in the literature, which use either classical nonlinear analysis techniques or numerical schemes. In the paper \\cite{colombo} a similar model is studied in the context of pedestrian movements, and the existence and uniqueness of entropy solutions is proven. We also mention \\cite{defilippis_goatin}, which covers a more general class of problems, and \\cite{amadori_shen} covering a similar model in the context of granular media. A quite general result was obtained in \\cite{piccoli_rossi} in which the velocity map $\\rho\\mapsto v[\\rho]$ is required to be Lipschitz continuous as a map from the space of probability measures (equipped with some $p$-Wasserstein distance) with values in $C(\\mathbb R^d)$, and the authors prove convergence of a time-discretised Lagrangian scheme. We also mention \\cite{betancourt}, in which a special class of local-nonlocal dependencies has been considered, however in a different numerical framework. We also recall at this stage the related results in \\cite{Bu_DiF_Dol,Bu_Dol_Sch} on the overcrowding preventing version of the Keller-Segel system for chemotaxis, in which the existence and uniqueness of entropy solutions is proven. To our knowledge, no papers in the literature provide (so far) a rigorous result of convergence of a deterministic particle system of the form \\eqref{eq:intro_discrete} towards a PDE of the form \\eqref{eq:intro_continuum} in the case of local-nonlocal dependence $v=v[\\rho]$. Indeed, the result in \\cite{piccoli_rossi} does not apply to this case in view of the Lipschitz continuity assumption on the velocity field, see also a similar result in \\cite{goatin_rossi}.\n\nIn this paper we aim at providing, for the first time, a rigorous deterministic many-particle limit for the one-dimensional \\emph{nonlocal interaction equation} with \\emph{nonlinear mobility}\n\\begin{equation}\\label{eq:intro_PDE}\n \\partial_t \\rho -\\partial_x (\\rho v(\\rho) K\\ast \\rho) = 0,\n\\end{equation}\nin which $v$ and $K$ satisfy the following set of assumptions:\n\\begin{itemize}\n\\item[(Av)] $v \\in C^1([0,+\\infty))$ is a decreasing function such that $v(0)=v_{max}>0$, $v(M)=0$ for some $M>0$, $v'<0$ on interval $(0,M]$, $v\\equiv 0$ on $[M,+\\infty)$.\n\\item[(AK)] $K \\in C^2(\\mathbb R)$, $K(0)=0$ (without restriction), $K(x)=K(-x)$ for all $x\\in \\mathbb R$, $K'(x)>0$ for $x>0$, $K'' \\in \\mathrm{Lip}_{loc}(\\mathbb R)$.\n\\end{itemize}\nAlso in view of the applications in mind, the unknown $\\rho=\\rho(x,t)$ in \\eqref{eq:intro_PDE} will be assumed to be non-negative throughout the whole paper. The PDE \\eqref{eq:intro_PDE} is coupled with an initial condition\n\\begin{equation}\\label{eq:intro_initial}\n \\rho(x,0)=\\bar{\\rho}(x),\\qquad \\bar{\\rho}\\in L^\\infty(\\mathbb R)\\cap BV(\\mathbb R),\\,\\,\\,0\\leq \\bar{\\rho}(x)\\leq M,\\,\\,\\,\\hbox{$\\mathrm{supp}(\\bar{\\rho})$ compact}.\n\\end{equation}\nThe constant $M$ here plays the role of a \\emph{maximal density}, which is supposed not to be exceeded by the density for all times. Clearly, the property $\\rho\\in [0,M]$ has to be proven to be invariant with respect to time. We notice that the total mass of $\\rho$ in \\eqref{eq:intro_PDE} is formally conserved. For simplicity, throughout the paper we shall set\n We set\n\\begin{equation*}\n \\| \\bar{\\rho} \\|_{L^1(\\mathbb R)}=1\\,.\n\\end{equation*}\nWe set $[\\bar{x}_{min}, \\bar{x}_{max}]$ as the closed convex hull of $\\mathrm{supp}\\bar{\\rho}$.\n\nOur goal is to approximate rigorously the solution $\\rho$ to \\eqref{eq:intro_PDE} with initial datum $\\bar{\\rho}$ via a set of moving particles. More precisely, we aim to proving that the \\emph{entropy solution} of the Cauchy problem for \\eqref{eq:intro_PDE} can be obtained as the large particle limit of a discrete Lagrangian approximation of the form \\eqref{eq:intro_discrete}. Such a Lagrangian approximation can be introduced as follows as a reasonable generalization of particle approximations considered previously in the literature in \\cite{DiFra-Rosini,DiFra-Fagioli-Rosini,DiFra-Fagioli-Rosini-Russo1,DiFra-Fagioli-Rosini-Russo2}. For a fixed integer $N$ sufficiently large, we split $[\\bar{x}_{min}, \\bar{x}_{max}]$ into $N$ intervals such that\nthe integral of the restriction of $\\bar{\\rho}$ over each interval equals $1\/N$. More precisely, we let $\\bar{x}_0= \\bar{x}_{min}$ and $\\bar{x}_N = \\bar{x}_{max}$, and define recursively the points $\\bar{x}_i$ for $i \\in \\{ 1,\\, \\ldots,\\, N-1\\}$ as\n\\begin{equation}\\label{eq:dscr_IC}\n\\bar{x}_i = \\sup \\left\\lbrace x \\in \\mathbb R : \\int_{\\bar{x}_{i-1}}^x \\bar{\\rho}(x) dx < \\frac{1}{N} \\right\\rbrace\\,.\n\\end{equation}\nIt is clear from the construction that $\\int_{\\bar{x}_{N-1}}^{\\bar{x}_N} \\bar{\\rho}(x) dx = 1\/N$ and $\\bar{x}_0 < \\bar{x}_1 < \\ldots\\, < \\bar{x}_{N-1} < \\bar{x}_N$.\nConsider then $N+1$ particles located at initial time at the positions $\\bar{x}_i$ and let them evolve accordingly to the following system ODEs\n\\begin{equation}\\label{Odes}\n\\dot{x}_i(t) = - \\frac{v(R_i(t))}{N} \\sum_{j > i} K'(x_i(t) - x_j(t)) - \\frac{v(R_{i-1}(t))}{N} \\sum_{j < i} K'(x_i(t) - x_j(t))\\,,\n\\end{equation}\nwith $i\\in\\{0,\\ldots,N\\}$, where the discrete density $R_i(t)$ is defined as follows\n\\[ R_i (t):= \\frac{1}{N(x_{i+1}(t) - x_i(t))},\\qquad i=0,\\ldots, N-1. \\]\nIn \\eqref{Odes}, each particle $x_i$ has mass $1\/N$. We are then in position to define the $N$-discrete density\n\\begin{equation}\\label{eq:discrete_density}\n\\rho^N(t,\\,x):= \\sum_{i=0}^{N-1} R^N_i (t) \\chi_{[x_i(t),\\,x_{i+1}(t))}(x).\n\\end{equation}\nWe observe that $\\rho^N(t,\\cdot)$ has total mass equal to $1$ for all times. We refer to system \\eqref{Odes} as \\emph{non-local Follow-the-leader} scheme, as in fact this system is a non-local extension of the classical Follow-the-leader scheme previously considered in the literature. More in detail, system \\eqref{Odes} is motivated as follows. The right-hand side of \\eqref{Odes} represents the velocity of each particle. Therefore, it has to be reminiscent of a discrete Lagrangian formulation of the Eulerian velocity $-v(\\rho)K'\\ast \\rho$ in the continuity equation \\eqref{eq:intro_PDE}. Now, since we are in one-space dimension, the discrete density $R_i$ is a totally reasonable replacement for the continuum density $\\rho$, except that one has to decide whether the discrete density should be constructed in a forward, backward, or centred fashion. Our choice of splitting the velocity $\\dot{x}_i$ into a backward and forward term is motivated by the sign of the nonlocal interaction $K'(x_i-x_j)$, which is concordant with the sign of $x_i - x_j$. Hence, since $K'(x)$ is negative on $x<0$, particles labelled by $x_j$ with $x_j>x_i$ yield a drift on $x_i$ oriented towards the \\emph{positive} direction. Since the role of the nonlinear mobility term $\\rho v(\\rho)$ is that of preventing overcrowding at high densities (consistently with the assumption of $v$ being monotone decreasing), such a drift term should be ``tempered\" by the position of the $(i+1)$-th particle. This motivates the use or $v(R_i)$ in the sum with $x_j>x_i$. A symmetric argument justifies the use of $v(R_{i-1})$ in the remaining part of the sum with $x_j0$, the discrete density $\\rho^N$ constructed in \\eqref{eq:discrete_density} converges almost everywhere and in $L^1([0,\\,T] \\times \\mathbb R)$ to the unique entropy solution $\\rho$ of the Cauchy problem\n\\begin{equation}\\label{CauchyProblem}\n\\left\\lbrace \\begin{array}{ll}\n\\partial_t \\rho = \\partial_x(\\rho v(\\rho) K' \\ast \\rho) &(t,\\,x) \\in (0,\\,T] \\times \\mathbb R\\,,\\\\\n\\rho(0,\\,x) = \\bar{\\rho}(x) &x \\in \\mathbb R\\,.\n\\end{array}\\right.\n\\end{equation}\n\\end{theorem}\n\nAs a by-product, the above result also imply existence of entropy solutions for \\eqref{CauchyProblem}, a task which has been touched in other papers previously \\cite{colombo,defilippis_goatin,Bu_DiF_Dol,betancourt}.\nImplicitly, our results also asserts the uniqueness of entropy solutions for \\eqref{eq:intro_PDE}, a side result that we shall prove as well in the paper, similarly to what done in \\cite{Bu_DiF_Dol,Bu_Dol_Sch}.\n\nThe need of the entropy condition to define a suitable notion of solution semigroup for \\eqref{eq:intro_PDE} is not only motivated by the possibility of proving its uniqueness. We actually prove in the paper that a mere notion of weak solution does not infer the well-posedness of the semigroup as multiple weak solution can be produced with the same initial condition. \n\nOur paper is structured as follows. In Section \\ref{sec:2} we introduce the nonlocal follow-the-leader particle scheme and prove that it satisfies a discrete maximum principle, a crucial ingredient in order to deal with the particle approximation in the sequel of the paper. In Section \\ref{sec:convergence} we prove all the estimates needed in order to detect strong $L^1$ compactness for the approximating sequence $\\rho^N$. The main ingredient of this section is the $BV$ estimate proven in Proposition \\ref{totalvariation}. We emphasize that the presence of an \\emph{attractive} interaction potential in the particle system implies most likely a \\emph{growth} w.r.t. time of the total variation. Therefore, one has to check that the blow-up in finite time of the total variation is avoided. In Section \\ref{sec:consi}, we prove that the limit of the approximating sequence is an entropy solution in the sense of Definition \\ref{solentropicadef}. This task is quite technical as it requires checking a discrete version of Kruzkov's entropy condition. In Section \\ref{sec:discussion} we provide an explicit example of non uniqueness of weak solutions, which has links with the admissibility of steady states. Finally, in Section \\ref{sec:numerics} we complement our results with numerical simulations.\n\n\\section{The non-local Follow-the-leader scheme}\\label{sec:2}\n\nIn this section we introduce and analyse in detail our approximating particle scheme \\eqref{Odes}. Here the macroscopic variable $\\rho$ does not need to be labelled by $N$, as $N$ is supposed to be fixed throughout the whole section. The regularity assumptions on $v$ and $K$ in (Av) and (AK) imply that the right-hand side of \\eqref{Odes} is locally Lipschitz with respect to the $N+1$-tuple $(x_0,x_1,\\ldots,x_N)$ as long as we can guarantee that the denominator in $R_i$ does not vanish. Such a property is a consequence of the following \\emph{Discrete Maximum Principle}, ensuring that the particles cannot touch each other at any time. This implies both the (global-in-time) existence of solutions of the system~\\eqref{Odes} for all times $t>0$, and the conservation of the initial particle ordering during the evolution.\n\n\\begin{lemma}[Discrete Maximum Principle]\\label{lem:maximum}\nLet $N\\in\\mathbb N$ be fixed and assume that (Av) and (AK) hold. In particular, let $M>0$ be as in assumption (Av). Let $\\bar{x}_0<\\bar{x}_1<\\ldots<\\bar{x}_N$ be the initial positions for \\eqref{Odes}, and assume that\n\\begin{equation}\\label{eq:MPcondition}\n\\bar{x}_{i+1}-\\bar{x}_i \\geq \\frac{1}{MN}\n\\end{equation}\nThen every solution $x_i(t)$ to the system~\\eqref{Odes} satisfies\n\\begin{equation}\\label{MaxPrinc}\n\\frac{1}{MN} \\leq x_{i+1}(t) - x_i(t) \\qquad \\hbox{for all $i \\in \\{0,\\,\\ldots,\\,N-1\\}$ and for all $t \\in [0,\\,+\\infty)$}.\n\\end{equation}\nConsequently, the unique solution $(x_0(t),\\ldots,x_N(t))$ to \\eqref{Odes} with initial condition $(\\bar{x}_0,\\ldots,\\bar{x}_N)$ exists globally in time.\n\\end{lemma}\n\n\\begin{proof}\nLet $T_{max}>0$ be the maximal existence time for \\eqref{Odes}. Due to the assumptions (Av) and (AK), the local-in-time solution $(x_0(t),\\ldots,x_N(t))$ is $C^1$ on $[0,T_{max})$. If we prove that \\eqref{MaxPrinc} holds on $[0,T_{max})$, this will automatically prove global existence by a simple continuation principle. Arguing by contradiction, assume that $t_1< T_{max}$ is the first instant where two consecutive particles are the distance $1\/MN$ and get closer afterwards, i.e.\n\\[t_1=\\inf\\{ t \\in [0,\\,T] : \\,\\,\\hbox{there exists}\\, i\\, : x_{i+1}(t) - x_i(t) = 1\/MN \\},\\]\nand there exists $t_2 \\in (t_1,\\,T]$ such that\n\\[x_{i+1}(t) - x_i(t) < \\frac{1}{MN} \\qquad \\forall t \\in (t_1,\\, t_2]\\,. \\]\nNotice that the minimality of $t_1$ ensures that all particles maintain their initial order for all $t \\in [0,\\,t_1)$. At time $t_1$ we have $R_i(t_1)=0$ due to (Av). Substituting this value in the equation \\eqref{Odes} for $x_i$, we easily see that only the terms $j0$ or some index $h\\leq i$ such that $\\dot{x}_k(t_1)<0$, otherwise any two consecutive particles would be placed at distance $1\/MN$ and the system would be static for all $t \\in (t_1,\\,T]$, which would contradict the existence of $t_2$.\n\nThe above considerations imply that we can assume, without loss of generality, that\n\\[ \\dot{x}_{i+1}(t_1) >0,\\quad \\mbox{ and } \\quad\\dot{x}_i(t_1) \\leq 0\\,. \\]\nLet $\\varepsilon_{i+1}>0$ be small enough such that $t_1+ \\varepsilon_{i+1} < t_2$, then by Taylor expansion one has\n\\[ x_{i+1}(t) = x_{i+1}(t_1) + \\dot{x}_{i+1}(t_1)(t-t_1) + o(|t-t_1|)\\,, \\]\nwhere, up to taking $\\varepsilon_{i+1}$ even smaller, the contribute $o(t-t_1)$ does not affect the sign of $\\dot{x}_{i+1}(t_1)(t-t_1)$. As a consequence, $x_{i+1}(t) > x_{i+1}(t_1)$ for all $t\\in (t_1,t_1 +\\varepsilon_{i+1})$ and a symmetric argument gives also $x_i(t) \\leq x_i(t_1)$ for all $t\\in (t_1,t_1+\\varepsilon_{i})$. In particular, we deduce that\n\\[ x_{i+1}(t) - x_i(t) \\geq x_{i+1}(t_1) - x_i(t_1) = \\frac{1}{MN} \\quad \\forall t \\in (t_1,\\, t_1 + \\min\\{\\varepsilon_{i},\\,\\varepsilon_{i+1} \\}) \\]\nand this contradicts the existence of $t_2$. This argument ensures both the validity of~\\eqref{MaxPrinc} and the existence of solutions for all times $t>0$.\n\\end{proof}\n\nLet us consider the discrete density\n\\begin{equation*}\n\\rho(t,\\,x):= \\sum_{i=0}^{N-1} R_i (t) \\chi_{[x_i(t),\\,x_{i+1}(t))}(x).\n\\end{equation*}\nA straightforward consequence of Lemma \\ref{lem:maximum} is that\n\\[\\rho(t,x)\\leq M\\qquad \\hbox{for all $(t,x)\\in [0,+\\infty)\\times\\mathbb R$}.\\]\nMoreover, we observe that $\\rho$ has unit mass on $\\mathbb R$ for all times.\n\nAs already mentioned before, a straightforward consequence of the above Maximum Principle is that the particles can never touch or cross each other. In particular, the particle $x_0$ will have no particles at its left for all times, which means that the ODE for $x_0$ will only feature terms with $j>0$ on the nonlocal sum. A symmetric statement holds for $x_N$. As a consequence of that $\\dot{x}_0(t) \\geq 0$ and $\\dot{x}_N(t) \\leq 0$ for all $t$, thus the support of $\\rho^N(t,\\,\\cdot)$ is bounded by $\\ell$ uniformly in $N$ and $t$.\nWe summarize this property in the next lemma.\n\n\\begin{lemma}\\label{lem:support}\nUnder the same assumptions of Lemma \\ref{lem:maximum}, the support of $\\rho(t,\\cdot)$ is contained in the interval $[\\bar{x}_0,\\bar{x}_N]$ for all times $t\\in [0,+\\infty)$.\n\\end{lemma}\n\n\\section{Convergence of particle scheme}\\label{sec:convergence}\n\nWe now focus on the converge of the particle scheme \\eqref{Odes}, where the initial condition \\eqref{eq:dscr_IC} is constructed from an $L^\\infty(\\mathbb R)$ initial density $\\bar\\rho$ having compact support and finite total variation.\n\nThe proof of Theorem~\\ref{main} relies on two main steps: the first one consists in proving that the discrete density $(\\rho^N)$ defined in \\eqref{eq:discrete_density} is strongly convergent (up to a subsequence) to a limit $\\rho$ in $L^1([0,T] \\times \\mathbb R)$, the second one is to show that the limit $\\rho$ is a weak entropy solution of~\\eqref{CauchyProblem} according to Definition \\ref{solentropicadef}. In this section we take care of the former step. As we will show in Propositions~\\ref{totalvariation} and~\\ref{continuitytime} below, the sequence $(\\rho_N)_{N\\in \\mathbb N}$ satisfies good compactness properties with respect to the space variables but, on the other hand, we cannot reach a uniform $L^1$ control on the time oscillations. In our case, we are only able to prove a uniform time continuity estimate with respect to the $1$-Wasserstein distance (see \\cite{villani_book}), which nevertheless will suffice to achieve the required compactness in the product space. Such a strategy recalls the one used in \\cite{DiFra-Rosini} for the case of a scalar conservation law. The main result of this section is the content of the following\n\n\\begin{theorem}\\label{convergence}\nUnder the assumptions of Theorem~\\ref{main}, the sequence $\\rho^N$ is strongly relatively compact in $L^1([0,T] \\times \\mathbb R)$\n\\end{theorem}\n\nThe proof of Theorem~\\ref{convergence} relies on a generalized statement of the celebrated Aubin-Lions Lemma (see \\cite{RS,DiFra-Matthes,DiFra-Fagioli-Rosini}) that we recall here for the reader's convenience. In what follows, $d_1$ is the $1$-Wasserstein distance.\n\n\\begin{theorem}[Generalized Aubin-Lions Lemma]\\label{thm:aubin}\nLet $\\tau>0$ be fixed. Let $\\eta^N$ be a sequence in $L^{\\infty}((0,\\,\\tau); L^1(\\mathbb R))$ such that\n$\\eta^N(t,\\,\\cdot) \\geq 0$ and $\\| \\eta^N(t,\\,\\cdot) \\|_{L^1(\\mathbb R)}=1$ for every $N\\in\\mathbb N$ and $t\\in [0,\\,\\tau]$.\nIf the following conditions hold\n\\begin{enumerate}\n\\item[I)] $\\sup_{N} \\int_0^{\\tau} \\left[\\|\\eta^N(t,\\,\\cdot)\\|_{L^1(\\mathbb R)}dt + TV\\big[ \\eta^N(t,\\,\\cdot)\\big]+ \\mathrm{meas}(\\mathrm{supp}[\\eta^N(t,\\cdot)])\\right]dt < \\infty$,\n\\item[II)] there exists a constant $C>0$ independent from $N$ such that $d_{W^1}\\big( \\eta^N(t,\\,\\cdot), \\eta^N(s,\\,\\cdot) \\big) < C |t-s|$ for all $s,\\,t \\in (0,\\,\\tau)$,\n\\end{enumerate}\nthen $\\eta^N$ is strongly relatively compact in $L^1([0,\\,\\tau]\\times \\mathbb R)$.\n\\end{theorem}\n\nIn view of Theorem \\ref{thm:aubin}, the result in Theorem \\ref{convergence} will follow as a consequence of the following two propositions.\n\n\\begin{prop}\\label{totalvariation}\nLet $\\bar{\\rho},\\,v,\\,K$ and $T$ be as in the statement of Theorem~\\ref{main}. Then, there exists a positive constant $C>0$ (only depending on $K$, $v$, and on $\\mathrm{supp}(\\bar\\rho)$) such that for every $N \\in \\mathbb N$ one has\n\\begin{equation}\\label{TV}\nTV[\\rho^N(t,\\,\\cdot)] \\leq TV[\\bar{\\rho}] e^{Ct} \\qquad \\hbox{for all $t \\in [0,T]$}\\,.\n\\end{equation}\n\\end{prop}\n\n\\begin{prop}\\label{continuitytime}\nLet $\\bar{\\rho},\\,v,\\,K$ and $T$ be as in the statement of Theorem~\\ref{main}. Then, there exists a positive constant $C>0$ (only depending on $K$) such that\n\\begin{equation}\\label{contime}\nd_{W^1}\\big( \\rho^N(t,\\,\\cdot), \\rho^N(s,\\,\\cdot) \\big) < C |t-s| \\quad \\hbox{for all $s,\\,t \\in (0,\\,T)$, and for all $N \\in \\mathbb N$}\\,.\n\\end{equation}\n\\end{prop}\n\n\nThe remaining part of this section is devoted to prove Propositions~\\ref{totalvariation} and~\\ref{continuitytime}.\nFor future use we compute\n\\begin{align}\n\\dot{R}_i(t) =& -N(R_i)^2 (\\dot{x}_{i+1} - \\dot{x}_i) = -N(R_i)^2 \\Big[ -2v(R_i)\\frac{1}{N}K'(x_{i+1} - x_i) \\nonumber \\\\\n& -(v(R_{i+1}) - v(R_i))\\frac{1}{N}\\sum_{j > i+1} K'(x_{i+1} - x_j)\\nonumber\\\\\n& - v(R_i) \\frac{1}{N}\\sum_{j > i+1}\\big( K'(x_{i+1} - x_j) - K'(x_i - x_j) \\big) \\nonumber\\\\\n& -(v(R_i) - v(R_{i-1}))\\frac{1}{N}\\sum_{j0$. The total variation of $\\rho^N$ at time $t$ is given by\n\\begin{align*}\nT&V[\\rho^N(t,\\,\\cdot)] = R_0(t) + R_{N}(t) + \\sum_{i=0}^{N-1} |R_{i+1}(t) - R_i(t)| \\\\\n&=\\sum_{i=1}^{N-1}R_i [\\mathrm{sign}(R_i - R_{i-1}) - \\mathrm{sign}(R_{i+1}-R_i)] - R_0 (\\mathrm{sign}(R_1-R_0) -1)\\\\\n&\\, + R_N( \\mathrm{sign}(R_N - R_{N-1}) + 1) \\\\\n&=\\mu_0(t)R_0(t)+\\mu_N(t)R_N + \\sum_{i=1}^{N-1}R_i \\mu_i,\n\\end{align*}\nwhere we set for brevity\n\\begin{align*}\n & \\mu_i(t):=\\mathrm{sign}(R_i(t) - R_{i+1}(t)) - \\mathrm{sign}(R_{i-1}(t) - R_i(t))\\qquad i=1,\\ldots,N-1,\\\\\n & \\mu_0(t)= \\big( 1- \\mathrm{sign}(R_1 -R_0) \\big),\\\\\n & \\mu_N(t)=\\big(1+ \\mathrm{sign}(R_N -R_{N-1}) \\big).\n\\end{align*}\nThen we can compute\n\\begin{align*}\n \\frac{d}{dt} TV[\\rho^N(t,\\,\\cdot)] &= \\dot{R}_0(t) +\\dot{R}_{N}(t) + \\sum_{i=0}^{N-1} \\mathrm{sign}\\big(R_{i+1}(t) - R_i(t)\\big)\\big( \\dot{R}_{i+1}(t) - \\dot{R}_i(t) \\big) \\\\\n&= \\mu_0(t)\\dot{R}_0(t) + \\mu_N(t) \\dot{R}_N(t) + \\sum_{i=1}^{N-1} \\mu(R_i(t))\\dot{R}_i(t)\\,.\n\\end{align*}\nThe value of the coefficient $\\mu_i(t)$ clearly depends on the positions of the consecutive particles, it is easy to see that for $i \\in \\{ 1,\\,\\ldots,\\,N-1\\}$\n\\begin{equation*}\n\\mu_i(t)= \\left\\lbrace\\begin{array}{lll}\n-2 \\quad &\\mbox{if $R_{i+1} > R_i$ and $R_{i-1}> R_i$},\\\\\n2 \\quad &\\mbox{if $R_{i+1} < R_i$ and $R_{i-1}< R_i$},\\\\\n0 \\quad &\\mbox{if $R_{i+1} \\geq R_i \\geq R_{i-1}$ or $R_{i-1}\\geq R_i \\geq R_{i+1}$,}\n\\end{array}\n\\right.\n\\end{equation*}\nmoreover\n\\begin{equation*}\n\\mu_0(t)=\\left\\lbrace \\begin{array}{ll}\n0 \\quad \\mbox{if $R_1 < R_0$,}\\\\\n2 \\quad \\mbox{if $R_1 > R_0$,}\n\\end{array} \\right.\n\\qquad\n\\mu_N(t)= \\left\\lbrace\\begin{array}{ll}\n0 \\quad \\mbox{if $R_{N-1} > R_N$,}\\\\\n2 \\quad \\mbox{if $R_{N-1} < R_N$.}\n\\end{array}\\right.\n\\end{equation*}\nRecalling \\eqref{eq:explcit_Rdot}, we can rewrite\n\\begin{equation}\\label{eq:ddt_estimate}\n\\frac{d}{dt} TV[\\rho^N(t,\\,\\cdot)] = \\mu_0(t)\\dot{R}_0(t) + \\mu_N(t) \\dot{R}_N(t) - \\sum_{i=1}^{N-1} \\mu_i(t)(R_i(t))^2 \\emph{I}_i - \\sum_{i=1}^{N-1} \\mu_i(t)R_i(t) \\emph{II}_i\\,,\n\\end{equation}\nwhere\n\\[ \\emph{I}_i = -\\big(v(R_{i+1}(t)) - v(R_i(t))\\big)\\sum_{j > i+1} K'(x_{i+1}(t) - x_j(t)) - \\big(v(R_i(t)) - v(R_{i-1}(t))\\big)\\sum_{j < i} K'(x_i(t) - x_j(t))\\,, \\]\nand\n\\begin{align*}\n & \\emph{II}_i = -R_i(t)v(R_i(t)) \\sum_{j \\neq i,\\,i+1} \\big( K'(x_{i+1}(t) - x_j(t)) - K'(x_i(t) - x_j(t)) \\big)\\\\\n & \\,\\, - 2R_i(t)v(R_i(t))K'(x_{i+1}(t)-x_i(t))\\,.\n\\end{align*}\nLet us first estimate $-\\sum_{i=1}^{N-1}\\mu_i(t)(R_i(t))^2\\emph{I}_i$ in \\eqref{eq:ddt_estimate}. Clearly, the only relevant contributions in the sum come from the particles $x_i$ for which $\\mu_i(t)\\neq 0$. However, if the index $i$ is such that $\\mu_i(t)=-2$, then $R_{i+1}, R_{i-1} > R_i$ and the monotonicity of $v$ implies\n\\[ v(R_{i+1}(t)) - v(R_i(t)) <0\\,, \\quad\\mbox{ and }\\quad v(R_i(t))-v(R_{i-1}(t)) >0\\,. \\]\nThe assumption (AK) on $K$ ensures that $\\emph{I}_i < 0$, thus, on the other hand, $\\mu_i(t)(R_i (t))^2 \\emph{I}_i <0$.\nAn analogous argument implies that, if $i$ such that $\\mu_i(t)=2$, then $\\emph{I}_i > 0$ and $2(R_i (t))^2 \\emph{I}_i >0$. These considerations lead immediately to\n\\begin{equation}\\label{stuck[I]}\n-\\sum_{i=1}^{N-1}\\mu_i(t)(R_i(t))^2\\emph{I}_i < 0\\,.\n\\end{equation}\nLet us now focus on $-\\sum_{i=1}^{N-1}\\mu_i(t)R_i(t)\\emph{II}_i$. In this case, we would like to obtain an upper bound in terms of $TV[\\rho^N(t,\\,\\cdot)]$ and for this purpose we need to estimate $| II_i |$. We recall that $K'$ is locally Lipschitz and that $v(\\rho)\\in [0,v_{max}]$. The former in particular implies that $K'$ has finite Lipschitz constant on the compact interval $[-2\\mathrm{meas}(\\mathrm{supp}(\\bar\\rho)),2\\mathrm{meas}(\\mathrm{supp}(\\bar\\rho))]$, we name such a constant $L=L(\\bar\\rho)$. We get\n\\begin{align*}\n|\\emph{II}_i| &= R_i(t)|v(R_i(t))| \\left| -\\sum_{j \\neq i,\\,i+1} \\big(K'(x_{i+1}(t) - x_j(t)) - K'(x_i(t) - x_j(t)) \\big) -2K'(x_{i+1}(t) - x_i(t))\\right| \\\\\n&\\leq R_i(t)\\,L\\,v_{max} \\frac{N-2}{N} \\frac{1}{R_i(t)} + 2v_{max}\\,L\\frac{1}{N} \\leq L\\,v_{max}\\,,\n\\end{align*}\nand this gives\n\\begin{equation}\\label{stuck[II]}\n\\left| -\\sum_{i=1}^{N-1}\\mu_i(t)R_i(t)\\emph{II}_i \\right| \\leq L\\,v_{max}\\left| \\sum_{i=1}^{N-1}\\mu_i(t)R_i(t) \\right| \\leq L\\,v_{max}\\,TV[\\rho^N(t,\\,\\cdot)].\n\\end{equation}\nWe can now focus on $\\dot{R}_0$ and $\\dot{R}_N$. Since the setting is symmetric, we only present the argument for $\\mu_0(t)\\dot{R}_0$ and leave the one for $\\mu_N(t)\\dot{R}_N$ to the reader. Since $\\mu_0(t)\\neq 0$ only if $R_1(t)>R_0(t)$, without restriction we can assume $(v(R_1) - v(R_0)) \\leq 0$ and can compute\n\\begin{align*}\n\\mu_0\\dot{R}_0 &= \\mu_0R_0 [R_0v(R_1)\\sum_{j>1}\\big(K'(x_1-x_j) - K'(x_0-x_j)\\big) +2R_0v(R_0)K'(x_1-x_0)] \\\\\n&\\quad + \\mu_0(R_0)^2(v(R_1) - v(R_0))\\sum_{j>1} K'(x_0 - x_j) \\\\\n&\\leq \\mu_0R_0 [R_0v(R_1)\\sum_{j>1}\\big(K'(x_1-x_j) - K'(x_0-x_j)\\big) +2R_0v(R_0)K'(x_1-x_0)]\\,.\n\\end{align*}\nMoreover,\n\\[ \\left| R_0v(R_1)\\sum_{j>1}\\big(K'(x_1-x_j) - K'(x_0-x_j)\\big) +2R_0v(R_0)K'(x_1-x_0)\\right| \\leq v_{max}\\,L\\frac{N-1}{N} + \\frac{2v_{max}\\,L}{N}\\,. \\]\nIn particular, $\\mu_0\\dot{R}_0 \\leq (3CL) R_0$ and\n\\begin{equation}\\label{primoeultimotermine}\n\\mu_0\\dot{R}_0 + \\mu(R_N)\\dot{R}_N \\leq 3v_{max}\\,L\\, (R_0 + R_N) \\leq 3v_{max}\\,L\\, TV[\\rho^N(t,\\,\\cdot)]\\,.\n\\end{equation}\nBy putting together~\\eqref{stuck[I]},~\\eqref{stuck[II]} and~\\eqref{primoeultimotermine} we get estimate~\\eqref{dTVlimitata} and~\\eqref{TV} follows as a consequence of Gronwall Lemma.\n\\end{proof}\n\nWe now prove the equi-continuity w.r.t. time with respect to the $1$-Wasserstein distance for $\\rho^N$.\n\n\\proofof{Proposition~\\ref{continuitytime}}\nAssume without loss of generality that $0< s0$ independent of $N$ such that\n\\[ \\| X_{\\rho^N(t,\\cdot)} - X_{\\rho^N(s,\\,\\cdot)} \\|_{L^1([0,1])} < C|t-s|, \\]\nfor all $s,\\,t \\in (0,T)$.\nBy the definition of $\\rho^N$ we can explicitly compute\n\\[ X_{\\rho^N(t,\\,\\cdot)}(z) = \\sum_{i=0}^{N-1} \\left(x_i^N(t) + \\left(z-i\\frac{1}{N}\\right) \\frac{1}{R_i^N(t)}\\right) \\textbf{1}_{[i\\frac{1}{N},\\,(i+1)\\frac{1}{N})}(z)\\,. \\]\nTherefore,\n\\begin{align*}\nd_{1}\\big( \\rho^N(t,\\,\\cdot), \\rho^N(s,\\,\\cdot) \\big) &= \\| X_{\\rho^N(t,\\,\\cdot)} - X_{\\rho^N(s,\\,\\cdot)} \\|_{L^1([0,\\,1])} \\\\\n&\\leq \\sum_{i=0}^{N-1} \\int_{i\/N}^{(i+1)\/N} \\left| x_i^N(t) - x_i^N(s) + \\left(z - \\frac{i}{N} \\right) \\left(\\frac{1}{R_i^N(t)} - \\frac{1}{R_i^N(s)} \\right) \\right| dz \\\\\n&\\leq \\sum_{i=0}^{N-1} \\frac{1}{N} |x_i^N(t) - x_i^N(s)| + \\sum_{i=0}^{N-1} \\left|\\frac{1}{R_i^N(t)} - \\frac{1}{R_i^N(s)} \\right| \\int_{i\/N}^{(i+1)\/N} \\left(z - \\frac{i}{N} \\right)dz \\\\\n&= \\sum_{i=0}^{N-1} \\frac{1}{N} |x_i^N(t) - x_i^N(s)| + \\sum_{i=0}^{N-1} \\frac{1}{2N^2} \\int_s^t \\left| \\frac{d}{d\\tau} \\frac{1}{R_i^N(\\tau)}\\right| d\\tau \\\\\n&\\leq 3 \\sum_{i=0}^{N} \\frac{1}{N} \\int_s^t \\left|\\dot{x}_i^N(\\tau)\\right| d\\tau\\,,\n\\end{align*}\nwhere in the last inequality we used that\n\\[ \\left| \\frac{d}{d\\tau} \\frac{1}{R_i^N(\\tau)} \\right| = N |\\dot{x}_{i+1}^N(\\tau) - \\dot{x}_i^N(\\tau)| \\leq N |\\dot{x}_{i+1}^N(\\tau)| + N|\\dot{x}_i^N(\\tau)|\\,. \\]\nNotice that we can control $|\\dot{x}_i^N(\\tau)|$ uniformly in $N$ and in $\\tau$. Indeed, recalling the assumption (AK), setting $L$ as the Lipschitz constant of $K'$ on the interval $[-2\\mathrm{meas}(\\mathrm{supp}(\\bar\\rho)),2\\mathrm{meas}(\\mathrm{supp}(\\bar\\rho))]$ as in the proof of Proposition \\ref{totalvariation}, we have\n\\[ |\\dot{x}_i^N(\\tau)| = \\frac{1}{N}\\left|-v(R_i(t)) | \\sum_{j>i} K'(x_i - x_j) - v(R_{i-1})\\sum_{j0$ depending only on $\\bar\\rho$ such that\n \\[\\sup_{t\\geq 0}\\|W\\ast \\rho^N(t,\\cdot)-W\\ast \\hat{\\rho}^N(t)\\|_{L^1}\\leq \\frac{C}{N},\\]\n for all $N\\in \\mathbb{N}$. To prove this, let $\\gamma^N_o(t)$ be an optimal plan between $\\rho^N(t,\\cdot)$ and $\\hat{\\rho}^N(t,\\cdot)$ with respect to the cost $c(x)=|x|$. We then estimate, for all $t\\geq 0$,\n \\begin{align*}\n & \\|W\\ast \\rho^N-W\\ast \\hat{\\rho}^N\\|_{L^1(\\mathbb R)} = \\int_\\mathbb R \\left|\\int_\\mathbb R W(x-y)\\, d\\rho^N(t,\\cdot)(y)-\\int_\\mathbb R W(x-y)\\,d\\hat{\\rho}^N(t)(y)\\right|\\, dx\\\\\n & \\ =\\int_\\mathbb R \\left|\\iint_{\\mathbb R^2} \\left(W(x-y)-W(x-z)\\right)\\, d\\gamma_0^N(t)(y,z)\\right|\\, dx\\\\\n & \\ \\leq C \\int_\\mathbb R \\iint_{\\mathbb R^2}|y-z|d\\gamma_0^N(t)(y,z),dx,\n \\end{align*}\n where we have used that the supports of $\\rho^N$ and $\\hat{\\rho}^N$ are contained in $\\mathrm{supp}(\\bar\\rho)$ which is bounded and independent of time. By definition of $1$-Wasserstein distance we therefore have\n \\[\\|W\\ast \\rho^N-W\\ast \\hat{\\rho}^N\\|_{L^1(\\mathbb R)} \\leq C d_1(\\rho^N(t,\\cdot),\\hat{\\rho}^N(t)) \\leq \\frac{\\tilde{C}}{N},\\]\n for some suitable constant $\\tilde{C}>0$ in view of Lemma \\ref{lem:empirical1}.}\n\\end{remark}\n\nOur next goal is to prove that the entropy inequality\n\\[0 \\leq \\int_0^T \\int_{\\mathbb R} |\\rho^N - c|\\varphi_t - \\mathrm{sign}(\\rho^N - c)[(f(\\rho^N) - f(c))K' \\star \\hat{\\rho}^N \\varphi_x - f(c)K'' \\ast \\hat{\\rho}^N \\varphi]dxdt\\]\nholds for every non-negative test function $\\varphi$ with compact support in $\\mathcal{C}^{\\infty}_c ((0,+\\infty)\\times \\mathbb R)$, every constant $c\\geq 0$ and every $N$ \\emph{large enough}. Such a goal, which requires some tedious calculations, is however not enough to prove that the limit $\\rho$ of the previous section is an entropy solution because the of the discontinuity of the sign function in the above inequality, which does not allow to pass to the limit for $\\rho^N\\rightarrow \\rho$ almost everywhere and in $L^1$. To bypass this problem we shall then introduce a $\\delta$-regularization of the sign function in order to first let $N\\rightarrow +\\infty$ and then $\\delta\\searrow 0$. In the last part of the section we prove the uniqueness of entropy solutions, which allows to conclude that the whole approximating sequence $\\rho^N$ converges to $\\rho$, thus completing the proof of our main Theorem~\\ref{main}.\n\n\n\\begin{lemma}\\label{entropiarhoN}\nFor every non negative $\\varphi \\in C^{\\infty}_c ((0,+\\infty)\\times \\mathbb R),\\,c\\geq 0$ and $N \\in \\mathbb N$ the following inequality holds\n\\begin{equation}\\label{entropiaN}\n\\liminf_{N\\rightarrow +\\infty}\\int_0^T \\int_{\\mathbb R} |\\rho^N - c| \\varphi_t - \\mathrm{sign}(\\rho^N -c)[(f(\\rho^N) - f(c))K' \\ast \\hat{\\rho}^N \\varphi_x - f(c)K'' \\ast \\hat{\\rho}^N \\varphi] dx dt \\geq 0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nLet $T>0$ such that $\\mathrm{supp}\\varphi\\subset[0,T]\\times \\mathbb R$. The basic idea of the proof is rather simple, although the computations are quite technical: we need to rewrite the left hand side of the inequality so that it is possible to isolate a term with positive sign and then show that the remaining terms give negligible contributions as $N \\to \\infty$.\nBy definition of $\\rho^N$ and $\\hat{\\rho}^N$ we obtain\n\\[ \\int_0^T \\int_{\\mathbb R} |\\rho^N - c| \\varphi_t - \\mathrm{sign}(\\rho^N -c)[(f(\\rho^N) - f(c))K' \\ast \\varphi_x - f(c)K'' \\ast \\rho^N \\varphi] dx dt = B.T._1 + \\sum_{i=0}^{N-1} I_i + \\sum_{i=0}^{N-1} II_i, \\]\nwhere\n\\begin{align*}\n& I_i := \\int_0^T \\int_{x_i}^{x_{i+1}} |R_i^N -c| \\varphi_t\\,dxdt ,\\\\\n& II_i := - \\int_0^T \\int_{x_i}^{x_{i+1}} \\mathrm{sign}(R_i^N - c)(f(R_i^N) - f(c))K' \\ast \\hat{\\rho}^N \\varphi_x\\,dxdt \\\\\n&\\hspace*{1.2cm}+ \\int_0^T \\int_{x_i}^{x_{i+1}} f(c) \\mathrm{sign}(R_i^N - c) K'' \\ast \\hat{\\rho}^N \\varphi\\, dxdt, \\\\\n& B.T._1 := \\int_0^T \\int_{-\\infty}^{x_0} c\\varphi_t - f(c)[K' \\ast \\hat{\\rho}^N \\varphi_x + K'' \\ast \\hat{\\rho}^N \\varphi]\\,dxdt \\\\\n&\\hspace*{1.2cm}+ \\int_0^T \\int_{x_N}^{\\infty} c\\varphi_t - f(c)[K' \\ast \\hat{\\rho}^N \\varphi_x + K'' \\ast \\hat{\\rho}^N \\varphi]\\,dxdt.\n\\end{align*}\nFor simplicity of notation we set $S^N_i : = \\mathrm{sign}(R_i^N - c)$ and we omit the dependence on $N$ and $t$ wherever it is clear from the context.\nIntegrating by parts and recalling the definition of $\\hat{\\rho}^N$ and the expression for $\\dot{R}_i$, we can rewrite $I_i$ as\n\\begin{align*}\nI_i=& \\int_0^T S_i R_i (\\dot{x}_{i+1} - \\dot{x}_i) \\left(\\hbox{\\ }\\Xint{\\hbox{\\vrule height -0pt width 10pt depth 1pt}}_{x_{i}}^{x_{i+1}} \\varphi(t,x) dx - \\varphi(t,x_{i+1})\\right)dt \\\\\n&+ \\int_0^T S_i [R_i (\\dot{x}_{i+1} - \\dot{x}_i) \\varphi(t,x_{i+1}) - (R_i -c) (\\dot{x}_{i+1}\\varphi(t,x_{i+1}) - \\dot{x}_i \\varphi(t,x_i))]dt,\n\\end{align*}\nand $II_i$ as\n\\begin{align*}\nII_i =& -\\int_0^T S_i\\frac{(f(R_i) - f(c))}{N}\\sum_{j=0}^N (K'(x_{i+1} - x_j)\\varphi(t,x_{i+1}) - K'(x_i - x_j)\\varphi(t,x_i))dt \\\\\n&+ \\int_0^T S_i \\frac{f(R_i)}{N} \\sum_{j=0}^N \\int_{x_i}^{x_{i+1}} K''(x-x_j)\\varphi(t,x)dxdt\\,.\n\\end{align*}\nThen the sum $I_i + II_i$ becomes\n\\[I_i + II_i = A^1_i + A^2_i + Z_i, \\]\nwhere we set\n\\begin{align*}\nA^1_i &= \\int_0^T S_i R_i (\\dot{x}_{i+1} - \\dot{x}_i) \\left(\\hbox{\\ }\\Xint{\\hbox{\\vrule height -0pt width 10pt depth 1pt}}_{x_{i}}^{x_{i+1}} \\varphi(t,x) dx - \\varphi(t,x_{i+1})\\right)dt, \\\\\nA^2_i &= \\int_0^T S_i \\frac{f(R_i)}{N} \\sum_{j=0}^N \\int_{x_i}^{x_{i+1}} K''(x-x_j)\\varphi(t,x)dxdt,\n\\end{align*}\nand\n\\begin{align*}\nZ_i = &- \\sum_{i=0}^{N-1} \\int_0^T S_i\\varphi(t,x_{i+1})[R_i \\dot{x}_i + \\frac{f(R_i)}{N}\\sum_{j=0}^N K'(x_{i+1}-x_j)]dt \\\\\n&+ \\sum_{i=0}^{N-1} \\int_0^T S_i\\varphi(t,x_{i+1})[c \\dot{x}_{i+1} + \\frac{f(c)}{N}\\sum_{j=0}^N K'(x_{i+1}-x_j)]dt \\\\\n&+ \\sum_{i=0}^{N-1} \\int_0^T S_i\\varphi(t,x_{i})[R_i \\dot{x}_i + \\frac{f(R_i)}{N}\\sum_{j=0}^N K'(x_{i}-x_j)]dt \\\\\n&- \\sum_{i=0}^{N-1} \\int_0^T S_i\\varphi(t,x_{i})[c \\dot{x}_i + \\frac{f(c)}{N}\\sum_{j=0}^N K'(x_{i}-x_j)]dt.\n\\end{align*}\nBy performing a summation by parts, we get\n\\begin{align*}\n\\sum_{i=0}^{N-1} Z_i &= B.T._2 + \\sum_{i=1}^{N-1} \\int_0^T \\varphi(t,x_i) S_i\\left(R_i \\dot{x}_i +\\frac{f(R_i)}{N} \\sum_{j=0}^N K'(x_i -x_j) \\right)dt \\\\\n&\\quad - \\sum_{i=1}^{N-1} \\int_0^T \\varphi(t,x_i) S_{i-1}\\left(R_{i-1}\\dot{x}_{i-1} + \\frac{f(R_{i-1})}{N}\\sum_{j=0}^N K'(x_i-x_j) \\right)dt \\\\\n&\\quad +\\sum_{i=1}^{N-1} \\int_0^T \\varphi(t,x_i)(S_{i-1}- S_i)\\left(c\\dot{x}_i + \\frac{f(c)}{N} \\sum_{j=0}^N K'(x_i-x_j) \\right)dt \\\\\n&= B.T._2 + B.T._3 + \\sum_{i=1}^{N-2} (A_i^3 + A_i^4) + \\sum_{i=1}^{N-1}B_i.\n\\end{align*}\nwhere $B.T._2$ and $B.T._3$ regard the external particles. More precisely, $B.T._2= B.T._{21}+ B.T._{22}$, where\n\\begin{align*}\nB.T._{21} =& c \\int_0^T \\varphi(t,x_N) S_{N-1} \\frac{v(c)-v(R_{N-1})}{N} \\sum_{j=0}^N K'(X_N - x_j)dt \\\\\n& -c \\int_0^T \\varphi(t,x_0) S_{0}\\frac{v(c)-v(R_{0})}{N}\\sum_{j=0}^N K'(X_0 - x_j)dt, \\\\\nB.T._{22} =& \\int_0^T \\varphi(t,x_0)S_{0}R_0 \\left(\\dot{x}_0 + \\frac{v(R_0)}{N}\\sum_{j=0}^N K'(X_0 - x_j)\\right)dt \\\\\n&- \\int_0^T \\varphi(t,x_N)S_{N-1} R_{N-1} \\left(\\dot{x}_{N-1} + \\frac{v(R_{N-1})}{N}\\sum_{j=0}^N K'(X_N - x_j)\\right)dt,\n\\end{align*}\nand $B.T._3$ corresponds to\n\\begin{align*}\nB.T._3 =& \\int_0^T \\varphi(t,x_{N-1}) S_{N-1}\\left(R_{N-1}\\dot{x}_{N-1} + \\frac{f(R_{N-1})}{N} \\sum_{j=0}^N K'(x_{N-1}-x_j)\\right)dt \\\\\n&- \\int_0^T \\varphi(t,x_{0})S_0 \\left( R_{0}\\dot{x}_{0} + \\frac{f(R_{0})}{N} \\sum_{j=0}^N K'(x_{1}-x_j)\\right)dt.\n\\end{align*}\nThe terms $A_i^3,\\,A_i^4$ and $B_i$ regards, instead, the internal particles and they are defined as follows\n\\begin{align*}\nA_i^3 =& \\int_0^T \\varphi(t,x_i) S_i \\frac{f(R_i)}{N}\\sum_{j=0}^N [K'(x_i-x_j)-K'(x_{i+1}-x_j)]dt, \\\\\nA_i^4 =& \\int_0^T (\\varphi(t,x_i) - \\varphi(t,x_{i+1}))S_i R_i \\left( \\dot{x}_i + \\frac{v(R_i)}{N} \\sum_{j=0}^N K'(x_{i+1}-x_j) \\right)dt,\\\\\nB_i =& \\int_0^T \\varphi(t,x_i)(S_{i-1}- S_i))\\left(c\\dot{x}_i + \\frac{f(c)}{N} \\sum_{j=0}^N K'(x_i-x_j) \\right)dt.\n\\end{align*}\nSummarizing, we can rewrite $B.T._1 + \\sum_{i=0}^{N-1} (I_i + II_i)$ as\n\\[ B.T._1 + B.T_{21} + B.T._{22} + B.T._3 + \\sum_{i=0}^{N-1} (A^1_i + A^2_i) + \\sum_{i=1}^{N-2} (A^3_i + A^4_i) + \\sum_{i=1}^{N-1} B_i, \\]\nthen estimate \\eqref{entropiaN} follows if we prove that such sum is non negative when $N \\gg 1$, and this can be done by showing that\n\\begin{equation}\\label{partepositiva}\nB.T._1 + B.T._{21} + \\sum_{i=1}^{N-1}B_i > 0,\n\\end{equation}\nwhile\n\\begin{equation}\\label{ordine1\/N}\n\\left| B.T._{22} + B.T._3 + \\sum_{i=0}^{N-1}(A_i^1 + A_i^2) + \\sum_{i=1}^{N-2} (A_i^3 + A_i^4) \\right| \\leq \\frac{C}{N}\n\\end{equation}\nfor a positive constant $C= C(\\varphi,K,\\bar{\\rho},v,T)$.\nThe remaining part of the proof is devoted to show the validity of \\eqref{partepositiva} and \\eqref{ordine1\/N}.\nWe focus first on \\eqref{partepositiva}. Integrating by parts, recalling that $\\varphi(0,\\cdot)=\\varphi(T,\\cdot)=0$, $\\varphi(t,\\cdot) \\geq 0$ and the assumption (AK), we immediately obtain\n\\begin{equation}\\label{BT1}\nB.T._1 = -\\frac{f(c)}{N} \\int_0^T \\left(\\varphi(t,x_N)\\sum_{j=0}^N K'(x_N-x_j) + \\varphi(t,x_0)\\sum_{j=0}^N K'(x_0-x_j)\\right) > 0.\n\\end{equation}\nBecause of the monotonicity of $v$ (see (Av)), for all times $t$ we know that\n\\[ S_{0}(t)(v(c) - v(R_{0}(t))) \\geq 0,\\quad \\mbox{ and }\\quad S_{N-1}(t)(v(c) - v(R_{N-1}(t))) \\geq 0 \\]\nthus, recalling again (AK), we deduce\n\\begin{equation}\\label{BT21}\nB.T._{21} \\geq 0.\n\\end{equation}\nLet us now consider the generic term $B_i$. Substituting the expression of $\\dot{x}_i$, we get\n\\[ B_i = \\int_0^T\\varphi(t,x_i) (S_{i-1} - S_i)\\left[\\frac{v(c)-v(R_i)}{N} \\sum_{j>i} K'(x_i-x_j) + \\frac{v(c)-v(R_{i-1})}{N} \\sum_{j0$ if $x>0$, for all times holds\n\\[ (S_{i-1}- S_i)\\left[\\frac{v(c)-v(R_i)}{N} \\sum_{j>i} K'(x_i-x_j) + \\frac{v(c)-v(R_{i-1})}{N} \\sum_{j0$ such that\n\\[ L=\\sup\\left\\{ |K''(x)|\\,,\\,\\, x\\in[-(\\bar{x}_{max}-\\bar{x}_{min}),(\\bar{x}_{max}-\\bar{x}_{min})]\\right\\}. \\]\nSince the argument is quite technical, it is more convenient to split the left hand side of \\eqref{ordine1\/N} in three parts:\n\\[ \\Gamma_1 = B.T._{22} + B.T._3 + A_0^2 + A_{N-1}^2,\\quad \\Gamma_2 = \\sum_{i=0}^{N-1} A_i^1 + \\sum_{i=1}^{N-2} A_i^4,\\quad \\Gamma_3 = \\sum_{i=1}^{N-2} (A_i^2 + A_i^3). \\]\nRecalling that $K',\\,\\varphi$ and $v$ are uniformly bounded and Lipschitz, we get\n\\begin{align}\\label{Gamma1}\n\\notag|\\Gamma_1| \\leq\\, &4 L\\,\\|\\varphi\\|_{L^{\\infty}} \\|v\\|_{L^{\\infty}} \\int_0^T (R_0(x_1 - x_0) + R_{N-1}(x_{N}-x_{N-1}))dt \\\\\n&+ 2 L\\,\\|v\\|_{L^{\\infty}}Lip[\\varphi] \\int_0^T R_{N-1}(x_N - x_{N-1})dt \\leq \\frac{C(\\varphi,v,L,T)}{N}\n\\end{align}\nThen, inserting the expression of $\\dot{x}_i$, we can rearrange $\\Gamma_2$ in such a way that\n\\begin{align*}\n|\\Gamma_2| &\\leq 3\\sum_{i=0}^{N-1} \\int_0^T R_i \\left|\\hbox{\\ }\\Xint{\\hbox{\\vrule height -0pt width 10pt depth 1pt}}_{x_i}^{x_{i+1}}\\varphi(t,x) - \\varphi(t,x_{i+1})\\right|\\frac{|v(R_{i+1})-v(R_i)|}{N}\\sum_{j=0}^N |K'(x_{i+1}-x_j)| dt \\\\\n&\\quad + \\sum_{i=0}^{N-1} \\int_0^T R_i \\left|\\hbox{\\ }\\Xint{\\hbox{\\vrule height -0pt width 10pt depth 1pt}}_{x_i}^{x_{i+1}}\\varphi(t,x) - \\varphi(t,x_{i+1})\\right| \\frac{v(R_i)}{N} \\sum_{j=0}^N |K'(x_i-x_j)- K'(x_{i+1}-x_j)|dt \\\\\n&\\quad + \\sum_{i=1}^{N-2} \\int_0^T R_i |\\varphi(t,x_i) - \\varphi(t,x_{i+1})|\\frac{|v(R_{i-1})-v(R_i)|}{N}\\sum_{j=0}^N |K'(x_{i}-x_j)|dt \\\\\n&\\quad + \\sum_{i=1}^{N-2} \\int_0^T R_i |\\varphi(t,x_i) - \\varphi(t,x_{i+1})|\\frac{v(R_i)}{N} \\sum_{j=0}^N |K'(x_i-x_j)- K'(x_{i+1}-x_j)|dt\n\\end{align*}\nand using the Lipschitz and the uniform regularity of $K',\\,\\varphi,\\,v$, estimate \\eqref{TV} and the uniform bound on the support of $\\rho^N$, it is easy to see that\n\\begin{align}\\label{Gamma2}\n\\notag|\\Gamma_2| \\leq& \\,4L\\, Lip[\\varphi]Lip[v] TV[\\bar{\\rho}] \\int_0^T e^{Ct} \\sum_{i=0}^{N-1} R_i (x_{i+1}-x_i)dt \\\\\n& + 2 L\\,\\|v\\|_{L^{\\infty}} Lip[\\varphi] \\int_0^T \\sum_{i=0}^{N-1} R_i (x_{i+1}-x_i)^2 dt \\leq \\frac{C(\\varphi,v,K,\\bar{\\rho},T)}{N}.\n\\end{align}\nIt remains to show that also $\\Gamma_3$ vanishes as $N \\to \\infty$. In this case, the uniform bound on $K''$ implies\n\\begin{align}\\label{Gamma3}\n\\notag |\\Gamma_3| &\\leq \\sum_{i=1}^{N-2} \\int_0^T \\frac{|f(R_i)|}{N} \\int_{x_i}^{x_{i+1}}|\\varphi(t,x)-\\varphi(t,x_i)| \\sum_{j=0}^N |K''(x-x_j)|dxdt \\\\\n&\\leq L\\,\\|v\\|_{L^{\\infty}} Lip[\\varphi] \\int_0^T R_i \\int_{x_i}^{x_{i+1}} (x-x_i)dxdt \\leq \\frac{C(\\varphi,v,K,\\bar{\\rho},T)}{N}.\n\\end{align}\nFinally, by putting together \\eqref{Gamma1},\\eqref{Gamma2} and \\eqref{Gamma3}, we obtain \\eqref{ordine1\/N} and, recalling also \\eqref{partepositiva}, \\eqref{entropiaN}.\n\\end{proof}\n\nWe are now in position to prove that the large particle limit $\\rho$ that we obtained in the previous section is an entropy solution for the PDE.\n\n\\begin{lemma}\\label{entropiarho}\nLet $\\rho$ be the limit of $\\rho^N$ up to a subsequence. For every non negative $\\varphi \\in C^{\\infty}_c ([0,+\\infty)\\times \\mathbb R)$ and $c\\geq 0$, one has\n\\begin{equation}\\label{entropia}\n0 \\leq \\int_\\mathbb R |\\bar\\rho - c|\\varphi(0,x)dx + \\int_0^{+\\infty} \\int_{\\mathbb R} |\\rho - c| \\varphi_t - \\mathrm{sign}(\\rho -c)[(f(\\rho) - f(c))K' \\ast \\rho\\, \\varphi_x - f(c)K'' \\ast \\rho\\, \\varphi] dx dt.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nLet $T>0$ be such that $\\mathrm{supp}(\\varphi)\\subset [0,T)$. Roughly speaking, the statement holds provided we can show that it is possible to pass to the limit as $N \\to \\infty$ in the inequality~\\eqref{entropiaN}. More precisely,\nwe need to prove the following\n\\begin{align*}\n& \\lim_{N \\to \\infty} \\int_\\mathbb R |\\rho^N(0,x) - c|\\varphi(0,x)dx = \\int_\\mathbb R |\\bar\\rho - c|\\varphi(0,x)dx\\\\\n& \\lim_{N \\to \\infty} \\int_0^T \\int_{\\mathbb R} |\\rho^N-c|\\,\\varphi_t\\, dx dt = \\int_0^T \\int_{\\mathbb R} |\\rho-c|\\,\\varphi_t \\, dx dt \\\\\n& \\lim_{N \\to \\infty} \\int_0^T \\int_{\\mathbb R} \\mathrm{sign}(\\rho^N-c)(f(\\rho^N)-f(c))K'\\ast \\hat{\\rho}^N\\,\\varphi_x \\, dx dt\\\\\n & \\qquad = \\int_0^T \\int_{\\mathbb R} \\mathrm{sign}(\\rho-c)(f(\\rho)-f(c))K'\\ast \\rho \\,\\varphi_x \\, dx dt\\\\\n& \\lim_{N \\to \\infty} \\int_0^T \\int_{\\mathbb R} f(c) \\mathrm{sign}(\\rho^N - c)K'' \\ast \\hat{\\rho}^N\\,\\varphi \\, dx dt = \\int_0^T \\int_{\\mathbb R} f(c) \\mathrm{sign}(\\rho - c)K'' \\ast \\rho\\,\\varphi\\, dx dt\n\\end{align*}\nThe first two limits are immediate in view of the strong $L^1$-convergence of $\\rho^N(0,x)$ to $\\bar\\rho$ and to the convergence of $\\rho^N$ to $\\rho$ almost everywhere in $L^1([0,T] \\times \\mathbb R)$ respectively.\nNotice now that the continuity of $f$ ensures the continuity of the function $h(\\mu):= \\mathrm{sign}(\\mu - c)(f(\\mu)-f(c))$. We have\n\\begin{align*}\n\\int_0^T \\int_{\\mathbb R}& [\\mathrm{sign}(\\rho^N-c)(f(\\rho^N)-f(c))K'\\ast \\hat{\\rho}^N - \\mathrm{sign}(\\rho-c)(f(\\rho)-f(c))K'\\ast \\rho ]\\,\\varphi_x\\,dx dt\\\\\n&= \\int_0^T \\int_{\\mathbb R}(h(\\rho)-h(\\rho^N))K'\\ast \\rho\\, \\varphi_x\\,dx dt + \\int_0^T \\int_{\\mathbb R} h(\\rho^N) K'\\ast(\\rho -\\hat{\\rho}^N)\\,\\varphi_x\\,dx dt,\\\\\n\\end{align*}\nthen the regularity of $h$ and $K'$ required in the assumptions (Av) and (AK), the convergence of $\\rho^N$ to $\\rho$ almost everywhere in $[0,T] \\times \\mathbb R$ and the strong $L^1$-convergence of $K' \\ast \\hat{\\rho}^N$ to $K' \\ast \\rho$ established in Remark \\ref{rem:empirical} imply that\n\\begin{equation}\\label{bla2}\n\\int_0^T \\int_{\\mathbb R}|[(h(\\rho)-h(\\rho^N))K'\\ast \\rho + h(\\rho^N) K'\\ast(\\rho -\\hat{\\rho}^N)]\\,\\varphi_x|\\,dx dt \\to 0\n\\end{equation}\nas $N$ tends to $+\\infty$. Concerning the fourth limit, instead, we can see that\n\\begin{align*}\n\\int_0^T \\int_{\\mathbb R}& f(c)[\\mathrm{sign}(\\rho^N-c)K''\\ast \\hat{\\rho}^N - \\mathrm{sign}(\\rho-c)K''\\ast\\rho]\\,\\varphi\\,dx dt \\\\\n&= \\int_0^T \\int_{\\mathbb R} f(c)\\mathrm{sign}(\\rho^N-c)K''\\ast (\\hat{\\rho}^N - \\rho)\\,\\varphi\\,dx dt \\\\\n&\\quad + \\int_0^T \\int_{\\mathbb R} f(c)(\\mathrm{sign}(\\rho^N-c) -\\mathrm{sign}(\\rho-c))K''\\ast \\rho\\,\\varphi\\,dx dt.\n\\end{align*}\nThe first of the two terms can be handled as above. By using Remark \\ref{rem:empirical} and Lemma \\ref{lem:empirical1}, we get \n\\begin{equation}\\label{bla3}\n\\int_0^T \\int_{\\mathbb R} |f(c)\\mathrm{sign}(\\rho^N-c)K''\\ast (\\hat{\\rho}^N - \\rho)\\,\\varphi\\,dx| dt \\leq \\frac{C(K'',\\|f\\|_{\\infty},\\varphi)}{N}\\,.\n\\end{equation}\nOn the other hand, passing to the limit in the terms including the difference $\\mathrm{sign}(\\rho^N-c)- \\mathrm{sign}(\\rho-c)$ is less straightforward because of the discontinuity of the sign function.\nLet us then focus on the proof of\n\\[ \\lim_{N \\to \\infty} \\int_0^T \\int_{\\mathbb R} f(c)(\\mathrm{sign}(\\rho^N-c)-\\mathrm{sign}(\\rho-c))K''\\ast \\rho\\,\\varphi\\,dxdt = 0\\,. \\]\nIn order to get rid of the discontinuity, we need to introduce two smooth approximations of the \\emph{sign} function, we call them $\\eta_{\\delta}^{\\pm}$, so that\n\\[ \\mathrm{sign}(z) - \\eta_{\\delta}^+(z) \\geq 0 \\quad \\mbox{ and } \\quad \\mathrm{sign}(z)-\\eta_{\\delta}^-(z) \\leq 0. \\]\nLet us recall that the regularity of $K$ ensures the existence of a constant $L>0$ such that $|K''(z)| \\leq L$ for every $z \\in [-2\\mathrm{meas}(\\mathrm{supp}(\\rho^N)), 2\\mathrm{meas}(\\mathrm{supp}(\\rho^N))]$ and every $N$.\nThen we can estimate\n\\begin{align*}\n\\int_0^T\\int_{\\mathbb R}f(c)\\mathrm{sign}&(\\rho^N-c) K''\\ast \\rho \\varphi \\\\\n&= \\int_0^T\\int_{\\mathbb R}f(c)\\mathrm{sign}(\\rho^N-c)(K''-L)\\ast\\rho\\,\\varphi + \\int_0^T\\int_{\\mathbb R}f(c)\\mathrm{sign}(\\rho^N-c)L\\ast \\rho\\,\\varphi \\\\\n&\\leq \\int_0^T\\int_{\\mathbb R}f(c)\\eta_{\\delta}^+(\\rho^N-c)(K''-L)\\ast \\rho\\,\\varphi + \\int_0^T\\int_{\\mathbb R}f(c)\\eta_{\\delta}^-(\\rho^N-c)L\\ast \\rho\\,\\varphi\\,.\n\\end{align*}\nwhere the inequality holds because\n\\begin{align*}\n&(\\mathrm{sign}(\\rho^N-c)-\\eta_{\\delta}^+(\\rho^N-c))(K''-L)\\ast \\rho \\leq 0 \\\\\n&(\\mathrm{sign}(\\rho^N-c)-\\eta_{\\delta}^-(\\rho^N-c)) L\\ast \\rho \\leq 0.\n\\end{align*}\nNow, observe that\n\\begin{align*}\n\\lim_{N\\to \\infty}& f(c) \\int_0^T \\int_{\\mathbb R} (\\eta_{\\delta}^+(\\rho^N -c) - \\eta_{\\delta}^+(\\rho -c)) (K''-L)\\ast \\rho\\,\\varphi\\\\\n&\\leq \\lim_{N\\to \\infty} f(c)\\int_0^T\\int_{\\mathbb R} |\\eta_{\\delta}^+(\\rho^N -c) -\\eta_{\\delta}^+(\\rho -c)| |(K''-L)\\ast\\rho\\,\\varphi| \\\\\n&\\leq \\lim_{N\\to \\infty} f(c) 2L \\| \\varphi\\|_{\\infty} Lip[\\eta_{\\delta}^+] \\int_0^T\\int_{\\mathbb R} |\\rho^N-\\rho| \\\\\n&\\leq C(L, \\varphi, \\eta_{\\delta}^+) \\lim_{N\\to \\infty} \\| \\rho^N - \\rho\\|_{L^1([0,T]\\times \\mathbb R)}= 0\n\\end{align*}\nand in a similar way we get also\n\\[ \\lim_{N \\to \\infty} \\int_0^T\\int_{\\mathbb R} f(c)(\\eta_{\\delta}^-(\\rho^N-c) - \\eta_{\\delta}^-(\\rho-c))L\\ast \\rho\\,\\varphi = 0\\,, \\]\nthus implying that\n\\begin{equation*}\n\\limsup_{N\\to\\infty} \\int_0^T\\int_{\\mathbb R}f(c)\\mathrm{sign}(\\rho^N-c) K''\\ast \\rho\\,\\varphi \\leq \\int_0^T\\int_{\\mathbb R} f(c)[\\eta_{\\delta}^+ (\\rho-c)(K''-L)\\ast\\rho + \\eta_{\\delta}^-(\\rho-c)L\\ast \\rho]\\,\\varphi\n\\end{equation*}\nOnce here, the dominated convergence Theorem ensures that we can pass to the limit in $\\delta$ to get\n\\[\\limsup_{N \\to \\infty} f(c)\\int_0^T\\int_{\\mathbb R} \\mathrm{sign}(\\rho^N-c) K'' \\ast \\rho\\,\\varphi \\leq \\int_0^T\\int_{\\mathbb R} \\mathrm{sign}(\\rho-c) K'' \\ast \\rho\\,\\varphi.\\]\nA symmetric argument provides the inverse inequality with $\\liminf$ replacing $\\limsup$, hence we obtain\n\\begin{equation}\\label{limiteinN}\n\\lim_{N \\to \\infty} f(c)\\int_0^T\\int_{\\mathbb R} (\\mathrm{sign}(\\rho^N-c)-\\mathrm{sign}(\\rho-c)) K'' \\ast \\rho\\,\\varphi = 0.\n\\end{equation}\nThe above argument, together with~\\eqref{bla2}-\\eqref{limiteinN}, implies estimate~\\eqref{entropia}, and the proof is complete.\n\\end{proof}\n\nWe now tackle another crucial task for our result, namely the \\emph{uniqueness of the entropy solution} for a fixed initial datum. To perform this task we rely on a stability result due to Karlsen and Risebro \\cite{KarlsenRiebro}, that we report here for sake of completeness in an adapted version.\n\n\\begin{theorem}\\label{KarlsenRiebro}\nLet $f,P,Q$ be such that\n\\[ f \\quad \\mbox{ is locally Lipschitz, }\\qquad P,Q \\in W^{1,1}(\\mathbb R) \\cap \\mathcal{C}(\\mathbb R), \\qquad P_x,Q_x \\in L^{\\infty}(\\mathbb R), \\]\nand let $p,q \\in L^\\infty([0,T]; BV(\\mathbb R))$ be respectively entropy solutions to\n\\[ \\left\\lbrace \\begin{array}{ll}\n&p_t = (f(p)P(x))_x \\quad p(0,x)=p_0(x),\\\\\n&q_t = (f(q)Q(x))_x \\quad q(0,x)=q_0(x),\n\\end{array}\\right.\\]\nwhere the initial data $(p_0,q_0)$ are in $L^1(\\mathbb R) \\cap L^{\\infty}(\\mathbb R) \\cap BV(\\mathbb R)$.\nThen for almost every $t \\in (0,T)$ one has\n\\begin{equation}\\label{contrattivitasol}\n\\| p(t) - q(t) \\|_{L^1(\\mathbb R)} \\leq \\| p_0 - q_0 \\|_{L^1(\\mathbb R)} + t(C_1 \\| P-Q\\|_{L^{\\infty}(\\mathbb R)} + C_2\\| P-Q\\|_{BV(\\mathbb R)})\n\\end{equation}\nwhere $C_1 = Lip[f] \\min\\{ \\|P\\|_{BV(\\mathbb R)}, \\|Q \\|_{BV(\\mathbb R)}\\}$ and $C_2 = \\| f\\|_{L^{\\infty}}$.\n\\end{theorem}\n\nWe are now ready to prove our main theorem.\n\n\\proofof{Theorem~\\ref{main}}\nThe results in Theorem \\ref{convergence} and Lemma \\ref{entropiarho} imply that there exist a subsequence of $\\rho^N$ converging almost everywhere on $[0,+\\infty)\\times \\mathbb R$ and in $L^1_{loc}$ to an entropy solution $\\rho$ to \\eqref{CauchyProblem} in the sense of Definition \\ref{solentropicadef}. Therefore, the proof of Theorem~\\ref{main} is concluded once we show that $\\rho$ is the unique entropy solution. We argue by contradiction. Assume that there exist two different functions $\\rho$ and $\\varrho$ satisfying Definition~\\ref{solentropicadef} with $\\rho(0,\\cdot) = \\varrho(0,\\cdot) = \\bar{\\rho}$, then we can define two vector fields $P(x) = K' \\ast \\rho (x)$ and $Q(x)= K'\\ast \\varrho(x)$.\nIn order to apply Theorem~\\ref{KarlsenRiebro} to $P$ and $Q$, let us check that all assumptions therein are satisfied.\nFirst of all, $P$ and $Q$ are locally Lipschitz in $\\mathbb R$ thanks to the assumption (AK), thus $P_x, Q_x \\in L^{\\infty}_{loc}(\\mathbb R)$. Then, we observe that\n\\begin{align*}\n|P(t,x) - Q(t,x)| &= \\left| \\int_{\\mathbb R} K'(x-y)\\rho(t,y)dy - \\int_{\\mathbb R} K'(x-y)\\varrho(t,y)dy \\right| \\\\\n&\\leq \\int_{\\mathbb R} |K'(x-y)(\\rho(t,y) - \\varrho(t,y))|dy \\leq L_{\\bar{\\rho}} \\| \\rho - \\varrho\\|_{L^{\\infty}([0,T];L^1(\\mathbb R))},\n\\end{align*}\nand\n\\begin{align*}\n\\int_{\\mathbb R} |P_x(s,x) - Q_x(s,x)|dx &= \\int_{\\mathbb R} |K'' \\ast \\rho(t,x) - K'' \\ast \\varrho(t,x)| dx \\\\\n&= \\int_{\\mathbb R} |K''\\ast(\\rho - \\varrho)(t,x)|dx \\leq L_{\\bar{\\rho}} \\|\\rho - \\varrho\\|_{L^{\\infty}([0,T];L^1(\\mathbb R))},\n\\end{align*}\nwhere $L_{\\bar\\rho}=\\max\\{\\|K'\\|_{L^\\infty(I_{\\bar\\rho})},\\|K''\\|_{L^1(I_{\\bar\\rho})}\\}$, and $I_{\\bar\\rho}=[-2\\mathrm{meas}(\\mathrm{supp}(\\bar{\\rho})), 2\\mathrm{meas}(\\mathrm{supp}(\\bar{\\rho}))]$.\nAs a consequence\n\\begin{align*}\n& \\| P-Q\\|_{L^{\\infty}([0,T] \\times \\mathbb R)} \\leq L_{\\bar{\\rho}} \\| \\rho - \\varrho\\|_{L^{\\infty}([0,T];L^1(\\mathbb R))}\\\\\n & \\| P-Q\\|_{L^{\\infty}([0,T] ; BV(\\mathbb R))} \\leq L_{\\bar{\\rho}} \\| \\rho - \\varrho\\|_{L^{\\infty}(0,T;L^1(\\mathbb R))}.\n\\end{align*}\nBy applying Theorem~\\ref{KarlsenRiebro} to $\\rho,\\varrho, P$ and $Q$ we obtain\n\\begin{equation}\\label{assurdo}\n\\| \\rho(t) - \\varrho(t) \\|_{L^1(\\mathbb R)} \\leq C(K,\\bar{\\rho}) t \\| \\rho(t) - \\varrho(t) \\|_{L^1(\\mathbb R)}.\n\\end{equation}\nAssume that there exists an open interval $(t_1,t_2)\\subset [0,T]$ such that $\\rho(t,\\cdot)$ and $\\varrho(t,\\cdot)$ differ in $L^1(\\mathbb R)$ on $t\\in (t_1,t_2)$. Then, due to the fact that \\eqref{eq:intro_PDE} is invariant with respect to time-translations, the inequality ~\\eqref{assurdo} implies\n\\begin{equation}\\label{assurdovero}\n\\| \\rho(t,\\cdot) - \\varrho(t,\\cdot) \\|_{L^1(\\mathbb R)} \\leq C(K,\\bar{\\rho}) (t-t_1) \\| \\rho(t,\\cdot) - \\varrho(t,\\cdot) \\|_{L^1(\\mathbb R)}\\quad \\forall\\,t \\in (t_1,t_2).\n\\end{equation}\nClearly can always consider $t\\in (t_1,t_2)$ small enough such that $C(K,\\bar{\\rho}) (t-t_1)< 1$, but this is in contradiction with~\\eqref{assurdovero}. In conclusion, $\\rho(t,\\cdot)\\equiv \\varrho(t,\\cdot)$ on $[0,T]$ and the proof is complete.\n\\end{proof}\n\n\\section{Non-uniqueness of weak solutions and steady states}\\label{sec:discussion}\n\nThe use of the notion of entropy solution in the present context is not merely motivated by the technical need of identifying a notion of solution (stronger than weak solutions) allowing to prove uniqueness. Similarly to what happens for scalar conservation laws, we prove that there are explicit examples of initial data in $BV$ for which there exists two weak solutions to the Cauchy problem \\eqref{CauchyProblem}. \n\nFor simplicity, we use\n\\[v(\\rho)=(1-\\rho)_{+}.\\]\nConsider the initial condition\n\\[\\bar\\rho(x)=\\mathbf{1}_{[-1,-1\/2]}+\\mathbf{1}_{[1\/2,1]}.\\]\nClearly, the stationary function\n\\[\\rho_s(t,x)=\\mathbf{1}_{[-1,-1\/2]}+\\mathbf{1}_{[1\/2,1]}\\]\nis a weak solution to \\eqref{CauchyProblem} with initial condition $\\bar\\rho$. To see this, let $\\varphi\\in C^1_c([0,+\\infty)\\times \\mathbb R)$. We have\n\\begin{align*}\n & \\int_0^{+\\infty}\\int_\\mathbb R\\left[\\rho_s\\varphi_t + \\rho_s v(\\rho_s)K'\\ast \\rho\\varphi_x\\right] dx dt + \\int_\\mathbb R\\bar\\rho\\varphi(0,x) dx \\\\\n & \\ = \\int_0^{+\\infty}\\frac{d}{dt}\\left(\\int_{[-1,-1\/2]\\cup [1\/2,1]}\\varphi dx\\right)dt +\\int_{[-1,-1\/2]\\cup [1\/2,1]}\\varphi(0,x) dx = 0.\n\\end{align*}\n\nWe now prove that $\\rho_s$ is not an entropy solution, in that it does not satisfy the entropy condition in Definition \\ref{solentropicadef}. Let $\\psi\\in C^\\infty_c(\\mathbb R)$ be a standard non-negative mollifier supported on $[-1\/4,1\/4]$. Let $T>0$ and consider the test function $\\varphi(t,x)=\\phi(x)\\xi(t)$ with\n\\[\\phi(x)=\n\\begin{cases}\n\\psi(x+1\/2) & \\hbox{if $-3\/4\\leq x\\leq -1\/4$} \\\\\n\\psi(x-1\/2) & \\hbox{if $1\/4\\leq x\\leq 3\/4$} \\\\\n0 & \\hbox{otherwise},\n\\end{cases}\n\\]\nand $\\xi\\in C^\\infty([0,+\\infty))$ with $\\xi(t)=1$ for $t\\leq T$, $\\xi(t)=0$ for $t\\geq T+1$ and $\\xi$ non-increasing. Let us set $c=1\/2$, $I=[1\/4,3\/4]$, and compute\n\\begin{align*}\n & \\int_\\mathbb R |\\rho_s -c|\\phi dx +\\int_0^{+\\infty}\\int_\\mathbb R\\left[|\\rho_s-c|\\phi(x)\\xi'(t) -\\mathrm{sign}(\\rho_s-c)(f(\\rho)-f(c))K'\\ast\\rho_s\\phi'(x)\\xi(t) \\right.\\\\\n & \\qquad \\left.-f(c)K''\\ast \\rho_s \\phi(x)\\xi(t)\\right]\\,dxdt\\\\\n & \\leq 2\\int_I \\varphi dx + \\frac{1}{4}\\int_0^{T+1}\\xi(t)dt\\left[\\int_{(-I)\\cap(-\\infty,1\/2]}K'\\ast\\rho_s \\varphi_x dx - \\int_{(-I)\\cap[1\/2,+\\infty)}K'\\ast\\rho_s \\varphi_x dx \\right.\\\\\n & \\quad \\left.- \\int_{I\\cap(-\\infty,1\/2]}K'\\ast\\rho_s \\varphi_x dx +\\int_{I\\cap[1\/2,+\\infty)}K'\\ast\\rho_s \\varphi_x dx- \\int_{(-I)\\cup I}K''\\ast \\rho_s \\varphi dx\\right]\\\\\n & = 2\\int_I \\varphi dx - \\frac{1}{4}\\int_0^{T+1}\\xi(t)dt\\left[\\int_{(-I)\\cap(-\\infty,1\/2]}K''\\ast\\rho_s \\varphi dx - \\int_{(-I)\\cap[1\/2,+\\infty)}K''\\ast\\rho_s \\varphi dx \\right.\\\\\n & \\quad \\left.- \\int_{I\\cap(-\\infty,1\/2]}K''\\ast\\rho_s \\varphi dx +\\int_{I\\cap[1\/2,+\\infty)}K''\\ast\\rho_s \\varphi dx+\\int_{(-I)\\cup I}K''\\ast \\rho_s \\varphi dx\\right].\n\\end{align*}\nNow, since $K''$ and $\\rho_s$ are even, the same holds for $K''\\ast \\rho_s$. Therefore we get\n\\begin{align}\n & \\int_\\mathbb R |\\rho_s -c|\\varphi dx +\\int_0^T\\int_\\mathbb R\\left[|\\rho_s-c|\\varphi_t -\\mathrm{sign}(\\rho_s-c)(f(\\rho)-f(c))K'\\ast\\rho_s\\varphi_x -f(c)K''\\ast \\rho_s \\varphi\\right]\\,dxdt\\nonumber\\\\\n & \\ \\leq 2\\int_I \\varphi dx - \\frac{1}{2}\\int_0^{T+1}\\xi(t)dt\\iint_{I\\times I} \\left(K''(x-y)+K''(x+y)\\right) \\varphi(x)dy dx.\\label{eq:nonunique1}\n\\end{align}\nLet us now require for simplicity the following additional assumption:\n\\begin{equation}\\label{eq:nonunique_assumption}\n K''(x)>0\\qquad \\hbox{for all $x\\in \\mathbb R$}.\n\\end{equation}\nActually, such assumption can be relaxed, see remark \\ref{relaxed} below. Then, the last integral in \\eqref{eq:nonunique1} is clearly positive, and recalling that $\\xi(t)=1$ on $t\\in[0,T]$, we can choose $T$ large enough so that the whole right-hand side of \\eqref{eq:nonunique1} is strictly negative, thus contradicting Definition \\ref{solentropicadef}.\n\nThe above argument shows that $\\rho_s$ is a weak solution but not an entropy solution. On the other hand, the initial condition $\\rho_s$ is $L^\\infty$ and $BV$, therefore it must generate an entropy solution according to our main Theorem \\ref{main}. Clearly, such solution cannot coincide with $\\rho_s$. We have therefore proven the following theorem.\n\n\\begin{theorem}\\label{thm:nonuniqueness}\n Assume (Av), (AK), and \\eqref{eq:nonunique_assumption} are satisfied. Then, there exists an initial condition $\\bar\\rho \\in L^\\infty(\\mathbb R)\\cap BV(\\mathbb R)$ such that the Cauchy problem \\eqref{CauchyProblem} has more than one distributional weak solution.\n\\end{theorem}\n\n\\begin{remark}\\label{relaxed}\n\\emph{The assumption \\eqref{eq:nonunique_assumption} can be relaxed to include also Gaussian kernels $K(x)=- A e^{-B x^2}$ with $A,B>0$. Indeed, in order to fulfil \n\\[\\iint_{I\\times I} \\left(K''(x-y)+K''(x+y)\\right) \\varphi(x)dy dx>0\\]\none has to choose the size of the interval $I$ small enough. We omit the details.}\n\\end{remark}\n\n\\begin{remark}\n\\emph{The fact that the initial condition $\\rho_s$ will not give rise to a stationary solution can also be seen intuitively by using the result in Theorem \\ref{main}. Let us approximate $\\bar\\rho$ with $2(N+1)$ particles with mass $1\/(2(N+1))$, with $N$ integer, and with the particles located at $\\bar{x}_i$, $i=1,\\ldots,2(N+1)$, with\n\\begin{align*}\n & \\bar{x}_i=-1+\\frac{i}{2(N+1)}\\,,\\qquad i=0,\\ldots,N\\\\\n & \\bar{x}_i=1\/2+\\frac{i-N}{2(N+1)}\\,,\\qquad i=N+1,\\ldots,2N+1.\n\\end{align*}\nLet now evolve the particles' positions with the usual ODE system\n\\[\\dot{x}_i=-\\frac{v(R_i)}{N}\\sum_{j>i}(x_i-x_j) -\\frac{v(R_{i-1})}{N}\\sum_{jN}K'(x_j(0)-x_N(0)) \\geq v(1\/N)\\frac{N+1}{N}K'(2)>v(1\/2)K'(2)>0.\\]\nSimilarly, one can show that all particles $i=0,\\ldots,N-1$ `move' from their initial position, although their initial speed is zero. A numerical simulation performed in Section \\ref{sec:numerics} actually show that for large $N$ the discrete density tends to form a unique bump for large times. Hence, since Theorem \\ref{main} shows that the particle solution is arbitrarily close in $L^1_{loc}$ to the entropy solution, this argument supports the evidence that the entropy solution is not stationary. }\n\\end{remark}\n\nApart from producing an explicit example of non-uniqueness of weak solutions, the above example shows that there are stationary weak solutions that are not entropy solutions, and therefore cannot be considered as stationary solutions to our problem according to Definition \\ref{solentropicadef}. This raises the following natural question: what are the steady states of \\eqref{eq:intro_PDE} in the entropy sense? Before asking this question, it will be useful to tackle another task: as the approximating particle system converges to the entropy solutions, detecting the \\emph{steady states of \\eqref{Odes}} will give us a useful insight about the steady states at the continuum level.\n\nLet us restrict, for simplicity, to the case of an even initial condition $\\bar{\\rho}$, such that $\\|\\bar{\\rho}\\|_{L^1}=1$ and $N\\in\\mathbb N$ fixed. We assume here that $K'$ is supported on the whole $\\mathbb R$. Consider the following particle configuration,\n\\begin{equation}\\label{stable_conf}\n\\begin{cases}\n\\tilde{x}_1=-\\frac{1}{2}+\\frac{1}{2N},\\\\\n\\tilde{x}_{i+1}=\\tilde{x}_1+\\frac{i}{N},\\quad i=1,...,N-2,\\\\\n\\tilde{x}_N=\\tilde{x}_1+\\frac{N-1}{N}=\\frac{1}{2}-\\frac{1}{2N}.\n\\end{cases}\n\\end{equation}\nWith this choice we get\n\\[R_i = \\frac{1}{N(\\tilde{x}_{i+1}-\\tilde{x}_i)}=1, \\quad v(R_i)=0, \\quad \\forall i=1,...,N-1,\n\\]\nand it is easy to show that this configuration is a stationary solution for system \\eqref{Odes}. Actually, up to space translations, this is the \\emph{only} possible stationary solution. In order to prove that, assume that we have a particle configuration as in \\eqref{stable_conf} but with only one particle labelled $I$ such that\n\\[\n \\tilde{x}_I=\\tilde{x}_{1}+\\frac{I-1}{N}, \\quad \\tilde{x}_{I+1}=\\bar{x}>\\tilde{x}_{I}+\\frac{1}{N}.\n\\]\nFor such a configuration\n\\[\nR_I = \\frac{m}{N(\\tilde{x}_{I+1}-\\tilde{x}_I)}<1, \\quad v(R_I)>0,\\mbox{ and } \\quad v(R_i)=0 \\quad\\forall i\\neq I,\n\\]\nand the $I$ particles evolves according to\n\\begin{align*}\n & \\dot{\\tilde{x}}_I = -\\frac{v(R_I)}{N}\\sum_{j>I}K'(\\tilde{x}_I-\\tilde{x}_j) =-\\frac{v(R_I)}{N}\\sum_{j>I}K'\\left(\\frac{1}{N}(I-j)\\right)>0,\n\\end{align*}\nand then $\\tilde{x}_I$ moves with positive velocity.\n\nWe observe that, as $N\\to\\infty$, the piecewise constant density reconstructed by configuration \\eqref{stable_conf} converges in $L^1$ to the step function\n\\[\n\\rho_S=\\mathbf{1}_{[-\\frac{1}{2},\\frac{1}{2}]}.\n\\]\nThe above discussion suggests that all initial data with multiple bumps only attaining the values $0$ and $1$ are (weak solutions but) not entropy solutions except $\\rho_S$. Actually, this statement can be proven exactly in the same way as we proved Theorem \\ref{thm:nonuniqueness}, as it is clear that the position of the decreasing discontinuity at $x=-1\/2$ and of the increasing discontinuity at $x=1\/2$ do not play an essential role. By choosing the test function $\\varphi$ suitably, one can easily show that the entropy condition can be contradicted by suitably centring $\\varphi$ around the non-admissible discontinuities. We omit the details. As a consequence, we can assert that $\\rho_S$ is the \\emph{only} stationary solution to \\eqref{eq:intro_PDE} in the sense of Definition \\ref{solentropicadef}.\n\n\\section{Numerical simulations}\\label{sec:numerics}\nThe last section of the paper is devoted to present some numerical experiments based on the particle methods presented in the paper, supporting the results in the previous sections. The qualitative property that emerges more clearly in the simulations below is that solutions tend to aggregate and narrow their support. However, the maximal density constraint avoids the blow-up, and the density profile tends for large times towards the non-trivial stationary pattern presented at the end of the previous section. We compare our particle method with a classical Godunov method for \\eqref{eq:intro_continuum}.\n\n\\subsection*{Particle simulations}\n\nWe first test the particle method introduced in Section \\ref{sec:2}. We proceed as follows: we set the number of particles as $N$ and we reconstruct the initial distribution according to \\eqref{eq:dscr_IC} (for step functions we simply set the particles initially at distance $\\frac{\\ell}{N}$ from each other where $\\ell$ is the length of the support). Once we have defined the initial distribution, we solve the system \\eqref{Odes} with a MATLAB solver and we reconstruct the discrete density as\n\\begin{equation}\\label{eq:central}\n R_i(t)=\\frac{m}{2N(x_{i+1}(t)-x_{i-1}(t))}, \\quad i=2,N-1.\n\\end{equation}\n\nThe choice of central differences does not effect the particle evolution, since in solving system \\eqref{Odes} we define $R_i$ with forward differences. The choice in\\eqref{eq:central} is only motivated by the symmetry of the patterns we expect to achieve for large times.\n\\begin{remark}\\label{rem_zero_den}\n\\emph{In the construction of the discrete densities we get the problem of giving density to the first and the last particles (or only to the last one if we use forward differences). Among all the possible choices we set at zero this two densities, namely\n\\[\n R_1(t)=R_N(t)=0.\n\\]\nThis is a natural choice if we are dealing with step functions but it is not suitable with more general initial conditions, see \\figurename~\\ref{fig:cup_IF}.}\n\\end{remark}\n\nIn all the simulations we set\n\\[\n v(\\rho)=1-\\rho, \\quad K(x)=\\frac{C}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}} \\mbox{ and } N=300.\n\\]\nIn the particles evolution we don't fix any time step that is automatically determined by the solver.\n\nThe first example we furnish is the case of a single step function with symmetric support,\n\\begin{equation}\\label{eq:sing_step}\n \t\\bar{\\rho}(x) = 0.3 \\quad x\\in\\left[-1,1\\right].\n\\end{equation}\nFor this initial condition $m=0.6$, so the final configuration will be a step function of value $\\rho=1$ supported in $\\left[-0.3,0.3\\right]$. In \\figurename~\\ref{fig:1s_IF} we plot initial (left) and final (right) configurations, while in \\figurename~\\ref{fig:1s_Ev} evolution in time is plotted.\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL1S03_300_1}\n\\end{minipage}\n\\hspace{3mm}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL1S03_300_601}\n\\end{minipage}\n\\end{center}\n\\caption{On the left: initial condition as in \\eqref{eq:sing_step}; on the right: the final stationary configuration. We plot the discrete density in (red)-continuous line and the particles positions in (blue)-circles on the bottom of the picture.}\n\\label{fig:1s_IF}\n\\end{figure}\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{minipage}[c]{.70\\textwidth}\n\\includegraphics[width=.95\\textwidth]{Time-evolution_1s}\n\\end{minipage}\n\\end{center}\n\\caption{Evolution of the discrete density for the initial configuration \\eqref{eq:sing_step}.}\n\\label{fig:1s_Ev}\n\\end{figure}\n\nNext we show the evolution corresponding to the following initial condition,\n\\begin{equation}\\label{eq:id_parabola}\n\\bar{\\rho}(x) = \\frac{3}{4}(1-x^2), \\quad x\\in\\left[-1,1\\right].\n\\end{equation}\nEven in this case the function is symmetric with respect to the origin so it will converge to the unitary step function supported in $\\left[-0.5,0.5\\right]$ since $\\bar{\\rho}$ has normalized mass. As in the previous example initial and final configurations and time evolution of the solution are plotted in \\figurename~\\ref{fig:cup_IF}.\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL_CUP_300_1}\n\\end{minipage}\n\\hspace{3mm}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL_CUP_300_601}\n\\end{minipage}\n\\hspace{3mm}\n\\begin{minipage}[c]{.70\\textwidth}\n\\vspace{3mm}\n\\includegraphics[width=.95\\textwidth]{Time-evolution_G}\n\\end{minipage}\n\\end{center}\n\\caption{For the initial condition \\eqref{eq:id_parabola} the initial particle configuration is obtained thanks to \\eqref{eq:dscr_IC}. The discrete density behaves suitably around all the particles except the first and the last one. See Remark \\ref{rem_zero_den}. }\n\\label{fig:cup_IF}\n\\end{figure}\n\nWe conclude with step functions with disconnected support. We first study the case\n\\begin{equation}\\label{eq:2s_0206}\n \t\\bar{\\rho}(x) = \\begin{cases}\n \t 0.2 \\quad x\\in\\left[-0.5,0\\right]\\\\\n 0.6 \\quad x\\in\\left[0.5,1\\right]\n \t\\end{cases},\n\\end{equation}\nshowing that the two bumps merge into a single step. Since symmetry is lost, it is not straightforward to determine where this final configuration will stabilize, but in \\figurename~\\ref{fig:0206_Ev} we can see that they still aggregate in a step of unitary density and support of length $m$.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL2S0206_300_1}\n\\end{minipage}\n\\hspace{3mm}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL2S0206_300_601}\n\\end{minipage}\n\\hspace{3mm}\n\\begin{minipage}[c]{.70\\textwidth}\n\\vspace{3mm}\n\\includegraphics[width=.95\\textwidth]{Time-evolution_2s0206}\n\\end{minipage}\n\\end{center}\n\\caption{Evolution of a the two steps initial condition \\eqref{eq:2s_0206}. The pattern on the left is the one with less density and moves faster attracted by the one on the right and they merge in a single step of unitary density.}\n\\label{fig:0206_Ev}\n\\end{figure}\n\nMore interesting is the case of the following initial condition:\n\\begin{equation}\\label{eq:2s_11}\n \\bar{\\rho}(x) = \\begin{cases}\n \t 1 \\quad x\\in\\left[-0.5,0\\right]\\\\\n 1 \\quad x\\in\\left[0.5,1\\right]\n \t\\end{cases}.\n\\end{equation}\nNote that this is a stationary weak solution to \\eqref{eq:intro_continuum} but it is not an entropy solution.\nIn \\figurename~\\ref{fig:11_IF} we plot the time evolution of this initial configuration.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL2S11_300_1}\n\\end{minipage}\n\\hspace{3mm}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL2S11_300_31}\n\\end{minipage}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL2S11_300_121}\n\\end{minipage}\n\\hspace{3mm}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL2S11_300_601}\n\\end{minipage}\n\\end{center}\n\\caption{Solution to \\eqref{CauchyProblem} with initial condition \\eqref{eq:2s_11}. The initial condition is a weak stationary solution to \\eqref{eq:intro_PDE}. However, the particle scheme converges to another solution, actually the unique entropy solution to \\eqref{CauchyProblem}. The picture shows how that two `internal' discontinuities are not admissible in the entropy sense, and they are therefore `smoothed' immediately after $t=0$.}\n\\label{fig:11_IF}\n\\end{figure}\n\n\n\\subsection*{Comparison with classical Godunov method}\nIn order to validate the previous simulations we compare the results with a classical Godunov method. The main issue in this case is to dealing with the two directions in the transport term. More precisely, since the kernel $K$ is an even function, we can rephrase \\eqref{eq:intro_continuum} as\n\\begin{equation}\\label{eq:god1}\n \\partial_t \\rho = \\partial_x(\\rho v(\\rho) )K_\\rho^{+}(x)+\\partial_x(\\rho v(\\rho)) K_\\rho^{-}(x)+\\rho v(\\rho) K''\\ast\\rho,\n\\end{equation}\nwhere\n\\begin{align*}\n & K_\\rho^{+}(x)=\\int_{x\\geq y}K'(x-y)\\rho(y)dy\\geq 0,\\\\\n & K_\\rho^{-}(x)=\\int_{x < y}K'(x-y)\\rho(y)dy\\leq 0.\n \\end{align*}\n The evolution of $\\rho$ is driven by two transport fields: $K_\\rho^{+}$ pushing the density from left to right $K_\\rho^{-}$ pushing the density from right to left. The third term on the r.h.s. in \\eqref{eq:god1} plays the role of a source term. Following the standard finite volume approximation procedure on $N$ cells $\\left[x_{j-\\frac12},x_{j+\\frac12}\\right]$, the discrete equation reads as\n\\[\n \\frac{d}{dt}\\tilde{\\rho}_j = K_\\rho^{+}(x_j)\\frac{F_{j+\\frac12}^{+}-F_{j-\\frac12}^{+}}{\\Delta x} + K_\\rho^{-}(x_j)\\frac{F_{j+\\frac12}^{-}-F_{j-\\frac12}^{-}}{\\Delta x} + \\tilde{\\rho}_j v(\\tilde{\\rho}_j)dK_j\n\\]\nwhere $F_{j+\\frac12}^{+}$ and $F_{j+\\frac12}^{-}$ are the Godunov approximations of the fluxes and $dK_j$ is an approximation of the convolution in the reaction term obtained via a quadrature formula. We integrate in time with a time step satisfying the CFL condition of the method. In \\figurename~\\ref{fig:Test} we compare the solutions obtained with the two methods in all the examples illustrated above.\n\n\n \\begin{figure}[!ht]\n\\begin{center}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL_God_1S03_300_601}\n\\end{minipage}\n\\hspace{3mm}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL_God_CUP_300_601}\n\\end{minipage}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL_God_2S0206_300_601}\n\\end{minipage}\n\\hspace{3mm}\n\\begin{minipage}[c]{.45\\textwidth}\n\\includegraphics[width=.95\\textwidth]{PLNL_God_2S11_300_601}\n\\end{minipage}\n\\end{center}\n\\caption{Comparison between particles (red stars) and Godunov (green continuous line) methods at final time $t=1$. On the top: solutions corresponding to initial condition \\eqref{eq:sing_step} (left) and \\eqref{eq:id_parabola}(right). On the bottom: final configurations for \\eqref{eq:2s_0206} (left) and \\eqref{eq:2s_11}.}\n\\label{fig:Test}\n\\end{figure}\n\n\\section*{Acknowledgements}\nThe authors acknowledge support from the EU-funded Erasmus Mundus programme `MathMods - Mathematical models in engineering: theory, methods, and applications' at the University of L'Aquila, from the Italian GNAMPA mini-project `Analisi di modelli matematici della fisica,\ndella biologia e delle scienze sociali', and from the local fund of the University\nof L'Aquila `DP-LAND (Deterministic Particles for Local And Nonlocal Dynamics).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nExperiments at the Fermilab Tevatron and the CERN Large Hadron Collider are engaged in searches\nfor the Higgs boson, a heavy scalar resonance predicted by\nthe Standard Model (SM). SM Higgs bosons are excitations of the neutral $CP$-even component\nof an $SU(2)_L$ weak isospin doublet field $H$ carrying unit hypercharge under $U(1)_Y$, whose\nvacuum expectation value (VEV) $v\/\\sqrt{2}$ is responsible for electroweak symmetry breaking \n(for reviews, see \\cite{Gunion:1989we,Djouadi:2005gi}).\n\nIf one or more new heavy resonances are discovered at the LHC, it will be imperative\nto pin down their quantum numbers relative to the expected properties of the SM Higgs.\nDetermination of the spin and $CP$ properties of a new resonance will be \nchallenging, although recent studies indicate that definitive results could be obtained\nat or around the moment of discovery, if the decay mode to $ZZ$ is observable \\cite{Cao:2009ah, DeRujula:2010ys,Gao:2010qx}.\n\nGiven a neutral $CP$-even spin 0 resonance $S$, one still needs to establish its \nelectroweak quantum numbers in order to reveal any possible connection to electroweak\nsymmetry breaking. This in turn requires information about the couplings between\n$S$ and pairs of vector bosons, which can be extracted from observations of $S$\ndecaying via $W^+W^-$, $ZZ$, $\\gamma\\gamma$, or $Z\\gamma$.\nTo an excellent approximation\nthe couplings of the SM Higgs boson to $WW$ and $ZZ$ derive from the dimension-four Higgs kinetic terms\nin the SM effective action, and are thus directly related to both the strength of electroweak\nsymmetry breaking and the electroweak quantum numbers of the Higgs field.\nThe couplings of the SM Higgs boson to $\\gamma\\gamma$, $Z\\gamma$, or a pair of gluons\nare elegantly derived from the observation that Higgs couplings in the SM \nare identical to those of a conformal-compensating dilaton in a theory where\nscale invariance is violated by the trace anomaly \\cite{Adler:1976zt, Djouadi:1993ji, Goldberger:2007zk, Bai:2009ms}.\nThus these couplings\nappear first at dimension five, with coefficients related to SM gauge group beta functions.\n\nIn this paper we exhibit a general analysis, up to operators of dimension five, of\nthe relation between the electroweak properties of $S$ and its decay branchings\nto $V_1V_2 = WW$, $ZZ$, $\\gamma\\gamma$, and $Z\\gamma$. We ignore decays into two\ngluons because of the folklore that these are unobservable, and postpone until\nthe end a discussion of extracting complementary information from \nvector boson fusion production of $S$ \\cite{Plehn:2001nj, Hankele:2006ma}. Nevertheless, we should emphasize that our analysis only involves the decays of the scalar into electroweak vector bosons, and hence is independent of the production mechanism of the scalar. \n\nA key feature of our analysis is the classification of Higgs look-alikes according\nto the custodial symmetry $SU(2)_C$. In the SM this global symmetry is the diagonal remnant\nafter electroweak symmetry breaking of an accidental global $SU(2)_L\\times SU(2)_R$,\nin which $SU(2)_L$ and the $U(1)_Y$ subgroup of $SU(2)_R$ are gauged. \nCustodial symmetry implies $\\rho\\equiv m_W^2\/(m_Zc_w)^2 = 1$ \\cite{Sikivie:1980hm}, where $c_w$ is\nthe cosine of the weak mixing angle. Experimentally $\\rho$ is constrained to be\nvery close to one \\cite{Amsler:2008zzb}, implying either that the full scalar sector\nrespects $SU(2)_C$, or that there are percent-level cancellations unmotivated by\nsymmetry arguments. In our analysis we will assume that unbroken\n$SU(2)_C$ is built into the scalar sector.\n\nWe consider $S$ arising from one of the neutral $CP$-even components of arbitrary \nspin 0 multiplets of $SU(2)_L\\times SU(2)_R$. The case of a singlet under\n$SU(2)_L\\times SU(2)_R$ is special, since then no $SV_1V_2$ couplings can\nappear from operators of dimension four. All other cases can be grouped\naccording to whether the neutral scalar components transform as a singlet\nor a 5-plet under $SU(2)_C$. Again under the assumption that the full scalar\npotential respects the custodial symmetry, these three ``pure cases'' only give\nrise to one nontrivial mixed case, i.e. when $S$ from a $SU(2)_L\\times SU(2)_R$\nsinglet mixes with another $S$ from the $SU(2)_C$ singlet part of a\n$SU(2)_L\\times SU(2)_R$ nonsinglet.\n\nGiven the framework just described, we are able to enumerate all possible\ndeviations from the SM expectations for decays of a Higgs look-alike into\npairs of electroweak bosons. These deviations are typically quite large,\nand thus accessible to experiment at the LHC. Furthermore\nthe deviations exhibit patterns that point towards\nparticular non-SM scenarios. It would therefore be\npossible with LHC data to rule out a new scalar resonance as the agent (or the\nsole agent) of electroweak symmetry breaking.\nThis possibility emphasizes the importance of observing all\nfour $V_1V_2$ decay channels at the LHC with maximum sensitivity.\nWe give examples of Higgs imposters that meet SM expectations\nfor branching fractions into two of the electroweak $V_1V_2$ modes,\nonly revealing their ersatz nature in the other two $V_1V_2$ decay modes. The approach taken here is complimentary to that in Refs.~\\cite{Cao:2009ah, DeRujula:2010ys,Gao:2010qx}, where angular correlations and total decay width were used to distinguish Higgs look-alikes. A fully global analysis using all of the available decay and production observables in each channel will of course give superior results to the simple counting experiments described here.\n\n\nIn Sect. \\ref{sect:section2} we describe the dimension four couplings\nof an arbitrary neutral $CP$-even scalar charged under $SU(2)_L\\times U(1)_Y$\nto $WW$ or $ZZ$; we also describe the general dimension five couplings\nof a $SU(2)_L\\times U(1)_Y$ singlet to two electroweak vector bosons.\nSect. \\ref{sect:section3} contains the general framework based on custodial\nsymmetry. In Sect. \\ref{sect:section4} we provide general results on the patterns\nof $S\\to V_1V_2$ branching fractions, as well as discussing some interesting special cases.\nFurther discussion and outlook are in Sect. \\ref{sect:section5}, with some general formulae\nfor off-shell decays relegated to an appendix.\n\n\n\\section{Scalar Couplings with $V_1V_2$}\n\\label{sect:section2}\n\nIn this section we consider scalar couplings with two electroweak gauge bosons $V_1V_2$, where $V_1V_2=\\{WW, ZZ, Z\\gamma, \\gamma\\gamma\\}$. Such couplings are dictated by the electroweak quantum numbers of the scalar $S$. We will write down $SU(2)_L\\times U(1)_Y$ invariant operators giving rise to the $SV_1V_2$ couplings at the leading order. For an electroweak nonsinglet, the leading operator is the kinetic term of the scalar, assuming $S$ receives a VEV, while for the singlet scalar the leading operator starts at dimension five. \n\n\nFor nonsinglet scalars, the leading contribution to the $SV_1V_2$ coupling arises from spontaneous breaking of $SU(2)_L\\times U(1)_Y$ down to $U(1)_{em}$ via the Higgs mechanism, when $S$ develops a VEV. It is possible to derive the general coupling when there are multiple scalars in arbitrary representations of the $SU(2)_L$ group \\cite{Tsao:1980em, Haber:1999zh}. Using the notation $\\phi_k$ for scalars in the complex representations and $\\eta_i$ for scalars in the real representations\\footnote{A real representation is defined as a real multiplet with integer weak isospin and $Y=0$. }, the kinetic terms are\n\\begin{equation}\n\\sum_k {\\rm Tr} (D_\\mu\\phi_k)^\\dagger (D^\\mu\\phi_k) + \\frac12 \\sum_i{\\rm Tr} (D_\\mu \\eta_i)(D^\\mu\\eta_i) \\ ,\n\\end{equation}\nwhere \n\\begin{equation}\nD_\\mu = \\partial_\\mu -ig W_\\mu^a T^a - \\frac{i}2 g' B_\\mu Y \n\\end{equation}\nis the covariant derivative. In the above\n$W_\\mu^a$ and $g$ are the $SU(2)_L$ gauge bosons and gauge coupling, respectively, while $B_\\mu$ and $g'$ are the $U(1)_Y$ gauge boson and gauge coupling. In addition, $T^a$ are the $SU(2)_L$ generators in the corresponding representation of the scalar, and $Y$ is the hypercharge generator. For complex representations we work in the basis where $T^3$ and $Y$ are diagonal. After shifting the scalar fields by their VEV's: $\\phi_k\\to \\phi_k +\\langle \\phi_k \\rangle$ and $\\eta_i\\to \\eta_i + \\langle \\eta_i \\rangle$, where the VEV's are normalized as follows\n\\begin{equation}\n\\label{eq:vevnorm}\n{\\rm Tr}(\\langle \\phi_k \\rangle^\\dagger \\langle \\phi_k \\rangle)=\\frac12 v_k^2 \\ , \\qquad {\\rm Tr}(\\langle \\eta_i \\rangle^\\dagger \\langle \\eta_i \\rangle)= \\tilde{v}_i^2 \\ ,\n\\end{equation}\nelectroweak symmetry is broken and $W$ and $Z$ bosons become massive. The mass eigenstates are defined as\n\\begin{eqnarray}\nW^\\pm &=& \\frac1{\\sqrt{2}}(W^1 \\mp i W^2)\\ , \\nonumber \\\\ \n\\label{eq:eweigen}\n\\left( \\begin{array}{cc}\n W^3\\\\\n B \n \\end{array}\\right) &=& \n \\left( \\begin{array}{cc}\n c_w & s_w \\\\\n -s_w & c_w\n \\end{array}\\right)\n \\left( \\begin{array}{c}\n Z\\\\\n A\n \\end{array}\\right) ,\n\\end{eqnarray}\nwhere the sine and cosine of the weak mixing angle are $c_w = {g}\/{\\sqrt{g^2+g^{\\prime 2}}}$ and $s_w = {g^\\prime}\/{\\sqrt{g^2+g^{\\prime 2}}}$, respectively. Notice the unbroken $U(1)_{em}$ leads to the conditions\n\\begin{equation}\n\\left(T^3+\\frac12 Y\\right) \\langle \\phi_k \\rangle= 0 \\ , \\qquad T^3 \\langle \\eta_i \\rangle = 0 \\ .\n\\end{equation}\n Using $T^3 \\langle \\phi_k \\rangle = -Y \\langle \\phi_k \\rangle\/2$ it is possible to express the mass terms of the $W$ and $Z$ in terms of the eigenvalues $T^2 \\langle \\phi_k\\rangle \\equiv T^aT^a \\langle \\phi_k\\rangle = T_k(T_k+1) \\langle \\phi_k\\rangle$:\n\\begin{eqnarray}\n\\label{eq:mwgen}\nm_W^2&=&\\frac18 g^2\\sum_{k} \\left[4T_k(T_k+1) - Y_k^2\\right] v_k^2 + \\frac1{2} g^2\\sum_{i} T_i(T_i+1) \\tilde{v}_i^2 \\ , \\\\\n\\label{eq:mzgen}\nm_Z^2 &=&\\frac14\\, \\frac{g^2}{c_w^2} \\sum_k Y_k^2 v_k^2 \\ ,\n\\end{eqnarray}\nwhere $Y_k$ and $Y_i$ are the hypercharges of $\\phi_k$ and $\\eta_i$. Couplings of the real component of the neutral scalar with the $W$ and $Z$ can be read off by the replacement $v_k \\to v_k(1+\\phi_k^0\/v_k)$ and $\\tilde{v}_i \\to \\tilde{v}_i(1+ \\eta_i^0\/\\tilde{v}_i)$ in the mass terms:\n\\begin{equation}\n\\Gamma^{\\mu\\nu}_{SV_1V_2}= g_{SV_1V_2}\\, g^{\\mu\\nu} \\ ,\n\\end{equation}\nwhere\\footnote{We include a factor of 2! when there are two identical particles in the vertex.}\n\\begin{equation}\n\\label{eq:genwwzzcoup}\n\\begin{array}{ll}\n\\displaystyle g_{\\phi_k WW} = \\frac14 g^2 \\left[4T_k(T_k+1) - Y_k^2\\right] v_k \\ , \\phantom{ccc} &\n\\displaystyle g_{\\phi_k ZZ} = \\frac12\\, \\frac{g^2}{c_w^2} Y_k^2 v_k \\ , \\\\\n\\displaystyle g_{\\eta_i WW} = g^2 T_i(T_i+1) \\tilde{v}_i \\ , &\n\\displaystyle g_{\\eta_i ZZ} = 0 \\ .\n\\end{array}\n\\end{equation}\nNotice that a scalar in a real representation only couples to $WW$ but not $ZZ$. Moreover,\nat this order there is no scalar coupling with $Z\\gamma$ and $\\gamma\\gamma$, which are only induced at the loop level.\n\nAt this point it is worth discussing a few examples of the $SU(2)_L$ representations appearing in the literature. The benchmark is of course the doublet Higgs scalar $H$ with $(T,Y)=(1\/2,1)$. Couplings of the $CP$-even neutral Higgs $h$ with two electroweak bosons are\n\\begin{equation} \n\\label{eq:hwwzz}\ng_{hWW} = \\frac12 g^2 v_h \\ , \\quad g_{hZZ}= \\frac12\\, \\frac{g^2}{c_w^2} v_h \\ , \\quad g_{hZ\\gamma}= g_{h\\gamma\\gamma} = 0 \\ .\n\\end{equation}\nTwo more popular examples are the real triplet scalar $\\phi$ and the complex triplet scalar $\\Phi$ with $(T,Y)=(1,0)$ and $(T,Y)=(1,2)$, respectively, for which the couplings are\n\\begin{eqnarray}\n\\label{eq:phiwwzz}\n&& g_{\\phi^0WW} = 2 g^2 v_\\phi \\ , \\quad g_{\\phi^0ZZ}= g_{\\phi^0Z\\gamma}= g_{\\phi^0\\gamma\\gamma} = 0 \\ , \\\\\n\\label{eq:Phiwwzz}\n&& g_{\\Phi^0WW} = g^2 v_\\Phi \\ , \\quad g_{\\Phi^0ZZ}= 2 \\frac{g^2}{c_w^2} v_\\Phi \\ , \\quad g_{\\Phi^0Z\\gamma}= g_{\\Phi^0\\gamma\\gamma} = 0 \\ .\n\\end{eqnarray}\nWe see that the $SV_1V_2$ couplings are distinctly different for scalars carrying different electroweak quantum numbers, which would give rise to different patterns of decay branching ratios into two electroweak vector bosons. However, it is well known that $\\phi$ and $\\Phi$ individually violate the custodial symmetry and leads to unacceptably large corrections to the $\\rho$ parameter unless the VEV is extremely small, on the order of a few GeV \\cite{Amsler:2008zzb, Boughezal:2004ef, Awramik:2006uz}.\n\n\n\nFor a singlet scalar $s$, the $sV_1V_2$ couplings do not come from the Higgs mechanism. Instead, they originate from the following two dimension-five operators at the leading order:\n\\begin{equation}\n\\label{eq:singletsu2}\n\\kappa_2 \\frac{s}{4m_s} W_{\\mu\\nu}^a W^{a\\, \\mu\\nu} + \\kappa_1\\frac{s}{4m_s} B_{\\mu\\nu} B^{\\mu\\nu} \\ ,\n\\end{equation}\nwhere the singlet $s$ is assumed to be $CP$-even.\n We have normalized the dimensionful couplings to the mass of the singlet $m_s$, although in general an unrelated mass scale could enter.\nIn terms of the mass eigenstate in Eq.~(\\ref{eq:eweigen}), the operators become\n\\begin{eqnarray}\n&& \\kappa_2 \\frac{s}{2m_s} W_{\\mu\\nu}^+ W^{-\\, \\mu\\nu} + (\\kappa_2 c_w^2 + \\kappa_1 s_w^2) \\frac{s}{4m_s} Z_{\\mu\\nu}Z^{\\mu\\nu} \\nonumber \\\\\n&& \\quad \n + 2 c_w s_w \\frac{s}{4m_s} (\\kappa_2 - \\kappa_1) Z_{\\mu\\nu}F^{\\mu\\nu} + (\\kappa_2 s_w^2 +\\kappa_1 c_w^2) \\frac{s}{4m_s} F_{\\mu\\nu} F^{\\mu\\nu} \\ .\n\\end{eqnarray}\nfrom which we obtain the following couplings:\n\\begin{eqnarray}\n\\label{eq:stensor}\n&& \\Gamma^{\\mu\\nu}_{sV_1V_2}= \\frac{g_{sV_1V_2}}{m_s} (p_{V_1}\\cdot p_{V_2} g^{\\mu\\nu} -p_{V_1}^\\nu p_{V_2}^\\mu) \\ , \\\\\n \\label{eq:singscoup}\n&& \\begin{array}{ll}\ng_{sWW}=\\kappa_2 \\ , \\phantom{ccc} & g_{sZZ}=(\\kappa_2 c_w^2 + \\kappa_1 s_w^2)\\ , \\\\\ng_{sZ\\gamma}= c_w s_w (\\kappa_2-\\kappa_1) \\ , \\phantom{ccc} & g_{s\\gamma\\gamma} = (\\kappa_2 s_w^2 + \\kappa_1 c_w^2)\\ .\n\\end{array}\n\\end{eqnarray}\nOne sees immediately that branching ratios following from these couplings are distinctly different from those coming from the Higgs mechanism. Moreover, the four couplings are controlled by only two unknown coefficients $\\kappa_2$ and $\\kappa_1$. So measurements of any two couplings would allow us to predict the remaining couplings, which, if verified experimentally, would be a striking confirmation of the singlet nature of the scalar resonance.\n\n\nIt is worth commenting that the coefficients $\\kappa_2$ and $\\kappa_1$ are related to the one-loop beta functions of $SU(2)_L$ and $U(1)_Y$ gauge groups, respectively, via the Higgs low-energy theorem \\cite{Ellis:1975ap,Shifman:1979eb}:\n\\begin{eqnarray}\n \\beta_2(g) &=& - \\frac{g^3}{(4\\pi)^2} \\left(\\frac{11}3 C_2(G) - \\frac13 n_s C(r)\\right) \\ , \\\\\n \\beta_1(g') &=& +\\frac{g'^3}{(4\\pi)^2} \\frac13 Y^2 n_s^\\prime \\ .\n \\end{eqnarray}\n In the above the Casmir invariants are defined as\n \\begin{equation}\n {\\rm Tr}[t^a_r t^b_r] = C(r)\\delta^{ab} \\ , \\qquad t_G^a t_G^b = C_2(G) \\cdot \\mathbf{1} \\ ,\n \\end{equation}\nwhile $n_s$ is the number of scalars in the complex representation $r$ and $n_s^\\prime$ is the number of scalars charged under $U(1)_Y$. \n Such a connection has been exploited to compute that partial width of $h\\to gg$ and $h\\to \\gamma\\gamma$ in the standard model \\cite{Ellis:1975ap,Shifman:1979eb}, as well as to derive the constraints on the Higgs effective couplings \\cite{Low:2009di}. For our purpose such relations serve to demonstrate that the special case of $\\kappa_2=\\kappa_1$, where the ratio of singlet couplings with $WW$ and $ZZ$ coincides with the standard model expectation, in general requires a conspiracy between the two one-loop beta functions to cancel each other. In this case, however, the coupling to $\\gamma\\gamma$ is identical to the coupling to $ZZ$. On the other hand, depending on whether the $SU(2)_L$ running is asymptotically free, $\\kappa_2$ and $\\kappa_1$ could have either the same or opposite sign, resulting in a reduction (same sign) or enhancement (opposite sign) of the $Z\\gamma$ width relative to $ZZ$ and $\\gamma\\gamma$ channels. It is also possible that $\\kappa_2=0$, resulting in a very suppressed decay width into $WW$. We will discuss further these special cases in Sect.~\\ref{sect:section4}.\n\n\n\\section{Implications of Custodial Invariance}\n\\label{sect:section3}\n\nWe have seen in the previous section that scalar couplings with two electroweak bosons are uniquely determined by the $SU(2)_L\\times U(1)_Y$ quantum number of the scalar involved. For nonsinglet scalars the leading contribution to the $SV_1V_2$ couplings come from the kinetic terms via the Higgs mechanism, which in turn are related to the contribution of each scalar VEV to the masses of the $W$ and $Z$ bosons. However, the ratio of the $W$ and $Z$ masses are measured very precisely and related to the precision electroweak observable $\\rho=m_W^2\/(m_Z c_w)^2$, which is determined at the tree-level by the structure of the scalar sector in a model. Experimentally $\\rho$ is very close to 1 at the percent level \\cite{Amsler:2008zzb}, which severely constrains the electroweak quantum number of any scalar which develops a VEV. \n\nIt has been known for a long time that the Higgs sector in the standard model possesses an accidental global symmetry $SU(2)_L\\times SU(2)_R$, in which the $SU(2)_L$ and $T_R^3$ are gauged and identified with the weak isospin and the hypercharge, respectively. After electroweak symmetry breaking the global symmetry is broken down to the diagonal $SU(2)$, which remains unbroken. The unbroken $SU(2)$ is dubbed the custodial symmetry in Ref.~\\cite{Sikivie:1980hm}, where it was shown the relation $\\rho=1$ is protected by the custodial symmetry $SU(2)_C$. In this section we classify scalar interactions with two electroweak vector bosons according to the $SU(2)_C$ quantum number of the scalar.\\footnote{In the SM the custodial invariance is explicitly broken by fermion masses, since the up-type and down-type fermions have different masses. However, this breaking is oblique in nature and only feeds into the gauge boson masses at the loop-level. Thus we do not include this particular effect in our discussion.}\n\nThere are two possibilities for the scalar sector of a model to preserve the $SU(2)_C$ symmetry. One could find a single irreducible representation of $SU(2)_L\\times U(1)_Y$ which realizes $\\rho=1$. In this case there is only one neutral $CP$-even scalar and the $W$ and $Z$ obtain masses from a single source, the VEV of the neutral scalar $S^0$.\nFrom Eqs.~(\\ref{eq:mwgen}) and (\\ref{eq:mzgen}) we see the condition to realize this possibility is\n\\begin{equation}\n(2T+1)^2-3Y^2 =1\\ .\n\\end{equation}\nAn obvious solution is the Higgs doublet $(T,Y)=(1\/2,1)$, beyond which the next simplest case is $(T,Y)=(3,4)$ \\cite{Tsao:1980em}. However, it is clear that, since there is only one source for the masses of $W$ and $Z$ bosons, the $SV_1V_2$ couplings are derived by replacing $m_V\\to m_V(1+S^0\/v)$ in the mass term, which results in\n\\begin{equation}\n\\label{eq:smwwzz}\ng_{S^0WW}= 2\\frac{m_W^2}{v} \\ , \\quad g_{S^0ZZ}=2\\frac{m_Z^2}{v} , \\quad g_{S^0Z\\gamma}=g_{S^0\\gamma\\gamma}=0 \\ .\n\\end{equation}\nIn other words, when there is only a single source for the mass of electroweak bosons, the custodial symmetry uniquely determines the ratio of the scalar couplings to $WW$ and $ZZ$ to be\n\\begin{equation}\n\\label{eq:gswwzz}\n\\frac{g_{S^0WW}}{g_{S^0ZZ}} = \\frac{m_W^2}{m_Z^2} = c_w^2\\ ,\n\\end{equation}\nregardless of the $SU(2)_L\\times U(1)_Y$ quantum number of the scalar involved. In the next section we will see that Eq.~(\\ref{eq:gswwzz}) predicts the ratio of the decay branching fractions into $WW$ and $ZZ$ to be roughly two-to-one, which is the case in the SM with a Higgs doublet. \n\n\nThe second possibility is to consider multiple scalars all contributing to the $W$ and $Z$ masses through the Higgs mechanism in such a way that, although individually the custodial invariance is not respected, the $\\rho$ parameter remains 1 due to cancellations between the multiple scalars. This would happen if the scalars sits in a complete multiplet $(\\mathbf{M}_L, \\mathbf{N}_R)$ of the full $SU(2)_L\\times SU(2)_R$ group, where $\\mathbf{M}$ and $\\mathbf{N}$ are positive integers labeling the $M$-dimensional and $N$-dimensional irreducible representations of $SU(2)_L$ and $SU(2)_R$, respectively. Recall that $SU(2)_L$ is fully gauged and identified with the weak isospin, while $T_R^3$ is gauged and corresponds to the $U(1)_Y$ such that $T^3_R=Y\/2$, which implies the electric charge is exactly $T_C^3$:\n\\begin{equation}\nQ = T_L^3 +\\frac{Y}2 = T_L^3+T_R^3 = T_C^3 \\ .\n\\end{equation}\nTherefore, all neutral components in the scalar multiplets have $T_C^3=0$. On the other hand, unbroken custodial symmetry requires that only $SU(2)_C$ singlets are allowed to have a VEV. In other words, the scalar representation $(\\mathbf{M}_L,\\mathbf{N}_R)$ must contain a state with $T_C=0$, where $T_C$ is the eigenvalue labeling the quadratic Casmir operator $T_C^aT_C^a = T_C (T_C+1) \\openone$. Since $T_C$ satisfies\n\\begin{equation}\n|M-N| \\le T_C \\le M+N\\ ,\n\\end{equation}\nwe conclude that $\\rho=1$ is possible only when $M=N$ and the scalar must furnish the $(\\mathbf{N}_L, \\mathbf{N}_R)$ representation. \n\nThe trivial representation $(\\mathbf{1}_L, \\mathbf{1}_R)$ is a singlet scalar under $SU(2)_L\\times U(1)_Y$, which was considered in the previous section. In the following we focus on the non-trivial representations, in which the $SV_1V_2$ couplings arise from the Higgs mechanism after the electroweak symmetry breaking. We will represent a scalar $\\Phi_N$ in the $(\\mathbf{N}_L, \\mathbf{N}_R)$ multiplet in a $N\\times N$ matrix whose column vectors are $N$-plets under $SU(2)_L$. The kinetic term of $\\Phi_N$ is \n\\begin{eqnarray}\n\\label{eq:phinkin}\n&& \\frac12 {\\rm Tr}\\left[ (D^\\mu\\Phi_N)^\\dagger D_\\mu\\Phi_N\\right] \\ ,\\\\ \n&& D_\\mu\\Phi_N = \\partial_\\mu \\Phi_N + i gW_\\mu^a T^a \\Phi_N - i g' B_\\mu \\Phi_N T^3 \\ ,\n\\end{eqnarray}\nwhere $T^a$ are generators of $SU(2)$ in the $N$-plet representation.\nWhen $\\Phi_N$ develops a VEV in a custodially invariant fashion\\footnote{When $N$ is an odd integer, $\\Phi_N$ contains a real $SU(2)_L$ $N$-plet with zero hypercharge, whose VEV has a different normalization from that in Eq.~(\\ref{eq:vevnorm}): $\\tilde{v}=v\/\\sqrt{2}$.}\n\\begin{equation}\n\\label{eq:phinvev}\n\\langle \\Phi_N \\rangle = \\frac{v}{\\sqrt{2}}\\,\\openone \\ ,\n\\end{equation}\nelectroweak symmetry breaking occurs and $\\rho=1$ at the tree-level.\n\nIn general various scalars in $\\Phi_N$ could mix with one another and the mass eigenstates do not necessarily have well-defined $SU(2)_L\\times U(1)_Y$ quantum numbers. However, it is highly desirable that the scalar potential respects the custodial symmetry so as to be consistent with $\\rho=1$, which we assume to be the case. \nThen scalars with different $SU(2)_C$ quantum numbers do not mix and all the mass eigenstates have definite $SU(2)_C$ quantum numbers, according to which\nwe will proceed to classify the $SV_1V_2$ interactions. The $(\\mathbf{N}_L, \\mathbf{N}_R)$ representation decomposes under the unbroken $SU(2)_C$ as\n\\begin{equation}\n(\\mathbf{N}_L, \\mathbf{N}_R) = \\mathbf{1} \\oplus \\mathbf{3} \\oplus \\cdots \\oplus \\mathbf{2N-3} \\oplus \\mathbf{2N-1} \\ .\n\\end{equation}\nScalars in the $(4k+1)$-plet are $CP$-even and those in the $(4k+3)$-plet are $CP$-odd. We assume no $CP$-violation in the scalar sector and neglect the $CP$-odd scalar interactions. Since we are interested in interactions with two electroweak gauge bosons, it is worth recalling that $W_\\mu^a$ and $B_\\mu$ transform as (part of) $(\\mathbf{3}_L, \\mathbf{3}_R)$ under $SU(2)_L\\times SU(2)_R$. Therefore the only possible $SU(2)_C$ quantum numbers of a system of two electroweak gauge bosons are a singlet, a triplet, or a 5-plet, which implies the scalar must also be in one of the above three representations in order to have a non-zero coupling with two electroweak bosons. We conclude that $CP$-even $SV_1V_2$ interactions are allowed only when the scalar is either a $SU(2)_C$ singlet or a 5-plet. This is equivalent to saying two spin-1 objects can only couple to either a spin-0 or a spin-2 object. Interactions of two electroweak bosons with scalars in higher representations of $SU(2)_C$ all vanish.\n\n\nLet's define the the neutral component of a custodial $n$-plet as $H_n^0= h_n^0 X_n^0$, where $h_n^0$ is the neutral scalar field and $X_n^0$ is a $N\\times N$ diagonal matrix satisfying\\footnote{Recall that neutral scalars have $T_C^3=T_L^3+T_R^3=0$ and hence belong to the diagonal entries in $\\Phi_N$.} \n\\begin{equation}\n[T^a T^a, X_n^0] = n(n+1) X_n^0 \\ , \\qquad \\qquad [T^3, X_n^0] = 0 \\ , \\qquad \\qquad {\\rm Tr}(X_n^0 X_n^0) = 1 \\ .\n\\end{equation}\nAs emphasized already, only $h_1^0$ is allowed to develop a VEV. From Eq.~(\\ref{eq:phinvev}) we see that $\\langle h_1^0\\rangle = \\sqrt{N\/2}\\,v$ and $X_1^0=\\openone\/\\sqrt{N}$, which implies all other neutral components must be (diagonal) traceless matrices:\n\\begin{equation}\n{\\rm Tr}(X_n^0 X_1^0) = {\\rm Tr} (X_n^0) = 0 \\ , \\qquad n \\ge 2 \\ .\n\\end{equation}\nThe VEV of $h_1^0$ gives rise to the following masses from the kinetic term of $\\Phi_N$ :\n\\begin{eqnarray}\n\\label{eq:mwcus}\nm_W^2&=&\\frac14 g^2 v^2\\, {\\rm Tr}\\left[T^aT^a - T^3T^3\\right] = \\frac1{24} g^2 v^2 N(N^2-1) \\ , \\\\\n\\label{eq:mzcus}\nm_Z^2&=&\\frac12 \\, \\frac{g^2}{c_w^2} v^2\\, {\\rm Tr}\\left[ T^3 T^3 \\right] =\\frac1{24} \\frac{g^2}{c_w^2} v^2 N(N^2-1) \\ ,\n\\end{eqnarray}\nwhich exhibits $\\rho=1$. It can be verified explicitly that Eqs.~(\\ref{eq:mwcus}) and (\\ref{eq:mzcus}) are consistent with Eqs.~(\\ref{eq:mwgen}) and (\\ref{eq:mzgen}). Interactions of $h_{n}^0$, $n=1,5$, with electroweak bosons can be obtained by setting $\\Phi_N = (v\/\\sqrt{2})\\openone + H_{n}^0$ in Eq.~(\\ref{eq:phinkin}):\n\\begin{eqnarray}\ng_{h_{n}^0WW} &=& \\frac1{\\sqrt{2}} \\, g^2 v\\, {\\rm Tr}\\left[X_{n}^0 (T^aT^a - T^3T^3)\\right] \\ , \\\\\ng_{h_{n}^0ZZ} &=& \\sqrt{2}\\, \\frac{g^2}{c_w^2} v\\, {\\rm Tr}\\left[X_{n}^0\\, T^3 T^3 \\right] \\ .\n\\end{eqnarray}\nFor the custodial singlet, $n=1$ and $X_1^0 = \\openone\/\\sqrt{N}$, we obtain\n\\begin{eqnarray}\n\\label{eq:h10ww}\ng_{h_{1}^0WW} &=& \\frac1{\\sqrt{2N}} \\, g^2 v\\, {\\rm Tr}\\left[ (T^aT^a - T^3T^3)\\right] = 2\\sqrt{\\frac{2}N}\\frac{m_W^2}{v} \\ , \\\\\n\\label{eq:h10zz}\ng_{h_{1}^0ZZ} &=& \\sqrt{\\frac{2}N}\\, \\frac{g^2}{c_w^2} v\\, {\\rm Tr}\\left[T^3 T^3 \\right] =2\\sqrt{\\frac{2}N} \\frac{m_Z^2}{v} \\ ,\n\\end{eqnarray}\nwhich is a demonstration of the statement that any custodial singlet (apart from the one in the trivial representation $(\\mathbf{1}_L,\\mathbf{1}_R)$) must have couplings to the $WW$ and $ZZ$ bosons with a fixed ratio as in Eq.~(\\ref{eq:gswwzz}). On the other hand, since $X_{5}^0$ is a traceless diagonal matrix, we have\n\\begin{equation}\n{\\rm Tr}[X_{5}^0 T^a T^a] \\propto {\\rm Tr}[X_{5}^0 \\openone] = 0 \\ .\n\\end{equation}\nThen the couplings are\n\\begin{eqnarray}\n\\label{eq:g2kww}\ng_{h_{5}^0WW} &=& - \\frac1{\\sqrt{2}} \\, g^2 v\\, {\\rm Tr}\\left[X_{5}^0 T^3T^3\\right] \\ , \\\\\n\\label{eq:g2kzz}\ng_{h_{5}^0ZZ} &=& \\sqrt{2}\\, \\frac{g^2}{c_w^2} v\\, {\\rm Tr}\\left[X_{5}^0 T^3 T^3 \\right] \\ ,\n\\end{eqnarray}\nwhich turn out to have a ratio\n\\begin{equation} \n\\frac{g_{h_{5}^0WW}}{g_{h_{5}^0ZZ}} = - \\frac{c_w^2}2 \\ \n\\end{equation}\nthat is different from the ratio of $c_w^2$ for the custodial singlet $h_1^0$. We emphasize that the ratios of the couplings only depend on the $SU(2)_C$ quantum numbers, and not on the particular $(\\mathbf{N}_L, \\mathbf{N}_R)$ representation.\n\n\n\nAgain we discuss a few examples. The canonical example is the familiar Higgs doublet: $(\\mathbf{2}_L, \\mathbf{2}_R)=\\mathbf{1}\\oplus\\mathbf{3}$, where the complex $SU(2)_L$ doublet decomposes into a singlet and a triplet under $SU(2)_C$. The $SU(2)_C$ singlet is the neutral $CP$-even Higgs, $h$, which develops a VEV and breaks the electroweak symmetry, while the triplet contains the Goldstone bosons eaten by the $W$ and $Z$. Our general expressions in Eqs.~(\\ref{eq:h10ww}) and (\\ref{eq:h10zz}) are consistent with those in Eq.~(\\ref{eq:gswwzz}) for $N=2$.\nAnother example appearing in the literature \\cite{Georgi:1985nv, Chanowitz:1985ug, Gunion:1989ci} is the $(\\mathbf{3}_L, \\mathbf{3}_R)$ representation. Under $SU(2)_L\\times U(1)_Y$ it consists of a real electroweak triplet with $(T,Y)=(1,0)$ and a complex electroweak triplet with $(T,Y)=(1,2)$, whose individual couplings to two electroweak bosons were summarized in Eqs.~(\\ref{eq:phiwwzz}) and (\\ref{eq:Phiwwzz}). In this case, the $SU(2)_C$ quantum numbers are $(\\mathbf{3}_L, \\mathbf{3}_R)=\\mathbf{1}\\oplus\\mathbf{3} \\oplus \\mathbf{5}$, which contains two $CP$-even neutral scalars in the singlet and the 5-plet and one $CP$-odd scalar in the triplet \\cite{Georgi:1985nv}. Our expressions for couplings of the singlet and the 5-plet with $WW$ and $ZZ$ are consistent with those in Refs.~\\cite{Georgi:1985nv, Chanowitz:1985ug, Gunion:1989ci}.\\footnote{Although $\\rho=1$ at the tree-level in this model, constraints from $Zb\\bar{b}$ vertex require $v \\sim 50$ GeV \\cite{Haber:1999zh}.}\n\nIt is also possible that the scalar sector of a model has multiple neutral scalar particles. In this case only scalars within the same $SU(2)_C$ multiplet are allowed to mix in order to preserve $\\rho=1$. Then the ratio of the $SV_1V_2$ couplings in the mass eigenstate depends only on the $SU(2)_C$ quantum number and not on the mixing angle at all, except when there exists an electroweak singlet scalar $s$ which couples to $V_1V_2$ through the higher dimensional operators in Eq.~(\\ref{eq:singletsu2}). In this case, it is necessary to include the loop-induced couplings of $h_1^0$ with $Z\\gamma$ and $\\gamma\\gamma$ since they are in the same order as the $sV_1V_2$ couplings. Furthermore, there could be a higher dimensional operator of the form $s|D_\\mu \\Phi_N|^2$, with the coefficient $\\kappa_s\/m_s$, which gives rise to the coupling $sV_1^\\mu V_{2\\, \\mu}$ in addition to those in Eq.~(\\ref{eq:stensor}). Even so, there are only seven unknown parameters: $g_{h_1^0WW}$, $g_{h_1^0Z\\gamma}$, $g_{h_1^0\\gamma\\gamma}$, $\\kappa_1$, $\\kappa_2$, $\\kappa_s$, and the mixing angle between $h_1^0$ and $s$, while one could measure a total of eight branching fractions of two mass eigenstates decaying into $V_1V_2$. Therefore there are enough experimental measurements to not only solve for the seven unknowns, but also test the hypothesis of mixing between $h_1^0$ and $s$. If we observe multiple scalars whose couplings to two electroweak bosons do not follow from that of $h_1^0$ or $h_5^0$, one would be motivated to consider mixing of $h_1^0$ with an electroweak singlet scalar.\n\n\\section{Partial Widths of $S\\to V_1V_2^{(*)}$}\n\\label{sect:section4}\n\nIn this section we compute the partial decay width of $S\\to V_1 V_2^{(*)}$ using the couplings derived in the previous sections. Given that the mass of the scalar could be lighter than the $WW$ threshold, we include the case of $S\\to V_1V_2^*$ when one of the vector bosons is off-shell. Although decays of an electroweak doublet scalar into two electroweak bosons have been computed both in the on-shell \\cite{Lee:1977eg} and off-shell \\cite{Rizzo:1980gz, Keung:1984hn, Grau:1990uu} cases, off-shell decays of an electroweak singlet scalar into two electroweak bosons do not appear to have been considered to the best of our knowledge. In the appendix we compute the decay width of a massive spin-0 particle into two off-shell vector bosons, which serve as the basis of the discussion in what follows.\n\n\nFrom Eq.~(\\ref{eq:onshellfinal}) in the appendix decays of non-electroweak singlet scalars into $WW$ and $ZZ$ are given by\n\\begin{equation}\n\\label{eq:hvvgen}\n\\Gamma(S\\to V_1V_2) = \\delta_V \\frac1{128\\pi} \\frac{|\\tilde{g}_{hV_1V_2}|^2}{x^2 m_S} \\sqrt{1-4x}\\, (1-4x+12x^2) \\ ,\n\\end{equation}\nwhere $x={m_{V}^2}\/{m_S^2}$, $\\delta_W=2$ and $\\delta_Z=1$. In the limit $x^2 \\ll 1$, which is a good approximation if $m_S$ is much larger than the $ZZ$ threshold, the pattern of a scalar decaying into two electroweak vector bosons is \n\\begin{equation}\n\\label{eq:doubpatt}\n\\Gamma(S\\to WW) : \\Gamma(S\\to ZZ) :\\Gamma(S\\to Z\\gamma) : \\Gamma(S\\to \\gamma\\gamma) \\ \\ \\approx \\ 2\\frac{\\tilde{g}_{hWW}^2}{m_W^4} : \n \\frac{\\tilde{g}_{hZZ}^2}{m_Z^4} : 0 : 0 \\ .\n\\end{equation}\nIn terms of branching fractions, normalized to the branching ratio into $WW$, we have\n\\begin{eqnarray}\n&& {Br}_S(ZZ\/WW) = \\rho^2 c_w^4 {\\tilde{g}_{hZZ}^2}\/{\\tilde{g}_{hWW}^2}\\approx c_w^4 {\\tilde{g}_{hZZ}^2}\/{\\tilde{g}_{hWW}^2}\\ , \\\\\n&& {Br}_S(Z\\gamma\/WW) \\approx {Br}_S(\\gamma\\gamma\/WW) \\approx 0 \\ ,\n\\end{eqnarray}\nwhere $Br_S(V_1V_2\/WW)\\equiv Br(S\\to V_1V_2)\/Br(S\\to WW)$. Custodial symmetry then predicts unique patterns of decay branching fractions for $h_1^0$ and $h_5^0$:\n\\begin{eqnarray}\n\\label{eq:doubpatt1}\n&& {Br}_{h_1^0}(ZZ\/WW) \\approx \\frac12 \\ , \\qquad {Br}_{h_1^0}(Z\\gamma\/WW) \\approx {Br}_{h_1^0}(\\gamma\\gamma\/WW) \\approx 0 \\ , \\\\\n&& {Br}_{h_5^0}(ZZ\/WW) \\approx 2 \\ , \\qquad {Br}_{h_5^0}(Z\\gamma\/WW) \\approx {Br}_{h_5^0}(\\gamma\\gamma\/WW) \\approx 0 \\ .\n\\end{eqnarray}\nWe see that a simple counting experiment would allow us to infer the $SU(2)_C$ quantum number of the decaying scalar!\n\n\\begin{figure}[t]\n\\includegraphics[scale=1]{fig1.eps}\n\\caption{\\label{fig:fig1}\\it Ratio of branching fractions into $WW$ and $ZZ$, $Br(ZZ\/WW)$, for an $SU(2)_C$ singlet and a 5-plet, as a function of the scalar mass.}\n\\end{figure} \n\nIn Fig.~\\ref{fig:fig1} we plot the ratio $Br(ZZ\/WW)$ for an $SU(2)_C$ singlet and a 5-plet, including the full kinematic dependence of the gauge boson masses, for the scalar mass between 115 GeV and 1 TeV. We include the decay into off-shell vector bosons using the expression in Eq.~(\\ref{eq:offtotalwidth}) for the scalar mass below the $W$ and\/or $Z$ threshold. Fig.~\\ref{fig:fig1} is the unique prediction of custodial symmetry. Any deviation would imply either the electroweak singlet nature of the scalar or significant violation of custodial symmetry, which in turns suggest cancellation in the $\\rho$ parameter at the percent level.\n\n\n\\begin{figure}[t]\n\\includegraphics[scale=1]{fig2.eps}\n\\includegraphics[scale=1]{fig3.eps}\n\\caption{\\label{fig:fig2}\\it The predicted ratios of branchings, as a function of $Br(ZZ\/WW)$, for an electroweak singlet scalar. The red (gray) curves are for\n$Br(\\gamma\\gamma\/WW)$ and black curves for $Br(Z\\gamma\/WW)$. In this plot we assume the branching into $WW$ is nonzero.}\n\\end{figure} \n\nOn the other hand, using Eqs.~(\\ref{eq:onshellfinal}), (\\ref{eqn:onshellZgamma}), and (\\ref{eq:sgaga}) in the appendix, an electroweak singlet has the following the partial decay widths into two on-shell electroweak bosons \n\\begin{eqnarray}\n\\label{eq:sWWwid}\n\\Gamma(s\\to WW) &=& \\frac1{32\\pi} g_{sWW}^2\\ m_s \\sqrt{1-4x}(1-4x+6x^2) \\ , \\\\\n\\label{eq:sZZwid}\n\\Gamma(s\\to ZZ) &=& \\frac1{64\\pi} g_{sZZ}^2\\ m_s \\sqrt{1-4x}(1-4x+6x^2)\\ , \\\\\n\\label{eq:sggwid}\n\\Gamma(s\\to Z\\gamma) &=& \\frac1{32\\pi} g_{sZ\\gamma}^2\\ m_s (1-x^2)^3 \\ , \\\\\n\\Gamma(s\\to \\gamma\\gamma) &=& \\frac1{64\\pi} g_{s\\gamma\\gamma}^2\\, m_s \\ ,\n\\end{eqnarray}\nwhere the $g_{sV_1V_2}$ couplings are given in Eq.~(\\ref{eq:singscoup}).\nThe pattern of partial decay widths into two electroweak bosons is then, again ignoring the effect of gauge boson masses,\n\\begin{equation}\nBr_s(V_1V_2\/WW) = \\delta_{V_1V_2} \\frac{g_{sV_1V_2}^2}{2g_{sWW}^2}\\ .\n\\end{equation}\nwhere $V_1V_2 = \\{ZZ,Z\\gamma,\\gamma\\gamma\\}$, and $\\delta_{V_1V_2}$ is 2 for $Z\\gamma$ and 1 otherwise. This pattern is generically different from that in Eq.~(\\ref{eq:doubpatt}), where the couplings arise from the Higgs mechanism. More importantly, there are only two unknowns $\\kappa_1$ and $\\kappa_2$. So the branching fractions into $Z\\gamma$ and $\\gamma\\gamma$, normalized to $WW$ mode, could be predicted as follows:\n\\begin{eqnarray}\nBr_s(Z\\gamma\/WW)&\\approx& \\frac{c_w^2}{s_w^2} \\left[\\sqrt{2 Br_s(ZZ\/WW)} -1 \\right]^2 \\ ,\\\\\nBr_s(\\gamma\\gamma\/WW)&\\approx&\\frac12 \\left[ \\frac{c_w^2}{s_w^2} \\sqrt{2 Br_s(ZZ\/WW)} + 1- \\frac{c_w^2}{s_w^2} \\right]^2 \\ .\n\\end{eqnarray}\nIn Fig.~\\ref{fig:fig2} we plot the predicted $Br(Z\\gamma\/WW)$ and $Br(\\gamma\\gamma\/WW)$ branching fractions in terms of $Br(ZZ\/WW)$. Experimental verification of these relations would be a striking confirmation of the singlet nature of the scalar resonance.\n\n\nBy inspection of Eq.~(\\ref{eq:singscoup}) we see that a special case occurs when $\\kappa_2=\\kappa_1$, giving\n$Br_s(ZZ\/WW)=1\/2$, similar to that of $h_1^0$. However, in this case we have\n\\begin{equation}\nBr_s(Z\\gamma\/WW) \\approx 0 \\ , \\qquad Br_s(\\gamma\\gamma\/WW) \\approx \\frac12 \\ ,\n\\end{equation}\nup to corrections due to the mass of the $Z$ boson. By considering all four \npartial widths into the electroweak bosons it is still possible to distinguish \na singlet scalar from the Higgs doublet even in this special case. However, \nas commented in the end of Section \\ref{sect:section2}, such a scenario lacks\nany obvious physical motivation.\n\n\n\n\nAnother special case is when $\\kappa_1$=0, which occurs in the event that \nthe new fermions inducing the dimension-five operators in Eq.~(\\ref{eq:singletsu2}) \ncarry only hypercharge and no isospin. This case is not included in Fig.~\\ref{fig:fig2}\nsince the partial width of the scalar decaying into $WW$ vanishes! Nevertheless, \nthere would still be significant decay branching fractions into $ZZ$, $Z\\gamma$, \nand $\\gamma\\gamma$ states, as predicted by Eq.~(\\ref{eq:singscoup}).\n\n\n\n\\begin{table}[t]\n\\begin{tabular}{|c|c|c|c|} \\hline\n\\makebox[3cm]{$m_S$ (GeV)} & \\makebox[4cm]{$Br(ZZ\/WW)$} & \\makebox[4cm]{$Br(Z\\gamma\/WW)$} & \\makebox[4cm]{$Br(\\gamma\\gamma\/WW)$}\n\\\\ \\hline \\hline\n130 & 0.13 (0.13) & $4.3\\times 10^{-2}$ ($6.7\\times 10^{-3}$) & $3.8\\times 10^{2}$ ($7.8\\times 10^{-3}$) \\\\ \\hline\n150 & 0.12 (0.12) & $1.9\\times 10^{-2}$ ($3.5\\times 10^{-3}$) & 65 ($2.0\\times 10^{-3}$)\\\\ \\hline\n170 & $2.3 \\times 10^{-2}$ ($2.3 \\times 10^{-2}$) & $7.8\\times 10^{-2}$ ($4.1\\times 10^{-4}$) & 1.9 ($1.6\\times 10^{-4}$) \\\\ \\hline\n200 & 0.36 (0.36) & $7.3\\times 10^{-2}$ ($2.4\\times 10^{-4}$) & 3.3 ($\\alt 10^{-4}$) \\\\ \\hline\n300 & 0.44 (0.44) & $1.1\\times 10^{-3}$ ($\\alt 10^{-4}$) & $0.91$ ($\\alt 10^{-4}$) \\\\ \\hline\n400 & 0.47 (0.47) & $\\alt 10^{-4}$ ($\\alt 10^{-4}$) & $0.68$ ($\\alt 10^{-4}$) \\\\ \\hline\n\\end{tabular}\n\\caption{\\label{table1}\\em Ratios of branching fractions for an electroweak singlet scalar when $Br(ZZ\/WW)$ is tuned to the SM value. The value in the parenthesis is for the corresponding SM prediction.}\n\\end{table}\n\nIn Table I we list the ratios of branching fractions for an electroweak singlet, when $Br_s(ZZ\/WW)$ of the scalar is ``tuned'' to fake that\nof a SM Higgs doublet. We see in all cases $Br_s(Z\\gamma\/WW)$ and $Br_s(\\gamma\\gamma\/WW)$ are enhanced over that of the\nSM ratios, especially in the low mass region, when the difference could reach five orders of magnitude at $m_S=130$ GeV for $Br_s(\\gamma\\gamma\/WW)$. The reason behind the enhancement is quite easy to understand: the singlet coupling strengths to all four vector boson pairs are all in the same order. Thus decays into massive final states such as $ZZ$ and $WW$ are suppressed due to phase space and kinematic factors, especially in the low scalar mass region when $WW$ and $ZZ$ channels are off-shell.\n To the contrary, in the SM the Higgs couplings to $WW$ and $ZZ$ arise at the tree-level while the couplings to $Z\\gamma$ and $\\gamma\\gamma$ come from dimension-five operators at the one-loop level. So decays into massive final states could still dominate even below the kinematic threshold.\n\nAnother interesting case is exhibited in Table II, where $Br_s(\\gamma\\gamma\/WW)$ is dialed to fake that of the SM Higgs. In this case the $ZZ$ channel is suppressed relative to the $WW$ channel, while the $Z\\gamma$ channel is significantly enhanced. The importance of $Z\\gamma$ decays is notable, since this channel is so far neglected in the physics planning of the LHC experiments.\n\n\\begin{table}[t]\n\\begin{tabular}{|c|c|c|c|} \\hline\n\\makebox[3cm]{$m_S$ (GeV)} & \\makebox[4cm]{$Br(\\gamma\\gamma\/WW)$} & \\makebox[4cm]{$Br(ZZ\/WW)$} & \\makebox[4cm]{$Br(Z\\gamma\/WW)$}\n\\\\ \\hline \\hline\n115 & $2.7\\times 10^{-2}$ ($2.7\\times 10^{-2}$) & $5.1\\times10^{-2}$ (0.11) & 39 ($9.0\\times 10^{-3}$) \\\\ \\hline\n120 & $1.7\\times 10^{-2}$ ($1.7\\times 10^{-2}$) & $5.7\\times 10^{-2}$ (0.11) & 35 ($8.2\\times 10^{-3}$)\\\\ \\hline\n130 & $7.8\\times 10^{-3}$ ($7.8\\times 10^{-3}$) & $6.7\\times 10^{-2}$ (0.13) & 26 ($6.7\\times 10^{-3}$) \\\\ \\hline\n140 & $4.0\\times 10^{-3}$ ($4.0\\times 10^{-3}$) & $7.1\\times 10^{-2}$ (0.14) & 18 ($5.1\\times 10^{-3} $) \\\\ \\hline\n150 & $2.0\\times 10^{-3}$ ($2.0\\times 10^{-3}$) & $6.4\\times10^{-2}$ (0.12) & 10 ($3.5\\times 10^{-3}$) \\\\ \\hline\n170 & $1.6\\times 10^{-4}$ ($1.6\\times 10^{-4}$)& $1.4 \\times10^{-2}$ ($2.3 \\times 10^{-2}$) & $0.81$ ($4.1\\times 10^{-4}$) \\\\ \\hline\n\\end{tabular}\n\\caption{\\label{table2}\\em Ratios of branching fractions for an electroweak singlet scalar when $Br(\\gamma\\gamma\/WW)$ is tuned to the SM value. The value in the parenthesis is for the corresponding SM prediction.}\n\\end{table}\n\nIf one makes the assumption that the individual partial decay width of a scalar decaying into to $V_1V_2$ could be obtained, presumably in a future lepton collider or with a very high integrated luminosity at the LHC, then we could explore the possibility of determining the $(\\mathbf{N}_L,\\mathbf{N}_R)$ multiplet structure under $SU(2)_L \\times SU(2)_R$. The specific question one could ask, given that the $SU(2)_C$ singlet from all $(\\mathbf{N}_L,\\mathbf{N}_R)$ multiplet has the same ratio of couplings to $WW$ and $ZZ$, is whether it is possible to distinguish the $SU(2)_C$ singlet contained in a $(\\mathbf{2}_L,\\mathbf{2}_R)$ from that contained in a $(\\mathbf{3}_L,\\mathbf{3}_R)$. To this end we observe that the couplings, $g_{h_{1}^0WW}$ and $g_{h_{1}^0ZZ} $\nin Eqs.~(\\ref{eq:h10ww}) and (\\ref{eq:h10zz}), and the gauge boson masses in Eqs.~(\\ref{eq:mwcus}) and (\\ref{eq:mzcus}) are given by two parameters: $N$ and the scalar VEV $v$. Solving for $v$ in terms of the masses and $N$ we obtain\n\\begin{equation}\ng_{h_1^0WW}=g_{h_1^0ZZ}\\, c_w^2= \\sqrt{\\frac{N^2-1}3} \\, g m_W \\ ,\n\\end{equation}\nTherefore the coupling becomes stronger as $N$ increases. The Higgs doublet has $N=2$, while the coupling of the $h_1^0$ in the $(\\mathbf{N}_L,\\mathbf{N}_R)$ is $\\sqrt{(N^2-1)\/3}$ times larger than that in the Higgs doublet, resulting in a partial decay width that is $(N^2-1)\/3$ enhanced. Once $N$ is known, the complete $SU(2)_L\\times U(1)_Y$ quantum number of the scalar resonance is determined.\n\n\n\n\nAs an example, at the LHC one could consider the production of the scalar in the vector boson fusion channels\n$WW\/ZZ \\to S \\to WW$ and $WW\/ZZ \\to S \\to ZZ$, which provide estimates of\n\\begin{eqnarray}\n\\label{eq:vbfguys}\n(\\Gamma_{WW} + \\Gamma_{ZZ})\\frac{\\Gamma_{WW}}{\\Gamma_t}\n\\;\\;{\\rm and}\\; \\;\n(\\Gamma_{WW} + \\Gamma_{ZZ})\\frac{\\Gamma_{ZZ}}{\\Gamma_t}\n\\; .\n\\end{eqnarray}\nThe total width $\\Gamma_t$ could be extracted by measuring the Breit-Wigner shape of the invariant mass spectrum in the $ZZ$ channel. Then one could simply fit the partial widths $\\Gamma_{WW}$ and $\\Gamma_{ZZ}$ using the different hypothesis for $N$. Since the event rate in this case is proportional to $\\Gamma_{WW\/ZZ}^2$, if the total width remains the same the enhancement of a $N\\ge 3$ multiplet over the Higgs doublet is $(N^2-1)^2\/9\\ge 64\/9\\approx 7$, which is a significant enhancement.\n\n\n\n\\section{Discussion and outlook}\n\\label{sect:section5}\n\nWe have performed a general analysis up to dimension five of the couplings\nbetween electroweak vector boson pairs $V_1V_2$ and a Higgs look-alike $S$, assumed to\nbe a neutral $CP$-even scalar resonance. We used the framework of unbroken\ncustodial symmetry to group the possibilities into three ``pure cases'':\nscalars whose electroweak properties match a SM Higgs, scalars that are\n$SU(2)_L\\times SU(2)_R$ singlets and thus couple to $V_1V_2$ only at dimension\nfive, and scalars that couple to $V_1V_2$ as a 5-plet under custodial $SU(2)_C$.\n\nFig.~\\ref{fig:fig1} shows that it should be straightforward to experimentally\ndistinguish the 5-plet case from the SM-like case of a custodial singlet, using\njust the ratio of the $ZZ$ and $WW$ decay rates. \nFig.~\\ref{fig:fig2}\nillustrates that $SU(2)_L\\times SU(2)_R$ singlets produce distinctive\nrelations between the various ratios of $V_1V_2$ decay rates, emphasizing the\nimportance of detecting all four decay channels: $WW$, $ZZ$, $\\gamma\\gamma$,\nand $Z\\gamma$. \n\nTo implement our proposal one can either try to extract ratios of partial decay widths directly \\cite{Zeppenfeld:2002ng}, or measure the individual partial decay widths into pairs of electroweak vector bosons first \\cite{Duhrssen:2004cv, Lafaye:2009vr} and then take the ratios. In the first possibility the event rate measured in each decay channel of a scalar resonance $S$ is given by \n\\begin{equation}\nB\\sigma(V_1V_2)= \\sigma(S)\\times Br(S\\to V_1V_2) \\ .\n\\end{equation}\nTherefore one could approximate the ratio of partial decay widths by the ratio of event rates in each channel, which are measured directly in collider experiments. It would be interesting to study ways to improve on the uncertainty arising from either possibilities.\n\n\nSince experimental analyses are often driven by final states observed, our study demonstrates the importance of having a correlated understanding of all decay channels into pairs of electroweak vector bosons to avoid misidentification. Tables I and II show how one can be badly fooled by measuring only two of the\nelectroweak $V_1V_2$ decay channels for a candidate Higgs. The tables were generated from\nthe predicted properties of a neutral $CP$-even spin 0 ``Higgs'' that\nis in fact an $SU(2)_L \\times SU(2)_R$ singlet imposter. In Table 1 the coefficients\n$\\kappa_1$, $\\kappa_2$ of the dimension-five operators in Eq.~(\\ref{eq:singletsu2})\nhave been adjusted so that the ratio of branching fractions of $S\\to ZZ$ over\n$S\\to WW$ coincides with the SM value for the given masses $m_S$.\nIn Table II the same coefficients\nhave been adjusted so that the branching ratio of $S\\to \\gamma\\gamma$ over\n$S\\to WW$ coincides with the SM value.\nIn both cases measurement of the two remaining $V_1V_2$ decay rates\nunmasks the Higgs imposter in dramatic fashion.\n\n\n\nIn a real experiment, the analysis suggested here could be folded into\nhypothesis testing based on likelihood ratios designed to expose the spin and $CP$ properties of new\nheavy resonances \\cite{DeRujula:2010ys,Gao:2010qx}.\nHigher order effects could be included, as well as the uncertainties\nassociated with unfolding the experimental data to extract the $S\\to V_1V_2$ \nproduction and decay properties.\n\n\n\\begin {acknowledgements}\nWe are grateful to Marcela Carena, Riccardo Rattazzi, and Maria Spiropulu for interesting discussions, and\nto Alvaro De R\\'ujula for coining the phrase ``Higgs imposters''.\nI.~L. was supported in part by the U.S. Department of Energy under\ncontracts No. DE-AC02-06CH11357 and No. DE-FG02-91ER40684.\nFermilab is operated by the Fermi\nResearch Alliance LLC under contract DE-AC02-07CH11359 with the\nU.S. Department of Energy.\n\\end{acknowledgements}\n\n\n\\section*{Appendix}\n\nWe consider a massive spin-0 particle $S$ decaying to two off-shell vector bosons $V_1^*$, $V_2^*$. In the rest\nframe of $S$, and choosing the positive $z$-axis along the direction of $V_2$, the 4-momenta can be written:\n\\begin{equation}\np_S = (m_S,0,0,0) \\, \\quad p_1 = m_1(\\gamma_1, 0, 0, -\\beta_1 \\gamma_1) \\ , \\quad\np_2 = m_2(\\gamma_2, 0, 0, \\beta_2 \\gamma_2) \\ ,\n\\end{equation}\nwhere $m_1$, $m_2$ are the off-shell vector boson masses, and the boosts\nfactors $\\gamma_1$, $\\gamma_2$, $\\beta_1$, $\\beta_2$ are defined by\n\\begin{eqnarray}\n\\label{eqn:cha}\n\\gamma_1 &=& \\frac{m_S}{2m_1}\\left( 1 + \\frac{m_1^2 - m_2^2}{m_S^2} \\right) \\; , \\quad\n\\gamma_2 = \\frac{m_S}{2m_2}\\left( 1 - \\frac{m_1^2 - m_2^2}{m_S^2} \\right) \\; ,\\\\\n\\label{eqn:sha}\n\\beta_1\\gamma_1&=& \\frac{m_S}{2m_1}\\sqrt{\\left( 1 - \\frac{(m_1+m_2)^2}{m_S^2} \\right)\\left( 1 - \\frac{(m_1-m_2)^2}{m_S^2} \\right)} \\; ,\\\\\n\\label{eqn:shb}\n\\beta_2\\gamma_2 &=& \\frac{m_S}{2m_2}\\sqrt{\\left( 1 - \\frac{(m_1+m_2)^2}{m_S^2} \\right)\\left( 1 - \\frac{(m_1-m_2)^2}{m_S^2} \\right)} \\; .\n\\end{eqnarray}\nWe will use the following convenient notation:\n\\begin{equation}\n\\gamma_a = \\gamma_1\\gamma_2(1 + \\beta_1\\beta_2) = \\ch(y_2 - y_1)\\; , \\qquad\n\\gamma_b = \\gamma_1\\gamma_2(\\beta_1 + \\beta_2) = \\sh{(y_2 - y_1)} \\; ,\n\\end{equation}\nwhere $y_1$ and $y_2$ are the vector boson rapidities, as well as the following useful identities:\n\\begin{equation}\n \\gamma_a^2 - \\gamma_b^2 = 1 \\; , \\qquad \n \\gamma_a = \\frac{1}{2m_1m_2}\\left[ m_S^2 - (m_1^2+m_2^2) \\right] \\; , \\qquad\n\\label{eqn:bgamident}\n \\gamma_b = \\frac{m_S}{m_1}\\beta_2\\gamma_2 \\; .\n\\end{equation}\n\nIt is very convenient to compute the decay widths using helicity amplitudes.\nFor this purpose we need to choose a consistent basis for the polarization vectors\nof the vector bosons:\n\\begin{eqnarray}\n\\epsilon_2(\\lambda_2=\\pm) &=& \\pm\\frac{1}{\\sqrt{2}}(0, 1, \\pm i, 0) \\ , \\quad \\epsilon_2(\\lambda_2=0) = (\\beta_2\\gamma_2, 0, 0, \\gamma_2) \\\\\n\\epsilon_1(\\lambda_1=\\mp) &=& \\pm \\frac{1}{\\sqrt{2}}(0, 1,\\pm i, 0) \\ , \\quad\n\\epsilon_1(\\lambda_1=0) = (\\beta_1\\gamma_1, 0, 0, -\\gamma_1) \n\\end{eqnarray}\nwhere $\\lambda_1$, $\\lambda_2$ label the transverse and longitudinal polarizations.\n\nLast but not least we will also need an expression for the two-body phase space:\n\\begin{eqnarray}\nd\\Phi_2(p_S; p_1,p_2) &=& \\frac{d^3p_1 d^3p_2}{(2\\pi)^3 2E_1 (2\\pi)^3 2E_2} (2\\pi)^4 \\delta^4(p_S-p_1-p_2) \\\\\n&=& \\frac{1}{16\\pi^2}\\frac{\\vert \\vec{p}_1 \\vert}{m_S} \\; d{\\rm cos}\\,\\theta \\; d\\phi\n\\end{eqnarray}\nwhere $\\theta$, $\\phi$ are the polar and azimuthal angles between the direction of $V_2$ and\nsome other reference direction, e.g. the direction of the boost from the lab frame to the $S$ rest\nframe, or the direction of the beam. Note that\n\\begin{equation}\n\\vert \\vec{p}_1 \\vert = \\vert \\vec{p}_2\\vert = m_1\\beta_1\\gamma_1 = m_2\\beta_2\\gamma_2\n=\\frac{m_1m_2}{m_S}\\gamma_b \\; .\n\\end{equation}\n\nIt is important to remember that when $V_1$, $V_2$ are distinguishable particles,\nwe integrate $\\theta$, $\\phi$ over the full $4\\pi$ solid angle. However when $V_1$,\n$V_2$ are identical particles (e.g. two $Z$'s or two $\\gamma$'s) we should only\nintegrate $\\theta$ from zero to $\\pi\/2$, to avoid counting the same final state\nconfiguration twice. Thus the angular integration gives $2\\pi$ in this case, not\n$4\\pi$. \n\nThe differential off-shell decay width can be written:\n\\begin{equation}\n\\frac{d^2\\Gamma (S\\to V_1^* V_2^*)}{dm_1^2 dm_2^2}\n= \\frac{2\\pi \\delta_V}{2 m_S} \\frac{m_1m_2\\gamma_b}{16\\pi^2m_S^2} \n\\, P_1P_2 \\hspace*{-10pt}\n\\sum_{\\lambda_1,\\lambda_2 = \\pm,0} \n\\left\\vert \\Gamma^{\\mu\\nu}_{SV_1V_2} \\epsilon^*_{\\mu}(\\lambda_1)\\epsilon^*_{\\nu}(\\lambda_2)\n\\right\\vert^2\n\\end{equation}\nwhere $\\delta_V =1$ for identical vector bosons and 2 otherwise. Here $\\Gamma^{\\mu\\nu}_{SV_1V_2} $\nis the $SV_1V_2$ coupling tensor that can be read off from the Lagrangian. The propagator\nfactors\n\\begin{equation}\nP_i = \\frac{M_{V_i}\\Gamma_{V_i}}{\\pi}\\frac{1}\n{(m_i^2-M_{V_i}^2)^2+M_{V_i}^2\\Gamma_{V_i}^2}\n\\end{equation}\nbecome just $\\delta(m_i^2 - M_{V_i}^2)$ in the narrow width approximation. We will write the coupling tensor as\n\\begin{equation}\n\\Gamma^{\\mu\\nu}_{SV_1V_2} = \\left( \\tilde{g}_{hV_1V_2} + \\frac{\\tilde{g}_{sV_1V_2}}{m_S} p_1\\cdot p_2 \\right) \\; g^{\\mu\\nu} - \\frac{\\tilde{g}_{sV_1V_2}}{m_S}\\, p_1^\\nu p_2^\\mu \\ ,\n\\end{equation}\nwhere the coupling constants $g_h$ and $g_s$ are defined as coefficients of the following operators\n\\begin{equation} \n \\frac{\\delta_V}2 \\left( \\tilde{g}_{hV_1V_2}\\,S\\, V_1^\\mu V_{2\\, \\mu} + \\frac{\\tilde{g}_{sV_1V_2}}{2m_S} S\\, V_1^{\\mu\\nu} V_{2\\, \\mu\\nu} \\right) \\ .\n \\end{equation}\nIn the standard model $\\tilde{g}_{hV_1V_2}^2=8 m_1^2 m_2^2 G_F\/\\sqrt{2}$ for $WW$ and $ZZ$ channels and all other couplings vanish at the tree-level, while for an electroweak singlet scalar $\\tilde{g}_{hV_1V_2}=0$. By angular momentum conservation the only nonvanishing contributions\nfrom the helicity sums are for $(\\lambda_1,\\lambda_2) = (\\pm,\\pm)$, or $(0,0)$:\n\\begin{eqnarray}\n\\sum_{(\\lambda_1,\\lambda_2)} \\left| \\Gamma^{\\mu\\nu} \\epsilon_\\mu^*(\\lambda_1) \\epsilon_\\nu^*(\\lambda_2)\\right|^2 &=& \n\\left|\\tilde{g}_{hV_1V_2}\\right|^2 (2+ \\gamma_a^2) +\\frac{m_1^2 m_2^2}{m_S^2} |\\tilde{g}_{sV_1V_2}|^2 (2\\gamma_a^2+1)\\nonumber \\\\\n&& +\\frac{6m_1m_2\\gamma_a}{m_S}\\, \\Re(\\tilde{g}_{hV_1V_2}\\, \\tilde{g}_{sV_1V_2}^*) ,\n\\end{eqnarray}\nwhere $\\Re(c)$ is the real part of the complex number $c$. Then the off-shell decay width is\n\\begin{eqnarray}\n\\frac{d\\Gamma(S\\to V_1^*V_2^*)}{dm_1^2dm_2^2}& =& \\frac{2\\pi \\delta_V}{2m_S} \\frac{m_1m_2\\gamma_b}{16\\pi^2m_S^2}\\left[ \\left|\\tilde{g}_{hV_1V_2}\\right|^2(2+ \\gamma_a^2) +\n |\\tilde{g}_{sV_1V_2}|^2\\,\\frac{m_1^2 m_2^2}{m_S^2} (2\\gamma_a^2+1) \\right. \\nonumber \\\\\n && \\qquad \\left.+ \\Re(\\tilde{g}_{hV_1V_2}\\, \\tilde{g}_{sV_1V_2}^*)\\, \\frac{6m_1m_2\\gamma_a}{m_S} \\right] P_1 P_2\\ .\n \\end{eqnarray}\n The total decay width of $S\\to V_1^*V_2^*$ is given by\n \\begin{equation}\n \\label{eq:offtotalwidth}\n \\Gamma(S\\to V_1^*V_2^*) = \\int_0^{m_S^2} dm_1^2 \\int_0^{\\left(m_S-\\sqrt{m_1^2}\\right)^2} dm_2^2\\, \\frac{d\\Gamma(S\\to V_1^*V_2^*)}{dm_1^2dm_2^2} \\ .\n \\end{equation}\nThe above formula is valid even when the scalar mass crosses the mass thresholds of $W$ and $Z$ bosons. More explicitly, when both vector bosons are on-shell, $m_1 \\to m_V$, $m_2 \\to m_V$, we have\n \\begin{eqnarray}\n \\label{eq:onshellfinal}\n\\Gamma(S\\to V_1V_2) &=& \\frac{\\delta_V}{32\\pi m_S}\\sqrt{1-4x} \\left\\{\\left|\\tilde{g}_{hV_1V_2}\\right|^2 \\frac1{4x^2}(1-4x+12x^2) \\right. \\nonumber \\\\\n&& \\qquad \\left.+ |\\tilde{g}_{sV_1V_2}|^2 \\frac{m_S^2}2(1-4x+6x^2) + \\Re(\\tilde{g}_{hV_1V_2}\\, \\tilde{g}_{sV_1V_2}^*)\\, 3m_S(1-2x) \\right \\}.\n\\end{eqnarray}\nFor a standard model Higgs boson, $h$, we recover the well-known expression \\cite{Lee:1977eg}\n\\begin{equation}\n\\Gamma (h\\to V_1 V_2) = \\delta_V \\frac{G_F}{\\sqrt{2}}\n\\frac{m_h^3}{16\\pi} \\sqrt{1-4x} (1-4x+12x^2) \\ .\n\\end{equation}\n\nIn the case of $S \\to Z^*\\gamma$, we have to take into account that only the\ntransverse polarizations contribute, and take the limit $m_2 \\to 0$.\nAs $m_2 \\to 0$ \n\\begin{equation}\n\\frac{d\\Gamma (s\\to Z^* \\gamma)}{dm_1^2}\n= \\frac{1}{32\\pi} \\, \\vert \\tilde{g}_{sZ\\gamma} \\vert^2\\,\nm_S\\, \\left(1-\\frac{m_1^2}{m_S^2}\\right)^3 \\,P_1 \\ .\n\\label{eqn:offshellZgamma}\n\\end{equation}\nWhen the $Z$ is on-shell this becomes\n\\begin{equation}\n\\Gamma (S\\to Z \\gamma)\n= \\frac{1}{32\\pi} \\, \\vert \\tilde{g}_{sZ\\gamma} \\vert^2\\,\nm_S\\, (1-x)^3 \\ .\n\\label{eqn:onshellZgamma}\n\\end{equation}\nThe width for $S\\to \\gamma\\gamma$ follows from this\n(note we divide by 2 to get the correct phase space):\n\\begin{equation}\n\\label{eq:sgaga}\n\\Gamma (S\\to \\gamma\\gamma)\n= \\frac{1}{64\\pi} \\, \\vert \\tilde{g}_{s\\gamma\\gamma} \\vert^2\\,\nm_S \\ .\n\\end{equation}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzecqe b/data_all_eng_slimpj/shuffled/split2/finalzzecqe new file mode 100644 index 0000000000000000000000000000000000000000..ebed8cc828c5a1e3f00ee34d714a85d5d6c8e1b7 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzecqe @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nGiven a set of tasks and a set of resources, \nthe problem of task allocation \nis to determine the allocation of resources (e.g., robots) to tasks so as to \nmaximize the overall utility achieved. \nThe task allocation problem has been well studied in the robotics community. \\cite{gerkey_taxonomy_2004} classifies the problem according to three categories: Single-Task\/Multi-Task (ST\/MT) robot, Single-Robot\/Multi-Robot (SR\/MR) task, and Instantaneous\/Time-Extended Assignment (IA\/TA).\n\\cite{korash_taxonomy_2013} extends this taxonomy by considering interrelated utilities and constraints among the tasks.\nIn this paper, we focus on the allocation of multi-robot tasks with\nsingle-task robots and instantaneous assignments. \nThis problem has many applications in the real-world,\nsuch as for urban search and rescue,\nautomated manufacturing and warehousing, etc. \nIn this formulation, each task has a set of pre-specified resource requirements and \neach robot is associated with a set of resources (a.k.a. capabilities). \nA task can be satisfied if the set of robots\nassigned to it satisfy the resource requirements. \n\n\nOne assumption made in prior problem formulations is that \neach task is associated with a unique set of resource requirements.\nThis however may not always be the case for real-world applications. \nConsider a monitoring task for an open area. \nIt may be achieved by multiple mobile robots with cameras\nor a single UAV. \nThe resources required for each way of achieving the task\nare very different. \nIn this paper, we set out to address the multi-robot\ntask allocation problem with task variants,\nwhich represent different ways to achieve a task. \nWe first theoretically prove that this extension\ndoes not impact the complexity class,\nwhich is still NP-complete. \nTo provide a solution, \nwe adapt two previous greedy methods that are introduced\nfor the multi-robot task allocation problem. \nWe show that it is not difficult to compile the \nnew problem into a ``flattened'' problem,\nfor which the previous method would apply. \nWith slight modifications to the proofs, we show that the solution bounds carry over to the new problem. \nFinally, we thoroughly evaluate these two methods along with a random baseline to demonstrate their efficacy for the new problem. \n\n\n\n\n\n\n\n\\section{Related Work}\n\nThe multi-robot task allocation problem is known to be NP-complete \\cite{gerkey_taxonomy_2004}, and is closely related to the coalition formation problem~\\cite{SANDHOLM1999209} in the multi-agent community. \nIn fact,\n\\cite{shehory_methods_1998} first looked at the task allocation problem via coalition formation \nand provided a greedy method based on\nthe set covering problem. \n\\cite{service_coalition_2011} studied the task allocation problem that maximizes utility rather than minimizing cost, and showed that this seemingly innocuous change resulted in very different solution bounds. \nA greedy heuristic was provided to solve this problem. \\cite{zhang_considering_2013} further analyzed \nthis problem and proposed a new heuristic that incorporates the influence of resource requirements between tasks when making assignments. \nOur work adapted the heuristics in these earlier works to solve the new problem with task variants. \n\n\nOur work falls in line with many prior approaches that aimed at extending the applicability of the task allocation problem. \n\\cite{vig_coalition_2006,vig_coalition_2007} adapted prior task allocation methods to work in multi-robot systems with additional constraints and preferences (e.g., balanced workload) that are present in physical robotic systems.\n\\cite{699077} extended the problem to work with decentralized task allocation. \n\\cite{6224910} considered dynamic and environmental influences for allocation in distributed robot systems. \n\\cite{LIEMHETCHARAT201441} adapted task allocation to accommodate synergies between tasks, and \\cite{5980500} studied the problem with precedence constraints between the tasks. \nThe Complex Dependencies category in \\cite{korash_taxonomy_2013} (e.g. CD[ST-MR-IA]) is also of interest. It describes problems in which a set of subtasks (or task decompositions) must be chosen in addition to the problem of choosing optimal assignments. The optimal task decompositions are not known prior to assigning robots.\nA recent work~\\cite{cano_solving_2018} that studied the task allocation problem with task variants, applying to the domain of process scheduling. \nHowever, the problem studied was single-robot tasks with multi-task robots (i.e., MT-SR)\nwhile we are addressing the multi-robot task allocation problem (i.e., ST-MR). \n\nOn the aspect of task variants,\nthe information invariant theory~\\cite{donald1995information} discussed different ways that a task may be achieved\nby different sensori-computational systems,\nwhich are considered equivalent for achieving the task. \n\\cite{1570327,6381528} applied this idea\nto the problem of coalition formation,\nresulting in greater flexibility in dynamic environments compared\nto traditional approaches. \nThe task variants in our work can be \nconsidered as static ways of capturing information invariant for tasks. \n\n\n\n\n\\section{Problem Formulation}\nFollowing prior work, we formulate our variation of the ST-MR-IA problem below, only redefining tasks as sets of task configurations (i.e., task variants). A multi-robot task allocation problem with task variants is a tuple ($R$, $C$, $T$, \\textbf{W}, \\textbf{V}, $Cost$, $U$):\n\\begin{itemize}\n\\item A set of robots $R = \\{r_1, r_2, ...\\}$. Each robot $r_i$ is associated with a vector $B_i$ of H real non-negative capabilities, in which H is assumed to be a constant that specifies the maximum number of capabilities for a domain.\n\\item A set of coalitions, $C = \\{c_1, c_2, ...\\}$. Each coalition $c_j$ satisfies $c_j \\subseteq R$.\n\\item A set of tasks to be assigned $T = \\{t_1, t_2, ...\\}$. Each task $t_k$ is associated with a set of task configurations $T_k = \\{\\tau_{k,1}, \\tau_{k,2} ...\\}$, where each task configuration $\\tau_{k, l}$ requires a vector $P_{k, l}$ of $H$ real non-negative capabilities for achieving task $t_k$ using configuration $\\tau_{k, l}$.\n\\item A vector \\textbf{W} of real non-negative costs for capabilities: the use of the capability indexed by $h$ incurs \\textbf{W}$[h]$ cost per unit.\n\\item A vector \\textbf{V} of real positive rewards for tasks: achieving task $t_k$ with any of its configurations receives \\textbf{V}$[k]$ reward.\n\\item A function Cost: $C \\times \\tau \\rightarrow \\mathcal{R}^0$ that computes real non-negative communication and coordination costs for an assignment based on the coalition and task configuration pair,\nwhere $\\tau$ is used above to denote the union of task configurations for all the tasks.\n\\item A utility function $U$ for assignments, defined as:\n\n\\noindent\n\\begin{eqnarray}\nU_s(m_{jk,l}) = \\textbf{V}[k] - \\sum_h P_{k,l}[h]\\textbf{W}[h] - Cost(c_j, \\tau_{k,l}) \\\\\nU(m_{jk, l}) =\n\\begin{cases}\nU_s(m_{jk,l}) & \\forall h: \\sum_{r_i \\in c_j} B_i[h] \\geq P_{k,l}[h] \\\\\n0 & \\text{otherwise}\n\\end{cases}\n\\end{eqnarray}\n\nin which $m_{jk,l}$ represents an assignment of a coalition $c_j$ to a task configuration $\\tau_{k,l}$.\n\\end{itemize}\n\nThe problem is then to search for a set of assignments $S$ that maximizes:\n\\begin{equation}\n \\underset{m_{jk,l} \\in S}{\\sum}U(m_{jk,l})\n\\end{equation}\nsubject to the constraints\nthat no assignments must have overlapping robots\nand any task must have at most one task configuration assigned in the solution. \nNext, we analyze the complexity of this new formulation\n\n\\begin{theorem}\nThe decision problem of whether there exists an assignment of no less than a given utility value for the multi-robot task allocation problem with task variants is NP-complete. \n\\end{theorem}\n\nThe proof is straightforward as verifying the solution of this problem would only take polynomial time,\nso the problem is in NP. \nFurthermore, since the task allocation problem without task variants is clearly a special case of this new formulation, which is NP-complete, this new problem must also be NP-complete.\n\n\n\n\\section{Solution Methods}\n\nSince the new problem is NP-complete, instead of looking for exact solutions, \nwe propose to study approximate solutions. \n\n\\subsubsection{Random Task Configuration:}\nThe first thought is to randomly pick from the set of task configurations for each task,\nwhich essentially turns the new problem into a task allocation problem without variants. \nWe can then apply any of the state-of-the art task allocation methods. \nThis also becomes our baseline approach for comparison.\nThis method clearly would perform poorly in situations\nwhere a very bad task configuration has been chosen, e.g., \nit renders all the remaining tasks achievable. \n\n\n\\subsubsection{Flattening Formulation:}\n\nA better idea is to try to convert the new problem into a problem without task variants such that prior task allocation solutions can be applied. \nAn obvious solution we consider here is a flattening approach that\ntreats every task configuration as an independent task.\nThis ``flattens'' the extra dimension of task variants and allows us to consider all possible task configurations at once while allowing prior methods to be directly applied.\nA remaining problem, of course, is \nthat this formulation can potentially lead to invalid solutions,\nsince the same task may be assigned multiple times as different task configurations.\nThis seems to imply that we cannot consider different task configurations at the same time. \nThe dilemma here, hence, is\nto incorporate the influences among the task variants when making assignments\nwhile preserving the validity of solutions. \nWe will show soon that this in fact is not difficult at all. \n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{res\/flattening.png}\n\\caption{An illustration of how our methods maintain valid solutions after flattening. Circles on the left hand side represent task configurations and squares represent coalitions. Circles on the right represent assignments. Arrows indicate a feasible assignment and\ndotted lines represent conflicts, e.g., $c_1$ and $c_2$ above conflict (due to overlapping coalitions). The top on the left is a problem without flattening while the bottom with flattening. \nTo maintain valid solutions, \nthe intuition is to maintain the conflicts in the new problem formulation by updating the definition of $\\mathcal{M}$. \n}\n\\label{fig1}\n\\end{figure}\n\n\\subsection{Maximum Utility with Flattening (FlatMaxUtil)}\nFirst, we introduce a natural greedy heuristic that selects an assignment that maximizes the utility among those remaining at every greedy step,\nsimilar to those in~\\cite{service_coalition_2011}.\nSince we consider all task variants as separate tasks,\nwe need to ensure the validity of our solution. A simple way to achieve this is to eliminate assignments to all variants of an assigned task at each greedy iteration.\nAt each step, the following metric is to be maximized by the chosen assignment:\n\n\\begin{equation}\nm^\\lambda = \\underset{m_{xy} \\in \\mathcal{M'}(\\lambda)}{\\max} U(m_{xy})\n\\label{eqfmu}\n\\end{equation}\nwhere $m_{xy}$ refers to the assignment of coalition $x$ to task $y$. Note that given the flattened formulation, we no longer need to consider the task configuration. \nA special note on the definition of $\\mathcal{M}'(\\lambda)$ above,\nwhere $\\lambda$ refers to the greedy step,\n$\\mathcal{M}'(\\lambda)$ represents the remaining valid assignments to be considered,\nand $m^\\lambda$ refers to the assignment chosen at the greedy step. \nIn~\\cite{service_coalition_2011},\nthere is a definition of $\\mathcal{M}(\\lambda)$,\nwhere assignments that have coalitions overlapping with the chosen assignment $m^\\lambda$ or for the same task will be removed for the next iteration. \nIn our formulation, in order to maintain the validity\nof the solution, we change $\\mathcal{M}(\\lambda)$ to $\\mathcal{M}'(\\lambda)$, which additionally removes assignments that represent different task configurations for the chosen task. \nIn this way, we have preserved all the task configurations to be considered at any greedy step\nwhile ensuring that no invalid solution will be produced. \nFig. \\ref{fig1} provides an illustration of this intuition. \n\\begin{theorem}\nApplying FlatMaxUtil to the ST-MR-IA problem \\textbf{with task variants} without restricting the maximum coalition size yields a worst case ratio $\\theta = |R + T|$, while restricting the maximum coalition size to be k yields a worst case ratio of $\\theta = k+2$.\n\\end{theorem}\n\n\\begin{proof}\nGiven a task allocation problem with task variants, first, for each task $t_k$, we change the problem by adding a robot $r^k$ that is shared among all the assignments for all the task configurations for $t_k$.\nFurthermore, we modify the problem such that each $r^k$ has only a unique capability that is not used by any task. \nThis essentially allows at most one of the assignments for a task being made,\nwhich is exactly how we ensure a valid solution. \nAs a result, \nthe bounds in~\\cite{service_coalition_2011} are directly applicable to the flattened problem\nafter this modification (which are $|R|$ and $k+1$ above, respectively, without task variants). \nSince we add a total of $T$ robots and the maximum coalition size is increased by $1$, \nwe have the bounds holds. \n\\end{proof}\n\n\\subsection{Resource Centric with Flattening (FlatRC)}\nThe \\textit{FlatMaxUtil} method is expected to perform poorly in many scenarios, as it only considers the utility of assignment for each greedy choice and does not consider the influences of assignments on each other.\nThis effect is first observed in~\\cite{zhang_considering_2013}.\n\n\\subsubsection{Motivating Example:}\nAs a motivating example, consider a task allocation problem with 3 tasks, 2 variants per task:\n$T = \\{t_1, t_2, t_3\\}, T_1 = \\{\\tau_{1,1}, \\tau_{1,2}\\}, T_2 = \\{\\tau_{2,1}, \\tau_{2,2}\\}$ and $T_3 = \\{\\tau_{3,1}, \\tau_{3,2}\\}$,\nwith capability requirements:\n$P_{1,1} = (2, 0, 0, 0), P_{1,2} = (1, 1, 0, 1), P_{2,1} = (1, 1, 1, 0), P_{2,2} = (1, 1, 0, 1).$\nSuppose we only have two robots with the first capability, but sufficient robots with the other three capabilities. Also assume that robots have at most one unit of each capability, all tasks have equal rewards, all capabilities have equal costs, and $Cost$ always returns $0$ for all assignments.\nMaximizing solely on utility will cause $\\tau_{1,1}$ to be chosen, preventing assignment of either variant of $t_2$, reducing the utility of the final solution.\n\n\n\nSimilar to the \\textit{ResourceCentric} heuristic in~\\cite{zhang_considering_2013},\nwe use a similar heuristic that maximizes the following metric after flattening:\n\n\n\\begin{equation}\n \\rho_{xy} = U(m_{xy}) - \\underset{m_{jl} \\in M_{xy}'(\\lambda)}{\\sum} \\frac{1}{|M'_{jl}(\\lambda)|} \\cdot U(m_{jl})\n\\end{equation}\nwhere $M'_{jl}(\\lambda)$ represents the set of assignments conflicting with $m_{jl}$ (assignment of $c_j$ to task $t_l$ after flattening), with conflicts defined similarly to how we remove conflicting assignments in $\\mathcal{M}'$ in Eq. \\eqref{eqfmu}, differing from $\\mathcal{M}_{jl}$ in~\\cite{zhang_considering_2013}.\nIt follows that the approximation bounds remain similar to those in~\\cite{zhang_considering_2013}:\n\\begin{coro}\nApplying FlatRC to the ST-MR-IA problem \\textbf{with task variants} while restricting the maximum coalition size to be k yields a worst case ratio of $\\theta = min(2k+4, max_{m_{jl} \\in S^*}(|M_{jl}'(1)|))$, in which $S^*$ the optimal solution.\n\\end{coro}\nThe proof proceeds nearly identically to that shown for FlaxMaxUtil given the bound in~\\cite{zhang_considering_2013} (which is $min(2k+2, max_{m_{jl} \\in S^*}(|M_{jl}(1)|))$) .\n\n\n\n\\subsubsection{Complexity Analysis:}\nThe algorithm \nfor FlatRC follows almost\nidentically to \\textit{ResourceCentric} in \\cite{zhang_considering_2013}. \nAs we now have multiple configurations per task, the worst case complexity is increased, but only linearly.\nFor clarity, let $|T_{max}| = \\underset{t_k \\in T}{max}(|t_k|)$, the size of the largest task configuration set. Then,\nthe complexity is bounded by $O(|T||C||T_{max}||\\mathcal{M}|)$, where $\\mathcal{M}$ is the set of assignments. Each greedy step is bounded by $O(|\\mathcal{M}|^2|)$. As there can be at most $min(|R|, |T||T_{max}|)$ assignments, the overall complexity is bounded by $O(min(|R|, |T||T_{max}|) \\cdot |T|^2|T_{max}|^2|C|^2)$.\n\n\n\n\n\\subsection{Approximated FlatRC (FlatRCA)}\nTo improve the computational performance,\nwe also adapt the \\textit{ResourceCentricApprox} heuristic in~\\cite{zhang_considering_2013} to our problem, after flattening. Following a similar reasoning, we wish to reduce the complexity of our algorithm as $|C|$ grows exponentially with $|R|$.\nTo this end, we compute $\\beta_{il}$ = , which measures how much task $t_l$ (note that this is after flattening) depends on robot $r_i$. Then we compute the average expected loss for each \\textit{task} $t_l$ due to the assignment of $r_i$, $\\varphi_{il}$. Finally, we compute the greedy criteria $\\hat\\rho_{xy}$ from this value and the utility of each remaining assignment:\n\n\\begin{eqnarray}\n\\beta_{il} = \\frac{ |M'_{il}| }{ |M'_{l}| } \\\\\n\\varphi_{il} = \\overline{ \\beta_{il} \\cdot U(m_{jl}) }_{m_{jl} \\in \\mathcal{M}'_{i}(\\lambda)} \\\\\n\\hat\\rho_{xy} = U(m_{xy}) - \\underset{r_{i} \\in c_{x}}{\\sum} \\underset{l \\neq y}{\\sum} \\varphi_{il} \n\\end{eqnarray}\n\n\n\\section{Simulation Results}\nIn this section, we provide simulation results for the task variant problem. We focus mainly on randomly generated allocation scenarios, varying key parameters. In all cases when evaluating performance ratios we compare against the upper bound of the optimal solution as \\cite{shehory_methods_1998}: the sum of the feasible assignments with the maximum utility for each task without checking for conflicts. The costs of each capability (i.e. \\textbf{W}) are randomly generated from [0.0, 1.0]. Each task or robot has a 50\\% chance to need\/provide any capability. Capability values, unless specified otherwise, are generated from [0, 8]. The number of capabilities \\textbf{H} is fixed at 7. The maximum size of coalitions is fixed at 5 ($k=5$).\nTask rewards (i.e. \\textbf{V}) are generated randomly from [100, 200]. \\textit{Cost} is defined as a linear function of coalition size, $4n$. Measurements are made over 1000 runs.\n\nWe make two comparisons: varying the number of robots and tasks. We also compare time when varying robots. Varying the number of task variants showed similar trends to varying robots and is not shown. Our time analysis we only consider the time required to assign coalitions to tasks.\nNote that in most of our results, \\textit{FlatRC} and \\textit{FlatRCA} overlap significantly.\nWe show a clear improvement in applying \\textit{FlatRC} and \\textit{FlatRCA} over the simple greedy heuristic for varying numbers of robots, tasks.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{res\/plot-last-varied-r.png}\n\\caption{Results varying \\# of robots available. $|T|$ is fixed at 10 and the maximum \\# of configurations per task is 5.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{res\/plot-last-varied-r-time.png}\n\\caption{Time comparison for Figure~\\ref{fig2}.}\n\\label{fig3}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{res\/plot-last-varied-T.png}\n\\caption{Results for varying \\# of tasks, $|R|$ is fixed at 8 and the maximum \\# of configurations per task is 5.}\n\\label{fig4}\n\\end{figure}\n\n\n\\section{Conclusion and Future Work}\nFirst, we introduced a new formulation of the ST-MR-IA problem that allows for more realistic and flexible scenarios of achieving tasks in the form of task configuration variants. A simple but effective method of solving this problem is to ``flatten'' it into a task allocation problem without the variants. With slight modifications, this allows the application of existing greedy heuristics that provide good approximation bounds.\nHowever, this method effectively discards some finer information about the interaction between task variants. It is clear that improved methods that utilize this information may be devised, but the increased complexity of the problem do not make it trivial to do so. In future work, we plan to investigate such a method if it does exist and compare its performance with those discussed in this work.\n\n\n\n\\section{Introduction}\nGiven a set of tasks and a set of resources, \nthe problem of task allocation \nis to determine the allocation of resources (e.g., robots) to tasks so as to \nmaximize the overall utility achieved. \nThe task allocation problem has been well studied in the robotics community. \\cite{gerkey_taxonomy_2004} classifies the problem according to three categories: Single-Task\/Multi-Task (ST\/MT) robot, Single-Robot\/Multi-Robot (SR\/MR) task, and Instantaneous\/Time-Extended Assignment (IA\/TA).\n\\cite{korash_taxonomy_2013} extends this taxonomy by considering interrelated utilities and constraints among the tasks.\nIn this paper, we focus on the allocation of multi-robot tasks with\nsingle-task robots and instantaneous assignments. \nThis problem has many applications in the real-world,\nsuch as for urban search and rescue,\nautomated manufacturing and warehousing, etc. \nIn this formulation, each task has a set of pre-specified resource requirements and \neach robot is associated with a set of resources (a.k.a. capabilities). \nA task can be satisfied if the set of robots\nassigned to it satisfy the resource requirements. \n\n\nOne assumption made in prior problem formulations is that \neach task is associated with a unique set of resource requirements.\nThis however may not always be the case for real-world applications. \nConsider a monitoring task for an open area. \nIt may be achieved by multiple mobile robots with cameras\nor a single UAV. \nThe resources required for each way of achieving the task\nare very different. \nIn this paper, we set out to address the multi-robot\ntask allocation problem with task variants,\nwhich represent different ways to achieve a task. \nWe first theoretically prove that this extension\ndoes not impact the complexity class,\nwhich is still NP-complete. \nTo provide a solution, \nwe adapt two previous greedy methods that are introduced\nfor the multi-robot task allocation problem. \nWe show that it is not difficult to compile the \nnew problem into a ``flattened'' problem,\nfor which the previous method would apply. \nWith slight modifications to the proofs, we show that the solution bounds carry over to the new problem. \nFinally, we thoroughly evaluate these two methods along with a random baseline to demonstrate their efficacy for the new problem. \n\n\n\n\n\n\n\n\\section{Related Work}\n\nThe multi-robot task allocation problem is known to be NP-complete \\cite{gerkey_taxonomy_2004}, and is closely related to the coalition formation problem~\\cite{SANDHOLM1999209} in the multi-agent community. \nIn fact,\n\\cite{shehory_methods_1998} first looked at the task allocation problem via coalition formation \nand provided a greedy method based on\nthe set covering problem. \n\\cite{service_coalition_2011} studied the task allocation problem that maximizes utility rather than minimizing cost, and showed that this seemingly innocuous change resulted in very different solution bounds. \nA greedy heuristic was provided to solve this problem. \\cite{zhang_considering_2013} further analyzed \nthis problem and proposed a new heuristic that incorporates the influence of resource requirements between tasks when making assignments. \nOur work adapted the heuristics in these earlier works to solve the new problem with task variants. \n\n\nOur work falls in line with many prior approaches that aimed at extending the applicability of the task allocation problem. \n\\cite{vig_coalition_2006,vig_coalition_2007} adapted prior task allocation methods to work in multi-robot systems with additional constraints and preferences (e.g., balanced workload) that are present in physical robotic systems.\n\\cite{699077} extended the problem to work with decentralized task allocation. \n\\cite{6224910} considered dynamic and environmental influences for allocation in distributed robot systems. \n\\cite{LIEMHETCHARAT201441} adapted task allocation to accommodate synergies between tasks, and \\cite{5980500} studied the problem with precedence constraints between the tasks. \nThe Complex Dependencies category in \\cite{korash_taxonomy_2013} (e.g. CD[ST-MR-IA]) is also of interest. It describes problems in which a set of subtasks (or task decompositions) must be chosen in addition to the problem of choosing optimal assignments. The optimal task decompositions are not known prior to assigning robots.\nA recent work~\\cite{cano_solving_2018} that studied the task allocation problem with task variants, applying to the domain of process scheduling. \nHowever, the problem studied was single-robot tasks with multi-task robots (i.e., MT-SR)\nwhile we are addressing the multi-robot task allocation problem (i.e., ST-MR). \n\nOn the aspect of task variants,\nthe information invariant theory~\\cite{donald1995information} discussed different ways that a task may be achieved\nby different sensori-computational systems,\nwhich are considered equivalent for achieving the task. \n\\cite{1570327,6381528} applied this idea\nto the problem of coalition formation,\nresulting in greater flexibility in dynamic environments compared\nto traditional approaches. \nThe task variants in our work can be \nconsidered as static ways of capturing information invariant for tasks. \n\n\n\n\n\\section{Problem Formulation}\nFollowing prior work, we formulate our variation of the ST-MR-IA problem below, only redefining tasks as sets of task configurations (i.e., task variants). A multi-robot task allocation problem with task variants is a tuple ($R$, $C$, $T$, \\textbf{W}, \\textbf{V}, $Cost$, $U$):\n\\begin{itemize}\n\\item A set of robots $R = \\{r_1, r_2, ...\\}$. Each robot $r_i$ is associated with a vector $B_i$ of H real non-negative capabilities, in which H is assumed to be a constant that specifies the maximum number of capabilities for a domain.\n\\item A set of coalitions, $C = \\{c_1, c_2, ...\\}$. Each coalition $c_j$ satisfies $c_j \\subseteq R$.\n\\item A set of tasks to be assigned $T = \\{t_1, t_2, ...\\}$. Each task $t_k$ is associated with a set of task configurations $T_k = \\{\\tau_{k,1}, \\tau_{k,2} ...\\}$, where each task configuration $\\tau_{k, l}$ requires a vector $P_{k, l}$ of $H$ real non-negative capabilities for achieving task $t_k$ using configuration $\\tau_{k, l}$.\n\\item A vector \\textbf{W} of real non-negative costs for capabilities: the use of the capability indexed by $h$ incurs \\textbf{W}$[h]$ cost per unit.\n\\item A vector \\textbf{V} of real positive rewards for tasks: achieving task $t_k$ with any of its configurations receives \\textbf{V}$[k]$ reward.\n\\item A function Cost: $C \\times \\tau \\rightarrow \\mathcal{R}^0$ that computes real non-negative communication and coordination costs for an assignment based on the coalition and task configuration pair,\nwhere $\\tau$ is used above to denote the union of task configurations for all the tasks.\n\\item A utility function $U$ for assignments, defined as:\n\n\\noindent\n\\begin{eqnarray}\nU_s(m_{jk,l}) = \\textbf{V}[k] - \\sum_h P_{k,l}[h]\\textbf{W}[h] - Cost(c_j, \\tau_{k,l}) \\\\\nU(m_{jk, l}) =\n\\begin{cases}\nU_s(m_{jk,l}) & \\forall h: \\sum_{r_i \\in c_j} B_i[h] \\geq P_{k,l}[h] \\\\\n0 & \\text{otherwise}\n\\end{cases}\n\\end{eqnarray}\n\nin which $m_{jk,l}$ represents an assignment of a coalition $c_j$ to a task configuration $\\tau_{k,l}$.\n\\end{itemize}\n\nThe problem is then to search for a set of assignments $S$ that maximizes:\n\\begin{equation}\n \\underset{m_{jk,l} \\in S}{\\sum}U(m_{jk,l})\n\\end{equation}\nsubject to the constraints\nthat no assignments must have overlapping robots\nand any task must have at most one task configuration assigned in the solution. \nNext, we analyze the complexity of this new formulation\n\n\\begin{theorem}\nThe decision problem of whether there exists an assignment of no less than a given utility value for the multi-robot task allocation problem with task variants is NP-complete. \n\\end{theorem}\n\nThe proof is straightforward as verifying the solution of this problem would only take polynomial time,\nso the problem is in NP. \nFurthermore, since the task allocation problem without task variants is clearly a special case of this new formulation, which is NP-complete, this new problem must also be NP-complete.\n\n\n\n\\section{Solution Methods}\n\nSince the new problem is NP-complete, instead of looking for exact solutions, \nwe propose to study approximate solutions. \n\n\\subsubsection{Random Task Configuration:}\nThe first thought is to randomly pick from the set of task configurations for each task,\nwhich essentially turns the new problem into a task allocation problem without variants. \nWe can then apply any of the state-of-the art task allocation methods. \nThis also becomes our baseline approach for comparison.\nThis method clearly would perform poorly in situations\nwhere a very bad task configuration has been chosen, e.g., \nit renders all the remaining tasks achievable. \n\n\n\\subsubsection{Flattening Formulation:}\n\nA better idea is to try to convert the new problem into a problem without task variants such that prior task allocation solutions can be applied. \nAn obvious solution we consider here is a flattening approach that\ntreats every task configuration as an independent task.\nThis ``flattens'' the extra dimension of task variants and allows us to consider all possible task configurations at once while allowing prior methods to be directly applied.\nA remaining problem, of course, is \nthat this formulation can potentially lead to invalid solutions,\nsince the same task may be assigned multiple times as different task configurations.\nThis seems to imply that we cannot consider different task configurations at the same time. \nThe dilemma here, hence, is\nto incorporate the influences among the task variants when making assignments\nwhile preserving the validity of solutions. \nWe will show soon that this in fact is not difficult at all. \n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.6\\columnwidth]{res\/flattening.png}\n\\caption{An illustration of how our methods maintain valid solutions after flattening. Circles on the left hand side represent task configurations and squares represent coalitions. Circles on the right represent assignments. Arrows indicate a feasible assignment and\ndotted lines represent conflicts, e.g., $c_1$ and $c_2$ above conflict (due to overlapping coalitions). The top on the left is a problem without flattening while the bottom with flattening. \nTo maintain valid solutions, \nthe intuition is to maintain the conflicts in the new problem formulation by updating the definition of $\\mathcal{M}$. \n}\n\\label{fig1}\n\\end{figure}\n\n\\subsection{Maximum Utility with Flattening (FlatMaxUtil)}\nFirst, we introduce a natural greedy heuristic that selects an assignment that maximizes the utility among those remaining at every greedy step,\nsimilar to those in~\\cite{service_coalition_2011}.\nSince we consider all task variants as separate tasks,\nwe need to ensure the validity of our solution. A simple way to achieve this is to eliminate assignments to all variants of an assigned task at each greedy iteration.\nAt each step, the following metric is to be maximized by the chosen assignment:\n\n\\begin{equation}\nm^\\lambda = \\underset{m_{xy} \\in \\mathcal{M'}(\\lambda)}{\\max} U(m_{xy})\n\\label{eqfmu}\n\\end{equation}\nwhere $m_{xy}$ refers to the assignment of coalition $x$ to task $y$. Note that given the flattened formulation, we no longer need to consider the task configuration. \nA special note on the definition of $\\mathcal{M}'(\\lambda)$ above,\nwhere $\\lambda$ refers to the greedy step,\n$\\mathcal{M}'(\\lambda)$ represents the remaining valid assignments to be considered,\nand $m^\\lambda$ refers to the assignment chosen at the greedy step. \nIn~\\cite{service_coalition_2011},\nthere is a definition of $\\mathcal{M}(\\lambda)$,\nwhere assignments that have coalitions overlapping with the chosen assignment $m^\\lambda$ or for the same task will be removed for the next iteration. \nIn our formulation, in order to maintain the validity\nof the solution, we change $\\mathcal{M}(\\lambda)$ to $\\mathcal{M}'(\\lambda)$, which additionally removes assignments that represent different task configurations for the chosen task. \nIn this way, we have preserved all the task configurations to be considered at any greedy step\nwhile ensuring that no invalid solution will be produced. \nFig. \\ref{fig1} provides an illustration of this intuition. \n\\begin{theorem}\nApplying FlatMaxUtil to the ST-MR-IA problem \\textbf{with task variants} without restricting the maximum coalition size yields a worst case ratio $\\theta = |R + T|$, while restricting the maximum coalition size to be k yields a worst case ratio of $\\theta = k+2$.\n\\end{theorem}\n\n\\begin{proof}\nGiven a task allocation problem with task variants, first, for each task $t_k$, we change the problem by adding a robot $r^k$ that is shared among all the assignments for all the task configurations for $t_k$.\nFurthermore, we modify the problem such that each $r^k$ has only a unique capability that is not used by any task. \nThis essentially allows at most one of the assignments for a task being made,\nwhich is exactly how we ensure a valid solution. \nAs a result, \nthe bounds in~\\cite{service_coalition_2011} are directly applicable to the flattened problem\nafter this modification (which are $|R|$ and $k+1$ above, respectively, without task variants). \nSince we add a total of $T$ robots and the maximum coalition size is increased by $1$, \nwe have the bounds holds. \n\\end{proof}\n\n\\subsection{Resource Centric with Flattening (FlatRC)}\nThe \\textit{FlatMaxUtil} method is expected to perform poorly in many scenarios, as it only considers the utility of assignment for each greedy choice and does not consider the influences of assignments on each other.\nThis effect is first observed in~\\cite{zhang_considering_2013}.\n\n\\subsubsection{Motivating Example:}\nAs a motivating example, consider a task allocation problem with 3 tasks, 2 variants per task:\n$T = \\{t_1, t_2, t_3\\}, T_1 = \\{\\tau_{1,1}, \\tau_{1,2}\\}, T_2 = \\{\\tau_{2,1}, \\tau_{2,2}\\}$ and $T_3 = \\{\\tau_{3,1}, \\tau_{3,2}\\}$,\nwith capability requirements:\n$P_{1,1} = (2, 0, 0, 0), P_{1,2} = (1, 1, 0, 1), P_{2,1} = (1, 1, 1, 0), P_{2,2} = (1, 1, 0, 1).$\nSuppose we only have two robots with the first capability, but sufficient robots with the other three capabilities. Also assume that robots have at most one unit of each capability, all tasks have equal rewards, all capabilities have equal costs, and $Cost$ always returns $0$ for all assignments.\nMaximizing solely on utility will cause $\\tau_{1,1}$ to be chosen, preventing assignment of either variant of $t_2$, reducing the utility of the final solution.\n\n\n\nSimilar to the \\textit{ResourceCentric} heuristic in~\\cite{zhang_considering_2013},\nwe use a similar heuristic that maximizes the following metric after flattening:\n\n\n\\begin{equation}\n \\rho_{xy} = U(m_{xy}) - \\underset{m_{jl} \\in M_{xy}'(\\lambda)}{\\sum} \\frac{1}{|M'_{jl}(\\lambda)|} \\cdot U(m_{jl})\n\\end{equation}\nwhere $M'_{jl}(\\lambda)$ represents the set of assignments conflicting with $m_{jl}$ (assignment of $c_j$ to task $t_l$ after flattening), with conflicts defined similarly to how we remove conflicting assignments in $\\mathcal{M}'$ in Eq. \\eqref{eqfmu}, differing from $\\mathcal{M}_{jl}$ in~\\cite{zhang_considering_2013}.\nIt follows that the approximation bounds remain similar to those in~\\cite{zhang_considering_2013}:\n\\begin{coro}\nApplying FlatRC to the ST-MR-IA problem \\textbf{with task variants} while restricting the maximum coalition size to be k yields a worst case ratio of $\\theta = min(2k+4, max_{m_{jl} \\in S^*}(|M_{jl}'(1)|))$, in which $S^*$ the optimal solution.\n\\end{coro}\nThe proof proceeds nearly identically to that shown for FlaxMaxUtil given the bound in~\\cite{zhang_considering_2013} (which is $min(2k+2, max_{m_{jl} \\in S^*}(|M_{jl}(1)|))$) .\n\n\n\n\\subsubsection{Complexity Analysis:}\nThe algorithm \nfor FlatRC follows almost\nidentically to \\textit{ResourceCentric} in \\cite{zhang_considering_2013}. \nAs we now have multiple configurations per task, the worst case complexity is increased, but only linearly.\nFor clarity, let $|T_{max}| = \\underset{t_k \\in T}{max}(|t_k|)$, the size of the largest task configuration set. Then,\nthe complexity is bounded by $O(|T||C||T_{max}||\\mathcal{M}|)$, where $\\mathcal{M}$ is the set of assignments. Each greedy step is bounded by $O(|\\mathcal{M}|^2|)$. As there can be at most $min(|R|, |T||T_{max}|)$ assignments, the overall complexity is bounded by $O(min(|R|, |T||T_{max}|) \\cdot |T|^2|T_{max}|^2|C|^2)$.\n\n\n\n\n\\subsection{Approximated FlatRC (FlatRCA)}\nTo improve the computational performance,\nwe also adapt the \\textit{ResourceCentricApprox} heuristic in~\\cite{zhang_considering_2013} to our problem, after flattening. Following a similar reasoning, we wish to reduce the complexity of our algorithm as $|C|$ grows exponentially with $|R|$.\nTo this end, we compute $\\beta_{il}$ = , which measures how much task $t_l$ (note that this is after flattening) depends on robot $r_i$. Then we compute the average expected loss for each \\textit{task} $t_l$ due to the assignment of $r_i$, $\\varphi_{il}$. Finally, we compute the greedy criteria $\\hat\\rho_{xy}$ from this value and the utility of each remaining assignment:\n\n\\begin{eqnarray}\n\\beta_{il} = \\frac{ |M'_{il}| }{ |M'_{l}| } \\\\\n\\varphi_{il} = \\overline{ \\beta_{il} \\cdot U(m_{jl}) }_{m_{jl} \\in \\mathcal{M}'_{i}(\\lambda)} \\\\\n\\hat\\rho_{xy} = U(m_{xy}) - \\underset{r_{i} \\in c_{x}}{\\sum} \\underset{l \\neq y}{\\sum} \\varphi_{il} \n\\end{eqnarray}\n\n\n\\section{Simulation Results}\nIn this section, we provide simulation results for the task variant problem. We focus mainly on randomly generated allocation scenarios, varying key parameters. In all cases when evaluating performance ratios we compare against the upper bound of the optimal solution as \\cite{shehory_methods_1998}: the sum of the feasible assignments with the maximum utility for each task without checking for conflicts. The costs of each capability (i.e. \\textbf{W}) are randomly generated from [0.0, 1.0]. Each task or robot has a 50\\% chance to need\/provide any capability. Capability values, unless specified otherwise, are generated from [0, 8]. The number of capabilities \\textbf{H} is fixed at 7. The maximum size of coalitions is fixed at 5 ($k=5$).\nTask rewards (i.e. \\textbf{V}) are generated randomly from [100, 200]. \\textit{Cost} is defined as a linear function of coalition size, $4n$. Measurements are made over 1000 runs.\n\nWe make two comparisons: varying the number of robots and tasks. We also compare time when varying robots. Varying the number of task variants showed similar trends to varying robots and is not shown. Our time analysis we only consider the time required to assign coalitions to tasks.\nNote that in most of our results, \\textit{FlatRC} and \\textit{FlatRCA} overlap significantly.\nWe show a clear improvement in applying \\textit{FlatRC} and \\textit{FlatRCA} over the simple greedy heuristic for varying numbers of robots, tasks.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{res\/plot-last-varied-r.png}\n\\caption{Results varying \\# of robots available. $|T|$ is fixed at 10 and the maximum \\# of configurations per task is 5.}\n\\label{fig2}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{res\/plot-last-varied-r-time.png}\n\\caption{Time comparison for Figure~\\ref{fig2}.}\n\\label{fig3}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.8\\columnwidth]{res\/plot-last-varied-T.png}\n\\caption{Results for varying \\# of tasks, $|R|$ is fixed at 8 and the maximum \\# of configurations per task is 5.}\n\\label{fig4}\n\\end{figure}\n\n\n\\section{Conclusion and Future Work}\nFirst, we introduced a new formulation of the ST-MR-IA problem that allows for more realistic and flexible scenarios of achieving tasks in the form of task configuration variants. A simple but effective method of solving this problem is to ``flatten'' it into a task allocation problem without the variants. With slight modifications, this allows the application of existing greedy heuristics that provide good approximation bounds.\nHowever, this method effectively discards some finer information about the interaction between task variants. It is clear that improved methods that utilize this information may be devised, but the increased complexity of the problem do not make it trivial to do so. In future work, we plan to investigate such a method if it does exist and compare its performance with those discussed in this work.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{The ALICE Collaboration}\n\\begingroup\n\\small\n\\begin{flushleft}\nS.~Acharya\\Irefn{org141}\\And \nD.~Adamov\\'{a}\\Irefn{org93}\\And \nS.P.~Adhya\\Irefn{org141}\\And \nA.~Adler\\Irefn{org73}\\And \nJ.~Adolfsson\\Irefn{org79}\\And \nM.M.~Aggarwal\\Irefn{org98}\\And \nG.~Aglieri Rinella\\Irefn{org34}\\And \nM.~Agnello\\Irefn{org31}\\And \nN.~Agrawal\\Irefn{org48}\\textsuperscript{,}\\Irefn{org10}\\And \nZ.~Ahammed\\Irefn{org141}\\And \nS.~Ahmad\\Irefn{org17}\\And \nS.U.~Ahn\\Irefn{org75}\\And \nA.~Akindinov\\Irefn{org90}\\And \nM.~Al-Turany\\Irefn{org105}\\And \nS.N.~Alam\\Irefn{org141}\\And \nD.S.D.~Albuquerque\\Irefn{org122}\\And \nD.~Aleksandrov\\Irefn{org86}\\And \nB.~Alessandro\\Irefn{org58}\\And \nH.M.~Alfanda\\Irefn{org6}\\And \nR.~Alfaro Molina\\Irefn{org71}\\And \nB.~Ali\\Irefn{org17}\\And \nY.~Ali\\Irefn{org15}\\And \nA.~Alici\\Irefn{org10}\\textsuperscript{,}\\Irefn{org53}\\textsuperscript{,}\\Irefn{org27}\\And \nA.~Alkin\\Irefn{org2}\\And \nJ.~Alme\\Irefn{org22}\\And \nT.~Alt\\Irefn{org68}\\And \nL.~Altenkamper\\Irefn{org22}\\And \nI.~Altsybeev\\Irefn{org112}\\And \nM.N.~Anaam\\Irefn{org6}\\And \nC.~Andrei\\Irefn{org47}\\And \nD.~Andreou\\Irefn{org34}\\And \nH.A.~Andrews\\Irefn{org109}\\And \nA.~Andronic\\Irefn{org144}\\And \nM.~Angeletti\\Irefn{org34}\\And \nV.~Anguelov\\Irefn{org102}\\And \nC.~Anson\\Irefn{org16}\\And \nT.~Anti\\v{c}i\\'{c}\\Irefn{org106}\\And \nF.~Antinori\\Irefn{org56}\\And \nP.~Antonioli\\Irefn{org53}\\And \nR.~Anwar\\Irefn{org125}\\And \nN.~Apadula\\Irefn{org78}\\And \nL.~Aphecetche\\Irefn{org114}\\And \nH.~Appelsh\\\"{a}user\\Irefn{org68}\\And \nS.~Arcelli\\Irefn{org27}\\And \nR.~Arnaldi\\Irefn{org58}\\And \nM.~Arratia\\Irefn{org78}\\And \nI.C.~Arsene\\Irefn{org21}\\And \nM.~Arslandok\\Irefn{org102}\\And \nA.~Augustinus\\Irefn{org34}\\And \nR.~Averbeck\\Irefn{org105}\\And \nS.~Aziz\\Irefn{org61}\\And \nM.D.~Azmi\\Irefn{org17}\\And \nA.~Badal\\`{a}\\Irefn{org55}\\And \nY.W.~Baek\\Irefn{org40}\\And \nS.~Bagnasco\\Irefn{org58}\\And \nX.~Bai\\Irefn{org105}\\And \nR.~Bailhache\\Irefn{org68}\\And \nR.~Bala\\Irefn{org99}\\And \nA.~Baldisseri\\Irefn{org137}\\And \nM.~Ball\\Irefn{org42}\\And \nS.~Balouza\\Irefn{org103}\\And \nR.C.~Baral\\Irefn{org84}\\And \nR.~Barbera\\Irefn{org28}\\And \nL.~Barioglio\\Irefn{org26}\\And \nG.G.~Barnaf\\\"{o}ldi\\Irefn{org145}\\And \nL.S.~Barnby\\Irefn{org92}\\And \nV.~Barret\\Irefn{org134}\\And \nP.~Bartalini\\Irefn{org6}\\And \nK.~Barth\\Irefn{org34}\\And \nE.~Bartsch\\Irefn{org68}\\And \nF.~Baruffaldi\\Irefn{org29}\\And \nN.~Bastid\\Irefn{org134}\\And \nS.~Basu\\Irefn{org143}\\And \nG.~Batigne\\Irefn{org114}\\And \nB.~Batyunya\\Irefn{org74}\\And \nP.C.~Batzing\\Irefn{org21}\\And \nD.~Bauri\\Irefn{org48}\\And \nJ.L.~Bazo~Alba\\Irefn{org110}\\And \nI.G.~Bearden\\Irefn{org87}\\And \nC.~Bedda\\Irefn{org63}\\And \nN.K.~Behera\\Irefn{org60}\\And \nI.~Belikov\\Irefn{org136}\\And \nF.~Bellini\\Irefn{org34}\\And \nR.~Bellwied\\Irefn{org125}\\And \nV.~Belyaev\\Irefn{org91}\\And \nG.~Bencedi\\Irefn{org145}\\And \nS.~Beole\\Irefn{org26}\\And \nA.~Bercuci\\Irefn{org47}\\And \nY.~Berdnikov\\Irefn{org96}\\And \nD.~Berenyi\\Irefn{org145}\\And \nR.A.~Bertens\\Irefn{org130}\\And \nD.~Berzano\\Irefn{org58}\\And \nM.G.~Besoiu\\Irefn{org67}\\And \nL.~Betev\\Irefn{org34}\\And \nA.~Bhasin\\Irefn{org99}\\And \nI.R.~Bhat\\Irefn{org99}\\And \nM.A.~Bhat\\Irefn{org3}\\And \nH.~Bhatt\\Irefn{org48}\\And \nB.~Bhattacharjee\\Irefn{org41}\\And \nA.~Bianchi\\Irefn{org26}\\And \nL.~Bianchi\\Irefn{org125}\\textsuperscript{,}\\Irefn{org26}\\And \nN.~Bianchi\\Irefn{org51}\\And \nJ.~Biel\\v{c}\\'{\\i}k\\Irefn{org37}\\And \nJ.~Biel\\v{c}\\'{\\i}kov\\'{a}\\Irefn{org93}\\And \nA.~Bilandzic\\Irefn{org117}\\textsuperscript{,}\\Irefn{org103}\\And \nG.~Biro\\Irefn{org145}\\And \nR.~Biswas\\Irefn{org3}\\And \nS.~Biswas\\Irefn{org3}\\And \nJ.T.~Blair\\Irefn{org119}\\And \nD.~Blau\\Irefn{org86}\\And \nC.~Blume\\Irefn{org68}\\And \nG.~Boca\\Irefn{org139}\\And \nF.~Bock\\Irefn{org94}\\textsuperscript{,}\\Irefn{org34}\\And \nA.~Bogdanov\\Irefn{org91}\\And \nL.~Boldizs\\'{a}r\\Irefn{org145}\\And \nA.~Bolozdynya\\Irefn{org91}\\And \nM.~Bombara\\Irefn{org38}\\And \nG.~Bonomi\\Irefn{org140}\\And \nH.~Borel\\Irefn{org137}\\And \nA.~Borissov\\Irefn{org144}\\textsuperscript{,}\\Irefn{org91}\\And \nM.~Borri\\Irefn{org127}\\And \nH.~Bossi\\Irefn{org146}\\And \nE.~Botta\\Irefn{org26}\\And \nL.~Bratrud\\Irefn{org68}\\And \nP.~Braun-Munzinger\\Irefn{org105}\\And \nM.~Bregant\\Irefn{org121}\\And \nT.A.~Broker\\Irefn{org68}\\And \nM.~Broz\\Irefn{org37}\\And \nE.J.~Brucken\\Irefn{org43}\\And \nE.~Bruna\\Irefn{org58}\\And \nG.E.~Bruno\\Irefn{org33}\\textsuperscript{,}\\Irefn{org104}\\And \nM.D.~Buckland\\Irefn{org127}\\And \nD.~Budnikov\\Irefn{org107}\\And \nH.~Buesching\\Irefn{org68}\\And \nS.~Bufalino\\Irefn{org31}\\And \nO.~Bugnon\\Irefn{org114}\\And \nP.~Buhler\\Irefn{org113}\\And \nP.~Buncic\\Irefn{org34}\\And \nZ.~Buthelezi\\Irefn{org72}\\And \nJ.B.~Butt\\Irefn{org15}\\And \nJ.T.~Buxton\\Irefn{org95}\\And \nD.~Caffarri\\Irefn{org88}\\And \nA.~Caliva\\Irefn{org105}\\And \nE.~Calvo Villar\\Irefn{org110}\\And \nR.S.~Camacho\\Irefn{org44}\\And \nP.~Camerini\\Irefn{org25}\\And \nA.A.~Capon\\Irefn{org113}\\And \nF.~Carnesecchi\\Irefn{org10}\\And \nJ.~Castillo Castellanos\\Irefn{org137}\\And \nA.J.~Castro\\Irefn{org130}\\And \nE.A.R.~Casula\\Irefn{org54}\\And \nF.~Catalano\\Irefn{org31}\\And \nC.~Ceballos Sanchez\\Irefn{org52}\\And \nP.~Chakraborty\\Irefn{org48}\\And \nS.~Chandra\\Irefn{org141}\\And \nB.~Chang\\Irefn{org126}\\And \nW.~Chang\\Irefn{org6}\\And \nS.~Chapeland\\Irefn{org34}\\And \nM.~Chartier\\Irefn{org127}\\And \nS.~Chattopadhyay\\Irefn{org141}\\And \nS.~Chattopadhyay\\Irefn{org108}\\And \nA.~Chauvin\\Irefn{org24}\\And \nC.~Cheshkov\\Irefn{org135}\\And \nB.~Cheynis\\Irefn{org135}\\And \nV.~Chibante Barroso\\Irefn{org34}\\And \nD.D.~Chinellato\\Irefn{org122}\\And \nS.~Cho\\Irefn{org60}\\And \nP.~Chochula\\Irefn{org34}\\And \nT.~Chowdhury\\Irefn{org134}\\And \nP.~Christakoglou\\Irefn{org88}\\And \nC.H.~Christensen\\Irefn{org87}\\And \nP.~Christiansen\\Irefn{org79}\\And \nT.~Chujo\\Irefn{org133}\\And \nC.~Cicalo\\Irefn{org54}\\And \nL.~Cifarelli\\Irefn{org10}\\textsuperscript{,}\\Irefn{org27}\\And \nF.~Cindolo\\Irefn{org53}\\And \nJ.~Cleymans\\Irefn{org124}\\And \nF.~Colamaria\\Irefn{org52}\\And \nD.~Colella\\Irefn{org52}\\And \nA.~Collu\\Irefn{org78}\\And \nM.~Colocci\\Irefn{org27}\\And \nM.~Concas\\Irefn{org58}\\Aref{orgI}\\And \nG.~Conesa Balbastre\\Irefn{org77}\\And \nZ.~Conesa del Valle\\Irefn{org61}\\And \nG.~Contin\\Irefn{org59}\\textsuperscript{,}\\Irefn{org127}\\And \nJ.G.~Contreras\\Irefn{org37}\\And \nT.M.~Cormier\\Irefn{org94}\\And \nY.~Corrales Morales\\Irefn{org58}\\textsuperscript{,}\\Irefn{org26}\\And \nP.~Cortese\\Irefn{org32}\\And \nM.R.~Cosentino\\Irefn{org123}\\And \nF.~Costa\\Irefn{org34}\\And \nS.~Costanza\\Irefn{org139}\\And \nJ.~Crkovsk\\'{a}\\Irefn{org61}\\And \nP.~Crochet\\Irefn{org134}\\And \nE.~Cuautle\\Irefn{org69}\\And \nL.~Cunqueiro\\Irefn{org94}\\And \nD.~Dabrowski\\Irefn{org142}\\And \nT.~Dahms\\Irefn{org103}\\textsuperscript{,}\\Irefn{org117}\\And \nA.~Dainese\\Irefn{org56}\\And \nF.P.A.~Damas\\Irefn{org137}\\textsuperscript{,}\\Irefn{org114}\\And \nS.~Dani\\Irefn{org65}\\And \nM.C.~Danisch\\Irefn{org102}\\And \nA.~Danu\\Irefn{org67}\\And \nD.~Das\\Irefn{org108}\\And \nI.~Das\\Irefn{org108}\\And \nP.~Das\\Irefn{org3}\\And \nS.~Das\\Irefn{org3}\\And \nA.~Dash\\Irefn{org84}\\And \nS.~Dash\\Irefn{org48}\\And \nA.~Dashi\\Irefn{org103}\\And \nS.~De\\Irefn{org84}\\textsuperscript{,}\\Irefn{org49}\\And \nA.~De Caro\\Irefn{org30}\\And \nG.~de Cataldo\\Irefn{org52}\\And \nC.~de Conti\\Irefn{org121}\\And \nJ.~de Cuveland\\Irefn{org39}\\And \nA.~De Falco\\Irefn{org24}\\And \nD.~De Gruttola\\Irefn{org10}\\And \nN.~De Marco\\Irefn{org58}\\And \nS.~De Pasquale\\Irefn{org30}\\And \nR.D.~De Souza\\Irefn{org122}\\And \nS.~Deb\\Irefn{org49}\\And \nH.F.~Degenhardt\\Irefn{org121}\\And \nK.R.~Deja\\Irefn{org142}\\And \nA.~Deloff\\Irefn{org83}\\And \nS.~Delsanto\\Irefn{org131}\\textsuperscript{,}\\Irefn{org26}\\And \nP.~Dhankher\\Irefn{org48}\\And \nD.~Di Bari\\Irefn{org33}\\And \nA.~Di Mauro\\Irefn{org34}\\And \nR.A.~Diaz\\Irefn{org8}\\And \nT.~Dietel\\Irefn{org124}\\And \nP.~Dillenseger\\Irefn{org68}\\And \nY.~Ding\\Irefn{org6}\\And \nR.~Divi\\`{a}\\Irefn{org34}\\And \n{\\O}.~Djuvsland\\Irefn{org22}\\And \nU.~Dmitrieva\\Irefn{org62}\\And \nA.~Dobrin\\Irefn{org34}\\textsuperscript{,}\\Irefn{org67}\\And \nB.~D\\\"{o}nigus\\Irefn{org68}\\And \nO.~Dordic\\Irefn{org21}\\And \nA.K.~Dubey\\Irefn{org141}\\And \nA.~Dubla\\Irefn{org105}\\And \nS.~Dudi\\Irefn{org98}\\And \nM.~Dukhishyam\\Irefn{org84}\\And \nP.~Dupieux\\Irefn{org134}\\And \nR.J.~Ehlers\\Irefn{org146}\\And \nD.~Elia\\Irefn{org52}\\And \nH.~Engel\\Irefn{org73}\\And \nE.~Epple\\Irefn{org146}\\And \nB.~Erazmus\\Irefn{org114}\\And \nF.~Erhardt\\Irefn{org97}\\And \nA.~Erokhin\\Irefn{org112}\\And \nM.R.~Ersdal\\Irefn{org22}\\And \nB.~Espagnon\\Irefn{org61}\\And \nG.~Eulisse\\Irefn{org34}\\And \nJ.~Eum\\Irefn{org18}\\And \nD.~Evans\\Irefn{org109}\\And \nS.~Evdokimov\\Irefn{org89}\\And \nL.~Fabbietti\\Irefn{org117}\\textsuperscript{,}\\Irefn{org103}\\And \nM.~Faggin\\Irefn{org29}\\And \nJ.~Faivre\\Irefn{org77}\\And \nA.~Fantoni\\Irefn{org51}\\And \nM.~Fasel\\Irefn{org94}\\And \nP.~Fecchio\\Irefn{org31}\\And \nA.~Feliciello\\Irefn{org58}\\And \nG.~Feofilov\\Irefn{org112}\\And \nA.~Fern\\'{a}ndez T\\'{e}llez\\Irefn{org44}\\And \nA.~Ferrero\\Irefn{org137}\\And \nA.~Ferretti\\Irefn{org26}\\And \nA.~Festanti\\Irefn{org34}\\And \nV.J.G.~Feuillard\\Irefn{org102}\\And \nJ.~Figiel\\Irefn{org118}\\And \nS.~Filchagin\\Irefn{org107}\\And \nD.~Finogeev\\Irefn{org62}\\And \nF.M.~Fionda\\Irefn{org22}\\And \nG.~Fiorenza\\Irefn{org52}\\And \nF.~Flor\\Irefn{org125}\\And \nS.~Foertsch\\Irefn{org72}\\And \nP.~Foka\\Irefn{org105}\\And \nS.~Fokin\\Irefn{org86}\\And \nE.~Fragiacomo\\Irefn{org59}\\And \nU.~Frankenfeld\\Irefn{org105}\\And \nG.G.~Fronze\\Irefn{org26}\\And \nU.~Fuchs\\Irefn{org34}\\And \nC.~Furget\\Irefn{org77}\\And \nA.~Furs\\Irefn{org62}\\And \nM.~Fusco Girard\\Irefn{org30}\\And \nJ.J.~Gaardh{\\o}je\\Irefn{org87}\\And \nM.~Gagliardi\\Irefn{org26}\\And \nA.M.~Gago\\Irefn{org110}\\And \nA.~Gal\\Irefn{org136}\\And \nC.D.~Galvan\\Irefn{org120}\\And \nP.~Ganoti\\Irefn{org82}\\And \nC.~Garabatos\\Irefn{org105}\\And \nE.~Garcia-Solis\\Irefn{org11}\\And \nK.~Garg\\Irefn{org28}\\And \nC.~Gargiulo\\Irefn{org34}\\And \nA.~Garibli\\Irefn{org85}\\And \nK.~Garner\\Irefn{org144}\\And \nP.~Gasik\\Irefn{org103}\\textsuperscript{,}\\Irefn{org117}\\And \nE.F.~Gauger\\Irefn{org119}\\And \nM.B.~Gay Ducati\\Irefn{org70}\\And \nM.~Germain\\Irefn{org114}\\And \nJ.~Ghosh\\Irefn{org108}\\And \nP.~Ghosh\\Irefn{org141}\\And \nS.K.~Ghosh\\Irefn{org3}\\And \nP.~Gianotti\\Irefn{org51}\\And \nP.~Giubellino\\Irefn{org105}\\textsuperscript{,}\\Irefn{org58}\\And \nP.~Giubilato\\Irefn{org29}\\And \nP.~Gl\\\"{a}ssel\\Irefn{org102}\\And \nD.M.~Gom\\'{e}z Coral\\Irefn{org71}\\And \nA.~Gomez Ramirez\\Irefn{org73}\\And \nV.~Gonzalez\\Irefn{org105}\\And \nP.~Gonz\\'{a}lez-Zamora\\Irefn{org44}\\And \nS.~Gorbunov\\Irefn{org39}\\And \nL.~G\\\"{o}rlich\\Irefn{org118}\\And \nS.~Gotovac\\Irefn{org35}\\And \nV.~Grabski\\Irefn{org71}\\And \nL.K.~Graczykowski\\Irefn{org142}\\And \nK.L.~Graham\\Irefn{org109}\\And \nL.~Greiner\\Irefn{org78}\\And \nA.~Grelli\\Irefn{org63}\\And \nC.~Grigoras\\Irefn{org34}\\And \nV.~Grigoriev\\Irefn{org91}\\And \nA.~Grigoryan\\Irefn{org1}\\And \nS.~Grigoryan\\Irefn{org74}\\And \nO.S.~Groettvik\\Irefn{org22}\\And \nJ.M.~Gronefeld\\Irefn{org105}\\And \nF.~Grosa\\Irefn{org31}\\And \nJ.F.~Grosse-Oetringhaus\\Irefn{org34}\\And \nR.~Grosso\\Irefn{org105}\\And \nR.~Guernane\\Irefn{org77}\\And \nB.~Guerzoni\\Irefn{org27}\\And \nM.~Guittiere\\Irefn{org114}\\And \nK.~Gulbrandsen\\Irefn{org87}\\And \nT.~Gunji\\Irefn{org132}\\And \nA.~Gupta\\Irefn{org99}\\And \nR.~Gupta\\Irefn{org99}\\And \nI.B.~Guzman\\Irefn{org44}\\And \nR.~Haake\\Irefn{org34}\\textsuperscript{,}\\Irefn{org146}\\And \nM.K.~Habib\\Irefn{org105}\\And \nC.~Hadjidakis\\Irefn{org61}\\And \nH.~Hamagaki\\Irefn{org80}\\And \nG.~Hamar\\Irefn{org145}\\And \nM.~Hamid\\Irefn{org6}\\And \nR.~Hannigan\\Irefn{org119}\\And \nM.R.~Haque\\Irefn{org63}\\And \nA.~Harlenderova\\Irefn{org105}\\And \nJ.W.~Harris\\Irefn{org146}\\And \nA.~Harton\\Irefn{org11}\\And \nJ.A.~Hasenbichler\\Irefn{org34}\\And \nH.~Hassan\\Irefn{org77}\\And \nD.~Hatzifotiadou\\Irefn{org10}\\textsuperscript{,}\\Irefn{org53}\\And \nP.~Hauer\\Irefn{org42}\\And \nS.~Hayashi\\Irefn{org132}\\And \nS.T.~Heckel\\Irefn{org68}\\And \nE.~Hellb\\\"{a}r\\Irefn{org68}\\And \nH.~Helstrup\\Irefn{org36}\\And \nA.~Herghelegiu\\Irefn{org47}\\And \nE.G.~Hernandez\\Irefn{org44}\\And \nG.~Herrera Corral\\Irefn{org9}\\And \nF.~Herrmann\\Irefn{org144}\\And \nK.F.~Hetland\\Irefn{org36}\\And \nT.E.~Hilden\\Irefn{org43}\\And \nH.~Hillemanns\\Irefn{org34}\\And \nC.~Hills\\Irefn{org127}\\And \nB.~Hippolyte\\Irefn{org136}\\And \nB.~Hohlweger\\Irefn{org103}\\And \nD.~Horak\\Irefn{org37}\\And \nS.~Hornung\\Irefn{org105}\\And \nR.~Hosokawa\\Irefn{org133}\\And \nP.~Hristov\\Irefn{org34}\\And \nC.~Huang\\Irefn{org61}\\And \nC.~Hughes\\Irefn{org130}\\And \nP.~Huhn\\Irefn{org68}\\And \nT.J.~Humanic\\Irefn{org95}\\And \nH.~Hushnud\\Irefn{org108}\\And \nL.A.~Husova\\Irefn{org144}\\And \nN.~Hussain\\Irefn{org41}\\And \nS.A.~Hussain\\Irefn{org15}\\And \nT.~Hussain\\Irefn{org17}\\And \nD.~Hutter\\Irefn{org39}\\And \nD.S.~Hwang\\Irefn{org19}\\And \nJ.P.~Iddon\\Irefn{org127}\\textsuperscript{,}\\Irefn{org34}\\And \nR.~Ilkaev\\Irefn{org107}\\And \nM.~Inaba\\Irefn{org133}\\And \nM.~Ippolitov\\Irefn{org86}\\And \nM.S.~Islam\\Irefn{org108}\\And \nM.~Ivanov\\Irefn{org105}\\And \nV.~Ivanov\\Irefn{org96}\\And \nV.~Izucheev\\Irefn{org89}\\And \nB.~Jacak\\Irefn{org78}\\And \nN.~Jacazio\\Irefn{org27}\\And \nP.M.~Jacobs\\Irefn{org78}\\And \nM.B.~Jadhav\\Irefn{org48}\\And \nS.~Jadlovska\\Irefn{org116}\\And \nJ.~Jadlovsky\\Irefn{org116}\\And \nS.~Jaelani\\Irefn{org63}\\And \nC.~Jahnke\\Irefn{org121}\\And \nM.J.~Jakubowska\\Irefn{org142}\\And \nM.A.~Janik\\Irefn{org142}\\And \nM.~Jercic\\Irefn{org97}\\And \nO.~Jevons\\Irefn{org109}\\And \nR.T.~Jimenez Bustamante\\Irefn{org105}\\And \nM.~Jin\\Irefn{org125}\\And \nF.~Jonas\\Irefn{org144}\\textsuperscript{,}\\Irefn{org94}\\And \nP.G.~Jones\\Irefn{org109}\\And \nA.~Jusko\\Irefn{org109}\\And \nP.~Kalinak\\Irefn{org64}\\And \nA.~Kalweit\\Irefn{org34}\\And \nJ.H.~Kang\\Irefn{org147}\\And \nV.~Kaplin\\Irefn{org91}\\And \nS.~Kar\\Irefn{org6}\\And \nA.~Karasu Uysal\\Irefn{org76}\\And \nO.~Karavichev\\Irefn{org62}\\And \nT.~Karavicheva\\Irefn{org62}\\And \nP.~Karczmarczyk\\Irefn{org34}\\And \nE.~Karpechev\\Irefn{org62}\\And \nU.~Kebschull\\Irefn{org73}\\And \nR.~Keidel\\Irefn{org46}\\And \nM.~Keil\\Irefn{org34}\\And \nB.~Ketzer\\Irefn{org42}\\And \nZ.~Khabanova\\Irefn{org88}\\And \nA.M.~Khan\\Irefn{org6}\\And \nS.~Khan\\Irefn{org17}\\And \nS.A.~Khan\\Irefn{org141}\\And \nA.~Khanzadeev\\Irefn{org96}\\And \nY.~Kharlov\\Irefn{org89}\\And \nA.~Khatun\\Irefn{org17}\\And \nA.~Khuntia\\Irefn{org118}\\textsuperscript{,}\\Irefn{org49}\\And \nB.~Kileng\\Irefn{org36}\\And \nB.~Kim\\Irefn{org60}\\And \nB.~Kim\\Irefn{org133}\\And \nD.~Kim\\Irefn{org147}\\And \nD.J.~Kim\\Irefn{org126}\\And \nE.J.~Kim\\Irefn{org13}\\And \nH.~Kim\\Irefn{org147}\\And \nJ.~Kim\\Irefn{org147}\\And \nJ.S.~Kim\\Irefn{org40}\\And \nJ.~Kim\\Irefn{org102}\\And \nJ.~Kim\\Irefn{org147}\\And \nJ.~Kim\\Irefn{org13}\\And \nM.~Kim\\Irefn{org102}\\And \nS.~Kim\\Irefn{org19}\\And \nT.~Kim\\Irefn{org147}\\And \nT.~Kim\\Irefn{org147}\\And \nS.~Kirsch\\Irefn{org39}\\And \nI.~Kisel\\Irefn{org39}\\And \nS.~Kiselev\\Irefn{org90}\\And \nA.~Kisiel\\Irefn{org142}\\And \nJ.L.~Klay\\Irefn{org5}\\And \nC.~Klein\\Irefn{org68}\\And \nJ.~Klein\\Irefn{org58}\\And \nS.~Klein\\Irefn{org78}\\And \nC.~Klein-B\\\"{o}sing\\Irefn{org144}\\And \nS.~Klewin\\Irefn{org102}\\And \nA.~Kluge\\Irefn{org34}\\And \nM.L.~Knichel\\Irefn{org34}\\And \nA.G.~Knospe\\Irefn{org125}\\And \nC.~Kobdaj\\Irefn{org115}\\And \nM.K.~K\\\"{o}hler\\Irefn{org102}\\And \nT.~Kollegger\\Irefn{org105}\\And \nA.~Kondratyev\\Irefn{org74}\\And \nN.~Kondratyeva\\Irefn{org91}\\And \nE.~Kondratyuk\\Irefn{org89}\\And \nP.J.~Konopka\\Irefn{org34}\\And \nL.~Koska\\Irefn{org116}\\And \nO.~Kovalenko\\Irefn{org83}\\And \nV.~Kovalenko\\Irefn{org112}\\And \nM.~Kowalski\\Irefn{org118}\\And \nI.~Kr\\'{a}lik\\Irefn{org64}\\And \nA.~Krav\\v{c}\\'{a}kov\\'{a}\\Irefn{org38}\\And \nL.~Kreis\\Irefn{org105}\\And \nM.~Krivda\\Irefn{org109}\\textsuperscript{,}\\Irefn{org64}\\And \nF.~Krizek\\Irefn{org93}\\And \nK.~Krizkova~Gajdosova\\Irefn{org37}\\And \nM.~Kr\\\"uger\\Irefn{org68}\\And \nE.~Kryshen\\Irefn{org96}\\And \nM.~Krzewicki\\Irefn{org39}\\And \nA.M.~Kubera\\Irefn{org95}\\And \nV.~Ku\\v{c}era\\Irefn{org60}\\And \nC.~Kuhn\\Irefn{org136}\\And \nP.G.~Kuijer\\Irefn{org88}\\And \nL.~Kumar\\Irefn{org98}\\And \nS.~Kumar\\Irefn{org48}\\And \nS.~Kundu\\Irefn{org84}\\And \nP.~Kurashvili\\Irefn{org83}\\And \nA.~Kurepin\\Irefn{org62}\\And \nA.B.~Kurepin\\Irefn{org62}\\And \nS.~Kushpil\\Irefn{org93}\\And \nJ.~Kvapil\\Irefn{org109}\\And \nM.J.~Kweon\\Irefn{org60}\\And \nJ.Y.~Kwon\\Irefn{org60}\\And \nY.~Kwon\\Irefn{org147}\\And \nS.L.~La Pointe\\Irefn{org39}\\And \nP.~La Rocca\\Irefn{org28}\\And \nY.S.~Lai\\Irefn{org78}\\And \nR.~Langoy\\Irefn{org129}\\And \nK.~Lapidus\\Irefn{org34}\\textsuperscript{,}\\Irefn{org146}\\And \nA.~Lardeux\\Irefn{org21}\\And \nP.~Larionov\\Irefn{org51}\\And \nE.~Laudi\\Irefn{org34}\\And \nR.~Lavicka\\Irefn{org37}\\And \nT.~Lazareva\\Irefn{org112}\\And \nR.~Lea\\Irefn{org25}\\And \nL.~Leardini\\Irefn{org102}\\And \nS.~Lee\\Irefn{org147}\\And \nF.~Lehas\\Irefn{org88}\\And \nS.~Lehner\\Irefn{org113}\\And \nJ.~Lehrbach\\Irefn{org39}\\And \nR.C.~Lemmon\\Irefn{org92}\\And \nI.~Le\\'{o}n Monz\\'{o}n\\Irefn{org120}\\And \nE.D.~Lesser\\Irefn{org20}\\And \nM.~Lettrich\\Irefn{org34}\\And \nP.~L\\'{e}vai\\Irefn{org145}\\And \nX.~Li\\Irefn{org12}\\And \nX.L.~Li\\Irefn{org6}\\And \nJ.~Lien\\Irefn{org129}\\And \nR.~Lietava\\Irefn{org109}\\And \nB.~Lim\\Irefn{org18}\\And \nS.~Lindal\\Irefn{org21}\\And \nV.~Lindenstruth\\Irefn{org39}\\And \nS.W.~Lindsay\\Irefn{org127}\\And \nC.~Lippmann\\Irefn{org105}\\And \nM.A.~Lisa\\Irefn{org95}\\And \nV.~Litichevskyi\\Irefn{org43}\\And \nA.~Liu\\Irefn{org78}\\And \nS.~Liu\\Irefn{org95}\\And \nW.J.~Llope\\Irefn{org143}\\And \nI.M.~Lofnes\\Irefn{org22}\\And \nV.~Loginov\\Irefn{org91}\\And \nC.~Loizides\\Irefn{org94}\\And \nP.~Loncar\\Irefn{org35}\\And \nX.~Lopez\\Irefn{org134}\\And \nE.~L\\'{o}pez Torres\\Irefn{org8}\\And \nP.~Luettig\\Irefn{org68}\\And \nJ.R.~Luhder\\Irefn{org144}\\And \nM.~Lunardon\\Irefn{org29}\\And \nG.~Luparello\\Irefn{org59}\\And \nM.~Lupi\\Irefn{org73}\\And \nA.~Maevskaya\\Irefn{org62}\\And \nM.~Mager\\Irefn{org34}\\And \nS.M.~Mahmood\\Irefn{org21}\\And \nT.~Mahmoud\\Irefn{org42}\\And \nA.~Maire\\Irefn{org136}\\And \nR.D.~Majka\\Irefn{org146}\\And \nM.~Malaev\\Irefn{org96}\\And \nQ.W.~Malik\\Irefn{org21}\\And \nL.~Malinina\\Irefn{org74}\\Aref{orgII}\\And \nD.~Mal'Kevich\\Irefn{org90}\\And \nP.~Malzacher\\Irefn{org105}\\And \nA.~Mamonov\\Irefn{org107}\\And \nV.~Manko\\Irefn{org86}\\And \nF.~Manso\\Irefn{org134}\\And \nV.~Manzari\\Irefn{org52}\\And \nY.~Mao\\Irefn{org6}\\And \nM.~Marchisone\\Irefn{org135}\\And \nJ.~Mare\\v{s}\\Irefn{org66}\\And \nG.V.~Margagliotti\\Irefn{org25}\\And \nA.~Margotti\\Irefn{org53}\\And \nJ.~Margutti\\Irefn{org63}\\And \nA.~Mar\\'{\\i}n\\Irefn{org105}\\And \nC.~Markert\\Irefn{org119}\\And \nM.~Marquard\\Irefn{org68}\\And \nN.A.~Martin\\Irefn{org102}\\And \nP.~Martinengo\\Irefn{org34}\\And \nJ.L.~Martinez\\Irefn{org125}\\And \nM.I.~Mart\\'{\\i}nez\\Irefn{org44}\\And \nG.~Mart\\'{\\i}nez Garc\\'{\\i}a\\Irefn{org114}\\And \nM.~Martinez Pedreira\\Irefn{org34}\\And \nS.~Masciocchi\\Irefn{org105}\\And \nM.~Masera\\Irefn{org26}\\And \nA.~Masoni\\Irefn{org54}\\And \nL.~Massacrier\\Irefn{org61}\\And \nE.~Masson\\Irefn{org114}\\And \nA.~Mastroserio\\Irefn{org138}\\And \nA.M.~Mathis\\Irefn{org103}\\textsuperscript{,}\\Irefn{org117}\\And \nP.F.T.~Matuoka\\Irefn{org121}\\And \nA.~Matyja\\Irefn{org118}\\And \nC.~Mayer\\Irefn{org118}\\And \nM.~Mazzilli\\Irefn{org33}\\And \nM.A.~Mazzoni\\Irefn{org57}\\And \nA.F.~Mechler\\Irefn{org68}\\And \nF.~Meddi\\Irefn{org23}\\And \nY.~Melikyan\\Irefn{org91}\\And \nA.~Menchaca-Rocha\\Irefn{org71}\\And \nE.~Meninno\\Irefn{org30}\\And \nM.~Meres\\Irefn{org14}\\And \nS.~Mhlanga\\Irefn{org124}\\And \nY.~Miake\\Irefn{org133}\\And \nL.~Micheletti\\Irefn{org26}\\And \nM.M.~Mieskolainen\\Irefn{org43}\\And \nD.L.~Mihaylov\\Irefn{org103}\\And \nK.~Mikhaylov\\Irefn{org90}\\textsuperscript{,}\\Irefn{org74}\\And \nA.~Mischke\\Irefn{org63}\\Aref{org*}\\And \nA.N.~Mishra\\Irefn{org69}\\And \nD.~Mi\\'{s}kowiec\\Irefn{org105}\\And \nC.M.~Mitu\\Irefn{org67}\\And \nA.~Modak\\Irefn{org3}\\And \nN.~Mohammadi\\Irefn{org34}\\And \nA.P.~Mohanty\\Irefn{org63}\\And \nB.~Mohanty\\Irefn{org84}\\And \nM.~Mohisin Khan\\Irefn{org17}\\Aref{orgIII}\\And \nM.~Mondal\\Irefn{org141}\\And \nM.M.~Mondal\\Irefn{org65}\\And \nC.~Mordasini\\Irefn{org103}\\And \nD.A.~Moreira De Godoy\\Irefn{org144}\\And \nL.A.P.~Moreno\\Irefn{org44}\\And \nS.~Moretto\\Irefn{org29}\\And \nA.~Morreale\\Irefn{org114}\\And \nA.~Morsch\\Irefn{org34}\\And \nT.~Mrnjavac\\Irefn{org34}\\And \nV.~Muccifora\\Irefn{org51}\\And \nE.~Mudnic\\Irefn{org35}\\And \nD.~M{\\\"u}hlheim\\Irefn{org144}\\And \nS.~Muhuri\\Irefn{org141}\\And \nJ.D.~Mulligan\\Irefn{org78}\\textsuperscript{,}\\Irefn{org146}\\And \nM.G.~Munhoz\\Irefn{org121}\\And \nK.~M\\\"{u}nning\\Irefn{org42}\\And \nR.H.~Munzer\\Irefn{org68}\\And \nH.~Murakami\\Irefn{org132}\\And \nS.~Murray\\Irefn{org72}\\And \nL.~Musa\\Irefn{org34}\\And \nJ.~Musinsky\\Irefn{org64}\\And \nC.J.~Myers\\Irefn{org125}\\And \nJ.W.~Myrcha\\Irefn{org142}\\And \nB.~Naik\\Irefn{org48}\\And \nR.~Nair\\Irefn{org83}\\And \nB.K.~Nandi\\Irefn{org48}\\And \nR.~Nania\\Irefn{org53}\\textsuperscript{,}\\Irefn{org10}\\And \nE.~Nappi\\Irefn{org52}\\And \nM.U.~Naru\\Irefn{org15}\\And \nA.F.~Nassirpour\\Irefn{org79}\\And \nH.~Natal da Luz\\Irefn{org121}\\And \nC.~Nattrass\\Irefn{org130}\\And \nR.~Nayak\\Irefn{org48}\\And \nT.K.~Nayak\\Irefn{org141}\\textsuperscript{,}\\Irefn{org84}\\And \nS.~Nazarenko\\Irefn{org107}\\And \nR.A.~Negrao De Oliveira\\Irefn{org68}\\And \nL.~Nellen\\Irefn{org69}\\And \nS.V.~Nesbo\\Irefn{org36}\\And \nG.~Neskovic\\Irefn{org39}\\And \nB.S.~Nielsen\\Irefn{org87}\\And \nS.~Nikolaev\\Irefn{org86}\\And \nS.~Nikulin\\Irefn{org86}\\And \nV.~Nikulin\\Irefn{org96}\\And \nF.~Noferini\\Irefn{org10}\\textsuperscript{,}\\Irefn{org53}\\And \nP.~Nomokonov\\Irefn{org74}\\And \nG.~Nooren\\Irefn{org63}\\And \nJ.~Norman\\Irefn{org77}\\And \nP.~Nowakowski\\Irefn{org142}\\And \nA.~Nyanin\\Irefn{org86}\\And \nJ.~Nystrand\\Irefn{org22}\\And \nM.~Ogino\\Irefn{org80}\\And \nA.~Ohlson\\Irefn{org102}\\And \nJ.~Oleniacz\\Irefn{org142}\\And \nA.C.~Oliveira Da Silva\\Irefn{org121}\\And \nM.H.~Oliver\\Irefn{org146}\\And \nC.~Oppedisano\\Irefn{org58}\\And \nR.~Orava\\Irefn{org43}\\And \nA.~Ortiz Velasquez\\Irefn{org69}\\And \nA.~Oskarsson\\Irefn{org79}\\And \nJ.~Otwinowski\\Irefn{org118}\\And \nK.~Oyama\\Irefn{org80}\\And \nY.~Pachmayer\\Irefn{org102}\\And \nV.~Pacik\\Irefn{org87}\\And \nD.~Pagano\\Irefn{org140}\\And \nG.~Pai\\'{c}\\Irefn{org69}\\And \nP.~Palni\\Irefn{org6}\\And \nJ.~Pan\\Irefn{org143}\\And \nA.K.~Pandey\\Irefn{org48}\\And \nS.~Panebianco\\Irefn{org137}\\And \nV.~Papikyan\\Irefn{org1}\\And \nP.~Pareek\\Irefn{org49}\\And \nJ.~Park\\Irefn{org60}\\And \nJ.E.~Parkkila\\Irefn{org126}\\And \nS.~Parmar\\Irefn{org98}\\And \nA.~Passfeld\\Irefn{org144}\\And \nS.P.~Pathak\\Irefn{org125}\\And \nR.N.~Patra\\Irefn{org141}\\And \nB.~Paul\\Irefn{org24}\\textsuperscript{,}\\Irefn{org58}\\And \nH.~Pei\\Irefn{org6}\\And \nT.~Peitzmann\\Irefn{org63}\\And \nX.~Peng\\Irefn{org6}\\And \nL.G.~Pereira\\Irefn{org70}\\And \nH.~Pereira Da Costa\\Irefn{org137}\\And \nD.~Peresunko\\Irefn{org86}\\And \nG.M.~Perez\\Irefn{org8}\\And \nE.~Perez Lezama\\Irefn{org68}\\And \nV.~Peskov\\Irefn{org68}\\And \nY.~Pestov\\Irefn{org4}\\And \nV.~Petr\\'{a}\\v{c}ek\\Irefn{org37}\\And \nM.~Petrovici\\Irefn{org47}\\And \nR.P.~Pezzi\\Irefn{org70}\\And \nS.~Piano\\Irefn{org59}\\And \nM.~Pikna\\Irefn{org14}\\And \nP.~Pillot\\Irefn{org114}\\And \nL.O.D.L.~Pimentel\\Irefn{org87}\\And \nO.~Pinazza\\Irefn{org53}\\textsuperscript{,}\\Irefn{org34}\\And \nL.~Pinsky\\Irefn{org125}\\And \nS.~Pisano\\Irefn{org51}\\And \nD.B.~Piyarathna\\Irefn{org125}\\And \nM.~P\\l osko\\'{n}\\Irefn{org78}\\And \nM.~Planinic\\Irefn{org97}\\And \nF.~Pliquett\\Irefn{org68}\\And \nJ.~Pluta\\Irefn{org142}\\And \nS.~Pochybova\\Irefn{org145}\\And \nM.G.~Poghosyan\\Irefn{org94}\\And \nB.~Polichtchouk\\Irefn{org89}\\And \nN.~Poljak\\Irefn{org97}\\And \nW.~Poonsawat\\Irefn{org115}\\And \nA.~Pop\\Irefn{org47}\\And \nH.~Poppenborg\\Irefn{org144}\\And \nS.~Porteboeuf-Houssais\\Irefn{org134}\\And \nV.~Pozdniakov\\Irefn{org74}\\And \nS.K.~Prasad\\Irefn{org3}\\And \nR.~Preghenella\\Irefn{org53}\\And \nF.~Prino\\Irefn{org58}\\And \nC.A.~Pruneau\\Irefn{org143}\\And \nI.~Pshenichnov\\Irefn{org62}\\And \nM.~Puccio\\Irefn{org34}\\textsuperscript{,}\\Irefn{org26}\\And \nV.~Punin\\Irefn{org107}\\And \nK.~Puranapanda\\Irefn{org141}\\And \nJ.~Putschke\\Irefn{org143}\\And \nR.E.~Quishpe\\Irefn{org125}\\And \nS.~Ragoni\\Irefn{org109}\\And \nS.~Raha\\Irefn{org3}\\And \nS.~Rajput\\Irefn{org99}\\And \nJ.~Rak\\Irefn{org126}\\And \nA.~Rakotozafindrabe\\Irefn{org137}\\And \nL.~Ramello\\Irefn{org32}\\And \nF.~Rami\\Irefn{org136}\\And \nR.~Raniwala\\Irefn{org100}\\And \nS.~Raniwala\\Irefn{org100}\\And \nS.S.~R\\\"{a}s\\\"{a}nen\\Irefn{org43}\\And \nB.T.~Rascanu\\Irefn{org68}\\And \nR.~Rath\\Irefn{org49}\\And \nV.~Ratza\\Irefn{org42}\\And \nI.~Ravasenga\\Irefn{org31}\\And \nK.F.~Read\\Irefn{org130}\\textsuperscript{,}\\Irefn{org94}\\And \nK.~Redlich\\Irefn{org83}\\Aref{orgIV}\\And \nA.~Rehman\\Irefn{org22}\\And \nP.~Reichelt\\Irefn{org68}\\And \nF.~Reidt\\Irefn{org34}\\And \nX.~Ren\\Irefn{org6}\\And \nR.~Renfordt\\Irefn{org68}\\And \nA.~Reshetin\\Irefn{org62}\\And \nJ.-P.~Revol\\Irefn{org10}\\And \nK.~Reygers\\Irefn{org102}\\And \nV.~Riabov\\Irefn{org96}\\And \nT.~Richert\\Irefn{org79}\\textsuperscript{,}\\Irefn{org87}\\And \nM.~Richter\\Irefn{org21}\\And \nP.~Riedler\\Irefn{org34}\\And \nW.~Riegler\\Irefn{org34}\\And \nF.~Riggi\\Irefn{org28}\\And \nC.~Ristea\\Irefn{org67}\\And \nS.P.~Rode\\Irefn{org49}\\And \nM.~Rodr\\'{i}guez Cahuantzi\\Irefn{org44}\\And \nK.~R{\\o}ed\\Irefn{org21}\\And \nR.~Rogalev\\Irefn{org89}\\And \nE.~Rogochaya\\Irefn{org74}\\And \nD.~Rohr\\Irefn{org34}\\And \nD.~R\\\"ohrich\\Irefn{org22}\\And \nP.S.~Rokita\\Irefn{org142}\\And \nF.~Ronchetti\\Irefn{org51}\\And \nE.D.~Rosas\\Irefn{org69}\\And \nK.~Roslon\\Irefn{org142}\\And \nP.~Rosnet\\Irefn{org134}\\And \nA.~Rossi\\Irefn{org29}\\And \nA.~Rotondi\\Irefn{org139}\\And \nF.~Roukoutakis\\Irefn{org82}\\And \nA.~Roy\\Irefn{org49}\\And \nP.~Roy\\Irefn{org108}\\And \nO.V.~Rueda\\Irefn{org79}\\And \nR.~Rui\\Irefn{org25}\\And \nB.~Rumyantsev\\Irefn{org74}\\And \nA.~Rustamov\\Irefn{org85}\\And \nE.~Ryabinkin\\Irefn{org86}\\And \nY.~Ryabov\\Irefn{org96}\\And \nA.~Rybicki\\Irefn{org118}\\And \nH.~Rytkonen\\Irefn{org126}\\And \nS.~Sadhu\\Irefn{org141}\\And \nS.~Sadovsky\\Irefn{org89}\\And \nK.~\\v{S}afa\\v{r}\\'{\\i}k\\Irefn{org37}\\textsuperscript{,}\\Irefn{org34}\\And \nS.K.~Saha\\Irefn{org141}\\And \nB.~Sahoo\\Irefn{org48}\\And \nP.~Sahoo\\Irefn{org49}\\And \nR.~Sahoo\\Irefn{org49}\\And \nS.~Sahoo\\Irefn{org65}\\And \nP.K.~Sahu\\Irefn{org65}\\And \nJ.~Saini\\Irefn{org141}\\And \nS.~Sakai\\Irefn{org133}\\And \nS.~Sambyal\\Irefn{org99}\\And \nV.~Samsonov\\Irefn{org91}\\textsuperscript{,}\\Irefn{org96}\\And \nA.~Sandoval\\Irefn{org71}\\And \nA.~Sarkar\\Irefn{org72}\\And \nD.~Sarkar\\Irefn{org143}\\And \nN.~Sarkar\\Irefn{org141}\\And \nP.~Sarma\\Irefn{org41}\\And \nV.M.~Sarti\\Irefn{org103}\\And \nM.H.P.~Sas\\Irefn{org63}\\And \nE.~Scapparone\\Irefn{org53}\\And \nB.~Schaefer\\Irefn{org94}\\And \nJ.~Schambach\\Irefn{org119}\\And \nH.S.~Scheid\\Irefn{org68}\\And \nC.~Schiaua\\Irefn{org47}\\And \nR.~Schicker\\Irefn{org102}\\And \nA.~Schmah\\Irefn{org102}\\And \nC.~Schmidt\\Irefn{org105}\\And \nH.R.~Schmidt\\Irefn{org101}\\And \nM.O.~Schmidt\\Irefn{org102}\\And \nM.~Schmidt\\Irefn{org101}\\And \nN.V.~Schmidt\\Irefn{org94}\\textsuperscript{,}\\Irefn{org68}\\And \nA.R.~Schmier\\Irefn{org130}\\And \nJ.~Schukraft\\Irefn{org34}\\textsuperscript{,}\\Irefn{org87}\\And \nY.~Schutz\\Irefn{org34}\\textsuperscript{,}\\Irefn{org136}\\And \nK.~Schwarz\\Irefn{org105}\\And \nK.~Schweda\\Irefn{org105}\\And \nG.~Scioli\\Irefn{org27}\\And \nE.~Scomparin\\Irefn{org58}\\And \nM.~\\v{S}ef\\v{c}\\'ik\\Irefn{org38}\\And \nJ.E.~Seger\\Irefn{org16}\\And \nY.~Sekiguchi\\Irefn{org132}\\And \nD.~Sekihata\\Irefn{org132}\\textsuperscript{,}\\Irefn{org45}\\And \nI.~Selyuzhenkov\\Irefn{org105}\\textsuperscript{,}\\Irefn{org91}\\And \nS.~Senyukov\\Irefn{org136}\\And \nD.~Serebryakov\\Irefn{org62}\\And \nE.~Serradilla\\Irefn{org71}\\And \nP.~Sett\\Irefn{org48}\\And \nA.~Sevcenco\\Irefn{org67}\\And \nA.~Shabanov\\Irefn{org62}\\And \nA.~Shabetai\\Irefn{org114}\\And \nR.~Shahoyan\\Irefn{org34}\\And \nW.~Shaikh\\Irefn{org108}\\And \nA.~Shangaraev\\Irefn{org89}\\And \nA.~Sharma\\Irefn{org98}\\And \nA.~Sharma\\Irefn{org99}\\And \nM.~Sharma\\Irefn{org99}\\And \nN.~Sharma\\Irefn{org98}\\And \nA.I.~Sheikh\\Irefn{org141}\\And \nK.~Shigaki\\Irefn{org45}\\And \nM.~Shimomura\\Irefn{org81}\\And \nS.~Shirinkin\\Irefn{org90}\\And \nQ.~Shou\\Irefn{org111}\\And \nY.~Sibiriak\\Irefn{org86}\\And \nS.~Siddhanta\\Irefn{org54}\\And \nT.~Siemiarczuk\\Irefn{org83}\\And \nD.~Silvermyr\\Irefn{org79}\\And \nC.~Silvestre\\Irefn{org77}\\And \nG.~Simatovic\\Irefn{org88}\\And \nG.~Simonetti\\Irefn{org103}\\textsuperscript{,}\\Irefn{org34}\\And \nR.~Singh\\Irefn{org84}\\And \nR.~Singh\\Irefn{org99}\\And \nV.K.~Singh\\Irefn{org141}\\And \nV.~Singhal\\Irefn{org141}\\And \nT.~Sinha\\Irefn{org108}\\And \nB.~Sitar\\Irefn{org14}\\And \nM.~Sitta\\Irefn{org32}\\And \nT.B.~Skaali\\Irefn{org21}\\And \nM.~Slupecki\\Irefn{org126}\\And \nN.~Smirnov\\Irefn{org146}\\And \nR.J.M.~Snellings\\Irefn{org63}\\And \nT.W.~Snellman\\Irefn{org126}\\And \nJ.~Sochan\\Irefn{org116}\\And \nC.~Soncco\\Irefn{org110}\\And \nJ.~Song\\Irefn{org60}\\textsuperscript{,}\\Irefn{org125}\\And \nA.~Songmoolnak\\Irefn{org115}\\And \nF.~Soramel\\Irefn{org29}\\And \nS.~Sorensen\\Irefn{org130}\\And \nI.~Sputowska\\Irefn{org118}\\And \nJ.~Stachel\\Irefn{org102}\\And \nI.~Stan\\Irefn{org67}\\And \nP.~Stankus\\Irefn{org94}\\And \nP.J.~Steffanic\\Irefn{org130}\\And \nE.~Stenlund\\Irefn{org79}\\And \nD.~Stocco\\Irefn{org114}\\And \nM.M.~Storetvedt\\Irefn{org36}\\And \nP.~Strmen\\Irefn{org14}\\And \nA.A.P.~Suaide\\Irefn{org121}\\And \nT.~Sugitate\\Irefn{org45}\\And \nC.~Suire\\Irefn{org61}\\And \nM.~Suleymanov\\Irefn{org15}\\And \nM.~Suljic\\Irefn{org34}\\And \nR.~Sultanov\\Irefn{org90}\\And \nM.~\\v{S}umbera\\Irefn{org93}\\And \nS.~Sumowidagdo\\Irefn{org50}\\And \nK.~Suzuki\\Irefn{org113}\\And \nS.~Swain\\Irefn{org65}\\And \nA.~Szabo\\Irefn{org14}\\And \nI.~Szarka\\Irefn{org14}\\And \nU.~Tabassam\\Irefn{org15}\\And \nG.~Taillepied\\Irefn{org134}\\And \nJ.~Takahashi\\Irefn{org122}\\And \nG.J.~Tambave\\Irefn{org22}\\And \nS.~Tang\\Irefn{org134}\\textsuperscript{,}\\Irefn{org6}\\And \nM.~Tarhini\\Irefn{org114}\\And \nM.G.~Tarzila\\Irefn{org47}\\And \nA.~Tauro\\Irefn{org34}\\And \nG.~Tejeda Mu\\~{n}oz\\Irefn{org44}\\And \nA.~Telesca\\Irefn{org34}\\And \nC.~Terrevoli\\Irefn{org125}\\textsuperscript{,}\\Irefn{org29}\\And \nD.~Thakur\\Irefn{org49}\\And \nS.~Thakur\\Irefn{org141}\\And \nD.~Thomas\\Irefn{org119}\\And \nF.~Thoresen\\Irefn{org87}\\And \nR.~Tieulent\\Irefn{org135}\\And \nA.~Tikhonov\\Irefn{org62}\\And \nA.R.~Timmins\\Irefn{org125}\\And \nA.~Toia\\Irefn{org68}\\And \nN.~Topilskaya\\Irefn{org62}\\And \nM.~Toppi\\Irefn{org51}\\And \nF.~Torales-Acosta\\Irefn{org20}\\And \nS.R.~Torres\\Irefn{org120}\\And \nS.~Tripathy\\Irefn{org49}\\And \nT.~Tripathy\\Irefn{org48}\\And \nS.~Trogolo\\Irefn{org26}\\textsuperscript{,}\\Irefn{org29}\\And \nG.~Trombetta\\Irefn{org33}\\And \nL.~Tropp\\Irefn{org38}\\And \nV.~Trubnikov\\Irefn{org2}\\And \nW.H.~Trzaska\\Irefn{org126}\\And \nT.P.~Trzcinski\\Irefn{org142}\\And \nB.A.~Trzeciak\\Irefn{org63}\\And \nT.~Tsuji\\Irefn{org132}\\And \nA.~Tumkin\\Irefn{org107}\\And \nR.~Turrisi\\Irefn{org56}\\And \nT.S.~Tveter\\Irefn{org21}\\And \nK.~Ullaland\\Irefn{org22}\\And \nE.N.~Umaka\\Irefn{org125}\\And \nA.~Uras\\Irefn{org135}\\And \nG.L.~Usai\\Irefn{org24}\\And \nA.~Utrobicic\\Irefn{org97}\\And \nM.~Vala\\Irefn{org116}\\textsuperscript{,}\\Irefn{org38}\\And \nN.~Valle\\Irefn{org139}\\And \nS.~Vallero\\Irefn{org58}\\And \nN.~van der Kolk\\Irefn{org63}\\And \nL.V.R.~van Doremalen\\Irefn{org63}\\And \nM.~van Leeuwen\\Irefn{org63}\\And \nP.~Vande Vyvre\\Irefn{org34}\\And \nD.~Varga\\Irefn{org145}\\And \nZ.~Varga\\Irefn{org145}\\And \nM.~Varga-Kofarago\\Irefn{org145}\\And \nA.~Vargas\\Irefn{org44}\\And \nM.~Vargyas\\Irefn{org126}\\And \nR.~Varma\\Irefn{org48}\\And \nM.~Vasileiou\\Irefn{org82}\\And \nA.~Vasiliev\\Irefn{org86}\\And \nO.~V\\'azquez Doce\\Irefn{org117}\\textsuperscript{,}\\Irefn{org103}\\And \nV.~Vechernin\\Irefn{org112}\\And \nA.M.~Veen\\Irefn{org63}\\And \nE.~Vercellin\\Irefn{org26}\\And \nS.~Vergara Lim\\'on\\Irefn{org44}\\And \nL.~Vermunt\\Irefn{org63}\\And \nR.~Vernet\\Irefn{org7}\\And \nR.~V\\'ertesi\\Irefn{org145}\\And \nM.G.D.L.C.~Vicencio\\Irefn{org9}\\And \nL.~Vickovic\\Irefn{org35}\\And \nJ.~Viinikainen\\Irefn{org126}\\And \nZ.~Vilakazi\\Irefn{org131}\\And \nO.~Villalobos Baillie\\Irefn{org109}\\And \nA.~Villatoro Tello\\Irefn{org44}\\And \nG.~Vino\\Irefn{org52}\\And \nA.~Vinogradov\\Irefn{org86}\\And \nT.~Virgili\\Irefn{org30}\\And \nV.~Vislavicius\\Irefn{org87}\\And \nA.~Vodopyanov\\Irefn{org74}\\And \nB.~Volkel\\Irefn{org34}\\And \nM.A.~V\\\"{o}lkl\\Irefn{org101}\\And \nK.~Voloshin\\Irefn{org90}\\And \nS.A.~Voloshin\\Irefn{org143}\\And \nG.~Volpe\\Irefn{org33}\\And \nB.~von Haller\\Irefn{org34}\\And \nI.~Vorobyev\\Irefn{org103}\\And \nD.~Voscek\\Irefn{org116}\\And \nJ.~Vrl\\'{a}kov\\'{a}\\Irefn{org38}\\And \nB.~Wagner\\Irefn{org22}\\And \nY.~Watanabe\\Irefn{org133}\\And \nM.~Weber\\Irefn{org113}\\And \nS.G.~Weber\\Irefn{org144}\\textsuperscript{,}\\Irefn{org105}\\And \nA.~Wegrzynek\\Irefn{org34}\\And \nD.F.~Weiser\\Irefn{org102}\\And \nS.C.~Wenzel\\Irefn{org34}\\And \nJ.P.~Wessels\\Irefn{org144}\\And \nE.~Widmann\\Irefn{org113}\\And \nJ.~Wiechula\\Irefn{org68}\\And \nJ.~Wikne\\Irefn{org21}\\And \nG.~Wilk\\Irefn{org83}\\And \nJ.~Wilkinson\\Irefn{org53}\\And \nG.A.~Willems\\Irefn{org34}\\And \nE.~Willsher\\Irefn{org109}\\And \nB.~Windelband\\Irefn{org102}\\And \nW.E.~Witt\\Irefn{org130}\\And \nY.~Wu\\Irefn{org128}\\And \nR.~Xu\\Irefn{org6}\\And \nS.~Yalcin\\Irefn{org76}\\And \nK.~Yamakawa\\Irefn{org45}\\And \nS.~Yang\\Irefn{org22}\\And \nS.~Yano\\Irefn{org137}\\And \nZ.~Yasin\\Aref{orgV}\\And \nZ.~Yin\\Irefn{org6}\\And \nH.~Yokoyama\\Irefn{org63}\\And \nI.-K.~Yoo\\Irefn{org18}\\And \nJ.H.~Yoon\\Irefn{org60}\\And \nS.~Yuan\\Irefn{org22}\\And \nA.~Yuncu\\Irefn{org102}\\And \nV.~Yurchenko\\Irefn{org2}\\And \nV.~Zaccolo\\Irefn{org58}\\textsuperscript{,}\\Irefn{org25}\\And \nA.~Zaman\\Irefn{org15}\\And \nC.~Zampolli\\Irefn{org34}\\And \nH.J.C.~Zanoli\\Irefn{org121}\\And \nN.~Zardoshti\\Irefn{org34}\\And \nA.~Zarochentsev\\Irefn{org112}\\And \nP.~Z\\'{a}vada\\Irefn{org66}\\And \nN.~Zaviyalov\\Irefn{org107}\\And \nH.~Zbroszczyk\\Irefn{org142}\\And \nM.~Zhalov\\Irefn{org96}\\And \nX.~Zhang\\Irefn{org6}\\And \nZ.~Zhang\\Irefn{org6}\\textsuperscript{,}\\Irefn{org134}\\And \nC.~Zhao\\Irefn{org21}\\And \nV.~Zherebchevskii\\Irefn{org112}\\And \nN.~Zhigareva\\Irefn{org90}\\And \nD.~Zhou\\Irefn{org6}\\And \nY.~Zhou\\Irefn{org87}\\And \nZ.~Zhou\\Irefn{org22}\\And \nJ.~Zhu\\Irefn{org6}\\And \nY.~Zhu\\Irefn{org6}\\And \nA.~Zichichi\\Irefn{org27}\\textsuperscript{,}\\Irefn{org10}\\And \nM.B.~Zimmermann\\Irefn{org34}\\And \nG.~Zinovjev\\Irefn{org2}\\And \nN.~Zurlo\\Irefn{org140}\\And\n\\renewcommand\\labelenumi{\\textsuperscript{\\theenumi}~}\n\n\\section*{Affiliation notes}\n\\renewcommand\\theenumi{\\roman{enumi}}\n\\begin{Authlist}\n\\item \\Adef{org*}Deceased\n\\item \\Adef{orgI}Dipartimento DET del Politecnico di Torino, Turin, Italy\n\\item \\Adef{orgII}M.V. Lomonosov Moscow State University, D.V. Skobeltsyn Institute of Nuclear, Physics, Moscow, Russia\n\\item \\Adef{orgIII}Department of Applied Physics, Aligarh Muslim University, Aligarh, India\n\\item \\Adef{orgIV}Institute of Theoretical Physics, University of Wroclaw, Poland\n\\item \\Adef{orgV}PINSTECH, Islamabad, Pakistan\n\\end{Authlist}\n\n\\section*{Collaboration Institutes}\n\\renewcommand\\theenumi{\\arabic{enumi}~}\n\\begin{Authlist}\n\\item \\Idef{org1}A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute) Foundation, Yerevan, Armenia\n\\item \\Idef{org2}Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, Kiev, Ukraine\n\\item \\Idef{org3}Bose Institute, Department of Physics and Centre for Astroparticle Physics and Space Science (CAPSS), Kolkata, India\n\\item \\Idef{org4}Budker Institute for Nuclear Physics, Novosibirsk, Russia\n\\item \\Idef{org5}California Polytechnic State University, San Luis Obispo, California, United States\n\\item \\Idef{org6}Central China Normal University, Wuhan, China\n\\item \\Idef{org7}Centre de Calcul de l'IN2P3, Villeurbanne, Lyon, France\n\\item \\Idef{org8}Centro de Aplicaciones Tecnol\\'{o}gicas y Desarrollo Nuclear (CEADEN), Havana, Cuba\n\\item \\Idef{org9}Centro de Investigaci\\'{o}n y de Estudios Avanzados (CINVESTAV), Mexico City and M\\'{e}rida, Mexico\n\\item \\Idef{org10}Centro Fermi - Museo Storico della Fisica e Centro Studi e Ricerche ``Enrico Fermi', Rome, Italy\n\\item \\Idef{org11}Chicago State University, Chicago, Illinois, United States\n\\item \\Idef{org12}China Institute of Atomic Energy, Beijing, China\n\\item \\Idef{org13}Chonbuk National University, Jeonju, Republic of Korea\n\\item \\Idef{org14}Comenius University Bratislava, Faculty of Mathematics, Physics and Informatics, Bratislava, Slovakia\n\\item \\Idef{org15}COMSATS University Islamabad, Islamabad, Pakistan\n\\item \\Idef{org16}Creighton University, Omaha, Nebraska, United States\n\\item \\Idef{org17}Department of Physics, Aligarh Muslim University, Aligarh, India\n\\item \\Idef{org18}Department of Physics, Pusan National University, Pusan, Republic of Korea\n\\item \\Idef{org19}Department of Physics, Sejong University, Seoul, Republic of Korea\n\\item \\Idef{org20}Department of Physics, University of California, Berkeley, California, United States\n\\item \\Idef{org21}Department of Physics, University of Oslo, Oslo, Norway\n\\item \\Idef{org22}Department of Physics and Technology, University of Bergen, Bergen, Norway\n\\item \\Idef{org23}Dipartimento di Fisica dell'Universit\\`{a} 'La Sapienza' and Sezione INFN, Rome, Italy\n\\item \\Idef{org24}Dipartimento di Fisica dell'Universit\\`{a} and Sezione INFN, Cagliari, Italy\n\\item \\Idef{org25}Dipartimento di Fisica dell'Universit\\`{a} and Sezione INFN, Trieste, Italy\n\\item \\Idef{org26}Dipartimento di Fisica dell'Universit\\`{a} and Sezione INFN, Turin, Italy\n\\item \\Idef{org27}Dipartimento di Fisica e Astronomia dell'Universit\\`{a} and Sezione INFN, Bologna, Italy\n\\item \\Idef{org28}Dipartimento di Fisica e Astronomia dell'Universit\\`{a} and Sezione INFN, Catania, Italy\n\\item \\Idef{org29}Dipartimento di Fisica e Astronomia dell'Universit\\`{a} and Sezione INFN, Padova, Italy\n\\item \\Idef{org30}Dipartimento di Fisica `E.R.~Caianiello' dell'Universit\\`{a} and Gruppo Collegato INFN, Salerno, Italy\n\\item \\Idef{org31}Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy\n\\item \\Idef{org32}Dipartimento di Scienze e Innovazione Tecnologica dell'Universit\\`{a} del Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy\n\\item \\Idef{org33}Dipartimento Interateneo di Fisica `M.~Merlin' and Sezione INFN, Bari, Italy\n\\item \\Idef{org34}European Organization for Nuclear Research (CERN), Geneva, Switzerland\n\\item \\Idef{org35}Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Split, Croatia\n\\item \\Idef{org36}Faculty of Engineering and Science, Western Norway University of Applied Sciences, Bergen, Norway\n\\item \\Idef{org37}Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Prague, Czech Republic\n\\item \\Idef{org38}Faculty of Science, P.J.~\\v{S}af\\'{a}rik University, Ko\\v{s}ice, Slovakia\n\\item \\Idef{org39}Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-Universit\\\"{a}t Frankfurt, Frankfurt, Germany\n\\item \\Idef{org40}Gangneung-Wonju National University, Gangneung, Republic of Korea\n\\item \\Idef{org41}Gauhati University, Department of Physics, Guwahati, India\n\\item \\Idef{org42}Helmholtz-Institut f\\\"{u}r Strahlen- und Kernphysik, Rheinische Friedrich-Wilhelms-Universit\\\"{a}t Bonn, Bonn, Germany\n\\item \\Idef{org43}Helsinki Institute of Physics (HIP), Helsinki, Finland\n\\item \\Idef{org44}High Energy Physics Group, Universidad Aut\\'{o}noma de Puebla, Puebla, Mexico\n\\item \\Idef{org45}Hiroshima University, Hiroshima, Japan\n\\item \\Idef{org46}Hochschule Worms, Zentrum f\\\"{u}r Technologietransfer und Telekommunikation (ZTT), Worms, Germany\n\\item \\Idef{org47}Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest, Romania\n\\item \\Idef{org48}Indian Institute of Technology Bombay (IIT), Mumbai, India\n\\item \\Idef{org49}Indian Institute of Technology Indore, Indore, India\n\\item \\Idef{org50}Indonesian Institute of Sciences, Jakarta, Indonesia\n\\item \\Idef{org51}INFN, Laboratori Nazionali di Frascati, Frascati, Italy\n\\item \\Idef{org52}INFN, Sezione di Bari, Bari, Italy\n\\item \\Idef{org53}INFN, Sezione di Bologna, Bologna, Italy\n\\item \\Idef{org54}INFN, Sezione di Cagliari, Cagliari, Italy\n\\item \\Idef{org55}INFN, Sezione di Catania, Catania, Italy\n\\item \\Idef{org56}INFN, Sezione di Padova, Padova, Italy\n\\item \\Idef{org57}INFN, Sezione di Roma, Rome, Italy\n\\item \\Idef{org58}INFN, Sezione di Torino, Turin, Italy\n\\item \\Idef{org59}INFN, Sezione di Trieste, Trieste, Italy\n\\item \\Idef{org60}Inha University, Incheon, Republic of Korea\n\\item \\Idef{org61}Institut de Physique Nucl\\'{e}aire d'Orsay (IPNO), Institut National de Physique Nucl\\'{e}aire et de Physique des Particules (IN2P3\/CNRS), Universit\\'{e} de Paris-Sud, Universit\\'{e} Paris-Saclay, Orsay, France\n\\item \\Idef{org62}Institute for Nuclear Research, Academy of Sciences, Moscow, Russia\n\\item \\Idef{org63}Institute for Subatomic Physics, Utrecht University\/Nikhef, Utrecht, Netherlands\n\\item \\Idef{org64}Institute of Experimental Physics, Slovak Academy of Sciences, Ko\\v{s}ice, Slovakia\n\\item \\Idef{org65}Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India\n\\item \\Idef{org66}Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic\n\\item \\Idef{org67}Institute of Space Science (ISS), Bucharest, Romania\n\\item \\Idef{org68}Institut f\\\"{u}r Kernphysik, Johann Wolfgang Goethe-Universit\\\"{a}t Frankfurt, Frankfurt, Germany\n\\item \\Idef{org69}Instituto de Ciencias Nucleares, Universidad Nacional Aut\\'{o}noma de M\\'{e}xico, Mexico City, Mexico\n\\item \\Idef{org70}Instituto de F\\'{i}sica, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil\n\\item \\Idef{org71}Instituto de F\\'{\\i}sica, Universidad Nacional Aut\\'{o}noma de M\\'{e}xico, Mexico City, Mexico\n\\item \\Idef{org72}iThemba LABS, National Research Foundation, Somerset West, South Africa\n\\item \\Idef{org73}Johann-Wolfgang-Goethe Universit\\\"{a}t Frankfurt Institut f\\\"{u}r Informatik, Fachbereich Informatik und Mathematik, Frankfurt, Germany\n\\item \\Idef{org74}Joint Institute for Nuclear Research (JINR), Dubna, Russia\n\\item \\Idef{org75}Korea Institute of Science and Technology Information, Daejeon, Republic of Korea\n\\item \\Idef{org76}KTO Karatay University, Konya, Turkey\n\\item \\Idef{org77}Laboratoire de Physique Subatomique et de Cosmologie, Universit\\'{e} Grenoble-Alpes, CNRS-IN2P3, Grenoble, France\n\\item \\Idef{org78}Lawrence Berkeley National Laboratory, Berkeley, California, United States\n\\item \\Idef{org79}Lund University Department of Physics, Division of Particle Physics, Lund, Sweden\n\\item \\Idef{org80}Nagasaki Institute of Applied Science, Nagasaki, Japan\n\\item \\Idef{org81}Nara Women{'}s University (NWU), Nara, Japan\n\\item \\Idef{org82}National and Kapodistrian University of Athens, School of Science, Department of Physics , Athens, Greece\n\\item \\Idef{org83}National Centre for Nuclear Research, Warsaw, Poland\n\\item \\Idef{org84}National Institute of Science Education and Research, Homi Bhabha National Institute, Jatni, India\n\\item \\Idef{org85}National Nuclear Research Center, Baku, Azerbaijan\n\\item \\Idef{org86}National Research Centre Kurchatov Institute, Moscow, Russia\n\\item \\Idef{org87}Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark\n\\item \\Idef{org88}Nikhef, National institute for subatomic physics, Amsterdam, Netherlands\n\\item \\Idef{org89}NRC Kurchatov Institute IHEP, Protvino, Russia\n\\item \\Idef{org90}NRC Kurchatov Institute - ITEP, Moscow, Russia\n\\item \\Idef{org91}NRNU Moscow Engineering Physics Institute, Moscow, Russia\n\\item \\Idef{org92}Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom\n\\item \\Idef{org93}Nuclear Physics Institute of the Czech Academy of Sciences, \\v{R}e\\v{z} u Prahy, Czech Republic\n\\item \\Idef{org94}Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States\n\\item \\Idef{org95}Ohio State University, Columbus, Ohio, United States\n\\item \\Idef{org96}Petersburg Nuclear Physics Institute, Gatchina, Russia\n\\item \\Idef{org97}Physics department, Faculty of science, University of Zagreb, Zagreb, Croatia\n\\item \\Idef{org98}Physics Department, Panjab University, Chandigarh, India\n\\item \\Idef{org99}Physics Department, University of Jammu, Jammu, India\n\\item \\Idef{org100}Physics Department, University of Rajasthan, Jaipur, India\n\\item \\Idef{org101}Physikalisches Institut, Eberhard-Karls-Universit\\\"{a}t T\\\"{u}bingen, T\\\"{u}bingen, Germany\n\\item \\Idef{org102}Physikalisches Institut, Ruprecht-Karls-Universit\\\"{a}t Heidelberg, Heidelberg, Germany\n\\item \\Idef{org103}Physik Department, Technische Universit\\\"{a}t M\\\"{u}nchen, Munich, Germany\n\\item \\Idef{org104}Politecnico di Bari, Bari, Italy\n\\item \\Idef{org105}Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f\\\"ur Schwerionenforschung GmbH, Darmstadt, Germany\n\\item \\Idef{org106}Rudjer Bo\\v{s}kovi\\'{c} Institute, Zagreb, Croatia\n\\item \\Idef{org107}Russian Federal Nuclear Center (VNIIEF), Sarov, Russia\n\\item \\Idef{org108}Saha Institute of Nuclear Physics, Homi Bhabha National Institute, Kolkata, India\n\\item \\Idef{org109}School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom\n\\item \\Idef{org110}Secci\\'{o}n F\\'{\\i}sica, Departamento de Ciencias, Pontificia Universidad Cat\\'{o}lica del Per\\'{u}, Lima, Peru\n\\item \\Idef{org111}Shanghai Institute of Applied Physics, Shanghai, China\n\\item \\Idef{org112}St. Petersburg State University, St. Petersburg, Russia\n\\item \\Idef{org113}Stefan Meyer Institut f\\\"{u}r Subatomare Physik (SMI), Vienna, Austria\n\\item \\Idef{org114}SUBATECH, IMT Atlantique, Universit\\'{e} de Nantes, CNRS-IN2P3, Nantes, France\n\\item \\Idef{org115}Suranaree University of Technology, Nakhon Ratchasima, Thailand\n\\item \\Idef{org116}Technical University of Ko\\v{s}ice, Ko\\v{s}ice, Slovakia\n\\item \\Idef{org117}Technische Universit\\\"{a}t M\\\"{u}nchen, Excellence Cluster 'Universe', Munich, Germany\n\\item \\Idef{org118}The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of Sciences, Cracow, Poland\n\\item \\Idef{org119}The University of Texas at Austin, Austin, Texas, United States\n\\item \\Idef{org120}Universidad Aut\\'{o}noma de Sinaloa, Culiac\\'{a}n, Mexico\n\\item \\Idef{org121}Universidade de S\\~{a}o Paulo (USP), S\\~{a}o Paulo, Brazil\n\\item \\Idef{org122}Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil\n\\item \\Idef{org123}Universidade Federal do ABC, Santo Andre, Brazil\n\\item \\Idef{org124}University of Cape Town, Cape Town, South Africa\n\\item \\Idef{org125}University of Houston, Houston, Texas, United States\n\\item \\Idef{org126}University of Jyv\\\"{a}skyl\\\"{a}, Jyv\\\"{a}skyl\\\"{a}, Finland\n\\item \\Idef{org127}University of Liverpool, Liverpool, United Kingdom\n\\item \\Idef{org128}University of Science and Techonology of China, Hefei, China\n\\item \\Idef{org129}University of South-Eastern Norway, Tonsberg, Norway\n\\item \\Idef{org130}University of Tennessee, Knoxville, Tennessee, United States\n\\item \\Idef{org131}University of the Witwatersrand, Johannesburg, South Africa\n\\item \\Idef{org132}University of Tokyo, Tokyo, Japan\n\\item \\Idef{org133}University of Tsukuba, Tsukuba, Japan\n\\item \\Idef{org134}Universit\\'{e} Clermont Auvergne, CNRS\/IN2P3, LPC, Clermont-Ferrand, France\n\\item \\Idef{org135}Universit\\'{e} de Lyon, Universit\\'{e} Lyon 1, CNRS\/IN2P3, IPN-Lyon, Villeurbanne, Lyon, France\n\\item \\Idef{org136}Universit\\'{e} de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France, Strasbourg, France\n\\item \\Idef{org137}Universit\\'{e} Paris-Saclay Centre d'Etudes de Saclay (CEA), IRFU, D\\'{e}partment de Physique Nucl\\'{e}aire (DPhN), Saclay, France\n\\item \\Idef{org138}Universit\\`{a} degli Studi di Foggia, Foggia, Italy\n\\item \\Idef{org139}Universit\\`{a} degli Studi di Pavia, Pavia, Italy\n\\item \\Idef{org140}Universit\\`{a} di Brescia, Brescia, Italy\n\\item \\Idef{org141}Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata, India\n\\item \\Idef{org142}Warsaw University of Technology, Warsaw, Poland\n\\item \\Idef{org143}Wayne State University, Detroit, Michigan, United States\n\\item \\Idef{org144}Westf\\\"{a}lische Wilhelms-Universit\\\"{a}t M\\\"{u}nster, Institut f\\\"{u}r Kernphysik, M\\\"{u}nster, Germany\n\\item \\Idef{org145}Wigner Research Centre for Physics, Hungarian Academy of Sciences, Budapest, Hungary\n\\item \\Idef{org146}Yale University, New Haven, Connecticut, United States\n\\item \\Idef{org147}Yonsei University, Seoul, Republic of Korea\n\\end{Authlist}\n\\endgroup\n\n\\section*{Acknowledgements}\n\\end{acknowledgement}\n\n\n\\newpage\n\n\n\n\\section{Introduction}\nThe energy densities reached in the collisions of ultra-relativistic particles lead to a significant production of complex \\mbox{(anti-)}\\mbox{(hyper-)}nuclei. The high yield of anti-quarks produced in these reactions has led to the first observation of the anti-alpha particle~\\cite{Agakishiev:2011ib} as well as of the anti-hyper-triton~\\cite{Abelev:2010} by the STAR collaboration, and to detailed measurements by the ALICE collaboration \\cite{nuclei,Adam:2015yta,Acharya:2017dmc,Acharya:2019rgc} at energies reached at the CERN LHC. However, the production mechanism is not fully understood. In a more general context, these measurements also provide input for the background determination in searches for anti-nuclei in space. Such an observation of anti-deuterons or $^{3}\\overline{\\rm He}$\\ of cosmic origin could carry information on the existence of large amounts of anti-matter in our universe or provide a signature of the annihilation of dark matter particles~\\cite{Blum:2017qnn,Poulin:2018wzu,Tomassetti:2017qjk,Korsmeier:2017xzj,Cui:2010ud}.\n\nRecent data in pp and in heavy-ion collisions provide evidence for an interesting observation regarding the production mechanism of \\mbox{(anti-)}nuclei~\\cite{nuclei,Acharya:2017fvb,Acharya:2017bso,Acharya:2017dmc,Acharya:2019rgc}: in \\PbPb{} interactions, the d\/p ratio does not vary with the collision centrality and the value agrees with expectations from thermal-statistical models which feature a common chemical freeze-out temperature of all hadrons around 156 MeV~\\cite{nuclei,Andronic:2010qu,Cleymans:2011pe}.\nIn inelastic pp collisions, the corresponding ratio is a factor 2.2 lower than in Pb--Pb collisions \\cite{nuclei, Acharya:2017fvb}. With respect to these measurements, the results of d and $^3$He produced in \\pPb{} collisions at \\snn{} = 5.02~TeV, being a system in between the two extremes of pp and Pb--Pb collisions, are of prominent interest and they are the subject of this letter. \nWhile deuterons have been measured differentially in multiplicity, the $^{3}\\overline{\\rm He}${} ($^{3}{\\mathrm{He}}$){} spectrum was only obtained inclusively for all non-single diffractive events because of their low production rate.\n\nIn addition to the evolution of the integrated d\/p ratio for various multiplicity classes, the question whether the transverse momentum distribution of deuterons is consistent with a collective radial expansion together with the non-composite hadrons is of particular interest. Such behaviour has been observed for light nuclei in \\PbPb{} collisions~\\cite{nuclei,Acharya:2017dmc}. The presence of collective effects in \\pPb\\ collisions at LHC energies has recently been supported by several experimental findings (see for instance~\\cite{Abelev:2012ola,ABELEV:2013wsa,Abelev:2013haa,CMS:2012qk,Aad:2012gla,Aad:2013fja,Chatrchyan:2013nka} and recent reviews in~\\cite{Loizides:2016tew,Nagle:2018nvi}). \nThese include a clear mass ordering of the mean transverse momenta of light flavoured hadrons in p--Pb collisions as expected from hydrodynamical models~\\cite{Abelev:2013haa}.\n\n\n\n\n\\section{Analysis}\n\nThe results presented here are based on a low pile-up p--Pb data sample collected with the ALICE detector during the LHC running campaign at \\snn~=~5.02~TeV in 2013. A detailed description of the detector is available in~\\cite{Alessandro:2006yt,Carminati:2004fp,Aamodt:2008zz,Aamodt:2010dx,Abelev:2014ffa}. The main detectors used in this analysis are the Inner Tracking System (ITS)~\\cite{Aamodt:2010aa}, the Time Projection Chamber (TPC)~\\cite{Alme:2010ke}, and the Time-Of-Flight detector (TOF)~\\cite{Akindinov:2010zzb,Akindinov:2013tea}. The two innermost layers of the ITS consist of Silicon Pixel Detectors (SPD), followed by two layers of Silicon Drift Detectors (SDD), and two layers of Silicon Strip Detectors (SSD). As the main tracking device, the TPC provides full azimuthal acceptance for tracks in the pseudo-rapidity region $|\\hlab| <$ 0.8. In addition, it provides particle identification via the measurement of the specific energy loss d$E$\/d$x$.\nThe TOF array is located at about 3.7 m from the beam line and provides particle identification by measuring the particle speed with the time-of-flight technique.\nIn p-Pb collisions, the overall time resolution is about 85~ps for high multiplicity events.\nIn peripheral events, where multiplicities are similar to pp, it decreases to about 120~ps due to a worse start-time (collision-time) resolution~\\cite{Adam:2016ilk}. All detectors are positioned in a solenoidal magnetic field of $B$ = 0.5~T.\n\nThe event sample used for the analysis presented in this letter \nwas collected exclusively in the beam configuration where the proton travels towards negative \\hlab. The minimum-bias trigger signal and the definition of the multiplicity classes was provided by the V0 detector consisting of two arrays of 32 scintillator tiles each covering the full azimuth within $2.8 < \\hlab < 5.1$ (V0A, Pb-beam direction) and $-3.7 < \\hlab < -1.7$ (V0C, p-beam direction). The event selection was performed in a similar way to that described in Ref.~\\cite{Abelev:2013haa}. A coincidence of signals in both V0A and V0C was required online in order to remove background from single diffractive and electromagnetic events. In the offline analysis, further background suppression was achieved by requiring that the arrival time of the signals in the two neutron Zero Degree Calorimeters (ZDC), which are located $\\pm$112.5~m from the interaction point, is compatible with a nominal \\pPb{} collision. \nThe contamination from pile-up events was reduced to a negligible level ($<1\\%$) by rejecting events in which more than one primary vertex was reconstructed either from SPD tracklets or from tracks reconstructed in the whole central barrel.\nThe position of the reconstructed primary vertex was required to be located within $\\pm 10$~cm of the nominal interaction point in the longitudinal direction. In total, an event sample of about 100 million minimum-bias (MB) events after all selections was analysed. The corresponding integrated luminosity, $L_{\\mathrm{int}} = N_\\mathrm{MB}\/\\sigma_\\mathrm{MB}$, where $\\sigma_\\mathrm{MB}$ is the MB trigger cross-section measured with van-der-Meer scans, amounts to 47.8~$\\mu\\mathrm{b}^{-1}$ with a relative uncertainty of 3.7\\%~\\cite{Abelev:2014epa}.\n\nThe final results are given normalised to the total number of non-single diffractive (NSD) events. Therefore, a correction of 3.6\\%$\\pm$3.1\\% \\cite{ALICE:2012xs} is applied to the minimum-bias results, which corresponds to the trigger and vertex reconstruction inefficiency for this selection. For the study of d and $\\overline{\\rm d}${}, the sample is divided into five multiplicity classes, which are defined as percentiles of the V0A signal. This signal is proportional to the charged-particle multiplicity in the corresponding pseudo-rapidity region in the direction of the Pb-beam. \nFollowing the approach in~\\cite{Adam:2016dau}, the multiplicity dependent results are normalized to the number of events $N_{\\rm ev}$ corresponding to the visible (triggered) cross-section. The event sample is corrected for the vertex reconstruction efficiency. This correction is of the order of 4\\% for the lowest V0A multiplicity class (60-100\\%) and negligible ($<$1\\%) for the other multiplicity classes. \nThe chosen selection and the corresponding charged-particle multiplicity at mid-rapidity are summarized in Table~\\ref{tab:mult-bins}.\n\n\\begin{table}[hb]\n \\centering\n \\begin{tabular}[hb]{rc}\n \\hline\n \\hline\n V0A Class & $\\left. \\langle {\\rm d}N_{\\rm ch}\/{\\rm d}\\eta_{\\rm lab} \\rangle \\right|_{\\left|\\eta_{\\rm lab}\\right| < 0.5}$ \\\\\n \\hline \n 0--10\\% & 40.6 $\\pm$ 0.9 \\\\\n 10--20\\% & 30.5 $\\pm$ 0.7 \\\\\n 20--40\\% & 23.2 $\\pm$ 0.5 \\\\\n 40--60\\% & 16.1 $\\pm$ 0.4 \\\\\n 60--100\\% & 7.1 $\\pm$ 0.2 \\\\\n \\hline\n \\hline\n \\end{tabular}\n \\caption{Multiplicity intervals and the corresponding charged-particle multiplicities at mid-rapidity. The uncertainties reported for the $\\langle {\\rm d}N_{\\rm ch}\/{\\rm d}\\eta_{\\rm lab} \\rangle |_{\\left|\\eta_{\\rm lab}\\right| < 0.5}$ are the systematic ones, statistical uncertainties are negligible. Values are taken from \\cite{Abelev:2013haa}.}\n \\label{tab:mult-bins}\n\\end{table}\n\nIn this analysis, the production of primary deuterons and $^{3}{\\mathrm{He}}${}-nuclei and that of their respective anti-particles are measured in a rapidity window $-1 < y < 0$ in the centre-of-mass system. Since the energy per nucleon of the proton beam is higher than that of the Pb beam, the nucleon-nucleon system moves in the laboratory frame with a rapidity of -0.465.\n Potential differences of the spectral shape or normalisation due to the larger $y$-range with respect to the measurement of $\\pi$, K, and p \\cite{Abelev:2013haa} are found to be negligible for the (anti-)deuteron and $^{3}{\\mathrm{He}}$\\ minimum-bias spectra with respect to the overall statistical and systematic uncertainties.\nIn order to select primary tracks of suitable quality, various track selection criteria are applied. At least 70 clusters in the TPC and two hits in the ITS (out of which at least one in the SPD) are required. These selections guarantee a track momentum resolution of 2\\% in the relevant \\PT-range and a d$E$\/d$x$ resolution of about~6\\% for minimum ionising particles. The maximum allowed Distance-of-Closest-Approach (DCA) to the primary collision vertex is \n0.12~cm in the transverse (\\dcaxy) and 1.0~cm in the longitudinal (\\dcaz) plane. Furthermore, it is required that the $\\chi^2$ per TPC cluster is less than 4 and tracks of weak-decay products with kink topology are rejected \\cite{Abelev:2014ffa}, as they cannot originate from the tracks of primary nuclei.\n\nThe particle identification performance of the TPC and TOF detectors in \\pPb{} collisions is shown in Fig.~\\ref{fig:tpctofperf}. For the mass determination with the TOF detector, the contribution of tracks with a wrongly assigned TOF cluster is largely reduced by a 3$\\sigma$ pre-selection in the TPC d$E$\/d$x$, where $\\sigma$ corresponds to the TPC d$E$\/d$x$ resolution. Nevertheless, due to the small abundance of deuterons the background is still significant and it is removed using a fit to the squared mass distribution. An example of a fit for anti-deuterons with transverse momenta $2.2~\\textrm{GeV}\/c < \\PT{} < 2.4~\\textrm{GeV}\/c$ is shown in the right panel of Fig.~\\ref{fig:tpctofperf}. The squared rest mass of the deuteron has been subtracted to simplify the fitting function. The signal has a Gaussian shape with an exponential tail on the right side. This tail is necessary to describe the time-signal shape of the TOF detector~\\cite{Akindinov:2013tea}. For the background, the sum of two exponential functions is used. One of the exponential functions accounts for the mismatched tracks and the other accounts for the tail of the proton peak. For \\mbox{(anti-)}$^{3}{\\mathrm{He}}${} nuclei, the \\dedx{} is sufficient for a clean identification using only this technique over the entire momentum range $1.5~\\gevc<$~\\PT{} $< 5~\\gevc$ as the atomic number $Z=2$ for $^3$He leads to a clear separation from other particles.\n\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true, width = 0.49\\textwidth]{img\/FinalVersion\/dEdxPerformancePlot.png} \n \\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true, width = 0.49\\textwidth]{img\/FinalVersion\/m2TofPerf.eps} \n \\caption{Energy loss \\dedx{} in the TPC and the corresponding expected energy loss from a parametrization of the Bethe-Bloch curve (left). Example of the fit to the squared TOF mass difference which shows separately the signal and the background from the exponential tail of protons and from mismatched tracks (right).}\n \\label{fig:tpctofperf}\n \\end{center}\n\\end{figure} \n\nThe tracking acceptance $\\times$ efficiency determination is based on a Monte-Carlo simulation using the DPMJET event generator~\\cite{Roesler:2000he} and a full detector description in GEANT3~\\cite{Geant:1994zzo}. As discussed in~\\cite{nuclei}, the hadronic interaction of \\mbox{(anti-)}nuclei with detector material is not fully described in GEANT3, therefore two additional correction factors are applied. Firstly, in order to account for the material between the collision vertex and the TPC, the track reconstruction efficiencies extracted from GEANT3 are scaled to match those from GEANT4~\\cite{Agostinelli:2002hh,Uzhinsky:2011zz}. Secondly, for tracks which cross in addition the material between the TPC and the TOF detectors, a data-driven correction factor has been evaluated by comparing the matching efficiency of tracks to TOF hits in data and Monte Carlo simulation. Since the TRD was not fully installed in 2013, this study was repeated for regions in azimuth with and without installed TRD modules. The matching efficiencies for tracks crossing the TRD material were then scaled such that the corrected yield agrees with the one obtained for tracks that are not crossing any TRD material. This procedure results in a further reduction of the acceptance $\\times$ efficiency of 6\\% for deuterons and 11\\% for anti-deuterons. \nThe acceptance and efficiency corrections are found to be independent of the event multiplicity and are shown in Fig.~\\ref{fig:eff} for primary deuterons and anti-deuterons, with and without requiring a TOF match, as well as for $^{3}{\\mathrm{He}}${} and $^{3}\\overline{\\rm He}${}. \n\n\n\\begin{figure}[ht]\n \\begin{center}\n \\includegraphics[width = 0.49\\textwidth]{figures\/EfficiencyPlot_deuterons.pdf} \n \\includegraphics[width = 0.49\\textwidth]{figures\/EfficiencyPlot_He3.pdf}\n \\caption{Tracking acceptance $\\times$ efficiency correction for (anti-)deuterons (left) and for $^3$He and $^{3}\\overline{\\rm He}$ (right) in the minimum-bias class. \n The efficiencies for anti-nuclei are lower due to the larger cross-section for hadronic interactions.}\n \\label{fig:eff}\n \\end{center}\n\\end{figure} \n\n\nThe raw yields of deuterons and $^{3}{\\mathrm{He}}${} also include secondary particles which stem \nfrom the interactions of primary particles with the detector material.\nTo subtract this contribution, a data-driven approach as in~\\cite{Abelev:2013haa,nuclei} is used. The distribution of the \\dcaxy{} is fitted with two distributions (called \"templates\" in the following) obtained from Monte-Carlo simulations describing primary and secondary deuterons, respectively. The fit is performed in the range $|\\dcaxy| < 0.5$~cm which allows the contribution from material to be constrained by the plateau of the distribution at larger distances ($|\\dcaxy| > 0.15$~cm). The contamination of secondaries amounts to about 45\\% to 55\\% in the lowest \\PT-interval and decreases exponentially towards higher \\PT{} until it becomes negligible ($<$ 1\\%) above 2~\\gevc{}. The limited number \nof $^{3}{\\mathrm{He}}${} candidate tracks does not allow a background subtraction based on templates, instead a bin counting procedure in the aforementioned \\dcaxy{} signal and background regions is used.\n\nThe systematic uncertainties of the measurement are summarised for deuterons and $^{3}{\\mathrm{He}}${} as well as for their antiparticles in Table~\\ref{tab:systematics}. For deuterons, the uncertainty related to the secondary correction is estimated by repeating the template fit procedure under a variation of the \\dcaz{} cut. The corresponding uncertainty for $^{3}{\\mathrm{He}}${} nuclei is determined by varying the ranges in \\dcaxy{} for the signal and background regions in the bin counting procedure.\nFor d and $^{3}{\\mathrm{He}}${} the systematic uncertainty on the cross-section for hadronic interaction is determined by a systematic comparison of different propagation codes (GEANT3 and GEANT4).\nThe material between TPC and TOF needs to be considered only for the (anti-)deuteron spectrum and increases the uncertainty by additional 3\\% and 5\\% for deuterons and anti-deuterons, respectively. This corresponds to the half of the observed discrepancy in the TPC-TOF matching efficiencies evaluated in data and Monte Carlo.\nFor both deuterons and anti-deuterons, the particle identification procedure introduces only a small uncertainty which slightly increases at high \\PT{} and is estimated based on the variation of the $n\\sigma$-cuts in the TPC d$E$\/d$x$ as well as on a variation of the signal extraction in the TOF with different fit functions. The PID related uncertainties for $^{3}{\\mathrm{He}}${} and $^{3}\\overline{\\rm He}${} remain negligible over the entire \\PT-range due to the background-free identification based on the TPC d$E$\/d$x$. \nFeed-down from weakly decaying hyper-tritons ($^{3}_{\\Lambda}{\\mathrm H}$) is negligible for deuterons~\\cite{Adam:2015yta,nuclei}. Since only about 4-8\\% of all $^{3}_{\\Lambda}{\\mathrm H}$ decaying into $^{3}{\\mathrm{He}}${} pass the track selection criteria for primary $^{3}{\\mathrm{He}}${}, the remaining contamination has not been subtracted and the uncertainty related to it was further investigated by a variation of the \\dcaxy{}-cut in data and a final uncertainty of 5\\% is assigned.\nThe influence of uncertainties in the material budget on the reconstruction efficiency has been studied by simulating events varying the amount of material by $\\pm $10\\%. The estimates of the uncertainties related to the tracking and ITS-TPC matching are based on a variation of the track cuts and are found to be approximately 5\\%.\nThe uncertainties related to tracking, transport code, material budget and TPC-TOF matching are fully correlated across different multiplicity intervals.\n\n\n\\begin{table}[hb]\n\\caption{Main sources of systematic uncertainties for deuterons and $^{3}{\\mathrm{He}}${} as well as their anti-particles for low and high \\PT{}.}\n \\centering\n \\begin{tabular}[hb]{l|cccc|cccc}\n \\hline\n \\hline \n\t&&&\\\\[-1.0em]\n & \\multicolumn{2}{c}{d} &\\multicolumn{2}{c}{$\\overline{\\rm d}$} & \\multicolumn{2}{c}{$^{3}{\\mathrm{He}}$} &\\multicolumn{2}{c}{$^{3}\\overline{\\rm He}$} \\\\\n \\hline\n \\PT{} (\\gevc) & 0.9 & 2.9 & 0.9 & 2.9 & 2.2 & 5.0 & 1.8 & 5.0 \\\\\n \\hline\n Tracking (ITS-TPC matching) & 5\\% & 5\\% & 5\\% & 5\\% & 6\\% & 4\\% & 6\\% & 4\\% \\\\\n Secondaries material & 1\\% & negl. & negl. & negl. & 20\\% & 1\\% & negl. & negl. \\\\\n Secondaries weak decay & negl. & negl. & negl. & negl. & 5\\% & negl. & 5\\% & negl. \\\\\n Material budget & 3\\% & 3\\% & 3\\% & 3\\% & 3\\% & 1\\% & 3\\% & 1\\% \\\\\n Particle identification & 1\\% & 3\\% & 1\\% & 3\\% & 3\\% & 3\\% & 3\\% & 3\\% \\\\\n Transport code & 3\\% & 3\\% & 3\\% & 3\\% & 6\\% & 6\\% & 18\\% & 11\\%\\\\\n TPC-TOF matching & 3\\% & 3\\% & 5\\% & 5\\% & - & - & - & -\\\\\n \\hline\n Total\t\t\t & 7\\% & 8\\% & 8\\% & 9\\% & 23\\% & 8\\% & 20\\% & 12\\% \\\\\n \\hline\n \\hline\n \\end{tabular}\n \\label{tab:systematics}\n\\end{table}\n\n\n\\section{Results and Discussion}\n\n\n\\subsection{Spectra and yields}\n\n\\begin{figure}[htb]\n\t\\begin{center}\n\t \\includegraphics[width = 0.49\\textwidth]{figures\/spectraTOFM.pdf} \n\t \\includegraphics[width = 0.49\\textwidth]{figures\/spectraTOFA.pdf}\n\t \\caption{Transverse momentum distributions of deuterons (left) and anti-deuterons (right) for various multiplicity classes. The multiplicity class definition is based on the signal amplitude observed in the V0A detector located on the Pb-side. The vertical bars represent the statistical errors,\n\t the empty boxes show the systematic uncertainty. The lines represent individual fits using a $m_{\\rm T}$-exponential function. }\n\t \\label{fig:spectra}\n\t\\end{center}\n\\end{figure} \n\n\nThe transverse momentum spectra of deuterons and anti-deuterons in the rapidity range $-1~<~y~<~0$ are presented in Fig.~\\ref{fig:spectra} for several multiplicity classes. The spectra show a hardening with increasing event multiplicity. This behaviour was already observed for lower mass particles in \\pPb{} collisions \\cite{Abelev:2013haa}. For the extraction of \\avpT{} and \\PT-integrated yields \\dndy{}, the spectra are fitted individually using a $m_{\\rm T}$-exponential function \\cite{Hagedorn:1965st}. \n\n\\begin{figure}[htbp]\n\t\\begin{center}\n\t\t\\includegraphics[width = 0.55\\textwidth]{figures\/PlusMinusRatiosDeuteron.pdf} \n\t\t\\caption{Anti-deuteron to deuteron production ratio for the five multiplicity classes. All ratios are compatible with unity, indicated as a dashed grey line. \n\t\tThe vertical bars represent the statistical errors while the empty boxes show the total systematic uncertainty.\n\t\t}\n\t\t\\label{fig:ratios}\n\t\\end{center}\n\\end{figure} \n\n\nThe values obtained for \\dndy{} for (anti-)deuterons are summarized in Table~\\ref{tab:results}. They have been calculated by summing up the \\PT-differential yield in the region where the spectrum is measured and by integrating the fit result in the unmeasured region at low and high transverse momenta. \nWhile the fraction of the extrapolated yield at high \\PT{} is negligible, the fraction at low \\PT{} ranges from 23\\% at high to 38\\% at low multiplicities. \nThe uncertainty introduced by this extrapolation is estimated by comparing the result obtained with the $m_{\\rm T}$-exponential fit to fit results from several alternative functional forms (Boltzmann, Blast-wave~\\cite{Schnedermann:1993ws}, and \\PT-exponential).\n\n\n\\begin{table}[htbp]\n\t\\caption[Integrated]{Integrated yields \\dndy{} of (anti-)deuterons. The first value is the statistical and the second is the total systematic uncertainty which includes both the systematic uncertainty on the measured spectra and the uncertainty of the extrapolation to low and high \\PT.} \n\n\t\\centering\n\t{\\small\n\t\\begin{tabular}{ccc}\n\t\n\t\t\\hline\n\t\t\\hline\n\t\t&&\\\\[-0.7em]\n\tMultiplicity classes & \\dndy~(d) & \\dndy~($\\overline{\\rm d}$) \\\\\n\t\t\\hline\n\t\t&&\\\\[-0.7em]\n0-10\\% & $(2.86\\pm0.03\\pm0.30)\\times10^{-3} $ & $(2.83\\pm0.03\\pm0.35)\\times10^{-3}$ \\\\\n10-20\\% & $(2.08\\pm0.02\\pm0.22)\\times10^{-3} $ & $(1.94\\pm0.03\\pm0.24)\\times10^{-3}$ \\\\\n20-40\\% & $(1.43\\pm0.01\\pm0.15)\\times10^{-3} $ & $(1.43\\pm0.02\\pm0.17)\\times10^{-3}$ \\\\\n40-60\\% & $(8.93\\pm0.08\\pm0.93)\\times10^{-4} $ & $(9.06\\pm0.15\\pm1.09)\\times10^{-4}$ \\\\\n60-100\\% & $(2.89\\pm0.05\\pm0.30)\\times10^{-4} $ & $(3.02\\pm0.07\\pm0.36)\\times10^{-4}$ \\\\\t&&\\\\[-0.7em]\n\t\\hline\n\t\\hline\n \\end{tabular}\n}\n \\label{tab:results}\n\\end{table}\n\n\n\nFigure~\\ref{fig:ratios} shows the $\\mathrm{\\overline{d}}\/\\rm d$ ratios as a function of \\PT{} for all multiplicity intervals. The ratios are found to be consistent with unity within uncertainties. This behaviour is expected, since thermal and coalescence models predict that the $\\mathrm{\\overline{d}}\/\\rm d$ ratio is given by $(\\pbarp)^2$ (see for instance~\\cite{Cleymans:2011pe}) and the $\\pbarp$ ratio measured in \\pPb{} collisions is consistent with unity for all multiplicity intervals~\\cite{Abelev:2013haa}.\n\n\n\nThe rare production of $A>2$ nuclei only allows the extraction of minimum-bias spectra for $^{3}{\\mathrm{He}}${} and $^{3}\\overline{\\rm He}${} with the available statistics and thus the result is normalised to all non-single diffractive (NSD) events. In total, 40 $^{3}\\overline{\\rm He}${} nuclei are observed, while about 29400 tracks from $\\overline{\\rm d}${} are reconstructed in the same data sample. The corresponding spectra are shown in Fig.~\\ref{fig:spectrum-3} together with a $m_{\\rm T}$-exponential fit which is used for the extraction of the \\dndy{} and \\avpT{} of the spectra. The fit is performed such that the residuals to both the $^{3}{\\mathrm{He}}${} and $^{3}\\overline{\\rm He}${} spectrum are minimised simultaneously. The fraction of the extrapolated yield corresponds to about 58\\%. The uncertainty introduced by this extrapolation is also estimated by comparing the result obtained with the $m_{\\rm T}$-exponential fit to fit results from several alternative functional forms (Boltzmann, Blast-wave~\\cite{Schnedermann:1993ws}, and \\PT-exponential). A \\PT-integrated yield of \\dndy~=~$(1.36 \\pm 0.16 (\\rm stat) \\pm 0.52 (\\rm syst))\\times 10^{-6}$ and an average transverse momentum of \\avpT{}~=~$(1.78 \\pm 0.11 (\\rm stat) \\pm 0.77 (\\rm syst))$~\\gevc{} are obtained.\n\n\n\\begin{figure}[htbp]\n\t\\begin{center}\n\n\t\t\\includegraphics[width = 0.8\\textwidth]{figures\/spectra_he3.pdf}\n\t\t\\caption{Transverse momentum distribution of $^{3}{\\mathrm{He}}${} and $^{3}\\overline{\\rm He}$ for all NSD collisions ($N_{\\mathrm{NSD}}$). \n\t\t\t\tThe vertical bars represent the statistical errors while the empty boxes show the total systematic uncertainty.\n\t\tThe line represents a $\\chi^2$ fit with a $m_{\\rm T}$-exponential function (see text for details).}\n\t\t\\label{fig:spectrum-3}\n\t\\end{center}\n\\end{figure}\n\nThe yields of p, d and $^{3}{\\mathrm{He}}${} for NSD p--Pb events and normalised to their spin degeneracy are shown in Fig.~\\ref{fig:yield_mass} as a function of the mass number $A$ together with results for inelastic pp collisions and central Pb-Pb collisions. An exponential decrease with increasing $A$ is observed in all cases, yet with different slopes. \nThe penalty factor, i.e. the reduction of the yield for each additional nucleon, is obtained from a fit to the data and a value of \n$635 \\pm 90$ in p-Pb collisions is found which is significantly \nlarger than the factor of $359 \\pm 41$ \nwhich was observed for central \\PbPb{} collisions~\\cite{nuclei}. \nThe penalty factor obtained for the inelastic pp collisions \\cite{Acharya:2017fvb} is found to be $942 \\pm 107$.\nSuch an exponential decrease of the \\mbox{(anti-)}nuclei yield with mass number has also been observed at lower incident energies in heavy-ion \\cite{Barrette:1994tw, BraunMunzinger:1994iq, Arsenescu:2003eg, Agakishiev:2011ib} as well as in p--A collisions \\cite{Antipov:1970uc}.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true, width = 0.7\\textwidth]{figures\/PenaltyFactor.pdf} \t\n\t\t\\caption{Production yield $\\ensuremath{\\mathrm{d}N\/\\mathrm{d}y}$ normalised by the spin degeneracy as a function of the mass number for inelastic pp collisions, minimum-bias p-Pb and central Pb-Pb collisions \\cite{Acharya:2017bso,Acharya:2017fvb,Abelev:2013vea,Abelev:2013haa,Adam:2015qaa}. The empty boxes represent the total systematic uncertainty while the statistical errors are shown by the vertical bars. The lines represent fits with an exponential function.}\n\t\t\\label{fig:yield_mass}\n\t\\end{center}\n\\end{figure} \n\n\n\n\\subsection{Coalescence parameter}\n\nIn the traditional coalescence model, deuterons and other light nuclei are formed by protons and neutrons, which are close in phase space. In this picture, the deuteron momentum spectra are related to those of its constituent nucleons via~\\cite{Butler:1963pp,Scheibl:1998tk} \n\n\\begin{equation}\nE_{\\rm d} \\, \\frac{{\\rm d^3} N_{\\rm d}}{{\\rm d} p_{\\rm d}^3} = B_2 \\, \\left(\nE_{\\rm p} \\, \\frac{{\\rm d^3} N_{\\rm p}}{{\\rm d} p_{\\rm p}^3}\n\\right)^2,\n\\label{coal}\n\\end{equation}\n\n\nwhere the momentum of the deuteron is given by $p_{\\mathrm{d}} = 2 p_{\\mathrm{p}}$. Since the neutron spectra are experimentally not accessible, they are approximated by the proton spectra. The value of \\Btwo{} is computed as a function of event multiplicity and transverse momentum as the ratio between the deuteron yield measured at $p_{\\mathrm{T}}=p_{\\mathrm{T,d}}$ and the square of the proton yield at $p_{\\mathrm{T,p}} = 0.5p_{\\mathrm{T,d}}$. The obtained \\Btwo{}-values are shown in Fig.~\\ref{fig:b2}. In its simplest implementation, the coalescence model for uncorrelated particle emission from a point-like source predicts that the observed \\Btwo{}-values are independent of \\PT{} and of event multiplicity (called \"simple coalescence\" in the following). \nWithin uncertainties and given the current width of the multiplicity classes, the observed \\PT{} dependence is still compatible with the expected flat behaviour (for a detailed discussion see \\cite{Acharya:2019rgc}).\nMoreover, a decrease of the measured \\Btwo{} parameter with increasing event multiplicity for a fixed \\PT{} is observed. This effect is even more pronounced in \\PbPb{} collisions \\cite{nuclei} and a possible explanation is an increasing source volume, which can effectively reduce the coalescence probability~\\cite{Scheibl:1998tk,Blum:2017qnn}.\n\n\\begin{figure}[htbp]\t\n \\begin{center}\n \\includegraphics[width = 0.6\\textwidth]{figures\/B2plot.pdf} \n \\caption{Coalescence parameter $B_2$ as a function of \\PT{} for different V0A multiplicity classes. \n \t\tThe vertical lines represent the statistical errors and the empty boxes show the total systematic uncertainty.\n\t}\n \\label{fig:b2}\n \\end{center}\n\\end{figure} \n\n\\subsection{Mean transverse momenta}\n\nIn Fig.~\\ref{fig:meanpt} (left), the mean values of the transverse momenta of deuterons are compared with the corresponding results for \n$\\pi^{\\pm}$, K$^{\\pm}$, p($\\rm \\bar{p}$), and $\\Lambda$($\\overline{\\Lambda}$)~\\cite{Abelev:2013haa}. \nAs for all other particles, the \\avpT{} of deuterons shows an increase with increasing event multiplicity, which reflects the observed hardening of the spectra. However, it is striking that deuterons violate the mass ordering which was observed for non-composite particles~\\cite{Abelev:2013haa,Adam:2016bpr}: despite their much larger mass, the \\avpT{} values are similar to those of $\\Lambda$($\\overline{\\Lambda}$) and only slightly higher than those of p($\\rm \\bar{p}$).\n\nNote that simple coalescence models give a significantly different prediction for the \\avpT{} of deuterons with respect to hydrodynamical models.\nThis can be best illustrated with two simplifying requirements which are approximately fulfilled in data. Firstly, the coalescence parameter is assumed flat in \\PT{} and secondly the proton spectrum can be described by an exponential shape, i.e. $C \\exp(-\\PT\/T)$ with two parameters $C$ and $T$. In this case, the shape of the deuteron spectrum can be analytically calculated based on the definition of \\Btwo{}. Due to the self-similarity feature of the exponential function, $ \\left(\\exp(x\/a)\\right)^{a} = \\exp(x) $, the spectral shape of the proton and the deuteron are then found to be identical:\n\n\n\\begin{equation}\n \\frac{1}{2\\pi p_{\\rm T}^{d}} \\frac{{\\rm d}^{2}N^{d}}{{\\rm d}y \\,{\\rm d}p_{\\rm T}^{d}} = B_{2} \n\\Bigl( \\frac{1}{ 2\\pi p_{\\rm T}^{p}} \\frac{{\\rm d}^{2}N^{p} }{ {\\rm d}y \\,{\\rm d}p_{\\rm T}^{p}} \\Bigr)^{2} =\nB_{2} \\Bigl( C \\exp(-\\frac{p_{\\rm T}^{p} }{ T}) \\Bigr)^{2} = B_{2} \\Bigl( C \\exp(-\\frac{p_{\\rm T}^{d} }{ 2T}) \\Bigr)^{2} = B_{2}~C^{2}\\exp(-\\frac{p_{\\rm T}^{d} }{ T})\n\\;.\n\\label{eq.:B2expression}\n\\end{equation}\n\n\n\nThus, the same \\avpT{} for both particles is expected and the behaviour observed in p--Pb collisions is well described by simple coalescence models. This finding can be even further substantiated by directly calculating the \\avpT{} of deuterons assuming a constant value of \\Btwo{} and using the measured proton spectrum as input. As shown in Fig.~\\ref{fig:meanpt} (right), in this case, a good agreement with the data is found considering that a large fraction of the systematic uncertainty is correlated among different multiplicity bins. The Blast-Wave model~\\cite{Schnedermann:1993ws} fails to describe the \\avpT{} values for deuterons using the common kinetic freeze-out parameters from~\\cite{Abelev:2013haa}, which describe simultaneously the spectra of pions, kaons, and protons.\n\n\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true, width = 1.0\\textwidth]{figures\/meanpt.pdf}\n \\caption{Mean \\PT{} of various particle species as a function of the mean charged-particle density at mid-rapidity for different V0A multiplicity classes. The empty boxes show the total systematic uncertainty while the shaded boxes indicate the contribution which is uncorrelated across multiplicity intervals (left). \n Comparison of \\avpT{} of protons and deuterons with the simple coalescence and the Blast-Wave model expectations. The shaded areas show the expected \\avpT{} for deuterons from a simple coalescence model assuming a \\PT-independent \\Btwo{} as well as the calculated \\avpT{} for protons and deuterons from the Blast-Wave model~\\cite{Schnedermann:1993ws} using the kinetic freeze-out parameters for pions, kaons, protons and $\\Lambda$ from~\\cite{Abelev:2013haa} (right).}\n \\label{fig:meanpt}\n \\end{center}\n\\end{figure} \n\n\\subsection{Deuteron-over-proton ratio}\n\nThe deuteron-over-proton ratio is shown in Fig.~\\ref{fig:doverp} for three collision systems as a function of the charged-particle density at mid-rapidity. In \\PbPb{} collisions it has been observed that the d\/p ratio does not vary \nwith centrality within uncertainties \n(red symbols). Such a trend is consistent with a thermal-statistical approach and the magnitude of the measured values agree with freeze-out temperatures in the range of 150-160~MeV \\cite{nuclei}. \nThe d\/p ratio obtained in inelastic \\pp{} collisions increases with multiplicity~\\cite{Acharya:2019rgc}.\nThe results in \\pPb{} collisions bridge the two measurements in terms of multiplicity and system size and show an increase of the d\/p ratio with multiplicity. Here, the low (high) multiplicity value is compatible with the result from pp (\\PbPb) collisions. \nNote that the experimental significance of this enhancement is further substantiated by considering only the part of the systematic uncertainty which is uncorrelated across multiplicity intervals.\n\n\\begin{figure}[htbp]\n \\begin{center}\n \\includegraphics[trim=0cm 0cm 0cm 0cm, clip=true, width = 0.6\\textwidth]{figures\/doverp.pdf} \n \\caption{Deuteron-over-proton ratio as a function of charged-particle multiplicity at mid-rapidity for \\pp, \\pPb{} and \\PbPb{} collisions \\cite{Acharya:2019rgc,nuclei,Acharya:2017fvb}. The empty boxes show the systematic uncertainty while the vertical lines represent the statistical uncertainty.}\n \\label{fig:doverp}\n \\end{center}\n\\end{figure} \n\nA similar rise with multiplicity is observed for the ratios of the yields of multi-strange particles to that of pions in \\pPb{} collisions~\\cite{multistrange-p-Pb}. In this case the canonical suppression due to exact strangeness conservation in smaller systems gives a qualitative explanation \\cite{Acharya:2018orn}. An interpretation of the d\/p ratio within thermal models is difficult, since the measured \\ppi{} ratio in these three systems is about the same \\cite{Abelev:2013haa}. Therefore, the available parameter space for a change in the freeze-out temperature or a suppression due to exact conservation of baryon number is limited \\cite{Vovchenko:2018fiy}.\nCoalescence models are able to explain such an observation. The probability of forming a deuteron increases with the nucleon density and thus also with the charged-particle density. The results from pp and \\pPb{} collisions at low charged-particle density fit with this concept.\n\n\n\\section{Conclusions}\n\nThe production of deuterons and $^{3}{\\mathrm{He}}${} and their antiparticles in \\pPb{} collisions at \\snn{} = 5.02 TeV has been studied at mid-rapidity. The results on deuteron production in \\pPb{} collisions exhibit a continuous evolution with multiplicity between \\pp{} and \\PbPb{} collisions. \nThe production of complex nuclei shows an exponential decrease with mass (number). The penalty factor (decrease of yield for each additional nucleon) is larger than the one observed in central Pb--Pb collisions and smaller than the one measured in pp collisions. The transverse momentum distributions of deuterons become harder with increasing multiplicity. Two intriguing observations that have been recently reported by ALICE \\cite{Acharya:2019rgc} in high multiplicity pp collisions are confirmed in the present paper.\nFirstly, the \\avpT{} values of deuterons are comparable to those of the much lighter $\\Lambda$ baryons and thus do not follow a mass ordering.\nThis behaviour is observed for all multiplicity intervals and it is in contrast to the expectation from simple hydrodynamical models. \nThese observations made in \\pPb{} collisions support a coalescence mechanism, while in \\PbPb{} collisions the deuteron seems to follow the collective expansion of the fireball. Secondly, the d\/p ratio rises strongly with multiplicity, while this ratio remains approximately constant as a function of multiplicity in \\PbPb{} collisions, where its value agrees with thermal-model predictions. \n\n\\newenvironment{acknowledgement}{\\relax}{\\relax}\n\\begin{acknowledgement}\n\\section*{Acknowledgements}\n\\input{fa_2019-05-30.tex} \n\\end{acknowledgement}\n\n\\bibliographystyle{utphys} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nQuantum computing (QC) and machine learning (ML) have changed our ways to process data fundamentally in the last years \\cite{Jordan2015,LeCun2015,Preskill2018,Deutsch2020}. Quantum algorithms accelerated the data search \\cite{Grover2001} or improved the sampling of probability distributions \\cite{Arute2019,Wu2021}. These quantum advantages found already their way to various applications \\cite{Orus2019,Ajagekar2019,Cao2019}, even though we are still in the era of noisy intermediate scale quantum (NISQ) devices that suffer from decoherence and are limited to qubit numbers $\\sim 10^2$ with resulting shallow quantum circuit depths \\cite{Altman2021}. \n\nMeanwhile, ML algorithms in the form of deep convolutional neural networks extract features effectively and classify big data bases \\cite{Hinton2012,Ronneberger2015,Goodfellow2016,Fonda2019}. Quantum machine learning ports such methods to a quantum computer \\cite{Biamonte2017,Ciliberto2017,Benedetti2019} with the prospect that particularly high-dimensional problems can be solved much faster than with their classical counterparts. This expectation arises from two facts, (i) the data space dimension grows exponentially as $2^n$ with the number of qubits $n$, the smallest unit of information in QC; (ii) the entanglement of qubits creates highly correlated tensor product states that can represent complex features in the data effectively \\cite{Nielsen2010}. Thus{\\color{red},} for example, quantum support vector machines are expected to have the potential to determine nonlinear decision boundaries of classification problems in high-dimensional quantum enhanced feature Hilbert spaces more efficiently \\cite{Rebentrost2014,Schuld2019,Blank2020}. \n\nRecurrent neural networks (RNN) are specific ML algorithms with internal feedback loops which predict the time evolution of dynamical systems without knowing the underlying nonlinear ordinary or partial differential equations; they can be implemented either as gated RNNs in the form of long short-term memory networks \\cite{Schmidhuber1997} or as reservoir computing models (RCM) \\cite{Jaeger2001,Maass2002,Jaeger2004,Lukosevicius2009,Pathak2018}. As a consequence, RNNs have been used for the description of chaotic dynamics, fluid mechanical problems, and even turbulence \\cite{Kutz2017,Srinivasan2019,Brunton2020,Pandey2020a}. RCMs were also applied to represent low-dimensional chaotic models, one-dimensional Kuramoto-Sivashinsky equations \\cite{Lu2017,Vlachas2020}, or even turbulent Rayleigh-B\\'{e}nard convection \\cite{Pandey2020,Heyder2021,Pandey2022,Valori2022}. At the center of the RCM is the reservoir, a randomly initialized and fixed high-dimensionl network of perceptrons which is represented by an adjacency matrix. This specific implementation of an RNN requires only an optimization of the output layer, which maps the reservoir state back to the data space, and avoids costly backpropagation as required in most other ML algorithms \\cite{Goodfellow2016}. \n\nIn this work, we combine quantum algorithms with reservoir computing to a gate-based quantum reservoir computing model (QRCM) for a universal quantum computer to predict and reconstruct the dynamics of a thermal convection flow in the weakly nonlinear regime. The algorithm is of {\\em hybrid quantum-classical nature} since the optimization of the output map is done by a classical ridge regression. The {\\em quantum reservoir} is composed of a sequence of elementary single and two-qubit quantum gates which form a complex quantum circuit. Following the axioms of quantum mechanics, an elementary quantum gate performs a unitary transformation to a single- or two-qubit state. As a consequence, a highly entangled multi-quibit state will result which is a tensor product state of single qubits. \n\nOur first contribution is to demonstrate the feasibility of such QRCM to describe the classical chaotic dynamics of a thermal convection flow on an actual NISQ device. The description of the thermal convection flow is based on Lorenz-type Galerkin models with $N_{\\rm dof}\\le 8$ degrees of freedom \\cite{Lorenz1963,Howard1986,Thiffeault1996,Gluhovsky2002,Moon2017}. This class of models is directly derived from the Boussinesq equations of two-dimensional thermal convection between two impermeable parallel plates, heated uniformly from below and cooled from above with free-slip boundary conditions for the velocity field \\cite{Koschmieder1993,Chilla2012}. Here, we explore QRCMs in two different modes of operation \\cite{Lukosevicius2021}: \n\\begin{enumerate}\n\\item {\\em Closed--loop scenario}: a fully autonomous prediction of the temporal dynamics of all degrees of freedom of a Lorenz 63 model with $N_{\\rm dof}=3$. This study is done by a quantum computation applying the ideal Qiskit quantum simulator \\cite{Qiskit}, see the sketch in Fig. \\ref{fig0}(a). \n\\item {\\em Open--loop scenario}: a reconstruction of the temporal dynamics of a Lorenz-type model with $N_{\\rm dof}=8$. In this case, one or two degrees of freedom are continually fed into the quantum reservoir and the remaining degrees of freedom are obtained by the QRCM evolution, see Fig. \\ref{fig0}(b). This investigation is done in two two different ways. First, we reconstruct the whole model from a single degree of freedom ($N_{\\rm in}=1$) by means of the open loop structure in an ideal Qiskit simulator. Secondly, we strongly reduce the number of quantum gates to even demonstrate the feasibility of QRCM on a real noisy IBM quantum computer. \n\\end{enumerate}\n\\begin{figure}\n\\includegraphics[scale=0.45]{Figure1.png}\n\\caption{Sketch of the two scenarios in which the reservoir computing model is run. (a) Closed-loop scenario for autonomous prediction of the dynamics. (b) Open-loop scenario for reconstruction of dynamics from continually available data. The matrix $W^{\\rm out}_{\\ast}$ stands for the classically optimized output layer. $N_{\\rm dof}$ is the number of degrees of freedom of the dynamical system, $N_{\\rm in}$ is the number of continually available components of the system state vector. The dimensionality of the reservoir is $N_{\\rm res}\\gg N_{\\rm dof}$. For (a), $N_{\\rm in} = N_{\\rm dof}$ and for (b), $N_{\\rm in}\\ll N_{\\rm dof}$.} \n\\label{fig0}\n\\end{figure}\n\nSecondly, we directly compare the results of the QRCM to its classical counterpart for the same flow. We identify hyperparameters in both approaches that can be related to each other. Note that classical and quantum reservoir computing models differ essentially, which is primarily a consequence of the linearity and unitarity of the quantum dynamics \\cite{Nielsen2010}. The present work also extends previous QRCM investigations, such as in the form of spin ensembles \\cite{Fujii2017,Nakajima2019,Kutvonen2020}, single nonlinear oscillators \\cite{Govia2021}, and smaller universal quantum circuits \\cite{Chen2020,Dasgupta2020} that have been applied for the one-step prediction of one-dimensional time series or solutions of the Mackey-Glass time-delay differential equation, see also ref. \\cite{Markovic2020} for a compact review. This is similar to the open--loop scenario with $N_{\\rm in}=N_{\\rm dof}$. \n\nIt is finally demonstrated that a systematic reduction of the degree of entanglement in the quantum reservoir by a stepwise transition from a fully to a weakly entangled configuration reduces the performance of the present QRCM algorithm. More detailed, this is done by the division of an $n$-qubit reservoir state into blocks of entangled $p$-qubit states, so called $p$-blocks \\cite{Josza2003}. The strong encoding capabilities of fully entangled quantum reservoirs are demonstrated in the present flow case by runs with qubit numbers $n0$. The convection flow domain is $A=[0,\\Gamma]\\times [0,1]$. The velocity ${\\bm u}({\\bm x},t)=(u_x(x,z,t), u_z(x,z,t))$ and (total) temperature $T(x,z,t)$ are coupled by the balances of mass, momentum, and energy. The fluid is incompressible and the mass density $\\rho$ depends linearly on $\\theta$ in the buoyancy term only, known as the Boussinesq approximation in thermal convection \\cite{Chilla2012}. The total temperature is decomposed into $T(x,z,t)=1-z+\\theta(x,z,t)$ where $T_{\\rm eq}(z)=1-z$ is the static equilibrium profile and $\\theta(x,z,t)$ is the temperature deviation. The non-dimensional equations are then given by\n\\begin{align}\n{\\bm \\nabla \\cdot u}&= 0 \\,,\n\\label{eq:NS1}\\\\\n\\frac{\\partial {\\bm u}}{\\partial t} + ({\\bm u} \\cdot {\\bm \\nabla}) {\\bm u} &= - {\\bm \\nabla} p + \\sigma {\\bm \\nabla}^2 {\\bm u} + {\\rm Ra} \\sigma \\theta {\\bm e}_z \\,,\\label{eq:NS2b}\\\\\n\\frac{\\partial \\theta}{\\partial t} + ({\\bm u}\\cdot {\\bm \\nabla})\\theta &= {\\bm \\nabla}^2 \\theta + u_z\\,.\n\\label{eq:NS3}\n\\end{align}\nThe Rayleigh number ${\\rm Ra}$ and the Prandtl number $\\sigma$ are the two parameters that characterize the strength of the thermal driving via the temperature difference $\\Delta T$ and the ratio of momentum to temperature diffusion, respectively. The boundary conditions in $x$-direction are periodic. At $z=0,1$, one takes\n\\begin{align}\nu_z\\big|_{z=0,1}=0,\\quad \\frac{\\partial u_x}{\\partial z}\\Bigg|_{z=0,1}=0\\quad\\mbox{and}\\quad \\theta\\big|_{z=0,1}=0\\,.\\label{bc1}\n\\end{align}\nThey correspond to isothermal, impermeable, free-slip walls. Incompressibility and two-dimensionality allow to reduce the velocity vector field further to a scalar stream function $\\zeta(x,z,t)$ by \n\\begin{equation}\nu_x=-\\frac{\\partial \\zeta}{\\partial z} \\quad \\mbox{and} \\quad u_z=\\frac{\\partial \\zeta}{\\partial x}\\,. \n\\end{equation}\nThis ansatz satisfies \\eqref{eq:NS1} automatically and the equations of motion \\eqref{eq:NS2b}--\\eqref{eq:NS3} are now given by \n\\begin{align}\n\\frac{\\partial \\nabla^2\\zeta}{\\partial t}&=\\frac{\\partial \\zeta}{\\partial z} \\frac{\\partial \\nabla^2\\zeta}{\\partial x}-\\frac{\\partial \\zeta}{\\partial x} \\frac{\\partial \\nabla^2\\zeta}{\\partial z} + \\sigma \\nabla^4\\zeta +{\\rm Ra} \\sigma\\frac{\\partial \\theta}{\\partial x}\\,,\\label{2d1}\\\\ \n\\frac{\\partial \\theta}{\\partial t}&=\\frac{\\partial \\zeta}{\\partial z} \\frac{\\partial \\theta}{\\partial x}-\\frac{\\partial \\zeta}{\\partial x} \\frac{\\partial \\theta}{\\partial z} + \\nabla^2\\theta +\\frac{\\partial \\zeta}{\\partial x}\\label{2d2}\\,,\n\\end{align}\nwith boundary conditions in the vertical direction\n\\begin{align}\n\\zeta\\big|_{z=0,1}=0,\\quad \\frac{\\partial^2\\zeta}{\\partial z^2}\\Bigg|_{z=0,1}=0\\quad\\mbox{and}\\quad \\theta\\big|_{z=0,1}=0\\,.\\label{bc}\n\\end{align}\nEquations \\eqref{2d1} and \\eqref{2d2} are then reduced by an expansion into trigonometric Fourier modes which satisfy the boundary conditions for the stream function and temperature and encode the spatial structure of the thermal convection flow, see appendix A for further technical details. A subsequent truncation to $N$ and $M$ real time-dependent amplitudes is done for the stream function, $\\{A_1(\\tau), \\dots, A_N(\\tau)\\}$, and the temperature, $\\{B_1(\\tau), \\dots, B_M(\\tau)\\}$, respectively. \n\nThis step leads to a class of low-dimensional Lorenz-type Galerkin models of the thermal convection flow starting with the original three-dimensional Lorenz 63 model \\cite{Lorenz1963} for $N=1$ and $M=2$ (where $N_{\\rm dof}=N+M$). The resulting coupled nonlinear system of ordinary differential equations is given by \n\\begin{align}\n\\frac{dA_i}{d\\tau}&=F_i(A_j, B_k, \\sigma, r, b)\\,,\\label{Lo1}\\\\\n\\frac{dB_k}{d\\tau}&=G_k(B_l, A_i, \\sigma, r, b)\\,, \\label{Lo2}\n\\end{align}\nfor $i,j=1\\dots N$ and $k,l=1\\dots M$. Here, $\\sigma$ is again the Prandtl number, $r$ the {\\em relative} Rayleigh number, and $b$ an aspect ratio parameter, see appendix A. Furthermore, $F_i$ and $G_i$ are quadratic nonlinear functions of the amplitudes $A_i(\\tau)$ and $B_k(\\tau)$. We will consider two implementations, the Lorenz 63 model (L63) \\cite{Lorenz1963} with $N=1$ and $M=2$ and an extended 8-dimensional model \\cite{Gluhovsky2002} with $N=M=4$ that introduces shear in the flow and causes tilted convection rolls and shearing motion. It thus displays a more complex fluid motion further away from the primary instability point at $r=1$ or $Ra_c=27\\pi^4\/4$ \\cite{Koschmieder1993}. \n\nFigures \\ref{fig1}(a) shows two instances of the temperature and velocity fields with the counter-rotating circulation rolls that cause a rise of warm and a descent of cold fluid. These two flow states corresponds to trajectory points of L63 in each of the two butterfly-like wings in panel (c) of the same figure. \n\n\\section{Closed-loop scenario for three-dimensional Lorenz model} \n\\subsection{Quantum reservoir and classical data input}\nThe design of our time-discrete and gate-based QRCM builds on a $n$-qubit tensor product state at time $t$. In appendix B, we provide a compact primer on qubits and tensor product spaces. The $n$-qubit state in Dirac notation \\cite{Nielsen2010} is given by\n\\begin{equation}\n |{\\bm \\psi}^t\\rangle=\\sum_{k=1}^{N_{\\rm res}} a^t_k |k\\rangle \\quad\\mbox{with}\\quad a_k^t\\in \\mathbb{C}\\,,\n\\end{equation}\nwith $N_{\\rm res}=2^n$. Here $|k\\rangle$ is the standard basis of the $n$-qubit quantum register. The pure state density operator $\\rho^t$ is given by the outer product of the quantum state with itself,\n\\begin{equation}\n\\rho^t=|{\\bm \\psi}^t\\rangle \\langle{\\bm \\psi}^t| =\\sum_{k=1}^{2^n} p^t_k |k\\rangle \\langle k| \\quad\\mbox{with}\\quad \\sum_{k=1}^{2^n}p_k^t=1\\,. \n\\label{Rho_and_P}\n\\end{equation}\nThe reservoir state evolves from time step $t$ to $t+1$ as follows. First, the dynamical part is updated by three blocks of unitary linear transformations\n\\begin{equation}\n|\\tilde{\\bm \\psi}^{t+1}\\rangle= U({\\bm \\beta})U(4\\pi{\\bm x}^t)U(4\\pi{\\bm p}^t) |0\\rangle\\,, \\label{unitary}\n\\end{equation}\nwith rotation angles ${\\bm \\beta}=(\\beta_1, ..., \\beta_n)$, reservoir state probability amplitudes ${\\bm p}^t=(p_1^t, ..., p_{2^n}^t)$, and the past system state vector ${\\bm x}^t=(x_1^t, ..., x_{N+M}^t)$, the latter of which summarizes $(A_1,...,B_M)$. The $n$-qubit state vector $|0\\rangle$ is the ground state of the quantum register. With eq. (\\ref{Rho_and_P}) for the probability amplitudes $\\tilde{p}_k^{\\ t+1}$ out of $|\\tilde{\\bm \\psi}^{t+1}\\rangle$ from eq. (\\ref{unitary}), the RCM update step outside the quantum reservoir is given by the following iteration \n\\begin{equation}\np_k^{t+1} = (1-\\varepsilon) \\ p_k^{t} + \\varepsilon \\ \\tilde{p}_k^{\\ t+1}\\,.\n\\label{unitary1}\n\\end{equation}\nSee also equation \\eqref{eq:rcm_activation} in appendix C. The equation contains a leaking rate $0\\le \\varepsilon\\le 1$ that blends update and past state and thus represents a short-term memory, see also appendix B. More detailed, the unitary transformation consist of single-qubit rotation gates $R_Y$ and subsequent two-qubit controlled NOT (in short CNOT) gates. The $R_Y$-gate is defined by $$ R_Y(x) = \\begin{pmatrix} \\cos(x\/2) & -\\sin(x\/2) \\\\ \\sin(x\/2) & \\;\\;\\;\\cos(x\/2) \\end{pmatrix}.$$\n\nFigure \\ref{fig1}(b) shows the corresponding circuit diagram of the quantum reservoir which consists of three circuit blocks as in eq. \\eqref{unitary}. The first block of unitary transformations $U(4\\pi{\\bm p}^t)$ loads the reservoir state probability amplitudes of the previous time step $t$ (indicated as the blue box). This is done by rotation gates $R_Y(4\\pi p^t_k)$ where $p_k^t=|a_k^t|^2$. In the circuit diagram of Fig.\\ref{fig1}, these $2^9=512$ probabilities with $0\\le p_k^t\\le 1$ are summarized to a vector to keep the notation less crowded. For this as for all the following blocks, the combination of $R_Y$ and CNOT gates is continued until the last qubit is reached. There, the CNOT is applied to the previous qubit and if not yet finished, the constructor starts at the upper qubit again.\n\nThe application of an $R_Y$ rotation gate, which is parametrized by the input value, is a {\\em nonlinear} operation in terms of the amplitudes \\cite{Dasgupta2020,Govia2022}. It can be considered as an analogy to the nonlinear activation in a classical RCM, e.g., by $\\tanh(\\cdot)$, see also table \\ref{table1} where we summarize hyperparameters of the classical and quantum RCMs. Note also that all $2^9$ amplitudes $p_k^t$ are loaded into the reservoir in this case.\n\nSimilarly, the degrees of freedom $x_i^t$ at time $t$ are loaded into the quantum reservoir in the second block $U({4\\pi\\bm x}^t)$ before the model advances further (indicated as the yellow box) followed by some additional rotations in the last block (indicated as the gray box). This third block $U({\\bm \\beta})$ performs additional rotations by angles which are randomly chosen but fixed at the beginning of the evolution. It corresponds to a random seed initialization of a classical reservoir.\n\\begin{table*}\n \\renewcommand{\\arraystretch}{1.1}\n \\centering\n \\begin{tabular}{lccccc}\n \\hline\\hline\n Quantity & $\\;\\;$ Classical RCM $\\;\\;$& Optimal value & $\\quad$ & $\\;\\;$ Quantum RCM $\\;\\;$ & Optimal value \\\\\n \\hline\n Reservoir dimension & $N_{\\rm res}$ & 2048 & & $N_{\\rm res}=2^n$ & 512\\\\\n Leaking rate & $\\varepsilon$ & 0.1 & & $\\varepsilon$ & 0.05 \\\\\n Spectral radius of reservoir & $\\rho(W_r)$ & 0.99 & & $\\rho(U)$ & 1.0 \\\\\n Reservoir state at time $t$& ${\\bm \\psi}^t\\in \\mathbb{R}^{N_{\\rm res}}$& & & $|{\\bm \\psi}^t\\rangle\\in \\mathbb{C}^{N_{\\rm res}}$ & \\\\\n Training steps & $N_{\\rm train}$ & 2000 & & $N_{\\rm train}$ & 2000 \\\\\n Reservoir model nonlinearity & $\\tanh(\\cdot)$ & & & $R_Y(\\cdot)$ & \\\\\n \\hline\\hline\n \\end{tabular}\n \\caption{Comparison of classical and quantum reservoir computing models. Different essential quantities including optimal hyperparameters for the Lorenz 63 model are listed. The spectral radius $\\rho(W^r)$ in the quantum case is always equal to 1 since unitary transformations are norm-preserving. The number of qubits is $n$. Two additional hyperparameters are used in the classical RCM: a reservoir density $D=0.2$ which determines the percentage of active nodes in the matrix $W^r$ and an additional Tikhonov regularization term with a parameter $\\gamma=10^{-3}$ in the cost function $C(W^{\\rm out})$, see appendix C.}\n \\label{table1}\n\\end{table*}\n\n\\subsection{Quantum reservoir readout and classical optimization}\nA projection-valued measure in the standard basis of the Pauli-$Z$ operator provides the probabilities $p_k^t$ from $K\\gg 2^n$ independent circuit simulations, know as shots. These probabilities are mapped to the updated dynamical system state by the output matrix, \n\\begin{align}\nx_i^{t} &= \\sum_{k=1}^{2^n} W^{{\\rm out}\\ast}_{ik} p^t_k\\,, \n\\label{outp}\n\\end{align} \nwith the optimized weights which are summarized in the matrix $W^{{\\rm out}\\ast}\\in \\mathbb{R}^{(N+M)\\times N_{\\rm res}}$. We note once more that the output matrix is optimized by a classical algorithm similar to the classical RCM case. This optimization seeks a minimum of the cost function $C(W^{\\rm out})$ which is given in appendix C. \n\nPanel (d) of figure \\ref{fig1} and table \\ref{table1} compare the classical and quantum RCM with the numerical simulation of the equations of motion obtained by a 4th-order Runge-Kutta method. The integration time is rescaled by the largest Lyapunov exponent of the system, $\\lambda_1=0.9056$, which quantifies the deterministic chaos of the model \\cite{Geurts2020}. The training phase comprises $N_{\\rm train}=2000$ integration time steps, both for the classical and quantum case. For times $t\\ge 0$ the reservoirs are exposed to unseen test data predicting the dynamics autonomously. It is seen that the prediction horizon of the QRCM with 9 qubits is about 1.5 $\\lambda_1 t$. The noise in the Qiskit simulator causes a switch of the trajectory into the other wing of the butterfly-like Lorenz attractor. The leaking rate in this example is $\\varepsilon=0.05$. \n\nTwo hyperparameters are varied, the leaking rate $\\varepsilon$ and the number of qubits $n$ that determines the reservoir dimension $N_{\\rm res}$. We identify a minimum of the cost function in the form of a mean squared error (MSE) around $\\varepsilon=0.025$. The larger the number of qubits the smaller MSE, although the improvements in the Qiskit simulations remain small (and thus the difference of the displayed to the optimal case). A small leaking rate implies that the reservoir dynamics is memory-dominated blended with a small nonlinear contribution \\cite{Inubushi2017}. We have listed all details on the classical reservoir computing model, the hyperparameters, and the cost function in appendix C in order to keep the work self-contained. \n\n\\begin{figure*}\n\\includegraphics[scale=0.6]{Figure3.png}\n\\caption{Comparison of the 8-dimensional Lorenz-type model, time-integrated system of equations as ground truth (GT), classical reservoir computing model (CRC), and quantum reservoir computing model (QRC). Panel (a) displays the time evolution of all variables. (b) Reconstruction of the flow and temperature fields at times $t_1$ to $t_4$ which are indicated in $A_1(t)$. The model parameters are $\\sigma=10$, $b=8\/3$, and $r=28$. Here, $N=4$ and $M=4$. Mode $A_4$ is the only input and always given accurately into each reservoir.}\n\\label{fig2}\n\\end{figure*}\n\n\\section{Open-loop scenario for 8-dimensional Lorenz-type model}\n\\subsection{Quantum reservoir with one continually available degree of freedom}\nWe proceed from the standard L63 model to an extension with 8 degrees of freedom, which is listed and explained further in appendix A. As shown by Gluhovsky {\\it et al.} \\cite{Gluhovsky2002}, this extension can be decomposed into subgroups, so-called gyrostats. The model conserves total energy and vorticity. They are given by \n\\begin{equation}\n E(t)=\\frac{1}{2A}\\int_A ((\\nabla\\zeta)^2-z\\theta) dA\\,,\n\\end{equation}\nwith a kinetic and potential energy term and\n\\begin{equation}\n\\Omega(t)=\\frac{1}{A}\\int_A \\omega dA\\,, \n\\end{equation}\nwith the vorticity $\\omega=-\\nabla^2\\zeta$ and the convection domain size $A$. The open-loop scenario of the QRCM implies that a subset of the degrees of freedom will remain continually available in the reconstruction phase after the training phase. In this subsection, we will take $N_{\\rm in}=1$ which will be $A_4(t)$. The leading Lyapunov exponent was computed numerically by a method proposed in~\\cite{Sprott2003} and turned out to be $\\lambda_1= 0.825$.\n \nFigure \\ref{fig2} displays the results for the 8--dimensional Lorenz-type model which receives the time series $A_4(t)$ to reconstruct the remaining 7 degrees of freedom of the thermal convection flow model. Panel (a) compares the times series of the ground truth (GT) with the results of a classical RCM (CRCM), and a QRCM which was run on $n=7$ qubits on an ideal Qiskit simulator. We see that the data remain closely together for the displayed interval of over 16 Lyapunov time units. The QRCM runs through a training phase of $N_{\\rm train}=2000$ integration time steps and a leaking rate $\\varepsilon=0.05$. Figure \\ref{fig2}(b) displays the reconstructed convection flow at four instants. The 8-dimensional model incorporates the shearing modes which are missing in the lower-dimensional Lorenz 63 model and lead to tilted convection cells, as can be seen in the panels.\n\nThe dimension of the quantum reservoir is $N_{\\rm res}=128$, while the one of the CRCM in Fig. \\ref{fig2} is $N_{\\rm res}=1024$. We have thus reduced the dimension of the CRCM and compare the mean squared error (MSE) as a function of $N_{\\rm res}$ in table \\ref{table2}. The MSE is here given as\n\\begin{equation}\n \\mbox{MSE}=\\frac{1}{T_{\\rm test}}\\sum_{t=1}^{T_{\\rm test}}|{\\bm x}^t-{\\bm x}_{\\rm tg}^t|^2\\,,\n\\end{equation}\nsee also eq. \\eqref{outp} and $T_{\\rm test}=2000$. Subscript tg stands for target and denotes the ground truth (GT) which is obtained by time integration of the eqns. \\eqref{Lo1} and \\eqref{Lo2}. As the table shows, the MSE of the CRCM increases with decreasing reservoir size. The MSE of the QRCM corresponds to a classical reservoir size between $N_{\\rm res} = 512$ and 1024, which is almost one order of magnitude larger which clearly confirms the encoding capabilities of the QRCM. \n\\begin{table}\n \\renewcommand{\\arraystretch}{1.1}\n \\centering\n \\begin{tabular}{lcc}\n \\hline\\hline\n Model & $\\quad N_{\\rm res} \\quad$& MSE \\\\\n \\hline\n CRCM1 & 2048 & $0.8 \\cdot 10^{-3}$\\\\\n CRCM2 & 1024 & $1.2 \\cdot 10^{-3}$\\\\\n CRCM3 & 512 & $2.3 \\cdot 10^{-3}$\\\\\n CRCM4 & 128 & $3.5 \\cdot 10^{-3}$\\\\\n QRCM & 128 & $1.6 \\cdot 10^{-3}$\\\\\n \\hline\\hline\n \\end{tabular}\n \\caption{Comparison of classical and quantum reservoir computing models in the reconstruction phase. The remaining hyperparameters remain fixed.}\n \\label{table2}\n\\end{table}\n\\begin{figure*}\n\\includegraphics[scale=0.5]{Figure4.png}\n\\caption{Quantum reservoir computing model run of the 8-dimensional time-integrated Lorenz-type model an actual quantum device. (a) Sketch of the quantum reservoir which had to be reduced due to decoherence in comparison with the one that is displayed in Fig. \\ref{fig1}. (b) Time series of the extended Lorenz model. We compare the ground truth (Model) with an ideal and noisy Qiskit simulator and the 7--qubit quantum computer (IBM Q). The number of training steps was again $N_{\\rm train}=2000$ and the leaking rate $\\varepsilon=0.2$. The two degrees of freedom that are continually available in the reconstruction phase are indicated. (c) Connection of the 7 qubits on the {\\em ibm\\_perth} quantum computer.}\n\\label{fig3}\n\\end{figure*}\n\n\\subsection{Implementation on an actual quantum device}\n\nThe 8-dimensional convection flow model is finally implemented on an actual noisy quantum device. Figure \\ref{fig3}(a) displays the quantum reservoir for the implementation. The circuit depth on the real devices is still rather limited by the decoherence of the elementary quantum gates that are installed in the form of microwave-controlled superconducting SQUIDs. The figure shows that we had to reduce the original three-block-structure of the quantum reservoir to one block. Two further steps were necessary: (1) Instead of one continually available variable in the reconstruction phase, we provide now $A_4(t)$ and $B_3(t)$. (2) Only 12 out of the 128 components of the reservoir state measurement vector ${\\bm p}^t$ are fed back into reservoir together with the two degrees of freedom. The total qubit number was limited to $n=7$. The studies were conducted on two devices, {\\em ibmq\\_ehningen}, a 27--qubit quantum computer in Germany, and {\\em ibmq\\_perth}, a 7-qubit machine. Figure \\ref{fig3}(c) displays the arrangement of the 7 qubits on {\\em ibmq\\_perth}. Entanglement operations, e.g. by C-NOT gates, are only possible for qubits which are connected by the bars in the panel. No error correction was performed. \n\nWe backed up this investigation by two runs on the Qiskit simulator with the same configuration. One is the ideal simulator that has been used already before. The other simulation was done on a noisy Qiskit simulator for which you can prescribe the probabilities of measurement errors, here $p_m=0.05$, gate errors, here $p=0.1$, and qubit resets, here $p_r=0.03$. Values have been chosen such that they come close to those on the real devices. All environments are compared in Fig. \\ref{fig3}(b). The data from the noisy Qiskit simulator and the real quantum device do partly deviate, but are found to follow the overall trend fairly well. This proves the concept of an QRCM for a classical dynamical system on a NISQ device. \n\\begin{figure*}\n\\includegraphics[scale=0.55]{Figure5.png}\n\\caption{Performance of the quantum reservoir computing model for different reservoir architectures. (a) Mean squared error on a logarithmic scale as a function of the total number of qubits and the size of the blocks of entangled qubits. The dark cells in the lower left stand for impossible decompositions. (b) Sketch of an example case. Fully entangled 4-qubit-quantum circuit which is the normal setting. (c) Two 2-qubit blocks ($p=2$) build the 4-qubit-quantum circuit.}\n\\label{fig5}\n\\end{figure*}\n \n\\subsection{Stepwise reduction of reservoir entanglement and quantum advantage}\nFinally, we investigated if a simulation of the Lorenz-type model with the simpler quantum reservoir than from Fig. \\ref{fig3}(a) is successful when the corresponding quantum circuit is decomposed into several $p$-qubit-blocks which are disentangled. If $p=n$ the circuit is fully entangled, for $p=1$ the $n$-qubit quantum state is separable; see appendix B for the defintions of both possible multi-qubit quantum states. The decomposition is illustrated in Figs. \\ref{fig5}(b,c). The rational behind this analysis, which we did with the ideal Qiskit simulator for $n=8$, is that a $p$-blocked structure might be simulated efficiently on a classical computer loosing the quantum advantage \\cite{Josza2003}. \n\nIn Fig. \\ref{fig5}(a), we summarize the MSE in a diagram for circuits with $3\\le n\\le 8$ and possible block size $2\\le p\\le 8$. For example, $n=4$ and $p=3$ imply that a single qubit remains which is disentangled from the 3-qubit-block. In general, the number of blocks of size $p$ follows by $n_p=\\lfloor n\/p \\rfloor$. All reservoirs were trained, the simulations were carried out for different 100 seeds of the quantum reservoir. The block diagram shows that the MSE decreases when block size is increased. We can conclude from this analysis that the entanglement of the qubits in the reservoir is essential for the performance of the QRCM. This is different compared to the classical reservoir which is a sparse network for which 20\\% of the network nodes are actively only, but the number of perceptrons is by 2 to 3 order of magnitude larger in comparison to the number of qubits of the quantum reservoir. \n \nIn a final discussion of a prospective quantum advantage of the algorithm, we stress that the main computational effort of a single time step lies in the calculation of the leaking rate equation (\\ref{Rho_and_P}). This effort scales linearly with the number of reservoir states $N=2^n$. In detail, one needs $3N$ operations to multiply each vector with the corresponding prefactor to add them subsequently. The dynamical part in its simplest form can be seen in the circuit scheme of Fig. \\ref{fig3}(a). There, the 7 qubit circuit with $N=128$ states needs 20 operations only, whereas a classical counterpart, cf. eq. (\\ref{eq:rcm_activation}), would require more operations than states to produce a comparable dynamical counterpart. For a further more comprehensive analysis, it would be necessary to determine if the layer depth -- 3 gate operations per qubit in our case -- can be held constant for a larger number of qubits. This would result in a clear quantum advantage for the dynamical part of the reservoir update equation.\n\n\\section{Summary and outlook}\nThe main objective of our present work was to show the feasibility of a quantum reservoir computing model to predict and to reproduce the dynamical evolution of a classical, nonlinear thermal convection flow, on an actual quantum computer with up to 7 superconducting qubits. In a nutshell, quantum reservoir computing models are recurrent machine learning algorithms for which the reservoir state is built by a highly entangled tensor product quantum state that grows exponentially in dimension with the number of qubits. \n \nOur work showed that a quantum reservoir has a qubit number that is by 2 to 3 orders smaller than that of the perceptrons in a classical one. On the one hand, we could thus take advantage of the data compression capabilities of quantum machine learning algorithms where the dimension of the data space grows exponentially with the number of qubits which is essential for the modeling of higher-dimensional nonlinear dynamical systems. On the other hand, it shows that a classical reservoir state which is caused by a sparsely occupied network matrix of dimension $\\gtrsim 10^3$ can be substituted by a highly entangled quantum state that is caused by the application of unitary transformations. The qubit number was $n<10$ in the present case. \n\nThe study can be extended into several directions. It is clear that the present thermal convection flow model is still very low-dimensional and thus far away from convective turbulence. Our efforts should be considered as one first step to model real fluid flows on a quantum computer, a possible route beside other directions, such as quantum embeddings of nonlinear dynamical systems by the Koopman operator framework \\cite{Giannakis2022} or variational quantum algorithms for the direct solution of the equations of motion \\cite{Gourianov2022}, see also ref. \\cite{Bharadwaj2020} for further directions such as lattice Boltzmann methods. Extensions to higher-dimensional models are currently still limited by the technological capabilities of quantum computers. As the technological progress in this field is very rapidly, it can be expected that Galerkin models with significantly more modes will be modeled on upcoming devices with chips with a higher noise-resilience and lower error rates at the gates. The model that we applied here can be systematically extended towards turbulent convection, as discussed in detail in refs. \\cite{Moon2017,Park2021}. A QRCM with $n\\sim 10$ might thus be able to run a two-dimensional turbulent convection flow usable as a subgrid-scale superparametrization in a global circulation model \\cite{Grabowski1999}. A further possible route of research is to compose the quantum reservoir from first principles, e.g. in the form of a multilayer tensorial network that potentially improve the performance of quantum algorithms on NISQ devices \\cite{Barratt2021}. \n \n\\acknowledgments\nThis work is supported by the project no. P2018-02-001 \"DeepTurb -- Deep Learning in and of Turbulence\" of the Carl Zeiss Foundation and the Deutsche Forschungsgemeinschaft under grant no. DFG-SPP 1881. We acknowledge the use of IBM Quantum services for this work. The views expressed are those of the authors, and do not reflect the official policy or position of IBM or the IBM Quantum team. In this paper we used {\\em ibmq\\_ehningen} and {\\em ibmq\\_perth}, which are run with the IBM Quantum Falcon Processor. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\protect\\bigskip Introduction}\n\nIn 1892, in search for special algebras, Corrado Segre $\\left\\{ \\text{cf. \n\\cite{Segre}}\\right\\} $ published a paper in which he treated an infinite\nfamily of algebras whose elements are commutative generalization of complex\nnumbers called bicomplex numbers, tricomplex numbers,.....etc. Segre \n\\left\\{ \\text{cf. \\cite{Segre}}\\right\\} $ defined a bicomplex number $\\xi $\nas follows:\n\n\\begin{equation*}\n\\xi=x_{1}+i_{1}x_{2}+i_{2}x_{3}+i_{1}i_{2}x_{4},\n\\end{equation*}\n\nwhere $x_{1},$ $x_{2},x_{3},x_{4}$ are real numbers, $i_{1}^{2}=i_{2}^{2}=-1$\nand $i_{1}i_{2}=i_{2}i_{1}.$ The set of bicomplex numbers, complex numbers\nand real numbers are respectively denoted by \n\\mathbb{C}\n_{2},$ \n\\mathbb{C}\n_{1}$ and \n\\mathbb{C}\n_{0}$ . Thus\n\n\\begin{equation*}\n\\mathbb{C}\n_{2}=\\{\\xi:\\xi=x_{1}+i_{1}x_{2}+i_{2}x_{3}+i_{1}i_{2}x_{4},\\text{ \nx_{1},x_{2},x_{3},x_{4}\\i\n\\mathbb{C}\n_{0}\\}\n\\end{equation*}\n\n\\begin{equation*}\n\\text{i.e., \n\\mathbb{C}\n_{2}=\\{\\xi=z_{1}+i_{2}z_{2}:\\text{ \nz_{1}(=x_{1}+i_{1}x_{2}),z_{2}(=x_{3}+i_{1}x_{4})\\i\n\\mathbb{C}\n_{1}\\}.\n\\end{equation*}\n\nThere are \\ two non trivial elements $e_{1}=\\frac{1+i_{1}i_{2}}{2}$ and \ne_{2}=\\frac{1-i_{1}i_{2}}{2}$ \\ in \n\\mathbb{C}\n_{2}$\\ with the properties $\\ e_{1}^{2}=e_{1},e_{2}^{2}=e_{2},e_{1}\\cdot\ne_{2}=e_{2}\\cdot e_{1}=0$ and $\\ e_{1}+e_{2}=1$ which means that $\\ e_{1}$\nand $e_{2}$ are idempotents alternatively called orthogonal idempotents. By\nthe help of the idempotent elements $\\ e_{1}$ and $e_{2},$ any bicomplex\nnumbe\n\\begin{equation*}\n\\xi\n=a_{0}+i_{1}a_{1}+i_{2}a_{2}+i_{1}i_{2}a_{3}=(a_{0}+i_{1}a_{1})+i_{2}(a_{2}+i_{1}a_{3})=z_{1}+i_{2}z_{2}\n\\end{equation*\n\\ where \\ $a_{0,}$ $a_{1},a_{2},a_{3}$ $\\in \n\\mathbb{C}\n_{0},$ \n\\begin{equation*}\nz_{1}(=a_{0}+i_{1}a_{1})\\text{ and }z_{2}(=a_{2}+i_{1}a_{3})\\text{ }\\in \n\\mathbb{C}\n_{1}\n\\end{equation*\ncan be expressed as $\\ \n\\begin{equation*}\n\\xi =z_{1}+i_{2}z_{2}=\\xi _{1}e_{1}+\\xi _{2}e_{2}\n\\end{equation*\nwhere $\\xi _{1}(=z_{1}-i_{1}z_{2})\\in \n\\mathbb{C}\n_{1}$ and $\\xi _{2}(=z_{1}+i_{1}z_{2})$ $\\in \n\\mathbb{C}\n_{1}.$\n\n\\section{Fourier Transform}\n\nLet $f(t)$ be a real valued continuous function in \\ $(-\\infty ,\\infty )$\nwhich satisfies the estimate\n\\begin{align}\n|f(t)|& \\leq C_{1}\\exp (-\\alpha t),t\\geq 0,\\alpha >0 \\notag \\\\\n\\text{and }|f(t)|& \\leq C_{2}\\exp (-\\beta t),t\\leq 0,\\beta >0. \\label{6.4.5}\n\\end{align\nThen the bicomplex Fourier transform $\\left\\{ \\text{cf. \\cite{Banerjee}\n\\right\\} $ of $f(t)$ can be defined as\n\n\\begin{equation*}\n\\widehat{f}(\\omega )=\\ \\tciFourier \\{f(t)\\}=\\tint\\limits_{-\\infty }^{\\infty\n}\\exp (i_{1}\\omega t)f(t)dt,\\omega \\in \n\\mathbb{C}\n_{2}.\n\\end{equation*\nThe Fourier transform $\\widehat{f}(\\omega )$\\ exists and holomorphic for all \n$\\omega \\in $ $\\Omega $\\ where\n\n\\begin{equation*}\n\\Omega =\\{\\omega =a_{0}+i_{1}a_{1}+i_{2}a_{2}+i_{1}i_{2}a_{3}\\in \n\\mathbb{C}\n_{2}:-\\infty \\beta$\nwith centre at the origin. Then all possible singularities (if exists) of \n\\widehat{f_{1}}(\\omega_{1})$ must lie in the region above the horizontal\nline $x_{2}=\\beta$ . At this stage we now consider the following two cases:\n\n\\textbf{CaseI:}We assume that $\\widehat{f_{1}}(\\omega _{1})$ is holomorphic\nin $x_{2}>\\beta $ except for having a finite number of poles $\\omega\n_{1}^{(k)}$ \\ for $k=1,2,...n$ therein (See Figure 2 in Appendix). By taking \n$R\\rightarrow \\infty ,$\\ we can guarantee that all these poles lie inside\nthe contour $\\Gamma _{R}$. Since $exp(-i_{1}\\omega _{1}t)$ never vanishes\nthen the status of these poles $\\omega _{1}^{(k)}$ of $\\widehat{f_{1}\n(\\omega _{1})$ is not affected by multiplication of it with \nexp(-i_{1}\\omega _{1}t)$.Then by Cauchy's residue theorem\n\\begin{align}\n& \\lim_{R\\longrightarrow \\infty }\\int_{\\Gamma _{R}}\\exp (-i_{1}\\omega _{1}t\n\\widehat{f_{1}}(\\omega _{1})d\\omega _{1} \\notag \\\\\n& =2\\pi i_{1}\\sum_{Im(\\omega _{1}^{(k)})>0}\\func{Re}s\\{\\exp (-i_{1}\\omega\n_{1}t)\\widehat{f_{1}}(\\omega _{1}):\\omega _{1}=\\omega _{1}^{(k)}\\}.\n\\label{6.11}\n\\end{align}\n\nFurthermore as $x_{2}\\geq 0,$ we can get $|\\exp (-i_{1}\\omega _{1}t)|\\leq 1$\n\\ for $\\omega _{1}\\in C_{R}$ only when $t\\leq 0$. In particular for $t<0,$\n\n\\begin{align*}\nM(R) & =\\max_{\\omega_{1}\\in C_{R}}|\\widehat{f_{1}}(\\omega_{1})|=\\max\n_{\\omega_{1}\\in C_{R}}|\\tint\n\\limits_{-\\infty}^{0}\\exp(i_{1}\\omega_{1}t)f(t)dt| \\\\\n& \\leq C_{2}\\max_{\\omega_{1}\\in C_{R}}|\\tint\n\\limits_{-\\infty}^{0}\\exp\\{(\\beta+i_{1}\\omega_{1})t\\}dt|=C_{2}\\max_\n\\omega_{1}\\in C_{R}}|\\frac {1}{\\beta+i_{1}\\omega_{1}}| \\\\\n& \\leq C_{2}\\max_{\\omega_{1}\\in C_{R}}\\frac{1}{\\beta+|i_{1}||\\omega_{1}|}\n\\end{align*}\nwhere we use the estimate $\\ref{6.4.5}$. Now for \\TEXTsymbol{\\vert}\n\\omega_{1}|=R\\rightarrow\\infty$ , we obtain that $M(R)\\rightarrow0$. Thus\nthe conditions for Jordan's lemma $\\left\\{ \\text{cf. \\cite{Sid}}\\right\\} $\nare met and so employing it we get tha\n\\begin{equation}\n\\lim_{R\\longrightarrow\\infty}\\int_{C_{R}}\\exp(-i_{1}\\omega_{1}t)\\widehat \nf_{1}}(\\omega_{1})d\\omega_{1}=0. \\label{6.12}\n\\end{equation}\nFinally as,\n\n\\begin{equation*}\n\\lim_{R\\longrightarrow\\infty}\\int_{\\Gamma_{R}}\\exp(-i_{1}\\omega_{1}t\n\\widehat{f_{1}}(\\omega_{1})d\\omega_{1}\n\\end{equation*}\n\n\\begin{equation*}\n=\\int_{C_{R}}\\exp(-i_{1}\\omega_{1}t)\\widehat{f_{1}}(\\omega_{1})d\\omega_{1}\n\\tint \\limits_{-R+i_{1}x_{2}}^{R+i_{1}x_{2}}\\exp(-i_{1}\\omega_{1}t)\\widehat\nf_{1}}(\\omega_{1})d\\omega_{1}\n\\end{equation*}\nthen for $R\\rightarrow\\infty,$ on using ($\\ref{6.11})$ and ($\\ref{6.12})$ we\nobtain that\n\n\\begin{equation*}\n\\tint \\limits_{-\\infty+i_{1}x_{2}}^{\\infty+i_{1}x_{2}}\\exp(-i_{1}\\omega_{1}t\n\\widehat{f_{1}}(\\omega_{1})d\\omega_{1}\n\\end{equation*}\n\n\\begin{equation*}\n=2\\pi i_{1}\\sum_{Im(\\omega _{1}^{(k)})>0}\\func{Re}s\\{\\exp (-i_{1}\\omega\n_{1}t)\\widehat{f_{1}}(\\omega _{1}):\\omega _{1}=\\omega _{1}^{(k)}\\}\\text{ for \n}t<0\n\\end{equation*}\n\nand so\n\n\\begin{equation*}\n\\text{ }f(t)=i_{1}\\sum_{Im(\\omega_{1}^{(k)})>0}\\func{Re}s\\{\\exp\n(-i_{1}\\omega_{1}t)\\widehat{f_{1}}(\\omega_{1}):\\omega_{1}=\\omega_{1}^{(k)}\\\n\\text{ for }t<0.\n\\end{equation*}\n\n\\textbf{Case II: }Suppose $\\widehat{f_{1}}(\\omega _{1})$ has infinitely many\npoles $\\omega _{1}^{(k)}$ \\ for $k=1,2,...n$ in $x_{2}>\\beta $ and we\narrange them in such a way that \\TEXTsymbol{\\vert}$\\omega _{1}^{(1)}\n\\TEXTsymbol{\\vert}$\\leq |\\omega _{1}^{(2)}|\\leq |\\omega _{1}^{(3)}|.....\nwhere \\TEXTsymbol{\\vert} $\\omega _{1}^{(k)}|\\rightarrow \\infty $ as \nk\\rightarrow \\infty $. We then consider a sequence of contours $\\Gamma _{k}$\nconsisting of the segments $[-$ $x_{1}^{(k)}+i_{1}x_{2},$ \nx_{1}^{(k)}+i_{1}x_{2}]$ and the semicircles $C_{k}$ of radii r$_{k}$ = \n\\TEXTsymbol{\\vert}$\\omega _{1}^{(k)}|>\\beta $ enclosing the first k poles \n\\omega _{1}^{(1)}$, $\\omega _{1}^{(2)},\\omega _{1}^{(3)},.......\\omega\n_{1}^{(k)}$ (See Figure 3 in Appendix).Then by Cauchy's residue theorem we\nget that\n\n\\begin{equation*}\n2\\pi i_{1}\\sum_{Im(\\omega_{1}^{(k)})>0}\\func{Re}\\text{s}\\{\\exp\n(-i_{1}\\omega_{1}t)\\widehat{f_{1}}(\\omega_{1}):\\omega_{1}=\\omega_{1}^{(k)}\\}\n\\end{equation*}\n\n\\begin{equation*}\n=\\int_{\\Gamma_{R}}\\exp(-i_{1}\\omega_{1}t)\\widehat{f_{1}}(\\omega_{1})\n\\omega_{1}\n\\end{equation*}\n\n\\begin{equation*}\n=\\int_{C_{R}}\\exp(-i_{1}\\omega_{1}t)\\widehat{f_{1}}(\\omega_{1})d\\omega_{1}\n\\end{equation*}\n\n\\begin{equation}\n+\\tint\\limits_{-x_{1}^{(k)}+i_{1}x_{2}}^{x_{1}^{(k)}+i_{1}x_{2}}\\exp\n(-i_{1}\\omega _{1}t)\\widehat{f_{1}}(\\omega _{1})d\\omega _{1}. \\label{6.13}\n\\end{equation\nNow for t \\TEXTsymbol{<} 0, in the procedure similar to Case I, employing\nJordan lemma here also we may deduce tha\n\\begin{equation*}\n\\lim_{|\\omega _{1}^{(k)}|\\longrightarrow \\infty }\\int_{C_{R}}\\exp\n(-i_{1}\\omega _{1}t)\\widehat{f_{1}}(\\omega _{1})d\\omega _{1}=0.\n\\end{equation*}\n\nHence in the limit \\ $|\\omega _{1}^{(k)}|\\longrightarrow \\infty $ ( which\nimplies that $|x_{1}^{(k)}|\\longrightarrow \\infty )$ , ($\\ref{6.13})$ leads\nt\n\\begin{align*}\n& \\tint\\limits_{-\\infty +i_{1}x_{2}}^{\\infty +i_{1}x_{2}}\\exp (-i_{1}\\omega\n_{1}t)\\widehat{f_{1}}(\\omega _{1})d\\omega _{1} \\\\\n& =2\\pi i_{1}\\sum_{Im(\\omega _{1}^{(k)})>0}\\func{Re}\\text{s}\\{\\exp\n(-i_{1}\\omega _{1}t)\\widehat{f_{1}}(\\omega _{1}):\\omega _{1}=\\omega\n_{1}^{(k)}\\}\\text{ for }t<0\n\\end{align*\nand as its consequenc\n\\begin{equation*}\n\\text{ }f(t)=i_{1}\\sum_{Im(\\omega _{1}^{(k)})>0}\\func{Re}\\text{s}\\{\\exp\n(-i_{1}\\omega _{1}t)\\widehat{f_{1}}(\\omega _{1}):\\omega _{1}=\\omega\n_{1}^{(k)}\\}\\text{ for }t<0.\n\\end{equation*\nThus for $x_{2}\\geq 0$\\ , whatever the number of poles is finite or\ninfinite, from the above two cases we obtain the complex version of Fourier\ninverse transform as \n\\begin{equation}\n\\text{ }f(t)=i_{1}\\sum_{Im(\\omega _{1}^{(k)})>0}\\func{Re}\\text{s}\\{\\exp\n(-i_{1}\\omega _{1}t)\\widehat{f_{1}}(\\omega _{1}):\\omega _{1}=\\omega\n_{1}^{(k)}\\}\\text{ for }t<0\\text{.} \\label{6.14}\n\\end{equation\nWe now consider the Case Im\\textbf{\\ (}$\\omega _{1}$\\textbf{) = }$x_{2}\\leq\n0 $. The complex valued function $\\widehat{f_{1}}(\\omega _{1})$ is\ncontinuous for $x_{2}\\leq 0$ and holomorphic in $-\\alpha \\alpha $\nwith centre at the origin, we see that all possible singularities (if\nexists) of $\\widehat{f_{1}}(\\omega _{1})$ must lie in the region below the\nhorizontal line $x_{2}=-\\alpha $ . If $\\overline{\\omega }_{1}^{(k)}$ for \nk=1,2...$ are the poles in $x_{2}<\\alpha $, whatever the number of poles are\nfinite or not for $R^{\\prime }\\rightarrow \\infty $, in similar to the\nprevious consideration for $x_{2}\\geq 0$ we see that for t \\TEXTsymbol{>} 0\nthe conditions for Jordan lemma are met and s\n\\begin{equation}\n\\text{ }f(t)=-i_{1}\\sum_{Im(\\omega _{1}^{(k)})<0}\\func{Re}\\text{s}\\{\\exp\n(-i_{1}\\omega _{1}t)\\widehat{f_{1}}(\\omega _{1}):\\omega _{1}=\\overline\n\\omega }_{1}^{(k)}\\}\\text{ for }t>0\\text{.} \\label{6.15}\n\\end{equation\nWe then assign the value of f(t) at t = 0 fulfilling the requirement of\ncontinuity of it in $-\\infty 0}\\func{Re}\\text{s}\\{\\exp\n(-i_{1}\\omega _{2}t)\\widehat{f_{2}}(\\omega _{2}):\\omega _{2}=\\omega\n_{2}^{(k)}\\}\\text{ for }t<0 \\notag \\\\\n& =-i_{1}\\sum_{Im(\\omega _{2}^{(k)})<0}\\func{Re}\\text{s}\\{\\exp (-i_{1}\\omega\n_{2}t)\\widehat{f_{2}}(\\omega _{2}):\\omega _{2}=\\omega _{2}^{(k)}\\}\\text{ for \n}t>0 \\label{6.17}\n\\end{align\nand the value of f(t) at t = 0 can be assigned fulfilling the requirement of\ncontinuity of it in $-\\infty 0}\\func{Re}\\text{s\n\\{\\exp(-i_{1}\\omega_{1}t)\\widehat{f_{1}}(\\omega\n_{1}):\\omega_{1}=\\omega_{1}^{(k)}\\}\n\\end{equation*}\n\n\\bigski\n\\begin{equation}\n+i_{1}e_{2}\\tsum \\limits_{Im(\\omega_{2}^{(k)})>0}\\func{Re}\\text{s\n\\{\\exp(-i_{1}\\omega_{2}t)\\widehat{f_{2}}(\\omega\n_{2}):\\omega_{2}=\\omega_{2}^{(k)}\\}\\text{ for }t<0 \\label{6.21}\n\\end{equation}\n\nand\n\n\\begin{equation*}\nf(t)=-i_{1}e_{1}\\sum_{Im(\\omega_{1}^{(k)})<0}\\func{Re}\\text{s\n\\{\\exp(-i_{1}\\omega_{1}t)\\widehat{f_{1}}(\\omega_{1}):\\omega_{1}=\\overline \n\\omega}_{1}^{(k)}\\}\n\\end{equation*}\n\n\\begin{equation}\n-i_{1}e_{2}\\sum_{Im(\\omega_{2}^{(k)})<0}\\func{Re}\\text{s}\\{\\exp\n(-i_{1}\\omega_{2}t)\\widehat{f_{2}}(\\omega_{2}):\\omega_{2}=\\omega_{2}^{(k)}\\\n\\text{ for }t>0\\text{.} \\label{6.22}\n\\end{equation}\nWe assign the value of $f(t)$ at $t=0$ fulfilling the requirement of\ncontinuity of it in the whole real line $(-\\infty}0\\ \\textbf{\\ }the\n\\begin{equation*}\n\\widehat{f_{1}}(\\omega_{1})=\\frac{2a}{a^{2}+\\omega_{1}^{2}},\n\\end{equation*\n\\begin{equation*}\n\\mathbf{\\ \\ }\\widehat{f_{2}}(\\omega_{2})=\\frac{2a}{a^{2}+\\omega_{2}^{2}}\n\\end{equation*}\n\\end{example}\n\nand in each of $\\omega _{1}$ and $\\omega _{2}$ planes the poles are simple\nat $\\ i_{1}a$ and $i_{1}a$ . Now employing $\\ref{6.21}$ and $\\ref{6.22},$\nfor t \\TEXTsymbol{<} 0 we\\ obtain that \n\\begin{equation*}\nf(t)=i_{1}e_{1}\\func{Re}\\text{s}\\{\\exp (-i_{1}\\omega _{1}t)\\frac{2a}\na^{2}+\\omega _{1}^{2}}:\\omega _{1}=i_{1}a\\}\n\\end{equation*}\n\n\\begin{equation*}\n+i_{1}e_{2}\\func{Re}\\text{s}\\{\\exp(-i_{1}\\omega_{2}t)\\frac{2a}\na^{2}+\\omega_{2}^{2}}:\\omega_{2}=i_{1}a\\}\n\\end{equation*}\n\n\\begin{equation*}\n=i_{1}e_{1}\\{-i_{1}\\exp(at)\\}+i_{1}e_{2}\\{-i_{1}\\exp(at)\\}\n\\end{equation*}\n\n\\begin{equation*}\n=\\exp(-a|t|)\n\\end{equation*}\n\nand for \\ t \\TEXTsymbol{>}0\n\\begin{equation*}\nf(t)=-i_{1}e_{1}\\func{Re}\\text{s}\\{\\exp(-i_{1}\\omega_{1}t)\\frac {2a}\na^{2}+\\omega_{1}^{2}}:\\omega_{1}=i_{1}a\\}\n\\end{equation*}\n\n\\begin{equation*}\n-i_{1}e_{2}\\func{Re}\\text{s}\\{\\exp(-i_{1}\\omega_{2}t)\\frac{2a}\na^{2}+\\omega_{2}^{2}}:\\omega_{2}=i_{1}a\\}\n\\end{equation*}\n\n\\begin{equation*}\n=-i_{1}e_{1}\\{i_{1}\\exp(-at)\\}-i_{1}e_{2}\\{i_{1}\\exp(at)\\}\n\\end{equation*}\n\n\\begin{equation*}\n=\\exp(-a|t|).\n\\end{equation*}\nNow for the continuity of t in \\ the real line, we find $f(0)=1$. Thus the\nFourier inverse transform of $\\widehat{f}(\\omega)$ is $f(t)=\\exp(-a|t|)$.\n\n\\begin{example}\n\\textbf{2. }If \n\\begin{equation*}\n\\widehat{f}(\\omega)=\\frac{1}{2}[\\frac{1}{\\omega+\\omega_{0}+\\frac{i_{1}}{T}}\n\\frac{1}{\\omega-\\omega_{0}+\\frac{i_{1}}{T}}]\\text{ for }T,\\omega_{0}>0\n\\end{equation*}\n\\end{example}\n\nthen in each of $\\omega _{1}$ and $\\omega _{2}$\\ plane the poles are at (\n\\omega _{0}-\\frac{i_{1}}{T})$ and ($-\\omega _{0}-\\frac{i_{1}}{T})$\\ . For\nboth the poles the imaginary components are negative and so the poles are in\nlower half of both the planes. In otherwords, no poles exist in upper half\nof $\\omega _{1}$ or $\\omega _{2}$\\ planes and as its consequence $f(t)=0$\nfor t \\TEXTsymbol{<} 0. Now at t \\TEXTsymbol{>} 0\n\\begin{equation*}\nf(t)=-i_{1}e_{1}\\func{Re}\\text{s}\\{\\exp (-i_{1}\\omega _{1}t)\\widehat{f_{1}\n(\\omega _{1}):\\omega _{1}=-\\omega _{0}-\\frac{i_{1}}{T}\\}\n\\end{equation*}\n\n\\begin{equation*}\n-i_{1}e_{1}\\func{Re}\\text{s}\\{\\exp(-i_{1}\\omega_{1}t)\\widehat{f_{1}\n(\\omega_{1}):\\omega_{1}=\\omega_{0}-\\frac{i_{1}}{T}\\}\n\\end{equation*}\n\n\\begin{equation*}\n-i_{1}e_{2}\\func{Re}\\text{s}\\{\\exp(-i_{1}\\omega_{2}t)\\widehat{f_{2}\n(\\omega_{2}):\\omega_{2}=-\\omega_{0}-\\frac{i_{1}}{T}\\}\n\\end{equation*}\n\n\\begin{equation*}\n-i_{1}e_{2}\\func{Re}\\text{s}\\{\\exp(-i_{1}\\omega_{2}t)\\widehat{f_{2}\n(\\omega_{2}):\\omega_{2}=\\omega_{0}-\\frac{i_{1}}{T}\\}\n\\end{equation*}\n\n\\begin{align*}\n& =-i_{1}e_{1}\\frac{1}{2}\\exp (-\\frac{t}{T})\\exp (i_{1}\\omega _{0}t) \\\\\n& +i_{1}e_{1}\\frac{1}{2}\\exp (-\\frac{t}{T})\\exp (-i_{1}\\omega _{0}t) \\\\\n& -i_{1}e_{2}\\frac{1}{2}\\exp (-\\frac{t}{T})\\exp (i_{1}\\omega _{0}t) \\\\\n& +i_{1}e_{2}\\frac{1}{2}\\exp (-\\frac{t}{T})\\exp (-i_{1}\\omega _{0}t) \\\\\n& =-i_{1}\\frac{1}{2}\\exp (-\\frac{t}{T})\\exp (i_{1}\\omega _{0}t) \\\\\n& +i_{1}\\frac{1}{2}\\exp (-\\frac{t}{T})\\exp (-i_{1}\\omega _{0}t) \\\\\n& =\\exp (-\\frac{t}{T})\\sin (\\omega _{0}t).\n\\end{align*\nFinally, the continuity of $f(t)$ in the whole real line implies that \nf(0)=0 $.\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Introduction}\n\n\n\nOne of the most basic methods of transforming an old 3-manifold\ninto a new 3-manifold is by performing surgery along a knot in\nthe old manifold. Given a 3-manifold $Y$ and a knot $K \\subset Y$\n(which is just an embedding of $S^1$ into $Y$), one performs\nDehn surgery by cutting out a solid torus neigborhood\n$n_K \\subset Y$ of $K$, performing a Dehn twist on $n_K$,\nand then gluing this Dehn-twisted version of $n_K$\nback into $Y \\setminus n_K$ to form a new 3-manifold, $Y^{\\prime}$.\nOne can perform a similar procedure with a link in $Y$ (an embedding\nof a disjoint union of $S^1$'s into $Y$), but this is equivalent\nto the iterated procedure of performing surgery along each knot\nthat forms a component of the link.\n\nAny 3-manifold can be obtained via integer Dehn surgery on a link in $S^3$.\nMuch is still unknown, however, about the problem of classifying\nnon-hyperbolic integer surgeries on knots in $S^3$.\nOne of the early lines of progress towards answering this question was\ninitiated by Berge, who constructed a list of knots in $S^3$ admitting\ninteger lens space surgeries \\cite{Berge},\na list which Gordon later conjectured was exhaustive.\n\nReversing our point of view, we could equivalently ask which knots,\nin which lens spaces, have integer $S^3$ surgeries. This change of perspective\nsends Berge's knots in $S^3$ to the knot cores of their associated lens space fillings.\nIt turns out that all of these resulting knots are {\\em simple}, where a\nsimple knot in a lens space $L(p,q)$\nis a knot obtained by placing two basepoints inside the standard\ngenus one Heegaard diagram of $L(p,q)$.\nSince there is a unique simple\nknot in each homology class $k \\in H_1(L(p,q)) \\cong {{\\mathbb Z}}\/p$,\nany simple knot in a lens space can be described by a triple\n$(p, q, k)$ with $k \\in {{\\mathbb Z}}\/p$.\n\n\n\nWhile the conjectured simplicity of all lens space knots admitting integer $S^3$\nsurgeries is a key part of the Berge-Gordon conjecture,\nand a difficult question of ongoing interest (see, {\\em e.g.}, \\cite{BGH}),\nwe shall not address it in this paper.\nInstead, we focus on classifying which simple knots in lens spaces\nadmit integer $S^3$ surgeries. From this standpoint, the conjecture of the\nBerge and Gordon could be phrased as follows.\n\\begin{conj}[Berge, Gordon]\n\\label{conj: berge}\nIf a knot $K$ in a lens space $L(p,q)$ is simple,\nhence parameterized by the triple $(p,q,k)$, then\n$K$ admits an integer $S^3$ surgery if and only if\n$q \\equiv k^2 (\\mod p)$\nand one or more of\nthe following is true:\n\n{\\noindent{Solutions when $p > k^2$:}}\n\\begin{align*}\n\\text{ I and II:}\\;\\;\n&\n\\begin{cases}\n p \\equiv ik \\pm 1 \\; (\\mod k^2),\n & \\gcd (i,k) = 1, 2\n\\end{cases}\n \\\\\n\\text{ III:}\\;\\;\n&\n\\begin{cases}\n p \\equiv \\pm d(2k+1) \\; (\\mod k^2),\n & d \\vert k-1,\\; 2\\!\\! \\not\\vert\\, \\frac{k-1}{d}\n \\\\\n p \\equiv \\pm d(2k-1) \\; (\\mod k^2),\n & d \\vert k+1,\\; 2\\!\\! \\not\\vert\\, \\frac{k+1}{d}\n\\end{cases}\n \\\\\n\\text{ IV:}\\;\\;\n&\n\\begin{cases}\n p \\equiv \\pm d(k+1) \\; (\\mod k^2),\n & d \\vert 2k-1,\\; 2\\!\\! \\not\\vert\\, \\frac{2k-1}{d}\n \\\\\n p \\equiv \\pm d(k-1) \\; (\\mod k^2),\n & d \\vert 2k+1,\\; 2\\!\\! \\not\\vert\\, \\frac{2k+1}{d}\n\\end{cases}\n \\\\\n\\text{ V:}\\;\\;\n&\n\\begin{cases}\n p \\equiv \\pm d(k+1) \\; (\\mod k^2),\n & d \\vert k+1,\\; 2\\!\\! \\not\\vert\\, d\n \\\\\n p \\equiv \\pm d(k-1) \\; (\\mod k^2),\n & d \\vert k-1,\\; 2\\!\\! \\not\\vert\\, d\n\\end{cases}\n \\\\\n\\text{ VI:}\\;\\;\n&\n\\begin{cases}\n \\text{Special case of V}.\n\\end{cases}\n\\end{align*}\n\n{\\noindent{Solutions when $p < k^2$:}}\n\\begin{align*}\n\\text{ VII and VIII:}\n&\n\\begin{cases}\n k^2\\pm k \\pm 1 \\equiv 0 \\; (\\mod p)\n\\end{cases}\n \\\\\n\\text{ IX, X, XI, and XII:}\n&\n\\begin{cases}\np = \\frac{1}{11}(2k^2 + k + 1).\n\\end{cases}\n\\end{align*}\n\\end{conj}\n\nIn \\cite{Rasmussen}\nJacob Rasmussen analyzed the above conjecture by studying\nthe Heegaard Floer homology of knots in lens spaces\nwith L-space homology-sphere surgeries. An L-space\nis a 3-manifold whose Heegaard Floer homology\nhas the smallest possible rank, in a certain precise sense.\nA homology sphere has the same ordinary homology as a sphere.\nThe only known 3-manifolds satisfying both of these properties\nare the Poincar{\\'e} sphere and $S^3$.\n\n\nRasmussen showed in \\cite{Rasmussen}\nthat if $K \\subset L(p,q)$\nadmits an L-space surgery, then the unique simple knot $K^{\\prime}$\nin the same homology class as $K$ satisfies \n\\begin{equation}\n\\label{genus bound}\n\\mathrm{genus}(K^{\\prime}) < \\frac{p+1}{2}.\n\\end{equation}\n\nOn the other hand, the genus of a simple knot is easily calculated\n\\cite{Rasmussen},\n\\cite{OSLens}.\nThe standard doubly-pointed Heegaard diagram for a simple knot\nprovides a simple presentation for the fundamental group of the knot,\nfrom which one can compute its Alexander polynomial using Fox calculus.\nThe simple knot $K^{\\prime} \\subset L(p,q)$ \nof homology class $k \\in H_1(L(p,q)) \\cong {{\\mathbb Z}}\/p$\nhas Alexander polynomial\n\\begin{equation}\n {\\Delta}_{K^{\\prime}}\n= \\left(\\frac{t - 1}{t^p - 1}\\right) {\\bar{\\Delta}}_{K^{\\prime}},\n\\end{equation}\nwhere\n\\begin{equation}\n{\\bar{\\Delta}}_{K^{\\prime}} = \\sum_{i=0}^{p-1} t^{f(i)},\n\\end{equation}\nand $f(i)$ is defined recursively by\n\\begin{equation}\n f(i+1) - f(i) \n:= \\begin{cases}\n k-p\n & iq \\in \\{0, \\ldots, k-1\\} \\subset {{\\mathbb Z}}\/p\n \\\\\n k\n & \\text{otherwise}\n \\end{cases}\\;\\;.\n\\end{equation}\nThe genus of $K^{\\prime}$ is then given by\n\\begin{equation}\n \\mathrm{genus}(K^{\\prime})\n= \\frac{\\mathrm{deg}{\\bar{\\Delta}}_{K^{\\prime}} - p + 1}{2},\n\\end{equation}\nwhere\n\\begin{equation}\n \\mathrm{deg}{\\bar{\\Delta}}_{K^{\\prime}}\n= \\max_{i \\in {{\\mathbb Z}}\/p} f(i) - \\min_{i \\in {{\\mathbb Z}}\/p} f(i).\n\\end{equation}\nLetting $G(p,q,k)$ denote this degree $\\mathrm{deg}{\\bar{\\Delta}}_{K^{\\prime}}$,\none can then translate the condition (\\ref{genus bound}) on the genus of\n$K^{\\prime}$ into a condition on $p$, $q$, and $k$:\n\\begin{equation}\n\\mathrm{genus}(K^{\\prime}) < \\frac{p+1}{2}\n\\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;\nG(p,q,k) < 2p.\n\\end{equation}\n\nOn the other hand, a knot determined by\n$p$, $q$, $k$ only admits a homology-sphere surgery\nif $q \\equiv k^2\\;(\\mod p)$.\nThus, the knot determined by $p$, $q$, and $k$\nadmits an L-space homology-sphere surgery if and only if\nthe following two conditions hold:\n\\begin{itemize}\n\\item[(i)]$G(p,q,k) < 2p$,\n\\item[(ii)]$q \\equiv k^2\\;(\\mod p)$.\n\\end{itemize}\nThis translates the topological problem of classifying which\nsimple knots in $L(p,q)$ admitting $S^3$ surgeries into\nthe number theoretical and combinatorial problem on which\nthe present paper focuses.\nIn this paper, we classify all solutions $(p,q,k)$\nof conditions (i) and (ii)\nin the case of $p > k^2$, and observe that they\nmatch with Berge's prediction. That is, we prove:\n\\begin{theorem}\n\\label{thm: intro main thm}\nConjecture \\ref{conj: berge} holds in the case of $p > k^2$.\n\\end{theorem}\n\nIn the case of $p < k^2$, however, the above strategy generates\nnearly 20 different families of lens space knots with Poincar{\\'e} sphere\nsurgeries, making our methods less tractable.\n\nOn the other hand,\nthe case of $p > k^2$ is interesting in its own right.\nFor example,\none way to construct a knot in a lens space with an\n$S^3$ surgery is to start with a knot $K^{\\prime} \\in S^1 \\times D^2$\nwith a nontrivial $S^1 \\times D^2$ surgery.\nIf $\\alpha \\in H_1\\!\\left( \\partial\\!\\left(S^1 \\times D^2\\right)\\right)$\nis the class which bounds in the surgery,\nthen any Dehn filling of $S^1 \\times D^2$\nalong a curve $\\beta$ with $\\beta \\cdot \\alpha = \\pm 1$\ngives a knot in a lens space with an $S^3$ surgery.\nIn \\cite{GabaiTorus} and \\cite{BergeTorus},\nGabai and Berge\nclassified all knots in $S^1 \\times D^2$ with\nnontrivial $S^1 \\times D^2$ surgeries.\nAs observed by Berge in \\cite{Berge},\nthe knots obtained by this construction are precisely \nthe Berge knots of types I through VI\nlisted in Conjecture \\ref{conj: berge}.\nThus, Theorem \\ref{thm: intro main thm} implies the\nthe following result.\n\n\\begin{cor}\nIf $K \\subset L(p,q)$ has an integral $S^3$ surgery, and $p > k^2$, where\n$k \\in H_1(L(p,q)) \\cong {{\\mathbb Z}}\/p$ is the homology class of $K$,\nthen the unique simple knot in the same homology class\nis obtained as described above from a knot $K^{\\prime} \\in S^1 \\times D^2$ which\nadmits a nontrivial $S^1 \\times D^2$ surgery.\n\\end{cor}\n\nIn Section \\ref{s:defs},\nwe define an invariant $\\bar{G}\\left(p,q ,k\\right)$ of\n$(p,q,k)$ which is easier to work with, satisfying\n\\begin{equation}\nG\\left(p,q,k\\right) = \\bar{G}\\left(p,q^{-1}(\\mod p),k\\right).\n\\end{equation}\nWe also state a few simple properties of $\\bar{G}\\left(p,q ,k\\right)$.\n\nThe $G$--triple $(p,q,k)$ then satisfies $q = k^2 \\in {{\\mathbb Z}}\/p$\nif and only if the corresponding $\\bar{G}$--triple satisfies\n$q = k^{-2} \\in {{\\mathbb Z}}\/p$.\nIn Proposition \\ref{prop:section q=k^-2, (p,k^-2,k) is like (k^2, p^-1, k)}\nof Section \\ref{s:q=k-2}, we prove that\nif $p > k^2$, then the $\\bar{G}$--triple $(p, k^{-2}, k)$ \nis a solution if and only if\nthe $\\bar{G}$--triple $(k^2, p^{-1}(\\mod k^2), k)$ is solution,\nthereby reducing our classification problem to the study of\n$\\bar{G}$--triples of the form $(k^2, q, k)$.\n\nThus, the bulk of the work in this paper resides in\nSection \\ref{s:p=k2}, which classifies $\\bar{G}$--triples of the form\n$(k^2, q, k)$ satisfying $\\bar{G}(k^2,q,k) < 2p$.\nThe calculation of $\\bar{G}$ requires the bookkeeping of\ncertain marked elements of ${{\\mathbb Z}}\/p$. The main strategy of\nSection \\ref{s:p=k2} is to use a special choice (\\ref{eq: definition of z_i^j})\nof ordering of these marked elements, in order to exploit some\nuseful combinatorial properties of the problem at hand.\n \\\\\n\n\n\\noindent{\\bf Acknowledgements:}\n\nI would like to thank Jacob Rasmussen for introducing me to this problem,\nCliff Taubes for advising me from Harvard, Zoltan Szab{\\'o} for advising\nme from Princeton, Mike Hopkins for advising my minor thesis,\nand Richard Taylor for his endless academic and moral support\nas director of graduate studies.\nI am very grateful to my readers, Cliff Taubes,\nPeter Kronheimer, and Noam Elkies,\nand to the extremely patient referee who volunteered to check my paper.\n\nI would also like to thank Irene Minder---for\nrepeatedly going above and\nbeyond the call of duty to carry out my administrative obligations for me\nin my absence---and Susan Gilbert for all her work in the past year\nto help me meet graduation requirements in absentia.\n\n\nMany people have supported me during my graduate school years.\nAt Princeton, Zoltan's students took me in as one of their own.\nIn my year between Harvard and Princeton, the Rutgers string theorists\ngave me an office and let me participate in their seminars and discussions.\nAt Harvard, the students in my alcove, in addition to ``honorary'' alcove members,\nwere quick to include me in social gatherings. In my year at the physics\ndepartment at Stanford, Simeon Hellerman was generous about\nanswering questions and suggesting papers to read, while\nJacob Shapiro, Kevin Purbhoo, and Tom Coates helped me to retain my\nidentity as a mathematician.\nIn the mean time, Catherine and Christian Le Cocq\nstood in as surrogate parents and kept me well fed.\nOf course, I am also grateful\nto the National Defense Science and Engineering Fellowship and the\nNational Science Foundation's Graduate Research Fellowship for\nfinancing my graduate study.\n\n\nAs an undergraduate, I greatly benefited from interactions with the \nmathematicians and string theorists at Duke. From the mathematics\ndepartment, I would especially like to thank Chad Schoen,\nRobert Bryant, David Morrison, Bill Pardon, David Kraines, Eric Sharpe, and my advisor,\nPaul Aspinwall. I would also like to thank my dear friends\nIlarion Melnikov, Sven Rinke, and Ronen Plesser from the physics department.\n\nFrom my precollegiate years of education, I would like to thank\nJohn Kolena, John Goebel, Dan Teague, and Kevin Bartkovich\nfrom the NC School of Science and Mathematics,\nKen Collins, Rudine Marlowe, Jeff Knull, and Caroline Huggins\nfrom Charlotte Latin School,\nBill Cross and Dana Mackenzie from Duke T.I.P.,\nand Harold Reiter from the Charlotte Math Club.\n\n\nLast but not least, I want to thank my family.\nMy parents and brother nurtured my love of\nmental challenge from an early age,\nand have unfailingly supported me through the years.\nMore recently, my little daughter Katie\nhas reminded me of how much fun can be found in\nexploration and discovery.\nMost of all, I am indebted to my husband, Jake.\nHe took care of me during my long recovery from catching fire\nwhile cooking (oops), and helped to clean up after me during five\nmonths of constant morning sickness, not to mention\nall the cooking, cleaning, dishwashing, and\n(more recently) baby care he has always done without being asked.\nI cannot express how much his love, support, \nand patience have sustained me these last years.\n \n \n\n\\newpage\n\n\n\\section{General Case: Definitions and Basic Properties}\n\\label{s:defs}\n\nIn the following, we define an invariant $\\bar{G}$ related to\nthe invariant $G$---defined in the Introduction---associated\nto simple knots in lens spaces. We also provide a minor shortcut\nfor calculating $\\bar{G}$ and observe some of its basic symmetries.\n\n\n\n\\begin{definition}\n\\label{def:v}\nSuppose $p \\in {{\\mathbb Z}}$ with $p \\geq 2$, and $k, q \\in {{\\mathbb Z}}\/p$ with $k \\neq 0$\nand $q$ primitive. \nThen the triple $(p,q,k)$ determines a map\n$v_{(p,q,k)} : {{\\mathbb Z}}\/p \\times {{\\mathbb Z}}\/p \\rightarrow {{\\mathbb Z}}$, given by\n\\begin{equation*}\nv_{(p,q,k)}(x,y) := \\#\\!\\left({{\\mathbb Z}}\\cap \\left\\langle\\tilde{x},\\tilde{y}\\right]\\right)\\left[k\\right]_p \\;\\; - \\;\\;\n \\#\\!\\left(\\tilde{Q}\\cap \\left\\langle\\tilde{x},\\tilde{y}\\right]\\right)p,\n\\end{equation*} \nwhere $\\left[k\\right]_p$ denotes the representative of $k$ in $\\{0, \\ldots, p-1\\}$,\n$\\tilde{Q} := {\\pi}^{-1}(Q)$ is the preimage under \n${{\\mathbb Z}} \\stackrel{\\pi}{\\to} {{\\mathbb Z}}\/p$ of\n\\begin{equation*}\nQ := \\left\\{aq\\in{{\\mathbb Z}}\/p \\left|\\; a \\in \\{0, \\ldots, \\left[k\\right]_p-1\\} \\right. \\right\\},\n\\end{equation*}\nand $\\tilde{x}, \\tilde{y} \\in {{\\mathbb Z}}$ are any $\\pi$-lifts of $x$ and $y$ \nsatisfying $x < y$.\n\\end{definition}\n\n\n\nSince the difference between any two lifts of $(x,y)$ contributes a multiple of\n$p [k]_p - [k]_p p = 0$ to $v_{(p,q,k)}(x,y)$, this definition is independent\nof the choice of lift of $(x,y)$. Note that $v_{(p,q,k)}$ is antisymmetric:\n\\begin{equation}\nv_{(p,q,k)}(x,y) = -v_{(p,q,k)}(y,x).\n\\end{equation}\n\n\\begin{definition}\nThe triple $(p,q,k)$ determines a positive integer $\\bar{G}$, given by\n\\begin{equation*}\n\\bar{G}(p,q,k) := \\max_{x,y \\in {{\\mathbb Z}}\/p} v_{(p,q,k)}(x,y).\n\\end{equation*}\n\\end{definition}\n\\noindent A little thought shows that $\\bar{G}$ is related to the invariant\n$G$ defined in the Introduction by\n\\begin{equation}\n\\bar{G}(p,q,k) = G(p,q^{-1},k),\n\\end{equation}\nso that $\\bar{G}\\left(p,q,k\\right)$ gives the degree of the rescaled\nAlexander polynomial of the simple knot in $L(p, q^{-1}(\\mod p))$\nof homology class $k \\in H_1\\!\\left(L(p, q^{-1})\\right) \\cong {{\\mathbb Z}}\/p$.\nThus, misleadingly, the $q$ used in most of this paper is {\\em not}\nthe $\\tilde{q}$ defining the lens space $L(p,\\tilde{q})$ harboring\nthe simple knot, but is rather the inverse modulo $p$ of that $\\tilde{q}$.\nIf confusion is likely to occur, we shall distinguish between the\narguments of $\\bar{G}$ and $G$ by calling them $\\bar{G}$--triples\nand $G$--triples, respectively.\n\n\nThe following proposition somewhat simplifies the computation of $\\bar{G}$:\n\\begin{prop}\n\\label{prop:G from aqs}\n\\begin{equation*}\n\\bar{G}(p,q,k) =\n \\max_{x,y \\in Q} \\left|v_{(p,q,k)}(x,y)\\right|\n + p - [k]_p.\n\\end{equation*}\n\\end{prop}\n\n\\begin{proof}\nFor brevity, write $v$ for $v_{(p,q,k)}$.\nFirst, note that since $v(x,y) = -v(y,x)$, the above proposition would be\nequivalent if we removed the absolute value sign, but the absolute value sign\nwill be convenient in later arguments.\n\nLet $(x_*,y_*)$ denote an element of ${{\\mathbb Z}}\/p \\times {{\\mathbb Z}}\/p$ at which a maximum of\n$v$ (and thus also of $|v|$) occurs.\nThere must then exist $a_1\\in\\{0,\\ldots,[k]_p-1\\}$ such that\n$x_* \\equiv {a_1}q\\; (\\mod p)$. Otherwise $v(x_* -1,y_*) = v(x_*,y_*) + [k]_p$,\ncontradicting the maximality of $v(x_*,y_*)$.\nSimilarly, there must exist $a_2\\in\\{0,\\ldots,[k]_p-1\\}$ such that\n$y_* \\equiv {a_2}q - 1\\; (\\mod p)$. Otherwise, \n$v(x_*,y_*+1) = v(x_*,y_*) + [k]_p$. Thus,\n\\begin{equation}\n v(x_*,y_*) = v({a_1}q,{a_2}q - 1) = v({a_1}q, {a_2}q) + p - [k]_p.\n\\end{equation}\n\\end{proof}\n\n\n\\begin{definition}\nWe say that a triple $(p,q ,k)$ is \n{\\em genus-minimizing} if \n\\begin{equation*}\n\\bar{G}(p,q,k) < 2p.\n\\end{equation*}\n\\end{definition}\n\\noindent The choice of the term ``genus-minimizing'' is intended to reflect the\nfact that $\\bar{G}(p,q,k) < 2p$ if and only if the knot\ndetermined by the $\\bar{G}$--triple $(p,q,k)$ satisfies the\ngenus bound necessary for the knot to admit an $S^3$ surgery.\n\nWhen actually trying to determine if $(p, q, k)$ is genus-minimizing, we\noften exploit Proposition \\ref{prop:G from aqs} to use the following\nsimpler criterion.\n\\begin{cor}\n\\label{cor: intro defs, genus-minimizing if v < p + k}\nThe triple $(p, q, k)$ is genus-minimizing if and only if\n\\begin{equation*}\n\\max_{x,y \\in Q} \\left|v_{(p,q,k)}(x,y)\\right| < p + [k]_p.\n\\end{equation*}\n\\end{cor}\n\n\n\nLastly, we observe a few basic symmetries of $\\bar{G}$.\n\\begin{prop}\n\\label{prop:G properties}\nThe invariant $\\bar{G}$ obeys the following three identities:\n\\begin{itemize}\n\\item[\\em{(i)}] $\\bar{G}(p,-q,k) = \\bar{G}(p,q,k)$,\n\\item[\\em{(ii)}] $\\bar{G}(p,q,-k) = \\bar{G}(p,q,k)$,\n\\item[\\em{(iii)}] $\\bar{G}(p,q^{-1},qk) = \\bar{G}(p,q,k)$,\n\\end{itemize}\n\\end{prop}\n\\begin{proof}[Proof of (i)]\n\nIt is clear from the original definition of $v$ (Definition \\ref{def:v}) that\n\\begin{equation}\nv_{(p,q,k)}(x,y) = -v_{(p,-q,k)}(-x,-y) = v_{(p,-q,k)}(-y,-x).\n\\end{equation}\nBut $(x,y) \\mapsto (-y,-x)$ is just an involution on ${{\\mathbb Z}}\/p \\times {{\\mathbb Z}}\/p$,\nso $\\bar{G}(p,-q,k) = \\bar{G}(p,q,k)$.\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of (ii)]\nWe begin by remarking on the effect on $v$ of translating $Q$.\nFor any $a_0 \\in {{\\mathbb Z}}\/p$, define $Q_k^0$ and $Q_k^{a_0}$ by\n\\begin{align}\n Q_k^0 \n&:= \\left\\{aq\\in{{\\mathbb Z}}\/p \\left|\\; a \\in \\{0, \\ldots, \\left[k\\right]_p-1\\} \\right. \\right\\},\n \\\\\n Q_k^{a_0}\n&:= \\left\\{aq\\in{{\\mathbb Z}}\/p \\left|\\; a \\in \\{a_0 + 0, \\ldots, a_0+\\left[k\\right]_p-1\\} \\right. \\right\\},\n\\end{align}\nso that $Q_k^{a_0} = {a_0}q + Q_k^0$. This translation of $Q_k^0$ has the\neffect of translating the domain of $v$. That is,\n\\begin{equation}\nv^{Q_k^0}_{(p,q,k)}(x,y) = v^{Q_k^{a_0}}_{(p,q,k)}(x+{a_0}q, y+{a_0}q),\n\\end{equation}\nwhere $v^q$ denotes the result of replacing $Q$ with $q$\nin the definition of $v$.\n\nReturning to the problem at hand, set\n\\begin{equation}\nQ_{p-k}^0 = \\left\\{ aq \\in {{\\mathbb Z}}\/p | a \\in \\{0, \\ldots, p-[k]_p - 1\\}\\right\\}.\n\\end{equation}\nThen ${{\\mathbb Z}}\/p = Q^0_{p-k} \\coprod Q^{p-k}_k$, so if we let\n${\\tilde{Q}}^0_{p-k} := {\\pi}^{-1}\\left(Q^0_{p-k}\\right)$ and \n${\\tilde{Q}}^{p-k}_k := {\\pi}^{-1}\\left(Q^{p-k}_k\\right)$\ndenote the preimages of $Q^0_{p-k}$ and $Q^{p-k}_k$ under \n${{\\mathbb Z}} \\stackrel{\\pi}{\\to} {{\\mathbb Z}}\/p$, then we have\n\\begin{equation}\n{\\tilde{Q}}^0_{p-k} = {{\\mathbb Z}} \\; \\setminus \\; {\\tilde{Q}}^{p-k}_k.\n\\end{equation}\nApplying this relation to the definition of $v$ gives\n\\begin{align}\nv_{(p,q,-k)}(x,y) \n :\\!\\!&= \\#\\left({{\\mathbb Z}}\\cap (\\tilde{x},\\tilde{y}]\\right) (p\\!-\\![k]_p)\\; - \\;\n \\#\\!\\left( {\\tilde{Q}}^0_{p-k} \\cap (\\tilde{x},\\tilde{y}]\\right)p\n \\\\ \\nonumber\n &= \\#\\left({{\\mathbb Z}}\\cap (\\tilde{x},\\tilde{y}]\\right)(p\\!-\\![k]_p) \\; - \\;\n \\left[ \\#\\left( {{\\mathbb Z}} \\cap (\\tilde{x},\\tilde{y}]\\right) \\;-\\;\n \\#\\left( {\\tilde{Q}}^{p-k}_k \\!\\cap (\\tilde{x},\\tilde{y}]\\right) \\right]p\n \\\\ \\nonumber\n &= - \\#\\left({{\\mathbb Z}}\\cap (\\tilde{x},\\tilde{y}]\\right)[k]_p \\; + \\;\n \\#\\!\\left({\\tilde{Q}}^{p-k}_k \\!\\cap (\\tilde{x},\\tilde{y}]\\right)p\n \\\\ \\nonumber\n &= -v_{(p,q,k)}(x-(p\\!-\\!k)q,\\;y-(p\\!-\\!k)q).\n\\end{align}\nThen, since\n\\begin{equation}\n\\max_{x,y \\in {{\\mathbb Z}}\/p} -v_{(p,q,k)}(x-(p\\!-\\!k)q,\\;y-(p\\!-\\!k)q)\n= \\max_{x,y\\in{{\\mathbb Z}}\/p} v_{(p,q,k)}(x,y),\n\\end{equation}\nwe have $\\bar{G}(p,q,-k) = \\bar{G}(p,q,k)$.\n\\end{proof}\n\n\\begin{proof}[Proof of (iii)]\nIn Lemma 2.5 of \\cite{Rasmussen}, Jacob Rasmussen proves that\nunder the identification $L(p,q) \\cong L(p,q^{-1})$\nobtained by exchanging the roles of $\\alpha$ and $\\beta$\nin the Heegaard diagram,\nthe $G$--triples $(p,q,k)$ and $(p,q^{-1}, q^{-1}k)$\nspecify the same simple knot.\n\nThis implies that the $G$--triples\n$(p, q^{-1}, k)$ and $(p, q, qk)$ specify the same knot,\nwhich in turn implies that the $\\bar{G}$--triples\n$(p, q, k)$ and $(p, q^{-1}, qk)$ specify the same knot\n(recalling that $\\bar{G}(p,q,k) = G(p,q^{-1},k)$ for all $p$, $q$, and $k$).\nIn particular, this means that the knots corresponding to the $\\bar{G}$--triples\n$(p, q, k)$ and $(p, q^{-1}, qk)$ have the same Alexander polynomial.\nThus $\\bar{G}(p,q,k) = \\bar{G}(p,q^{-1}, qk)$.\n\\end{proof}\n\n\n\n\nFrom here on, we restrict our attention to special cases \nrelevant to Berge's Conjecture.\nBerge's Conjecture involves the classification of genus-minimizing\n$\\bar{G}$--triples for which $q = k^{-2}$ in ${{\\mathbb Z}}\/p$, and of those,\nwe are interested in the case in which $p > \\left([k]_p\\right)^2$. As shown in\nSection \\ref{s:q=k-2}, this classification problem is equivalent to\nclassifying genus-minimizing $\\bar{G}$--triples of the form $(k^2, q, k)$,\n{\\em i.e.}, with $p = k^2$, where here, we take $k$ to be a positive integer.\nThere is no loss of generality in taking $k$ to be positive, since\nby Proposition \\ref{prop:G properties}, $(k^2, q, -k)$ is genus-minimizing\nif and only if $(k^2, q, k)$ is genus-minimizing. We focus exclusively on\nthis latter case in Section \\ref{s:p=k2}.\n\n\n\n\n\\section{Case $p=k^2$}\n\\label{s:p=k2}\n\nFor the entirety of this section, we take $k$ to be a fixed positive integer\nand set $p = k^2$. Then the fact that we need $k \\in \\{0, \\ldots, p-1\\}$\nimplies that $k < k^2$, so we know that $k \\geq 2$.\nFurthermore, any primitive element $q \\in {{\\mathbb Z}}\/k^2$ defines an entire\ntriple $(k^2, q, k)$, so we shall often abbreviate notation\nand write $v_q$ for $v_{(k^2, q, k)}$ and $\\bar{G}(q)$ for\n$\\bar{G}(k^2, q, k)$. We shall also say that $q$ is (or is not)\ngenus-minimizing to convey that the triple $(k^2, q, k)$\nis (or is not) genus-minimizing. Lastly, we shall write\n$Q_q$ to denote the set \n$Q_q := \\left\\{ aq | a \\in \\{0, \\ldots, k-1\\}\\right\\} \\subset {{\\mathbb Z}}\/k^2$,\nand ${\\tilde{Q}}_q := {\\pi}^{-1}\\left(Q_q\\right) \\subset {{\\mathbb Z}}$ to denote the \npreimage of $Q_q$ under ${{\\mathbb Z}} \\stackrel{\\pi}{\\rightarrow} {{\\mathbb Z}}\/k^2$.\n\nThe goal of this section is to classify all genus-minimizing\nelements $q \\in {{\\mathbb Z}}\/k^2$. To determine if a particular $q$ is genus-minimizing,\nwe shall apply the following criterion:\n\\begin{prop}\n\\label{prop: not minimizing if v >= k(k+1)}\nFor any primitive $q \\in {{\\mathbb Z}}\/k^2$, \n$q$ is genus-minimizing if and only if\n$v_q(x,y) \\leq k(k-1)$ for all $x,y \\in Q_q$, and \n$q$ is not genus-minimizing\nif and only if\nthere exist $x_*, y_* \\in Q_q$ such that $\\left|v_q(x_*, y_*)\\right| \\geq k(k+1)$.\n\\end{prop}\n\\begin{proof}\nCorollary \\ref{cor: intro defs, genus-minimizing if v < p + k} states that a\ntriple $(p, q, k)$ is genus-minimizing if and only if\n\\begin{equation}\n\\max_{x,y \\in Q} \\left|v_{(p,q,k)}(x,y)\\right| < p + k.\n\\end{equation}\nHere, $p+k = k^2 + k$, and so it remains to show that $v_q(x,y) \\neq k^2$ for all $x, y \\in Q_q$.\nThis is true because if there exist $a_1, a_2 \\in \\{0, \\ldots, k-1\\}$ such that\n$\\frac{v_q(a_1 q, a_2 q)}{k} \\equiv 0 \\;(\\mod k)$, then $a_2 q - a_1 q \\equiv 0\\; (\\mod k)$,\nimplying $a_1 = a_2$, so that $v_q(a_1 q, a_2 q) = 0$.\n\n\\end{proof}\n\n\n\n\\subsection{Notation}\nBefore proceeding further, we should establish some modular\narithmetic notation. Most of these notations are completely\nconventional. For $x \\in {{\\mathbb Q}}$, we use ${\\lfloor}x{\\rfloor}$ to\nindicate the greatest integer less than or equal to $x$, and ${\\lceil}x{\\rceil}$\nto indicate the least integer greater than or equal to $x$.\nEqually conventional, for $x,y,N \\in {{\\mathbb Z}}$, is our use of\na vertical bar to indicate divisibility ($x|y$ denotes that\n$x$ divides $y$), and the notation \n$x \\equiv y\\; (\\mod N)$ to indicate that $N|x - y$.\nWhat is less conventional is our choice of notation to pick out\na particular representative of a congruence class modulo $N$.\nFor any $N >0$, we define the map\n\\begin{equation}\n\\left[\\cdot\\right]_N : {{\\mathbb Z}}\/N \\rightarrow \\{0, \\ldots, N-1\\} \\subset {{\\mathbb Z}}\n\\end{equation}\nby setting $\\left[x\\right]_N$ equal to the unique integer in $\\{0, \\dots, N-1\\}$\nsuch that $\\left[x\\right]_N \\equiv x\\; (\\mod N)$. We shall also sometimes \nuse $[\\cdot]_{N}$ to denote the composition of the quotient map\n${{\\mathbb Z}} \\rightarrow {{\\mathbb Z}}\/N$ with the representative selection map \n${{\\mathbb Z}}\/N \\stackrel{[\\cdot]_N}{\\rightarrow} {{\\mathbb Z}}$.\n\n\n\n\n\n\\subsection{Definition of Parameters}\n\\label{ss: definition of d, c, m, mu, gamma, alpha, and types}\nFrom now on, we take $k$ to satisfy $k > 100$,\nto avoid special cases that arise when $k$ is small.\nIt is easy to check by computer\nthat the genus-minimizing solutions for $q$ match those in \nProposition \\ref{prop: bookkeeping for q genus-minimizing}\nwhen $2 \\leq k \\leq 100$.\n\n\nAs will soon become clear in the proof of Proposition \\ref{prop: G = 2k(k-1) and unique}\nand in the definition of mobile points, we shall spend much more\ntime using $[\\pm q^{-1}]_k q$ than $q$ itself.\nOur goal is therefore the following.\nAfter choosing an appropriate\nsign $\\xi \\in \\{\\pm1\\}$, we want to choose integers\n$d \\in \\{ [\\pm q^{-1}]_k \\}$,\n$m, c \\in \\{0, \\ldots, k-1\\}$, and\n$\\alpha, \\gamma, \\mu \\in \\{\\pm1\\}$ to parameterize $\\xi q$, such that \n\\begin{equation}\n\\left[d{\\xi}q\\right]_{k^2} = ({\\mu}m+ {\\gamma}c)k + \\alpha < \\frac{k^2}{2}.\n\\end{equation}\nFor notational convenience, we first\ndefine the function \n${\\sigma}_n : \\{\\pm 1 \\in {{\\mathbb Z}}\/n \\} \\rightarrow \\{\\pm 1 \\in {{\\mathbb Z}} \\}$,\nfor $n\\in{{{\\mathbb Z}}}_{>0}$,\nby ${\\sigma}_n(+1)=+1$ and ${\\sigma}_n(-1)=-1$ when\n$n>2$, by ${\\sigma}_2 \\equiv -1$ when $n=2$, and by\n${\\sigma}_1 \\equiv +1$ when $n=1$. We then parameterize $q$ as follows.\n\n\\begin{definition}\n\\label{def: parameters assigned to q}\nSuppose that $k>100$.\nWe then take the following steps to associate\nthe parameters $d, c, m, \\xi, \\alpha, \\gamma, \\mu \\in {{\\mathbb Z}}$\nto any given primitive $q \\in {{\\mathbb Z}}\/k^2$. Set\n\\begin{align}\n\\label{eq: parameter definitions, d}\nd :=& \\min\\left\\{ \\left[q^{-1}\\right]_k, \\left[-q^{-1}\\right]_k \\right\\},\n \\\\\n\\label{eq: parameter definitions, xi}\n\\xi :=& \\begin{cases}\n +1 & \\left[dq\\right]_{k^2} < \\textstyle{\\frac{k^2}{2}}\n \\\\\n -1 & \\left[dq\\right]_{k^2} > \\textstyle{\\frac{k^2}{2}}\n \\end{cases}\n \\;\\;\\;\\text{(so that}\\;\n \\left[d{\\xi}q\\right]_{k^2} = \\min\\left\\{ \\left[dq\\right]_{k^2}, \\left[-dq\\right]_{k^2} \\right\\}\n \\text{)},\n \\\\\n\\label{eq: parameter definitions, alpha}\n\\alpha :=& {\\sigma}_k(dq)\\xi,\n \\\\\n\\label{eq: parameter definitions, c}\nc :=& \\min\\left\\{ \\left[k^{-1}\\right]_d, \\left[-k^{-1}\\right]_d \\right\\},\n \\\\\n\\label{eq: parameter definitions, gamma}\n\\gamma :=& {\\sigma}_d(-ck)\\alpha,\n \\\\\n\\label{eq: parameter definitions, mu}\n\\mu :=& \\begin{cases}\n +1 & m^{\\prime} \\geq 0\n \\\\\n -1 & m^{\\prime} < 0\n \\end{cases},\n \\;\\;\\;\\;\\text{where}\\;m^{\\prime} := \\left[\\frac{d{\\xi}q - \\alpha}{k}\\right]_k - {\\gamma}c,\n \\\\\n\\label{eq: parameter definitions, m}\nm :=& \\left[{\\mu}m^{\\prime}\\right]_k.\n\\end{align}\n\\end{definition}\nIt is then straightforward to show that the above definitions imply that\n\\begin{equation}\n\\xi q = \\alpha\\gamma\\mu \\frac{ck+\\alpha\\gamma}{d}\\left(mk+\\alpha\\mu\\right) \\;\\;\\in{{\\mathbb Z}}\/k^2\n\\end{equation}\nand that $\\left[d{\\xi}q\\right]_{k^2} = ({\\mu}m+ {\\gamma}c)k + \\alpha$.\nThe definition of $\\xi$ in (\\ref{eq: parameter definitions, xi}) then implies that\n$\\left[d{\\xi}q\\right]_{k^2} < \\frac{k^2}{2}$.\nNote that, since (\\ref{eq: parameter definitions, gamma}) implies\n$d$ divides $ck +\\alpha\\gamma$, the fraction $\\frac{ck + \\alpha\\gamma}{d}$\ndenotes an integer. This also implies that \n$c = 0$ if and only if $d = 1$, which is true if and only if\n$q \\equiv \\pm 1\\;(\\mod k)$.\n\n\nWe next consider the possible values of $c, d,$ and $m$\nwhen $c \\neq 0$. So far, we know that $c>0$ and $d > 1$.\nMoreover, (\\ref{eq: parameter definitions, gamma})\nand the definition of ${\\sigma}_2$ imply that $d=2$ only if $\\alpha\\gamma = -1$.\nSince $k > 2$, (\\ref{eq: parameter definitions, d}) tells us that\n$d < \\frac{k}{2}$. Similarly, (\\ref{eq: parameter definitions, c}) \ntells us that $c=1$ when $d=2$, and otherwise $1 \\leq c < \\frac{d}{2}$.\nThis fact, combined with (\\ref{eq: parameter definitions, gamma}),\nimplies that $0 < \\frac{ck+\\alpha\\gamma}{d} < \\frac{k}{2}$.\nIn addition, since $1 < d < \\frac{k}{2}$, we know that\n$\\frac{ck+\\alpha\\gamma}{d} \\neq 1$, and so\n$\\frac{ck+\\alpha\\gamma}{d} \\geq 2$. The possible values for $m$\ndepend on $(\\mu, \\gamma) \\in \\{(1,1), (1,-1), (-1,1)\\}$ (with $(-1,-1)$ excluded\nbecause (\\ref{eq: parameter definitions, mu}) implies $\\mu = +1$ when\n$\\gamma = -1$).\nSince (\\ref{eq: parameter definitions, xi}) and (\\ref{eq: parameter definitions, alpha}) \nensure that $\\left[\\frac{d\\xi q - \\alpha}{k}\\right]_k \\leq \\frac{k}{2}$, we have\n$0 \\leq m \\leq \\frac{k}{2} - c < \\frac{k}{2}$\nwhen $(\\mu,\\gamma) = (1,1)$, and\n$1 \\leq c \\leq m \\leq \\frac{k}{2} + c < \\frac{3k}{4}$\nwhen $(\\mu, \\gamma) = (1,-1)$.\nWhen\n$(\\mu, \\gamma) = (-1,1)$, we have $m = c - \\left[\\frac{d\\xi q - \\alpha}{k}\\right]_k > 0$,\nand so $0 < m \\leq c < \\frac{k}{4}$.\n\n\nIt turns out that many properties of $v_q$ depend on whether\n$c=0$ and on the value of $(\\mu, \\gamma)$, motivating the\nfollowing definitions.\n\n\n\n\\begin{definition}\nSuppose that $k>100$ and $q$ is primitive in ${{\\mathbb Z}}\/k^2$.\nIf $q \\equiv \\pm 1\\; (\\mod k)$, or equivalently, if $c=0$,\nthen we say that $q$ is of {\\em type 0}.\n\\end{definition}\n\n\\begin{definition}\nSuppose that $k>100$, that $q$ is primitive in ${{\\mathbb Z}}\/k^2$, and that\n$c \\neq 0$, so that $q$ is not of type 0.\nWe then say that $q$ is of {\\em positive type} if \n$q = +\\xi q$, and of {\\em negative type} if $q = -\\xi q$.\nIn either case, $[d\\xi q]_{k^2} = (\\mu m+ \\gamma c)k + \\alpha$,\nwith $(\\mu, \\gamma) \\in \\{(1,1), (1,-1), (-1,1)\\}$.\n\\end{definition}\n\n\nNote that Proposition \\ref{prop:G properties} implies that $q$ is\ngenus-minimizing if and only if $\\xi q$ is genus-minimizing.\n\n\n\\begin{prop}\n\\label{prop: properties of parameters d, m, c, alpha, mu, gamma}\nSuppose that $q$ is genus-minimizing and of positive or negative type.\nThen $d$, $m$, $c$, $\\mu$, $\\gamma$, and $\\alpha \\in {{\\mathbb Z}}$\nsatisfy the following properties.\n\\begin{itemize}\n\\item[(i)]\n$\\xi q = \\alpha\\gamma\\mu \\frac{ck+\\alpha\\gamma}{d}\\left(mk+\\alpha\\mu\\right) \\in {{\\mathbb Z}}\/k^2$,\nwith $\\frac{ck+\\alpha\\gamma}{d} \\in {{\\mathbb Z}}$.\n\n\\item[(ii)]\n$\\left[d\\xi q\\right]_{k^2} = ({\\mu}m+ {\\gamma}c)k + \\alpha < \\frac{k^2}{2}$.\n\n\\item[(iii)]\n$2 \\leq d < \\frac{k}{2}$, and $d \\geq 3$ when $\\alpha\\gamma = +1$.\n\n\\item[(iv)]\n$1 \\leq c < \\frac{d}{2}$ (unless $d=2$, in which case $c=1$).\n\n\\item[(v)]\n$2 \\leq \\frac{ck+\\alpha\\gamma}{d} < \\frac{k}{2}$.\n\n\\item[(vi)]\n$1 \\leq m \\leq \\frac{k}{2} - c < \\frac{k}{2}$ when $(\\mu,\\gamma) = (1,1)$, \n{\\em i.e.}, when $d\\xi q = (m+c)k + \\alpha$.\n\n\\noindent $1 \\leq c < m \\leq \\frac{k}{2} + c < \\frac{3k}{4}$ when $(\\mu,\\gamma) = (1,-1)$,\n{\\em i.e.}, when $d\\xi q = (m-c)k + \\alpha$.\n\n\\noindent $1 \\leq m < c < \\frac{k}{4}$ when $(\\mu,\\gamma) = (-1,1)$,\n{\\em i.e.}, when $d\\xi q = (c-m)k + \\alpha$.\n\\end{itemize}\n\\end{prop}\n\\begin{proof}\nWe have already discussed the properties as listed above, except\nfor some changes made to the inequalities in (vi), due to the fact that\nwe now know that $q$ is genus-minimizing. The only extra information\nwe added is the fact that\n$m\\neq c$ if $q$ is genus-minimizing and $(\\mu, \\gamma) \\in \\{(1,-1), (-1,1)\\}$,\nand the fact that $q$ is not genus-minimizing when $m=0$.\n\nSuppose that $m=c$ and $(\\mu, \\gamma) \\in \\{(1,-1), (-1,1)\\}$.\nThen, setting $q^{\\prime}:= \\alpha\\xi q$, we have\n$(0q^{\\prime}, dq^{\\prime}, 2dq^{\\prime}) = (0,1,2)$. Thus\n\\begin{align}\n -v_{q^{\\prime}}(0{q^{\\prime}},2d{q^{\\prime}})\n&= -(2-0)k \\;\\;+\\;\\;\n \\# \\left({\\tilde{Q}}_{q^{\\prime}} \\cap \\left\\langle 0,2\\right] \\right) k^2\n \\\\ \\nonumber\n&= -2k + 2k^2\n \\\\ \\nonumber\n&\\geq k(k+1) \\;\\;\\mathrm{when}\\;\\; k \\geq 3,\n\\end{align}\nand so, by Proposition \n\\ref{prop: not minimizing if v >= k(k+1)},\n$q^{\\prime}$, hence $q$,\nis not genus-minimizing when $m=c$ and $(\\mu, \\gamma) \\in \\{(1,-1), (-1,1)\\}$.\n\n\n\nSuppose $m = 0$.\nThen setting $q^{\\prime} := \\gamma\\xi q = \\frac{ck+{\\alpha\\gamma}}{d}$ makes\n$[q^{\\prime}]_{k^2} < \\frac{k}{2}$, and so\n$[aq^{\\prime}]_{k^2} = a[q^{\\prime}]_{k^2}$\nfor all $a \\in \\{0, \\ldots, k-1\\}$. Thus\n\\begin{align}\n \\left|v_{q^{\\prime}}(0{q^{\\prime}},(k-1){q^{\\prime}})\\right|\n&= \\left|\\left((k-1)[q^{\\prime}]_{k^2} - 0[q^{\\prime}]_{k^2}\\right)k \\;\\;-\\;\\; \n \\# \\left({\\tilde{Q}}_q \\cap \\left\\langle 0, (k-1)[q^{\\prime}]_{k^2}\\right] \\right) k^2\\right|\n \\\\ \\nonumber\n&= \\left| (k-1)[q^{\\prime}]_{k^2}k \\;\\;-\\;\\; (k-1) k^2\\right|\n \\\\ \\nonumber\n&= k(k-1)(k-[q^{\\prime}]_{k^2})\n \\\\ \\nonumber\n&> k(k-1)\\textstyle{\\frac{k}{2}}\n \\\\ \\nonumber\n&\\geq k(k+1) \\;\\;\\mathrm{when}\\;\\; k \\geq 4,\n\\end{align}\nand so, by Proposition \n\\ref{prop: not minimizing if v >= k(k+1)},\n$q^{\\prime}$, hence $q$,\nis not genus-minimizing when $m=0$.\n\\end{proof}\n\n\n\n\n\\subsection{Genus-Minimizing $q$ of Type $0$}\n\\label{ss: q of type 0}\n\nWe begin by classifying the genus-minimizing solutions for\n$q$ of type 0, since this requires no additional machinery.\nRecall that $q$ is of type 0 if and only if\n$q \\equiv \\pm 1\\; (\\mod k)$. Thus we may write\n$q = nk \\pm 1$, for some $n \\in \\{0, \\ldots, k-1\\} \\subset {{\\mathbb Z}}$.\n\n\\begin{prop}\n\\label{prop: type 0 genus-minimizing classification}\nIf $q$ is of type 0, so that we may write $q=nk\\pm1$,\nthen $q$ is genus-minimizing if and only if $\\gcd(n,k) \\in \\{1,2\\}$.\nIf such $q$ is genus-minimizing, then $\\bar{G}(q) = 2k(k-1)$,\nand the maximum is attained uniquely.\n\\end{prop}\n\\begin{proof}\nAs observed in the proof of Proposition \\ref{prop:G properties}.(i),\n$v_{-q}(x,y) = v_q(-y,-x)$. Thus $\\bar{G}(q) := \\max_{x,y \\in {{\\mathbb Z}}\/k^2} v_q(x,y)$\nsatisfies $\\bar{G}(q) = \\bar{G}(-q)$, and the maximum is attained\nuniquely for $q$ if and only if it is attained uniquely for $-q$.\nIt therefore suffices to take $q = nk + 1$,\nsince any $q^{\\prime} = n^{\\prime}k -1$ satisfies\n$-q^{\\prime} = (k-n^{\\prime})k + 1$.\n\nWe first show that $q$ is not genus-minimizing when $\\gcd(n,k) \\geq 3$.\nLet $\\delta = \\gcd(n,k)$, so that $q = \\textstyle{\\frac{n}{\\delta}}{\\delta}k + 1$,\nand suppose that $\\delta \\geq 3$. Then\n\\begin{equation}\n \\left(0q, {\\textstyle{\\frac{k}{\\delta}}}q, 2{\\textstyle{\\frac{k}{\\delta}}}q \\right)\n= \\left(0, {\\textstyle{\\frac{k}{\\delta}}}, 2{\\textstyle{\\frac{k}{\\delta}}}\\right)\n\\end{equation}\nin $\\left({{\\mathbb Z}}\/k^2\\right)^3$, and so\n\\begin{align}\n \\left| v_q \\left(0q, 2{\\textstyle{\\frac{k}{\\delta}}}q \\right) \\right|\n&= \\left|\\left( 2{\\textstyle{\\frac{k}{\\delta}}} - 0\\right)k \\;\\;-\\;\\;\n \\#\\left({\\tilde{Q}}_q \\cap \\left\\langle 0, 2{\\textstyle{\\frac{k}{\\delta}}}\\right] \\right)k^2\\right|\n \\\\ \\nonumber\n&\\geq -2{\\textstyle{\\frac{k}{\\delta}}}k \\;\\;+\\;\\; 2k^2\n \\\\ \\nonumber\n&= k(2k - 2{\\textstyle{\\frac{k}{\\delta}}})\n \\\\ \\nonumber\n&\\geq k(2k - (k-1)) \\;\\;\\;(\\mathrm{since}\\; \\delta \\geq 3)\n \\\\ \\nonumber\n&= k(k+1).\n\\end{align}\nThus, by Proposition \\ref{prop: not minimizing if v >= k(k+1)},\n$q$ is not genus-minimizing.\n\n\nNext, suppose that $\\gcd(n,k)=1$.\nSince $n^{-1} \\in {{\\mathbb Z}}\/k$ exists, we have\n\\begin{equation}\n {\\left[ jn^{-1} \\right]_k} q \n= jk + \\left[ jn^{-1} \\right]_k \\;\\; \\in {{\\mathbb Z}}\/k^2\n \\forall j \\in \\{0, \\ldots, k-1\\}.\n\\end{equation}\nThus for each $j \\in \\{0, \\ldots, k-1\\}$, \nwe observe that ${\\tilde{Q}}_q \\cap \\left[jk, (j+1)k\\right\\rangle$ contains\nprecisely one element, namely, $\\left[{\\left[ jn^{-1} \\right]_k}q\\right]_{k^2}$.\nThis means that, for any $j_1 < j_2$ with $j_1, j_2 \\in \\{0, \\ldots, k-1\\}$,\n\\begin{equation}\n\\# \\left({\\tilde{Q}}_q \\cap\n \\left\\langle \\left[\\left[{j_1}n^{-1} \\right]_k q\\right]_{k^2} ,\n \\left[\\left[{j_2}n^{-1} \\right]_k q\\right]_{k^2} \\right] \\right)\n \\;=\\; j_2 - j_1.\n\\end{equation}\nWe now compute $v_q\\left({a_1}q, {a_2}q\\right)$ for an arbitrary\npair $a_1, a_2 \\in \\{0, \\ldots, k-1\\}$, ordered such that\n$\\left[{a_1}n\\right]_k < \\left[{a_2}n\\right]_k$:\n\\begin{align}\n v_q\\left({a_1}q, {a_2}q\\right)\n&= k \\cdot \\left[ \\left({\\left[{a_2}n\\right]_k}k + a_2\\right) - \n \\left({\\left[{a_1}n\\right]_k}k + a_1\\right) \\right] \\;\\;-\\;\\;\n k^2\\cdot \\left({\\left[{a_2}n\\right]_k} - {\\left[{a_1}n\\right]_k} \\right)\n \\\\ \\nonumber\n&= k\\left(a_2-a_1\\right).\n\\end{align}\nThus $\\left|v_q\\left({a_1}q, {a_2}q\\right)\\right| \\leq k(k-1)$, and the maximum\nvalue of $v_q\\left({a_1}q, {a_2}q\\right)$ is attained uniquely when \n$(a_1,a_2) = (0,k-1)$, yielding $v_q\\left(0q, (k-1)q\\right) = k(k-1)$.\nThus by Proposition \\ref{prop:G from aqs},\n\\begin{equation}\n\\bar{G}(q) = k(k-1) + k^2 - k = 2k(k-1),\n\\end{equation}\nand the maximum is attained uniquely.\n\n\nFinally, suppose that $\\gcd(n,k)=2$.\nLet $s = \\frac{n}{2}$, so that $q = 2sk + 1$.\nThen $\\gcd\\left(s, \\frac{k}{2}\\right) = 1$ implies\n$s^{-1} \\in {{\\mathbb Z}}\/\\frac{k}{2}$ exists, and so\nfor each $j \\in \\{0, \\ldots, \\frac{k}{2}-1\\}$, we have\n\\begin{align}\n \\left[ js^{-1} \\right]_{\\frac{k}{2}} q \n&\\equiv 2jk + \\left[ js^{-1} \\right]_{\\frac{k}{2}} \\; (\\mod k^2),\n \\\\\n \\left(\\left[ js^{-1} \\right]_{\\frac{k}{2}} \n + \\textstyle{\\frac{k}{2}} \\right) q \n&\\equiv 2jk + \\left[ js^{-1} \\right]_{\\frac{k}{2}} + \\textstyle{\\frac{k}{2}} \\; (\\mod k^2).\n\\end{align}\nThus for each $j \\in \\{0, \\ldots, \\frac{k}{2}-1\\}$, the set \n${\\tilde{Q}}_q \\cap \\left[2jk, 2(j +1)k\\right\\rangle$ contains\nprecisely two elements, namely, \n$\\left[ \\left[ js^{-1} \\right]_{\\frac{k}{2}} q \\right]_{k^2}$ and\n$\\left[ \\left(\\left[ js^{-1} \\right]_{\\frac{k}{2}} + \\frac{k}{2} \\right)q \\right]_{k^2}$.\nThis is similar to the case in which $\\gcd(n,k)=1$, but this time\nthere are two distinct types of element in each interval of length $2k$,\nso when we go to compute $v_q(x,y)$ for arbitrary elements $x,y\\in Q_q$,\nthere will be four types of pairs $(x,y)$ to consider.\n\nConsider an arbitrary pair $a_1,a_2 \\in \\{0, \\ldots, \\frac{k}{2}-1\\}$,\nordered such that \n$\\left[{a_1}s\\right]_{\\frac{k}{2}} < \\left[{a_2}s\\right]_{\\frac{k}{2}}$.\nIn order to exhaust all possible pairs of elements in $Q_q$, we need\nto consider all four pairs,\n\\begin{equation}\n\\left({a_1}q, {a_2}q\\right),\\;\n\\left(\\left({a_1} + \\textstyle{\\frac{k}{2}}\\right)\\!q, \n \\left({a_2} + \\textstyle{\\frac{k}{2}}\\right)\\!q\\right),\\;\n\\left({a_1}q, \\left({a_2} + \\textstyle{\\frac{k}{2}}\\right)\\!q\\right),\\;\\mathrm{and}\\;\n\\left(\\left({a_1} + \\textstyle{\\frac{k}{2}}\\right)\\!q, {a_2}q\\right).\n\\end{equation}\nBy arguments similar to those used in the case of $\\gcd(n,k)=1$, we have\n\\begin{equation}\n v_q\\left({a_1}q, {a_2}q\\right)\n= v_q\\left(\\left({a_1} + \\textstyle{\\frac{k}{2}}\\right)\\!q, \n \\left({a_2} + \\textstyle{\\frac{k}{2}}\\right)\\!q\\right)\n= k\\left(a_2-a_1\\right),\n\\end{equation}\nso that\n\\begin{equation}\n\\left|v_q\\left({a_1}q, {a_2}q\\right)\\right|,\n\\left|v_q\\left(\\left({a_1} + \\textstyle{\\frac{k}{2}}\\right)\\!q, \n \\left({a_2} + \\textstyle{\\frac{k}{2}}\\right)\\!q\\right)\\right|\n\\leq k\\left(\\textstyle{\\frac{k}{2}}-1\\right).\n\\end{equation}\nWe compute the two remaining cases by hand.\n\\begin{align}\n v_q \\left({a_1}q, \\left({a_2} + \\textstyle{\\frac{k}{2}}\\right)\\!q\\right)\n&= k\\cdot\\left[ \\left( 2\\left[{a_2}s\\right]_{\\frac{k}{2}}k \n + a_2 + \\textstyle{\\frac{k}{2}} \\right) -\n \\left( 2\\left[{a_1}s\\right]_{\\frac{k}{2}}k + a_1\\right) \\right]\n \\\\ \\nonumber\n&\\;\\;\\;\\;\\;\\;\\;\\;-\\; k^2\\cdot \\left(2\\left(\\left[{a_2}s\\right]_{\\frac{k}{2}}\n - \\left[{a_1}s\\right]_{\\frac{k}{2}}\\right) + 1 \\right)\n \\\\ \\nonumber\n&= k\\left(a_2 - a_1 - \\textstyle{\\frac{k}{2}} \\right),\n\\end{align}\nso that \n$\\left|v_q \\left({a_1}q, \\left({a_2} + \\textstyle{\\frac{k}{2}}\\right)\\!q\\right)\\right| \n\\leq k(k-1)$, and\n\\begin{align}\n v_q \\left(\\left({a_1} + \\textstyle{\\frac{k}{2}}\\right)\\!q, {a_2}q\\right)\n&= k\\cdot\\left[ \\left( 2\\left[{a_2}s\\right]_{\\frac{k}{2}}k + a_2\\right) - \n \\left( 2\\left[{a_1}s\\right]_{\\frac{k}{2}}k + a_1 \n + \\textstyle{\\frac{k}{2}} \\right) \\right]\n \\\\ \\nonumber\n&\\;\\;\\;\\;\\;\\;\\;\\;-\\; k^2\\cdot \\left(2\\left(\\left[{a_2}s\\right]_{\\frac{k}{2}}\n - \\left[{a_1}s\\right]_{\\frac{k}{2}}\\right) - 1 \\right)\n \\\\ \\nonumber\n&= k\\left(a_2 - a_1 + \\textstyle{\\frac{k}{2}} \\right),\n\\end{align}\nso that\n$\\left|v_q \\left(\\left({a_1} + \\textstyle{\\frac{k}{2}}\\right)\\!q, {a_2}q\\right)\\right| \n\\leq k(k-1)$.\n\nThus $\\left|v_q(x,y)\\right| \\leq k(k-1)$ for all $x,y \\in Q_q$, and the\nmaximum value of $v_q(x,y)$ is attained uniquely when \n$(x,y) = \\left(0q,(k-1)q\\right)$. Proposition \\ref{prop:G from aqs} then gives\n\\begin{equation}\n\\bar{G}(q) = k(k-1) + k^2 - k = 2k(k-1).\n\\end{equation}\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{$\\bar{G}(q) = 2k(k-1)$ for Genus-Minimizing $q$}\n\nIn the previous section, we learned that \nany genus-minimizing $q$ of type 0 satisfies\n$\\bar{G}(q) = 2k(k-1)$, and the corresponding maximum\nis uniquely attained. In fact, this result is true for\n{\\em any} genus-minimizing $q \\in {{\\mathbb Z}}\/k^2$, regardless of \nthe form $q$ or $-q$ takes.\n\n\\begin{prop}\n\\label{prop: G = 2k(k-1) and unique}\nIf $q$ is genus-minimizing, then $\\bar{G} = 2k(k-1)$ and\nthe corresponding maximum is uniquely attained. \n\\end{prop}\n\\begin{proof}\nIt is instructive to begin with a more general question:\nGiven $l \\in \\{0, \\ldots, k-1\\}$, what are all the pairs\n$x,y \\in Q_q$ for which\n$\\frac{v_q(x,y)}{k} \\equiv l \\;(\\mod k)$?\nThe answer is straight-forward.\nWrite $(x,y) = \\left({a_1}q, {a_2}q\\right)$, with\n$a_1, a_2 \\in \\{0, \\ldots, k-1\\}$. Then\n\\begin{align}\n \\frac{v_q\\left({a_1}q,{a_2}q\\right)}{k} \n&\\equiv l \\;(\\mod k)\n \\\\\n {a_2}q - {a_1}q\n&\\equiv l \\;(\\mod k)\n \\\\\n {a_2}\n&\\equiv {a_1} + q^{-1}l \\;(\\mod k),\n\\end{align}\nso there are exactly $k$ such pairs, \n$(x,y) \\in\n\\left\\{ \\left. \\left(aq, {\\left[a + q^{-1}l\\right]_k}q\\right) \\right|\na \\in \\{0, \\ldots, k-1\\} \\right\\}$.\n\nNote that this answer implies that if $\\frac{v_q(x,y)}{k} \\equiv 0\\;(\\mod k)$,\nthen $v_q(x,y) = 0$.\nIn particular, $k(k) \\notin \\left\\{\\left. v_q(x,y) \\right| x,y \\in Q_q\\right\\}$.\n\nOn the other hand, if $\\frac{v_q(x,y)}{k} \\equiv l \\;(\\mod k)$ for some\n$l \\neq 0$, and $q$ is genus-minimizing,\nthen $v_q(x,y) \\in \\{k(l), k(l-k)\\}$. If, in addition, $l$ is primitive in ${{\\mathbb Z}}\/k$,\nthen it is easy to determine how many of these pairs $(x,y)$ satisfy $v_q(x,y) = k(l)$\nand how many satisfy $v_q(x,y) = k(l-k)$. First, note that is clear\nfrom the definition of $v$ that $v_q(u,v) + v_q(v,w) = v_q(u,w)$\nfor any $u,v,w \\in {{\\mathbb Z}}\/k^2$. Thus\n\\begin{align}\n \\sum_{j\\in \\{0,\\dots, k-1\\}} \n v_q\\left( {\\left[j(q^{-1}l)\\right]_k}q, {\\left[(j+1)(q^{-1}l)\\right]_k}q \\right)\n&= v_q\\left( {\\left[0(q^{-1}l)\\right]_k}q, {\\left[k(q^{-1}l)\\right]_k}q \\right)\n \\\\ \\nonumber\n&= v_q(0,0)\n \\\\ \\nonumber\n&= 0, \n\\end{align}\nbut we also know that $(x,y) \\in\n\\left\\{\\left.\\left( {\\left[j(q^{-1}l)\\right]_k}q, {\\left[(j+1)(q^{-1}l)\\right]_k}q \\right)\n \\right| j \\in \\{0, \\ldots k-1\\} \\right\\}$\nif and only if $v_q(x,y) \\in \\{k(l), k(l-k)\\}$.\nLet $t:= \\#\\left\\{(x,y)\\in Q_q \\times Q_q | v_q(x,y) = k(l)\\right\\}$ \n(which implies $k-t$ is the number of pairs $(x,y)$ with $v_q(x,y) = k(l-k)$).\nThen we can rewrite the above sum as\n\\begin{equation}\nt\\cdot k(l) + (k-t)\\cdot k(l-k) = 0.\n\\end{equation}\nThis linear equation has solution $t = k-l$. Thus \n\\begin{align}\n\\label{k-l guys = l}\n \\#\\left\\{(x,y)\\in Q_q \\times Q_q | v_q(x,y) = k(l)\\right\\}\n&= k-l,\\;\\;\\; \\mathrm{and}\n \\\\\n \\#\\left\\{(x,y)\\in Q_q \\times Q_q | v_q(x,y) = k(l-k)\\right\\}\n&= l.\n\\end{align}\n\nAt last, we address the proposition at hand.\nSuppose $q$ is genus-minimizing. Thus\nby Proposition \\ref{prop: not minimizing if v >= k(k+1)},\n$v_q(x,y) \\leq k(k-1)$ for all $x, y \\in Q_q$.\nNow, $\\frac{k(k-1)}{k} = k-1$ is primitive in ${{\\mathbb Z}}\/k$,\nso by Equation (\\ref{k-l guys = l}) the number of pairs \n$(x,y) \\in Q_q \\times Q_q$ satisfying $v_q(x,y) = k(k-1)$\nis precisely $k - (k-1) = 1$. Thus\n\\begin{equation}\n\\max_{(x,y) \\in Q_q \\times Q_q} v_q(x,y) = k(k-1),\n\\end{equation}\nthis maximum is attained uniquely, and by Proposition \\ref{prop:G from aqs},\n\\begin{equation}\n\\bar{G}(q) \\;=\\; k(k-1) \\;+\\; k^2 - k \\;=\\; 2k(k-1).\n\\end{equation}\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Notation and Definitions for $q$ of Positive Type}\n\\label{ss: Notation and Definitions for q of Positive Type}\n\n\nSince Proposition \\ref{prop:G properties} implies that $q$ is\ngenus-minimizing if and only if $\\xi q$ is genus-minimizing,\nwe lose no information by restricting ourselves to the case in which\n$q = \\xi q$, and we gain the advantage of being able to write\n$q$ instead of $\\xi q$ for the next several dozen pages.\nWe therefore take $q$ to be of positive type\n({\\em i.e.}, with $q = +\\xi q$) for the following section.\nWe also take $q$ to be genus-minimizing.\n\n\nThe strategy in the preceding proof of focusing on $v_q(x,y)$ for pairs\n\\begin{equation}\n\\label{eq: pairs of [jq^-1]_k q terms}\n(x,y) \\in \\left\\{\\left.\\left( {\\left[j(q^{-1}l)\\right]_k}q, {\\left[(j+1)(q^{-1}l)\\right]_k}q \\right)\n \\right| j \\in \\{0, \\ldots k-1\\} \\right\\}\n\\end{equation}\nturns out to be quite powerful. Indeed, this idea serves as the foundation\nfor a combinatorial framework that helps us to classify the \ngenus-minimizing solutions for $q$ of positive type.\nWe use a minor modification of (\\ref{eq: pairs of [jq^-1]_k q terms}),\nin that we replace $[q^{-1}]_k$ with the parameter $d$ assigned to $q$\nin Definition \\ref{def: parameters assigned to q};\nthat is, $d := \\min\\left\\{\\left[q^{-1}\\right]_{\\!k},\\, \\left[-q^{-1}\\right]_{\\!k}\\right\\}$.\nIt is for this reason that we chose to parameterize $q$ of in a way\nthat gives $[dq]_{k^2}$ the convenient form\n\\begin{equation}\n[dq]_{k^2} = (\\mu m + \\gamma c)k + \\alpha < \\textstyle{\\frac{k^2}{2}}\n\\end{equation}\nwhen $q$ is of positive type.\nSee Definition \\ref{def: parameters assigned to q}\nfor definitions of\n$\\mu, \\gamma, \\alpha \\in \\{\\pm 1\\}$ and $d,m,c \\in{{\\mathbb Z}}$,\nand see Proposition \\ref{prop: properties of parameters d, m, c, alpha, mu, gamma}\nfor a list of properties the parameters satisfy.\n\n\n\nWe now begin the construction\ninspired by the proof of Proposition \\ref{prop: G = 2k(k-1) and unique}.\nWe start by associating to $q$ a $k$-tuple\n${\\bf{z}} = \\left(z_0, \\ldots, z_{k-1}\\right) \\in \\left({{\\mathbb Z}}\/k^2\\right)^k$,\ndefined by $z_r := \\left[rd\\right]_k\\! q \\in {{\\mathbb Z}}\/k^2$.\nNote that this makes $Q_q = \\{z_0, \\ldots, z_{k-1}\\}$.\nThis k-tuple then satisfies\n\\begin{equation}\n\\label{v_qprime (z) sums to zero}\n\\sum_{r\\in \\{0, \\ldots, k-1\\}} v_q\\!\\left(z_r, z_{r+1}\\right) = 0,\n\\end{equation}\nwhere we define $z_k := z_0$. Moreover, since\n$dq \\equiv \\alpha\\;(\\mod k)$, we know that\n\\begin{equation}\n\\label{v_q = alpha k (mod k^2)}\nv_q\\!\\left(z_r, z_{r+1}\\right) \\equiv \\alpha k \\; (\\mod k^2)\n\\;\\;\\;\n\\forall \\; r \\in \\{0, \\ldots, k-1\\}.\n\\end{equation}\nThus, following the reasoning\nused in the proof of Proposition \\ref{prop: G = 2k(k-1) and unique},\nwe deduce the following.\n\n\n\n\\begin{prop}\n\\label{prop: unique v_q = alpha(k - k^2), and the rest are v_q = alpha (k)}\nWhen of positive type, $q$ is genus-minimizing \nif and only if there exists $r_* \\in \\{0, \\ldots, k-1\\}$ for which\n\\begin{align*}\n v_q\\!\\left(z_{r_*}, z_{r_*+1}\\right) \n&= {\\alpha} (k-k^2), \\;\\;\\;\\mathrm{but}\n \\\\\n v_q\\!\\left(z_r, z_{r+1}\\right) \n&= {\\alpha} (k) \\;\\;\\;\n\\forall r \\in \\{0, \\ldots, k-1\\} \\setminus \\left\\{r_*\\right\\}.\n\\end{align*}\n\\end{prop}\n\\begin{proof}\nThe ``only if'' statement follows from the reasoning used in the proof of\nProposition \\ref{prop: G = 2k(k-1) and unique}. That is,\nfor all $r \\in \\{0, \\ldots, k-1\\}$,\nProposition \\ref{prop: not minimizing if v >= k(k+1)} tells us that\n$|v_q\\!\\left(z_r, z_{r+1}\\right)| \\leq k(k-1)$,\nand (\\ref{v_q = alpha k (mod k^2)}) tells us that\n$v_q\\!\\left(z_r, z_{r+1}\\right) \\equiv \\alpha k \\; (\\mod k^2)$.\nFrom this, we deduce that, for all $r \\in \\{0, \\ldots, k-1\\}$, either\n$v_q\\!\\left(z_r, z_{r+1}\\right) = \\alpha (k)$ or\n$v_q\\!\\left(z_r, z_{r+1}\\right) = \\alpha(k - k^2)$.\nThus, equation (\\ref{v_qprime (z) sums to zero}), and the fact that\nit has $k$ summands, gives us two linear equations in two variables,\nfor which the solution is that $\\alpha(k - k^2)$ occurs once, and\n$\\alpha (k)$ occurs $k-1$ times.\n\n\nFor the ``if'' statement, \nsuppose that there exists such an $r_*$.\nThen for any $z_{i_1}, z_{i_2} \\in Q_q$ with, say, $i_1 < i_2$, we have\n\\begin{align}\n v_q\\!\\left( z_{i_1}, z_{i_2}\\right)\n&= \\sum_{r=\\{i_1, \\ldots, i_2-1\\}} v_q\\!\\left( z_r, z_{r+1}\\right)\n \\\\\n&= \\begin{cases}\n k(i_2 - i_1)\n & r_* \\notin \\{i_1, \\ldots, i_2-1\\}\n \\\\\n k(i_2 - i_1 - k)\n & r_* \\in \\{i_1, \\ldots, i_2-1\\}\n \\end{cases},\n\\end{align}\nso that $\\left\\vert v_q\\!\\left( z_{i_1}, z_{i_2}\\right)\\right\\vert \\leq k(k-1)$.\nThus, by Proposition \\ref{prop: not minimizing if v >= k(k+1)},\n$q$ is genus-minimizing.\n\n\\end{proof}\n\n\n\n\nGiven such control over the value of $v_q\\left(z_j, z_{j+1}\\right)$,\nit will be useful to focus our attention on the intervals\n$\\left\\langle {\\tilde{z}}_j, {\\tilde{z}}_{j+1}\\right]$, for appropriate lifts\n${\\tilde{z}}_j$ and ${\\tilde{z}}_{j+1}$ of $z_j$ and $z_{j+1}$ to the integers. \nIndeed, we shall make such frequent use of intervals of\ninteger lifts of elements of ${{\\mathbb Z}}\/k^2$ that, for brevity, we establish\nthe following conventions for notational abuses.\n\\begin{conv}\nFor any $x, y \\in {{\\mathbb Z}}\/k^2$, we shall write ``$\\left\\langle x, y \\right\\rangle$''\n(or ``$\\left\\langle x, y \\right]$'', ``$\\left[ x, y \\right\\rangle$'', or ``$\\left[ x, y \\right]$'')\nfor the interval $\\left\\langle \\tilde{x}, \\tilde{y} \\right\\rangle$\n(or $\\left\\langle \\tilde{x}, \\tilde{y} \\right]$, et cetera), where $\\tilde{x}, \\tilde{y} \\in {{\\mathbb Z}}$\nare respective lifts of $x$ and $y$ to the integers. If the precise\nlifts intended are not clear from context, then any lifts satisfying\n$0 < \\tilde{y}-\\tilde{x} < k^2$ will suffice.\nFor $w \\in {{\\mathbb Z}}\/k^2$, we shall write ``$w \\in \\left\\langle x, y \\right\\rangle$\"\n(or ``$w \\in \\left\\langle x, y \\right]$,\" et cetera)\nto signify that there exists a lift $\\tilde{w} \\in {{\\mathbb Z}}$ of $w$ satisfying\n$\\tilde{w} \\in \\left\\langle x, y \\right\\rangle$ (or $\\tilde{w} \\in \\left\\langle x, y \\right]$, et cetera).\nSimilarly, for $w \\in {{\\mathbb Z}}\/k^2$ and $s,t \\in {{\\mathbb Z}}$, we shall write\n``$s < w < t$'' (or ``$s < w \\leq t$'', et cetera) to signify that there exists a lift\n$\\tilde{w} \\in {{\\mathbb Z}}$ of $w$ satisfying $s < \\tilde{w} < t$ (or $s < \\tilde{w} \\leq t$, et cetera).\nIf we only write ``$w < t$'' or ``$s < w$'', then the precise lift of $w$ intended\nshould be clear from context.\n\\end{conv}\n\n\n\n\nCarrying on, we begin by examining the length of an interval\n$\\left\\langle z_j, z_{j+1} \\right]$.\nFor any $r \\in \\{0, \\ldots, k-1\\}$, we have\n\\begin{equation}\n z_{r+1} - z_r \n= \\begin{cases}\n dq & \\left[rd\\right]_k < k-d\n \\\\\n (d-k)q & \\left[rd\\right]_k \\ge k-d\n \\end{cases},\n\\end {equation}\nwhere \n$q= \\alpha\\gamma\\mu\\frac{ck + \\alpha\\gamma}{d}(mk + \\alpha\\mu)$\nimplies that $kq = \\gamma\\frac{ck + \\alpha\\gamma}{d}k$.\n\n\n\n\n\nIt is instructive to break $\\bf{z}$ up into $d$ consecutive \nsub-tuples ${\\bf{z}} = \\left({\\bf{z}}^0, \\ldots, {\\bf{z}}^{d-1}\\right)$\nin a manner that reflects the above structure of the differences\nof consecutive entries of $\\bf{z}$, {\\em i.e.}, such that\n\\begin{align}\n z^j_{i+1} - z^j_i \n&= dq \\;\\;\\; \n \\forall\\; j \\in \\{0, \\ldots, d-1\\},\\;\n i \\in \\{0, \\ldots, {{\\mathrm{len}}}({\\bf{z}}^j) - 2 \\};\n \\\\\n z^{j+1}_0 - z^j_{{{\\mathrm{len}}}({\\bf{z}}^j) - 1}\n&= (d-k)q\\;\\;\\;\n \\forall \\; j \\in \\{0, \\ldots, d-2\\}.\n\\end{align}\nMore explicitly, setting $\\epsilon := \\left[-k\\right]_d$ (and noting that this\nimplies $c = \\left[\\alpha\\gamma{\\epsilon}^{-1}\\right]_{d}$),\nfor each $j \\in \\{0, \\ldots, d-1\\}$, we define\n$z^j_i := \\left(\\left[j\\epsilon\\right]_d + id\\right)\\!q \\in {{\\mathbb Z}}\/k^2$,\nwhere we restrict\nthe domain of $i$ so that \n$\\left[j\\epsilon\\right]_d + id \\in \\{0, \\ldots, k-1\\}$. Thus\n\\begin{align}\n\\label{eq: definition of z_i^j}\n {\\bf{z}}^0\n&= \\left(\\; \\left(\\left[0\\epsilon\\right]_d + 0d\\right)\\!q,\\;\n \\left(\\left[0\\epsilon\\right]_d + 1d\\right)\\!q,\\; \\ldots,\\;\n \\left(\\left[0\\epsilon\\right]_d\n + \\left({\\left\\lceil\\!\\frac{k - \\left[0\\epsilon\\right]_d\\!}{d}\n \\right\\rceil} \\!-\\! 1\\!\\right)\\!d\n \\right)\\! q\n \\;\\right),\n \\\\ \\nonumber\n {\\bf{z}}^1\n&= \\left(\\; \\left(\\left[1\\epsilon\\right]_d + 0d\\right)\\!q,\\;\n \\left(\\left[1\\epsilon\\right]_d + 1d\\right)\\!q,\\; \\ldots,\\;\n \\left(\\left[1\\epsilon\\right]_d\n + \\left({\\left\\lceil\\!\\frac{k - \\left[1\\epsilon\\right]_d\\!}{d}\n \\right\\rceil} \\!-\\! 1\\!\\right)\\!d\n \\right)\\! q\n \\;\\right),\n \\\\ \\nonumber\n&\\;\\;\\vdots\n \\\\ \\nonumber\n {\\bf{z}}^{d-1}\n&= \\left(\\; \\left(\\left[(d\\!-\\!1)\\epsilon\\right]_d + 0d\\right)\\!q,\\;\n \\left(\\left[(d\\!-\\!1)\\epsilon\\right]_d + 1d\\right)\\!q,\\; \\ldots,\n \\right.\n \\\\ \\nonumber\n &\\left.\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n \\left(\\left[(d\\!-\\!1)\\epsilon\\right]_d\n + \\left({\\left\\lceil\\!\\frac{k - \\left[(d\\!-\\!1)\\epsilon\\right]_d\\!}{d}\n \\right\\rceil} \\!-\\! 1\\!\\right)\\!d\n \\right)\\! q\n \\;\\right).\n\\end{align}\nSuch ${\\bf{z}}^j$ then has length\n\\begin{align}\n {{\\mathrm{len}}}({\\bf{z}}^j)\n&= \\left(\\left(\\left\\lceil \\frac{k - \\left[j\\epsilon\\right]_d}{d}\\right\\rceil - 1\\right)\n - 0\\right) + 1\n \\\\ \\nonumber\n&= \\left\\lceil \\frac{k}{d} \\right\\rceil\n - \\begin{cases}\n 1 &\\left[k\\right]_d - \\left[j\\epsilon\\right]_d \\leq 0\n \\\\\n 0 &\\left[k\\right]_d - \\left[j\\epsilon\\right]_d > 0\n \\end{cases}\n \\\\ \\nonumber\n&= \\frac{k+\\epsilon}{d}\n - \\begin{cases}\n 1 &\\left[j\\epsilon\\right]_d \\geq d - \\epsilon\n \\\\\n 0 &\\left[j\\epsilon\\right]_d < d - \\epsilon\n \\end{cases}\n \\\\ \\nonumber\n&= \\frac{k+\\epsilon}{d} - {\\theta}^{d, \\epsilon}(j),\n\\end{align}\nwhere, for any relatively prime positive integers $d$ and $\\epsilon$\nwith $d \\geq 2$ and $\\epsilon < d$, and for any $j \\in {{\\mathbb Z}}$,\n${\\theta}^{\\cdot, \\cdot}(\\cdot)$ is a function\ndefined by\n\\begin{equation}\n\\label{eq: def of theta}\n {\\theta}^{d, \\epsilon}(j)\n:= \\begin{cases}\n 1 &\\left[j\\epsilon\\right]_d \\geq d - \\epsilon\n \\\\\n 0 &\\left[j\\epsilon\\right]_d < d - \\epsilon\n \\end{cases},\n\\end{equation}\nor, equivalently, by\n\\begin{equation}\n\\label{eq: def 2 of theta}\n \\theta^{d, \\epsilon}(j)\n= \\left\\lfloor \\frac{(j+1)\\epsilon}{d}\\right\\rfloor\n -\\left\\lfloor \\frac{j\\epsilon}{d}\\right\\rfloor.\n\\end{equation}\n\nSince the definition of ${\\bf{z}}^j$ depends only on the value of\n$j$ modulo $d$, it is natural to extend the definition\nof ${\\bf{z}}^j$ to be valid for all $j \\in {{\\mathbb Z}}$,\nsetting ${\\bf{z}}^j := {\\bf{z}}^{\\left[j\\right]_d}$.\nWe therefore henceforward regard the collection $\\left\\{{\\mathbf{z}}^j\\right\\}_{j\\in{{\\mathbb Z}}}$ \nas being indexed by elements of ${{\\mathbb Z}}\/d$.\nFor any $j \\in{{\\mathbb Z}}\/d$, we then compute\n$z^{j+1}_0 - z^j_0 \\in {{\\mathbb Z}}\/k^2$, as follows:\n\\begin{align}\n \\label{eq: z^{j+1}_0 - z^j_0}\n z^{j+1}_0 - z^j_0\n&= \\left(z^{j+1}_0 - z^j_{{{\\mathrm{len}}}({\\bf{z}}^j)-1}\\right)\n + \\left(z^j_{{{\\mathrm{len}}}({\\bf{z}}^j)-1} - z^j_0\\right)\n \\\\ \\nonumber\n&= (d-k)q + \\left({{\\mathrm{len}}}({\\bf{z}}^j)-1\\right)dq\n \\\\ \\nonumber\n&= -kq + {{\\mathrm{len}}}({\\bf{z}}^j)dq\n \\\\ \\nonumber\n&= -\\gamma\\frac{ck + \\alpha\\gamma}{d}k\n +\\left(\\frac{k+\\epsilon}{d} - {\\theta}^{d, \\epsilon}(j)\\right)\\left(({\\mu}m+{\\gamma}c)k+\\alpha\\right)\n \\\\ \\nonumber\n&= {\\mu}m\\frac{k^2}{d}\n + \\left(\\frac{\\epsilon}{d} - {\\theta}^{d, \\epsilon}(j)\\right)\\left(({\\mu}m+{\\gamma}c)k+\\alpha\\right)\n \\\\ \\nonumber\n&= {\\mu}m\\frac{k^2}{d}\n + \\left(\\frac{\\epsilon}{d} - {\\theta}^{d, \\epsilon}(j)\\right)[dq]_{k^2}\n\\end{align}\nOf course, the summand ``$\\mu m\\frac{k^2}{d}$''\ndoes not make much sense as an element of ${{\\mathbb Z}}\/k^2$.\nTo make sense of the last two lines of (\\ref{eq: z^{j+1}_0 - z^j_0}),\nand of similar expressions, one must first interpret the summands as elements of $\\frac{1}{d}{{\\mathbb Z}}$,\nnext interpret their sum as an element of ${{\\mathbb Z}}$, and then take the image\nof this element of ${{\\mathbb Z}}$ in ${{\\mathbb Z}}\/k^2$.\nEquation (\\ref{eq: z^{j+1}_0 - z^j_0}) further abuses notation in its usage\nof ${\\theta}^{d, \\epsilon}$. Since ${\\theta}^{d, \\epsilon}$ is periodic modulo $d$,\nit descends to a function on ${{\\mathbb Z}}\/d$, which we denote in the same way as the\noriginal function on ${{\\mathbb Z}}$.\n\n\n\n\nGiven, in addition, any $l \\in {{\\mathbb Z}}\/d$, we can use\n(\\ref{eq: z^{j+1}_0 - z^j_0})\nto compute $z^{j+l}_0 - z^j_0 \\in {{\\mathbb Z}}\/k^2$.\nLet $\\tilde{j}, \\tilde{l} \\in {{\\mathbb Z}}$ denote arbitrary lifts of $j$ and $l$ to the integers. Then\n\\begin{align}\n\\label{eq: z_0^j+l - z_0^j = mu ml k^2\/d + xi dq}\n z^{j+l}_0 - z^j_0\n&= \\sum_{i=\\tilde{j}}^{\\tilde{j} + \\tilde{l}-1} \\left(z^{i+1}_0 - z^i_0\\right)\n \\\\ \\nonumber\n&=\\sum_{i=\\tilde{j}}^{\\tilde{j} + \\tilde{l}-1}\n \\left( {\\mu}m\\frac{k^2}{d}\n + \\left(\\frac{\\epsilon}{d} - {\\theta}^{d, \\epsilon}(i)\\right) [dq]_{k^2}\n \\right)\n \\\\ \\nonumber\n&= {{\\mu}m\\tilde{l}}\\frac{k^2}{d}\n +\\left(\\frac{\\tilde{l}\\epsilon}{d}\n - \\sum_{i=\\tilde{j}}^{\\tilde{j} + \\tilde{l}-1} {\\theta}^{d, \\epsilon}(i)\\right) [dq]_{k^2}\n \\\\ \\nonumber\n&= \\left[{\\mu}ml\\right]_d\\frac{k^2}{d}\n \\;+\\; \\Xi^{d, \\epsilon}_l(j)\\, [dq]_{k^2},\n\\end{align}\nwhere the function $\\Xi^{\\cdot, \\cdot}_{\\cdot}(\\cdot)$ is defined such that,\nfor any relatively prime positive integers $d$ and $\\epsilon$ with\n$d \\ge 2$ and $\\epsilon < d$, and for any\n$j, l \\in {{\\mathbb Z}}\/d$, we have\n\\begin{equation}\n \\Xi^{d, \\epsilon}_l(j) \n\\;:=\\; \\frac{\\tilde{l}\\epsilon}{d}\n -\\sum_{i=\\tilde{j}}^{\\tilde{j} + \\tilde{l}-1} {\\theta}^{d, \\epsilon}(i)\\;\\;\n \\in \\textstyle{\\frac{1}{d}}{{\\mathbb Z}},\n\\end{equation}\nwhere $\\tilde{j}, \\tilde{l} \\in {{\\mathbb Z}}$ denote arbitrary lifts of $j$ and $l$ to ${{\\mathbb Z}}$.\nThe value of $\\Xi^{d, \\epsilon}_l(j)$ does not depend on the choice of lift.\nNote that since $\\sum_{j\\in{{\\mathbb Z}}\/d} {\\theta}^{d, \\epsilon}(j) = \\epsilon$,\n$\\Xi^{d, \\epsilon}_l$ has mean value zero.\nThe following lemma shows that $\\Xi^{d, \\epsilon}_l(j)$\n(and hence $z^{j+l}_0 - z^j_0$)\nstays as close to its mean value as possible.\n\n\n\\begin{lemma}\n\\label{lemma: q of positive type, xi lemma}\nSuppose that $d$ and $\\epsilon$ are relatively prime positive integers with\n$d \\ge 2$ and $\\epsilon < d$,\nand that $l \\in {{\\mathbb Z}}\/d$. Then, for any $j \\in {{\\mathbb Z}}\/d$, we have\n\\begin{equation}\n\\nonumber\n \\Xi^{d, \\epsilon}_l(j) \n\\in \\left\\{ \\frac{[l\\epsilon]_d}{d}, \\frac{-[-l\\epsilon]_d}{d}\\right\\}.\n\\end{equation}\nWhen $l \\neq 0$, the sets\n$\\left\\{ j \\in {{\\mathbb Z}}\/d \\left\\vert\\; \\Xi^{d, \\epsilon}_l(j) = \\frac{[l\\epsilon]_d}{d}\\right.\\right\\}$\nand \n$\\left\\{ j \\in {{\\mathbb Z}}\/d \\left\\vert\\; \\Xi^{d, \\epsilon}_l(j) = \\frac{-[-l\\epsilon]_d}{d}\\right.\\right\\}$\nhave $\\left[-l\\epsilon\\right]_d$ elements and \n$\\left[l\\epsilon\\right]_d$ elements, respectively.\n\\end{lemma} \n\n\\begin{proof}\nLet $\\tilde{l}, \\tilde{j} \\in {{\\mathbb Z}}$ denote any lifts of $j, l \\in {{\\mathbb Z}}\/d$\nto the integers. Then, using (\\ref{eq: def 2 of theta}), we have\n\\begin{align}\n \\Xi^{d, \\epsilon}_l(j) \n&= \\frac{\\tilde{l}\\epsilon}{d}\n -\\sum_{i=\\tilde{j}}^{\\tilde{j} + \\tilde{l}-1}\n \\left(\\left\\lfloor \\frac{(i+1)\\epsilon}{d}\\right\\rfloor\n -\\left\\lfloor \\frac{i\\epsilon}{d}\\right\\rfloor\\right)\n \\\\ \\nonumber\n&= \\frac{\\tilde{l}\\epsilon}{d}\n -\\left(\\left\\lfloor \\frac{(\\tilde{j} + \\tilde{l})\\epsilon}{d}\\right\\rfloor\n -\\left\\lfloor \\frac{\\tilde{j}\\epsilon}{d}\\right\\rfloor\\right)\n \\\\ \\nonumber\n&\\in \\left\\{ \\frac{\\tilde{l}\\epsilon}{d} - \n \\left\\lfloor \\frac{\\tilde{l}\\epsilon}{d}\\right\\rfloor,\\;\n \\frac{\\tilde{l}\\epsilon}{d} - \n \\left\\lceil\\frac{\\tilde{l}\\epsilon}{d}\\right\\rceil \\right\\}\n \\\\ \\nonumber\n&= \\left\\{ \\frac{[l\\epsilon]_d}{d}, \\frac{-[-l\\epsilon]_d}{d}\\right\\}.\n\\end{align}\n\n\nNext, suppose that $l \\neq 0$, and let\n$n_+$ (respectively, $n_-$) denote the number of\nvalues of $j \\in {{\\mathbb Z}}\/d$ for which\n$\\Xi^{d, \\epsilon}_l(j) = \\frac{[l\\epsilon]_d}{d}$\n(respectively, $\\Xi^{d, \\epsilon}_l(j) = \\frac{-[-l\\epsilon]_d}{d}$).\nThen $n_+$ and $n_-$ satisfy two independent linear equations,\n\\begin{align}\n d \n&\\;=\\; n_+ \\;+\\; n_-, \\; \\text{and}\n \\\\\n 0\n&\\;=\\; \\sum_{j \\in {{\\mathbb Z}}\/d} \\Xi^{d, \\epsilon}_l(j)\n \\;=\\; n_+ \\!\\left\\lfloor\\frac{l\\epsilon}{d}\\right\\rfloor\n \\;+\\; n_- \\!\\left\\lceil\\frac{l\\epsilon}{d}\\right\\rceil,\n\\end{align}\nwith unique solution\n$n_+ = \\left[-l\\epsilon\\right]_d$ and\n$n_- = \\left[l\\epsilon\\right]_d$.\n\n\\end{proof}\n\n\n\n\nBefore proceeding further, we need some new notation.\nFirst, we introduce the operations $\\minq$ and $\\maxq$,\nwhich are only defined on two-element subsets of ${{\\mathbb Z}}\/k^2$\ndiffering by $dq$. For any $x \\in {{\\mathbb Z}}\/k^2$, we say that\n\\begin{align}\n \\minq\\, \\left\\{x,\\; x + dq\\right\\} \n&= x,\n \\\\\n \\maxq\\, \\left\\{x,\\; x + dq\\right\\}\n&= x + dq.\n\\end{align}\nNext, for brevity, set\n\\begin{equation}\nn_j := {{\\mathrm{len}}}({\\mathbf{z}}^j) - 1 = \\left\\lfloor\\frac{k}{d}\\right\\rfloor - {\\theta}^{d, \\epsilon}(j)\n\\end{equation}\nfor each $j\\in {{\\mathbb Z}}\/d$, so that ${\\mathbf{z}}^j = (z_0^j, \\ldots, z_{n_j}^j)$.\nThat is, for each $j\\in{{\\mathbb Z}}\/d$, $n_j$ counts the number of intervals \n$\\left\\langle z_i^j, z_{i+1}^j\\right]$ for which $z_i^j$ and $z_{i+1}^j$ are defined.\nThe combination of \n(\\ref{eq: z_0^j+l - z_0^j = mu ml k^2\/d + xi dq})\nand Lemma \\ref{lemma: q of positive type, xi lemma}\nthen provides the following result.\n\n\n\n\\begin{cor}\n\\label{cor: q of positive type: combo of lemma and difference eq}\nIf $q$ is of positive type, then for any $l \\in {{\\mathbb Z}}\/d$ with $l \\neq 0$,\nthe sets $\\left\\{z_0^{j+l} - z_i^j\\right\\}_{j\\in{{\\mathbb Z}}\/d}$ and \n$\\left\\{z_{n_{j-l}}^{j-l} - z^j_{n_j - i}\\right\\}_{j \\in {{\\mathbb Z}}\/d}$\neach contain precisely two elements, which differ by $dq$.\nMore specifically,\n\\begin{align*}\n \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l} - z_i^j\\right)\n&= \\left[{\\mu}ml\\right]_d\\frac{k^2}{d} + \n \\left(\\frac{\\left[l\\epsilon\\right]_d}{d} - i - 1\\right)[dq]_{k^2},\n \\\\\n \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j-l}}^{j-l} - z^j_{n_j - i}\\right)\n&= -\\left[{\\mu}ml\\right]_d\\frac{k^2}{d} -\n \\left(\\frac{\\left[l\\epsilon\\right]_d}{d} - i - 1\\right)[dq]_{k^2}.\n\\end{align*}\nFor any $l \\in {{\\mathbb Z}}\/d$ with $l \\neq 1$, the sets\n$\\left\\{z_0^{j+l} - z_{n_j - (i+1)}^j\\right\\}_{j \\in {{\\mathbb Z}}\/d}$ and\n$\\left\\{z_{n_{j-l}}^{j-l} - z^j_{i+1}\\right\\}_{j \\in {{\\mathbb Z}}\/d}$\neach contain precisely two elements, which differ by $dq$.\nMore specifically,\n\\begin{align*}\n \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l} - z_{n_j - (i+1)}^j\\right)\n&= \\left[{\\mu}m(l-1)\\right]_d\\frac{k^2}{d} + \n \\left(\\frac{\\left[(l-1)\\epsilon\\right]_d}{d} + i\\right)[dq]_{k^2} + \\psi,\n \\\\\n \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j-l}}^{j-l} - z^j_{i+1}\\right)\n&= -\\left[{\\mu}m(l-1)\\right]_d\\frac{k^2}{d} - \n \\left(\\frac{\\left[(l-1)\\epsilon\\right]_d}{d} + i\\right)[dq]_{k^2} - \\psi,\n\\end{align*}\nwhere $\\psi := \\left[z^{j+1}_0 - z_{n_j}^j\\right]_{k^2} \n= \\left[dq - kq\\right]_{k^2} =\n \\left[(({\\mu}m + {\\gamma}c)k + \\alpha) - \\gamma\\frac{ck+\\alpha\\gamma}{d}k\\right]_{k^2}$.\n\\end{cor}\nThis result also uses the fact that\n$\\frac{-[-l\\epsilon]_d}{d} = \\frac{[l\\epsilon]_d}{d}-1$ when $l \\neq 0$.\nAgain, note that to make sense of the four above right-hand expressions\nas elements of ${{\\mathbb Z}}\/k^2$, one\nmust first interpret each expression as an integer, and then\ntake the image of that integer under the quotient\n${{\\mathbb Z}} \\rightarrow {{\\mathbb Z}}\/k^2$.\n\n\nThe above corollary plays an important role in studying any set of the form\n${\\tilde{Q}}_q \\cap \\left\\langle z_i^j, z_{i+1}^j\\right]$ or\n${\\tilde{Q}}_q \\cap \\left\\langle z_{n_j - (i+1)}^j, z_{n_j - i}^j\\right]$,\nfor fixed $i \\in \\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2\\}$,\nas a function of $j \\in {{\\mathbb Z}}\/d$. (We shall also use the\ncorollary to study sets of the form ${\\tilde{Q}}_q \\cap \\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$\nor ${\\tilde{Q}}_q \\cap \\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$\nas functions of $j \\in {{\\mathbb Z}}\/d$, but more on that later.)\nSuppose that for some $j^{\\prime} \\in {{\\mathbb Z}}\/d$ and\n$i \\in \\{0, \\ldots, n_{j^{\\prime}} \\}$ there is an element of \n${\\tilde{Q}}_q$ contained in the interval\n$\\left\\langle z_i^{j^{\\prime}}, z_{i+1}^{j^{\\prime}}\\right]$.\nThen we can write $z_{i^{\\prime}}^{j^{\\prime}+l}$ for the image of this element in ${\\tilde{Q}}_q$,\nfor some $l \\in {{\\mathbb Z}}\/d$ and $i^{\\prime} \\in \\{0, \\ldots, n_{j^{\\prime}+l} \\}$.\nLet $x := z_{i^{\\prime}}^{j^{\\prime}+l} - z_i^{j^{\\prime}}$, so that\n$z_{i^{\\prime}}^{j^{\\prime}+l} = z_i^{j^{\\prime}} + x$.\nThen if $i^{\\prime} \\notin \\{0, n_{j^{\\prime}+l} \\}$,\nCorollary \\ref{cor: q of positive type: combo of lemma and difference eq}\nimplies that\n$z_i^j + x \\in \\{z_{i-1}^j, z_i^j, z_{i+1}^j\\} \\subset {\\tilde{Q}}_q$ for all $j \\in {{\\mathbb Z}}\/d$.\nThat is, an element of ${\\tilde{Q}}_q$ is contained\nat the same relative position in $\\left\\langle z_i^j, z_{i+1}^j\\right]$\nfor every $j \\in {{\\mathbb Z}}\/d$.\nThus, if {\\em every} $z_{i^{\\prime}}^{j+l}$ contained in\n$\\left\\langle z_i^j, z_{i+1}^j \\right]$\nsatisfied $i^{\\prime} \\notin \\{0, n_{j+l}\\}$, for every $j \\in {{\\mathbb Z}}\/d$, then\n$\\#\\!\\left({\\tilde{Q}}_q \\cap \\left\\langle z_i^j, z_{i+1}^j \\right]\\right)$\nwould be constant in $j \\in {{\\mathbb Z}}\/d$.\n\n\nThis means that the only way for $\\#\\!\\left({\\tilde{Q}}_q \\cap \\left\\langle z_i^j, z_{i+1}^j \\right]\\right)$\nto change as $j$ varies in ${{\\mathbb Z}}\/d$ is for there to exist some\n$l \\in {{\\mathbb Z}}\/d$ for which either $z_0^{j^{\\prime}+l} \\in\n\\left\\langle z_i^{j^{\\prime}}, z_{i+1}^{j^{\\prime}}\\right]$\nor $z_{n_{j^{\\prime}-l}}^{j^{\\prime}-l} \\in\n\\left\\langle z_i^{j^{\\prime}}, z_{i+1}^{j^{\\prime}}\\right]$, for some $j^{\\prime} \\in {{\\mathbb Z}}\/d$.\nThis is the idea behind {\\em mobile points}, which we define as follows.\nFor any $i \\in \\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2\\}$\nand $l \\in {{\\mathbb Z}}\/d$ with $l \\neq 0$,\nwe say that $z_0^{j+l}$ is an R-mobile point in the interval\n$\\left\\langle z_i^j, z_{i+1}^j \\right]$ if\n\\begin{equation}\n0 < \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l} - z_i^j\\right) < \\left[dq\\right]_{k^2},\n\\end{equation}\nand that\n$z_{n_{j-l}}^{j-l}$ is an L-mobile point in the interval\n$\\left\\langle z_{n_j - (i+1)}^j, z_{n_j - i}^j \\right]$ if\n\\begin{equation}\n-\\left[dq\\right]_{k^2} < \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j-l}}^{j-l} - z_{n_j - i}^j\\right) < 0.\n\\end{equation}\n\n\nNote that since we take $i$ to be constant in $j$, we must demand\n$i \\in \\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2\\}$,\nas opposed to, for example, $i \\in \\{0, \\ldots, n_{j^{\\prime}}-1\\}$.\nThe ``R'' and ``L'' correspond to the fact that when an R-mobile point\n``escapes,'' it does so by a distance of $dq$ to the right, whereas, when an\nL-mobile point ``escapes,'' it does so by a distance of $dq$ to the left.\nWe say that the R-mobile point described in the above paragraph is\nR-mobile ``rel $z_0^j$'', since positions are measured relative to $z_0^j$.\nLikewise, we say that the above-described L-mobile point is L-mobile\n``rel $z_{n_j}^j$'', since positions are measured relative to $z_{n_j}^j$.\nRecall that Corollary \\ref{cor: q of positive type: combo of lemma and difference eq} shows that\n\\begin{equation}\n\\label{eq: mirror relation 1}\n \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l} - z_i^j\\right)\n= -\\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j-l}}^{j-l} - z^j_{n_j - i}\\right)\n\\end{equation}\nfor all $l\\in {{\\mathbb Z}}\/d$ with $l\\neq 0$. This equation establishes an isomorphism\nwhich we call a {\\em mirror relation}\nbetween R-mobile points rel $z_0^j$ and L-mobile points rel $z_{n_j}^j$.\nIn particular, \n$z_0^{j+l}$ is an R-mobile point in $\\left\\langle z_i^j, z_{i+1}^j \\right]$\nif and only if $z_{n_{j-l}}^{j-l}$ is L-mobile point in\n$\\left\\langle z_{n_j - (i+1)}^j, z_{n_j - i}^j \\right]$.\nWe call such pairs of mobile points {\\em mirror} mobile points.\n\n\n\n\nOne can also define R-mobile points rel $z_{n_j}^j$ and L-mobile points \nrel $z_0^j$, as follows. For any $l \\in {{\\mathbb Z}}$\/d with $l \\neq 1$, we say that $z_0^{j+l}$ \nis R-mobile in\n$\\left\\langle z_{n_j - (i+1)}^j, z_{n_j - i}^j \\right]$ \n(hence is R-mobile rel $z_{n_j}^j$) if\n\\begin{equation}\n0 < \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l} - z_{n_j - (i+1)}^j\\right) < \\left[dq\\right]_{k^2}.\n\\end{equation}\nLikewise, $z_{n_{j-l}}^{j-l}$ is L-mobile in\n$\\left\\langle z_i^j, z_{i+1}^j \\right]$ (hence is L-mobile rel $z_0^j$) if\n\\begin{equation}\n-\\left[dq\\right]_{k^2} < \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j-l}}^{j-l} - z_{i+1}^j\\right) < 0.\n\\end{equation}\nAgain, Corollary \\ref{cor: q of positive type: combo of lemma and difference eq} shows that\n\\begin{equation}\n\\label{eq: mirror relation 2}\n \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l} - z_{n_j - (i+1)}^j\\right)\n= - \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j-l}}^{j-l} - z^j_{i+1}\\right)\n\\end{equation}\nfor all $l \\in {{\\mathbb Z}}\/d$ with $l \\neq 1$. Thus \n$z_0^{j+l}$ is R-mobile in $\\left\\langle z_{n_j - (i+1)}^j, z_{n_j - i}^j \\right]$ \nif and only if $z_{n_{j-l}}^{j-l}$ is L-mobile in\n$\\left\\langle z_i^j, z_{i+1}^j \\right]$. We call these pairs mirror as well.\n\n\n\n\nWe shall use the term ``mobile point'' to describe a point which is either\nR-mobile or L-mobile. We say that a point is mobile (respectively\nR-mobile or L-mobile) in ${\\bf{z}}^j$ if it is mobile (respectively\nR-mobile or L-mobile) rel $z_0^j$ or rel $z_{n_j}^j$. Note that\nit is possible for a single mobile point to have both a ``rel $z_0^j$''\nand a ``rel $z_{n_j}^j$'' description. Indeed, this is always the case\nwhen $i \\notin \\left\\{0, \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2\\right\\}$ and $l \\notin \\{0,1\\}$.\n\nLet us pause here to explain what we mean by\nmeasuring positions relative to $z_0^j$ or to $z_{n_j}^j$.\nSuppose that for some nonzero $l \\in{{\\mathbb Z}}\/d$, we know that\n$z_0^{j+l}$ is R-mobile rel $z_0^j$. Then there exists some\n$i \\in \\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2\\}$ such that\n$z_0^{j+l}$ is R-mobile in $\\left\\langle z_i^j, z_{i+1}^j\\right]$, and so\n\\begin{equation}\n i\\left[dq\\right]_{k^2}\n< \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l} - z_0^j\\right) \n< (i+1)\\left[dq\\right]_{k^2}.\n\\end{equation}\nNote that if $\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\left((m\\!+\\!c)k\\!+\\!1\\right) > k^2$,\nthen more than one value of $i$ could satisfy this property.\nLikewise, if for some nonzero $l\\in{{\\mathbb Z}}\/d$ we know that\n$z_{n_{j-l}}^{j-l}$ is L-mobile rel $z_{n_j}^j$, then there exists some\n$i \\in \\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2\\}$ such that\n$z_{n_{j-l}}^{j-l}$ is L-mobile in\n$\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j \\right]$, and so\n\\begin{equation}\n -(i+1)\\left[dq\\right]_{k^2}\n< \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j-l}}^{j-l} - z_{n_j}^j\\right)\n< -i\\left[dq\\right]_{k^2}.\n\\end{equation}\nAgain, it is possible for more than one value of $i$ to satisfy this property.\n\n\n\nOne could justifiably argue that a ``mobile point''\nshould really be called a ``collection of points,''\nbut it is better to think of a mobile point $z_0^{j+l}$\nas an element of ${{\\mathbb Z}}\/k^2$ that is a function of $j$,\njust as the position $x(t)$ of a particle on a line\nis viewed as an element of ${{\\mathbb R}}$ that is a function of $t$.\nIndeed, it is perhaps useful to think of $j$ as a discrete\ntime variable. For example, suppose that $z_0^{j+l}$ is\nR-mobile rel $z_0^j$, hence R-mobile in some interval\n$\\left\\langle z_i^j, z_{i+1}^j \\right]$. This means\nthat as $j$ varies, the point $z_0^{j+l} - z_0^j$\nhops back and forth between $x$ and $x + dq$,\nfor some fixed $x \\in \n\\left\\langle i\\,dq, (i+1)dq \\right\\rangle$.\nIf $z_0^{j+l}$ is R-mobile rel $z_{n_j}^j$,\nhence R-mobile in some interval\n$\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j \\right]$,\nthen as $j$ varies, the point $z_0^{j+l} - z_{n_j}^j$\nhops back and forth between $x$ and $x+dq$\nfor some fixed $x \\in\n\\left\\langle -(i+1)dq, -i\\,dq \\right\\rangle$.\nThis is the type of motion indicated in the term ``mobile,''\nand the type of ``worldline'' viewpoint by which we call the collection\n$\\{z_0^{j+l}\\}_{j\\in{{{\\mathbb Z}}\/d}}$ a ``point.''\n\n\n\n\nWe shall use the terms {\\em active} and {\\em inactive} to describe\nwhether such an R-mobile point is occupying position $x$ or position\n$x + dq$ at a particular time $j=j^{\\prime}\\in{{\\mathbb Z}}\/d$.\nThat is, we say that an R-mobile point \n $z_0^{j+l}$ in $\\left\\langle z_i^j, z_{i+1}^j \\right]$\n is active at time $j=j^{\\prime} \\in {{\\mathbb Z}}\/d$ if\n\\begin{equation}\n z_0^{j^{\\prime}+l} - z_i^{j^{\\prime}}\n\\;=\\; \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l} - z_i^j\\right)\n\\end{equation}\nand inactive otherwise. Likewise, we say that\nan L-mobile point $z_{n_{j-l}}^{j-l}$ in $\\left\\langle z_{n_j - (i+1)}^j, z_{n_j - i}^j \\right]$\nis active at time $j = j^{\\prime} \\in {{\\mathbb Z}}\/d$ if\n\\begin{equation}\nz_{n_{j^{\\prime}-l}}^{j^{\\prime}-l} - z_{n_{j^{\\prime}} - i}^{j^{\\prime}}\n= \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j-l}}^{j-l} - z_{n_j - i}^j\\right)\n\\end{equation}\nand is inactive otherwise.\nThe obvious analogous definitions for active and inactive\nhold for an R-mobile point rel $z_{n_j}^j$\nand an L-mobile point rel $z_0^j$.\n\n\nThe fact that $[dq]_{k^2} < \\frac{k^2}{2}$ for $q$ of positive type\nimplies that an R-mobile point $z_0^{j+l}$ in $\\left\\langle z_i^j, z_{i+1}^j \\right]$\nsatisfies\n\\begin{equation}\n\\label{eq: active means in for R-mobile}\nz_0^{j+l}\\;\\text{is active at time}\\;j=j^{\\prime}\n\\;\\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;\\;\nz_0^{j^{\\prime}+l} \\in \\left\\langle z_i^{j^{\\prime}}, z_{i+1}^{j^{\\prime}} \\right]\n\\end{equation}\nfor all $j^{\\prime}\\in{{\\mathbb Z}}\/d$, and that an L-mobile point \n$z_{n_{j-l}}^{j-l}$ in $\\left\\langle z_{n_j - (i+1)}^j, z_{n_j - i}^j \\right]$ satisfies\n\\begin{equation}\n\\label{eq: active means in for L-mobile}\nz_{n_{j-l}}^{j-l}\\;\\text{is active at time}\\;j=j^{\\prime}\n\\;\\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;\\;\nz_{n_{j^{\\prime}-l}}^{j^{\\prime}-l}\n\\in \\left\\langle z_{n_{j^{\\prime}}-(i+1)}^{j^{\\prime}}, z_{n_{j^{\\prime}}-i}^{j^{\\prime}} \\right]\n\\end{equation}\nfor all $j^{\\prime}\\in{{\\mathbb Z}}\/d$ (and similarly for an R-mobile point rel $z_{n_j}^j$\nor an L-mobile point rel $z_0^j$). This is because, for example, if $z_0^{j+l}$ is\nR-mobile in $\\left\\langle z_i^j, z_{i+1}^j\\right]$, then\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right) \\in \\left\\langle 0, [dq]_{k^2} \\right\\rangle$,\nimplying $\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right) \\in \n\\left\\langle [dq]_{k^2}, 2[dq]_{k^2} \\right\\rangle$.\nOrdinarily, there would be nothing to prevent the intersection\n$\\left\\langle [dq]_{k^2}, 2[dq]_{k^2} \\right\\rangle \\cap\n\\left\\langle 0, [dq]_{k^2} \\right\\rangle$ from being nonempty,\nbut since $[dq]_{k^2} < \\frac{k^2}{2}$, we know that\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right) \\notin \\left\\langle 0, [dq]_{k^2} \\right\\rangle$.\nThus $z_0^{j^{\\prime}+l} \\in \\left\\langle z_i^{j^{\\prime}}, z_{i+1}^{j^{\\prime}} \\right]$\nwhen $z_0^{j+l}$ is inactive at time $j=j^{\\prime}$.\n\n\n\n\n\nThe notion of active versus inactive is important because it determines how\na mobile point contributes to the intersection of ${\\tilde{Q}}_q$ with the\ninterval in question at a particular time.\n\\begin{prop}\nFor any $i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor-2\\right\\}$,\nthere is some $C_i \\in {{\\mathbb Z}}$, constant in $j^{\\prime} \\in {{\\mathbb Z}}\/d$, such that\n\\begin{equation}\n\\label{eq: contribution of active mobile points to Q_q cap < z_i^j, z_{i+1}^j ] }\n \\#\\left({\\tilde{Q}}_q \\cap \\left\\langle z_i^{j^{\\prime}}, \n z_{i+1}^{{j^{\\prime}}} \\right] \\right)\n\\;=\\; C_i \\;+\\; \n \\#\\!\\left\\{\\begin{array}{cc}\n \\mathrm{mobile\\; points\\; in}\\; \\left\\langle z_i^j, z_{i+1}^j \\right]\n \\\\\n \\mathrm{which\\; are\\; active\\; at\\; time}\\;j^{\\prime}\n \\end{array}\n \\right\\}\n\\end{equation}\nfor each $j^{\\prime} \\in {{\\mathbb Z}}\/d$.\nLikewise, \nfor any $i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor-2\\right\\}$,\nthere is some ${\\bar{C}}_i \\in {{\\mathbb Z}}$, constant in $j^{\\prime} \\in {{\\mathbb Z}}\/d$, such that\n\\begin{equation}\n\\label{eq: contribution of active mobile points to Q_q cap < z_{n_j-(i+1)}^j, etc}\n \\#\\left({\\tilde{Q}}_q \\cap \\left\\langle z_{n_{j^{\\prime}}-(i+1)}^{j^{\\prime}}, \n z_{n_{j^{\\prime}}-i}^{j^{\\prime}} \\right] \\right)\n\\;=\\; {\\bar{C}}_i \\;+\\; \n \\#\\!\\left\\{\\begin{array}{cc}\n \\mathrm{mobile\\; points\\; in}\\; \\left\\langle z_{n_j-(i+1)}^j, z_{n_j - i}^j \\right]\n \\\\\n \\mathrm{which\\; are\\; active\\; at\\; time}\\;j^{\\prime}\n \\end{array}\n \\right\\}\n\\end{equation}\nfor each $j^{\\prime} \\in {{\\mathbb Z}}\/d$.\n\\end{prop}\n\\begin{proof}\nWe restrict ourselves to the proof of\n(\\ref{eq: contribution of active mobile points to Q_q cap < z_i^j, z_{i+1}^j ] }),\nsince\n(\\ref{eq: contribution of active mobile points to Q_q cap < z_{n_j-(i+1)}^j, etc})\nthen follows from a mirror argument.\nEquation (\\ref{eq: contribution of active mobile points to Q_q cap < z_i^j, z_{i+1}^j ] })\ncounts the number of elements of $Q_q$\ncontained in the interval $\\left\\langle z_i^{j^{\\prime}}, z_{i+1}^{j^{\\prime}} \\right]$,\nfor any given $j^{\\prime} \\in {{\\mathbb Z}}\/d$. Thinking of this interval as\n$z_i^{j^{\\prime}} + \\left\\langle 0, dq \\right]$,\nwe can describe all the elements of $\\left\\langle 0, dq \\right] \\cap {{\\mathbb Z}}$\nas belonging to one of three categories. For any\n$x \\in \\left\\langle 0, dq \\right] \\cap {{\\mathbb Z}}$, one of the following is true:\n\\begin{itemize}\n\\item[(i)]\n {$z_i^{j^{\\prime}} + x$ is never an element of $Q_q$,\n regardless of the value of $j^{\\prime} \\in {{\\mathbb Z}}\/d$.}\n\n\\item[(ii)]\n {$z_i^{j^{\\prime}} + x$ is always an element of $Q_q$,\n for any value of $j^{\\prime} \\in {{\\mathbb Z}}\/d$.}\n\n\\item[(iii)]\n {$z_i^{j^{\\prime}} + x$ is sometimes an element of $Q_q$,\n and sometimes not, depending on $j^{\\prime} \\in {{\\mathbb Z}}\/d$.}\n\\end{itemize}\nEquation (\\ref{eq: contribution of active mobile points to Q_q cap < z_i^j, z_{i+1}^j ] })\nthen records the contribution of $x$ to\n$\\#\\left({\\tilde{Q}}_q \\cap \\left\\langle z_i^{j^{\\prime}}, z_{i+1}^{{j^{\\prime}}} \\right] \\right)$\nas follows.\nThere are no contributions from $x$ of type (i);\nwe define $C_i$ to count the \ncontributions from $x$ of type (ii); and\nas explained below, the second summand\nof (\\ref{eq: contribution of active mobile points to Q_q cap < z_i^j, z_{i+1}^j ] })\ncounts the contributions from $x$ of type (iii).\n\nThe point of Corollary 3.10 is that, generically speaking, most $x$ which are not of\ntype (i) are of type (ii). That is, suppose, for a given\n$x \\in \\left\\langle 0, dq \\right] \\cap {{\\mathbb Z}}$, that there exists some $j_0 \\in {{\\mathbb Z}}\/d$ for which\n $z_i^{j_0} + x \\in Q_q$.\nThen there exist $l \\in {{\\mathbb Z}}\/d$ and $i_0 \\in \\left\\{0, ..., n_{j_0+l}\\right\\}$ for which\n $z_i^{j_0} + x = z_{i_0}^{j_0+l}$.\nThe following argument, which holds in all but two exceptional cases\ndescribed below, shows that $x$ must be of type (ii).\nBy Corollary 3.10, the fact that $z_i^{j_0} + x = z_{i_0}^{j_0+l}$ implies that either\n $x = \\minq_{j \\in {{\\mathbb Z}}\/d}\\left( z_{i_0}^{j+l} - z_i^j\\right)$,\nin which case\n $z_i^{j^{\\prime}} + x \\in \\left\\{ z_{i_0}^{j^{\\prime}+l}, z_{i_0-1}^{j^{\\prime}+1} \\right\\}$\nfor all $j^{\\prime} \\in {{\\mathbb Z}}\/d$, or\n $x = \\maxq_{j \\in {{\\mathbb Z}}\/d}\\left( z_{i_0}^{j+l} - z_i^j\\right)$,\nin which case\n $z_i^{j^{\\prime}} + x \\in \\left\\{ z_{i_0}^{j^{\\prime}+l}, z_{i_0+1}^{j^{\\prime}+1} \\right\\}$\nfor all $j^{\\prime} \\in {{\\mathbb Z}}\/d$.\nThus $z_i^{j^{\\prime}} + x \\in Q_q$ for all $j^{\\prime} \\in {{\\mathbb Z}}\/d$,\nand so $x$ is of type (ii).\n\nThe above argument fails in precisely two cases,\nin each of which, $x$ is of type (iii).\n\\begin{itemize}\n\\item[Case R.]\n{If $i_0 = 0$, and $x = \\minq_{j \\in {{\\mathbb Z}}\/d}\\left( z_{i_0}^{j+l} - z_i^j\\right)$,\nthen $z_{0-1}^{j^{\\prime}-1}$ does not exist. Thus\n $z_i^{j^{\\prime}} + x \\in Q_q$\nif and only if\n $z_i^{j^{\\prime}} + x = z_0^{j^{\\prime}+l}$,\nwhich is true if and only if\n $z_0^{j^{\\prime}+l} - z_i^{j^{\\prime}}\n = \\minq_{j \\in {{\\mathbb Z}}\/d}\\left( z_0^{j+l} - z_i^j\\right)$,\nwhich, by definition, is true if and only if\n $z_0^{j+l}$ is active as an R-mobile point\nin $\\left\\langle z_i^j, z_{i+1}^j \\right]$ at time $j=j^{\\prime}$.}\n\n\\item[Case L.]\n{If $i_0 = n_{j_0+l}$, and $x = \\maxq_{j \\in {{\\mathbb Z}}\/d}\\left( z_{i_0}^{j+l} - z_i^j\\right)$,\nand there exists $j''$ for which\n $z_i^{j''} + x = z_{n_{j''+l}}^{j''+l} + dq$ (which is not in $Q_q$).\nWe then say that $z_{n_{j+l}}^{j+l}$ is L-mobile\nin $\\left\\langle z_i^j, z_{i+1}^j \\right]$. In this case,\n $z_i^{j^{\\prime}} + x \\in Q_q$\nif and only if\n $z_i^{j^{\\prime}} + x = z_{n_{j^{\\prime}+l}}^{j^{\\prime}+l}$,\nwhich is true if and only if\n $z_{n_{j^{\\prime}+l}}^{j^{\\prime}+l} - z_i^{j^{\\prime}}\n = \\maxq_{j \\in {{\\mathbb Z}}\/d}\\left( z_{i_0}^{j+l} - z_i^j\\right)$,\nwhich, by definition, is true if and only if\n $z_{n_{j+l}}^{j+l}$ is active as an L-mobile point\nin $\\left\\langle z_i^j, z_{i+1}^j \\right]$ at time $j=j^{\\prime}$.}\n\\end{itemize}\nThus, the second summand of\n(\\ref{eq: contribution of active mobile points to Q_q cap < z_i^j, z_{i+1}^j ] })\ncounts all $x \\in \\left\\langle 0, dq \\right] \\cap {{\\mathbb Z}}$ of type (iii).\n\nNote that the above argument makes no use of equations\n(\\ref{eq: active means in for R-mobile}) and \n(\\ref{eq: active means in for L-mobile}).\nIn particular, the above argument would hold\neven if $[dq]_{k^2}$ failed to satisfy the condition $[dq]_{k^2} < \\frac{k^2}{2}$.\n\\end{proof}\n\n\n\nCorollary \\ref{cor: q of positive type: combo of lemma and difference eq}\nimplies that for each mobile point, there must be at least one time when\nthe mobile point is active and at least one time when the mobile point is inactive.\nIn fact, we can specify precisely how many times a given mobile point is active.\n\n\\begin{prop}\n\\label{prop: mobile point is active [lepsilon] times or [(l-1)epsilon] times}\nSuppose that $q$ of positive type is genus-minimizing.\nIf $z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_i^j, z_{i+1}^j \\right]$ (and hence\n$z_{n_{j-l}}^{j-l}$ is L-mobile in\n$\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j \\right]$), for some $l \\neq 0 \\in {{\\mathbb Z}}\/d$\nand $i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2\\right\\}$,\nthen each of the two mobile points is active precisely\n$[l\\epsilon]_d$ times.\nIf $z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j \\right]$\n(and hence $z_{n_{j-l}}^{j-l}$ is L-mobile in\n$\\left\\langle z_i^j, z_{i+1}^j \\right]$), for some $l \\neq 1 \\in {{\\mathbb Z}}\/d$\nand $i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2\\right\\}$,\nthen each of the two mobile points is active precisely\n$[(l-1)\\epsilon]_d$ times.\n\\end{prop}\n\\begin{proof}\nSuppose the hypothesis of the first statement is true.\nThen for all $j\\in {{\\mathbb Z}}\/d$, \n(\\ref{eq: z_0^j+l - z_0^j = mu ml k^2\/d + xi dq}) implies that\n\\begin{align}\n\\label{prop: mobile active le times. eq: explicit equation for z_0^j+l -z_0^j}\n z^{j+l}_0 - z^{j}_i\n&= \\left[{\\mu}ml\\right]_d\\frac{k^2}{d}\n \\;+\\; \\left(\\Xi^{d, \\epsilon}_l(j)-i\\right) [dq]_{k^2},\n \\\\ \\nonumber\n z_{n_{j-l}}^{j-l} - z_{n_j-i}^j\n&= -\\left[{\\mu}ml\\right]_d\\frac{k^2}{d}\n \\;-\\; \\left(\\Xi^{d, \\epsilon}_l(j-l+1)-i\\right) [dq]_{k^2},\n\\end{align}\nwhere, by\nLemma \\ref{lemma: q of positive type, xi lemma},\n$\\Xi^{d, \\epsilon}_l(j) \\in \\left\\{ \\frac{[l\\epsilon]_d}{d}, \\frac{[l\\epsilon]_d}{d}-1 \\right\\}$\nfor all $j\\in{{\\mathbb Z}}\/d$, with $\\frac{[l\\epsilon]_d}{d}$ occurring\n$[-l\\epsilon]_d$ times and $\\frac{[l\\epsilon]_d}{d} - 1$\noccurring $[l\\epsilon]_d$ times.\nSince (\\ref{prop: mobile active le times. eq: explicit equation for z_0^j+l -z_0^j})\nimplies $z_0^{j+l}$ is active in\n$\\left\\langle z_i^j, z_{i+1}^j\\right]$\n(respectively $z_{n_{j-l}}^{j-l}$ is active in\n$\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j \\right]$)\nat time $j = j^{\\prime}$ if and only if\n$\\Xi^{d, \\epsilon}_l(j^{\\prime}) = \\frac{[l\\epsilon]_d}{d}-1$\n(respectively $\\Xi^{d, \\epsilon}_l(j^{\\prime}-l+1) = \\frac{[l\\epsilon]_d}{d}-1$),\nwe conclude that each of the two mobile points is active\nprecisely $[l\\epsilon]_d$ times.\n\nNext, suppose the hypothesis of the second statement is true.\nThen for all $j\\in {{\\mathbb Z}}\/d$, \n(\\ref{eq: z_0^j+l - z_0^j = mu ml k^2\/d + xi dq}) implies that\n\\begin{align}\n\\label{prop: mobile active le times. eq: explicit equation for z_0^j+l -z_n_j^j}\n z^{j+l}_0 - z_{n_j -(i+1)}^j\n&= \\left[{\\mu}m(l-1)\\right]_d\\frac{k^2}{d}\n \\;+\\; \\left(\\Xi^{d, \\epsilon}_{l-1}(j+1)+(i+1)\\right) [dq]_{k^2} + \\psi,\n \\\\ \\nonumber\n z_{n_{j-l}}^{j-l} - z_{i+1}^j\n&= -\\left[{\\mu}m(l-1)\\right]_d\\frac{k^2}{d}\n \\;-\\; \\left(\\Xi^{d, \\epsilon}_{l-1}(j-l+1)+(i+1)\\right) [dq]_{k^2} - \\psi.\n\\end{align}\nHere,\n$\\Xi^{d, \\epsilon}_{l-1}(j) \\in \\left\\{ \\frac{[(l-1)\\epsilon]_d}{d}, \\frac{[(l-1)\\epsilon]_d}{d}-1 \\right\\}$\nfor all $j\\in{{\\mathbb Z}}\/d$, with $\\frac{[(l-1)\\epsilon]_d}{d}$ occurring\n$[-(l-1)\\epsilon]_d$ times and $\\frac{[(l-1)\\epsilon]_d}{d} - 1$\noccurring $[(l-1)\\epsilon]_d$ times.\nSince (\\ref{prop: mobile active le times. eq: explicit equation for z_0^j+l -z_n_j^j})\nimplies $z_0^{j+l}$ is active in\n$\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j \\right]$\n(respectively $z_{n_{j-l}}^{j-l}$ is active in\n$\\left\\langle z_i^j, z_{i+1}^j\\right]$)\nat time $j = j^{\\prime}$ if and only if\n$\\Xi^{d, \\epsilon}_{l-1}(j^{\\prime}+1) = \\frac{[(l-1)\\epsilon]_d}{d}-1$\n(respectively $\\Xi^{d, \\epsilon}_{l-1}(j^{\\prime}-l+1) = \\frac{[(l-1)\\epsilon]_d}{d}-1$),\nwe conclude that each of the two mobile points is active\nprecisely $[(l-1)\\epsilon]_d$ times.\n\n\\end{proof}\n\n\n\n\n\nWe next define what it means for a mobile point to be {\\em neutralized}.\nAn R-mobile point $z_0^{j+l}$ in \n$\\left\\langle z_i^j, z_{i+1}^j\\right]$, for some $l \\neq 0 \\in {{\\mathbb Z}}\/d$\n(respectively in $\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j\\right]$,\nfor some $l \\neq 1 \\in {{\\mathbb Z}}\/d$),\nis called neutralized if\n$z_{n_{j+l-1}}^{j+l-1}$ is also mobile in \n$\\left\\langle z_i^j, z_{i+1}^j\\right]$\n(respectively in $\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j\\right]$).\nLikewise, an L-mobile point $z_{n_{j-l}}^{j-l}$ in\n$\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j\\right]$, for some $l \\neq 0 \\in {{\\mathbb Z}}\/d$\n(respectively in $\\left\\langle z_i^j, z_{i+1}^j\\right]$,\nfor some $l \\neq 1 \\in {{\\mathbb Z}}\/d$),\nis called neutralized if\n$z_0^{j-l+1}$ is also mobile in \n$\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j\\right]$\n(respectively in $\\left\\langle z_i^j, z_{i+1}^j\\right]$).\nA pair $z_0^{j+l}$, $z_{n_{j+l-1}}^{j+l-1}$\nof neutralized mobile points in\n$\\left\\langle z_i^j, z_{i + 1}^j\\right]$\n(respectively in $\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j\\right]$)\n is so called because,\nas shown in Proposition \\ref{prop: active iff inactive true iff neutralized},\n\\begin{equation}\n\\label{main prop, part (i), neutralizing def, active iff inactive}\nz_{n_{j+l}}^{j+l}\\;\\text{is active}\n\\;\\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;\\;\nz_0^{j+l+1}\\;\\text{is inactive}\n\\end{equation}\nin $\\left\\langle z_i^j, z_{i + 1}^j\\right]$\n(respectively in $\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j\\right]$)\nat every time $j=j^{\\prime} \\in {{\\mathbb Z}}\/d$.\nThus, their combined contribution to\n$\\#\\!\\left({\\tilde{Q}}_q \\cap \\left\\langle z_i^{j^{\\prime}}, \n z_{i+1}^{j^{\\prime}} \\right] \\right)$\n(respectively to $\\#\\!\\left({\\tilde{Q}}_q \\cap \\left\\langle z_{n_{j^{\\prime}}-(i+1)}^{j^{\\prime}}, \n z_{n_{j^{\\prime}}-i}^{j^{\\prime}} \\right] \\right)$)\nis always constant in $j^{\\prime} \\in {{\\mathbb Z}}\/d$.\nIn other words, each member of the neutralized pair\n``neutralizes'' the nonconstancy of the other member's contribution\nto $\\#\\!\\left({\\tilde{Q}}_q \\cap \\left\\langle z_i^{j}, \nz_{i+1}^{j} \\right] \\right)$\n(respectively to \n$\\#\\!\\left({\\tilde{Q}}_q \\cap \\left\\langle z_{n_j-(i+1)}^{j}, \nz_{n_j-i}^{j} \\right] \\right)$).\n\n\n\nProperty \n(\\ref{main prop, part (i), neutralizing def, active iff inactive})\nis also sufficient condition for two mobile points to form a neutralized pair,\nas we now demonstrate.\n\\begin{prop}\n\\label{prop: active iff inactive true iff neutralized}\nSuppose that $q$ is of positive type, and that\n$z_0^{j+l_1}$ and $z_{n_{j-l_2}}^{j-l_2}$ are mobile in\n$\\left\\langle z_i^j, z_{i+1}^j\\right]$,\nfor some $l_1 \\neq 0, l_2\\neq 1 \\in {{\\mathbb Z}}\/d$\n(respectively in $\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j\\right]$,\nfor some $l_1 \\neq 1, l_2 \\neq 0 \\in {{\\mathbb Z}}\/d$). Then\n$z_0^{j+l_1}$ and $z_{n_{j-l_2}}^{j-l_2}$ form a neutralized pair\n({\\em i.e.}, $l_1 + l_2 = 1$) if and only if they satisfy\n\\begin{equation}\n\\nonumber\nz_{n_{j-l_2}}^{j-l_2}\\;\\text{is active}\n\\;\\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;\\;\nz_0^{j+l_1}\\;\\text{is inactive}\n\\end{equation}\nin $\\left\\langle z_i^j, z_{i+1}^j\\right]$\n(respectively in $\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j\\right]$)\nat all times $j=j^{\\prime} \\in {{\\mathbb Z}}\/d$.\n\\end{prop}\n\n\\begin{proof}\nWe begin with the ``only if'' statement.\nSuppose, for some $l \\neq 0 \\in {{\\mathbb Z}}\/d$,\nthat $z_0^{j+l}$ and $z_{n_{j+l-1}}^{j+l-1}$ are mobile in\n$\\left\\langle z_i^j, z_{i+1}^j\\right]$.\nThen for any $j^{\\prime}\\in{{\\mathbb Z}}\/d$, we have\n\\begin{align}\n z_{n_{j^{\\prime}+l-1}}^{j^{\\prime}+l-1} - z_{i+1}^{j^{\\prime}}\n&= \\left(z_0^{j^{\\prime}+l}-\\psi\\right) \n -\\left(z_i^{j^{\\prime}}+ dq\\right)\n \\\\ \\nonumber\n&= \\left(z_0^{j^{\\prime}+l} - z_i^{j^{\\prime}}\\right)\n -\\psi - dq.\n\\end{align}\nIn particular, $\\left(z_{n_{j^{\\prime}+l-1}}^{j^{\\prime}+l-1} - z_{i+1}^{j^{\\prime}}\\right)\n-\\left(z_0^{j^{\\prime}+l} - z_i^{j^{\\prime}}\\right)$ is constant in $j^{\\prime}\\in {{\\mathbb Z}}\/d$.\nThus, at any time $j=j^{\\prime} \\in {{\\mathbb Z}}\/d$,\n\\begin{equation}\nz_{n_{j^{\\prime}+l-1}}^{j^{\\prime}+l-1} - z_{i+1}^{j^{\\prime}}\n\\!= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1} - z_{i+1}^j\\right)\n\\;\\;\\;\\Leftrightarrow\\;\\;\\;\n z_0^{j^{\\prime}+l} - z_i^{j^{\\prime}}\n\\!= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}\\! - z_i^j\\right),\n\\end{equation}\nwhich means that at any time $j=j^{\\prime} \\in {{\\mathbb Z}}\/d$,\n$z_{n_{j+l-1}}^{j+l-1} $ is active in $\\left\\langle z_i^j, z_{i+1}^j\\right]$\nif and only if $z_0^{j+l}$ is inactive\nin $\\left\\langle z_i^j, z_{i+1}^j\\right]$.\n\n\nNext, we prove the ``if'' statement.\nSuppose we know, for every $j^{\\prime} \\in {{\\mathbb Z}}\/d$, that\n$z_{n_{j-l_2}}^{j-l_2}$ is active\nin $\\left\\langle z_i^j, z_{i+1}^j\\right]$\nat time $j=j^{\\prime}$ if and only if\n$z_0^{j+l_1}$ is inactive\nin $\\left\\langle z_i^j, z_{i+1}^j\\right]$\nat time $j=j^{\\prime}$.\nThen at any given time $j=j^{\\prime}\\in{{\\mathbb Z}}\/d$, either\n$z_0^{j+l_1}$ is active and $z_{n_{j-l_2}}^{j-l_2}$ is inactive, so that\n\\begin{align}\n z_0^{j^{\\prime}+l_1} - z_{n_{j^{\\prime}-l_2}}^{j^{\\prime}-l_2}\n&= \\left(z_0^{j^{\\prime}+l_1} - z_i^{j^{\\prime}}\\right)\n - \\left(z_{n_{j^{\\prime}-l_2}}^{j^{\\prime}-l_2} - z_{i+1}^{j^{\\prime}}\\right)\n - \\left(z_{i+1}^{j^{\\prime}} - z_i^{j^{\\prime}}\\right)\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l_1} - z_i^j\\right)\n -\\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j-l_2}}^{j-l_2} - z_{i+1}^j\\right)\n - dq,\n\\end{align}\nor $z_0^{j+l_1}$ is inactive and $z_{n_{j-l_2}}^{j-l_2}$ is active, so that\n\\begin{align}\n z_0^{j^{\\prime}+l_1} - z_{n_{j^{\\prime}-l_2}}^{j^{\\prime}-l_2}\n&= \\left(z_0^{j^{\\prime}+l_1} - z_i^{j^{\\prime}}\\right)\n - \\left(z_{n_{j^{\\prime}-l_2}}^{j^{\\prime}-l_2} - z_{i+1}^{j^{\\prime}}\\right)\n - \\left(z_{i+1}^{j^{\\prime}} - z_i^{j^{\\prime}}\\right)\n \\\\ \\nonumber\n&= \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l_1} - z_i^j\\right)\n -\\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j-l_2}}^{j-l_2} - z_{i+1}^j\\right)\n - dq\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l_1} - z_i^j\\right)\n -\\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j-l_2}}^{j-l_2} - z_{i+1}^j\\right)\n - dq.\n\\end{align}\nThus $z_0^{j+l_1} - z_{n_{j-l_2}}^{j-l_2}$ is constant in $j\\in{{\\mathbb Z}}\/d$,\nwhich means that $z_0^{j+l_1} - z_0^{j-l_2+1}$ is constant in $j\\in{{\\mathbb Z}}\/d$,\nwhich, by Corollary \\ref{cor: q of positive type: combo of lemma and difference eq},\nimplies that $(j+l_1) - (j-l_2+1) = 0$, and so $l_1+l_2 = 1$.\n \\\\\n\nThe analogous proof for mobile points in\n$\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j\\right]$ is the same, but with\n$l_1 \\neq 0$, $l_2 \\neq 1$, $z_i^j$, $z_{i+1}^j$, $z_i^{j^{\\prime}}$, and\n$z_{i+1}^{j^{\\prime}}$ replaced with\n$l_1 \\neq 1$, $l_2 \\neq 0$, $z_{n_j-(i+1)}^j$, $z_{n_j-i}^j$, \n$z_{n_{j^{\\prime}}-(i+1)}^{j^{\\prime}}$, and\n$z_{n_{j^{\\prime}}-i}^{j^{\\prime}}$, respectively.\n\\end{proof}\n \n \n\n\n\nLastly, we introduce the notion of the {\\em pseudomobile point},\nwhich plays a role analogous to that of an ordinary mobile point,\nbut in the interval\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$.\n{\\em A pseudomobile point is not a mobile point!}\nThe term mobile point {\\em only} pertains to intervals of the form\n$\\left\\langle z_i^j, z_{i+1}^j \\right]$ or\n$\\left\\langle z_{n_j - (i+1)}^j, z_{n_j - i}^j \\right]$\nfor some $i \\in \\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2 \\}$.\nThus, the question of whether ${\\bf{z}}^j$ is said to have mobile points\nhas nothing to do with whether\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$, which straddles \n${\\bf{z}}^{j-1}$ and ${\\bf{z}}^j$, is said to have\npseudomobile points.\n\n\nWith that caveat out of the way, we commence with a definition.\nFirst, for the remainder of Section \\ref{s:p=k2}, we fix\n\\begin{align}\n \\psi \n:=& \\left[z_0^j - z_{n_{j-1}}^{j-1}\\right]_{k^2}\\;\\;\\;\\text{(which is constant in}\\;j\\in{{\\mathbb Z}}\/d\\text{)}\n \\\\ \\nonumber\n=& \\left[dq - kq\\right]_{k^2}\n \\\\ \\nonumber\n=& \\left[(({\\mu}m+{\\gamma}c)k+\\alpha) \n \\;-\\; \\textstyle{\\gamma\\frac{ck+\\alpha\\gamma}{d}}k\\right]_{k^2}.\n\\end{align}\nThen, for any $l \\neq 0 \\in {{\\mathbb Z}}\/d$, we say that\n$z_0^{j+l}$ is an R-pseudomobile point in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$ if\n\\begin{equation}\n 0\n\\;<\\; \\minq_{j\\in{{\\mathbb Z}}\/d} \\left( z_0^{j+l} - z_{n_{j-1}}^{j-1} \\right)\n\\;<\\; \\psi,\n\\end{equation}\nand that $z_{n_{j-1-l}}^{j-1-l}$ is an\nL-pseudomobile point in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$ if\n\\begin{equation}\n -\\psi\n\\;<\\; \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left( z_{n_{j-1-l}}^{j-1-l}- z_0^j \\right)\n\\;<\\; 0.\n\\end{equation}\nWe say that a point is pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$ if it is\nR-pseudomobile or L-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$.\n\n\nUsing Corollary \\ref{cor: q of positive type: combo of lemma and difference eq},\nit is easy to calculate that\n\\begin{align}\n \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_{n_{j-1}}^{j-1}\\right)\n&= [{\\mu}ml]_d\\frac{k^2}{d} + \\left(\\frac{[l\\epsilon]_d}{d}-1\\right)[dq]_{k^2} + \\psi,\n \\\\\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-1-l}}^{j-1-l}-z_0^j\\right)\n&= -[{\\mu}ml]_d\\frac{k^2}{d} - \\left(\\frac{[l\\epsilon]_d}{d}-1\\right)[dq]_{k^2} - \\psi,\n\\end{align}\nfor all nonzero $l\\in{{\\mathbb Z}}\/d$. Thus\n\\begin{equation}\n \\minq_{j\\in{{\\mathbb Z}}\/d} \\left( z_0^{j+l} - z_{n_{j-1}}^{j-1} \\right)\n= -\\maxq_{j\\in{{\\mathbb Z}}\/d} \\left( z_{n_{j-1-l}}^{j-1-l} - z_0^j \\right)\n\\end{equation}\nfor all nonzero $l\\in{{\\mathbb Z}}\/d$.\nIn particular, $z_0^{j+l}$ is R-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$ if and only if\n$z_{n-1-l}^{j-1-l}$ is L-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$.\nWe therefore say that $z_0^{j+l}$ and\n$z_{n_{j-1-l}}^{j-1-l}$ are mirror pseudomobile points\nin $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$.\n\n\n\n\nNext, we discuss the notion of active and inactive\npseudomobile points.\nIf $z_0^{j+l}$ is R-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$,\nthen we say that $z_0^{j+l}$ is active\nat time $j=j^{\\prime} \\in {{\\mathbb Z}}\/d$ if \n\\begin{equation}\n z_0^{j^{\\prime}+l} - z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}\n\\;=\\; \\minq_{j\\in{{\\mathbb Z}}\/d} \\left( z_0^{j+l} - z_{n_{j-1}}^{j-1} \\right),\n\\end{equation}\nand is inactive otherwise. Likewise,\nif $z_{n_{j-1-l}}^{j-1-l}$ is L-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$,\nthen $z_{n_{j-1-l}}^{j-1-l}$ is active at time $j = j^{\\prime}\\in{{\\mathbb Z}}\/d$ if\n\\begin{equation}\n z_{n_{j^{\\prime}-1-l}}^{j^{\\prime}-1-l} - z_0^{j^{\\prime}}\n\\;=\\; \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left( z_{n_{j-1-l}}^{j-1-l} - z_0^j \\right),\n\\end{equation}\nand is inactive otherwise.\nSimilar to the case of mobile points,\nthere exists $C \\in {{\\mathbb Z}}$ constant in $j^{\\prime} \\in {{\\mathbb Z}}\/d$\nsuch that, for any $j^{\\prime} \\in {{\\mathbb Z}}\/d$, we have\n\\begin{equation}\n \\#\\left({\\tilde{Q}}_q \\cap \\left\\langle z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}, \n z_0^{{j^{\\prime}}} \\right] \\right)\n\\;=\\; C \\;+\\; \n \\#\\!\\left\\{\\begin{array}{cc}\n \\text{pseudomobile points in}\\;\n \\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]\n \\\\\n \\text{which are active at time}\\; j=j^{\\prime} \\end{array}\n \\right\\}.\n\\end{equation}\nThe following proposition specifies how many times a given\npseudomobile point is active.\n\n\n\n\\begin{prop}\n\\label{prop: pseudomobile point is active [lepsilon] times}\nSuppose that $q$ of positive type is genus-minimizing.\nIf $z_0^{j+l}$ is R-pseudomobile (and hence\n$z_{n_{j-1-l}}^{j-1-l}$ is L-pseudomobile)\nin $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$,\nthen each of the two pseudomobile points is active precisely\n$[l\\epsilon]_d$ times.\n\\end{prop}\n\\begin{proof}\nFor all $j\\in {{\\mathbb Z}}\/d$, \n(\\ref{eq: z_0^j+l - z_0^j = mu ml k^2\/d + xi dq}) implies that\n\\begin{align}\n\\label{prop: pseudomobile active le times. eq: explicit equation for z_0^j+l -z_0^j}\n z^{j+l}_0 - z_{n_{j-1}}^{j-1}\n&= \\left[{\\mu}ml\\right]_d\\frac{k^2}{d}\n \\;+\\; \\left(\\Xi^{d, \\epsilon}_l(j)\\right) [dq]_{k^2} +\\psi,\n \\\\ \\nonumber\n z_{n_{j-1-l}}^{j-1-l} - z_0^j\n&= -\\left[{\\mu}ml\\right]_d\\frac{k^2}{d}\n \\;-\\; \\left(\\Xi^{d, \\epsilon}_l(j-l)\\right) [dq]_{k^2} - \\psi,\n\\end{align}\nwhere, by\nLemma \\ref{lemma: q of positive type, xi lemma},\n$\\Xi^{d, \\epsilon}_l(j) \\in \\left\\{ \\frac{[l\\epsilon]_d}{d}, \\frac{[l\\epsilon]_d}{d}-1 \\right\\}$\nfor all $j\\in{{\\mathbb Z}}\/d$, with $\\frac{[l\\epsilon]_d}{d}$ occurring\n$[-l\\epsilon]_d$ times and $\\frac{[l\\epsilon]_d}{d} - 1$\noccurring $[l\\epsilon]_d$ times.\nSince (\\ref{prop: pseudomobile active le times. eq: explicit equation for z_0^j+l -z_0^j})\nimplies $z_0^{j+l}$ (respectively $z_{n_{j-1-l}}^{j-1-l}$)\nis active in $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$\nat time $j = j^{\\prime}$ if and only if\n$\\Xi^{d, \\epsilon}_l(j^{\\prime}) = \\frac{[l\\epsilon]_d}{d}-1$\n(respectively $\\Xi^{d, \\epsilon}_l(j^{\\prime}-l) = \\frac{[l\\epsilon]_d}{d}-1$),\nwe conclude that each of the two pseudomobile points is active\nprecisely $[l\\epsilon]_d$ times.\n\\end{proof}\n\n\n\n\n\n\nFinally, we say that an R-pseudomobile point $z_0^{j+l}$ in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$\nis {\\em neutralized} if\n$z_{n_{j+l-1}}^{j+l-1}$ is L-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$,\nand that an L-pseudomobile point $z_{n_{j+l}}^{j+l}$ in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$\nis neutralized if\n$z_0^{j+l+1}$ is R-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$.\nWe say a pseudomobile point is {\\em non-neutralized}\nif it is not neutralized.\nThe reason for this terminology is as follows.\nAs proven in\nProposition \\ref{prop: active iff inactive means neutralized for pseudomobile}\nbelow, if $z_0^{j+l}$ and $z_{n_{j+l-1}}^{j+l-1}$ are pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$, then\nat any time $j^{\\prime} \\in {{\\mathbb Z}}\/d$,\n\\begin{equation}\nz_0^{j^{\\prime}+l}\\;\\text{is active}\n\\;\\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;\\;\nz_{n_{j^{\\prime}+l-1}}^{j^{\\prime}+l-1}\\;\\text{is inactive}.\n\\end{equation}\nThus, their combined associated contribution to\n$\\#\\!\\left({\\tilde{Q}}_q \\cap \\left\\langle z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}, \n z_0^{{j^{\\prime}}} \\right] \\right)$\nis always constant in $j^{\\prime} \\in {{\\mathbb Z}}\/d$.\n \\\\\n\n\n\\begin{prop}\n\\label{prop: active iff inactive means neutralized for pseudomobile}\nSuppose that $q$ of positive type is genus-minimizing, and\nthat $z_{n_{j+l_1}}^{j+l_1}$ and $z_0^{j+l_2}\\!$ are pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$\nfor some $l_1, l_2 \\in {{\\mathbb Z}}\/d$. Then\n$z_{n_{j+l_1}}^{j+l_1}$ and $z_0^{j+l_2}$ form a neutralized pair\n({\\em i.e.}, $l_1 + 1 = l_2$) if and only if they satisfy\n\\begin{equation}\n\\nonumber\nz_{n_{j+l_1}}^{j+l_1}\\;\\text{is active}\n\\;\\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;\\;\nz_0^{j+l_2}\\;\\text{is inactive}\n\\end{equation}\nin $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$ at all times $j=j^{\\prime} \\in {{\\mathbb Z}}\/d$.\n\\end{prop}\n\\begin{proof}\nWe begin with the ``only if'' statement.\nFor any $j^{\\prime}\\in{{\\mathbb Z}}\/d$, since\n\\begin{align}\n z_{n_{j^{\\prime}+l}}^{j^{\\prime}+l} - z_0^{j^{\\prime}}\n&= \\left(z_0^{j^{\\prime}+l+1}-\\psi\\right) \n -\\left(z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}+\\psi\\right)\n \\\\ \\nonumber\n&= \\left(z_0^{j^{\\prime}+l+1} - z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}\\right) -2\\psi,\n\\end{align}\nwe know that\n\\begin{equation}\nz_{n_{j^{\\prime}\\!+l}}^{j^{\\prime}\\!+l}\\! - z_0^{j^{\\prime}}\n\\!= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l}}^{j+l} - z_0^{j}\\right)\n\\;\\;\\;\\Leftrightarrow\\;\\;\\;\n z_0^{j^{\\prime}\\!+l+1}\\! - z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}\n\\!= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l+1}\\! - z_{n_{j-1}}^{j-1}\\right).\n\\end{equation}\nThus, at any time $j=j^{\\prime} \\in {{\\mathbb Z}}\/d$,\n$z_{n_{j+l}}^{j+l}$ is active in $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$\nif and only if $z_0^{j+l+1}$ is inactive\nin $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$.\n\n\n\nNext, we prove the ``if'' statement.\nSuppose that at any given time $j=j^{\\prime}\\in{{\\mathbb Z}}\/d$, either\n$z_0^{j+l_2}$ is active and $z_{n_{j+l_1}}^{j+l_1}$ is inactive\nin $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$, so that\n\\begin{align}\n z_0^{j^{\\prime}+l_2} - z_{n_{j^{\\prime}+l_1}}^{j^{\\prime}+l_1}\n&= \\left(z_0^{j^{\\prime}+l_2} - z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}\\right)\n - \\left(z_{n_{j^{\\prime}+l_1}}^{j^{\\prime}+l_1} - z_0^{j^{\\prime}}\\right)\n - \\left(z_0^{j^{\\prime}} - z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}\\right)\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l_2} - z_{n_{j-1}}^{j-1}\\right)\n -\\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j+l_1}}^{j+l_1} - z_0^j\\right)\n - \\psi,\n\\end{align}\nor $z_0^{j+l_2}$ is inactive and $z_{n_{j+l_1}}^{j+l_1}$ is active, so that\n\\begin{align}\n z_0^{j^{\\prime}+l_2} - z_{n_{j^{\\prime}+l_1}}^{j^{\\prime}+l_1}\n&= \\left(z_0^{j^{\\prime}+l_2} - z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}\\right)\n - \\left(z_{n_{j^{\\prime}+l_1}}^{j^{\\prime}+l_1} - z_0^{j^{\\prime}}\\right)\n - \\left(z_0^{j^{\\prime}} - z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}\\right)\n \\\\ \\nonumber\n&= \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l_2} - z_{n_{j-1}}^{j-1}\\right)\n -\\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j+l_1}}^{j+l_1} - z_0^j\\right)\n - \\psi\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l_2} - z_{n_{j-1}}^{j-1}\\right)\n -\\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_{n_{j+l_1}}^{j+l_1} - z_0^j\\right)\n - \\psi,\n\\end{align}\nThus $z_0^{j+l_2} - z_{n_{j+l_1}}^{j+l_1}$ is constant in $j\\in{{\\mathbb Z}}\/d$,\nwhich means that $z_0^{j+l_2} - z_0^{j+l_1+1}$ is constant in $j\\in{{\\mathbb Z}}\/d$,\nwhich, by Corollary \\ref{cor: q of positive type: combo of lemma and difference eq},\nimplies that $(j+l_2) - (j+l_1+1) = 0$, and so $l_1+1 = l_2$.\n\\end{proof}\n \n\n\n\n\n\n\n\n\\subsection{Properties of ${\\mathbf{z}}^j$ for Genus-Minimizing $q$ of Positive Type}\n\\label{ss: Properties of z^j for Genus-Minimizing q of Positive Type}\n\nWe have finally introduced enough terminology to be able to\nstate some results. For an initial reading, the reader might\nobtain a clearer picture of the overall argument if he or she skips \nall treatments of the special case in which\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor = 2$.\n\n\n\\begin{prop}\n\\label{prop: positive type, main prop}\nSuppose $q$ of positive type is genus-minimizing,\nand let $x_*, y_*$ denote the unique elements of $Q_q$ for which\n$v_q(x_*, y_*) = {\\alpha}(k-k^2)$. Then the following are true:\n\\begin{itemize}\n\\item[(i)]\nIf an interval $\\left\\langle z_i^j, z_{i+1}^j\\right]$\n(respectively $\\left\\langle z_{n_j-(i+1)}^j, z_{n_j - i}^j\\right]$)\nhas any mobile points, then there exists a unique $j_* \\in {{\\mathbb Z}}\/d$\nsuch that $\\left(z_i^{j_*}, z_{i+1}^{j_*}\\right) = \\left(x_*, y_*\\right)$\n(respectively $\\left(z_{n_{j_*}-(i+1)}^{j_*}, z_{n_{j_*} - i}^{j_*}\\right) = \\left(x_*, y_*\\right)$).\n\n\\item[(i$\\psi$)]\nIf the interval\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$\nhas any non-neutralized pseudomobile points, then\nthere exists a unique $j_* \\in {{\\mathbb Z}}\/d$ such that\n$\\left(z_{n_{j_*-1}}^{j_*-1}, z_0^{j_*}\\right) = \\left(x_*, y_*\\right)$.\n\n\\item[(ii)]\nIf $v_q(z_{n_{j-1}}^{j-1}, z_0^j)$ is constant in $j\\in{{\\mathbb Z}}\/d$, then\nthere are precisely two mobile points in ${\\bf{z}}^j$,\nnamely, $z_0^{j+l}$ R-mobile in $\\left\\langle z_{i_*}^j, z_{i_*+1}^j \\right]$ and\n$z_0^{j-l}$ L-mobile in\n$\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$, for some nonzero $l\\in{{\\mathbb Z}}\/d$,\nwhere $i_*$ is the unique element of \n$\\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2 \\right\\}$ satisfying\n$\\left(x_*,y_*\\right) = \\left(z_{i_*}^{j_*},z_{i_*+1}^{j_*}\\right) \n= \\left(z_{n_{j_*} - (i_*+1)}^{j_*}, z_{n_{j_*} - i_*}^{j_*}\\right)$.\n\n\n\\item[(ii$\\psi$)]\nIf $v_q(z_{n_{j-1}}^{j-1}, z_0^j)$ is nonconstant in $j\\in{{\\mathbb Z}}\/d$, then\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$ has precisely one\nnon-neutralized R-pseudomobile point and precisely one\nnon-neutralized L-pseudomobile point, namely,\n$z_0^{j+l}$ and $z_{n_{j-1-l}}^{j-1-l}$ for some nonzero $l\\in {{\\mathbb Z}}\/d$.\n\n\n\\item[(iii)]\nIf $(\\mu,\\gamma) = (1,1)$, then\n$v_q(z_{n_{j-1}}^{j-1}, z_0^j) \\equiv {\\alpha}k$ for all $j \\in {{\\mathbb Z}}\/d$.\n\n\\item[(iv)]\nAll mobile points are non-neutralized.\nMoreover, $\\psi > 2\\left[dq\\right]_{k^2}$\nunless $(\\mu,\\gamma) = (1,1)$, $\\alpha = -1$, $m=2$, $c=1$,\nand $d= \\frac{k-1}{2} \\equiv 0\\; (\\mod 2)$. \n\\end{itemize}\n\\end{prop}\n\n\n\nBefore commencing with the proof,\nwe pause for a brief discussion that will facilitate the proofs\nof Parts (i), (i$\\psi$), (ii), and (ii$\\psi$).\n\nWe introduce some notation used to make lists of R-mobile points and L-mobile points,\nand sublists of mobile points which are active at a given time. In particular,\nwe show in Claim \\ref{claim: structure of L_R^j and L_L^j}\nthat an ordering on the list of R-mobile points endows the list of\nactive R-mobile points with a particular structure, and similarly for \nL-mobile points. The tools and terminology introduced in this discussion\nare not needed outside the proofs of Parts (i), (i$\\psi$), (ii), and (ii$\\psi$),\nsince the results of (i), (i$\\psi$), (ii), and (ii$\\psi$)\nobviate their utility.\n\n\n\nFor the following discussion and the proofs of Parts (i) and (ii), \nfix any $i \\in \\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2\\}$,\nand let $a_j$ denote one of two functions of $j\\in{{\\mathbb Z}}\/d$; either\n$a_j \\equiv i$ for all $j \\in {{\\mathbb Z}}\/d$, or $a_j = n_j - (i+1)$ for each $j \\in {{\\mathbb Z}}\/d$.\nWe do this so that we can speak of the interval\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$\nwithout needing to specify whether we are measuring positions \nrel $z_0^j$ or rel $z_{n_j}^j$.\nNote that when $a_j \\equiv i$,\nany mobile point $z_0^{j+l}$ (respectively $z_{n_{j-l}}^{j-l}$) in\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$ must satisfy $l \\neq 0$\n(respectively $l\\neq 1$),\nwhereas when $a_j = n_j - (i+1)$ for each $j \\in {{\\mathbb Z}}\/d$,\nany mobile point $z_0^{j+l}$ (respectively $z_{n_{j-l}}^{j-l}$) in\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$ must satisfy $l \\neq 1$\n(respectively $l\\neq 0$).\n\n\n\n\nLet us begin by listing the mobile points, if any, in \n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$.\nDefine $L_{\\mathrm{R}}$ and $L_{\\mathrm{L}}$ by\n\\begin{align}\nL_{\\mathrm{R}} := \\left\\{l \\in {{\\mathbb Z}}\/d \\left\\vert\\;\n z_0^{j+l}\\;\\text{is R-mobile in}\\;\n \\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]\\right.\\right\\},\n \\\\ \\nonumber\nL_{\\mathrm{L}} := \\left\\{l \\in {{\\mathbb Z}}\/d \\left\\vert\\;\n z_{n_{j+l}}^{j+l}\\;\\text{is L-mobile in}\\;\n \\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]\\right.\\right\\}.\n\\end{align}\nWe then define an order relation $<_{\\mathrm{R}}$ on $L_{\\mathrm{R}}$ as follows.\nFor any two distinct elements $l_1, l_2 \\in L_{\\mathrm{R}}$, we say that\n$l_1 <_{\\mathrm{R}} l_2$ if there exists $j^{\\prime} \\in {{\\mathbb Z}}\/d$ such that\n$z_0^{j+l_1}$ is active and\n$z_0^{j+l_2}$ is inactive in $\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$\nat time $j=j^{\\prime}$, or in other words, if\n\\begin{align}\n z_0^{j^{\\prime\\!}+l_1} - z_{a_{j^{\\prime}}}^{j^{\\prime}} \n&= \\minq_{j\\in{{\\mathbb Z}}\/d} \\left( z_0^{j+l_1}\\! - z_{a_j}^j\\right),\n \\\\ \\nonumber\n z_0^{j^{\\prime\\!}+l_2} - z_{a_{j^{\\prime}}}^{j^{\\prime}} \n&= \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left( z_0^{j+l_2}\\! - z_{a_j}^j\\right).\n\\end{align}\nSimilarly, for any two distinct elements $l_1, l_2 \\in L_{\\mathrm{L}}$, we say that\n$l_1 <_{\\mathrm{L}} l_2$ if there exists $j^{\\prime} \\in {{\\mathbb Z}}\/d$ such that\n$z_{n_{j+l_1}}^{j+l_1}$ is inactive and \n$z_{n_{j+l_2}}^{j+l_2}$ is active\nin $\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$ at time $j=j^{\\prime}$.\n\n\n\n\nWe claim that the relations $<_{\\mathrm{R}}$ and $<_{\\mathrm{L}}$ define\nvalid orderings on $L_{\\mathrm{R}}$ and $L_{\\mathrm{L}}$, respectively.\nWe begin by showing that $<_{\\mathrm{R}}$ and $<_{\\mathrm{L}}$\nare well defined.\nFocusing on the case of $<_{\\mathrm{R}}$, assume that\n$\\left| L_{\\mathrm{R}}\\right| \\geq 2$ (since otherwise the claim is trivial),\nand choose two arbitrary distinct elements $l_1, l_2 \\in L_{\\mathrm{R}}$.\nSuppose that there is no time $j=j^{\\prime}\\in{{\\mathbb Z}}\/d$ at which\n$z_0^{j+l_1}$ is active and\n$z_0^{j+l_2}$ is inactive in $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$, \nand that there is also no time\n$j=j^{\\prime}\\in{{\\mathbb Z}}\/d$ at which\n$z_0^{j+l_1}$ is inactive and\n$z_0^{j+l_2}$ is active in $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$.\nThen for every $j^{\\prime} \\in {{\\mathbb Z}}\/d$, either\n$z_0^{j+l_1}$ and $z_0^{j+l_2}$ are both active\nin $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$ at time $j=j^{\\prime}$, in which case\n\\begin{align}\n z_0^{j^{\\prime}\\!+l_1} - z_0^{j^{\\prime}\\!+l_2} \n&= \\left(z_0^{j^{\\prime}\\!+l_1} - z_{a_{j^{\\prime}}}^{j^{\\prime}}\\right)\n -\\left(z_0^{j^{\\prime}\\!+l_2} - z_{a_{j^{\\prime}}}^{j^{\\prime}}\\right)\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l_1} - z_{a_j}^j\\right)\n - \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l_2} - z_{a_j}^j\\right),\n\\end{align}\nor $z_0^{j+l_1}$ and $z_0^{j+l_2}$ are both inactive\nin $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$ at time $j=j^{\\prime}$, in which case\n\\begin{align}\n z_0^{j^{\\prime}\\!+l_1} - z_0^{j^{\\prime}\\!+l_2} \n&= \\left(z_0^{j^{\\prime}\\!+l_1} - z_{a_{j^{\\prime}}}^{j^{\\prime}}\\right)\n -\\left(z_0^{j^{\\prime}\\!+l_2} - z_{a_{j^{\\prime}}}^{j^{\\prime}}\\right)\n \\\\ \\nonumber\n&= \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l_1} - z_{a_j}^j\\right)\n - \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l_2} - z_{a_j}^j\\right)\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l_1} - z_{a_j}^j\\right)\n - \\minq_{j\\in{{\\mathbb Z}}\/d} \\left(z_0^{j+l_2} - z_{a_j}^j\\right).\n\\end{align}\nThus \n$z_0^{j^{\\prime}\\!+l_1} - z_0^{j^{\\prime}\\!+l_2}$\nis constant in $j^{\\prime}\\in{{\\mathbb Z}}\/d$,\ncontradicting Corollary \\ref{cor: q of positive type: combo of lemma and difference eq}.\n\n\nOn the other hand, suppose that there exist $j_1, j_2 \\in {{\\mathbb Z}}\/d$ such that\n$z_0^{j+l_1}$ is active and\n$z_0^{j+l_2}$ is inactive \nin $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$ at time $j=j_1$,\nbut\n$z_0^{j+l_1}$ is inactive and\n$z_0^{j+l_2}$ is active\nin $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$ at time $j=j_2$.\nThen the fact that $z_0^{j+l_1}$ is active when $j=j_1$\nbut inactive when $j=j_2$ implies that\n\\begin{equation}\n \\left(z_0^{j_1+l_1} - z_{a_{j_1}}^{j_1}\\right)\n -\\left(z_0^{j_2+l_1} - z_{a_{j_2}}^{j_2}\\right)\n= -dq,\n\\end{equation}\nwhereas the fact that $z_0^{j+l_2}$ is inactive when $j=j_1$\nbut active when $j=j_2$ implies that\n\\begin{equation}\n \\left(z_0^{j_1+l_2} - z_{a_{j_1}}^{j_1}\\right)\n -\\left(z_0^{j_2+l_2} - z_{a_{j_2}}^{j_2}\\right)\n= dq.\n\\end{equation}\nSubtracting these two equations gives\n\\begin{equation}\n \\left(z_0^{j_1+l_1} - z_0^{j_1+l_2}\\right)\n- \\left(z_0^{j_2+l_1} - z_0^{j_2+l_2}\\right)\n= -2dq,\n\\end{equation}\ncontradicting Corollary \\ref{cor: q of positive type: combo of lemma and difference eq}.\nThus $<_{\\mathrm{R}}$ is well-defined.\nA similar argument shows that $<_{\\mathrm{L}}$ is well-defined.\n\n\n\n\nNext, we show that $<_{\\mathrm{R}}$ defines a valid order relation on $L_{\\mathrm{R}}$.\nSuppose there exist distinct $l_1, l_2, l_3 \\in L_{\\mathrm{R}}$ such that\n$l_1 <_{\\mathrm{R}} l_2$, $l_2 <_{\\mathrm{R}} l_3$,\nand $l_3 <_{\\mathrm{R}} l_1$.\nThen there exists $j^{\\prime} \\in {{\\mathbb Z}}$ such that\n$z_0^{j+l_3}$ is active and $z_0^{j+l_1}$ is inactive \nin $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$ at time $j=j^{\\prime}$.\nNow, if $z_0^{j+l_2}$ is active in $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$\nwhen $j=j^{\\prime}$, then this\ncontradicts our assumption that $l_1 <_{\\mathrm{R}} l_2$,\nbut if $z_0^{j+l_2}$ is inactive in $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$\nwhen $j=j^{\\prime}$,\nthen this contradicts our assumption that $l_2 <_{\\mathrm{R}} l_3$.\nThus, if $l_1 <_{\\mathrm{R}} l_2$ and $l_2 <_{\\mathrm{R}} l_3$, \nthen we must have $l_1 <_{\\mathrm{R}} l_3$,\nand so $<_{\\mathrm{R}}$ defines an ordering on $L_{\\mathrm{R}}$.\nA similar argument shows that\n$<_{\\mathrm{L}}$ defines an ordering on $L_{\\mathrm{L}}$.\n\n\n\n\n\n\nWe therefore write \n$l_1^{\\mathrm{R}}, l_2^{\\mathrm{R}}, \\ldots, l_{|L_{\\mathrm{R}}|}^{\\mathrm{R}}$\nfor the elements of $L_{\\mathrm{R}}$ such that\n$l_1^{\\mathrm{R}} \n<_{\\mathrm{R}} l_2^{\\mathrm{R}}\n<_{\\mathrm{R}} \\ldots <_{\\mathrm{R}} l_{|L_{\\mathrm{R}}|}^{\\mathrm{R}}$,\nand likewise write\n$l_1^{\\mathrm{L}}, l_2^{\\mathrm{L}}, \\ldots, l_{|L_{\\mathrm{L}}|}^{\\mathrm{L}}$\nfor the elements of $L_{\\mathrm{L}}$ such that\n$l_1^{\\mathrm{L}} \n<_{\\mathrm{L}} l_2^{\\mathrm{L}}\n<_{\\mathrm{L}} \\ldots <_{\\mathrm{L}} l_{|L_{\\mathrm{L}}|}^{\\mathrm{L}}$.\nThus when, for each ${j^{\\prime}} \\in {{\\mathbb Z}}\/d$, we define\n\\begin{align}\n L^{{j^{\\prime}}}_{\\mathrm{R}}\n&:= \\left\\{l \\in L_{\\mathrm{R}} \\left\\vert\\; \n z_0^{j+l}\\;\\text{is active in}\n \\;\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]\\;\n \\text{when}\\;j=j^{\\prime}\n \\right.\\right\\},\n \\\\ \\nonumber\n L^{{j^{\\prime}}}_{\\mathrm{L}}\n&:= \\left\\{l \\in L_{\\mathrm{L}} \\left\\vert\\; \n z_{n_{j+l}}^{j+l}\\;\\text{is active in}\n \\;\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]\\;\n \\text{when}\\;j=j^{\\prime}\n \\right.\\right\\},\n\\end{align}\nthe order relations $<_{\\mathrm{R}}$ on $L_{\\mathrm{R}}$ and\n$<_{\\mathrm{L}}$ on $L_{\\mathrm{L}}$ make the elements of\n$L^{j^{\\prime}}_{\\mathrm{R}}$ and $L^{j^{\\prime}}_{\\mathrm{L}}$\neasy to enumerate:\n\\begin{equation}\n L^{j^{\\prime}}_{\\mathrm{R}}\n= \\left\\{ l_1^{\\mathrm{R}}, \\ldots, l_{|L_{\\mathrm{R}}^{j^{\\prime}}|}^{\\mathrm{R}} \\right\\}\n \\;\\;\\mathrm{and}\\;\\;\n L^{j^{\\prime}}_{\\mathrm{L}}\n= \\left\\{ l^{\\mathrm{L}}_{|L_{\\mathrm{L}}| - |L_{\\mathrm{L}}^{j^{\\prime}}|+1} \\ldots,\n l_{|L_{\\mathrm{L}}|}^{\\mathrm{L}} \\right\\}\n\\end{equation}\nfor any $j^{\\prime}\\in{{\\mathbb Z}}\/d$. In particular, the following is true.\n\\begin{claim}\n\\label{claim: structure of L_R^j and L_L^j}\nFor any $j^{\\prime} \\in {{\\mathbb Z}}\/d$ and $s\\in\\{1, \\ldots, |L_{\\mathrm{R}}|\\}$\nsuch that $ l_s^{\\mathrm{R}} \\in L^{{j^{\\prime}}}_{\\mathrm{R}}$, we have\n\\begin{equation*}\nl_1^{\\mathrm{R}}, \\ldots, l_s^{\\mathrm{R}} \\in L^{{j^{\\prime}}}_{\\mathrm{R}}.\n\\end{equation*}\nFor any $j^{\\prime} \\in {{\\mathbb Z}}\/d$ and $s\\in\\{1, \\ldots, |L_{\\mathrm{L}}|\\}$\nsuch that $ l_s^{\\mathrm{L}} \\in L^{{j^{\\prime}}}_{\\mathrm{L}}$, we have\n\\begin{equation*}\nl_s^{\\mathrm{L}}, \\ldots, l_{|L_{\\mathrm{L}}|}^{\\mathrm{L}}\n\\in L^{{j^{\\prime}}}_{\\mathrm{L}}.\n\\end{equation*}\n\\end{claim}\n\\begin{proof}\nThe result follows directly from the definitions of $<_{\\mathrm{R}}$ and $<_{\\mathrm{L}}$.\n\\end{proof}\n\n\n\n\n \n\n\n\\begin{proof}[Proof of (i)]\nThe proof of Part (i) relies on the result of Part (iv)\nthat neutralized mobile points do not exist when\n$q$ of positive type is genus-minimizing.\nWe therefore prove a modified result, which we call (i'),\nand which satisfies the property that the combined statements\nof (i') and (iv) imply Part (i).\n\\begin{itemize}\n\\item[(i')]\n{\\em If the interval\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$\nhas any non-neutralized mobile points, then\nthere exists $j_* \\in {{\\mathbb Z}}\/d$ such that\n$\\left(z_{a_{j_*}}^{j_*}, z_{a_{j_*+1}}^{j_*}\\right) = (x_*, y_*)$.}\n\\end{itemize}\n\n\n\n\n\\noindent{\\em{Proof of (i').}}\\;\nSuppose that $v_q(z_{a_j}^j, z_{a_j + 1}^j)$ is constant in $j \\in {{\\mathbb Z}}\/d$.\nThis implies that there is some constant $M \\in {{\\mathbb Z}}_{\\geq 0}$ for which\n\\begin{equation}\n \\left\\vert L^{j^{\\prime}}_{\\mathrm{R}}\\right\\vert\n + \\left\\vert L^{j^{\\prime}}_{\\mathrm{L}}\\right\\vert\n\\equiv M\\;\\;\\;\\;\\text{for all}\\; {j^{\\prime}}\\in {{\\mathbb Z}}\/d.\n\\end{equation}\nNow, since $z_0^{j+l_{|L_{\\mathrm{R}}|}}$\nis R-mobile in $\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$,\nthere must be some time $j=j_{\\mathrm{R}} \\in {{\\mathbb Z}}\/d$ at which\n$z_0^{j+l_{|L_{\\mathrm{R}}|}}$ is active \nin $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$.\nThus $l^{\\mathrm{R}}_{|L_{\\mathrm{R}}|} \\in L_{\\mathrm{R}}^{j_{\\mathrm{R}}}$,\nwhich, by Claim \\ref{claim: structure of L_R^j and L_L^j}, means that\n$l_1^{\\mathrm{R}}, \\ldots, \nl^{\\mathrm{R}}_{|L_{\\mathrm{R}}|} \\in L_{\\mathrm{R}}^{j_{\\mathrm{R}}}$,\nor in other words, $L_{\\mathrm{R}}^{j_{\\mathrm{R}}} = L_{\\mathrm{R}}$,\nimplying\n$M \\geq \\left\\vert L^{j_{\\mathrm{R}}}_{\\mathrm{R}}\\right\\vert\n = \\left\\vert L_{\\mathrm{R}}\\right\\vert$.\nOn the other hand, since $z_0^{j+l_1}$ is R-mobile in\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$,\nwe know there must be some time $j=j_{\\mathrm{R}}^{\\emptyset} \\in {{\\mathbb Z}}\/d$\nat which $z_0^{j+l_1}$ is inactive in $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$.\nThus $l^{\\mathrm{R}}_1 \\notin L_{\\mathrm{R}}^{j_{\\mathrm{R}}^{\\emptyset}}$,\nwhich, by the contrapositive of Claim \\ref{claim: structure of L_R^j and L_L^j}, \nmeans that $l_1^{\\mathrm{R}}, \\ldots, \nl^{\\mathrm{R}}_{|L_{\\mathrm{R}}|} \\notin L_{\\mathrm{R}}^{j_{\\mathrm{R}}}$,\nor in other words,\n$L_{\\mathrm{R}}^{j_{\\mathrm{R}}^{\\emptyset}} = \\emptyset$,\nimplying\n$M = \\left\\vert L^{j_{\\mathrm{R}}^{\\emptyset}}_{\\mathrm{L}}\\right\\vert\n \\leq \\left\\vert L_{\\mathrm{L}}\\right\\vert$.\nBy similar reasoning, there exists $j_{\\mathrm{L}} \\in {{\\mathbb Z}}\/d$ such that\n$L_{\\mathrm{L}}^{j_{\\mathrm{L}}} = L_{\\mathrm{L}}$, implying\n$M \\geq \\left\\vert L^{j_{\\mathrm{L}}}_{\\mathrm{L}}\\right\\vert\n = \\left\\vert L_{\\mathrm{L}}\\right\\vert$,\nand there exists $j_{\\mathrm{L}}^{\\emptyset} \\in {{\\mathbb Z}}\/d$ such that\n$L_{\\mathrm{L}}^{j_{\\mathrm{L}}^{\\emptyset}} = \\emptyset$, implying\n$M = \\left\\vert L^{j_{\\mathrm{L}}^{\\emptyset}}_{\\mathrm{R}}\\right\\vert\n \\leq \\left\\vert L_{\\mathrm{R}}\\right\\vert$.\nThus\n\\begin{equation}\n \\left\\vert L_{\\mathrm{R}} \\right\\vert\n\\leq M\n\\leq \\left\\vert L_{\\mathrm{R}} \\right\\vert\n\\;\\;\\;\\mathrm{and}\\;\\;\\;\n \\left\\vert L_{\\mathrm{L}} \\right\\vert\n\\leq M\n\\leq \\left\\vert L_{\\mathrm{L}} \\right\\vert,\n\\end{equation}\nand so\n$\\left\\vert L_{\\mathrm{R}} \\right\\vert = \\left\\vert L_{\\mathrm{L}} \\right\\vert = M$.\n\n\n\nThus, recalling that \n$\\left\\vert L^{j^{\\prime}}_{\\mathrm{R}}\\right\\vert\n+ \\left\\vert L^{j^{\\prime}}_{\\mathrm{L}}\\right\\vert = M$ for all $j^{\\prime}\\in {{\\mathbb Z}}\/d$,\nwe deduce that for any $j^{\\prime} \\in {{\\mathbb Z}}\/d$, we have\n\\begin{align}\n l_1^{\\mathrm{L}}, \\ldots, \n l_{\\left\\vert L_{\\mathrm{R}}^{j^{\\prime}} \\right\\vert}^{\\mathrm{L}}\n \\notin L^{j^{\\prime}}_{\\mathrm{L}},&\n&l_1^{\\mathrm{R}}, \\ldots, \n l_{\\left\\vert L_{\\mathrm{R}}^{j^{\\prime}} \\right\\vert}^{\\mathrm{R}}\n \\in L^{j^{\\prime}}_{\\mathrm{R}},&\n \\\\\n l_{\\left\\vert L_{\\mathrm{R}}^{j^{\\prime}} \\right\\vert + 1}^{\\mathrm{L}},\n \\ldots, l_M^{\\mathrm{L}} \\in L^{j^{\\prime}}_{\\mathrm{L}},&\n&l_{\\left\\vert L_{\\mathrm{R}}^{j^{\\prime}} \\right\\vert + 1}^{\\mathrm{R}},\n \\ldots, l_M^{\\mathrm{R}} \\notin L^{j^{\\prime}}_{\\mathrm{R}}.&\n\\end{align}\nThis means that, at any give time $j=j^{\\prime}\\in{{\\mathbb Z}}\/d$\nand for any $s \\in \\{1, \\ldots, M\\}$,\n$z_{n_{j+l_s^{\\mathrm{L}}}}^{j+l_s^{\\mathrm{L}}}$ is active\nin $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$\nif and only if \n$z_0^{j+l_s^{\\mathrm{R}}}$\nis inactive, and so\nProposition \\ref{prop: active iff inactive true iff neutralized} tells us that\n$z_{n_{j+l_s^{\\mathrm{L}}}}^{j+l_s^{\\mathrm{L}}}$ and \n$z_0^{j+l_s^{\\mathrm{R}}}$ form a neutralized pair in \n$\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$.\nIn other words, all mobile points in $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$\nmust be neutralized.\n\n\nWe have shown that if $v_q(z_{a_j}^j, z_{a_j + 1}^j)$ is constant\nin $j \\in {{\\mathbb Z}}\/d$, then all mobile points in\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$, if any,\nare neutralized. Thus, if\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$ has any\n{\\em{non}}-neutralized mobile points, then \n$v_q(z_{a_j}^j, z_{a_j + 1}^j)$ is {\\em not} constant\nin $j \\in {{\\mathbb Z}}\/d$, and so, by\nProposition \\ref{prop: unique v_q = alpha(k - k^2), and the rest are v_q = alpha (k)},\nthere must exist a unique\n$j_* \\in {{\\mathbb Z}}\/d$ for which\n$v_q(z_{a_{j_*}}^{j_*}, z_{a_{j_*} + 1}^{j_*}) = \\alpha(k-k^2)$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of (i$\\psi$)]\nThe proof of Part (i') works for Part (i$\\psi$), with only a few minor changes.\nOne must replace the word ``mobile'' with the word ``pseudomobile''\nand replace $z_{a_j}^j$ and $z_{a_{j+1}}^j$ with\n$z_{n_{j-1}}^{j-1}$ and $z_0^j$, respectively---both in the actual proof,\nand in the discussion of $L_{\\mathrm{R}}$, $L_{\\mathrm{L}}$, {\\em et cetera},\npreceding the proof of Part (i). One must also replace the reference to\nProposition \\ref{prop: active iff inactive true iff neutralized}\nwith a reference to \nProposition \\ref{prop: active iff inactive means neutralized for pseudomobile}.\n\\end{proof}\n\n\n\n\n\n\\begin{proof}[Proof of (ii)]\nHere again, the result of Part (ii) relies on the\nnonexistence of neutralized mobile points for genus-minimizing\n$q$ of positive type, proven in Part (iv).\nWe therefore prove a modified claim, Part (ii'), which\ndoes not rely on Part (iv), but which when combined with Part (iv)\nimplies (ii).\n\\begin{itemize}\n\\item[(ii')]\n{\\em If $v_q(z_{n_{j-1}}^{j-1}, z_0^j)$ is constant in $j\\in{{\\mathbb Z}}\/d$, then\nthere are precisely two non-neutralized mobile points in ${\\bf{z}}^j$,\nnamely, $z_0^{j+l}$ non-neutralized R-mobile in \n$\\left\\langle z_{i_*}^j, z_{i_*+1}^j \\right]$ and\n$z_0^{j+l}$ non-neutralized L-mobile in\n$\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$, for some $l\\in{{\\mathbb Z}}\/d$,\nwhere $i_*$ is the unique element of \n$\\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2 \\right\\}$ satisfying\n$(x_*,y_*) = \\left(z_{i_*}^{j_*},z_{i_*+1}^{j_*}\\right) \n= \\left(z_{n_{j_*} - (i_*+1)}^{j_*}, z_{n_{j_*} - i_*}^{j_*}\\right)$.}\n\\end{itemize}\n\n\n\n\n\\noindent{\\em{Proof of (ii').}}\\;\nSince $v_q\\!\\left(z_{n_{j-1}}^{j-1}, z_0^j\\right)$ is constant in $j \\in {{\\mathbb Z}}\/d$,\nProposition \\ref{prop: unique v_q = alpha(k - k^2), and the rest are v_q = alpha (k)} tells us that\nthere exist unique $j_* \\in {{\\mathbb Z}}\/d$ and $i_* \\in \\{0, \\ldots, n_{j_*}-1\\}$ for which\n$v_q\\!\\left(z_{i_*}^{j_*}, z_{i_*+1}^{j_*}\\right) = {\\alpha}(k-k^2)$.\nThus, for the function $a_j$ of $j \\in {{\\mathbb Z}}\/d$, we choose either\n$a_j \\equiv i_*$ for all $j \\in {{\\mathbb Z}}\/d$,\nor $a_j = n_j - (n_{j_*} - i_*)$ for each $j \\in {{\\mathbb Z}}\/d$,\nsince these are the only two valid choices for $a_j$ for which\n$\\left(z_{i_*}^{j_*}, z_{i_*+1}^{j_*}\\right) \\in \n\\left\\{\\left( z_{a_j}^j, z_{a_j+1}^j\\right)\\right\\}_{j \\in {{\\mathbb Z}}\/d}$.\nWe then have\n\\begin{equation}\n v_q\\!\\left(z_{a_j}^j, z_{a_j+1}^j\\right)\n= \\begin{cases}\n {\\alpha}(k-k^2)\n & j =j_*\n \\\\\n {\\alpha}k\n & j \\neq j_*\n \\end{cases},\n\\end{equation}\nand so there exists $M \\in {{\\mathbb Z}}$ with $M \\geq 1$ such that\n\\begin{equation}\n \\left\\vert L_{\\mathrm{R}}^j \\right\\vert\n+ \\left\\vert L_{\\mathrm{L}}^j \\right\\vert\n= \\begin{cases}\n M + \\alpha\n & j = j_*\n \\\\\n M\n & j \\neq j_*\n \\end{cases}.\n\\end{equation}\nWe next argue that $\\left\\langle z^j_{a_j}, z^j_{a_j + 1} \\right]$ has at most\none non-neutralized R-mobile point and at most one non-neutralized L-mobile point,\ntreating the cases of $\\alpha = +1$ and $\\alpha = -1$ separately.\n \\\\\n\n\n\n{\\noindent\\bf{Case \\!$\\mathbf{\\alpha = +1}$.}\\;}\nBy the same reasoning as used in Part (i'),\none can show there exist\n$j_{\\mathrm{R}}\\in{{\\mathbb Z}}\/d$ such that $L_{\\mathrm{R}}^{j_{\\mathrm{R}}} = L_{\\mathrm{R}}$,\n$j_{\\mathrm{R}}^{\\emptyset}\\in{{\\mathbb Z}}\/d$ such that \n$L_{\\mathrm{R}}^{j_{\\mathrm{R}}^{\\emptyset}} = \\emptyset$,\n$j_{\\mathrm{L}}\\in{{\\mathbb Z}}\/d$ such that $L_{\\mathrm{L}}^{j_{\\mathrm{L}}} = L_{\\mathrm{L}}$,\nand $j_{\\mathrm{L}}^{\\emptyset}\\in{{\\mathbb Z}}\/d$ such that \n $L_{\\mathrm{L}}^{j_{\\mathrm{L}}^{\\emptyset}} = \\emptyset$,\nfrom which we conclude, respectively, that\n$M+1 \\geq \\left\\vert L_{\\mathrm{R}} \\right\\vert$,\n$M \\leq \\left\\vert L_{\\mathrm{L}} \\right\\vert$,\n$M+1 \\geq \\left\\vert L_{\\mathrm{L}} \\right\\vert$, and\n$M \\leq \\left\\vert L_{\\mathrm{R}} \\right\\vert$.\nNow, if \n$\\left\\vert L_{\\mathrm{R}} \\right\\vert = \\left\\vert L_{\\mathrm{L}} \\right\\vert = M+1$,\nthen $j_{\\mathrm{R}} = j_{\\mathrm{L}} = j_*$,\nbut this implies that\n$M+1 = \\left\\vert L_{\\mathrm{R}}^{j_*} \\right\\vert\n+ \\left\\vert L_{\\mathrm{L}}^{j_*} \\right\\vert = 2(M+1)$, a contradiction.\nThus we are left with three possibilities:\n\\begin{equation}\n\\left\\vert L_{\\mathrm{R}} \\right\\vert = \\left\\vert L_{\\mathrm{L}} \\right\\vert = M;\n\\;\\;\\;\\;\\;\\;\n\\left\\vert L_{\\mathrm{R}} \\right\\vert = M+1,\\; \\left\\vert L_{\\mathrm{L}} \\right\\vert = M;\n\\;\\;\\;\\;\\mathrm{or}\\;\\;\\;\\;\n\\left\\vert L_{\\mathrm{R}} \\right\\vert = M,\\; \\left\\vert L_{\\mathrm{L}} \\right\\vert = M+1.\n\\end{equation}\n\n\n\n\nFirst, consider the case in which\n$\\left\\vert L_{\\mathrm{R}} \\right\\vert \n= \\left\\vert L_{\\mathrm{L}} \\right\\vert = M$.\nSince\n$ \\left\\vert L_{\\mathrm{R}}^{j_*} \\right\\vert\n+ \\left\\vert L_{\\mathrm{L}}^{j_*} \\right\\vert = M+1 > M$,\nthere must exist some $s_0 \\in \\{1, \\ldots, M\\}$ for which\n$l_{s_0}^{\\mathrm{R}} \\in L^{j_*}_{\\mathrm{R}}$ and\n$l_{s_0}^{\\mathrm{L}} \\in L^{j_*}_{\\mathrm{L}}$.\nClaim \\ref{claim: structure of L_R^j and L_L^j} then tells us that\n$l_1^{\\mathrm{R}}, \\ldots, l_{s_0}^{\\mathrm{R}} \\in L^{j_*}_{\\mathrm{R}}$ and\n$l_{s_0}^{\\mathrm{L}}, \\ldots, l_M^{\\mathrm{L}} \\in L^{j_*}_{\\mathrm{L}}$. But\n\\begin{equation}\n \\left|\\left\\{ l_1^{\\mathrm{R}}, \\ldots, l_{s_0}^{\\mathrm{R}}\\right\\}\\right|\n + \\left|\\left\\{ l_{s_0}^{\\mathrm{L}}, \\ldots, l_M^{\\mathrm{L}}\\right\\}\\right|\n= M+1,\n\\end{equation}\nand so we must have \n$L^{j_*}_{\\mathrm{R}} = \\left\\{l_1^{\\mathrm{R}}, \\ldots, l_{s_0}^{\\mathrm{R}}\\right\\}$ and\n$L^{j_*}_{\\mathrm{L}} = \\left\\{l_{s_0}^{\\mathrm{L}}, \\ldots, l_M^{\\mathrm{L}}\\right\\}$, implying\n$\\left|L^{j_*}_{\\mathrm{R}}\\right| = s_0$ and\n$\\left|L^{j_*}_{\\mathrm{R}}\\right| = M+1-s_0$.\nNote that this argument can also be used to show that if there is any\n$s \\in \\{1, \\ldots, M\\}$ and any $j^{\\prime} \\in {{\\mathbb Z}}\/d$ for which\n$l_{s}^{\\mathrm{R}} \\in L^{j^{\\prime}}_{\\mathrm{R}}$ and\n$l_{s}^{\\mathrm{L}} \\in L^{j^{\\prime}}_{\\mathrm{L}}$,\nthen $\\left|L^{j^{\\prime}}_{\\mathrm{R}}\\right| = s$ and\n$\\left|L^{j^{\\prime}}_{\\mathrm{R}}\\right| = M+1-s$,\nso that $j^{\\prime} = j_*$. \nThere are also, however, no $s\\in \\{1, \\ldots, M\\}$ and $j^{\\prime} \\in {{\\mathbb Z}}\/d$ for which \n$l_{s}^{\\mathrm{R}} \\notin L^{j^{\\prime}}_{\\mathrm{R}}$ and\n$l_{s}^{\\mathrm{L}} \\notin L^{j^{\\prime}}_{\\mathrm{L}}$, \nsince this would imply\n$\\left\\vert L_{\\mathrm{R}}^{j^{\\prime}} \\right\\vert\n+ \\left\\vert L_{\\mathrm{L}}^{j^{\\prime}} \\right\\vert\n< M$ when $j^{\\prime} \\neq j_*$, and since this would contradict the fact that \n$L^{j_*}_{\\mathrm{R}} = \\left\\{l_1^{\\mathrm{R}}, \\ldots, l_{s_0}^{\\mathrm{R}}\\right\\}$ and\n$L^{j_*}_{\\mathrm{L}} = \\left\\{l_{s_0}^{\\mathrm{L}}, \\ldots, l_M^{\\mathrm{L}}\\right\\}$\nwhen $j^{\\prime} = j_*$.\nThus, for any $s \\in \\{1, \\ldots, M\\} \\setminus \\{s_0\\}$,\nand at any time $j=j^{\\prime}\\in{{\\mathbb Z}}\/d$, we know that\n$z_{n_{j+l_s^{\\mathrm{L}}}}^{j+l_s^{\\mathrm{L}}}$\nis active in $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$\nif and only if\n$z_0^{j+l_s^{\\mathrm{R}}}$ is inactive.\nProposition \\ref{prop: active iff inactive true iff neutralized}\nthen tells us that \n$z_{n_{j+l_s^{\\mathrm{L}}}}^{j+l_s^{\\mathrm{L}}}$ and \n$z_0^{j+l_s^{\\mathrm{R}}}$ form a neutralized pair in\n$\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$, and so\nall mobile points in \n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j \\right]$\nare neutralized except\nexcept $z_0^{j+l_{s_0}^{\\mathrm{R}}}$ and\n$z_{n_{j+l_{s_0}^{\\mathrm{L}}}}^{j+l_{s_0}^{\\mathrm{L}}}$.\nThus, we have 1 non-neutralized R-mobile point\nand 1 non-neutralized L-mobile point in\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j \\right]$\nwhen $\\left\\vert L_{\\mathrm{R}} \\right\\vert \n= \\left\\vert L_{\\mathrm{L}} \\right\\vert = M$.\n\n\n\n\n\n\nNext, consider the case in which\n$\\left\\vert L_{\\mathrm{R}} \\right\\vert = M+1$ and \n$\\left\\vert L_{\\mathrm{L}} \\right\\vert = M$.\nSince $z_0^{j + l^{\\mathrm{R}}_{M+1}}$ is R-mobile in\n$\\left\\langle z_{a_j}^j, z_{a_j+1}^j\\right]$, there must be some time \nwhen it is active, so \nthere must exist some $j_{L_{\\mathrm{R}}} \\in {{\\mathbb Z}}\/d$ for which\n$l_{M+1}^{\\mathrm{R}} \\in L_{\\mathrm{R}}^{j_{L_{\\mathrm{R}}}}\\!$.\nBut this implies $L_{\\mathrm{R}}^{j_{L_{\\mathrm{R}}}} = L_{\\mathrm{R}}$,\nwhich means \n$\\left\\vert L_{\\mathrm{R}}^{j_{L_{\\mathrm{R}}}} \\right\\vert \n+ \\left\\vert L_{\\mathrm{L}}^{j_{L_{\\mathrm{R}}}} \\right\\vert \\geq M+1$,\nand so it must be the case that $j_{L_{\\mathrm{R}}} = j_*$.\nThus $L_{\\mathrm{R}}^{j_*} = L_{\\mathrm{R}} = \n\\left\\{l_1^{\\mathrm{R}}, \\ldots, l_{M+1}^{\\mathrm{R}}\\right\\}$ and \n$L_{\\mathrm{L}}^{j_*} = \\emptyset$.\nNow, suppose there exist $s\\in \\{1, \\ldots, M\\}$ and $j^{\\prime} \\in {{\\mathbb Z}}\/d$ for which \n$l_{s}^{\\mathrm{R}} \\in L^{j^{\\prime}}_{\\mathrm{R}}$ and\n$l_{s}^{\\mathrm{L}} \\in L^{j^{\\prime}}_{\\mathrm{L}}$.\nThen, just as in the $\\left\\vert L_{\\mathrm{R}} \\right\\vert \n= \\left\\vert L_{\\mathrm{L}} \\right\\vert = M$ case, we have\n$l_1^{\\mathrm{R}}, \\ldots, l_{s}^{\\mathrm{R}} \\in L^{j^{\\prime}}_{\\mathrm{R}}$ and\n$l_{s}^{\\mathrm{L}}, \\ldots, l_M^{\\mathrm{L}} \\in L^{j^{\\prime}}_{\\mathrm{L}}$, and then since\n$ \\left|\\left\\{ l_1^{\\mathrm{R}}, \\ldots, l_{s}^{\\mathrm{R}}\\right\\}\\right|\n + \\left|\\left\\{ l_{s}^{\\mathrm{L}}, \\ldots, l_M^{\\mathrm{L}}\\right\\}\\right|\n= M+1$,\nwe conclude that $j^{\\prime} = j_*$.\nBut this contradicts the fact that $L_{\\mathrm{L}}^{j_*} = \\emptyset$.\nThus there are no $s\\in \\{1, \\ldots, M\\}$ and $j^{\\prime} \\in {{\\mathbb Z}}\/d$ for which \n$l_{s}^{\\mathrm{R}} \\in L^{j^{\\prime}}_{\\mathrm{R}}$ and\n$l_{s}^{\\mathrm{L}} \\in L^{j^{\\prime}}_{\\mathrm{L}}$.\nThere are also, however, no $s\\in \\{1, \\ldots, M\\}$ and $j^{\\prime} \\in {{\\mathbb Z}}\/d$ for which \n$l_{s}^{\\mathrm{R}} \\notin L^{j^{\\prime}}_{\\mathrm{R}}$ and\n$l_{s}^{\\mathrm{L}} \\notin L^{j^{\\prime}}_{\\mathrm{L}}$, \nsince this would imply\n$\\left\\vert L_{\\mathrm{R}}^{j^{\\prime}} \\right\\vert\n+ \\left\\vert L_{\\mathrm{L}}^{j^{\\prime}} \\right\\vert\n< M$ when $j^{\\prime} \\neq j_*$, and since this would contradict\nthe fact that $L_{\\mathrm{R}}^{j_*} = L_{\\mathrm{R}} = \n\\left\\{l_1^{\\mathrm{R}}, \\ldots, l_{M+1}^{\\mathrm{R}}\\right\\}$\nwhen $j^{\\prime} = j_*$.\nThus, for any $s \\in \\{1, \\ldots, M\\}$,\nand at any time $j=j^{\\prime}\\in{{\\mathbb Z}}\/d$, we know that\n$z_{n_{j+l_s^{\\mathrm{L}}}}^{j+l_s^{\\mathrm{L}}}$\nis active in $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$\nif and only if\n$z_0^{j+l_s^{\\mathrm{R}}}$ is inactive.\nProposition \\ref{prop: active iff inactive true iff neutralized}\nthen tells us that \n$z_{n_{j+l_s^{\\mathrm{L}}}}^{j+l_s^{\\mathrm{L}}}$ and \n$z_0^{j+l_s^{\\mathrm{R}}}$ form a neutralized pair in\n$\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$, and so\nall mobile points in \n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j \\right]$\nare neutralized except $z_0^{j+l_{M+1}^{\\mathrm{R}}}$.\nThus, we have 1 non-neutralized R-mobile point\nand 0 non-neutralized L-mobile points in\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j \\right]$\nwhen $\\left\\vert L_{\\mathrm{R}} \\right\\vert = M+1$ and \n$\\left\\vert L_{\\mathrm{L}} \\right\\vert = M$.\n\n\n\nLastly, if\n$\\left\\vert L_{\\mathrm{R}} \\right\\vert = M$ and \n$\\left\\vert L_{\\mathrm{L}} \\right\\vert = M+1$,\nthen an argument similar to that used in the preceding paragraph\nshows that all mobile points in\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j \\right]$\nare neutralized except for one L-mobile point.\nThus, in all of the above three cases,\nthere is at most one non-neutralized R-mobile point\nand at most one non-neutralized L-mobile point\nin $\\left\\langle z_{a_j}^j, z_{a_j + 1}^j \\right]$.\n \\\\\n\n\n\n{\\noindent\\bf{Case \\!$\\mathbf{\\alpha = -1}$.}\\;}\nFollowing a strategy similar to that used in the case of $\\alpha = +1$,\nwe observe that, since $z_0^{j+l_{|L_{\\mathrm{R}}|}}$\nis R-mobile in $\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$,\nthere must be some time $j=j_{\\mathrm{R}} \\in {{\\mathbb Z}}\/d$ at which\n$z_0^{j+l_{|L_{\\mathrm{R}}|}}$ is active \nin $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$.\nThus $l^{\\mathrm{R}}_{|L_{\\mathrm{R}}|} \\in L_{\\mathrm{R}}^{j_{\\mathrm{R}}}$,\nwhich, by Claim \\ref{claim: structure of L_R^j and L_L^j}, means that\n$l_1^{\\mathrm{R}}, \\ldots, \nl^{\\mathrm{R}}_{|L_{\\mathrm{R}}|} \\in L_{\\mathrm{R}}^{j_{\\mathrm{R}}}$,\nor in other words, $L_{\\mathrm{R}}^{j_{\\mathrm{R}}} = L_{\\mathrm{R}}$, implying\n$M \\geq \\left\\vert L^{j_{\\mathrm{R}}}_{\\mathrm{R}}\\right\\vert\n = \\left\\vert L_{\\mathrm{R}}\\right\\vert$.\nOn the other hand, since $z_0^{j+l_1}$ is R-mobile in\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j\\right]$,\nwe know there must be some time $j=j_{\\mathrm{R}}^{\\emptyset} \\in {{\\mathbb Z}}\/d$\nat which $z_0^{j+l_1}$ is inactive in $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$.\nThus $l^{\\mathrm{R}}_1 \\notin L_{\\mathrm{R}}^{j_{\\mathrm{R}}^{\\emptyset}}$,\nwhich, by the contrapositive of Claim \\ref{claim: structure of L_R^j and L_L^j}, \nmeans that $l_1^{\\mathrm{R}}, \\ldots, \nl^{\\mathrm{R}}_{|L_{\\mathrm{R}}|} \\notin L_{\\mathrm{R}}^{j_{\\mathrm{R}}}$,\nor in other words,\n$L_{\\mathrm{R}}^{j_{\\mathrm{R}}^{\\emptyset}} = \\emptyset$,\nimplying\n$M-1 = \\left\\vert L^{j_{\\mathrm{R}}^{\\emptyset}}_{\\mathrm{L}}\\right\\vert\n \\leq \\left\\vert L_{\\mathrm{L}}\\right\\vert$.\nBy similar reasoning, there exist\n$j_{\\mathrm{L}}\\in{{\\mathbb Z}}\/d$ such that $L_{\\mathrm{L}}^{j_{\\mathrm{L}}} = L_{\\mathrm{L}}$,\nand $j_{\\mathrm{L}}^{\\emptyset}\\in{{\\mathbb Z}}\/d$ such that \n $L_{\\mathrm{L}}^{j_{\\mathrm{L}}^{\\emptyset}} = \\emptyset$,\nfrom which we conclude, respectively, that\n$M \\geq \\left\\vert L_{\\mathrm{L}} \\right\\vert$, and\n$M-1 \\leq \\left\\vert L_{\\mathrm{R}} \\right\\vert$.\nNow, if\n$\\left\\vert L_{\\mathrm{R}} \\right\\vert = \\left\\vert L_{\\mathrm{L}} \\right\\vert = M-1$,\nthen $j_{\\mathrm{R}} = j_{\\mathrm{L}} = j_*$. But this implies\n$M-1 = \\left\\vert L_{\\mathrm{R}}^{j_*} \\right\\vert\n+ \\left\\vert L_{\\mathrm{L}}^{j_*} \\right\\vert = 2(M-1)$, which means that\n$\\left\\vert L_{\\mathrm{R}} \\right\\vert = \\left\\vert L_{\\mathrm{L}} \\right\\vert = M-1 = 0$,\ncontradicting our assumption that at least one of\n$L_{\\mathrm{R}}$ and $L_{\\mathrm{L}}$ is nonempty.\nThus, we are left with three possibilities:\n\\begin{equation}\n\\left\\vert L_{\\mathrm{R}} \\right\\vert = \\left\\vert L_{\\mathrm{L}} \\right\\vert = M;\n\\;\\;\\;\\;\\;\\;\n\\left\\vert L_{\\mathrm{R}} \\right\\vert = M - 1,\\; \\left\\vert L_{\\mathrm{L}} \\right\\vert = M;\n\\;\\;\\;\\;\\mathrm{or}\\;\\;\\;\\;\n\\left\\vert L_{\\mathrm{R}} \\right\\vert = M,\\; \\left\\vert L_{\\mathrm{L}} \\right\\vert = M - 1.\n\\end{equation}\n\n\n\n\n\n\nFirst, consider the case in which\n$\\left\\vert L_{\\mathrm{R}} \\right\\vert \n= \\left\\vert L_{\\mathrm{L}} \\right\\vert = M$.\nSince\n$ \\left\\vert L_{\\mathrm{R}}^{j_*} \\right\\vert\n+ \\left\\vert L_{\\mathrm{L}}^{j_*} \\right\\vert = M-1 M$ when $j^{\\prime} \\neq j_*$, \nand since this would contradict the fact that\n$L^{j_*}_{\\mathrm{R}} = \\left\\{l_1^{\\mathrm{R}}, \\ldots, l_{s_0-1}^{\\mathrm{R}}\\right\\}$ and\n$L^{j_*}_{\\mathrm{L}} = \\left\\{l_{s_0+1}^{\\mathrm{L}}, \\ldots, l_M^{\\mathrm{L}}\\right\\}$\nwhen $j^{\\prime} = j_*$.\nThus, for any $s \\in \\{1, \\ldots, M\\} \\setminus \\{s_0\\}$,\nand at any time $j=j^{\\prime}\\in{{\\mathbb Z}}\/d$, we know that\n$z_{n_{j+l_s^{\\mathrm{L}}}}^{j+l_s^{\\mathrm{L}}}$\nis active in $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$\nif and only if\n$z_0^{j+l_s^{\\mathrm{R}}}$ is inactive.\nProposition \\ref{prop: active iff inactive true iff neutralized}\nthen tells us that \n$z_{n_{j+l_s^{\\mathrm{L}}}}^{j+l_s^{\\mathrm{L}}}$ and \n$z_0^{j+l_s^{\\mathrm{R}}}$ form a neutralized pair in\n$\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$, and so\nall mobile points in \n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j \\right]$\nare neutralized except\nexcept $z_0^{j+l_{s_0}^{\\mathrm{R}}}$ and\n$z_{n_{j+l_{s_0}^{\\mathrm{L}}}}^{j+l_{s_0}^{\\mathrm{L}}}$.\nThus, we have 1 non-neutralized R-mobile point\nand 1 non-neutralized L-mobile point in\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j \\right]$\nwhen $\\left\\vert L_{\\mathrm{R}} \\right\\vert \n= \\left\\vert L_{\\mathrm{L}} \\right\\vert = M$.\n\n\n\nNext, consider the case in which\n$\\left\\vert L_{\\mathrm{R}} \\right\\vert = M-1$ and \n$\\left\\vert L_{\\mathrm{L}} \\right\\vert = M$.\nSince \n$z_{n_{j + l^{\\mathrm{L}}_M}}^{j + l^{\\mathrm{L}}_M}$\nis L-mobile in\n$\\left\\langle z_{a_j}^j, z_{a_j+1}^j\\right]$,\nthere must be some time when it is inactive, so \nthere must exist some $j_{L_{\\mathrm{L}}}^{\\emptyset} \\in {{\\mathbb Z}}\/d$ for which\n$l_M^{\\mathrm{L}} \\notin L_{\\mathrm{L}}^{j_{L_{\\mathrm{L}}}^{\\emptyset}}\\!$.\nThe contrapositive of Claim \\ref{claim: structure of L_R^j and L_L^j}\nthen tells us that $L_{\\mathrm{L}}^{j_{L_{\\mathrm{L}}}^{\\emptyset}} = \\emptyset$,\nwhich means that\n$\\left\\vert L_{\\mathrm{R}}^{j_{L_{\\mathrm{L}}}^{\\emptyset}} \\right\\vert \n+ \\left\\vert L_{\\mathrm{L}}^{j_{L_{\\mathrm{L}}}^{\\emptyset}} \\right\\vert \\leq M-1$,\nand so it must be the case that $j_{L_{\\mathrm{L}}}^{\\emptyset} = j_*$.\nThus $L_{\\mathrm{L}}^{j_*} = \\emptyset$,\nand $l_M^{\\mathrm{L}} \\in L_{\\mathrm{L}}^{j^{\\prime}}$ whenever $j^{\\prime}\\neq j_*$.\nNow, suppose there exist $s\\in \\{1, \\ldots, M-1\\}$ and $j^{\\prime} \\in {{\\mathbb Z}}\/d$ for which \n$l_{s}^{\\mathrm{R}} \\notin L^{j^{\\prime}}_{\\mathrm{R}}$ and\n$l_{s}^{\\mathrm{L}} \\notin L^{j^{\\prime}}_{\\mathrm{L}}$.\nThe contrapositive of Claim \\ref{claim: structure of L_R^j and L_L^j}\nthen tells us that \n$l_{s}^{\\mathrm{R}}, \\ldots, l_{M-1}^{\\mathrm{R}} \\notin L^{j^{\\prime}}_{\\mathrm{R}}$ and\n$l_1^{\\mathrm{L}}, \\ldots, l_{s}^{\\mathrm{L}} \\notin L^{j^{\\prime}}_{\\mathrm{L}}$,\nand then since\n$ \\left|\\left\\{ l_1^{\\mathrm{R}}, \\ldots, l_{s-1}^{\\mathrm{R}}\\right\\}\\right|\n + \\left|\\left\\{ l_{s+1}^{\\mathrm{L}}, \\ldots, l_M^{\\mathrm{L}}\\right\\}\\right|\n= M-1$,\nwe conclude that $j^{\\prime} = j_*$.\nBut this contradicts the fact that $L_{\\mathrm{L}}^{j_*} = \\emptyset$.\nThus there are no $s\\in \\{1, \\ldots, M-1\\}$ and $j^{\\prime} \\in {{\\mathbb Z}}\/d$ for which \n$l_{s}^{\\mathrm{R}} \\notin L^{j^{\\prime}}_{\\mathrm{R}}$ and\n$l_{s}^{\\mathrm{L}} \\notin L^{j^{\\prime}}_{\\mathrm{L}}$.\nThere are also, however, no $s\\in \\{1, \\ldots, M-1\\}$ and $j^{\\prime} \\in {{\\mathbb Z}}\/d$ for which \n$l_{s}^{\\mathrm{R}} \\in L^{j^{\\prime}}_{\\mathrm{R}}$ and\n$l_{s}^{\\mathrm{L}} \\in L^{j^{\\prime}}_{\\mathrm{L}}$, \nsince this would imply\n$\\left\\vert L_{\\mathrm{R}}^{j^{\\prime}} \\right\\vert\n+ \\left\\vert L_{\\mathrm{L}}^{j^{\\prime}} \\right\\vert\n> M$ when $j^{\\prime} \\neq j_*$,\nand since this would contradict\nthe fact that $L_{\\mathrm{L}}^{j_*} = \\emptyset$\nwhen $j^{\\prime} = j_*$.\nThus, for any $s \\in \\{1, \\ldots, M-1\\}$,\nand at any time $j=j^{\\prime}\\in{{\\mathbb Z}}\/d$, we know that\n$z_{n_{j+l_s^{\\mathrm{L}}}}^{j+l_s^{\\mathrm{L}}}$\nis active in $\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$\nif and only if\n$z_0^{j+l_s^{\\mathrm{R}}}$ is inactive.\nProposition \\ref{prop: active iff inactive true iff neutralized}\nthen tells us that \n$z_{n_{j+l_s^{\\mathrm{L}}}}^{j+l_s^{\\mathrm{L}}}$ and \n$z_0^{j+l_s^{\\mathrm{R}}}$ form a neutralized pair in\n$\\left\\langle z_{a_j}^j, z_{a_j+1}^j \\right]$, and so\nall mobile points in \n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j \\right]$\nare neutralized except $z_{n_{j+l_M^{\\mathrm{L}}}}^{j+l_M^{\\mathrm{L}}}$.\nThus, we have 0 non-neutralized R-mobile points\nand 1 non-neutralized L-mobile point in\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j \\right]$\nwhen $\\left\\vert L_{\\mathrm{R}} \\right\\vert = M-1$ and \n$\\left\\vert L_{\\mathrm{L}} \\right\\vert = M$.\n\n\n\nLastly, if\n$\\left\\vert L_{\\mathrm{R}} \\right\\vert = M$ and \n$\\left\\vert L_{\\mathrm{L}} \\right\\vert = M-1$,\nthen an argument similar to that used in the preceding paragraph\nshows that all mobile points in\n$\\left\\langle z_{a_j}^j, z_{a_j + 1}^j \\right]$\nare neutralized except for one R-mobile point.\nIn conclusion, whether $\\alpha = +1$ or $\\alpha = -1$, there is always\nat most one non-neutralized R-mobile point and at most one\nnon-neutralized L-mobile point in $\\left\\langle z_{a_j}^j, z_{a_j + 1}^j \\right]$.\n \\\\\n\n\n\n\nWe next consider the existence of non-neutralized mobile points \nin all of ${\\bf{z}}^j$, as opposed to in just an interval in ${\\bf{z}}^j$.\nSince $v_q(z_{n_{j-1}}^{j-1}, z_0^j)$ is constant in $j\\in{{\\mathbb Z}}\/d$, we know\nby Part (i') that ${\\bf{z}}^j$ must have at least one non-neutralized\nmobile point. However, the existence of a non-neutralized R-mobile point\nimplies the existence of its mirror non-neutralized L-mobile point,\nand {\\em vice versa}, and so, in fact, ${\\bf{z}}^j$ must have at least one \nnon-neutralized R-mobile point and at least one non-neutralized L-mobile point.\n\n\nWrite $z_0^{j+l}$ for an arbitrary non-neutralized R-mobile point in ${\\bf{z}}^j$,\nand suppose that $z_0^{j+l}$ is {\\em not} non-neutralized R-mobile rel $z_0^j$.\nThen $z_0^{j+l}$ is non-neutralized R-mobile in \n$\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$\nfor some $i_* \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2 \\right\\}$,\nand $l \\neq 1$. Consider first the case in which $l \\neq 0$.\nIf $i_* > 0$, then\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor - (i_*+1),\n\\left\\lfloor\\frac{k}{d}\\right\\rfloor - (i_*+2)\n\\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2 \\right\\}$, and so\n$z_0^{j+l}$ is also R-mobile in either\n$\\left\\langle z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - (i_*+1)}^j,\nz_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - i_*}^j\\right]$ or\n$\\left\\langle z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - (i_*+2)}^j,\nz_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - (i_*+1)}^j\\right]$,\nhence is R-mobile rel $z_0^j$, a contradiction.\nThus we must have $i_*=0$,\nwith $z_0^{j+l}$ R-mobile in \n$\\left\\langle z_{n_j - 1}^j, z_{n_j}^j \\right]$.\nThis, in turn, implies the existence of \na mirror non-neutralized L-mobile point, namely,\n$z_{n_{j-l}}^{j-l}$ in $\\left\\langle z_0^j, z_1^j \\right]$.\nFrom Part (i'), we know that the existence of these non-neutralized \nmobile points implies that the sets\n$\\left\\{\\left(z_{n_j - 1}^j, z_{n_j}^j\\right)\\right\\}_{j\\in{{\\mathbb Z}}\/d}$\nand\n$\\left\\{\\left(z_0^j, z_1^j\\right)\\right\\}_{j\\in{{\\mathbb Z}}\/d}$\neach contain the pair $(x_*, y_*) \\in Q_q \\times Q_q$ for which\n$v_q\\!\\left(x_*, y_*\\right) = \\alpha(k-k^2)$.\nIn particular, these two sets must intersect, \nand so there exists some $j_*\\in {{\\mathbb Z}}\/d$ for which\n$n_{j_*} - 1 = 0$. Thus $n_{j_*}=1$ and\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor = 2$.\nWe can therefore express ${{\\mathbb Z}}\/d$ as the disjoint union of\n$J_0$, $J_2$, and $J_3$, where\n\\begin{align}\n J_0\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}} = 1;\\;\n z_0^{j+l}\\;\\text{is inactive in}\\;\\!\n \\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\}\\!,\n \\\\\n J_2\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}} = 2;\\;\n z_0^{j+l}\\;\\text{is inactive in}\\;\\!\n \\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\}\\!,\n \\\\\n J_3\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}} = 2;\\;\n z_0^{j+l}\\;\\text{is active in}\\;\\!\n \\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\}\\!.\n\\end{align}\nNote that we omitted the only other possibility,\n\\begin{equation}\n J_1\n:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}} = 1;\\;\n z_0^{j+l}\\;\\text{is active in}\\;\\!\n \\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\}\\!,\n\\end{equation}\nbecause the nonemptiness of $J_1$ would imply that \n$z_0^{j+l}$ was R-mobile in $\\left\\langle z_0^j, z_1^j\\right]$.\nWe know, however, that $J_0$, $J_2$, and $J_3$ are nonempty.\n$J_0$ is nonempty because $n_{j^{\\prime}}=1$ only if $j^{\\prime} \\in J_0$;\n$J_2$ is nonempty because otherwise $z_0^{j+l}-z_0^j$ would be constant\nin $j\\in{{\\mathbb Z}}\/d$, contradicting our assumption that $l \\neq 0$; and\n$J_3$ is nonempty because otherwise $z_0^{j+l}$ would never be active\nin $\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$.\nNow, the fact that $z_0^{j+l}$ is not R-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$ implies that its mirror mobile point,\n$z_{n_{j-l}}^{j-l}$, is not L-mobile in $\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$,\nand so $z_0^{j+l}$ is the only non-neutralized\nmobile point in $\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$.\nSince $v_q\\!\\left(z_{n_{j_*}-1}^{j_*}, z_{n_{j_*}}^{j_*}\\right) = \\alpha(k-k^2)$,\nand $v_q\\!\\left(z_{n_{j^{\\prime}}-1}^{j^{\\prime}}, z_{n_{j^{\\prime}}}^{j^{\\prime}}\\right) = \\alpha(k)$\nfor all $j^{\\prime} \\neq j_* \\in {{\\mathbb Z}}\/d$, \nand since any non-neutralized mobile point active in \n$\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$\nat time $j=j^{\\prime}$ contributes $-k^2$ to\n$v_q\\!\\left(z_{n_{j^{\\prime}}-1}^{j^{\\prime}}, z_{n_{j^{\\prime}}}^{j^{\\prime}}\\right)$,\nthis means that\n$z_0^{j+l}$ is inactive in $\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$\nprecisely once when $\\alpha = -1$, and active precisely once\nwhen $\\alpha = +1$. The fact that $J_0$ and $J_2$ are each nonempty\nmakes it impossible for $z_0^{j+l}$ to be inactive in \n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$ precisely once, so we must have $\\alpha = +1$,\nimplying $j_* \\in J_3$, since that is the only time when $z_0^{j+l}$ is active.\nBut this contradicts the fact that $n_{j_*} = 1$.\nThus, when $l \\neq 0$, any non-neutralized R-mobile point $z_0^{j+l}$ in\n${\\mathbf{z}}^j$ must be non-neutralized R-mobile rel $z_0^j$.\n\n\n\nThat leaves us with the case in which $l=0$: suppose that\n$z_0^j$ is non-neutralized R-mobile in ${\\mathbf{z}}^j$,\nhence is non-neutralized R-mobile in\n$\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$\nfor some $i_* \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2 \\right\\}$.\nThen its mirror mobile point, $z_{n_j}^j$, is non-neutralized\nL-mobile in $\\left\\langle z_{i_*}^j, z_{i_*+1}^j \\right]$\nand is not L-mobile in $\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$,\nso that $z_0^j$ is the only non-neutralized mobile point in\n$\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$.\nAgain, Part (i') implies there exists some $j_* \\in {{\\mathbb Z}}\/d$ for which\n\\begin{equation}\n\\label{main prop, part (ii'), eq: x,y = z_i,z_i+1, etc}\n\\left(z_{n_{j_*} - (i_*+1)}^{j_*}, z_{n_{j_*} - i_*}^{j_*}\\right)\n= \\left(z_{i_*}^{j_*}, z_{i_*+1}^{j_*}\\right) = (x_*, y_*),\n\\end{equation}\nso that $v_q\\!\\left(z_{n_{j_*} - (i_*+1)}^{j_*}, z_{n_{j_*} - i_*}^{j_*}\\right) = \\alpha(k-k^2)$, and\n$v_q\\!\\left(z_{n_{j^{\\prime}} - (i_*+1)}^{j^{\\prime}}, z_{n_{j^{\\prime}} - i_*}^{j^{\\prime}}\\right) = \\alpha(k)$\nfor all $j^{\\prime} \\neq j_* \\in {{\\mathbb Z}}\/d$.\nIf $\\alpha = -1$, then\n$z_0^j$ is inactive in $\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$\nat time $j=j_*$, and so\n$z_0^{j_*} \\in \\left\\langle z_{i_*}^{j_*}+dq, z_{i_*+1}^{j_*}+dq \\right]$.\nNow, the fact that \n$2[dq]_{k^2} < k^2$ implies that $i_* \\geq 2$,\nand (\\ref{main prop, part (ii'), eq: x,y = z_i,z_i+1, etc})\nimplies that $n_{j_*}-(i_*+1)=i_*$. Thus\n$i_*+2 \\leq 2i_* = n_{j_*}-1 \\leq \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 1$,\nso that $z_{i_*+2}^j$ is defined for all $j\\in {{\\mathbb Z}}\/d$,\nand so\n$z_0^{j_*} \\in \\left\\langle z_{i_*+1}^{j_*}, z_{i_*+2}^{j_*} \\right]$\nif $\\alpha = -1$.\nOn the other hand, if $\\alpha = +1$, then\n$z_0^j$ is active in $\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$\nat time $j=j_*$, and so\n$z_0^{j_*} \\in \\left\\langle z_{i_*}^{j_*}, z_{i_*+1}^{j_*} \\right]$.\nWe can summarize these two statements by saying that\n$z_0^{j_*}\\in \\left\\langle z_{i_*+\\frac{1-\\alpha}{2}}^{j_*}, z_{i_*+\\frac{3-\\alpha}{2}}^{j_*}\\right]$,\nwhich implies that\n$z_0^{j^{\\prime}}\\in \n\\left\\langle z_{i_*+\\frac{1-\\alpha}{2}}^{j^{\\prime}}, z_{i_*+\\frac{3-\\alpha}{2}}^{j^{\\prime}}\\right]$\nfor all $j^{\\prime}\\in {{\\mathbb Z}}\/d$.\nThus, for any $j_0 \\in {{\\mathbb Z}}\/d$,\n\\begin{equation}\n\\label{main prop, part (ii'), all of Q_q between the z_0^js}\n\\left[z_0^{j_0}\\right]_{k^2}\n\\;\\;<\\;\\; x \\;\\;<\\;\\; \\left[z_0^{j_0}\\right]_{k^2} + \\left(\\textstyle{{i_*+\\frac{3-\\alpha}{2}}}\\right)[dq]_{k^2}\n\\end{equation}\nfor {\\em every} $x \\neq z_0^{j^{\\prime}} \\in Q_q$.\nIn particular, (\\ref{main prop, part (ii'), all of Q_q between the z_0^js})\nholds for all $x \\in \\left\\{ z_{n_{j-1}}^{j-1} \\right\\}_{j\\in{{\\mathbb Z}}\/d}$.\nSince $z_{n_{j-1}}^{j-1} -z_0^j$ is constant in $j\\in{{\\mathbb Z}}\/d$, this implies that\n$z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}\n\\in \\left\\langle z_{i^{\\prime}}^{j^{\\prime}}, z_{{i^{\\prime}}+1}^{j^{\\prime}} \\right]$\nfor some $i^{\\prime} \\in \\left\\{0, \\ldots, i_*+\\frac{1-\\alpha}{2} \\right\\}$\nand for all $j^{\\prime} \\in {{\\mathbb Z}}\/d$,\nwhere we recall that \n$i_*+\\frac{1-\\alpha}{2} \\leq \\left\\lfloor\\frac{k}{d}\\right\\rfloor-2$.\nIf $i^{\\prime} \\neq 0$, this implies that $z_{n_{j-1}}^{j-1}$ is\nL-mobile rel $z_{n_j}^j$.\nIf $i^{\\prime}=0$, then this implies that\n$z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}\n\\in \\left\\langle z_{i_0}^{j^{\\prime}}, z_{i_0}^{j^{\\prime}} +dq \\right]$\nfor some $i_0 \\in \\left\\{i_* + \\frac{1-\\alpha}{2}, i_* + \\frac{3-\\alpha}{2} \\right\\}$,\nso that, again, $z_{n_{j-1}}^{j-1}$ is L-mobile rel $z_{n_j}^j$.\nIn either case,\n$z_{n_{j-1}}^{j-1}$ is L-mobile rel $z_{n_j}^j$, hence is L-mobile\nin $\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j\\right]$\nfor some $i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2 \\right\\}$.\nThus either $i=i_*$, so that\n$z_0^j$ is {\\em neutralized} R-mobile in\n$\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$,\nor $i \\neq i_*$, so that\n$z_{n_{j-1}}^{j-1}$ is non-neutralized L-mobile\nin $\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j\\right]$,\ncontradicting Part (i').\n\n\n\nThus any non-neutralized R-mobile point in ${\\bf{z}}^j$\nis non-neutralized R-mobile rel $z_0^j$, which also means that\nthe mirror statement must be true. That is, \nany non-neutralized L-mobile point in ${\\bf{z}}^j$\nis non-neutralized L-mobile rel $z_{n_j}^j$.\n\n\nWe therefore can express the unique non-neutralized R-mobile point\nin ${\\bf{z}}^j$ as $z_0^{j+l}$ in $\\left\\langle z_{i_*}^j, z_{i_*+1}^j\\right]$,\nfor some $l \\neq 0 \\in {{\\mathbb Z}}\/d$, where, as discussed earlier, \n$i_* \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2 \\right\\}$\nsatisfies $v_q\\!\\left(z_{i_*}^{j_*}, z_{i_*+1}^{j_*}\\right) = {\\alpha}(k-k^2)$\nfor some unique $j_* \\in {{\\mathbb Z}}\/d$. The unique non-neutralized L-mobile point\nis then the mirror mobile point,\n$z_{n_{j-l}}^{j-l}$ non-neutralized L-mobile in\n$\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$.\nPart (i') then tells us that\n$(x_*,y_*) = \\left(z_{i_*}^{j_*},z_{i_*+1}^{j_*}\\right) \n= \\left(z_{n_{j_*} - (i_*+1)}^{j_*}, z_{n_{j_*} - i_*}^{j_*}\\right)$.\n\n\\end{proof}\n\n\n\n\n\n\\begin{proof}[Proof of (ii$\\psi$)]\nFirst, to prove that there is at most one non-neutralized R-pseudomobile point\nand at most one non-neutralized L-pseudomobile point in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$, take the proof of the comparable\nstatement in Part (ii'), and make the following adaptations.\nReplace the word ``mobile'' with the word ``pseudomobile,''\nand replace $z_{a_j}^j$ and $z_{a_{j+1}}^j$ with\n$z_{n_{j-1}}^{j-1}$ and $z_0^j$, respectively---both in the actual proof of Part (ii'),\nand in the discussion of $L_{\\mathrm{R}}$, $L_{\\mathrm{L}}$, {\\em et cetera},\npreceding the proof of Part (i). In addition,\nreplace all references in Part (ii') to\nProposition \\ref{prop: active iff inactive true iff neutralized} with\nreferences to \nProposition \\ref{prop: active iff inactive means neutralized for pseudomobile}.\n\nNext, suppose there is a non-neutralized R-pseudomobile point,\nsay $z_0^{j+l}$ in $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$\nfor some $l\\neq 0 \\in {{\\mathbb Z}}\/d$. Then its mirror pseudomobile point,\n$z_{n_{j-1-l}}^{j-1-l}$, is also non-neutralized pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$.\nA similar result holds if we start with a non-neutralized L-pseudomobile point.\nThus $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$ has precisely one\nnon-neutralized R-pseudomobile point and precisely one\nnon-neutralized L-pseudomobile point, namely,\n$z_0^{j+l}$ and $z_{n_{j-1-l}}^{j-1-l}$ for some nonzero $l \\in {{\\mathbb Z}}\/d$.\n\\end{proof}\n\n\n\n\n\\begin{proof}[Proof of (iii)]\nFirst, recall that since $(\\mu,\\gamma)=(1,1)$, we have\n$\\left[dq\\right]_{k^2} = (m\\!+\\!c)k\\!+\\!\\alpha$, and\n$\\psi = \\left[(m\\!+\\!c)k\\!+\\!\\alpha - \\frac{ck + \\alpha}{d}k\\right]_{k^2}$,\nwhere we recall that \n$\\psi:= \\left[z_0^{j+1} - z_{n_j}^j\\right]_{k^2} = \\left[dq-kq\\right]_{k^2}$.\n\nConsider first the case in which \n$\\psi > 2\\left[dq\\right]_{k^2}$,\nwhich, as proven in Part (iv), implies that\nall mobile points are non-neutralized.\nThis means in particular that Part (i)---as\nopposed to merely Part (i')---holds.\n\n\nSince $v_q\\!\\left(z_{n_{j-1}}^{j-1}, z_0^j\\right) = -v_q\\!\\left(z_0^j, z_{n_{j-1}}^{j-1}\\right)$,\nit is sufficient to show that $\\# \\left( Q_q \\cap \\left\\langle z_0^j, z_{n_{j-1}}^{j-1}\\right] \\right)$\nis constant in $j\\in {{\\mathbb Z}}\/d$. To do this, our strategy will be to cover\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1}\\right]$ with sets for which we understand\nhow their intersection with $Q_q$ changes as $j$ varies in ${{\\mathbb Z}}\/d$.\nWe begin by examining the length of $\\left\\langle z_0^j, z_{n_{j-1}}^{j-1}\\right]$.\nSince $\\psi > 2\\left[dq\\right]_{k^2}$,\nwe know that $(m+c)k+\\alpha \\,-\\, \\frac{ck + \\alpha}{d}k < 0$. Thus,\nfor all $j\\in{{\\mathbb Z}}\/d$, we have\n\\begin{align}\n \\left[z_{n_{j-1}}^{j-1} - z_0^j\\right]_{k^2}\n&= \\textstyle{\\frac{ck+\\alpha}{d}}k - ((m\\!+\\!c)k\\!+\\!\\alpha)\n \\\\ \\nonumber\n&=((m\\!+\\!c)k\\!+\\!\\alpha) \\!\\left(\\textstyle{\\frac{k}{d}} - 1\\right) \\,-\\, m\\textstyle{\\frac{k^2}{d}}\n \\\\ \\nonumber\n&< \\textstyle{\\left\\lfloor \\frac{k}{d} \\right\\rfloor} \\left[dq\\right]_{k^2}.\n\\end{align}\nThis implies that\n\\begin{equation}\n\\left[ z^{j-1}_{n_{j-1} - 1} - z_0^j\\right]_{k^2} = \\left[z^{j-1}_{n_{j-1}} - z_1^j\\right]_{k^2}\n< \\left(\\textstyle{\\left\\lfloor \\frac{k}{d}\\right\\rfloor} -1\\right) \\left[dq\\right]_{k^2},\n\\end{equation}\nso that\n\\begin{align}\n\\left\\langle z_0^j, z^{j-1}_{n_{j-1} - 1} \\right] \n&\\subset \n\\left\\langle z_0^j, z_{{\\left\\lfloor \\frac{k}{d}\\right\\rfloor} -1}^j \\right],\n \\\\\n\\left\\langle z_1^j, z^{j-1}_{n_{j-1}} \\right] \n&\\subset \n\\left\\langle z_{n_{j-1} - \\left({\\left\\lfloor \\frac{k}{d}\\right\\rfloor} -1\\right)}^{j-1}, z_{n_{j-1}}^{j-1} \\right].\n\\end{align}\nThus, since\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1}\\right] =\n\\left\\langle z_0^j, z^{j-1}_{n_{j-1} - 1} \\right] \\cup\n\\left\\langle z_1^j, z^{j-1}_{n_{j-1}} \\right]$\nfor all $j \\in {{\\mathbb Z}}\/d$, we know that\n\\begin{equation}\n\\label{eq: main prop: part (iii), covering}\n \\left\\langle z_0^j, z_{n_{j-1}}^{j-1}\\right]\n\\;\\;\\subset\\;\\;\n \\left\\langle z_0^j, z_{{\\left\\lfloor \\frac{k}{d}\\right\\rfloor} -1}^j \\right]\n \\cup\n \\left\\langle z_{n_{j-1} - \\left({\\left\\lfloor \\frac{k}{d}\\right\\rfloor} -1\\right)}^{j-1}, z_{n_{j-1}}^{j-1} \\right]\n\\end{equation}\nfor all $j \\in {{\\mathbb Z}}\/d$.\n\n\n\nSuppose that $v_q\\!\\left( z_{n_{j-1}}^{j-1}, z_0^j \\right)$ is not constant\nin $j \\in {{\\mathbb Z}}\/d$. Then since $q$ is genus-minimizing, we know that\nfor any $i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2\\right\\}$,\nboth $v_q\\!\\left( z_i^j, z_{i+1}^j \\right)$ and\n$v_q\\!\\left( z_{n_j - (i+1)}^j, z_{n_j - i}^j \\right)$ are constant\nin $j\\in {{\\mathbb Z}}\/d$. Thus by Part (i), neither \n$\\left\\langle z_i^j, z_{i+1}^j \\right]$ nor\n$\\left\\langle z_{n_j - (i+1)}^j, z_{n_j - i}^j \\right]$\nhas any mobile points.\nThus, for all $j \\in {{\\mathbb Z}}\/d$ and $i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2\\right\\}$,\nwe know that\n\\begin{align}\n Q_q \\cap \\left\\langle z_i^j, z_{i+1}^j \\right]\n&= Q_q \\cap \\left\\langle z_i^0, z_{i+1}^0 \\right]+ \\left(z_0^j - z_0^0\\right),\n \\\\\n Q_q \\cap \\left\\langle z_{n_{j-1}-(i+1)}^{j-1}, z_{n_{j-1}-i}^{j-1} \\right]\n&= Q_q \\cap \\left\\langle z_{n_{-1}-(i+1)}^{-1}, z_{n_{-1}-i}^{-1} \\right] \n +\\left(z_{n_{j-1}}^{j-1} - z_{n_{-1}}^{-1}\\right),\n\\end{align}\nwhere we arbitrarily chose $j=0$ as a reference point.\nTaking the union over $i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2\\right\\}$,\nand using the fact that \n$z_0^j-z_0^0 = \\left(z_{n_{j-1}}^{j-1}+\\psi\\right) - \\left(z_{n_{-1}}^{-1} + \\psi\\right)\n= z_{n_{j-1}}^{j-1} - z_{n_{-1}}^{-1}$, we then obtain\n\\begin{align}\n Q_q \\cap \\left\\langle z_0^j, z_{{\\left\\lfloor \\frac{k}{d}\\right\\rfloor} -1}^j \\right]\n&= Q_q \\cap \\left\\langle z_0^0, z_{{\\left\\lfloor \\frac{k}{d}\\right\\rfloor} -1}^0 \\right]\n + \\left(z_0^j - z_0^0\\right)\n \\\\\n Q_q \\cap \\left\\langle z_{n_{j-1} - \\left({\\left\\lfloor \\frac{k}{d}\\right\\rfloor} -1\\right)}^{j-1}, \n z_{n_{j-1}}^{j-1} \\right]\n&= Q_q \\cap \\left\\langle z_{n_{-1} - \\left({\\left\\lfloor \\frac{k}{d}\\right\\rfloor} -1\\right)}^{-1},\n z_{n_{-1}}^{-1} \\right]\n + \\left(z_0^j - z_0^0\\right),\n\\end{align}\nand so, by equation (\\ref{eq: main prop: part (iii), covering}), we have\n\\begin{equation}\n Q_q \\cap \\left\\langle z_0^j, z_{n_{j-1}}^{j-1}\\right]\n= Q_q \\cap \\left\\langle z_0^{0}, z_{n_{-1}}^{-1}\\right] + \\left(z_0^j - z_0^0\\right).\n\\end{equation}\nThus,\n$\\#\\left(Q_q \\cap \\left\\langle z_0^j, z_{n_{j-1}}^{j-1}\\right]\\right)$, and hence\n$v_q\\!\\left(z_{n_{j-1}}^{j-1}, z_0^j\\right)$,\nis constant in $j\\in{{\\mathbb Z}}\/d$.\n \\\\\n\n\n\nWe next show that it is algebraically impossible for $\\psi$ to satisfy\n$\\left[dq\\right]_{k^2} < \\psi < 2\\left[dq\\right]_{k^2}$ when \n$(\\mu,\\gamma) = (1,1)$. Suppose that \n$\\left[dq\\right]_{k^2} < \\psi < 2\\left[dq\\right]_{k^2}$,\nso that \n$\\psi = (m+c)k+\\alpha \\;-\\; \\frac{ck+\\alpha}{d}k + k^2$.\nThen, since $0 < \\psi - \\left[dq\\right]_{k^2} < \\left[dq\\right]_{k^2}$,\nwe have\n\\begin{equation}\n0 \\;<\\; k^2 - \\textstyle{\\frac{ck+\\alpha}{d}}k \\;<\\; (m+c)k+\\alpha.\n\\end{equation}\nOn the other hand, since $(m\\!+\\!c)k\\!+\\!\\alpha \\;-\\; \\frac{ck+\\alpha}{d}k < 0$,\nwe know that\n$(m\\!+\\!c)k\\!+\\!\\alpha \\;<\\; \\textstyle{\\frac{ck+\\alpha}{d}}k$.\nCombining these two facts gives\n\\begin{equation}\nk^2 - \\textstyle{\\frac{ck+\\alpha}{d}}k < \\textstyle{\\frac{ck+\\alpha}{d}}k,\n\\end{equation}\nwhich implies that\n$\\frac{ck+\\alpha}{d} > \\frac{k}{2}$.\nThis, however, contradicts the constraint $\\frac{ck+\\alpha\\gamma}{d} < \\frac{k}{2}$ from\nProposition \\ref{prop: properties of parameters d, m, c, alpha, mu, gamma}.\nThus our initial supposition must be false, and so\n$\\psi \\notin \\left\\langle \\left[dq\\right]_{k^2} , 2\\left[dq\\right]_{k^2}\\right\\rangle$.\n \\\\\n\n\n\n\n\nThis leaves us with the case in which\n$\\psi < \\left[dq\\right]_{k^2}$.\nSuppose that\n$v_q\\!\\left(z_{n_{j-1}}^{j-1}, z_0^j\\right)$ is not constant in $j \\in {{\\mathbb Z}}\/d$,\nso that Part (ii$\\psi$) guarantees the existence of \na non-neutralized R-pseudomobile point, say, $z_0^{j+l}$ in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$,\nfor some $l \\neq 0 \\in {{\\mathbb Z}}\/d$.\nRecall that this implies that\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}- z_{n_{j-1}}^{j-1}\\right) \\in \\left\\langle 0, \\psi\\right\\rangle$,\nor equivalently, that\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}- z_0^j\\right) \\in \\left\\langle -\\psi, 0\\right\\rangle$.\nThis, in turn, implies that\n\\begin{align}\n\\label{eq: main prop (iii), max ineq for L-mobile}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1}- z_0^j\\right) \n&\\in \\left\\langle -\\psi -\\psi + dq,\\; 0 -\\psi + dq \\right\\rangle\n\\subset \\left\\langle -\\psi, dq\\right\\rangle,\\;\\mathrm{and}\n \\\\\n\\label{eq: main prop (iii), max ineq for R-mobile}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}- z_1^j\\right)\n&\\in \\left\\langle -\\psi - dq + dq,\\; 0 -dq + dq\\right\\rangle\n\\subset \\left\\langle -dq, 0\\right\\rangle.\n\\end{align}\nLine (\\ref{eq: main prop (iii), max ineq for L-mobile}) then tells us that\neither $z_{n_{j+l-1}}^{j+l-1}$ is L-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$---contradicting\nthe supposition that $z_0^{j+l}$ is non-neutralized pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$---or \n$z_{n_{j+l-1}}^{j+l-1}$ is L-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$.\nLine (\\ref{eq: main prop (iii), max ineq for R-mobile}),\non the other hand, implies that\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}- z_0^j\\right)\n\\in \\left\\langle -dq, 0\\right\\rangle$,\nwhich, since $2[dq]_{k^2} < k^2$, has no intersection\nwith $\\left\\langle 0, dq \\right\\rangle$, so that\n$z_0^{j+l}$ is not R-mobile in $\\left\\langle z_0^j, z_1^j \\right]$.\nThis means that if\n$z_{n_{j+l-1}}^{j+l-1}$ is L-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$, then\nit is non-neutralized L-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$,\ncontradicting our supposition that\n$v_q\\!\\left(z_{n_{j-1}}^{j-1}, z_0^j\\right)$ is not constant in $j \\in {{\\mathbb Z}}\/d$.\nThus, our original supposition must be false, \nand so $v_q\\!\\left(z_{n_{j-1}}^{j-1}, z_0^j\\right)$ must be constant in $j \\in {{\\mathbb Z}}\/d$.\n \\\\\n\n\n \nWe have now shown that \n$v_q\\!\\left(z_{n_{j-1}}^{j-1}, z_0^j\\right)$ is constant in $j \\in {{\\mathbb Z}}\/d$\nfor all possible values of $\\psi \\in {{\\mathbb Z}}\/k^2$.\nThus, by Proposition \\ref{prop: unique v_q = alpha(k - k^2), and the rest are v_q = alpha (k)},\n$v_q\\!\\left(z_{n_{j-1}}^{j-1}, z_0^j\\right) = \\alpha(k)$ for all $j \\in {{\\mathbb Z}}\/d$.\n\\end{proof}\n \n\n\n\n\n\n\n\\begin{proof}[Proof of (iv)]\nSuppose that\n$\\psi < \\left[dq\\right]_{k^2}$.\nThen, first of all, the conclusion of Part (iii) holds, \nbecause the $\\psi < \\left[dq\\right]_{k^2}$ case of the proof of Part (iii)\ndoes not use the hypothesis that $(\\mu,\\gamma) = (1,1)$.\nThus $v_q(z_{n_{j-1}}^{j-1}, z_0^j)$ is constant in $j \\in {{\\mathbb Z}}\/d$,\nand so by Part (ii'), we know that\nthere exist unique $l\\neq 0 \\in {{\\mathbb Z}}\/d$ and\n$i_* \\in \\left\\{ 0, \\ldots, \\left\\lfloor \\frac{k}{d} \\right\\rfloor - 2 \\right\\}$\nfor which $z_0^{j+l}$ is non-neutralized R-mobile in\n$\\left\\langle z_{i_*}^j, z_{{i_*}+1}^j \\right]$,\nand $z_{n_{j-l}}^{j-l}$ is non-neutralized L-mobile in\n$\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$, with\n$\\left(z_{i_*}^{j_*}, z_{{i_*}+1}^{j_*}\\right) \n= \\left(z_{n_j - (i_*+1)}^{j_*}, z_{n_j - i_*}^{j_*}\\right)\n= (x_*,y_*)$, where $x_*, y_* \\in Q_q$ are the unique elements of $Q_q$\nsatisfying $v_q(x_*,y_*) = \\alpha(k-k^2)$.\nNote that this implies $i_* = \\frac{n_{j_*}-1}{2}$.\n\n\nSince $z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_{i_*}^j, z_{{i_*}+1}^j \\right]$, we know that\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{i_*}^j\\right)\n\\in \\left\\langle 0, \\left[dq\\right]_{k^2}\\right\\rangle$.\nThe fact that\n$\\psi < \\left[dq\\right]_{k^2}$ then implies\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1}-z_{i_*}^j\\right)\n\\in \\left\\langle -[dq], \\left[dq\\right]_{k^2}\\right\\rangle$.\nHowever, since $z_0^{j+l}$ is non-neutralized R-mobile in\n$\\left\\langle z_{i_*}^j, z_{{i_*}+1}^j \\right]$, we know that\n$z_{n_{j+l-1}}^{j+l-1}$ is not L-mobile in\n$\\left\\langle z_{i_*}^j, z_{{i_*}+1}^j \\right]$, and so\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1}-z_{i_*}^j\\right)\n\\notin \\left\\langle -[dq], 0\\right\\rangle$. Thus\n\\begin{equation}\n\\label{eq: main prop (iv), type (1,1), chaperoned R-mobile, L min in}\n\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1}-z_{i_*}^j\\right)\n\\;\\in\\; \\left\\langle 0, \\left[dq\\right]_{k^2}\\right\\rangle.\n\\end{equation}\n\nSuppose that ${i_*} < \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2$.\nThen (\\ref{eq: main prop (iv), type (1,1), chaperoned R-mobile, L min in}) implies\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1}-z_{{i_*}+2}^j\\right)\n\\in \\left\\langle -\\left[dq\\right]_{k^2}, 0\\right\\rangle$,\nso that $z_{n_{j+l-1}}^{j+l-1}$ is L-mobile in\n$\\left\\langle z_{{i_*}+1}^j, z_{{i_*}+2}^j \\right]$.\nMoreover, the fact that\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{i_*}^j\\right)\n\\in \\left\\langle 0, \\left[dq\\right]_{k^2}\\right\\rangle$ implies\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{i_*+1}^j\\right)\n\\in \\left\\langle -\\left[dq\\right]_{k^2}, 0\\right\\rangle$, which, since\n$2[dq]_{k^2} < k^2$, has no intersection with\n$\\left\\langle 0, \\left[dq\\right]_{k^2}\\right\\rangle$,\nand so $z_0^{j+l}$ is {\\em not} R-mobile in\n$\\left\\langle z_{{i_*}+1}^j, z_{{i_*}+2}^j \\right]$.\nThus $z_{n_{j+l-1}}^{j+l-1}$ is non-neutralized L-mobile in\n$\\left\\langle z_{{i_*}+1}^j, z_{{i_*}+2}^j \\right]$,\ncontradicting the fact that\n$\\left\\langle z_{i_*}^j, z_{{i_*}+1}^j \\right]$\nalready has a non-neutralized mobile point.\nThus, ${i_*} = \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2$,\nimplying $\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\in \\{2,3\\}$,\nand $z_0^{j+l}$ is non-neutralized R-mobile in\n$\\left\\langle z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 2}^j, \nz_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 1}^j \\right]$.\n\n\n\n\nWe next determine which configurations of ${\\bf{z}}^j$\nadmit such an R-mobile point. We begin by partitioning\nthe values of\n$j^{\\prime}\\in{{\\mathbb Z}}\/d$ partition into four sets,\naccording to whether \n$n_{j^{\\prime}}=\\left\\lfloor\\frac{k}{d}\\right\\rfloor$\nor $n_{j^{\\prime}}=\\left\\lfloor\\frac{k}{d}\\right\\rfloor - 1$,\nand to whether the $z_0^{j+l}$ is active or inactive in\n$\\left\\langle z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 2}^j,\nz_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 1}^j \\right]$ at time $j=j^{\\prime}$\n(or equivalently, to whether\n$z_0^{j^{\\prime}+l}\n\\in \\left\\langle z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 2}^{j^{\\prime}}, \nz_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 1}^{j^{\\prime}} \\right]$ or\n$z_0^{j^{\\prime}+l}\n\\notin \\left\\langle z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 2}^{j^{\\prime}}, \nz_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 1}^{j^{\\prime}} \\right]$).\nWe begin by claiming that the set\n\\begin{equation}\n\\label{eq: part (iv), J_0 definition with i=k\/d-2}\nJ_0 := \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \nn_{j^{\\prime}}\\!=\\!\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}\\!-\\!1;\\;\nz_0^{j+l}\\;\\text{is inactive in}\\;\\!\n\\left\\langle z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-2}^j, z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-1}^j\\right]\n\\!\\;\\text{when}\\;j=j^{\\prime}\n\\right.\\!\\right\\}\n\\end{equation}\nis nonempty.\nSuppose that $J_0$ is empty.\nThen ${{\\mathbb Z}}\/d$ partitions\ninto the disjoint union of $J_1$, $J_2$, and $J_3$, defined as follows:\n\\begin{align}\n\\label{eq: part (iv), J_1 definition with i=k\/d-2}\n J_1\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}}\\!=\\!\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}\\!-\\!1;\\;\n z_0^{j+l}\\;\\text{is active in}\\;\\!\n \\left\\langle z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-2}^j, \n z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-1}^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\},\n \\\\\n\\label{eq: part (iv), J_2 definition with i=k\/d-2}\n J_2\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}}\\!=\\!\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor};\\;\n z_0^{j+l}\\;\\text{is inactive in}\\;\\!\n \\left\\langle z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-2}^j, \n z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-1}^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\},\n \\\\\n\\label{eq: part (iv), J_3 definition with i=k\/d-2}\n J_3\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}}\\!=\\!\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor};\\;\n z_0^{j+l}\\;\\text{is active in}\\;\\!\n \\left\\langle z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-2}^j, \n z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-1}^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\}.\n\\end{align}\nNote that $J_1$ and $J_2$ are nonempty. $J_1$ is nonempty because\n$n_{j^{\\prime}}=\\left\\lfloor\\frac{k}{d}\\right\\rfloor-1$ only if $j^{\\prime} \\in J_1$;\n$J_2$ is nonempty because $z_0^{j+l}$ is inactive in\n$\\left\\langle z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-2}^j, \nz_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-1}^j\\right]$\nat time $j=j^{\\prime}$ only if $j^{\\prime} \\in J_2$.\nThe question of whether or not $J_3$ is empty depends on whether or not $l=1$.\nSince $J_1$ is nonempty, we know that for any $j_1 \\in J_1$ we have\n$z_0^{j_1+l} \\in \\left\\langle z_{n_{j_1}-1}^{j_1}, z_{n_{j_1}}^{j_1}\\right]$,\nso that\n\\begin{align}\n z_0^{j_1+1}-z_0^{j_1+l}\n&= \\left(z_0^{j_1+1}-z_{n_{j_1}}^{j_1}\\right) + \\left(z_{n_{j_1}}^{j_1} - z_0^{j_1+l}\\right)\n \\\\ \\nonumber\n&=\\psi + \\left(z_{n_{j_1}}^{j_1} - z_0^{j_1+l}\\right)\n \\\\ \\nonumber\n&\\in \\left\\langle \\psi, \\psi + dq \\right\\rangle.\n\\end{align}\nThe fact that $0< \\psi + [dq]_{k^2} < k^2$ then implies that\n$l \\neq 1$. Thus $J_3$ is nonempty, since otherwise\n$z_{n_j}^j - z_0^{j+l}$ would be constant in ${{\\mathbb Z}}\/d$,\ncontradicting Corollary \\ref{cor: q of positive type: combo of lemma and difference eq}.\nThe definitions of $J_1$, $J_2$, and $J_3$, along with\nline (\\ref{eq: main prop (iv), type (1,1), chaperoned R-mobile, L min in}),\nthen show that\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1} - z_{n_j}^j\\right) \\in\n\\left\\langle -dq, 0\\right\\rangle$ and\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_{n_j}^j\\right) \\in\n\\left\\langle -dq, 0\\right\\rangle$,\nwhich tell us, respectively, that\n$z_{n_{j+l-1}}^{j+l-1}$ is L-mobile in\n$\\left\\langle z_{n_j - 1}^j, z_{n_j}^j\\right]$,\nand that\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_{n_j - 1}^j\\right) \\in\n\\left\\langle -dq, 0\\right\\rangle$,\nso that $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_{n_j - 1}^j\\right) \\notin\n\\left\\langle 0, dq\\right\\rangle$,\nmaking $z_{n_{j+l-1}}^{j+l-1}$ L-mobile non-neutralized L-mobile in\n$\\left\\langle z_{n_j - 1}^j, z_{n_j}^j\\right]$.\nPart (ii') then tells us that the non-neutralized \nR-mobile point $z_0^{j+l}$ in\n$\\left\\langle z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 2}^j, \nz_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 1}^j \\right]$ is the mirror of the\nnon-neutralized L-mobile point $z_{n_{j+l-1}}^{j+l-1}$ in\n$\\left\\langle z_{n_j - 1}^j, z_{n_j}^j\\right]$.\n\n\n\nThus $\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor = 2$, and\n$z_0^{j+l}$ is non-neutralized R-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$.\nWe claim $z_0^{j+l}$ is the only non-neutralized mobile point in\n$\\left\\langle z_0^j, z_1^j \\right]$.\nThat is, (\\ref{eq: main prop (iv), type (1,1), chaperoned R-mobile, L min in})\nimplies that\nthe $\\minq$ of $\\left\\{z^{j+l-1}_{n_{j+l-1}}-z_1^j\\right\\}_{j\\in{{\\mathbb Z}}\/d}$\noccurs when $j \\in J_2$, but for any $j_2 \\in J_2$,\n(\\ref{eq: main prop (iv), type (1,1), chaperoned R-mobile, L min in}) implies that\n$z^{j_2+l-1}_{n_{j_2+l-1}}-z_1^{j_2} \\in\n\\left\\langle 0, dq\\right\\rangle$, which, since $2[dq]_{k^2} < k^2$,\nhas no intersection with\n$\\left\\langle -dq, 0 \\right\\rangle$.\nThus $z^{j+l-1}_{n_{j+l-1}}$ is not L-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$, and so\n$z_0^{j+l}$ is the only non-neutralized mobile point in\n$\\left\\langle z_0^j, z_1^j \\right]$.\n\n\nSuppose that $\\alpha = +1$.\nThen by Part (i'), we know there is some $j_* \\in {{\\mathbb Z}}\/d$ such that\n$v_q(z_0^{j_*}, z_1^{j_*}) = k - k^2$, and \n$v_q(z_0^{j^{\\prime}}, z_1^{j^{\\prime}}) = k$ for all $j^{\\prime} \\neq j_*$ in ${{\\mathbb Z}}\/d$.\nThis means that there is one more active non-neutralized mobile point in\n$\\left\\langle z_0^j, z_1^j\\right]$ at time $j_*$ than at any other time, which\nin this case implies that the unique non-neutralized mobile point in\n$\\left\\langle z_0^j, z_1^j\\right]$, namely, $z_0^{j+l}$, must be active precisely once,\nbut this contradicts the fact that both $J_1$ and $J_3$ are nonempty.\nThus $J_0$ cannot be empty when $\\alpha = +1$.\n\n\nIf $\\alpha = -1$,\nthen Part (i') tells us there is some $j_* \\in {{\\mathbb Z}}\/d$ such that\n$v_q(z_0^{j_*}, z_1^{j_*}) = -k + k^2$, and \n$v_q(z_0^{j^{\\prime}}, z_1^{j^{\\prime}}) = -k$ for all $j^{\\prime} \\neq j_*$ in ${{\\mathbb Z}}\/d$.\nThis means that there is one fewer active non-neutralized mobile point in\n$\\left\\langle z_0^j, z_1^j\\right]$ at time $j_*$ than at any other time,\nand so we have $j_* \\in J_2$. But $j_* \\in J_2$ implies that\n$n_{j_*} = 2$, contradicting the fact that $i_* = \\frac{n_{j_*}-1}{2}\\in {{\\mathbb Z}}$\nimplies $n_{j_*}$ is odd.\nThus $J_0$ cannot be empty when $\\alpha = -1$,\nand so $J_0 \\neq \\emptyset$.\n\n\n\n\n\nWe next observe that the nonemptyness of $J_0$ implies that the set\n\\begin{equation}\n J_3\n:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}}\\!=\\!\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor};\\;\n z_0^{j+l}\\;\\text{is active in}\\;\\!\n \\left\\langle \\!z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-2}^j, \n z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-1}^j\\right]\n \\!\\;\\text{at time}\\;j=j^{\\prime}\n \\right.\\!\\right\\}\n\\end{equation}\nis empty, because if not, then for\narbitrary elements $j_0 \\in J_0$ and $j_3 \\in J_3$, we have\n\\begin{align}\n\\label{eq: part (iv), J_0 nonempty implies J_3 empty}\n z_0^{j_0+l} - z_{n_{j_0}}^{j_0}\n&= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 2}^j\\right) - dq\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 2}^j\\right)\n \\\\ \\nonumber\n&= z_0^{j_3+l}-z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 2}^{j_3}\n \\\\ \\nonumber\n&= z_0^{j_3+l} - z_{n_{j_3}}^{j_3} + 2dq,\n\\end{align}\ncontradicting Corollary \\ref{cor: q of positive type: combo of lemma and difference eq}.\n\n\n\nNote that we once again know only that\n$i_* = \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2$, with\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\in \\{2, 3\\}$.\nWriting $i_*$ instead of $\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 2$\nto simplify notation, we can now express ${{\\mathbb Z}}\/d$ is the disjoint union of\n\\begin{align}\n J_0\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}}\\!=\\!\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}\\!-\\!1;\\;\n z_0^{j+l}\\;\\text{is inactive in}\\;\\!\n \\left\\langle z_{i_*}^j, z_{i_* + 1}^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\},\n \\\\\n J_1\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}}\\!=\\!\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}\\!-\\!1;\\;\n z_0^{j+l}\\;\\text{is active in}\\;\\!\n \\left\\langle z_{i_*}^j, z_{i_* + 1}^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\},\\;\\;\\mathrm{and}\n \\\\\n J_2\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}}\\!=\\!\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor};\\;\n z_0^{j+l}\\;\\text{is inactive in}\\;\\!\n \\left\\langle z_{i_*}^j, z_{i_* + 1}^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\},\n\\end{align}\nall of which are nonempty. We have already proven that $J_0$ is nonempty.\n$J_1$ is nonempty because $z_0^{j+l}$ is active\nin $\\left\\langle z_{i_*}^j, z_{i_* + 1}^j\\right]$ at time $j=j^{\\prime}$\nonly if $j^{\\prime} \\in J_1$, and $J_2$ is nonempty because\n$n_{j^{\\prime}}= \\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor$ only if $j^{\\prime}\\in J_2$.\nThe above definitions of $J_0$, $J_1$, and $J_2$ imply\nthat $z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j \\right]$,\nand (\\ref{eq: main prop (iv), type (1,1), chaperoned R-mobile, L min in})\nimplies that $z^{j+l-1}_{n_{j+l-1}}$ is not L-mobile in\n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j \\right]$,\nso that $z_0^{j+l}$ is in fact non-neutralized R-mobile in\n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j \\right]$.\nThe mirror relation (\\ref{eq: mirror relation 2})\nthen tells us that $z_{n_{j-l}}^{j-l}$ is non-neutralized L-mobile in\n$\\left\\langle z_0^j, z_1^j\\right]$.\nThus, $i_* = 0$ and $\\left\\lfloor\\frac{k}{d}\\right\\rfloor =2$.\n\n\nWe next claim that\n\\begin{equation}\n\\label{eq: part (iv), type (1,1), min in (2psi, dq)}\n2 \\psi < \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right) < [dq]_{k^2}.\n\\end{equation}\nLine (\\ref{eq: main prop (iv), type (1,1), chaperoned R-mobile, L min in})\nalready implies\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right) \n\\in \\left\\langle \\psi, dq\\right\\rangle$.\nSuppose\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right)\n\\in \\left\\langle \\psi, 2\\psi \\right\\rangle$.\nThen, for arbitrary $j_0 \\in J_0$, we have\n\\begin{align}\n\\label{eq: part (iv), max = min - 2psi}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1} - z_0^{j+1}\\right)\n&= z_{n_{j_0+l-1}}^{j_0+l-1} - z_0^{j_0+1}\n \\\\ \\nonumber\n&= \\left(z_0^{j_0+l} - \\psi\\right) - \\left(z_0^{j_0} + dq+\\psi\\right)\n \\\\ \\nonumber\n&= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_0^j\\right)\n -dq - 2\\psi\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_0^j\\right) - 2\\psi\n \\\\ \\nonumber\n&\\in \\left\\langle -\\psi, 0\\right\\rangle,\n\\end{align}\nso that $z_{n_{j+l-1}}^{j+l-1}$ is L-pseudomobile in\n$\\left\\langle z_{n_j}^j, z_0^{j+1} \\right]$.\nMoreover, for arbitrary $j_1 \\in J_1$, we have\n\\begin{align}\n \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{n_j}^j\\right) \n&= z_0^{j_1+l}-z_1^{j_1}\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right) + dq\n \\\\ \\nonumber\n&\\in \\left\\langle \\psi + dq, 2dq\\right\\rangle,\n\\end{align}\nwhich, since $2[dq]_{k^2} < k^2$, has no intersection with\n$\\left\\langle 0, \\psi \\right\\rangle$. Thus \n$z_0^{j+l}$ is not R-pseudomobile in $\\left\\langle z_{n_j}^j, z_0^{j+1} \\right]$,\nbut this means that $z_{n_{j+l-1}}^{j+l-1}$ is L-pseudomobile in\n$\\left\\langle z_{n_j}^j, z_0^{j+1} \\right]$, a contradiction.\nThus (\\ref{eq: part (iv), type (1,1), min in (2psi, dq)}) must be true.\n\n\nThis, in turn, implies that\n$z_{n_{j+l-1}}^{j+l-1}$ is L-mobile in $\\left\\langle z_0^{j+1}, z_1^{j+1}\\right]$.\nThat is, \n\\begin{align}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1} - z_1^{j+1}\\right)\n&= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1} - z_0^{j+1}\\right) - dq\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_0^j\\right) - 2\\psi - dq\n \\\\ \\nonumber\n&\\in \\left\\langle -dq, -2\\psi\\right\\rangle,\n\\end{align}\nwhere the second line used (\\ref{eq: part (iv), max = min - 2psi})\nand the third line used (\\ref{eq: part (iv), type (1,1), min in (2psi, dq)}).\nMoreover, since\n\\begin{align}\n \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^{j+1}\\right)\n&= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_1^{j+1}\\right)\n \\\\ \\nonumber\n&= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1} - z_1^{j+1}\\right) + \\psi\n \\\\ \\nonumber\n&\\in \\left\\langle -dq + \\psi, -\\psi\\right\\rangle,\n\\end{align}\nwe know that $z_0^{j+l}$ is not R-mobile in\n$\\left\\langle z_0^{j+1}, z_1^{j+1}\\right]$.\nThus, $z_{n_{j+l-1}}^{j+l-1}$ is non-neutralized\nL-mobile in $\\left\\langle z_0^{j+1}, z_1^{j+1}\\right]$,\nor equivalently,\n$z_{n_{j+l-2}}^{j+l-2}$ is non-neutralized L-mobile in\n$\\left\\langle z_0^j, z_1^j\\right]$. We already know, however, that\n$z_{n_{j-l}}^{j-l}$ is non-neutralized L-mobile in\n$\\left\\langle z_0^j, z_1^j\\right]$. Thus,\n$l-2 \\equiv -l\\; (\\mod d)$, so that\n\\begin{equation}\n\\label{part iv: eq: 2l = 2}\n2l \\equiv 2\\;(\\mod d).\n\\end{equation}\n\n\n\n\nWe claim that this implies $c=1$.\nThat is, since $z_0^{j+l}$ and $z_{n_{j-l}}^{j-l}$ are mobile\nin $\\left\\langle z_0^j, z_1^j\\right]$, Part (i') tells us there is a unique\n$j_* \\in {{\\mathbb Z}}\/d$ such that\n$v_q\\!\\left( z_0^{j_*}, z_1^{j_*}\\right) = \\alpha(k-k^2)$, and\n$v_q\\!\\left( z_0^{j^{\\prime}}, z_1^{j^{\\prime}}\\right) = \\alpha(k)$\nfor all $j^{\\prime}\\neq j_* \\in {{\\mathbb Z}}\/k^2$.\nThus, if we define $\\chi_R(j^{\\prime})$ (respectively \n$\\chi_L(j^{\\prime})$) to be equal to 1 if $z_0^{j+l}$\n(respectively $z_{n_{j-l}}^{j-l}$) is active in \n$\\left\\langle z_0^j, z_1^j \\right]$ at time $j = j^{\\prime}$,\nand equal to 0 otherwise, then for any $j^{\\prime} \\in {{\\mathbb Z}}\/d$,\n\\begin{equation}\n\\label{part iv: eq: how many mobile points are active?}\n \\chi_R(j^{\\prime}) + \\chi_L(j^{\\prime})\n= \\begin{cases}\n 1 & j^{\\prime} \\neq j_*\n \\\\\n 2 & j^{\\prime} = j_*, \\;\\alpha = +1\n \\\\\n 0 & j^{\\prime} = j_*,\\;\\alpha = -1\n \\end{cases}.\n\\end{equation}\nThis is because a mobile point in\n$\\left\\langle z_0^j, z_1^j \\right]$ contributes\n$-k^2$ to $v_q\\!\\left( z_0^{j^{\\prime}}, z_1^{j^{\\prime}}\\right)$\nif it is active at time $j=j^{\\prime}$ and contributes zero otherwise.\nThus (\\ref{part iv: eq: how many mobile points are active?})\nimplies that\n\\begin{equation}\n\\label{part iv: eq: total actives is d+alpha}\n\\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_R(j) \\;+\\; \\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_L(j) = d+\\alpha.\n\\end{equation}\nOn the other hand, \nProposition \\ref{prop: mobile point is active [lepsilon] times or [(l-1)epsilon] times}\ntells us that\n\\begin{equation}\n\\label{part iv: eq: R active le, L active (l-1)e}\n\\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_R(j) = [l\\epsilon]_d,\\;\\;\\;\\;\\;\\;\n\\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_L(j) = [(l-1)\\epsilon]_d.\n\\end{equation}\nCombining \n(\\ref{part iv: eq: total actives is d+alpha}) and\n(\\ref{part iv: eq: R active le, L active (l-1)e}),\nwe then have\n\\begin{align}\n d+ \\alpha \n&= [l\\epsilon]_d + [(l-1)\\epsilon]_d\n \\\\ \\nonumber\n \\alpha\n&\\equiv l\\epsilon + (l-1)\\epsilon\\;\\; (\\mod d)\n \\\\ \\nonumber\n \\gamma\\left( \\alpha\\gamma\\,\\epsilon^{-1}\\right) \n&\\equiv 2l-1\\;\\; (\\mod d)\n \\\\ \\nonumber\n \\gamma c\n&\\equiv (2)-1\\;\\; (\\mod d)\n \\\\ \\nonumber\n c\n& = [\\gamma]_d,\n\\end{align}\nwhere the second to last line used the facts that\n$c = [\\alpha\\gamma\\,{\\epsilon}^{-1}]_d$ and that\n$2l \\equiv 2\\;(\\mod d)$, from (\\ref{part iv: eq: 2l = 2}).\nIf $\\gamma = -1$, then $c = d-1 \\leq \\frac{d}{2}$,\nimplying $d\\leq 2$, contradicting the fact that\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor = 2$. Thus $\\gamma = +1$ and $c=1$,\nThis, in turn, implies that $(\\mu,\\gamma) = (1,1)$ and $\\epsilon = [\\alpha]_d$.\n\n\n\nIf $\\alpha = +1$, then $\\epsilon = 1$. Now,\n$n_{j^{\\prime}} = \\left\\lfloor\\frac{k}{d}\\right\\rfloor - {\\theta}^{d, \\epsilon}(j^{\\prime})$\nfor all $j^{\\prime}\\in{{\\mathbb Z}}\/d$. Thus, we have\n\\begin{align}\n 1\n&= \\epsilon\n \\\\ \\nonumber\n&= \\#\\{j^{\\prime}\\in {{\\mathbb Z}}\/d\\left\\vert\\; {\\theta}^{d, \\epsilon}(j^{\\prime})=1 \\right.\\}\n \\\\ \\nonumber\n&= \\#\\{j^{\\prime}\\in {{\\mathbb Z}}\/d\\left\\vert\\; n_{j^{\\prime}}= \n \\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor} - 1\\right.\\}\n \\\\ \\nonumber\n&= \\#\\{J_0 \\cup J_1\\},\n\\end{align}\nbut this contradicts the fact that both $J_0$ and $J_1$ are nonempty.\nThus we are left with the case in which $\\alpha = -1$.\n\nThe case of $\\alpha = -1$ is somewhat more complicated.\nWe begin by computing $\\psi$:\n\\begin{align}\n \\psi\n&= \\left[(\\mu m + \\gamma c)k + \\alpha - \\gamma \\textstyle{\\frac{ck + \\alpha\\gamma}{d}}k\\right]_d\n \\\\ \\nonumber\n&= \\left[(m + 1)k -1 - \\textstyle{\\frac{k -1}{d}}k\\right]_d\n \\\\ \\nonumber\n&= \\left[(m + 1)k -1 - 2k\\right]_d\n \\\\ \\nonumber\n&= \\left[(m - 1)k -1\\right]_d,\n\\end{align}\nwhere the third line used the fact that $\\left\\lfloor\\frac{k}{d}\\right\\rfloor = 2$.\nIf $m=1$, then $\\psi = k^2-1$, contradicting the fact\nthat $\\psi < [dq]_k^2 < \\frac{k^2}{2}$. Thus $m > 1$ and\n$\\psi = (m-1)k-1$.\nNow, (\\ref{eq: part (iv), type (1,1), min in (2psi, dq)}) also tells us that\n$2\\psi < [dq]_{k^2}$. Thus,\n\\begin{align}\n \\nonumber\n 2\\psi\n&< (m+1)k -1\n \\\\ \\nonumber\n 2\\left((m-1)k-1\\right)\n&< (m+1)k -1\n \\\\ \\nonumber\n mk\n&< 3k +1\n \\\\ \n m\n&\\le 3.\n\\end{align}\nCombining this with the fact that $m>1$ tells us that $m \\in \\{2,3\\}$.\n\n\n\n\nFirst consider the case in which $m=3$, so that $q = -\\frac{k-1}{d}(3k-1) = -2(3k-1)$.\nNote that the fact that $\\frac{k-1}{d} = 2$ implies that $k$ is odd.\nWe claim that $q$ is not genus-minimizing, which we shall prove\nby showing that $q^{\\prime} := -q = 2(3k-1)$ is not genus-minimizing.\nRecall that $\\tilde{Q}_{q^{\\prime}}$ denotes the lift to the integers of the set\n$Q_{q^{\\prime}} := \\{a{q^{\\prime}}\\left\\vert\\;a\\in\\{0, \\ldots, k-1\\}\\right.\\}$.\nIf $k \\equiv 0 \\; (\\mod 3)$, then\n\\begin{equation}\n \\left(\\textstyle{\\frac{2k}{3}}{q^{\\prime}}, \\textstyle{\\frac{k}{3}}{q^{\\prime}}, 0{q^{\\prime}}, \\textstyle{\\frac{5k+3}{6}}{q^{\\prime}}, \n \\textstyle{\\frac{3k+3}{6}}{q^{\\prime}}, \\textstyle{\\frac{k+3}{6}}{q^{\\prime}}\\right)\n= \\left(-\\textstyle{\\frac{4k}{3}}, -\\textstyle{\\frac{2k}{3}}, 0, \\textstyle{\\frac{4k}{3}}\\!-\\!1, \n \\textstyle{\\frac{6k}{3}}\\!-\\!1, \\textstyle{\\frac{8k}{3}}\\!-\\!1\\right)\n\\end{equation}\nin $\\left({{\\mathbb Z}}\/k^2\\right)^6$, and so the interval\n$\\left\\langle -\\frac{4k}{3}, \\frac{8k}{3} - 1\\right]$ contains at least \n5 elements of $\\tilde{Q}_{q^{\\prime}}$. Thus\n\\begin{align}\n -v_{q^{\\prime}}\\left(\\textstyle{\\frac{2k}{3}}{q^{\\prime}},\\textstyle{\\frac{k+3}{6}}{q^{\\prime}}\\right)\n&= -\\left(\\left(\\textstyle{\\frac{8k}{3}}\\!-\\!1\\right)- \\left(-\\textstyle{\\frac{4k}{3}}\\right)\\right)k\n \\;\\;+\\;\\;\\#\\!\\left(\\left\\langle -\\textstyle{\\frac{4k}{3}}, \n \\textstyle{\\frac{8k}{3}} \\!-\\! 1\\right]\\cap\\tilde{Q}_{q^{\\prime}}\\right) k^2\n \\\\ \\nonumber\n&\\ge -(4k-1)k + 5k^2\n \\\\ \\nonumber\n&= k(k+1),\n\\end{align}\nand so Proposition \\ref{prop: not minimizing if v >= k(k+1)} implies\nthat ${q^{\\prime}}$ is not genus-minimizing.\nIf $k \\equiv -1 \\; (\\mod 3)$, then\n\\begin{equation}\n \\left(0{q^{\\prime}}, \\textstyle{\\frac{k+1}{6}}{q^{\\prime}}, \\textstyle{\\frac{k+1}{3}}{q^{\\prime}}, \\textstyle{\\frac{k+1}{2}}{q^{\\prime}}\\right)\n= \\left(0, \\textstyle{\\frac{2k-1}{3}}, \\textstyle{\\frac{4k-2}{3}}, 2k-1\\right)\n\\end{equation}\nin $\\left({{\\mathbb Z}}\/k^2\\right)^4$, and so the interval\n$\\left\\langle 0, 2k-1\\right]$ contains at least \n3 elements of $\\tilde{Q}_{q^{\\prime}}$. Thus\n\\begin{align}\n -v_{q^{\\prime}}\\left(0{q^{\\prime}},\\textstyle{\\frac{k+1}{2}}{q^{\\prime}}\\right)\n&= -\\left((2k-1) - (0)\\right)k\n \\;\\;+\\;\\;\\#\\!\\left(\\left\\langle 0, 2k-1\\right] \\cap \\tilde{Q}_{q^{\\prime}}\\right) k^2\n \\\\ \\nonumber\n&\\ge -(2k-1)k + 3k^2\n \\\\ \\nonumber\n&= k(k+1),\n\\end{align}\nand so ${q^{\\prime}}$ is not genus-minimizing. Lastly, if $k \\equiv 1 \\; (\\mod 3)$, then\n\\begin{equation}\n \\left(\\textstyle{\\frac{k-1}{6}}{q^{\\prime}}, \\textstyle{\\frac{5k+1}{6}}{q^{\\prime}}, 0{q^{\\prime}}, \\textstyle{\\frac{2k+1}{3}}{q^{\\prime}}\\right)\n= \\left(\\textstyle{\\frac{-4k+1}{3}}, \\textstyle{\\frac{-2k-1}{3}}, 0,\\textstyle{\\frac{2k-2}{3}}\\right)\n\\end{equation}\nin $\\left({{\\mathbb Z}}\/k^2\\right)^4$, and so the interval\n$\\left\\langle \\textstyle{\\frac{-4k+1}{3}}, \\textstyle{\\frac{2k-2}{3}}\\right]$ contains at least \n3 elements of $\\tilde{Q}_{q^{\\prime}}$. Thus\n\\begin{align}\n -v_{q^{\\prime}}\\left(\\textstyle{\\frac{k-1}{6}}{q^{\\prime}}, \\textstyle{\\frac{2k+1}{3}}{q^{\\prime}}\\right)\n&= -\\left(\\left(\\textstyle{\\frac{2k-2}{3}}\\right) - \\left(\\textstyle{\\frac{-4k+1}{3}}\\right)\\right)k\n \\;\\;+\\;\\;\\#\\!\\left(\\left\\langle \\textstyle{\\frac{-4k+1}{3}}, \\textstyle{\\frac{2k-2}{3}}\\right] \n \\cap \\tilde{Q}_{q^{\\prime}}\\right) k^2\n \\\\ \\nonumber\n&\\ge -(2k-1)k + 3k^2\n \\\\ \\nonumber\n&= k(k+1),\n\\end{align}\nand so ${q^{\\prime}}$ is not genus-minimizing. We have checked all three equivalence classes\nof $k$ modulo 3, and so $q= -q^{\\prime}$ is not genus-minimizing for any $k$.\n\n\nThis leaves us with the case in which $m=2$, so that\n$q = -\\frac{k-1}{d}(2k-1)= -2(2k-1)$.\nSince $\\frac{k-1}{d}=2$, we know that $k \\equiv 1\\;(\\mod 2)$.\nSuppose that $k \\equiv 3\\;(\\mod 4)$, and set $q^{\\prime} := -q = 4k-2$.\nThen\n\\begin{equation}\n \\left(0{q^{\\prime}}, \\textstyle{\\frac{k+1}{4}}{q^{\\prime}}, \\textstyle{\\frac{k+1}{2}}{q^{\\prime}}\\right)\n= \\left(0, \\textstyle{\\frac{k-1}{2}}, k-1 \\right)\n\\end{equation}\nin $\\left({{\\mathbb Z}}\/k^2\\right)^3$, and so the interval\n$\\left\\langle 0, k-1\\right]$ contains at least \n2 elements of $\\tilde{Q}_{q^{\\prime}}$. Thus\n\\begin{align}\n -v_{q^{\\prime}}\\left(0{q^{\\prime}},\\textstyle{\\frac{k+1}{2}}{q^{\\prime}}\\right)\n&= -\\left((k-1) - (0)\\right)k\n \\;\\;+\\;\\;\\#\\!\\left(\\left\\langle 0, k-1\\right] \\cap \\tilde{Q}_{q^{\\prime}}\\right) k^2\n \\\\ \\nonumber\n&\\ge -(k-1)k + 2k^2\n \\\\ \\nonumber\n&= k(k+1),\n\\end{align}\nand so ${q^{\\prime}}$ is not genus-minimizing.\n\n\nAt last, this leaves us with the case in which\n$(\\mu, \\gamma) = (1,1)$, $\\alpha = -1$,\n$m=2$, $c=1$, and $d=\\frac{k-1}{2} \\equiv 0\\; (\\mod 2)$,\nso that $q = -2(2k-1)$. As we shall show later,\nin Proposition \\ref{prop: bookkeeping for q genus-minimizing},\n$q$ is actually genus-minimizing in this case,\nso there is no argument we can make to explain it away.\n\nWe can still, however, prove that there are no\nnon-neutralized mobile points in this case.\nWe do this by showing that\n$\\left\\langle z_0^j, z_1^j\\right]$ has only one R-mobile point.\nNow, for any $l \\neq 0\\in{{\\mathbb Z}}\/d$, $z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_0^j, z_1^j\\right]$ if and only if\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right) \\in \\left\\langle 0, dq\\right\\rangle$,\nWe therefore proceed by using\nCorollary \\ref{cor: q of positive type: combo of lemma and difference eq}\nto compute $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right) \\in {{\\mathbb Z}}\/k^2$.\nNote that $\\epsilon = \\left[\\alpha\\gamma\\, c^{-1}\\right]_d = d-1$, implying\n$[l\\epsilon]_d = d-[l]_d$. We therefore have\n\\begin{align}\n \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right)\n&= [ml]_d\\textstyle{\\frac{k^2}{d}} + \\left(\\textstyle{\\frac{[l\\epsilon]_d}{d}}-1\\right)[dq]_{k^2}\n \\\\ \\nonumber\n&= [ml]_d\\textstyle{\\frac{k^2}{d}} - \\textstyle{\\frac{[l]_d}{d}}((m+c)k-1)\n \\\\ \\nonumber\n&= m[l]_dk\\left(\\textstyle{\\frac{k-1}{d}}\\right) - [l]_d\\textstyle{\\frac{ck-1}{d}}\n \\\\ \\nonumber\n&=\\left(\\textstyle{\\frac{k-1}{d}}\\right)[l]_d ( mk - 1)\n \\\\ \\nonumber\n&= [l]_d(4k-2).\n \\\\ \\nonumber\n\\end{align}\nThis leaves us with the task of determining which\n$[l]_d \\in \\{1, \\ldots, d-1\\} = \\left\\{1, \\ldots, \\frac{k-3}{2}\\right\\}$ satisfies\n$[l]_d(4k-2) \\in \\left\\langle 0, 3k-1\\right\\rangle$\n(since $[dq]_{k^2} = 3k-1$).\nNow, as integers,\n\\begin{equation}\n3k-1 < 4k-2 \\leq \\;\\;(4k-2)x \\;\\;\\leq k^2 - \\textstyle{\\frac{3k+1}{2}} < k^2\n\\end{equation}\nfor all $x \\in \\left\\{1, \\ldots, \\frac{k+3}{4}-1\\right\\}$, and \n\\begin{equation}\nk^2 + 3k-1 < k^2 + 6k + \\textstyle{\\frac{k-7}{2}} \n\\leq \\;\\;(4k-2)x \\;\\;\\leq 2k^2 - 7k + 3 < 2k^2\n\\end{equation}\nfor all $x \\in \\left\\{\\frac{k+3}{4}+1, \\ldots, \\frac{k-3}{2}\\right\\}$.\nThus, as elements of ${{\\mathbb Z}}\/k^2$, $[l]_d (4k-2) \\notin \\left\\langle 0, 3k-1 \\right\\rangle$\nfor all $l \\in {{\\mathbb Z}}\/d \\setminus \\left\\{0, \\frac{k+3}{4}\\right\\}$.\nOn the other hand, $\\frac{k+3}{4} (4k-2) = \\frac{5k-3}{2} \\in \\left\\langle 0, 3k-1 \\right\\rangle$.\nThus $z_0^{j+l}$ is genus-minimizing if and only if\n$l = \\frac{k+3}{4}$.\nThis is the answer for $l$ we should have expected, since\nby (\\ref{part iv: eq: 2l = 2}), we know that\n$2l \\equiv 2\\;(\\mod d)$, whose solutions are $l \\in \\left\\{1, \\frac{d}{2}+1\\right\\}$.\nWe already know that $l\\neq 1$, and $\\frac{d}{2} + 1 = \\frac{k+3}{4}$.\nMore to the point, we have shown that\n$\\left\\langle z_0^j, z_1^j\\right]$ has only one R-mobile point,\nand so all mobile points are non-neutralized in this case.\n\n\n\nOn the other hand, we have shown that\nwhen $\\psi < [dq]_{k^2}$ and we do {\\em not} have\n$(\\mu, \\gamma) = (1,1)$, $\\alpha = -1$,\n$m=2$, $c=1$, and $d=\\frac{k-1}{2} \\equiv 0\\; (\\mod 2)$,\nthen $q$ is not genus-minimizing.\n \\\\\n\n\nWe therefore assume for the remainder of the proof that\n$[dq]_{k^2} < \\psi < 2[dq]_{k^2}$.\nSuppose first that $(\\mu,\\gamma) \\in \\{(1,1), (-1,1)\\}$, so that $\\gamma = +1$.\nThen\n\\begin{equation}\n \\psi\n\\;=\\; \\left[dq - \\gamma\\textstyle{\\frac{ck+\\alpha\\gamma}{d}}k\\right]_{k^2}\n\\;=\\; \\left[dq - \\textstyle{\\frac{ck+\\alpha}{d}}k\\right]_{k^2}\n\\end{equation}\nand so the fact that $\\psi > [dq]_{k^2}$ implies \n$\\psi = [dq]_{k^2} - \\frac{ck+\\alpha}{d}k + k^2$.\nNow, Proposition \\ref{prop: properties of parameters d, m, c, alpha, mu, gamma}\ntells us that $[dq]_{k^2} < \\frac{k^2}{2}$ and $\\frac{ck+\\alpha}{d} < \\frac{k^2}{2}$.\nThus \n\\begin{align}\n \\psi\n&= [dq]_{k^2} - \\textstyle{\\frac{ck+\\alpha}{d}}k + k^2\n \\\\ \\nonumber\n&> [dq]_{k^2} + \\textstyle{\\frac{k^2}{2}}\n \\\\ \\nonumber\n&> 2[dq]_{k^2}.\n\\end{align}\n\n\n\nThis leaves us with the case in which \n$[dq]_{k^2} < \\psi < 2[dq]_{k^2}$ and\n$(\\mu,\\gamma) = (1,-1)$.\nWe first claim that $v_q(z_{n_{j-1}}^{j-1}, z_0^j)$ is constant in $j \\in {{\\mathbb Z}}\/d$.\nSuppose this is not the case.\nThen by Part (i$\\psi$) and Part (ii$\\psi$), there exists a non-neutralized \nR-pseudomobile point, say $z_0^{j+l}$,\nin $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$.\nIf $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_{n_{j-1}}^{j-1}\\right) \\in\n\\left\\langle \\psi - dq, \\psi \\right\\rangle$,\nthen $z^{j+l-1}_{n_{j+l-1}}$ is L-pseudomobile \nin $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$, contradicting the supposition\nthat $z_0^{j+l}$ is non-neutralized as an R-pseudomobile point\nin $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$.\nThus $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_{n_{j-1}}^{j-1}\\right) \\in\n\\left\\langle0, \\psi - dq \\right\\rangle$.\nThis, however, implies\n$z^{j+l-1}_{n_{j+l-1}}$ is L-mobile in\n$\\left\\langle z_{n_{j-1} - 1}^{j-1}, z_{n_{j-1}}^{j-1}\\right]$, since\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1} - z_{n_{j-1}}^{j-1}\\right)\n\\in \\left\\langle -\\psi + dq , 0\\right\\rangle$.\nMoreover, since \n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_{n_{j-1}-1}^{j-1}\\right)\n= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_{n_{j-1}}^{j-1}\\right) + dq\n\\in \\left\\langle dq, \\psi \\right\\rangle$,\nwe know that\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_{n_{j-1}-1}^{j-1}\\right) \\notin\n\\left\\langle0, dq\\right\\rangle$,\nand so $z_0^{j+l}$ is {\\em not} R-mobile in \n$\\left\\langle z_{n_{j-1} - 1}^{j-1}, z_{n_{j-1}}^{j-1}\\right]$.\nThus $z^{j+l-1}_{n_{j+l-1}}$ is in fact non-neutralized L-mobile in\n$\\left\\langle z_{n_{j-1} - 1}^{j-1}, z_{n_{j-1}}^{j-1}\\right]$, implying that\n$v_q(z_{n_{j-1}-1}^{j-1}, z_{n_{j-1}}^{j-1})$ is not constant in $j \\in {{\\mathbb Z}}\/d$.\nBut this contradicts our initial supposition that\n$v_q(z_{n_{j-1}}^{j-1}, z_0^j)$ is not constant in $j \\in {{\\mathbb Z}}\/d$.\nThus our initial supposition must have been false, and so\n$v_q(z_{n_{j-1}}^{j-1}, z_0^j)$ is constant in $j \\in {{\\mathbb Z}}\/d$.\n\n\n\nPart (ii') therefore tells us that there exist unique $l \\in {{\\mathbb Z}}\/d$ and\n$i_* \\in \\left\\{ 0, \\ldots, \\left\\lfloor \\frac{k}{d} \\right\\rfloor - 2 \\right\\}$\nfor which $z_0^{j+l}$ is non-neutralized R-mobile in\n$\\left\\langle z_{i_*}^j, z_{{i_*}+1}^j \\right]$,\nand $z_{n_{j-l}}^{j-l}$ is non-neutralized L-mobile in\n$\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$, with\n$\\left(z_{i_*}^{j_*}, z_{{i_*}+1}^{j_*}\\right) \n= \\left(z_{n_j - (i_*+1)}^{j_*}, z_{n_j - i_*}^{j_*}\\right)\n= (x_*,y_*)$, for some unique $j_* \\in {{\\mathbb Z}}\/d$,\nwhere $x_*, y_* \\in Q_q$ are the unique elements of $Q_q$\nsatisfying $v_q(x_*,y_*) = \\alpha(k-k^2)$.\nIn particular, $i_* = \\frac{n_{j_*}-1}{2}$.\nNow, if $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{i_*}^j\\right)\n\\in \\left\\langle \\psi - dq,\\, dq \\right\\rangle$, then\n\\begin{align}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1} - z_{{i_*}+1}^j\\right)\n&= dq + \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{i_*}^j\\right) - \\psi - dq\n \\\\ \\nonumber\n&\\in \\left\\langle - dq,\\, dq-\\psi \\right\\rangle\n \\\\ \\nonumber\n&\\subset \\left\\langle - dq,\\, 0\\right\\rangle,\n\\end{align}\nmaking $z_{n_{j+l-1}}^{j+l-1}$ L-mobile in\n$\\left\\langle z_{i_*}^j, z_{{i_*}+1}^j \\right]$, so that\n$z_0^{j+l}$ is in fact neutralized R-mobile in $\\left\\langle z_{i_*}^j, z_{{i_*}+1}^j \\right]$, \na contradiction.\nThus $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_{i_*}^j\\right) \\in\n\\left\\langle 0, \\psi - dq\\right\\rangle$, which implies\n\\begin{align}\n\\label{eq: part (iv), type (1,-1), max in (-dq, 0)}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1} - z_{i_*}^j\\right)\n&= dq + \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{i_*}^j\\right) - \\psi\n \\\\ \\nonumber\n&\\in \\left\\langle dq - \\psi, 0 \\right\\rangle\n \\\\ \\nonumber\n&\\subset \\left\\langle - dq,\\, 0\\right\\rangle.\n\\end{align}\nIf $\\left\\lfloor \\frac{k}{d} \\right\\rfloor > 2$, so that ${i_*} \\neq 0$,\nthen this makes $z_{n_{j+l-1}}^{j+l-1}$ non-neutralized L-mobile in\n$\\left\\langle z_{{i_*}-1}^j, z_{i_*}^j \\right]$ (a contradiction), since\n$z_{n_{j+l-1}}^{j+l-1}$ is L-mobile in $\\left\\langle z_{{i_*}-1}^j, z_{i_*}^j \\right]$\nand since\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{{i_*}-1}^j\\right)\n=\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_{i_*}^j\\right) + dq\n\\in \\left\\langle dq, \\psi \\right\\rangle$\nimplies that $z_0^{j+l}$ is {\\em{not}} R-mobile in \n$\\left\\langle z_{{i_*}-1}^j, z_{i_*}^j \\right]$.\nThus $\\left\\lfloor \\frac{k}{d} \\right\\rfloor = 2$ and $i_*=0$.\n\n\nEquation (\\ref{eq: part (iv), type (1,-1), max in (-dq, 0)})\nthen tells us that\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1} - z_0^j\\right) \\in\n\\left\\langle dq-\\psi,\\, 0\\right\\rangle\n\\subset \\left\\langle -\\psi, 0 \\right\\rangle$,\nso that $z_{n_{j+l-1}}^{j+l-1}$ is\nL-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$.\nWe then have\n\\begin{align}\n \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{n_{j-1}}^{j-1}\\right)\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(\\left(z_{n_{j+l-1}}^{j+l-1}+\\psi\\right)\n -\\left(z_0^j - \\psi\\right)\\right)\n \\\\ \\nonumber\n&= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1} -z_0^j\\right) + 2\\psi - dq\n \\\\ \\nonumber\n&\\in \\left\\langle \\psi, 2\\psi -dq\\right\\rangle.\n\\end{align}\nIf $2\\psi -[dq]_{k^2} < k^2$,\nthen $\\left\\langle 0, \\psi \\right\\rangle \\cap \\left\\langle \\psi, 2\\psi -dq\\right\\rangle = \\emptyset$,\nand so $z_0^{j+l}$ is not R-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$,\nwhich means that \n$z_{n_{j+l-1}}^{j+l-1}$ is\nnon-neutralized L-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$, a contradiction.\nThus $2\\psi -[dq]_{k^2} > k^2$, but this, in turn, implies that\n\\begin{equation}\nk^2 < \\psi + \\psi - [dq]_{k^2} \\;\\;<\\;\\; \\psi + [dq]_{k^2} \\;\\;<\\;\\; k^2 + [dq]_{k^2},\n\\end{equation}\nso that $\\psi + dq \\in \\left\\langle 0, dq \\right\\rangle$.\nNow, since $\\left\\lfloor \\frac{k}{d} \\right\\rfloor = 2$,\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+1}-z_0^j\\right) = \\psi + dq$,\nand so the fact that $\\psi + dq \\in \\left\\langle 0, dq \\right\\rangle$\nmeans that $z_0^{j+1}$ is R-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$.\nOn the other hand,\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_j}^j-z_1^j\\right) = dq \\notin \\left\\langle -dq, 0 \\right\\rangle$,\nand so $z_{n_j}^j$ is not L-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$.\nThus $z_0^{j+1}$ is non-neutralized R-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$.\n\n\n\n\nSince $z_0^{j+1}-z_{n_j}^j$ is constant in $j\\in{{\\mathbb Z}}\/d$,\nwe can then express ${{\\mathbb Z}}\/d$ as the disjoint union of $J_1$ and $J_2$, where\n\\begin{align}\n J_1\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}}=1;\\;\n z_0^{j^{\\prime}+1} \\in\n \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}}\\right]\n \\right.\\!\\right\\},\n \\\\\n J_2\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}} = 2;\\;\n z_0^{j^{\\prime}+1} \\in\n \\left\\langle z_1^{j^{\\prime}}, z_2^{j^{\\prime}}\\right]\n \\right.\\!\\right\\}.\n\\end{align}\nIn this case, $n_j := 2 - {\\theta}^{d, \\epsilon}(j)$,\nand the definition of ${\\theta}^{d, \\epsilon}(j)$, or alternatively, the $l=1$\ncase of Lemma \\ref{lemma: q of positive type, xi lemma}, implies that\n$\\#\\left\\{ j^{\\prime}\\in{{\\mathbb Z}}\/d\\left|\\; {\\theta}^{d, \\epsilon}(j^{\\prime})=1\\right.\\right\\} = \\epsilon$.\nThus $|J_1| = \\epsilon$.\nSince $z_{n_{j-1}}^{j-1} - z_0^j$ is constant in $j\\in{{\\mathbb Z}}\/d$,\n$z_{n_{j-1}}^{j-1}$ is not L-mobile in \n$\\left\\langle z_0^j, z_0^j\\right]$.\nThus $z_0^{j+1}$ is the only mobile point in\n$\\left\\langle z_0^j, z_1^j\\right]$,\nand it is active at time $j=j^{\\prime}\\in {{\\mathbb Z}}\/d$\nif and only if $j^{\\prime} \\in J_1$.\nThus, if $\\alpha = +1$, then\n$z_0^{j+l}$ is active in $\\left\\langle z_0^j, z_1^j \\right]$\nprecisely once, so that $\\epsilon = |J_1| = 1$.\nOn the other hand, if $\\alpha = -1$, then\n$z_0^{j+l}$ is inactive in $\\left\\langle z_0^j, z_1^j \\right]$\nprecisely once, so that $\\epsilon = |J_1| = d-1$;\n\n\nIn either case, \n$\\epsilon = [\\alpha]_d$, and so\n$c = \\left[\\alpha\\gamma\\, {\\epsilon}^{-1}\\right]_d = [\\gamma]_d = d-1$.\nThe fact that $c \\leq \\frac{d}{2}$ then implies $d=2$,\nbut this contradicts the fact that\n$\\left\\lfloor \\frac{k}{d} \\right\\rfloor = 2$,\nand so $\\psi \\notin \\left\\langle dq, 2dq\\right\\rangle$.\n \\\\\n\nNow that we have shown that \n$\\psi > 2[dq]_{k^2}$ except in the special case mentioned\nin the statement of Part (iv) (in which we have already shown that ${\\mathbf{z}}^j$ has\nno neutralized mobile points), it remains to show that\n$\\psi > 2[dq]_{k^2}$ implies\nthat all mobile points are non-neutralized.\n\n\nSuppose that $\\psi > 2[dq]_{k^2}$, and that\nthere exist $l \\neq 0 \\in {{\\mathbb Z}}\/d$ and\n$i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2\\right\\}$\nfor which $z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_i^j, z_{i+1}^j\\right]$.\nThen $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_i^j\\right) \\in \\left\\langle 0, dq\\right\\rangle$, and so\n\\begin{align}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1}-z_{i+1}^j\\right)\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(\\left(z_0^{j+l}-\\psi\\right)-\\left(z_i^j + dq\\right)\\right)+dq\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_i^j\\right) -\\psi\n \\\\ \\nonumber\n&\\in \\left\\langle -\\psi, dq - \\psi \\right\\rangle,\n\\end{align}\nwhich, since $[dq]_{k^2}-\\psi < -[dq]_{k^2}$,\nhas no intersection with $\\left\\langle -dq, 0 \\right\\rangle$.\nThus $z_{n_{j+l-1}}^{j+l-1}$ is not L-mobile in\n$\\left\\langle z_i^j, z_{i+1}^j\\right]$, and so\n$z_0^{j+l}$ is non-neutralized R-mobile in\n$\\left\\langle z_i^j, z_{i+1}^j\\right]$.\nA completely analogous argument shows that\nif $\\psi > 2[dq]_{k^2}$, and\n$z_0^{j+l}$ is R-mobile in \n$\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j \\right]$\nfor some $l \\neq 1 \\in {{\\mathbb Z}}\/d$ and\n$i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2\\right\\}$,\nthen $z_0^{j+l}$ is non-neutralized R-mobile in \n$\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j \\right]$.\nThus ${\\mathbf{z}}^j$ has no neutralized R-mobile points,\nwhich, in turn, implies that ${\\mathbf{z}}^j$ has no\nneutralized L-mobile points,\nand so all mobile points are non-neutralized.\n\n\\end{proof}\n\n\n\n\n\\begin{cor}\n\\label{cor: if z^j has mobile points, then (k\/d)dq < k^2}\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor\\! [dq]_{k^2} < k^2$.\n\\end{cor}\n\\begin{proof}\nSuppose that $\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\![dq]_{k^2} > k^2$.\nThen, since $2[dq]_{k^2} < k^2$, this implies that\n$z_0^{j^{\\prime}} \\in\n\\left\\langle z_0^{j^{\\prime}} + 2dq,\\,\nz_0^{j^{\\prime}} + \\left\\lfloor\\frac{k}{d}\\right\\rfloor\\! dq \\right\\rangle$\nfor all $j^{\\prime} \\in {{\\mathbb Z}}\/d$.\nThus, if we choose any $j_0 \\in {{\\mathbb Z}}\/d$ for which \n$n_{j_0} = \\left\\lfloor\\frac{k}{d}\\right\\rfloor$, then\nthere exists $i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor -2\\right\\}$\nfor which\n$z_0^{j_0} \\in\n\\left\\langle z_{n_{j_0}-(i+1)}^{j_0}, z_{n_{j_0}-i}^{j_0} \\right]$.\nThis, in turn, implies that\n$\\min_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^j - z_{n_j-(i+1)}^j\\right)\n\\in \\left\\langle 0, dq \\right\\rangle$, so that \n$z_0^j$ is R-mobile in \n$\\left\\langle z_{n_j-(i+1)}^j, z_{n_j-i}^j \\right]$.\nProposition \\ref{prop: positive type, main prop}.(ii)\nthen implies that\n$z_0^j$ is R-mobile rel $z_0^j$, but this is a contradiction,\nso our supposition that $\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\![dq]_{k^2} > k^2$\nmust have been false.\n\\end{proof}\n\n\n\n\n\nBefore preceding to other results, we pause to introduce one more\nitem of terminology, which we have postponed until now in order\nto keep Proposition \\ref{prop: positive type, main prop}\nfrom becoming any more unwieldy than it already is.\n\n\nSo far, we have only described pseudomobile points\nin terms of the interval $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$.\nIn some cases, however,\nit turns out to be more convenient to\nconsider the interval $\\left\\langle z_0^j, z_{n_{j-1}}^{j-1}\\right]$.\nWe therefore fix\n\\begin{equation}\n {{\\overline{\\psi}}}\n:= \\left[z_{n_{j-1}}^{j-1} - z_0^j\\right]_{k^2}\n= k^2 - \\psi,\n\\end{equation}\nand introduce the notion of an {\\em antipseudomobile} point.\nFor any $l \\neq 0 \\in {{\\mathbb Z}}\/d$, we say that\n$z_0^{j+l}$ is R-antipseudomobile (or R${{\\overline{\\psi}}}$-mobile) in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$ if\n\\begin{equation}\n 0\n\\;<\\; \\minq_{j\\in{{\\mathbb Z}}\/d} \\left( z_0^{j+l} - z_0^j \\right)\n\\;<\\; {{\\overline{\\psi}}},\n\\end{equation}\nand that $z_{n_{j-1-l}}^{j-1-l}$ is\nL-pseudomobile (or L${{\\overline{\\psi}}}$-mobile) in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$ if\n\\begin{equation}\n -{{\\overline{\\psi}}}\n\\;<\\; \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left( z_{n_{j-1-l}}^{j-1-l}- z_{n_{j-1}}^{j-1} \\right)\n\\;<\\; 0.\n\\end{equation}\nWe say that a point is antipseudomobile (or ${{\\overline{\\psi}}}$-mobile) in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$ if it is\nR${{\\overline{\\psi}}}$-mobile or L${{\\overline{\\psi}}}$-mobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$.\n(To be consistent in our method of abbreviation, we shall also sometimes say\n$\\psi$-mobile instead of pseudomobile.)\nCorollary \\ref{cor: q of positive type: combo of lemma and difference eq}\ntells us that\n\\begin{equation}\n\\label{eq: psibar mirror relation}\n \\minq_{j\\in{{\\mathbb Z}}\/d} \\left( z_0^{j+l} - z_0^j \\right)\n= -\\maxq_{j\\in{{\\mathbb Z}}\/d} \\left( z_{n_{j-1-l}}^{j-1-l} - z_{n_{j-1}}^{j-1}\\right)\n\\end{equation}\nfor all nonzero $l\\in{{\\mathbb Z}}\/d$.\nWe therefore say that $z_0^{j+l}$ and\n$z_{n_{j-1-l}}^{j-1-l}$ are mirror ${{\\overline{\\psi}}}$-mobile points\nin $\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$.\n\n\n\nIf $z_0^{j+l}$ is R${{\\overline{\\psi}}}$-mobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$,\nthen we say that $z_0^{j+l}$ is active\nat time $j=j^{\\prime} \\in {{\\mathbb Z}}\/d$ if \n\\begin{equation}\n z_0^{j^{\\prime}+l} - z_0^{j^{\\prime}}\n\\;=\\; \\minq_{j\\in{{\\mathbb Z}}\/d} \\left( z_0^{j+l} - z_0^j\\right),\n\\end{equation}\nand is inactive otherwise.\nIf $z_{n_{j-1-l}}^{j-1-l}$ is L${{\\overline{\\psi}}}$-mobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$,\nthen we say $z_{n_{j-1-l}}^{j-1-l}$ is active at time $j = j^{\\prime}\\in{{\\mathbb Z}}\/d$ if\n\\begin{equation}\n z_{n_{j^{\\prime}-1-l}}^{j^{\\prime}-1-l} - z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}\n\\;=\\; \\maxq_{j\\in{{\\mathbb Z}}\/d} \\left( z_{n_{j-1-l}}^{j-1-l} - z_{n_{j-1}}^{j-1} \\right),\n\\end{equation}\nand is inactive otherwise.\nWe say that an R${{\\overline{\\psi}}}$-mobile point $z_0^{j+l}$ in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$\nis {\\em neutralized} if\n$z_{n_{j+l-1}}^{j+l-1}$ is L${{\\overline{\\psi}}}$-mobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$,\nand that an L${{\\overline{\\psi}}}$-mobile point $z_{n_{j+l}}^{j+l}$ in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$\nis neutralized if\n$z_0^{j+l+1}$ is R${{\\overline{\\psi}}}$-mobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$.\nWe say a ${{\\overline{\\psi}}}$-mobile point is {\\em non-neutralized}\nif it is not neutralized.\n\n\nAll of our main results for pseudomobile points have analogs for\nantipseudomobile points.\n\n\\begin{prop}\n\\label{prop: antipseudomobile point is active [lepsilon] times}\nSuppose that $q$ of positive type is genus-minimizing.\nIf $z_0^{j+l}$ is R-antipseudomobile (and hence\n$z_{n_{j-1-l}}^{j-1-l}$ is L-antipseudomobile)\nin $\\left\\langle z_0^j , z_{n_{j-1}}^{j-1} \\right]$,\nthen each of the two antipseudomobile points is active precisely\n$[l\\epsilon]_d$ times.\n\\end{prop}\n\\begin{proof}\nFor all $j\\in {{\\mathbb Z}}\/d$, \n(\\ref{eq: z_0^j+l - z_0^j = mu ml k^2\/d + xi dq}) implies that\n\\begin{align}\n\\label{prop: antipseudomobile active le times. eq: explicit equation for z_0^j+l -z_0^j}\n z^{j+l}_0 - z_0^j\n&= \\left[{\\mu}ml\\right]_d\\frac{k^2}{d}\n \\;+\\; \\left(\\Xi^{d, \\epsilon}_l(j)\\right) [dq]_{k^2},\n \\\\ \\nonumber\n z_{n_{j-1-l}}^{j-1-l} - z_{n_{j-1}}^{j-1}\n&= -\\left[{\\mu}ml\\right]_d\\frac{k^2}{d}\n \\;-\\; \\left(\\Xi^{d, \\epsilon}_l(j-l)\\right) [dq]_{k^2},\n\\end{align}\nwhere, by\nLemma \\ref{lemma: q of positive type, xi lemma},\n$\\Xi^{d, \\epsilon}_l(j) \\in \\left\\{ \\frac{[l\\epsilon]_d}{d}, \\frac{[l\\epsilon]_d}{d}-1 \\right\\}$\nfor all $j\\in{{\\mathbb Z}}\/d$, with $\\frac{[l\\epsilon]_d}{d}$ occurring\n$[-l\\epsilon]_d$ times and $\\frac{[l\\epsilon]_d}{d} - 1$\noccurring $[l\\epsilon]_d$ times.\nSince (\\ref{prop: antipseudomobile active le times. eq: explicit equation for z_0^j+l -z_0^j})\nimplies $z_0^{j+l}$ (respectively $z_{n_{j-1-l}}^{j-1-l}$)\nis active in $\\left\\langle z_0^j , z_{n_{j-1}}^{j-1} \\right]$\nat time $j = j^{\\prime}$ if and only if\n$\\Xi^{d, \\epsilon}_l(j^{\\prime}) = \\frac{[l\\epsilon]_d}{d}-1$\n(respectively $\\Xi^{d, \\epsilon}_l(j^{\\prime}-l) = \\frac{[l\\epsilon]_d}{d}-1$),\nwe conclude that each of the two antipseudomobile points is active\nprecisely $[l\\epsilon]_d$ times.\n\\end{proof}\n\n\n\n\\begin{prop}\n\\label{prop: active iff inactive means neutralized for antipseudomobile}\nSuppose that $q$ of positive type is genus-minimizing, and\nthat $z_{n_{j+l_1}}^{j+l_1}$ and $z_0^{j+l_2}\\!$ are antipseudomobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$\nfor some $l_1, l_2 \\in {{\\mathbb Z}}\/d$. Then\n$z_{n_{j+l_1}}^{j+l_1}$ and $z_0^{j+l_2}$ form a neutralized pair\n({\\em i.e.}, $l_1 + 1 = l_2$) if and only if they satisfy\n\\begin{equation}\n\\nonumber\nz_{n_{j+l_1}}^{j+l_1}\\;\\text{is active}\n\\;\\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;\\;\nz_0^{j+l_2}\\;\\text{is inactive}\n\\end{equation}\nin $\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$ at all times $j=j^{\\prime} \\in {{\\mathbb Z}}\/d$.\n\\end{prop}\n\\begin{proof}\nThe result is proved by adapting\nProposition \\ref{prop: active iff inactive means neutralized for pseudomobile}---the\nanalogous result for pseudomobile points---in an obvious manner.\n\\end{proof}\n\n\n\n\\begin{prop}\n\\label{prop: antipseudomobile analog of main prop}\nSuppose $q$ of positive type is genus-minimizing,\nand let $x_*, y_*$ denote the unique elements of $Q_q$ for which\n$v_q(x_*, y_*) = {\\alpha}(k-k^2)$. Then the following are true:\n\\begin{itemize}\n\\item[(i${{\\overline{\\psi}}}$)]\nIf the interval\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$\nhas any non-neutralized antipseudomobile points, then\nthere exists a unique $j_* \\in {{\\mathbb Z}}\/d$ such that\n$\\left(z_0^{j_*}, z_{n_{j_*-1}}^{j_*-1} \\right) = \\left(y_*, x_*\\right)$.\n\n\\item[(ii${{\\overline{\\psi}}}$)]\nIf $v_q(z_0^j, z_{n_{j-1}}^{j-1})$ is nonconstant in $j\\in{{\\mathbb Z}}\/d$, then\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$ has precisely one\nnon-neutralized R${{\\overline{\\psi}}}$-mobile point and precisely one\nnon-neutralized L${{\\overline{\\psi}}}$-mobile point, namely,\n$z_0^{j+l}$ and $z_{n_{j-1-l}}^{j-1-l}$ for some nonzero $l\\in {{\\mathbb Z}}\/d$.\n\\end{itemize}\n\\end{prop}\n\\begin{proof}\nThe results of Parts (i${{\\overline{\\psi}}}$) and (ii${{\\overline{\\psi}}}$) are proved by taking the\nrespective proofs of Proposition \\ref{prop: positive type, main prop},\nParts (i$\\psi$) and (ii$\\psi$), and making the following adaptations.\nMostly, one must replace the word ``pseudomobile'' with the word ``antipseudomobile''\nand replace $z_{n_{j-1}}^{j-1}$ with $z_0^j$ and {\\em vice versa}.\nSince ${{\\overline{\\psi}}} \\equiv -dq \\equiv -\\alpha\\;(\\mod k^2)$,\none must also replace $\\alpha$ with $-\\alpha$.\nLastly, one must replace all references to\nProposition \\ref{prop: active iff inactive means neutralized for pseudomobile}.\nwith references to\nProposition \\ref{prop: active iff inactive means neutralized for antipseudomobile}.\n\\end{proof}\n\n\nNow that we have introduced the antipseudomobile point, \nwe resume our task of tabulating results useful for the\nclassfication of genus-minimizing $q$ of positive type.\nWe begin with a new result about antipseudomobile points.\n\n\n\n\\begin{prop}\n\\label{prop: psibar < k^2\/2 implies psibar-mobile are nn}\nSuppose $q$ of positive type is genus-minimizing.\nIf ${{\\overline{\\psi}}} < \\frac{k^2}{2}$, then all antipseudomobile points\nare non-neutralized.\n\\end{prop}\n\\begin{proof}\nWe begin by reducing the problem to the question of whether\n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$\nhas R-mobile points.\n\n\n\nSuppose, for some $l\\neq 0\\in {{\\mathbb Z}}\/d$, that $z_0^{j+l}$ is R${{\\overline{\\psi}}}$-mobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$, or in other words, that\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_0^j\\right) \\in \\left\\langle 0, {{\\overline{\\psi}}}\\right\\rangle$.\nIf $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_0^j\\right) \\in \\left\\langle 0, {{\\overline{\\psi}}}-dq\\right\\rangle$,\nthen\n\\begin{align}\n\\label{psibar < k^2\/2 implies NN, eq: min in <0,psibar-dq> implies NN}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1} - z_{n_{j-1}}^{j-1}\\right)\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(\\left(z_0^{j+l} + {{\\overline{\\psi}}}\\right) - \\left(z_0^j + {{\\overline{\\psi}}}\\right)\\right) + dq\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_0^j \\right) + dq\n \\\\ \\nonumber\n&\\in \\left\\langle dq, {{\\overline{\\psi}}} \\right\\rangle,\n\\end{align}\nwhich, since $2{{\\overline{\\psi}}} < k^2$, has no intersection with\n$\\left\\langle -{{\\overline{\\psi}}}, 0\\right\\rangle$, so that\n$z_{n_{j+l-1}}^{j+l-1}$ is not L${{\\overline{\\psi}}}$-mobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$.\nThus $z_0^{j+l}$ is non-neutralized R${{\\overline{\\psi}}}$-mobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$ if\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_0^j\\right) \\in \\left\\langle 0, {{\\overline{\\psi}}}-dq\\right\\rangle$.\nThis leaves us with the case in which\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_0^j\\right) \\in \n\\left\\langle {{\\overline{\\psi}}}-dq, {{\\overline{\\psi}}}\\right\\rangle$,\nbut this implies that\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{n_{j-1}-1}^{j-1}\\right) \\in\n\\left\\langle 0, dq \\right\\rangle$,\nso that\n$z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_{n_{j-1}-1}^{j-1}, z_{n_{j-1}}^{j-1} \\right]$.\nThus, if we can show that \n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j \\right]$\nhas no R-mobile points when ${{\\overline{\\psi}}} < \\frac{k^2}{2}$,\nthen we shall have proved the proposition.\n \\\\\n\n\n\nWe therefore assume, for the remainder of the proof,\nthat $\\left\\langle z_{n_j-1}^j, z_{n_j}^j \\right]$ has an R-mobile point.\nIt is important to note that this implies\n\\begin{equation}\n\\label{prop: psibar NN, eq: floor(k\/d) = 2}\n\\textstyle{\\left\\lfloor \\frac{k}{d} \\right\\rfloor} = 2.\n\\end{equation}\nThat is, the mirror relation (\\ref{eq: mirror relation 1}) tells us that\n$\\left\\langle z_0^j, z_1^j \\right]$ has an L-mobile point, and so\nProposition \\ref{prop: positive type, main prop}.(i)\nimplies that there exists $j_* \\in {{\\mathbb Z}}\/d$ for which\n$\\left\\langle z_0^{j_*}, z_1^{j_*} \\right] = \\left\\langle z_{n_{j_*}-1}^{j_*}, z_{n_{j_*}}^{j_*} \\right]$.\nIn particular, such $j_*$ must satisfy $n_{j_*} = 1$, implying\n$\\left\\lfloor \\frac{k}{d} \\right\\rfloor = 2$.\nNote as well that Proposition \\ref{prop: positive type, main prop}.(i)\ntells us that the R-mobile point in $\\left\\langle z_{n_j-1}^j, z_{n_j}^j \\right]$,\nsay, $z_0^{j+l}$, for some $l \\neq 1 \\in {{\\mathbb Z}}\/d$, must be R-mobile rel $z_0^j$,\nand so (\\ref{prop: psibar NN, eq: floor(k\/d) = 2}) implies that such\n$z_0^{j+l}$ is also R-mobile in $\\left\\langle z_0^j, z_1^j \\right]$.\n\n\n\nWe next claim that\n${{\\overline{\\psi}}} \\in \\left\\langle [dq]_{k^2}, 2[dq]_{k^2} \\right\\rangle$.\nRecalling that ${{\\overline{\\psi}}} :\\equiv z_{n_{j-1}}^{j-1}-z_0^j$ for all $j \\in {{\\mathbb Z}}\/d$,\nsuppose first that\n${{\\overline{\\psi}}} \\in \\left\\langle 0, dq \\right\\rangle$.\nWe then have\n\\begin{align}\n \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+1}-z_0^j\\right)\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(\\left(z_{n_j}^j - {{\\overline{\\psi}}}\\right) - z_0^j\\right)\n \\\\ \\nonumber\n&= z_1^j - {{\\overline{\\psi}}} - z_0^j\n \\\\ \\nonumber\n&= dq - {{\\overline{\\psi}}}\n \\\\ \\nonumber\n&\\in \\left\\langle 0, dq \\right\\rangle,\n\\end{align}\nwhere the second line used the fact that $\\left\\lfloor \\frac{k}{d} \\right\\rfloor = 2$.\nThis means that $z_0^{j+1}$ is R-mobile in $\\left\\langle z_0^j, z_1^j \\right]$,\nbut in the preceding paragraph, we showed that\n$\\left\\langle z_0^j, z_1^j \\right]$ already has an R-mobile point\n$z_0^{j+l}$ with $l \\neq 1$.\nSince Proposition \\ref{prop: positive type, main prop}.(ii) precludes the existence of\ntwo distinct R-mobile points in $\\left\\langle z_0^j, z_1^j \\right]$,\nour supposition that ${{\\overline{\\psi}}} \\in \\left\\langle 0, dq \\right\\rangle$\nmust have been false.\nOn the other hand, if $2[dq]_{k^2} < {{\\overline{\\psi}}} < \\frac{k^2}{2}$,\nthen this R-mobile point \n$z_0^{j+l}$ in $\\left\\langle z_0^j, z_1^j \\right]$ must satisfy\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right) \n\\in \\left\\langle 0, dq \\right\\rangle\n\\subset \\left\\langle 0, {{\\overline{\\psi}}} - dq \\right\\rangle$,\nand so the argument surrounding\n(\\ref{psibar < k^2\/2 implies NN, eq: min in <0,psibar-dq> implies NN})\nimplies that $z_0^{j+l}$ is non-neutralized\nR${{\\overline{\\psi}}}$-mobile in $\\left\\langle z_0^j, z_{n_{j-1}}^{j-1}\\right]$,\nanother contradiction. Thus ${{\\overline{\\psi}}} \\in \\left\\langle [dq]_{k^2}, 2[dq]_{k^2} \\right\\rangle$.\n\n\n\nWe next attempt to determine $m$ by interpreting $\\mu m$ as a sort of winding number\nof $\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l}}^{j-l}-z_0^j\\right)$ around ${{\\mathbb Z}}\/k^2$ as $l$ varies.\nTo make this notion more precise, we define the function\n$M : {{\\mathbb Z}} \\rightarrow {{\\mathbb Z}}$,\n\\begin{align}\n\\label{eq: M(l) definition}\n M(l) \n&:= -\\mu m(l-1)\\textstyle{\\frac{k^2}{d}}\n -\\left(\\textstyle{\\frac{(l-1)\\epsilon}{d} - \\left\\lceil\\!\\frac{(l-1)\\epsilon}{d}\\!\\right\\rceil}\\right)[dq]_{k^2}\n + {{\\overline{\\psi}}}\n \\\\ \\nonumber\n &\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n + l\\left({{\\overline{\\psi}}} - \\left(\\textstyle{\\frac{\\gamma c k + \\alpha}{d}}k - [dq]_{k^2}\\right)\\right).\n\\end{align}\nSince ${{\\overline{\\psi}}} = \\left[\\frac{\\gamma c k + \\alpha}{d}k - [dq]_{k^2}\\right]_{k^2}$,\nthe term on the second line vanishes when $\\gamma = +1$ and is equal to either zero\nor $lk^2$ when $\\gamma = -1$.\nThus, Corollary \\ref{cor: q of positive type: combo of lemma and difference eq}\ntells us that\n$M(l)$ is an integer lift of\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l}}^{j-l}-z_0^j\\right) \\in {{\\mathbb Z}}\/k^2$\nwhenever $l \\not\\equiv 1 \\; (\\mod d)$,\nand $M(l)$ is an integer lift of $z_{n_{j-1}}^{j-1}-z_0^j \\in {{\\mathbb Z}}\/k^2$\nwhen $l \\equiv 1 \\; (\\mod d)$.\nFor any $l \\in {{\\mathbb Z}}$, we can calculate $M(l+1)-M(l)$:\n\\begin{align}\n\\label{eq: M(l+1) - M(l)}\n M(l+1)-M(l)\n&= -\\mu m\\textstyle{\\frac{k^2}{d}}\n +\\left(-\\textstyle{\\frac{\\epsilon}{d} + \\left\\lceil\\frac{l\\epsilon}{d}\\right\\rceil\n - \\left\\lceil\\!\\frac{(l-1)\\epsilon}{d}\\!\\right\\rceil}\\right)[dq]_{k^2}\n \\\\ \\nonumber\n &\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n + {{\\overline{\\psi}}} - \\left(\\textstyle{\\frac{\\gamma c k + \\alpha}{d}}k - [dq]_{k^2}\\right)\n \\\\ \\nonumber\n&= -\\textstyle{\\frac{k+\\epsilon}{d}}\\left((\\mu m + \\gamma c)k+\\alpha\\right)\n + \\left( \\left\\lceil\\frac{l\\epsilon}{d}\\right\\rceil \n - \\left\\lceil\\!\\frac{(l-1)\\epsilon}{d}\\!\\right\\rceil + 1\\right)[dq]_{k^2} + {{\\overline{\\psi}}}\n \\\\ \\nonumber\n&= {{\\overline{\\psi}}} + \\textstyle{\\left( \\left\\lceil\\frac{l\\epsilon}{d}\\right\\rceil \n - \\left\\lceil\\!\\frac{(l-1)\\epsilon}{d}\\!\\right\\rceil\\;-\\;2\\right)} [dq]_{k^2}\n \\\\ \\nonumber\n&\\in \\left\\{ {{\\overline{\\psi}}} - 2[dq]_{k^2}, {{\\overline{\\psi}}} - [dq]_{k^2} \\right\\}.\n\\end{align}\nSince $[dq]_{k^2} < {{\\overline{\\psi}}} < 2[dq]_{k^2}$, (\\ref{eq: M(l+1) - M(l)})\nimplies in particular that\n\\begin{equation}\n\\left\\vert M(l+1)-M(l)\\right\\vert \\;<\\; [dq]_{k^2}\\;\\;\\text{for all}\\; l \\in {{\\mathbb Z}}.\n\\end{equation}\nSince $[M(1)]_{k^2} = \\psi \\notin \\left\\langle 0, [dq]_{k^2} \\right\\rangle$,\nthis means that the constraint\n\\begin{equation}\n[M(l)]_{k^2} \\in \\left\\langle 0, [dq]_{k^2} \\right\\rangle\n\\end{equation}\nhas at least $n$ solutions\nin $l \\neq 1 \\in \\{0, \\ldots, d-1\\} \\subset {{\\mathbb Z}}$,\nwhere $\\left| M(d) - M(0) \\right| = nk^2$.\nSince $z_{n_{j-l}}^{j-l}$ is L-mobile in\n$\\left\\langle z_0^j, z_1^j\\right]$ if and only if\n$[M(l)]_{k^2} \\in \\left\\langle 0, [dq]_{k^2} \\right\\rangle$,\nthis means that\n$M$ must satisfy\n$\\left|M(d)-M(0)\\right| \\leq (1)k^2$.\nUsing (\\ref{eq: M(l) definition}), we calculate that\n\\begin{align}\n\\label{eq: M(d) - M(0)}\n M(d)-M(0)\n&= -\\mu m(d)\\textstyle{\\frac{k^2}{d}}\n -\\left(\\textstyle{\\frac{(d)\\epsilon}{d}\n - \\left\\lceil\\!\\frac{(d-1)\\epsilon}{d}\\!\\right\\rceil\n + \\left\\lceil\\!\\frac{(0-1)\\epsilon}{d}\\!\\right\\rceil}\\right)[dq]_{k^2}\n \\\\ \\nonumber\n &\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n + (d)\\left({{\\overline{\\psi}}} - \\left(\\textstyle{\\frac{\\gamma c k + \\alpha}{d}}k - [dq]_{k^2}\\right)\\right)\n \\\\ \\nonumber\n&= -\\mu mk^2 - (\\epsilon - \\epsilon + 0)[dq]_{k^2}\n \\\\ \\nonumber\n &\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n + d\\left({{\\overline{\\psi}}} - \\left(\\textstyle{\\frac{\\gamma c k + \\alpha}{d}}k - [dq]_{k^2}\\right)\\right)\n \\\\ \\nonumber\n&= -\\mu mk^2 + d\\left({{\\overline{\\psi}}} - \\left(\\textstyle{\\frac{\\gamma c k + \\alpha}{d}}k - [dq]_{k^2}\\right) \\right)\n \\\\ \\nonumber\n&= \\begin{cases}\n -\\mu m k^2\n & {{\\overline{\\psi}}} = \\textstyle{\\frac{\\gamma c k + \\alpha}{d}}k - [dq]_{k^2}\n \\\\\n (-\\mu m + d)k^2\n & {{\\overline{\\psi}}} = \\textstyle{\\frac{\\gamma c k + \\alpha}{d}}k - [dq]_{k^2} + k^2\n \\end{cases}.\n\\end{align}\nThus, if $\\gamma = +1$, then\n${{\\overline{\\psi}}} = \\frac{\\gamma c k + \\alpha}{d}k - [dq]_{k^2}$, and so\n$|\\mu m| \\leq 1$, implying $m=1$.\nIf $\\gamma = -1$, so that $\\mu = +1$, then $m > c > 0$ implies $m \\neq 1$, \nso we must have $|d - m| \\leq 1$, implying\n$m \\in \\{d-1, d, d+1\\}$. If $m = d$, however, then\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l}}^{j-l}-z_0^j\\right)\n= \\frac{[-(l-1)\\epsilon]_d}{d}[dq]_{k^2} + {{\\overline{\\psi}}}\n\\in \\left\\langle 2dq, 3dq \\right\\rangle$ for all $l \\neq 1 \\in {{\\mathbb Z}}\/d$.\n(Here, again, we must first interpret $\\frac{[-(l-1)\\epsilon]_d}{d}[dq]_{k^2} + {{\\overline{\\psi}}}$\nas an integer---noting that in this case,\n$[dq]_{k^2} = dk - (ck + \\alpha\\gamma)$ is divisible by $d$---and\nonly then take its image in ${{\\mathbb Z}}\/k^2$.)\nSince Proposition \\ref{prop: positive type, main prop}.(iv)\ntells us that $\\psi > 2[dq]_{k^2}$, we know that\n$[dq]_{k^2} < {{\\overline{\\psi}}} < k^2 - 2[dq]_{k^2}$, implying\n$3[dq]_{k^2} < k^2$, so that\n$\\left\\langle 2dq, 3dq \\right\\rangle \\cap \\left\\langle 0, dq \\right\\rangle = \\emptyset$,\nwhich means that $\\left\\langle z_0^j, z_1^j \\right]$ has no L-mobile points,\nand so $\\left\\langle z_{n_j-1}^j, z_{n_j}^j \\right]$ has no R-mobile points,\na contradiction.\nThus,\n\\begin{equation}\n\\label{eq: m = 1 or m in d-1, d+1}\nm \\in \\begin{cases}\n \\{1\\}\n & \\gamma = +1\n \\\\\n \\{d-1, d+1\\}\n & \\gamma = -1\n \\end{cases},\n\\end{equation}\nand for either value of $\\gamma$, we have\n\\begin{equation}\n\\label{eq: (mu m)^2 equiv 1(mod d)}\n(\\mu m)^2 \\equiv 1\\; (\\mod d).\n\\end{equation}\n \n\nNow that we have almost determined $m$,\nwe turn our attention to $c$.\nWe shall momentarily do away with the case in which $\\gamma = -1$,\nbut when $\\gamma = +1$, we can find a lower bound for $c$\nthat helps us to prove the following claim.\n\n\\begin{claim}\n\\label{claim: k^2\/d < dq and l not 1,-1, 2, -2}\nSuppose that $\\gamma = +1$.\nThen $\\frac{k^2}{d} < [dq]_{k^2}$, and if\n$z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$,\nthen $l \\notin \\{0, \\pm 1, \\pm 2\\}$.\n\\end{claim}\n\n\\begin{proof}\nSuppose, for some $l \\neq 1 \\in {{\\mathbb Z}}\/d$, that\n$z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$.\nAs discussed in the paragraph surrounding (\\ref{prop: psibar NN, eq: floor(k\/d) = 2}),\nthis implies both that $\\left\\lfloor \\frac{k}{d} \\right\\rfloor = 2$ and that\n$z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$.\nWe can therefore partition ${{\\mathbb Z}}\/d$ as the disjoint union of $J_0$, $J_1$, and $J_2$,\nwhere \n\\begin{align}\n J_0\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}} = 1;\\;\n z_0^{j+l}\\;\\text{is inactive in}\\;\\!\n \\left\\langle z_0^j, z_1^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\}\\!,\n \\\\\n J_1\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}} = 1;\\;\n z_0^{j+l}\\;\\text{is active in}\\;\\!\n \\left\\langle z_0^j, z_1^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\}\\!,\n \\\\\n J_2\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}} = 2;\\;\n z_0^{j+l}\\;\\text{is inactive in}\\;\\!\n \\left\\langle z_0^j, z_1^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\}\\!.\n\\end{align}\nNote that we omitted the only other possibility,\n\\begin{equation}\n J_3\n:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}} = 2;\\;\n z_0^{j+l}\\;\\text{is active in}\\;\\!\n \\left\\langle z_0^j, z_1^j\\right]\n \\!\\;\\text{when}\\;j=j^{\\prime}\n \\right.\\!\\right\\}\\!,\n\\end{equation}\nbecause the nonemptiness of $J_3$ would imply that \n$z_0^{j+l}$ was not R-mobile in \n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$.\nObserve that $z_0^{j+l}$ is active in\n$\\left\\langle z_0^j, z_1^j\\right]$\nwhen and only when $j \\in J_1$. Thus,\nProposition \\ref{prop: mobile point is active [lepsilon] times or [(l-1)epsilon] times}\ntells us that\n$|J_1| = [l\\epsilon]_d$.\nSimilarly, $z_0^{j+l}$ is active in\n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$\nwhen and only when $j \\in \\left\\{J_1 \\cup J_2 \\right\\}$. Thus, by\nProposition \\ref{prop: mobile point is active [lepsilon] times or [(l-1)epsilon] times},\nwe have\n\\begin{align}\n\\label{eq: [(l-1)e] = [le] + d-e}\n [(l-1)\\epsilon]_d \n&= |J_1| + |J_2|\n \\\\ \\nonumber\n&= [l\\epsilon]_d + d-\\epsilon,\n\\end{align}\nwhere the second line can be deduced either from the fact that\n$[(l-1)\\epsilon]_d > [l\\epsilon]_d$, or from the fact that\n$d-\\epsilon = \\#\\left\\{j^{\\prime}\\in{{\\mathbb Z}}\/d\\left\\vert\\, \nn_{j^{\\prime}} = \\left\\lfloor\\frac{k}{d}\\right\\rfloor\\right.\\right\\} = |J_2|$.\n\n\n\nThe fact that $z_0^{j+l}$ is R-mobile in \n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$ implies\nthat its mirror mobile point,\n$z_{n_{j-l}}^{j-l}$, is L-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$.\nNow, by Proposition \\ref{prop: positive type, main prop}, we know that\nthere is a unique $j_* \\in {{\\mathbb Z}}\/d$ such that\n$v_q\\!\\left(z_0^{j_*}, z_1^{j_*}\\right) = \\alpha(k-k^2)$ and\n$v_q\\!\\left(z_0^{j^{\\prime}}, z_1^{j^{\\prime}}\\right) = \\alpha(k)$\nfor all $j^{\\prime} \\neq j_* \\in {{\\mathbb Z}}\/d$.\nThus, if we define $\\chi_R(j^{\\prime})$ (respectively \n$\\chi_L(j^{\\prime})$) to be equal to 1 if $z_0^{j+l}$\n(respectively $z_{n_{j-l}}^{j-l}$) is active in \n$\\left\\langle z_0^j, z_1^j \\right]$ at time $j = j^{\\prime}$,\nand equal to 0 otherwise, then for any $j^{\\prime} \\in {{\\mathbb Z}}\/d$, we must have\n\\begin{equation}\n\\label{eq: chi for mobile points in < z_0^j, z_1^j ]}\n \\chi_R(j^{\\prime}) + \\chi_L(j^{\\prime})\n= \\begin{cases}\n 1 & j^{\\prime} \\neq j_*\n \\\\\n 2 & j^{\\prime} = j_*, \\;\\alpha = +1\n \\\\\n 0 & j^{\\prime} = j_*,\\;\\alpha = -1\n \\end{cases}.\n\\end{equation}\nHere, the relative values of $\\chi_R(j^{\\prime}) + \\chi_L(j^{\\prime})$ are determined\nby the fact that a mobile point in\n$\\left\\langle z_0^j, z_1^j \\right]$ contributes\n$-k^2$ to $v_q\\!\\left( z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right)$\nif it is active at time $j=j^{\\prime}$ and contributes zero otherwise.\nThe exact values are then determined by the fact that\n$\\chi_R(j^{\\prime}), \\chi_L(j^{\\prime}) \\in \\{0,1\\}$, implying \n$\\chi_R(j^{\\prime}) + \\chi_L(j^{\\prime}) \\in \\{0,1,2\\}$.\nEquation (\\ref{eq: chi for mobile points in < z_0^j, z_1^j ]})\nthen implies that\n\\begin{equation}\n\\label{eq: total active mobile points is d+alpha}\n\\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_R(j) \\;+\\; \\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_L(j) = d+\\alpha.\n\\end{equation}\nOn the other hand, Proposition \\ref{prop: mobile point is active [lepsilon] times or [(l-1)epsilon] times}\ntells us that\n\\begin{equation}\n\\label{eq: one active le times, the other active (l-1)e times}\n\\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_R(j) = [l\\epsilon]_d,\\;\\;\\;\\;\n\\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_L(j) = [(l-1)\\epsilon]_d.\n\\end{equation}\nCombining \n(\\ref{eq: total active mobile points is d+alpha}) and\n(\\ref{eq: one active le times, the other active (l-1)e times}) yields\n\\begin{align}\n\\label{eq: le + (l-1)e = d+ alpha}\n d + \\alpha\n&= [l\\epsilon]_d + [(l-1)\\epsilon]_d \n \\\\ \\nonumber\n \\alpha \n&\\equiv (2l-1)\\epsilon\\; (\\mod d)\n \\\\ \\nonumber\n \\alpha\\gamma \\,\\epsilon^{-1} \n&\\equiv \\gamma(2l-1)\\; (\\mod d)\n \\\\\n\\label{eq: c = 2l-1}\n c\n&= [2l-1]_d.\n\\end{align}\n(Recall that $\\gamma = +1$.)\nAlternatively, we could use (\\ref{eq: [(l-1)e] = [le] + d-e})\nto solve (\\ref{eq: le + (l-1)e = d+ alpha}) for $[l\\epsilon]_d$,\nobtaining\n\\begin{align}\n \\nonumber\n [l\\epsilon]_d + [(l-1)\\epsilon]_d \n&= d + \\alpha\n \\\\ \\nonumber\n [l\\epsilon]_d + [l\\epsilon]_d + d-\\epsilon\n&= d + \\alpha\n \\\\\n\\label{eq: le = (e+a)\/2}\n [l\\epsilon]_d\n& = \\textstyle{\\frac{\\epsilon + \\alpha}{2}}.\n\\end{align}\n\n\n\nNow that we have derived \n(\\ref{eq: c = 2l-1})\nand\n(\\ref{eq: le = (e+a)\/2}),\nwe proceed with the task of finding a lower bound for $c$.\nSince $\\gamma = +1$, we have\n${{\\overline{\\psi}}} = \\frac{ck+\\alpha}{d}k - [dq]_{k^2}$, which means that the constraint\n${{\\overline{\\psi}}} \\in \\left\\langle [dq]_{k^2}, 2[dq]_{k^2}\\right\\rangle$\nimplies $\\frac{ck+\\alpha}{d}k \\in \\left\\langle 2[dq]_{k^2}, 3[dq]_{k^2}\\right\\rangle$.\nWe can then reexpress $\\frac{ck+\\alpha}{d}$ as\n$\\frac{ck+\\alpha}{d}\n= \\left(\\frac{k+\\epsilon}{d}\\right)c - \\frac{c\\epsilon - \\alpha}{d}\n= 3c - \\frac{c\\epsilon - \\alpha}{d}$,\nwhere the fact that $\\frac{ck+\\alpha}{d} \\in {{\\mathbb Z}}$ implies\n$\\frac{c\\epsilon - \\alpha}{d} \\in {{\\mathbb Z}}$, with\n$0 \\leq \\frac{c\\epsilon - \\alpha}{d} \\leq c$.\nSince (\\ref{eq: m = 1 or m in d-1, d+1}) tells us $m=1$,\nimplying $[dq]_{k^2} = (c+ \\mu)k + \\alpha$, we then have\n\\begin{equation}\n\\label{eq: 2dq < psibar < 3dq}\n(2c+2\\mu)k + 2\\alpha\n\\;\\;<\\;\\;\n\\left(3c - \\textstyle{\\frac{c\\epsilon - \\alpha}{d}}\\right)k\n\\;\\;<\\;\\;\n(3c+3\\mu)k + 3\\alpha,\n\\end{equation}\nwhich, when $\\mu = +1$, implies\n$c > 2\\mu + \\frac{c\\epsilon - \\alpha}{d} + \\frac{2\\alpha}{k} >1$,\nand when $\\mu = -1$, implies\n$\\frac{c\\epsilon - \\alpha}{d} > -3\\mu -\\frac{3\\alpha}{k} > 2$,\nso that $c \\geq \\frac{c\\epsilon - \\alpha}{d} > 2$.\nThus, in either case, $c> 1$. \nThis means that $\\frac{cd-1}{c} > d-1 \\geq \\epsilon$,\nso that $c > \\frac{c\\epsilon + 1}{d} \\geq \\frac{c\\epsilon - \\alpha}{d}$,\nimplying $c \\geq \\frac{c\\epsilon - \\alpha}{d} + 1$.\nIn addition, $c > 1$ implies $\\frac{c\\epsilon - 1}{d} > 0$,\nwhich, since $\\frac{c\\epsilon - \\alpha}{d} \\in {{\\mathbb Z}}$, implies\n$\\frac{c\\epsilon - \\alpha}{d} \\geq 1$.\nThus, when $\\mu = +1$, the left-hand inequality in (\\ref{eq: 2dq < psibar < 3dq})\ntells us that\n\\begin{align}\n\\label{prop: psibar NN, claim 2, eq: mu = +1, c > 2mu + ...}\n c\n&> 2\\mu + \\textstyle{\\frac{c\\epsilon - \\alpha}{d}} \\;+\\; \\textstyle{\\frac{2\\alpha}{k}}\n \\\\ \\nonumber\n&\\geq 3 - \\textstyle{\\frac{2}{k}},\n\\end{align}\nwhich, since $\\frac{2}{k} < 1$, implies $c \\geq 3$.\nWhen $\\mu = -1$, the right-hand inequality in (\\ref{eq: 2dq < psibar < 3dq}),\nalong with the fact that $c \\geq \\frac{c\\epsilon - \\alpha}{d} + 1$ (proved a few lines ago),\ntells us that\n\\begin{align}\n\\label{prop: psibar NN, claim 2, eq: mu = -1, c >= 4}\n c\n&\\geq \\textstyle{\\frac{c\\epsilon - \\alpha}{d}} + 1\n \\\\ \\nonumber\n&> \\left( - 3\\mu \\;-\\; \\textstyle{\\frac{3\\alpha}{k}}\\right) + 1\n \\\\ \\nonumber\n&\\geq 4 \\;-\\; \\textstyle{\\frac{3}{k}},\n\\end{align}\nwhich, since $\\frac{3}{k} < 1$, implies $c \\geq 4$.\nThus, for either value of $\\mu$, we have\n\\begin{align}\n [dq]_{k^2}\n&= (c+ \\mu)k + \\alpha\n \\\\ \\nonumber\n&\\geq \\begin{cases}\n (3 + 1)k + \\alpha & \\mu = +1\n \\\\\n (4 - 1)k + \\alpha & \\mu = -1\n \\end{cases}\n \\\\ \\nonumber\n&\\geq 3k + \\alpha\n \\\\ \\nonumber\n&= \\textstyle{\\frac{k+\\epsilon}{d}}k + \\alpha\n \\\\ \\nonumber\n&> \\textstyle{\\frac{k^2}{d}}.\n\\end{align}\n\n\nIt therefore remains to show that $l \\notin \\{0, \\pm 1, \\pm 2\\}$,\nrecalling that $z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$.\nThe R-mobility of $z_0^{j+l}$ in\n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j\\right]$\nrequires, by definition, that $l \\neq 1$.\nIt also implies, as shown in (\\ref{prop: psibar NN, eq: floor(k\/d) = 2}), that\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor = 2$, which, together\nwith Proposition \\ref{prop: positive type, main prop}.(ii), implies\nthat $z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$, requiring that $l \\neq 0$.\nSince we are taking $k > 100$, the fact that\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor = 2$ also implies that\n$d > \\frac{k}{3} > 33$. If $l = -1$, then by (\\ref{eq: c = 2l-1}),\n$c = [2l-1]_d = 2(-1)-1 + d = d-3 > \\frac{d}{2}$, a contradiction.\nIf $l = -2$, then $c = 2(-2)-1 + d = d-5 > \\frac{d}{2}$, another contradiction.\nThus, $l \\notin \\{0, 1,-1, -2\\}$.\n\nThis leaves the case of $l = 2$,\nwith $c = 2(2) -1 = 3$.\nAs shown in (\\ref{prop: psibar NN, claim 2, eq: mu = -1, c >= 4}),\n$c \\geq 4$ when $\\mu = -1$, so we must have $\\mu = +1$.\nThe first line of (\\ref{prop: psibar NN, claim 2, eq: mu = +1, c > 2mu + ...})\nthen gives\n\\begin{align}\n \\textstyle{\\frac{c\\epsilon - \\alpha}{d}}\n&< c - 2\\mu - \\textstyle{\\frac{2\\alpha}{k}}\n \\\\ \\nonumber\n&= 1 - \\textstyle{\\frac{2\\alpha}{k}},\n\\end{align}\nwhich, since $\\frac{c\\epsilon - \\alpha}{d} > 0$ is an integer,\nimplies $\\frac{c\\epsilon - \\alpha}{d} = 1$ and $\\alpha = -1$.\nThis, in turn, implies that $\\epsilon = \\frac{d-1}{3}$, so that\n$[l\\epsilon]_d = \\frac{2d-2}{3}$ and $\\frac{\\epsilon+\\alpha}{2} = \\frac{d-4}{6}$,\ncontradicting (\\ref{eq: le = (e+a)\/2}). Thus, $l \\neq 2$.\n\n\\end{proof}\n\n\n\n\n\nLastly, we attempt to further constrain the values of \n$l \\notin \\{0, \\pm1, \\pm2\\} \\in {{\\mathbb Z}}\/d$ for which\n$z_0^{j+l}$ can be R-mobile in both\n$\\left\\langle z_0^j, z_1^j \\right]$\nand $\\left\\langle z_{n_j-1}^j, z_{n_j}^j \\right]$.\nLet $s$ denote the smallest positive integer such that\n\\begin{equation}\n\\label{eq: q++, k\/d = 2, definition of s}\ns\\textstyle{\\frac{k^2}{d}} > [dq]_{k^2}.\n\\end{equation}\nSince $[dq]_{k^2} < \\frac{k^2}{2}$, we know that\n$\\left\\lceil\\frac{d}{2}\\right\\rceil \\frac{k^2}{d} > [dq]_{k^2}$,\nand so $s \\leq \\left\\lceil\\frac{d}{2}\\right\\rceil$.\n\n\n\n\\begin{claim}\n\\label{claim: m mu t_1 and m mu t_2 not R-mob implies m mu(t_1+t_2) R-mob}\nSuppose that $t_1$ and $t_2$ are positive integers such that\n$t_1< s$, $t_2 < s$, and $s \\leq t_1+t_2$. (Note that this implies $t_1 + t_2 < d$.)\nIf neither $z_0^{j+\\mu mt_1}$\nnor $z_0^{j + \\mu mt_2}$ is R-mobile in \n$\\left\\langle z_0^j, z_1^j \\right]$, then\n$z_0^{j + \\mu m(t_1+t_2)}$ is R-mobile in \n$\\left\\langle z_0^j, z_1^j \\right]$.\n\\end{claim}\n\n\\begin{proof}\nSuppose that $t_1$ and $t_2$ are positive integers such that\n$t_1< s$, $t_2 < s$, and $s \\leq t_1+t_2$, and that\nneither $z_0^{j+\\mu m t_1}$\nnor $z_0^{j + \\mu m t_2}$ is R-mobile in \n$\\left\\langle z_0^j, z_1^j \\right]$.\nRecalling that (\\ref{eq: (mu m)^2 equiv 1(mod d)})\ntells us that $(\\mu m)^2 \\equiv 1\\; (\\mod d)$,\nwe fix the lift\n\\begin{align}\n \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+\\mu mt}-z_0^j\\right)\n&= [\\mu m(\\mu m t)]_d \\frac{k^2}{d} + \\left(\\frac{[(\\mu mt)\\epsilon]_d}{d}-1\\right)[dq]_{k^2}\n \\\\ \\nonumber\n&= t \\frac{k^2}{d} - \\frac{[-(\\mu mt)\\epsilon]_d}{d}[dq]_{k^2}\n\\end{align}\nof $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+\\mu mt}-z_0^j\\right)$ to the integers\nfor any positive integer $t < d$. Thus, for any $t < s$,\nso that $t\\frac{k^2}{d} < [dq]_{k^2}$, we have\n$-[dq]_{k^2} < t \\frac{k^2}{d} - \\frac{[-(\\mu mt)\\epsilon]_d}{d}[dq]_{k^2} < [dq]_{k^2}$,\nand for any $t\\geq s$, so that $t\\frac{k^2}{d} > [dq]_{k^2}$, we have\n$0 < t \\frac{k^2}{d} - \\frac{[-(\\mu mt)\\epsilon]_d}{d}[dq]_{k^2} < k^2$.\n\n\n\n\nFor $i \\in \\{1,2\\}$, we know in addition that $z_0^{j+\\mu mt_i}$\nis not R-mobile in $\\left\\langle z_0^j, z_1^j \\right]$, and so\n\\begin{equation}\n-[dq]_{k^2} < t_i \\frac{k^2}{d} - \\frac{[-(\\mu mt_i)\\epsilon]_d}{d}[dq]_{k^2} < 0.\n\\end{equation}\nUsing the fact that\n\\begin{equation}\n [\\mu m(t_1\\!+t_2)\\epsilon]_d\n= \\begin{cases}\n [-(\\mu mt_1)\\epsilon]_d \\!+\\! [-(\\mu mt_2)\\epsilon]_d\n & [-(\\mu mt_1)\\epsilon]_d \\!+\\! [-(\\mu mt_2)\\epsilon]_d < d\n \\\\\n [-(\\mu mt_1)\\epsilon]_d \\!+\\! [-(\\mu mt_2)\\epsilon]_d \\!-\\!d\n & [-(\\mu m t_1)\\epsilon]_d \\!+\\! [-(\\mu mt_2)\\epsilon]_d \\geq d\n \\end{cases},\n\\end{equation}\nwe then obtain that\n\\begin{align}\n (t_1 + t_2)\\frac{k^2}{d} - \\frac{[-(\\mu m(t_1+t_2))\\epsilon]_d}{d}[dq]_{k^2}\n&\\leq \\left( t_1 \\frac{k^2}{d} - \\frac{[-(\\mu mt_1)\\epsilon]_d}{d}[dq]_{k^2}\\right)\n \\\\ \\nonumber\n &\\;\\;\\;\\;\n +\\left( t_2 \\frac{k^2}{d} - \\frac{[-(\\mu mt_2)\\epsilon]_d}{d}[dq]_{k^2}\\right)\n + \\frac{d}{d}[dq]_{k^2}\n \\\\ \\nonumber\n&< 0 \\;+\\; 0 \\;+\\; 1\\!\\cdot\\![dq]_{k^2}\n \\\\ \\nonumber\n&= [dq]_{k^2}.\n\\end{align}\nThus, $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+\\mu m( t_1+t_2)}-z_0^j\\right) \\in\n\\left\\langle 0, dq\\right\\rangle$, and so\n$z_0^{j + \\mu m(t_1+t_2)}$ is R-mobile in \n$\\left\\langle z_0^j, z_1^j \\right]$.\n\\end{proof}\n\n\n\n\n\n\nSuppose that $s \\geq 4$. Then\n$\\left\\lceil \\frac{s}{2} \\right\\rceil < s$,\n$s \\leq 2\\!\\left\\lceil \\frac{s}{2} \\right\\rceil < d$,\n$\\left\\lceil \\frac{s}{2} \\right\\rceil + 1 < s$, and\n$s \\leq 2\\!\\left(\\left\\lceil \\frac{s}{2} \\right\\rceil+1\\right) < d$,\nand so Claim \\ref{claim: m mu t_1 and m mu t_2 not R-mob implies m mu(t_1+t_2) R-mob}\nimplies that there exist\n$l_0 \\in \\left\\{ \\mu m\\! \\left\\lceil \\frac{s}{2} \\right\\rceil,\\, \n2\\mu m\\! \\left\\lceil \\frac{s}{2} \\right\\rceil \\right\\}$\nand\n$l_1 \\in \\left\\{ \\mu m\\!\\left(\\left\\lceil \\frac{s}{2} \\right\\rceil +1\\right),\\, \n2\\mu m\\!\\left(\\left\\lceil \\frac{s}{2} \\right\\rceil+1\\right) \\right\\}$\nfor which both\n$z_0^{j+l_0}$ and $z_0^{j+l_1}$ are R-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$, a contradiction.\nThus $s \\leq 3$.\nThis, in turn, implies that $3\\frac{k^2}{d} > [dq]_{k^2}$.\nThus, if $\\gamma = -1$, so that $\\mu = +1$ and $[dq]_{k^2} = (m-c)k + \\alpha$, \nthen we have\n\\begin{align}\n\\label{prop: psibar NN, claim 3, eq: mk < (d-1)k}\n mk \n&= [dq]_{k^2} + ck - \\alpha\n \\\\ \\nonumber\n&< 3\\textstyle{\\frac{k}{d}k} + ck - \\alpha\n \\\\ \\nonumber\n&= \\left(3\\left(3 - \\textstyle{\\frac{\\epsilon}{d}}\\right) + c\\right)k - \\alpha\n \\\\ \\nonumber\n&< \\left(9 + \\textstyle{\\frac{d}{2}}\\right)k\n \\\\ \\nonumber\n&< (d-1)k.\n\\end{align}\nHere, the third line uses the fact that\n$\\left\\lfloor \\frac{k}{d} \\right\\rfloor = 2$, implying $\\frac{k+\\epsilon}{d} = 3$;\nthe fourth line uses the fact that $c < \\frac{d}{2}$; and the fifth line\nuses the fact that $\\left\\lfloor \\frac{k}{d} \\right\\rfloor = 2$ and $k > 100$\nimply $d > 33$.\nSince (\\ref{prop: psibar NN, claim 3, eq: mk < (d-1)k})\ncontradicts\n(\\ref{eq: m = 1 or m in d-1, d+1}),\nour supposition that $\\gamma = -1$ must have been false,\nand so we are left with the case in which $\\gamma = +1$,\nwhich, by (\\ref{eq: m = 1 or m in d-1, d+1}), implies $m = 1$.\n\nIf $s=3$, then since\nClaim \\ref{claim: k^2\/d < dq and l not 1,-1, 2, -2} tells us that\nneither $z_0^{j + \\mu}$ nor $z_0^{j+2\\mu}$ is R-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$,\nClaim \\ref{claim: m mu t_1 and m mu t_2 not R-mob implies m mu(t_1+t_2) R-mob}\nimplies that both\n$z_0^{j+3\\mu}$ and $z_0^{j+4\\mu}$ are R-mobile in \n$\\left\\langle z_0^j, z_1^j \\right]$, a contradiction.\nIf $s=2$, then the fact that $z_0^{j+\\mu}$ is not R-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$ implies that\n $z_0^{j+2\\mu}$ is R-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$, again a contradiction.\nThus $s=1$, implying that $\\frac{k^2}{d} > [dq]_{k^2}$,\nbut this contradicts Claim \\ref{claim: k^2\/d < dq and l not 1,-1, 2, -2}.\n\nThus, it is impossible for \n$\\left\\langle z_{n_j-1}^j, z_{n_j}^j \\right]$\nto have R-mobile points, and so,\nby the argument at the beginning of the\nproof of this proposition,\nall ${{\\overline{\\psi}}}$-mobile points are non-neutralized.\n\n\\end{proof}\n\n\n\n\n\\begin{prop}\n\\label{prop: q of positive type. If mobile points exist, then l=1}\nSuppose $q$ of positive type is genus-minimizing,\nand $\\psi > 2[dq]_{k^2}$.\nIf $z_0^{j+l}$, and hence $z_{n_{j-l}}^{j-l}$, are mobile in ${\\mathbf{z}}^j$\nfor some nonzero $l \\in {{\\mathbb Z}}\/d$, then ${{\\overline{\\psi}}} < \\frac{k^2}{2}$,\n$l=1$, $c=1$, and $2 \\!\\!\\not\\vert\\, \\frac{k+\\alpha}{d}$,\nand either $(\\mu,\\gamma)=(1,-1)$ with $d=2$ and $\\alpha=+1$,\nor $(\\mu,\\gamma)=(1,1)$.\n\\end{prop}\n\\begin{proof}\nParts (i) and (ii) of Proposition \\ref{prop: positive type, main prop}\ntell us that $z_0^{j+l}$ and $z_{n_{j-l}}^{j-l}$ are respectively mobile in\n$\\left\\langle z_{i_*}^j, z_{i_*+1}^j\\right]$ and\n$\\left\\langle z_{n_j-(i_*+1)}^j, z_{n_j-i_*}^j \\right]$,\nwhere $i_*$ is the unique element of \n$\\left\\{ 0, \\ldots, \\left\\lfloor \\frac{k}{d} \\right\\rfloor - 2 \\right\\}$\nsatisfying\n$\\left(z_{i_*}^{j_*}, z_{{i_*}+1}^{j_*}\\right) \n= \\left(z_{n_j - (i_*+1)}^{j_*}, z_{n_j - i_*}^{j_*}\\right)\n= (x_*,y_*)$, for some unique $j_* \\in {{\\mathbb Z}}\/d$,\nwhere $x_*, y_* \\in Q_q$ are the unique elements of $Q_q$\nsatisfying $v_q(x_*,y_*) = \\alpha(k-k^2)$.\nNote that this implies $i_* = \\frac{n_{j_*}-1}{2}$.\nWe begin by showing that ${{\\overline{\\psi}}} < \\frac{k^2}{2}$.\n\n\n\n\nSuppose that $\\psi < \\frac{k^2}{2}$.\nSince $z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_{i_*}^j, z_{i_*+1}^j\\right]$,\nwe know that\n$z_0^j$ is R-mobile in\n$\\left\\langle z_{i_*}^{j-l}, z_{i_*+1}^{j-l}\\right]$.\nThus, \n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^j - z_0^{j-l}\\right) \\in\n\\left\\langle i_* dq, (i_* + 1)dq \\right\\rangle \\subset\n\\left\\langle 0, (i_* + 1)dq \\right\\rangle$,\nsince Corollary \\ref{cor: if z^j has mobile points, then (k\/d)dq < k^2}\ntells us that\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\! [dq]_{k^2} < k^2$.\nWe therefore have \n\\begin{align}\n \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j-l}-z_{n_{j-1}}^{j-1}\\right)\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j-l}-z_0^j\\right) + \\psi\n \\\\ \\nonumber\n&= -\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^j - z_0^{j-l}\\right) + \\psi\n \\\\ \\nonumber\n&= -\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^j - z_0^{j-l}\\right) -dq + \\psi\n \\\\ \\nonumber\n&\\in \\left\\langle - \\left(i_*+1\\right)dq,\\, 0 \\right\\rangle + \\psi-dq\n \\\\ \\nonumber\n&= \\left\\langle \\psi - \\left(i_*+2\\right)dq,\\, \\psi-dq \\right\\rangle.\n\\end{align}\nThis means that if\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j-l}-z_{n_{j-1}}^{j-1}\\right)\n\\in \\left\\langle 0, \\psi \\right\\rangle$,\nso that $z_0^{j-l}$ is R$\\psi$-mobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$,\nthen $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j-l}-z_{n_{j-1}}^{j-1}\\right)\n\\in \\left\\langle 0, \\psi - dq \\right\\rangle$, implying that\n\\begin{align}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l-1}}^{j-l-1}-z_0^j\\right)\n&= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(\\left(z_0^{j-l} - \\psi\\right)- \\left(z_{n_{j-1}}^{j-1}+\\psi\\right)\\right)\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j-l} - z_{n_{j-1}}^{j-1}\\right) -2\\psi +dq\n \\\\ \\nonumber\n&\\in \\left\\langle -2\\psi +dq, -\\psi \\right\\rangle,\n \\end{align}\nwhich, since $2\\psi < k^2$, has no intersection with\n$\\left\\langle -\\psi, 0\\right\\rangle$, so that\n$z_{n_{j-l-1}}^{j-l-1}$ is not L$\\psi$-mobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$, making \n$z_0^{j-l}$ non-neutralized R$\\psi$-mobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$, a contradiction.\nThus\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j-l}-z_{n_{j-1}}^{j-1}\\right) \\in\n\\left\\langle \\psi - \\left(i_*+2\\right)dq,\\, \\psi-dq \\right\\rangle\n\\setminus \\left\\langle 0, \\psi \\right\\rangle$.\nThe fact that $\\psi > 2[dq]_{k^2}$ implies that\n$\\psi - [dq]_{k^2} > 0$, which means that\n$\\left\\langle \\psi - \\left(i_*+2\\right)dq,\\, \\psi-dq \\right\\rangle\n\\subset \\left\\langle 0, \\psi \\right\\rangle$\nunless $\\psi - \\left(i_*+2\\right)[dq]_{k^2} < 0$.\nThus \n\\begin{equation}\n\\psi - \\left(i_*+2\\right)[dq]_{k^2}\n\\;<\\; \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j-l}-z_{n_{j-1}}^{j-1}\\right) \\;<\\; 0.\n\\end{equation}\nThis, in turn, implies that\n\\begin{align}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-1}}^{j-1}-z_0^{j-l}\\right)\n&= -\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j-l}-z_{n_{j-1}}^{j-1}\\right)\n \\\\ \\nonumber\n&\\in \\left\\langle 0, \\left(i_*+2\\right)[dq]_{k^2} - \\psi \\right\\rangle\n \\\\ \\nonumber\n&\\subset \\left\\langle 0, i_*[dq]_{k^2} \\right\\rangle,\n\\end{align}\nwhere the last line uses the fact that\n$\\psi > 2[dq]_{k^2}$.\nThus $z_{n_{j-1}}^{j-1}$ is L-mobile in\n$\\left\\langle z_i^{j-l}, z_{i+1}^{j-l}\\right]$\nfor some $i \\in \\{0, \\ldots, i_*-1\\}$,\nbut this means that both\n$\\left\\langle z_i^j, z_{i+1}^j\\right]$ and\n$\\left\\langle z_{i_*}^j, z_{i_*+1}^j\\right]$\nhave mobile points, even though $i \\neq i_*$.\nThis contradicts Proposition \\ref{prop: positive type, main prop}.(i),\nand so our supposition that $\\psi < \\frac{k^2}{2}$ must have been false.\n\n\n\n\n\nThus ${{\\overline{\\psi}}} < \\frac{k^2}{2}$,\nand so Proposition \\ref{prop: psibar < k^2\/2 implies psibar-mobile are nn}\ntells us that all ${{\\overline{\\psi}}}$-mobile points are non-neutralized.\nSince $z_0^{j+l}$ is R-mobile rel $z_0^j$, we know that\n\\begin{equation}\n\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_0^j\\right)\n\\in \\left\\langle 0, \\left(\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor} -1\\right)[dq]_{k^2} \\right\\rangle.\n\\end{equation}\nThus, if ${{\\overline{\\psi}}} > \\left(\\left\\lfloor\\frac{k}{d}\\right\\rfloor -1\\right)[dq]_{k^2}$,\nthen $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_0^j\\right) \\in\n\\left\\langle 0, {{\\overline{\\psi}}}\\right\\rangle$, so that\n$z_0^{j+l}$ is non-neutralized R${{\\overline{\\psi}}}$-mobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$, a contradiction.\nThus, ${{\\overline{\\psi}}} < \\left(\\left\\lfloor\\frac{k}{d}\\right\\rfloor -1\\right)[dq]_{k^2}$,\nbut this implies that\n$z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1} \\in \n\\left\\langle z_0^{j^{\\prime}}, z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-1}^{j^{\\prime}} \\right]$\nfor all $j^{\\prime} \\in {{\\mathbb Z}}\/d$,\nand so\n$z_{n_{j-1}}^{j-1}$ is L-mobile rel $z_{n_j}^j$, and $l=1$.\n\n\n\n\nSince $z_0^{j+1}-z_{n_j}^j$ is constant in $j\\in{{\\mathbb Z}}\/d$,\nwe can express ${{\\mathbb Z}}\/d$ as the disjoint union of $J_1$ and $J_2$, where\n\\begin{align}\n J_1\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}}\\!=\\!\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}\\!-\\!1;\\;\n z_0^{j^{\\prime}+1} \\in\n \\left\\langle z_{i_*}^{j^{\\prime}}, z_{i_* + 1}^{j^{\\prime}}\\right]\n \\right.\\!\\right\\},\n \\\\\n J_2\n&:= \\left\\{j^{\\prime} \\in {{\\mathbb Z}}\/d \\left|\\; \n n_{j^{\\prime}}\\!=\\!\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor};\\;\n z_0^{j^{\\prime}+1} \\in\n \\left\\langle z_{i_*+1}^{j^{\\prime}}, z_{i_* + 2}^{j^{\\prime}}\\right]\n \\right.\\!\\right\\}.\n\\end{align}\nRecall that $n_j := \\left\\lfloor\\frac{k}{d}\\right\\rfloor - {\\theta}^{d, \\epsilon}(j)$,\nand that the definition of ${\\theta}^{d, \\epsilon}(j)$, or alternatively, the $l=1$\ncase of Lemma \\ref{lemma: q of positive type, xi lemma}, implies that\n$\\#\\left\\{ j^{\\prime}\\in{{\\mathbb Z}}\/d\\left|\\; {\\theta}^{d, \\epsilon}(j^{\\prime})=1\\right.\\right\\} = \\epsilon$.\nThus $|J_1| = \\epsilon$.\nNow, since $z_{n_{j-1}}^{j-1} - z_0^j$ is constant in $j\\in{{\\mathbb Z}}\/d$,\n$z_{n_{j-1}}^{j-1}$ is not L-mobile in \n$\\left\\langle z_{i_*}^j, z_{i_*+1}^j\\right]$.\nThus $z_0^{j+1}$ is the only mobile point in\n$\\left\\langle z_{i_*}^j, z_{i_*+1}^j\\right]$,\nand it is active at time $j=j^{\\prime}\\in {{\\mathbb Z}}\/d$\nif and only if $j^{\\prime} \\in J_1$.\nThus, if $\\alpha = +1$, then\n$z_0^{j+l}$ is active in $\\left\\langle z_{i_*}^j, z_{i_*+1}^j \\right]$\nprecisely once, and so $\\epsilon = |J_1| = 1$; moreover, since $j_* \\in J_1$, we have\n$n_{j_*} = \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 1 = \\left(\\frac{k+\\epsilon}{d}-1\\right) -1\n= \\frac{k+1}{d}-2$.\nOn the other hand, if $\\alpha = -1$, then\n$z_0^{j+l}$ is inactive in $\\left\\langle z_{i_*}^j, z_{i_*+1}^j \\right]$\nprecisely once, and so $\\epsilon = |J_1| = d-1$;\nmoreover, since $j_* \\in J_2$, we have\n$n_{j_*} = \\left\\lfloor\\frac{k}{d}\\right\\rfloor = \\frac{k+\\epsilon}{d}-1\n= \\frac{k-1}{d}$.\nLastly, note that the equation $i_* = \\frac{n_{j_*} - 1}{2}$ implies that\n$n_{j_*} \\equiv 1\\;(\\mod 2)$. Thus, regardless of the value of $\\alpha$,\nwe have $\\epsilon = [\\alpha]_d$ and $2 \\!\\!\\not\\vert\\, \\frac{k+\\alpha}{d}$.\n\n\nSince $\\epsilon = [\\alpha]_d$,\nwe know that $c = \\left[\\alpha\\gamma\\, {\\epsilon}^{-1}\\right]_d = [\\gamma]_d$.\nIf $(\\mu,\\gamma) = (-1,1)$,\nthen $c>m>0$ implies $c>1$, contradicting the fact that\n$c = [\\gamma]_d = 1$. Thus $(\\mu,\\gamma) \\neq (-1,1)$.\nIf $(\\mu,\\gamma) = (1,-1)$, then\n$c = [\\gamma]_d = d-1$ contradicts the fact that $c \\leq \\frac{d}{2}$\nunless $d=2$. If $d=2$, then $c=d-1=1$, and the fact that\n$\\frac{ck-\\alpha}{d} < \\frac{k}{2}$ implies $\\alpha = +1$.\nIf $(\\mu,\\gamma)=(1,1)$, then $c=[\\gamma]_d = 1$.\n\n\\end{proof}\n\n\n\n\n\n\\begin{prop}\n\\label{prop: type (1,1), k\/d = 2 or 3}\nSuppose that $q$ of positive type is genus-minimizing\nand $\\psi > 2[dq]_{k^2}$.\nIf $(\\mu,\\gamma) = (1,1)$,\nand $\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\in \\{2,3\\}$, \nthen $m=c=1$.\n\\end{prop}\n\\begin{proof}\nSince $(\\mu, \\gamma) = (1,1)$,\nProposition \\ref{prop: positive type, main prop}\ntells us that ${\\mathbf{z}}^j$ has mobile points, and so\nthe conclusions of\nProposition \\ref{prop: q of positive type. If mobile points exist, then l=1} hold.\nAs we just saw in the last paragraph of the proof of \nProposition \\ref{prop: q of positive type. If mobile points exist, then l=1},\n$n_{j_*}$ is odd and is equal to\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor - 1$ (respectively $\\left\\lfloor\\frac{k}{d}\\right\\rfloor$)\nwhen $\\alpha = +1$ (respectively $\\alpha = -1$).\nThis means that\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\neq 3$ when\n$\\alpha = +1$, and $\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\neq 2$ when\n$\\alpha = -1$. Thus if $\\alpha = +1$, then\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor = 2$, $n_{j_*}=1$, and $i_*:= \\frac{n_{j_*}-1}{2} = 0$,\nwhereas if $\\alpha = -1$, then\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor = 3$, $n_{j_*}=3$, and $i_*:= \\frac{n_{j_*}-1}{2} = 1$.\nNote that in both cases, $\\left\\lfloor\\frac{k}{d}\\right\\rfloor - i_* = 2$ and\n$\\frac{k+\\alpha}{d} = 3$.\nProposition \\ref{prop: q of positive type. If mobile points exist, then l=1} \nalso tells us that $c=1$, and that\n$z_0^{j+1}$ is R-mobile in $\\left\\langle z_{i_*}^j, z_{i_*+1}^j\\right]$.\n\n\nWe begin by calculating $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+1} - z_{i_*}^j\\right)$.\nWe could use Corollary \\ref{cor: q of positive type: combo of lemma and difference eq}\nto do this, but it is easier to perform the calculation directly:\n\\begin{align}\n \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+1} - z_{i_*}^j\\right)\n&= \\left(\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}\\!-\\!1 - i_*\\right)dq \\;+\\; \\psi\n \\\\ \\nonumber\n&= \\left(\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}\\!-\\!1 - i_*\\right)dq\n \\;+\\; dq - \\textstyle{\\frac{ck+\\alpha}{d}}k\n \\\\ \\nonumber\n&= \\left(\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor} - i_*\\right)dq\n - \\textstyle{\\frac{k+\\alpha}{d}}k\n \\\\ \\nonumber\n&= 2dq - 3k.\n\\end{align}\nNow, the fact that $[dq]_{k^2}= (m+c)k+\\alpha \\geq 2k+\\alpha$ implies that\n$2[dq]_{k^2}-3k > 0$, and the fact that $[dq]_{k^2} < \\frac{k^2}{2}$\nimplies that $2[dq]_{k^2}-3k < k^2$. Thus, we in fact have\n$\\left[\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+1} - z_{i_*}^j\\right)\\right]_{k^2}\n=2[dq]_{k^2} - 3k$.\nThe fact that $z_0^{j+l}$ is\nR-mobile in $\\left\\langle z_{i_*}^j, z_{i_*+1}^j\\right]$ then implies that\n\\begin{align}\n \\left[\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+1} - z_{i_*}^j\\right)\\right]_{k^2}\n&< [dq]_{k^2}\n \\\\ \\nonumber\n 2[dq]_{k^2} - 3k\n&< [dq]_{k^2}\n \\\\ \\nonumber\n [dq]_{k^2}\n&< 3k\n \\\\ \\nonumber\n (m+1)k+\\alpha\n&< 3k\n \\\\ \\nonumber\n m\n&< 2 - \\textstyle{\\frac{\\alpha}{k}}.\n\\end{align}\nThus, $m=1$ unless $\\alpha = -1$ and $m=2$.\n\n\nSuppose that $\\alpha = -1$ and $m=2$. Then\n\\begin{align}\n q \n&= \\alpha\\gamma\\mu \\textstyle{\\frac{ck+\\alpha\\gamma}{d}}(mk + \\alpha\\mu)\n \\\\ \\nonumber\n&= - \\textstyle{\\frac{k-1}{d}}(2k -1)\n \\\\ \\nonumber\n&= -3(2k-1)\n \\\\ \\nonumber\n&= -6k+3.\n\\end{align}\nConsider the case in which $k \\equiv 1\\;(\\mod 2)$. Since $\\frac{k-1}{3} = d \\in {{\\mathbb Z}}$,\nwe know that $k \\equiv 1\\; (\\mod 6)$, and so\n$\\frac{2k+1}{3}, \\frac{k-1}{6}, \\frac{5k+1}{6} \\in {{\\mathbb Z}}$. Observe that\n\\begin{equation}\n \\left(0q, \\textstyle{\\frac{2k+1}{3}}q, \\textstyle{\\frac{k-1}{6}}q, \\textstyle{\\frac{5k+1}{6}}q \\right)\n= \\left(0, 1, \\textstyle{\\frac{3k-1}{2}}, \\textstyle{\\frac{3k+1}{2}}\\right) \\in \\left({{\\mathbb Z}}\/k^2\\right)^4,\n\\end{equation}\nwhich implies\n\\begin{align}\n \\left| v_q \\left(0q, \\textstyle{\\frac{5k+1}{6}}q\\right) \\right|\n&= \\left|\\left(\\textstyle{\\frac{3k+1}{2}} - 0\\right)k \\;\\;-\\;\\;\n \\#\\left({\\tilde{Q}}_q \\cap \\left\\langle 0, \\textstyle{\\frac{3k+1}{2}}\\right] \\right)k^2\\right|\n \\\\ \\nonumber\n&\\geq -\\textstyle{\\frac{3k+1}{2}}k\\;\\;+\\;\\; 3k^2\n \\\\ \\nonumber\n&> k(k+1),\n\\end{align}\nand so Proposition \\ref{prop: not minimizing if v >= k(k+1)} tells us that\n$q$ is not genus-minimizing.\nThis leaves us with the case in which $k \\equiv 0\\; (\\mod 2)$.\nIn this case, the fact that $\\frac{k-1}{3} \\in {{\\mathbb Z}}$ implies that\n$k \\equiv 4\\; (\\mod 6)$, and so\n$\\frac{k+2}{6}, \\frac{5k+4}{6}, \\frac{2k+1}{3} \\in {{\\mathbb Z}}$. Observe that\n\\begin{equation}\n \\left( \\textstyle{\\frac{k+2}{6}}q, \\textstyle{\\frac{5k+4}{6}}q, 0q, \\textstyle{\\frac{2k+1}{3}}q\\right)\n= \\left(-\\textstyle{\\frac{3k}{2}}+1, -\\textstyle{\\frac{3k}{2}}+2, 0, 1\\right) \\in \\left({{\\mathbb Z}}\/k^2\\right)^4,\n\\end{equation}\nwhich implies\n\\begin{align}\n \\left| v_q \\left(\\textstyle{\\frac{k+2}{6}}q, \\textstyle{\\frac{2k+1}{3}}q\\right) \\right|\n&= \\left|\\left(1 - \\left(-\\textstyle{\\frac{3k}{2}}+1\\right)\\right)k \\;\\;-\\;\\;\n \\#\\left({\\tilde{Q}}_q \\cap \\left\\langle -\\textstyle{\\frac{3k}{2}}+1, 1\\right] \\right)k^2\\right|\n \\\\ \\nonumber\n&\\geq -\\textstyle{\\frac{3k}{2}}k\\;\\;+\\;\\; 3k^2\n \\\\ \\nonumber\n&> k(k+1),\n\\end{align}\nand so $q$ is not genus-minimizing.\nSince $q$ is not genus-minimizing when $\\alpha = -1$ and $m=2$,\nwe must have $m=1$.\n\n\\end{proof}\n\n\n\n\n\\begin{prop}\n\\label{prop: mobile points implies mu m + gamma c = 2}\nSuppose that $q$ of positive type is genus-minimizing.\nIf ${\\mathbf{z}}^j$ has mobile points and \n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor > 3$, then\n$[dq]_{k^2}=2k+\\alpha$, so that $\\mu m + \\gamma c = 2$.\n\\end{prop}\n\\begin{proof}\nSince $\\left\\lfloor\\frac{k}{d}\\right\\rfloor > 3$,\nProposition \\ref{prop: positive type, main prop}.(iv) tells us that\n$\\psi > 2[dq]_{k^2}$.\nSince, in addition, $q$ of positive type is genus-minimizing\nand ${\\mathbf{z}}^j$ has mobile points, we know that the conclusions of\nProposition \\ref{prop: q of positive type. If mobile points exist, then l=1} hold,\nso that $z_0^{j+1}$ is R-mobile in $\\left\\langle z_{i_*}^j, z_{i_*+1}^j\\right]$\nand $z_{n_{j-1}}^{j-1}$ is L-mobile in $\\left\\langle z_{n_j - (i_*+1)}^j, z_{n_j - i_*}^j \\right]$,\nwhere $i_* = \\frac{n_{j_*}-1}{2}$, and $j_*\\in {{\\mathbb Z}}\/d$ is the unique element of ${{\\mathbb Z}}\/d$\nsatisfying $\\left( z_{i_*}^{j_*}, z_{i_*+1}^{j_*}\\right) = \n\\left( z_{n_{j_*} - (i_*+1)}^{j_*}, z_{n_{j_*} - i_*}^{j_*} \\right) = (x_*,y_*)$,\nwhere $x_*,y_* \\in Q_q$ are the unique elements of $Q_q$ satisfying\n$v_q(x_*,y_*) = \\alpha(k-k^2)$.\nIn particular, the last paragraph of the proof of \nProposition \\ref{prop: q of positive type. If mobile points exist, then l=1}\ntells us that $n_{j_*}$ is odd and is equal to\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor - 1$ (respectively $\\left\\lfloor\\frac{k}{d}\\right\\rfloor$)\nwhen $\\alpha = +1$ (respectively $\\alpha = -1$).\nThus, if $\\left\\lfloor\\frac{k}{d}\\right\\rfloor = 4$, then\n$\\alpha = +1$ and $n_{j_*} = 3$, so that\n$i_*:= \\frac{n_{j_*}-1}{2} = 1$ and\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor - i_* = 3$.\nOn the other hand, if $\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\geq 5$,\nthen $n_{j_*} \\geq 5$, and so\n$i_*:= \\frac{n_{j_*}-1}{2} \\geq \\frac{5-1}{2} = 2$, and\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor - i_*\n\\geq n_{j_*} - i_* = \\frac{n_{j_*}+1}{2} \\geq \\frac{5+1}{2} = 3$.\nThus, in all cases,\n$i_* \\in \\left\\{1, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 3\\right\\}$.\nIn particular, $i_* \\notin \\left\\{0, \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2\\right\\}$.\nThe fact that $z_0^{j+1} - z_{n_j}^j$ and $z_{n_{j-1}}^{j-1} - z_0^j$\nare constant in $j\\in {{\\mathbb Z}}\/d$ then tells us that $z_0^{j+1}$ is not R-mobile rel $z_{n_j}^j$\nand $z_{n_{j-1}}^{j-1}$ is not L-mobile rel $z_0^j$. Thus, \nthere are no R-mobile points rel $z_{n_j}^j$, and there are no L-mobile points rel $z_0^j$.\n\n\n\n\nSince $i_* > 0$, the interval\n$\\left\\langle z_0^j, z_1^j\\right]$ has no mobile points.\nThus, for any $j^{\\prime} \\in {{\\mathbb Z}}\/d$,\nProposition \\ref{prop: positive type, main prop}.(i) tells us that\n$v_q\\!\\left(z_0^{j^{\\prime}}, z_1^{j^{\\prime}}\\right) = \\alpha(k)$.\nIt is therefore sufficient to show that\n\\begin{equation}\n\\label{eq: mu m + gamma c = 2, Q_q cap blah has 2 elts}\n\\#\\left({\\tilde{Q}}_q \\cap \n\\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right]\\right)\n= 2\n\\end{equation}\nfor some $j^{\\prime}\\in{{\\mathbb Z}}\/d$,\nbecause this would imply that\n\\begin{align}\n \\nonumber\n v_q\\!\\left(z_0^{j^{\\prime}}, z_1^{j^{\\prime}}\\right)\n&= \\alpha(k)\n \\\\ \\nonumber\n \\left[ z_1^{j^{\\prime}} - z_0^{j^{\\prime}} \\right]_{k^2}\\!\\!k\n \\;-\\;\n \\#\\left({\\tilde{Q}}_q \\cap \n \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right]\\right) k^2\n&= \\alpha(k)\n \\\\ \\nonumber\n [dq]_{k^2}k - 2k^2\n&= \\alpha(k)\n \\\\\n\\label{eq: mu m + gamma c = 2, 2 elts of Q_q implies dq = 2k + alpha}\n [dq]_{k^2} \n&= 2k + \\alpha.\n\\end{align}\nWe therefore devote the remainder of the proof to showing that\n(\\ref{eq: mu m + gamma c = 2, Q_q cap blah has 2 elts}) is true.\n\n\n\n\n\nFirst, observe that\nCorollary \\ref{cor: if z^j has mobile points, then (k\/d)dq < k^2},\nwhich says that\n$\\left\\lfloor \\frac{k}{d}\\right\\rfloor \\! [dq]_{k^2} < k^2$,\nimplies that\n\\begin{equation}\n\\label{eq: mu m + gamma c = 2, z_i^j notin for all i}\nz_i^{j^{\\prime}} \\notin\n\\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right\\rangle\n\\text{for all}\\; j^{\\prime} \\in {{\\mathbb Z}}\/d,\\;\ni \\in \\left\\{0, \\ldots, n_{j^{\\prime}}\\right\\}.\n\\end{equation}\n\n\n\nWe next claim that\n\\begin{equation}\n\\label{eq: mu m + gamma c = 2, z_0^j+l notin for any l or j}\nz_0^{j^{\\prime}+l} \\notin\n\\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right]\n\\text{for all}\\; l, j^{\\prime} \\in {{\\mathbb Z}}\/d.\n\\end{equation}\nFirst, since $z_0^{j+1}$ is R-mobile in\n$\\left\\langle z_{i_*}^j, z_{i_*+1}^j\\right]$ and $i_* > 0$,\nCorollary \\ref{cor: if z^j has mobile points, then (k\/d)dq < k^2} implies that\n$z_0^{j^{\\prime}+1} \\notin\n\\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right]$ for all\n$j^{\\prime} \\in {{\\mathbb Z}}\/d$. This, in turn, implies that\n$z_0^{j^{\\prime}-1} \\notin\n\\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right]$ for all\n$j^{\\prime} \\in {{\\mathbb Z}}\/d$. Combining these two facts with\n(\\ref{eq: mu m + gamma c = 2, z_i^j notin for all i}),\nwe then have that\n\\begin{equation}\nz_0^{j^{\\prime}+l} \\notin\n\\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right]\n\\text{for all}\\; l\\in\\{-1,0,1\\},\\; j^{\\prime} \\in {{\\mathbb Z}}\/d.\n\\end{equation}\nIn addition, we know that\n\\begin{equation}\n\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_0^j\\right) \\notin \\left\\langle 0, dq\\right\\rangle\\;\n\\text{for all}\\; l \\in {{\\mathbb Z}}\/d \\setminus \\{-1,0,1\\},\\; j^{\\prime} \\in {{\\mathbb Z}}\/d,\n\\end{equation}\nsince $z_0^{j+1}$ is the only R-mobile point in ${\\mathbf{z}}^j$.\nSuppose there exists $l\\in {{\\mathbb Z}}\/d$, $l \\notin \\{-1,0,1\\}$, for which \n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right) \\in \\left\\langle -dq, 0\\right\\rangle$.\nSince $z_0^{j+1}$ is R-mobile in\n$\\left\\langle z_{i_*}^j, z_{i_*+1}^j\\right]$, we also know that\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z^{j+1}_0 - z^j_0\\right)\n\\in \\left\\langle i_* dq, (i_* + 1)dq\\right\\rangle$. Thus, for some $x \\in \\{0, dq\\}$,\nwe have\n\\begin{align}\n\\label{eq: mu m + gamma c = 2, z^{j+l+1}_0 is a new R-mobile point}\n \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z^{j+l+1}_0 - z^j_0\\right)\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(\\left(z^{j+l+1}_0 - z^{j+1}_0\\right) \n + \\left(z^{j+1}_0 - z^j_0\\right)\\right)\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z^{j+l}_0 - z^j_0\\right)\n + \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z^{j+1}_0 - z^j_0\\right) + x\n \\\\ \\nonumber\n&\\in \\left\\langle -dq + i_* dq + x,\\; 0 + (i_*+1)dq + x\\right\\rangle \n \\\\ \\nonumber\n&\\subset \\left\\langle (i_*-1) dq,\\; (i_*+2)dq \\right\\rangle\n \\\\ \\nonumber\n&\\subset \\left\\langle 0,\\; \\left(\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}-1\\right)dq \\right\\rangle,\n\\end{align}\nwhere the last line uses the fact that $1 \\leq i_* \\leq \\left\\lfloor\\frac{k}{d}\\right\\rfloor-3$.\nBut (\\ref{eq: mu m + gamma c = 2, z^{j+l+1}_0 is a new R-mobile point})\nis impossible, because\nit implies $z_0^{j+l+1}$ is R-mobile rel $z_0^j$, contradicting the\nfact that $z_0^{j+1}$ is the unique R-mobile point rel $z_0^j$.\nThus, $\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right) \\notin \\left\\langle 0, dq\\right\\rangle$\nfor all $l \\in {{\\mathbb Z}}\/d$, $l \\notin \\{-1,0,1\\}$, and so\n(\\ref{eq: mu m + gamma c = 2, z_0^j+l notin for any l or j}) must be true.\nThis, in turn, implies that\n\\begin{equation}\n\\label{eq: mu m + gamma c = 2, z_{n_{j-l}}^{j-l} not in z_n_j-1, z_n_j}\nz_{n_{j^{\\prime}-l}}^{j^{\\prime}-l} \\notin\n\\left\\langle z_{n_{j^{\\prime}}-1}^{j^{\\prime}}, z_{n_{j^{\\prime}}}^{j^{\\prime}}\\right]\n\\;\\;\\text{for all}\\;l, j^{\\prime}\\in{{\\mathbb Z}}\/d.\n\\end{equation}\n\n\n\nWe next claim, for all $j^{\\prime} \\in {{\\mathbb Z}}\/d$ and $i \\in \\{0, \\ldots, n_{j^{\\prime}}\\}$, that\n\\begin{equation}\n\\label{eq: mu m + gamma c = 2, l=1 implies just one element contained}\n z_{n_{j^{\\prime}-1}-i}^{j^{\\prime}-1}\n\\in \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right]\\;\n \\text{if and only if}\\,\\; i = i_* + \\textstyle{\\frac{\\alpha-1}{2}}.\n\\end{equation}\nWhen $\\alpha = +1$, $z_{n_{j-1}}^{j-1}$ is active in\n$\\left\\langle z_{n_j-(i_*+1)}^j, z_{n_j-i_*}^j\\right]$ at time $j=j_*$, and so\n$z_{n_{j_*-1}}^{j_*-1} \\in \n\\left\\langle z_{n_{j_*} - (i_*+1)}^{j_*}, z_{n_{j_*} - i_*}^{j_*} \\right]\n= \\left\\langle z_{i_*}^{j_*}, z_{i_*+1}^{j_*}\\right]$,\nimplying that\n$z_{n_{j_*-1}-i_*}^{j_*-1} \\in \\left\\langle z_0^{j_*}, z_1^{j_*}\\right]$.\nWhen $\\alpha = -1$, $z_{n_{j-1}}^{j-1}$ is inactive in\n$\\left\\langle z_{n_j-(i_*+1)}^j, z_{n_j-i_*}^j\\right]$ at time $j=j_*$, and so\n$z_{n_{j_*-1}}^{j_*-1} \\in \n\\left\\langle z_{i_*-1}^{j_*}, z_{i_*}^{j_*}\\right]$,\nimplying that\n$z_{n_{j_*-1}-(i_*-1)}^{j_*-1} \\in \\left\\langle z_0^{j_*}, z_1^{j_*}\\right]$.\nWe can summarize these two facts by saying that\n$z_{n_{j_*-1}-\\left(i_*+\\frac{\\alpha-1}{2}\\right)}^{j_*-1}\n \\in \\left\\langle z_0^{j_*}, z_1^{j_*}\\right]$, which, since\n $z_{n_{j-1}}^{j-1} - z_0^j$ is constant in $j\\in{{\\mathbb Z}}\/d$, implies that\n $z_{n_{j^{\\prime}-1}-\\left(i_*+\\frac{\\alpha-1}{2}\\right)}^{j^{\\prime}-1}\n \\in \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}}\\right]$ for all $j^{\\prime} \\in {{\\mathbb Z}}\/d$.\nThe fact that $\\left\\lfloor\\frac{k}{d}\\right\\rfloor [dq]_{k^2} < k^2$ then implies that\n$z_{n_{j^{\\prime}-1}-i}^{j^{\\prime}-1}\n\\notin \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}}\\right]$ for all $j^{\\prime} \\in {{\\mathbb Z}}\/d$\nand $i \\in \\left\\{0, \\ldots, n_{j^{\\prime}}\\right\\} \\setminus \n\\left\\{i_* + \\textstyle{\\frac{\\alpha-1}{2}}\\right\\}$.\n\n\n\n\nNext, we claim that\n\\begin{equation}\n\\label{eq: mu m + gamma c = 2, z_{n_{j-l}-k\/d-1}^{j-l} not in }\n z_{n_{j^{\\prime}-l}-\\left(\\left\\lfloor\\frac{k}{d}\\right\\rfloor-1\\right)}^{j^{\\prime}-l}\n\\notin \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right]\\;\n\\text{for all}\\; l \\neq 1, j^{\\prime} \\in {{\\mathbb Z}}\/d.\n\\end{equation}\nSuppose\n(\\ref{eq: mu m + gamma c = 2, z_{n_{j-l}-k\/d-1}^{j-l} not in }) fails\nfor some $l \\in {{\\mathbb Z}}\/d$, $l \\notin \\{0,1\\}$. Then\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l}-\\left(\\left\\lfloor\\frac{k}{d}\\right\\rfloor-1\\right)}^{j-l}-z_0^j\\right)\n\\notin \\left\\langle 0,dq\\right\\rangle$,\nsince otherwise\n(\\ref{eq: mu m + gamma c = 2, z_0^j+l notin for any l or j}) is contradicted,\nand so $\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l}-\\left(\\left\\lfloor\\frac{k}{d}\\right\\rfloor-1\\right)}^{j-l}-z_0^j\\right)\n\\in \\left\\langle 0,dq\\right\\rangle$, implying that\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l}}^{j-l}-z_{\\left\\lfloor\\frac{k}{d}\\right\\rfloor-1}^j\\right)\n\\in \\left\\langle 0,dq\\right\\rangle$. But this, in turn, implies that\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l}}^{j-l}-z_{n_j}^j\\right)\n\\in \\left\\langle -dq,dq\\right\\rangle$,\ncontradicting (\\ref{eq: mu m + gamma c = 2, z_{n_{j-l}}^{j-l} not in z_n_j-1, z_n_j}).\nThus (\\ref{eq: mu m + gamma c = 2, z_{n_{j-l}-k\/d-1}^{j-l} not in })\nmust be true.\n\n\n\nLastly, we claim that\n\\begin{equation}\n\\label{eq: mu m + gamma c = 2, z_{n_{j-l}-i} notin }\n z_{n_{j^{\\prime}-l}-i}^{j^{\\prime}-l}\n\\notin \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right]\\;\n\\text{for all}\\; l \\neq 1, j^{\\prime} \\in {{\\mathbb Z}}\/d,\\;\ni \\in \\left\\{0, \\ldots, \\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}-2\\right\\}.\n\\end{equation}\nFirst of all, we know that\n\\begin{equation}\n\\label{eq: mu m + gamma c = 2, max z_{n_{j-l}-i}-z_0^j notin}\n\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l}-i}-z_0^j\\right)\n\\notin \\left\\langle 0,dq \\right\\rangle\\;\n\\text{for all}\\; l\\neq 1 \\in {{\\mathbb Z}}\/d,\\;\ni \\in \\left\\{0, \\ldots, \\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}-2\\right\\},\n\\end{equation}\nsince otherwise we would have\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l}}-z_{i+1}^j\\right)\n\\in \\left\\langle -dq,0 \\right\\rangle$,\nmaking $z_{n_{j-l}}$ L-mobile in \n$\\left\\langle z_i^j, z_{i+1}^j\\right]$.\nLine (\\ref{eq: mu m + gamma c = 2, max z_{n_{j-l}-i}-z_0^j notin}),\nin turn, implies that\n\\begin{equation}\n\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l}-i}-z_0^j\\right)\n\\notin \\left\\langle 0,dq \\right\\rangle\\;\n\\text{for all}\\; l\\neq 1 \\in {{\\mathbb Z}}\/d,\\;\ni \\in \\left\\{0, \\ldots, \\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}-3\\right\\},\n\\end{equation}\nand (\\ref{eq: mu m + gamma c = 2, z_{n_{j-l}-k\/d-1}^{j-l} not in })\nimplies that\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l}-\\left(\\left\\lfloor\\frac{k}{d}\\right\\rfloor-2\\right)}-z_0^j\\right)\n\\notin \\left\\langle 0,dq \\right\\rangle$\nfor all $l\\neq 1 \\in {{\\mathbb Z}}\/d$. Thus\n(\\ref{eq: mu m + gamma c = 2, z_{n_{j-l}-i} notin })\nmust be true.\n\n\n\nTogether, lines\n(\\ref{eq: mu m + gamma c = 2, z_0^j+l notin for any l or j}),\n(\\ref{eq: mu m + gamma c = 2, z_{n_{j-l}-k\/d-1}^{j-l} not in }),\n(\\ref{eq: mu m + gamma c = 2, z_{n_{j-l}-i} notin }), and\n(\\ref{eq: mu m + gamma c = 2, l=1 implies just one element contained})\ntell us that\n\\begin{equation}\nQ_q \\cap \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}}\\right\\rangle\n= z_{n_{j^{\\prime}-1}-\\left(i_*+\\frac{\\alpha-1}{2}\\right)}^{j^{\\prime}-1}\\;\n\\text{for every}\\;j^{\\prime} \\in {{\\mathbb Z}}\/d.\n\\end{equation}\nSince $Q_q \\cap \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}}\\right]$\nalso contains $z_1^{j^{\\prime}}$, this implies that\n\\begin{equation}\n\\#\\left({\\tilde{Q}}_q \\cap \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}}\\right] \\right)\n=2\n\\end{equation}\nfor every $j^{\\prime} \\in {{\\mathbb Z}}\/d$.\nThus (\\ref{eq: mu m + gamma c = 2, 2 elts of Q_q implies dq = 2k + alpha})\ntells us that $[dq]_{k^2} = 2k + \\alpha$,\nso that ${\\mu}m + {\\gamma}c = 2$.\n\n\\end{proof}\n\n\n\nThis concludes our study of the case in which\n${\\mathbf{z}}^j$ has mobile points,\nsince we have now learned everything we need to know to\nclassify when such $q$ are genus-minimizing.\nFor the remainder of this section, we therefore focus\non the case in there are no mobile points, which means that\n$v_q\\!\\left(z_{n_{j-1}}^{j-1}, z_0^j\\right)$\nis nonconstant in $j\\in {{\\mathbb Z}}\/d$,\nand so there are non-neutralized pseudomobile points\nand non-neutralized antipseudomobile points.\n\n\n\n\n\\begin{prop}\n\\label{prop: no mobile points, psibar < k^2\/2, implies type (-1,1), m=1, c=2, d odd}\nSuppose that $q$ of positive type is genus-minimizing.\nIf ${\\mathbf{z}}^j$ has no mobile points\nand ${{\\overline{\\psi}}} < \\frac{k^2}{2}$, then\n$(\\mu, \\gamma) = (-1,1)$, $m=1$, $c=2$, and $2\\!\\! \\not\\vert \\,d$.\n\\end{prop}\n\n\\begin{proof}\nSuppose that $q$ is genus-minimizing and of positive type,\nthat ${\\mathbf{z}}^j$ has no mobile points,\nand that ${{\\overline{\\psi}}} < \\frac{k^2}{2}$,\nso that, by\nProposition \\ref{prop: psibar < k^2\/2 implies psibar-mobile are nn},\nall antipseudomobile points are non-neutralized.\n\n\nWe begin by claiming that if\n$z_0^{j+l}$ (hence $z_{n_{j-1-l}}^{j-1-l}$) is\nantipseudomobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$, then\n$[l\\epsilon]_d = \\frac{d-\\alpha}{2}$.\nSuppose that $z_0^{j+l}$ and $z_{n_{j-1-l}}^{j-1-l}$\nare ${{\\overline{\\psi}}}$-mobile in $\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$.\nThen the preceding paragraph tells us that they\nare non-neutralized ${{\\overline{\\psi}}}$-mobile, and\nProposition \\ref{prop: antipseudomobile analog of main prop}.(ii${{\\overline{\\psi}}}$)\ntells us that they are the only non-neutralized\n${{\\overline{\\psi}}}$-mobile points in $\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$.\nNow, by\nPropositions \\ref{prop: unique v_q = alpha(k - k^2), and the rest are v_q = alpha (k)}\nand \\ref{prop: antipseudomobile analog of main prop}.(i${{\\overline{\\psi}}}$),\nwe know that there is a unique $j_* \\in {{\\mathbb Z}}\/d$ such that\n$v_q\\!\\left( z_0^{j_*}, z_{n_{j_*-1}}^{j_*-1}\\right) = -\\alpha(k-k^2)$ and\n$v_q\\!\\left( z_0^{j^{\\prime}}, z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}\\right) = -\\alpha(k)$\nfor all $j^{\\prime}\\neq j_* \\in {{\\mathbb Z}}\/k^2$.\nThus, if we define $\\chi_R(j^{\\prime})$ (respectively \n$\\chi_L(j^{\\prime})$) to be equal to 1 if $z_0^{j+l}$\n(respectively $z_{n_{j-1-l}}^{j-1-l}$) is active in \n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$ at time $j = j^{\\prime}$,\nand equal to 0 otherwise, then for any $j^{\\prime} \\in {{\\mathbb Z}}\/d$,\n\\begin{equation}\n\\label{prop l=-1 in pseudomobile: eq: how many psibars are active?}\n \\chi_R(j^{\\prime}) + \\chi_L(j^{\\prime})\n= \\begin{cases}\n 1 & j^{\\prime} \\neq j_*\n \\\\\n 2 & j^{\\prime} = j_*, \\;\\alpha = -1\n \\\\\n 0 & j^{\\prime} = j_*,\\;\\alpha = +1\n \\end{cases}.\n\\end{equation}\nThis is because a ${{\\overline{\\psi}}}$-mobile point in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$ contributes\n$-k^2$ to $v_q\\!\\left( z_0^{j^{\\prime}}, z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}\\right)$\nif it is active at time $j=j^{\\prime}$ and contributes zero otherwise.\nLine (\\ref{prop l=-1 in pseudomobile: eq: how many psibars are active?})\nthen implies that\n\\begin{equation}\n\\label{prop l=-1 in pseudomobile: eq: total actives is d-alpha}\n\\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_R(j) \\;+\\; \\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_L(j) = d-\\alpha.\n\\end{equation}\nProposition \\ref{prop: antipseudomobile point is active [lepsilon] times}\nthen tells us that\n\\begin{equation}\n\\label{prop l=-1 in pseudomobile: eq: each is active le times}\n\\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_R(j) = \\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_L(j) = [l\\epsilon]_d.\n\\end{equation}\nThe combination of \n(\\ref{prop l=-1 in pseudomobile: eq: total actives is d-alpha}) and\n(\\ref{prop l=-1 in pseudomobile: eq: each is active le times}) then yields\n\\begin{equation}\n\\label{prop l=-1 in pseudomobile: eq: le = (d-alpha)\/2}\n[l\\epsilon]_d = \\textstyle{\\frac{d-\\alpha}{2}}.\n\\end{equation}\nNote that this implies $2 \\!\\! \\not\\vert \\, d$.\n\n\n\nWe next claim that $z_{n_j}^j$ is L-antipseudomobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$.\nSuppose not, so that\n\\begin{equation}\n\\label{eq: (k\/d-1)dq < psibar < k\/d dq}\n\\left(\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}-1\\right)[dq]_{k^2}\n\\;<\\; {{\\overline{\\psi}}} \\;<\\; \\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor} [dq]_{k^2}.\n\\end{equation}\nThat is, if ${{\\overline{\\psi}}} > \\left\\lfloor\\frac{k}{d}\\right\\rfloor [dq]_{k^2}$,\nthen $z_{n_{j^{\\prime}}}^{j^{\\prime}}\n\\in \\left\\langle z_0^{j^{\\prime}}, z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1} \\right]$\nfor all $j^{\\prime} \\in{{\\mathbb Z}}\/d$, making\n$z_{n_j}^j$ L${{\\overline{\\psi}}}$-mobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$.\nOn the other hand, if\n$\\left(\\left\\lfloor\\frac{k}{d}\\right\\rfloor-1\\right)[dq]_{k^2} > {{\\overline{\\psi}}}$, then\n$z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1} \\in\n\\left\\langle z_0^{j^{\\prime}},\nz_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor-1}^{j^{\\prime}} \\right]\n\\subset\n\\left\\langle z_0^{j^{\\prime}},\nz_{n_{j^{\\prime}}}^{j^{\\prime}} \\right]$ for all $j^{\\prime} \\in {{\\mathbb Z}}\/d$,\nmaking $z_{n_{j-1}}^{j-1}$ L-mobile in ${\\mathbf{z}}^j$, a contradiction,\nso (\\ref{eq: (k\/d-1)dq < psibar < k\/d dq}) must hold.\nBy Proposition \\ref{prop: antipseudomobile analog of main prop}.(ii${{\\overline{\\psi}}}$),\nwe know there exists,\nfor some $l \\neq 0 \\in {{\\mathbb Z}}\/d$, an L${{\\overline{\\psi}}}$-mobile point \n$z_{n_{j-1-l}}^{j-1-l}$ in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$.\nNow, (\\ref{eq: (k\/d-1)dq < psibar < k\/d dq}) implies that\n$z_{n_{j^{\\prime}}}^{j^{\\prime}}\n\\in \\left\\langle z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1} - dq,\nz_{n_{j^{\\prime}-1}}^{j^{\\prime}-1} + dq \\right\\rangle$\nfor all $j^{\\prime} \\in {{\\mathbb Z}}\/d$. Thus, if\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-1-l}}^{j-1-l} - z_{n_{j-1}}^{j-1}\\right)\n\\in \\left\\langle -{{\\overline{\\psi}}}, -dq\\right\\rangle$, then\n$z_{n_{j^{\\prime}-1-l}}^{j^{\\prime}-1-l} \\in\n\\left\\langle z_0^{j^{\\prime}}, z_{n_{j^{\\prime}}}^{j^{\\prime}} \\right]$\nfor all $j^{\\prime} \\in {{\\mathbb Z}}\/d$, implying\nthat $z_{n_{j-1-l}}^{j-1-l}$ is L-mobile rel $z_{n_j}^j$, a contradiction.\nThis means that instead, we must have\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-1-l}}^{j-l-l} - z_{n_{j-1}}^{j-1}\\right)\n\\in \\left\\langle -dq, 0\\right\\rangle$.\nThe mirror relation, (\\ref{eq: psibar mirror relation}), for\n${{\\overline{\\psi}}}$-mobile points\nthen tells us that the mirror ${{\\overline{\\psi}}}$-mobile point,\n$z_0^{j+l}$ R${{\\overline{\\psi}}}$-mobile in $\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$,\nsatisfies \n\\begin{align}\n \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_0^j\\right)\n&= -\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-1-l}}^{j-1-l} - z_{n_{j-1}}^{j-1}\\right)\n \\\\ \\nonumber\n&\\in \\left\\langle 0, dq\\right\\rangle,\n\\end{align}\nso that $z_0^{j+l}$ is R-mobile in\n$\\left\\langle z_0^j, z_1^j \\right]$,\na contradiction.\nThus, $z_{n_j}^j$ is L${{\\overline{\\psi}}}$-mobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$.\nNote that this implies\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\![dq]_{k^2} < {{\\overline{\\psi}}}$.\nSince ${{\\overline{\\psi}}} < \\frac{k^2}{2}$, we then have\n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\![dq]_{k^2} < \\frac{k^2}{2}$,\nimplying\n\\begin{equation}\n\\label{prop l=-1 in pseudomobile: eq: dq < dk}\n[dq]_{k^2} < dk.\n\\end{equation}\n\n\n\n\nNow, since $z_{n_j}^j$ and its mirror $z_0^{j-1}$ are ${{\\overline{\\psi}}}$-mobile in\n$\\left\\langle z_0^j, z_{n_{j-1}}^{j-1} \\right]$,\n(\\ref{prop l=-1 in pseudomobile: eq: le = (d-alpha)\/2})\ntells us that\n\\begin{equation}\n(-1)\\epsilon \\equiv \\textstyle{\\frac{d-\\alpha}{2}}\\; (\\mod d).\n\\end{equation}\nThus, since\n$c = \\left[\\alpha\\gamma {\\epsilon}^{-1}\\right]_d$, we have\n\\begin{align}\n c \n&= \\left[\\alpha\\gamma (-1)\\left(\\textstyle{\\frac{d-\\alpha}{2}}\\right)^{-1}\\right]_d\n \\\\ \\nonumber\n&= \\left[\\alpha\\gamma(-1)(-2\\alpha)\\right]_d\n \\\\ \\nonumber\n&= [2\\gamma]_d\n \\\\ \\nonumber\n&= \\begin{cases}\n 2 & \\gamma = +1\n \\\\\n d-2 & \\gamma = -1\n \\end{cases}.\n\\end{align}\nBy Proposition \\ref{prop: properties of parameters d, m, c, alpha, mu, gamma},\nwe know that $c \\leq \\frac{d}{2}$, with equality if and only if $d=2$ (which\ndoes not occur here, since $d$ must be odd). Thus if $\\gamma = -1$,\nthen $d-2 < \\frac{d}{2}$ implies $d=3$ and $c=1$.\nSince $\\gamma = -1$ implies $(\\mu, \\gamma) = (1,-1)$, we then have\n\\begin{align}\n\\label{prop l=-1 in pseudomobile: eq: psibar > k^2\/2}\n {{\\overline{\\psi}}}\n&= \\left[-dq - \\textstyle{\\frac{ck-\\alpha}{d}}k\\right]_{k^2}\n \\\\ \\nonumber\n&= k^2 -\\left([dq]_{k^2} + \\textstyle{\\frac{ck-\\alpha}{d}}k\\right)\n \\\\ \\nonumber\n&> k^2 -\\left(dk + \\textstyle{\\frac{ck-\\alpha}{d}}k\\right)\n \\\\ \\nonumber\n&= k^2 -\\left(3k + \\textstyle{\\frac{k-\\alpha}{3}}k\\right)\n \\\\ \\nonumber\n&> \\textstyle{\\frac{k^2}{2}},\n\\end{align}\nwhere the second line uses the facts that\n$[dq]_{k^2} < \\frac{k^2}{2}$ and $\\frac{ck-\\alpha}{d} < \\frac{k}{2}$,\nthe third line uses\n(\\ref{prop l=-1 in pseudomobile: eq: dq < dk}),\nand the last line uses the fact that $k > 100$.\nSince (\\ref{prop l=-1 in pseudomobile: eq: psibar > k^2\/2})\ncontradicts our supposition that ${{\\overline{\\psi}}} < \\frac{k^2}{2}$,\nwe conclude that $\\gamma \\neq -1$. Thus $\\gamma = 1$ and $c=2$.\nMoreover, since there are no mobile points, Proposition \\ref{prop: positive type, main prop}\ntells us that $(\\mu, \\gamma) \\neq (1,1)$.\nThus $(\\mu, \\gamma) = (-1, 1)$, and so $c > m > 0$\nimplies that $m=1$.\n\n\\end{proof}\n\n\n\n\n\\begin{prop}\n\\label{prop: no mobile points, psi < k^2\/2, implies type (1,-1), m=2, c=1, d odd}\nSuppose that $q$ of positive type is genus-minimizing.\nIf $\\psi < \\frac{k^2}{2}$ and $\\psi > 2[dq]_{k^2}$, then\n${\\mathbf{z}}^j$ has no mobile points,\nall pseudomobile points are non-neutralized,\n$(\\mu, \\gamma) = (1,-1)$, $m=2$, $c=1$, and $2\\!\\! \\not\\vert \\,d$.\n\\end{prop}\n\n\\begin{proof}\nSuppose that $q$ is genus-minimizing and of positive type,\nand that $2[dq]_{k^2}< \\psi < \\frac{k^2}{2}$.\nProposition \\ref{prop: q of positive type. If mobile points exist, then l=1}\ntells us that if $\\psi > 2[dq]_{k^2}$ and mobile points exist, then\n${{\\overline{\\psi}}} < \\frac{k^2}{2}$. Thus, the fact that $\\psi < \\frac{k^2}{2}$\nimplies that there are no mobile points.\nProposition \\ref{prop: positive type, main prop} then tells us that\n$v_q\\!\\left(z_{n_{j-1}}^{j-1}, z_0^j\\right)$ is nonconstant in $j\\in{{\\mathbb Z}}\/d$,\nand that $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$ has precisely one\nnon-neutralized R$\\psi$-mobile point and one\nnon-neutralized L$\\psi$-mobile point, namely,\n$z_0^{j+l}$ and $z_{n_{j-1-l}}^{j-1-l}$ for some $l\\neq 0\\in {{\\mathbb Z}}\/d$.\n\n\nWe begin by showing that\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, eq: all of z^{j+l} fits in < z_0^j, z_n>}\nz_0^{j^{\\prime}+l}, \\ldots, \nz_{n_{j^{\\prime}+l}}^{j^{\\prime}+l}\n\\in\\left\\langle z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}, z_0^{j^{\\prime}} \\right]\\;\n\\text{for all}\\; j^{\\prime} \\in {{\\mathbb Z}}\/d.\n\\end{equation}\nFirst, note that if $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{n_{j-1}}^{j-1}\\right)\n\\in \\left\\langle \\psi-dq, \\psi \\right\\rangle$, then\n\\begin{align}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l-1}}^{j+l-1}-z_0^j\\right)\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(\\left(z_0^{j+l}-\\psi\\right)-\\left(z_{n_{j-1}}^{j-1}+\\psi\\right)\\right) + dq\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l} - z_{n_{j-1}}^{j-1}\\right) + dq - 2\\psi\n \\\\ \\nonumber\n&\\in \\left\\langle -\\psi, dq-\\psi \\right\\rangle,\n\\end{align}\nso that $z_0^{j+l}$ is neutralized by\n$z_{n_{j+l-1}}^{j+l-1}$ L$\\psi$-mobile in $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$,\na contradiction. Thus\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{n_{j-1}}^{j-1}\\right)\n\\in \\left\\langle 0, \\psi-dq \\right\\rangle$, which implies that\n$z_0^{j^{\\prime}+l} \\in\n\\left\\langle z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}, z_0^{j^{\\prime}} \\right]$\nfor all $j^{\\prime} \\in {{\\mathbb Z}}\/d$.\nIn addition, observe that if \n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^j - z_0^{j+l}\\right)\n\\in \\left\\langle 0, \\left(\\left\\lfloor\\frac{k}{d}\\right\\rfloor -1 \\right)dq \\right\\rangle$,\nthen $z_0^j$ is R-mobile in $\\left\\langle z_i^{j+l}, z_{i+1}^{j+l} \\right]$,\nfor some $i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor-2\\right\\}$,\na contradiction.\nThus, for all $j^{\\prime} \\in {{\\mathbb Z}}\/d$, we have\n$\\left\\langle z_0^{j^{\\prime}+l} , \nz_0^{j^{\\prime}+l} + \\left(\\left\\lfloor\\frac{k}{d}\\right\\rfloor -1 \\right)dq \\right\\rangle\n\\subset\n\\left\\langle z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}, z_0^{j^{\\prime}} \\right]$,\nimplying\n$z_0^{j^{\\prime}+l}, \\ldots, \nz_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor -1}^{j^{\\prime}+l}\n\\in\\left\\langle z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}, z_0^{j^{\\prime}} \\right]$.\nNow, the fact that $z_{\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor -1}^{j^{\\prime}+l}\n\\in\\left\\langle z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}, z_0^{j^{\\prime}} \\right]$\nfor all $j^{\\prime} \\in {{\\mathbb Z}}\/d$ and that $\\psi,\\, [dq]_{k^2} < \\frac{k^2}{2}$\nimplies that $z_{n_{j^{\\prime}+l}}^{j^{\\prime}+l} \\neq z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}$,\nand so $l\\neq -1$. It is therefore safe to make our final observation that if\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^j - z_{n_{j+l}-1}^{j+l}\\right)\n\\in \\left\\langle 0, dq \\right\\rangle$, then\n$z_0^j$ is R-mobile in $\\left\\langle z_{n_{j+l}-1}^{j+l}, z_{n_{j+l}}^{j+l}\\right]$,\na contradiction. Thus \n$z_{n_{j^{\\prime}+l}}^{j^{\\prime}+l}\n\\in\\left\\langle z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}, z_0^{j^{\\prime}} \\right]$\nfor all $j^{\\prime} \\in {{\\mathbb Z}}\/d$, and so\n(\\ref{prop: type (1,-1), c=1, eq: all of z^{j+l} fits in < z_0^j, z_n>}) holds.\n\n\nOne implication of (\\ref{prop: type (1,-1), c=1, eq: all of z^{j+l} fits in < z_0^j, z_n>})\nis that $z_{n_{j+l}}^{j+l}$ is L$\\psi$-mobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$.\nWe claim that in fact, $z_{n_{j+l}}^{j+l}$ is non-neutralized\nL$\\psi$-mobile in $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$.\nThat is, since $\\psi < \\frac{k^2}{2}$,\n(\\ref{prop: type (1,-1), c=1, eq: all of z^{j+l} fits in < z_0^j, z_n>})\nimplies\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, k\/d dq < psi}\n\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor} [dq]_{k^2} <\n\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l}}^{j+l}-z_{n_{j-1}}^{j-1}\\right) < \\psi,\n\\end{equation}\nand so, recalling that $l\\neq -1$, we have\n\\begin{align}\n \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l+1} - z_{n_{j-1}}^{j-1}\\right)\n&= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(\\left(z_{n_{j+l}}^{j+l} + \\psi\\right) - z_{n_{j-1}}^{j-1}\\right) - dq\n \\\\ \\nonumber\n&= \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l}}^{j+l} - z_{n_{j-1}}^{j-1} \\right) - dq + \\psi\n \\\\ \\nonumber\n&= \\left\\langle \\left(\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor} - 1\\right) dq + \\psi, 2\\psi - dq \\right\\rangle,\n\\end{align}\nwhich has no intersection with $\\left\\langle 0, \\psi \\right\\rangle$.\nThus $z_0^{j+l+1}$ is not R$\\psi$-mobile in $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$,\nand so $z_{n_{j+l}}^{j+l}$ is non-neutralized\nL$\\psi$-mobile in $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$.\n\n\nSince $z_0^{j+l}$ and $z_{n_{j+l}}^{j+l}$ are non-neutralized\n$\\psi$-mobile in $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$,\nProposition \\ref{prop: positive type, main prop}.(ii$\\psi$) tells us that\n$z_0^{j+l}$ and $z_{n_{j+l}}^{j+l}$ must be mirror $\\psi$-mobile\nin $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$, or in other words,\n$z_{n_{j+l}}^{j+l} = z_{n_{j-1-l}}^{j-1-l}$.\nThus $j+l = j-1-l \\in {{\\mathbb Z}}\/d$, and so\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, eq: l = (d-1)\/2}\nl = \\textstyle{\\frac{d-1}{2}}\\;\\in{{\\mathbb Z}}\/d.\n\\end{equation}\nNote that this requires that $2 \\!\\! \\not\\vert \\, d$.\n\n\nOn the other hand, we can also show that\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, eq: le = (d+alpha)\/2}\n[l\\epsilon]_d = \\textstyle{\\frac{d+\\alpha}{2}}.\n\\end{equation}\nBy Propositions \\ref{prop: unique v_q = alpha(k - k^2), and the rest are v_q = alpha (k)}\nand \\ref{prop: positive type, main prop}.(i$\\psi$),\nwe know that there is a unique $j_* \\in {{\\mathbb Z}}\/d$ such that\n$v_q\\!\\left(z_{n_{j_*-1}}^{j_*-1}, z_0^{j_*} \\right) = \\alpha(k-k^2)$ and\n$v_q\\!\\left( z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}, z_0^{j^{\\prime}} \\right) = \\alpha(k)$\nfor all $j^{\\prime}\\neq j_* \\in {{\\mathbb Z}}\/k^2$.\nThus, if we define $\\chi_R(j^{\\prime})$ (respectively \n$\\chi_L(j^{\\prime})$) to be equal to 1 if $z_0^{j+l}$\n(respectively $z_{n_{j-1-l}}^{j-1-l}$) is active in \n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$ at time $j = j^{\\prime}$,\nand equal to 0 otherwise, then for any $j^{\\prime} \\in {{\\mathbb Z}}\/d$,\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, eq: how many psis are active?}\n \\chi_R(j^{\\prime}) + \\chi_L(j^{\\prime})\n= \\begin{cases}\n 1 & j^{\\prime} \\neq j_*\n \\\\\n 2 & j^{\\prime} = j_*, \\;\\alpha = +1\n \\\\\n 0 & j^{\\prime} = j_*,\\;\\alpha = -1\n \\end{cases}.\n\\end{equation}\nThis is because a $\\psi$-mobile point in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j \\right]$ contributes\n$-k^2$ to $v_q\\!\\left( z_{n_{j^{\\prime}-1}}^{j^{\\prime}-1}, z_0^{j^{\\prime}}\\right)$\nif it is active at time $j=j^{\\prime}$ and contributes zero otherwise.\nLine (\\ref{prop: type (1,-1), c=1, eq: how many psis are active?})\nthen implies that\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, eq: total actives is d+alpha}\n\\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_R(j) \\;+\\; \\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_L(j) = d+\\alpha,\n\\end{equation}\nand Proposition \\ref{prop: pseudomobile point is active [lepsilon] times}\ntells us that\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, eq: each is active le times}\n\\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_R(j) = \\sum_{j\\in {{\\mathbb Z}}\/d} \\!\\chi_L(j) = [l\\epsilon]_d.\n\\end{equation}\nLines\n(\\ref{prop: type (1,-1), c=1, eq: total actives is d+alpha}) and\n(\\ref{prop: type (1,-1), c=1, eq: each is active le times}) then tell us that\n(\\ref{prop: type (1,-1), c=1, eq: le = (d+alpha)\/2}) holds, and so\nlines \n(\\ref{prop: type (1,-1), c=1, eq: l = (d-1)\/2}) and\n(\\ref{prop: type (1,-1), c=1, eq: le = (d+alpha)\/2})\ntell us that \n\\begin{equation}\n\\epsilon = [-\\alpha]_d.\n\\end{equation}\nThis, in turn, allows us to compute $c$.\n\\begin{align}\n c\n&= \\left[\\alpha \\gamma \\, {\\epsilon}^{-1}\\right]_d\n \\\\ \\nonumber\n&= \\left[-\\gamma\\right]_d\n \\\\ \\nonumber\n&= \\begin{cases}\n 1 & \\gamma = -1\n \\\\\n d-1 & \\gamma = +1\n \\end{cases}.\n\\end{align}\nNow, Proposition \\ref{prop: properties of parameters d, m, c, alpha, mu, gamma}\ntells us that $c \\leq \\frac{d}{2}$. Thus, if $\\gamma = +1$, then\n$d-1 \\leq \\frac{d}{2}$, implying $d \\leq 2$, but the fact that \n$d > 1$ and $2\\!\\!\\not\\vert\\, d$ implies that $d \\geq 3$.\nThus $\\gamma = -1$, and so $(\\mu,\\gamma) = (1,-1)$\nand $c=1$.\n\n\n\n\nWe have now proven everything we wanted to prove except that\nall $\\psi$-mobile points are non-neutralized and that $m=2$,\nthe latter of which is now equivalent to showing that $[dq]_{k^2} = k+ \\alpha$.\nToward these ends, we claim that\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, eq: z_0^j+l_0 not in }\nz_0^{j^{\\prime}+l_0} \\notin \n\\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right\\rangle\\;\n\\text{for all}\\; l_0, j^{\\prime} \\in {{\\mathbb Z}}\/d.\n\\end{equation}\nSuppose not, so that there exist $l_0, j_0 \\in {{\\mathbb Z}}\/d$\nfor which $z_0^{j_0+l_0} \\in \n\\left\\langle z_0^{j_0}, z_1^{j_0} \\right\\rangle$.\nThen the fact that\n$z_0^{j^{\\prime}} \\notin \n\\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right\\rangle$\nimplies $l_0 \\neq 0$.\nMoreover, $l+l_0 \\neq 0$, because otherwise,\nsetting $j_1 = j_0-l$, we would have\n$z_0^{j_1} = z_0^{j_1 + l + l_0} \\in \n\\left\\langle z_0^{j_1 +l}, z_1^{j_1 + l} \\right\\rangle$,\ncontradicting (\\ref{prop: type (1,-1), c=1, eq: all of z^{j+l} fits in < z_0^j, z_n>}).\nWe also know that \n$\\minq_{j\\in {{\\mathbb Z}}\/d}\\left(z_0^{j+l_0}-z_0^j\\right) \\notin \\left\\langle 0, dq\\right\\rangle$,\nbecause otherwise $z_0^{j+l_0}$ would be R-mobile in\n$\\left\\langle z_0^j, z_1^j\\right]$.\nThus $\\minq_{j\\in {{\\mathbb Z}}\/d}\\left(z_0^{j+l_0}-z_0^j\\right) \\in \\left\\langle -dq, 0\\right\\rangle$,\nand so $\\minq_{j\\in {{\\mathbb Z}}\/d}\\left(z_0^{j+l+l_0}-z_0^{j+l}\\right) \\in \\left\\langle -dq, 0\\right\\rangle$.\nOn the other hand, because of (\\ref{prop: type (1,-1), c=1, eq: all of z^{j+l} fits in < z_0^j, z_n>}),\nwe know that\n\\begin{equation}\n0 < \\minq_{j\\in {{\\mathbb Z}}\/d}\\left(z_0^{j+l}-z_{n_{j-1}}^{j-1} \\right)\n< \\psi -\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor} [dq]_{k^2}.\n\\end{equation}\nThus, for some $x \\in \\{0, dq\\}$, we have\n\\begin{align}\n \\minq_{j\\in {{\\mathbb Z}}\/d}\\left(z_0^{j+l+l_0}-z_{n_{j-1}}^{j-1} \\right)\n&= \\minq_{j\\in {{\\mathbb Z}}\/d}\\left(\\left(z_0^{j+l+l_0}-z_0^{j+l}\\right) +\n \\left(z_0^{j+l}- z_{n_{j-1}}^{j-1}\\right) \\right)\n \\\\ \\nonumber\n&= \\minq_{j\\in {{\\mathbb Z}}\/d}\\left(z_0^{j+l+l_0}-z_0^{j+l}\\right) +\n \\minq_{j\\in {{\\mathbb Z}}\/d}\\left(z_0^{j+l}- z_{n_{j-1}}^{j-1}\\right) + x\n \\\\ \\nonumber\n&\\in \\left\\langle -dq + x,\\, \\psi -\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor} dq + x \\right\\rangle\n \\\\ \\nonumber\n&\\subset \\left\\langle -dq,\\, \n \\psi -\\left(\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}\\!-\\!1\\right)\\!dq \\right\\rangle.\n\\end{align}\nIf\n$\\minq_{j\\in {{\\mathbb Z}}\/d}\\left(z_0^{j+l+l_0}-z_{n_{j-1}}^{j-1}\\right) \\in \\left\\langle -dq, 0\\right\\rangle$,\nthen \n$\\minq_{j\\in {{\\mathbb Z}}\/d}\\left(z_0^{j+l+l_0}-z_{n_{j-1}-1}^{j-1}\\right) \\in \\left\\langle 0, dq\\right\\rangle$,\nso that $z_0^{j+l+l_0}$ is R-mobile in\n$\\left\\langle z_{n_{j-1}-1}^{j-1}, z_{n_{j-1}}^{j-1} \\right]$, a contradiction.\nThus instead, we have $\\minq_{j\\in {{\\mathbb Z}}\/d}\\left(z_0^{j+l+l_0}-z_{n_{j-1}}^{j-1}\\right) \n\\in \\left\\langle 0,\\, \\psi -\\left(\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}\\!-\\!1\\right)\\!dq \\right\\rangle$,\nso that $z_0^{j+l+l_0}$ is R-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$.\nWe then have\n\\begin{align}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l+l_0-1}}^{j+l+l_0-1} - z_0^j\\right)\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(\\left(z_0^{j+l+l_0}-\\psi\\right) - \\left(z_{n_{j-1}}^{j-1}+\\psi\\right)\\right) + dq\n \\\\ \\nonumber\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l+l_0} - z_{n_{j-1}}^{j-1}\\right) + dq -2\\psi\n \\\\ \\nonumber\n&= \\left\\langle -2\\psi + dq,\\, \n -\\psi -\\left(\\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor}\\!-\\!2\\right)\\!dq \\right\\rangle,\n\\end{align}\nwhich, since $\\left\\lfloor\\frac{k}{d}\\right\\rfloor\\!-\\!2 \\geq 0$ and $\\psi, [dq]_{k^2} < \\frac{k^2}{2}$,\nhas no intersection with $\\left\\langle -\\psi, 0\\right\\rangle$.\nThus $z_{n_{j+l+l_0-1}}^{j+l+l_0-1}$ is not L$\\psi$-mobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$, and so\n$z_0^{j+l+l_0}$ is non-neutralized R-pseudomobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$,\ncontradicting the uniqueness of $z_0^{j+l}$ as a non-neutralized\nR$\\psi$-mobile point in $\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$.\nThus (\\ref{prop: type (1,-1), c=1, eq: z_0^j+l_0 not in })\nmust be true.\n\n\n\n\nWe next claim that (\\ref{prop: type (1,-1), c=1, eq: z_0^j+l_0 not in })\nimplies that all pseudomobile points are non-neutralized.\nSuppose that for some nonzero $l_0 \\in {{\\mathbb Z}}\/d$,\nwe know that $z_0^{j+l_0}$ is R$\\psi$-mobile in \n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$.\nIf $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l_0}-z_{n_{j-1}}^{j-1}\\right)\n\\in \\left\\langle \\psi - 2dq, \\psi\\right\\rangle$, then there exists\n$j^{\\prime} \\in {{\\mathbb Z}}\/d$ for which\n$z_0^{j^{\\prime}+ l_0} \\in\n\\left\\langle z_0^{j^{\\prime}} - dq, z_0^{j^{\\prime}} \\right\\rangle$, implying that\n$z_0^{j^{\\prime}} \\in \\left\\langle z_0^{j^{\\prime}+l_0}, z_1^{j^{\\prime}+l_0} \\right\\rangle$,\nbut this contradicts (\\ref{prop: type (1,-1), c=1, eq: z_0^j+l_0 not in }).\nThus $\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l_0}-z_{n_{j-1}}^{j-1}\\right)\n\\in \\left\\langle 0, \\psi - 2dq \\right\\rangle$, and so\n\\begin{align}\n \\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j+l_0-1}}^{j+l_0-1}-z_0^j\\right)\n&= \\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_0^{j+l_0}-z_{n_{j-1}}^{j-1}\\right) + dq - 2\\psi\n \\\\ \\nonumber\n&\\in \\left\\langle -2\\psi + dq, -\\psi - dq \\right\\rangle,\n\\end{align}\nwhich, since $\\psi, [dq]_{k^2}< \\frac{k^2}{2}$, has no intersection with\n$\\left\\langle -\\psi, 0\\right\\rangle$.\nThus $z_{n_{j+l_0-1}}^{j+l_0-1}$ is not L$\\psi$-mobile in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$,\nand so $z_0^{j+l_0}$ is non-neutralized R$\\psi$-mobile in \n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$.\nThe fact that there are no neutralized R$\\psi$-mobile points in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$ implies that there are no\nneutralized L$\\psi$-mobile points in\n$\\left\\langle z_{n_{j-1}}^{j-1}, z_0^j\\right]$.\nThus all pseudomobile points are non-neutralized.\n\n\n\n\n\n\nLastly, we claim that $[dq]_{k^2} = k+ \\alpha$.\nTo prove this, we shall show that\n$Q_q \\cap \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right\\rangle = \\emptyset$\nfor all $j^{\\prime} \\in{{\\mathbb Z}}\/d$.\nFirst, since\n(\\ref{prop: type (1,-1), c=1, k\/d dq < psi}) implies both that\n$\\left\\lfloor \\frac{k}{d} \\right\\rfloor [dq]_{k^2} < \\psi$\nand that $\\left\\lfloor \\frac{k}{d} \\right\\rfloor [dq]_{k^2} + \\psi < k^2$,\nand since\n(\\ref{prop: type (1,-1), c=1, eq: z_0^j+l_0 not in })\ntells us in particular that\n$z_0^{j^{\\prime}-1} \\notin \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right]$,\nwe deduce that\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, eq: l=0 and l=1 not in}\nz_i^{j^{\\prime}-l_0}\n\\notin \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right\\rangle\\;\n\\text{for all}\\; l_0 \\in \\{0, 1\\},\\;\nj^{\\prime} \\in {{\\mathbb Z}}\/d,\\;\ni \\in \\left\\{0, \\ldots, n_{j^{\\prime}}\\right\\}\\!.\n\\end{equation}\nNext, note that (\\ref{prop: type (1,-1), c=1, eq: z_0^j+l_0 not in }),\nalong with the mirror relation (\\ref{eq: mirror relation 1}),\nimplies that\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, eq: z_nj-l notin z_nj}\nz_{n_{j^{\\prime}-l_0}}^{j^{\\prime}-l_0} \\notin\n\\left\\langle z_{n_{j^{\\prime}}-1}^{j^{\\prime}}, z_{n_{j^{\\prime}}}^{j^{\\prime}}\\right\\rangle\\;\n\\text{for all}\\;l_0, j^{\\prime}\\in{{\\mathbb Z}}\/d,\n\\end{equation}\nwhich is useful for showing that\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, eq: z_n_j - (k\/d -1) not in}\nz_{n_{j^{\\prime}-l_0} - \\left(\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - 1\\right)}^{j^{\\prime}-l_0}\n\\notin \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right\\rangle\\;\n\\text{for all}\\; l_0, j^{\\prime} \\in {{\\mathbb Z}}\/d.\n\\end{equation}\nSuppose\n(\\ref{prop: type (1,-1), c=1, eq: z_n_j - (k\/d -1) not in}) fails\nfor some $l_0 \\in {{\\mathbb Z}}\/d$, $l_0 \\notin \\{0,1\\}$. Then\n$\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l_0}-\\left(\\left\\lfloor\\frac{k}{d}\\right\\rfloor-1\\right)}^{j-l_0}-z_0^j\\right)\n\\notin \\left\\langle 0,dq\\right\\rangle$,\nsince otherwise\n(\\ref{prop: type (1,-1), c=1, eq: z_0^j+l_0 not in }) is contradicted,\nand so $\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l_0}-\\left(\\left\\lfloor\\frac{k}{d}\\right\\rfloor-1\\right)}^{j-l_0}-z_0^j\\right)\n\\in \\left\\langle 0,dq\\right\\rangle$, implying that\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l_0}}^{j-l_0}-z_{\\left\\lfloor\\frac{k}{d}\\right\\rfloor-1}^j\\right)\n\\in \\left\\langle 0,dq\\right\\rangle$. But this, in turn, implies that\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l_0}}^{j-l_0}-z_{n_j}^j\\right)\n\\in \\left\\langle -dq,dq\\right\\rangle$,\ncontradicting (\\ref{prop: type (1,-1), c=1, eq: z_nj-l notin z_nj}).\nThus \n(\\ref{prop: type (1,-1), c=1, eq: z_n_j - (k\/d -1) not in})\nmust be true.\nWe next claim that\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, eq: max n_j-i notin for i 0, ..., k\/d-2}\n\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l_0} - i}^{j-l_0}\n- z_1^j\\right) \\notin \\left\\langle -dq, 0\\right\\rangle\\;\n\\text{for all}\\;\ni \\in \\left\\{0, \\ldots, \\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor} - 2\\right\\}\\!,\\;\nl_0 \\neq 1 \\in {{\\mathbb Z}}\/d.\n\\end{equation}\nSuppose there exist\n$i \\in \\left\\{0, \\ldots, \\left\\lfloor\\frac{k}{d}\\right\\rfloor - 2\\right\\}$ and $l_0 \\notin \\{0,1\\} \\in {{\\mathbb Z}}\/d$\nfor which (\\ref{prop: type (1,-1), c=1, eq: max n_j-i notin for i 0, ..., k\/d-2})\ndoes not hold. Then, since \n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l_0}}^{j-l_0}\n- z_{i+1}^j\\right) \\in \\left\\langle -dq, 0\\right\\rangle$ and \n$n_j \\in \\left\\{\\left\\lfloor\\frac{k}{d}\\right\\rfloor -1, \\left\\lfloor\\frac{k}{d}\\right\\rfloor \\right\\}$\nfor all $j \\in {{\\mathbb Z}}\/d$, we know that either\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l_0}}^{j-l_0}\n- z_{n_j - \\left(\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - (i+2)\\right)}^j\\right) \n\\in \\left\\langle -dq, 0\\right\\rangle$, so that $z_{n_{j-l_0}}^{j-l_0}$\nis L-mobile in\n$\\left\\langle z_{n_j - \\left(\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - (i+1)\\right)}^j,\nz_{n_j - \\left(\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - (i+2)\\right)}^j \\right]$, or\n$\\maxq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l_0}}^{j-l_0}\n- z_{n_j - \\left(\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - (i+1)\\right)}^j\\right) \n\\in \\left\\langle -dq, 0\\right\\rangle$, so that $z_{n_{j-l_0}}^{j-l_0}$\nis L-mobile in\n$\\left\\langle z_{n_j - \\left(\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - i\\right)}^j,\nz_{n_j - \\left(\\left\\lfloor\\!\\frac{k}{d}\\!\\right\\rfloor - (i+1)\\right)}^j \\right]$,\neither of which is a contradiction.\nThus (\\ref{prop: type (1,-1), c=1, eq: max n_j-i notin for i 0, ..., k\/d-2})\nmust be true. This, combined with (\\ref{prop: type (1,-1), c=1, eq: z_n_j - (k\/d -1) not in}),\nthen implies that\n\\begin{equation}\n\\minq_{j\\in{{\\mathbb Z}}\/d}\\left(z_{n_{j-l_0} - i}^{j-l_0}\n- z_1^j\\right) \\notin \\left\\langle -dq, 0\\right\\rangle\\;\n\\text{for all}\\;\ni \\in \\left\\{0, \\ldots, \\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor} - 2\\right\\}\\!,\\;\nl_0 \\neq 1 \\in {{\\mathbb Z}}\/d,\n\\end{equation}\nand so, taking (\\ref{prop: type (1,-1), c=1, eq: l=0 and l=1 not in})\ninto account, we now know that\n\\begin{equation}\n\\label{prop: type (1,-1), c=1, eq: z_n_j-i notin for i 0, ..., k\/d-2}\nz_{n_{j^{\\prime}-l_0} - i}^{j^{\\prime}-l_0}\n\\notin \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right\\rangle\\;\n\\text{for all}\\;\ni \\in \\left\\{0, \\ldots, \\textstyle{\\left\\lfloor\\frac{k}{d}\\right\\rfloor} - 2\\right\\}\\!,\\;\nl_0 , j^{\\prime} \\in {{\\mathbb Z}}\/d.\n\\end{equation}\nThus, the combination of\n(\\ref{prop: type (1,-1), c=1, eq: z_0^j+l_0 not in }),\n(\\ref{prop: type (1,-1), c=1, eq: z_n_j - (k\/d -1) not in}), and\n(\\ref{prop: type (1,-1), c=1, eq: z_n_j-i notin for i 0, ..., k\/d-2})\ntells us that\n\\begin{equation}\nz_i^{j^{\\prime}+l_0}\n\\notin \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right\\rangle\\;\n\\text{for all}\\;\nl_0 , j^{\\prime} \\in {{\\mathbb Z}}\/d,\\;\ni \\in \\left\\{0, \\ldots, n_{j^{\\prime}} \\right\\},\n\\end{equation}\nor in other words,\n\\begin{equation}\nQ_q \\cap \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right\\rangle = \\emptyset\\;\n\\text{for all}\\;\nj^{\\prime} \\in {{\\mathbb Z}}\/d.\n\\end{equation}\n\n\n\nThus\n$Q_q \\cap \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right] = z_1^{j^{\\prime}}$\nfor any $j^{\\prime} \\in {{\\mathbb Z}}\/d$,\nand so we know in particular that\n\\begin{equation}\n\\#\\left( {\\tilde{Q}}_q \\cap \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right] \\right) = 1.\n\\end{equation}\nMoreover, since $v_q\\!\\left(z_{n_{j-1}}^{j-1}, z_0^j\\right)$ is not constant in $j\\in{{\\mathbb Z}}\/d$,\nwe know that $v_q\\!\\left(z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right) = \\alpha(k)$\nfor any $j^{\\prime} \\in {{\\mathbb Z}}\/d$.\nThus, for any $j^{\\prime} \\in {{\\mathbb Z}}\/d$, we have\n\\begin{align}\n \\nonumber\n v_q\\!\\left(z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right)\n&= \\alpha(k)\n \\\\ \\nonumber\n \\left[z_1^{j^{\\prime}} - z_0^{j^{\\prime}}\\right]_{k^2}\\!\\! k\n \\;-\\; \\#\\left( {\\tilde{Q}}_q \\cap \\left\\langle z_0^{j^{\\prime}}, z_1^{j^{\\prime}} \\right] \\right) k^2\n&= \\alpha(k)\n \\\\ \\nonumber\n [dq]_{k^2}k - (1)k^2\n&= \\alpha(k)\n \\\\\n [dq]_{k^2} \n&= k + \\alpha.\n\\end{align}\n\n\n\\end{proof}\n\n\nIn addition to concluding our study of the properties of\ngenus-minimizing $q$ of positive type,\nProposition \\ref{prop: no mobile points, psi < k^2\/2, implies type (1,-1), m=2, c=1, d odd}\nalso completes our survey of the ``non-neutralizedness''\nof (anti)(pseudo)mobile points, with the interesting result that all\n(anti)(pseudo)mobile points are as non-neutralized as possible.\nThat is, Proposition \\ref{prop: positive type, main prop}.(iv)\ntells us that all mobile points are non-neutralized,\nProposition \\ref{prop: psibar < k^2\/2 implies psibar-mobile are nn}\nstates that all antipseudomobile points are non-neutralized\nwhen ${{\\overline{\\psi}}} < \\frac{k^2}{2}$, and \nProposition \\ref{prop: no mobile points, psi < k^2\/2, implies type (1,-1), m=2, c=1, d odd}\ntells us that all pseudomobile points are non-neutralized when\n$2[dq]_{k^2}< \\psi < \\frac{k^2}{2}$. (It is easy to check that in\nthe exceptional case in which $\\psi < 2[dq]_{k^2}$, no pseudomobile points are present.)\nSince it is algebraically impossible\nfor a pseudomobile (respectively antipseudomobile) point to be non-neutralized\nwhen $\\psi < \\frac{k^2}{2}$ (respectively ${{\\overline{\\psi}}} < \\frac{k^2}{2}$),\nthis is the most non-neutralizedness that could have occurred.\nThe classification of genus-minimizing $q$ does not make use of\nthis observation, but it seems an interesting observation nonetheless.\n\n\n\n\\subsection{Classification of Genus-Minimizing Solutions for $q$}\n\n\nWe have now done all the work necessary to\nsay what the genus-minimizing solutions for $q$ are.\nIt is mainly a matter of bookkeeping to collect them all.\n\n\n\\begin{prop}\n\\label{prop: bookkeeping for q genus-minimizing}\nSuppose that $k$ is an integer $\\geq 2$, and that\n$q \\in {{\\mathbb Z}}\/k^2$ is primitive.\nThen the triple $(p=k^2, q, k)$ is genus-minimizing if and only if\n$q \\in {{\\mathbb Z}}\/k^2$ can be expressed in one or more of the following forms,\n\\begin{align*}\n\\text{0.}\\;\\;\n&\n\\begin{cases}\n q = ik \\pm 1, \\;\n & \\gcd (i,k) \\in \\{1, 2\\}\n\\end{cases}\n \\\\\n\\text{1.}\\;\\;\n&\n\\begin{cases}\n q = \\pm \\textstyle{\\frac{k+1}{d}}(k+1), \\; \n & 2\\!\\! \\not\\vert\\, \\textstyle{\\frac{k+1}{d}}\n \\\\\n q = \\pm \\textstyle{\\frac{k-1}{d}}(k-1), \\; \n & 2\\!\\! \\not\\vert\\, \\textstyle{\\frac{k-1}{d}}\n\\end{cases}\n \\\\\n\\text{2.}\\;\\;\n&\n\\begin{cases}\n q = \\pm \\textstyle{\\frac{k-1}{d}}(2k+1), \\; \n & 2\\!\\! \\not\\vert\\, d\n \\\\\n q = \\pm \\textstyle{\\frac{k+1}{d}}(2k-1), \\; \n & 2\\!\\! \\not\\vert\\, d\n\\end{cases}\n \\\\\n\\text{3.}\\;\\;\n&\n\\begin{cases}\n q = \\pm \\textstyle{\\frac{2k+1}{d}}(k-1), \\;\n & 2\\!\\! \\not\\vert\\, d\n \\\\\n q = \\pm \\textstyle{\\frac{2k-1}{d}}(k+1), \\;\n & 2\\!\\! \\not\\vert\\, d\n\\end{cases}\n\\end{align*}\nfor any positive integer $d$, where all fractions shown represent integers.\nIn case `3', the redundant condition $2 \\!\\! \\not\\vert \\, d$ is listed for\naesthetic reasons.\n\\end{prop}\n\n\\begin{proof}\nFor $k \\leq 100$, it is easy to check the proposition by computer.\n\nWe therefore take $k >100$ for the remainder of the proof.\nFor such $k$, Definition \\ref{def: parameters assigned to q}\nparameterizes all primitive $q$ in ${{\\mathbb Z}}\/k^2$, so that\n\\begin{equation}\n\\xi q = \\alpha\\gamma\\mu \\textstyle{\\frac{ck+\\alpha\\gamma}{d}}(mk + \\alpha\\mu).\n\\end{equation}\nIf $q \\equiv \\pm 1\\; (\\mod k)$, or equivalently, if\nthe parameter $c$ satisfies $c=0$,\nwe say that $q$ is of type 0.\nThe sign $\\xi \\in \\{\\pm1\\}$ is chosen in such a way as to make\n$\\xi q$ satisfy $[d\\xi q]_{k^2} < \\frac{k^2}{2}$.\nWhen $q$ is not of type 0, we say that\n$q$ is of positive type if $q = +\\xi q$,\nand that $q$ is of negative type if $q = -\\xi q$.\nIn order to avoid carrying around an extra $\\xi$ everywhere,\nwe restricted\nSections \\ref{ss: Notation and Definitions for q of Positive Type}\nand \\ref{ss: Properties of z^j for Genus-Minimizing q of Positive Type}\nto the case in which $q$ is of positive type.\nHowever, it is easy to see that the definitions and results of those\nsections also hold for $\\xi q$, for $q$ of negative type.\n\n\n\n\nProposition \\ref{prop: type 0 genus-minimizing classification}\nshows that if $q$ is of type 0, then\n$q$ is genus-minimizing if and only if\n$q$ is of the form shown in ``0'' above.\nThis leaves us with the case in which\n$q$ is of positive or negative type,\nor, for brevity, of {\\em nonzero type}.\nFor the reader's convenience, we pause to restate the\npropositions we shall use to classify genus-minimizing\n$q$ of nonzero type. Propositions\n\\ref{prop: q of positive type. If mobile points exist, then l=1},\n\\ref{prop: type (1,1), k\/d = 2 or 3}, and \n\\ref{prop: mobile points implies mu m + gamma c = 2}\ndeal only with cases in which ${\\mathbf{z}}^j$ has mobile points,\nwhereas Propositions\n\\ref{prop: no mobile points, psibar < k^2\/2, implies type (-1,1), m=1, c=2, d odd} and\n\\ref{prop: no mobile points, psi < k^2\/2, implies type (1,-1), m=2, c=1, d odd}\ndeal only with cases in which mobile points are not present.\n\\begin{itemize}\n\n\\item[\\ref{prop: positive type, main prop}.(iv)]\n{Suppose $q$ of positive type is genus-minimizing.\nThen all mobile points are non-neutralized.\nMoreover, $\\psi > 2\\left[dq\\right]_{k^2}$\nunless $(\\mu,\\gamma) = (1,1)$, $\\alpha = -1$, $m=2$, $c=1$,\nand $d= \\frac{k-1}{2} \\equiv 0\\; (\\mod 2)$.}\n\n\n\\item[\\ref{prop: q of positive type. If mobile points exist, then l=1}]\n{Suppose $q$ of positive type is genus-minimizing,\nand $\\psi > 2[dq]_{k^2}$.\nIf $z_0^{j+l}$, and hence $z_{n_{j-l}}^{j-l}$, are mobile in ${\\mathbf{z}}^j$\nfor some nonzero $l \\in {{\\mathbb Z}}\/d$, then ${{\\overline{\\psi}}} < \\frac{k^2}{2}$,\n$l=1$, $c=1$, and $2 \\!\\!\\not\\vert\\, \\frac{k+\\alpha}{d}$,\nand either $(\\mu,\\gamma)=(1,-1)$ with $d=2$ and $\\alpha=+1$,\nor $(\\mu,\\gamma)=(1,1)$.}\n\n\n\\item[\\ref{prop: type (1,1), k\/d = 2 or 3}]\n{Suppose $q$ of positive type is genus-minimizing\nand $\\psi > 2[dq]_{k^2}$.\nIf $(\\mu,\\gamma) = (1,1)$,\nand $\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\in \\{2,3\\}$, \nthen $m=c=1$.}\n\n\n\\item[\\ref{prop: mobile points implies mu m + gamma c = 2}]\n{Suppose $q$ of positive type is genus-minimizing.\nIf ${\\mathbf{z}}^j$ has mobile points and \n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor > 3$, then\n$[dq]_{k^2}=2k+\\alpha$, so that $\\mu m + \\gamma c = 2$.}\n\n\n\\item[\\ref{prop: no mobile points, psibar < k^2\/2, implies type (-1,1), m=1, c=2, d odd}]\n{Suppose $q$ of positive type is genus-minimizing.\nIf ${\\mathbf{z}}^j$ has no mobile points\nand ${{\\overline{\\psi}}} < \\frac{k^2}{2}$, then\n$(\\mu, \\gamma) = (-1,1)$, $m=1$, $c=2$, and $2\\!\\! \\not\\vert \\,d$.}\n\n\n\\item[\\ref{prop: no mobile points, psi < k^2\/2, implies type (1,-1), m=2, c=1, d odd}]\n{Suppose $q$ of positive type is genus-minimizing.\nIf $\\psi < \\frac{k^2}{2}$ and $\\psi > 2[dq]_{k^2}$, then\n${\\mathbf{z}}^j$ has no mobile points,\nall pseudomobile points are non-neutralized,\n$(\\mu, \\gamma) = (1,-1)$, $m=2$, $c=1$, and $2\\!\\! \\not\\vert \\,d$.}\n\\end{itemize}\n\n\n\nSuppose that $q$ of nonzero type is genus-minimizing---so that\n$\\xi q$ is of positive type and, by Proposition \\ref{prop:G properties},\nis also genus-minimizing---and consider the case in which\n${{\\overline{\\psi}}} < \\frac{k^2}{2}$ and \n${\\mathbf{z}}^j$ has mobile points.\nThen Proposition \\ref{prop: q of positive type. If mobile points exist, then l=1}\ntells us that $2 \\!\\!\\not\\vert\\, \\frac{k+\\alpha}{d}$, and\neither $(\\mu,\\gamma)=(1,1)$, or\n$(\\mu,\\gamma)=(1,-1)$ with $c=1$, $d=2$, and $\\alpha=+1$.\nIf $(\\mu,\\gamma) = (1,1)$, then either \n$\\left\\lfloor\\frac{k}{d}\\right\\rfloor \\leq 3$,\nso that Proposition \\ref{prop: type (1,1), k\/d = 2 or 3} tells us that\n$m=c=1$, or $\\left\\lfloor\\frac{k}{d}\\right\\rfloor > 3$, so that\nProposition \\ref{prop: mobile points implies mu m + gamma c = 2}\ntells us that $\\mu m + \\gamma c = m+c = 2$, implying $m=c=1$.\nThus, in either case, $(\\mu,\\gamma)=(1,1)$ implies\n$m=c=1$, so that\n$\\xi q = \\alpha \\frac{k + \\alpha}{d}(k + \\alpha)$, with\n$3 \\leq d \\leq \\frac{k+1}{3}$ when $\\alpha = +1$,\nand $2 \\leq d \\leq \\frac{k-1}{3}$ when $\\alpha = -1$.\n(Note that the fact that $2 \\!\\!\\not\\vert\\, \\frac{k+\\alpha}{d}$ implies that\n$d \\neq \\frac{k+\\alpha}{2}$.)\nOn the other hand, if $(\\mu,\\gamma)=(1,-1)$ with $c=1$, $d=2$, and $\\alpha=+1$,\nthen Proposition \\ref{prop: mobile points implies mu m + gamma c = 2}\ntells us that $\\mu m + \\gamma c = m - c = 2$, implying $m = 3$, so that\n$\\xi q = - \\frac{k - 1}{2}(3k + 1) = +\\frac{k+1}{2}(k+1)$.\nThus, if ${{\\overline{\\psi}}} < \\frac{k^2}{2}$ and ${\\mathbf{z}}^j$ has mobile points, then\n\\begin{equation}\nq = \\xi\\alpha \\textstyle{\\frac{k + \\alpha}{d}}(k + \\alpha),\\;\\;\n\\text{with}\\;\n\\textstyle{\\frac{k+\\alpha}{d}}\\in {{\\mathbb Z}},\\; 2 \\!\\!\\not\\vert\\, \\textstyle{\\frac{k+\\alpha}{d}},\\;\n2 \\leq d \\leq \\frac{k+\\alpha}{3}.\n\\end{equation}\nThese values of $q \\in {{\\mathbb Z}}\/k^2$ constitute a subset of the\nsolutions listed in ``1'' above. The complement of this subset consists of the\ncases in which $d \\in \\{1, k+\\alpha\\}$, which are simply the cases in which\nforms ``0'' and ``1'' intersect. We already classified them as\ngenus-minimizing solutions of type 0.\n\n\n\nNext, suppose that $q$ of nonzero type is genus-minimizing---so that\n$\\xi q$ is of positive type and is genus-minimizing---and consider the case in which\n$\\psi < \\frac{k^2}{2}$.\nProposition \\ref{prop: q of positive type. If mobile points exist, then l=1}\nthen tells us that ${\\mathbf{z}}^j$ has no mobile points unless\n$\\psi < 2[dq]_{k^2}$, which only occurs as the exceptional case of \nProposition \\ref{prop: positive type, main prop}.(iv), in which\n$(\\mu,\\gamma) = (1,1)$, $\\alpha = -1$, $m=2$, $c=1$,\nand $d= \\frac{k-1}{2} \\equiv 0\\; (\\mod 2)$.\nIn other words, \n$\\xi q = -\\frac{k-1}{\\left(\\frac{k-1}{2}\\right)}(2k-1) = -2(2k-1)$,\nwith $4 \\vert k-1$.\nFor reasons that will soon become clear, we choose to re-express this as\n$\\xi q = -\\frac{k-\\alpha}{\\left(\\frac{k-\\alpha}{2}\\right)}(2k+\\alpha)$, with $\\alpha=-1$ and\n$2 \\!\\! \\not\\vert \\frac{k-\\alpha}{2}$.\nThis leaves us with the case in which $2[dq]_{k^2}< \\psi < \\frac{k^2}{2}$,\nso that, by \nProposition \\ref{prop: no mobile points, psi < k^2\/2, implies type (1,-1), m=2, c=1, d odd},\n${\\mathbf{z}}^j$ has no mobile points,\n$(\\mu, \\gamma) = (1,-1)$, $m=2$, $c=1$, and $2\\!\\! \\not\\vert \\,d$.\nCombining this and the special case just mentioned yields\n\\begin{equation}\nq = -\\xi\\alpha \\textstyle{\\frac{k - \\alpha}{d}}(2k + \\alpha),\\;\\;\n\\text{with}\\;\n\\textstyle{\\frac{k-\\alpha}{d}}\\in {{\\mathbb Z}},\\; 2 \\!\\!\\not\\vert\\, d,\\;\n3 \\leq d \\leq \\frac{k-\\alpha}{2}.\n\\end{equation}\nThese values of $q \\in {{\\mathbb Z}}\/k^2$ constitute a subset of the\nsolutions listed in ``2'' above. The complement of this subset consists of the\ncases in which $d \\in \\{1, k-\\alpha\\}$, which are simply the cases in which\nforms ``0'' and ``2'' intersect. We already classified them as\ngenus-minimizing solutions of type 0.\n\n\nLastly, suppose that $q$ of nonzero type is genus-minimizing---so that\n$\\xi q$ is of positive type and is genus-minimizing---and\nconsider the only case that remains, namely, in which\n${{\\overline{\\psi}}} < \\frac{k^2}{2}$ and ${\\mathbf{z}}^j$ has no mobile points.\nProposition \\ref{prop: no mobile points, psibar < k^2\/2, implies type (-1,1), m=1, c=2, d odd}\nthen tells us that\n$(\\mu, \\gamma) = (-1,1)$, $m=1$, $c=2$, and $2\\!\\! \\not\\vert \\,d$, so that\n\\begin{equation}\nq = -\\xi\\alpha \\textstyle{\\frac{2k + \\alpha}{d}}(k - \\alpha),\\;\\;\n\\text{with}\\;\n\\textstyle{\\frac{2k+\\alpha}{d}}\\in {{\\mathbb Z}},\\; 2 \\!\\!\\not\\vert\\, d,\\;\n3 \\leq d \\leq \\frac{2k+\\alpha}{3}.\n\\end{equation}\nThese values of $q \\in {{\\mathbb Z}}\/k^2$ constitute a subset of the\nsolutions listed in ``3'' above. The complement of this subset consists of the\ncases in which $d \\in \\{1, 2k+\\alpha\\}$, which are simply the cases in which\nforms ``0'' and ``3'' intersect. We already classified them as\ngenus-minimizing solutions of type 0.\n \\\\\n\nWe have now shown that all genus-minimizing $q$ can be expressed in\none or more of forms ``0'', ``1'', ``2'', or ``3'', as shown above.\nIt remains to show that all such forms of $q$ are genus-minimizing.\nIt is straightforward but tedious to use the tools so far introduced to\nshow, using the original description for each $q$,\nthat each of the intervals of length $[dq]_{k^2}$ or of length\n$\\psi$ contains the correct number of elements of $q$.\nFortunately, we are not obligated to perform this task, because\nBerge has already provided us with a topological proof\n\\cite{Berge}\nthat all of the above forms of $q$ are genus-minimizing.\n\n\n\\end{proof}\n\n\n\n\nWe are almost done with this section, but it turns out that\nwe need to know the form of $q^{-1} \\in {{\\mathbb Z}}\/k^2$,\nrather than of $q \\in {{\\mathbb Z}}\/k^2$,\nfor genus-minimizing $q$.\n\n\n\n\\begin{prop}\n\\label{prop: classification of q^-1 for genus-minimizing q}\nSuppose that $k$ is an integer $\\geq 2$,\nand that $p \\in {{\\mathbb Z}}\/k^2$ is primitive.\nThe triple $(k^2, p^{-1}, k)$ is genus-minimizing if and only if\n$p \\in {{\\mathbb Z}}\/k^2$ can be expressed in one or more of the following forms,\n\\begin{align*}\n\\text{0.}\\;\\;\n&\n\\begin{cases}\n p = ik \\pm 1, \\;\n & \\gcd (i,k) \\in \\{1, 2\\}\n\\end{cases}\n \\\\\n\\text{1.}\\;\\;\n&\n\\begin{cases}\n p = \\pm d(2k+1), \\; \n & d\\vert k-1,\\; 2\\!\\! \\not\\vert\\, \\textstyle{\\frac{k-1}{d}}\n \\\\\n p = \\pm d(2k-1), \\; \n & d\\vert k+1,\\; 2\\!\\! \\not\\vert\\, \\textstyle{\\frac{k+1}{d}}\n\\end{cases}\n \\\\\n\\text{2.}\\;\\;\n&\n\\begin{cases}\n p = \\pm d(k+1), \\; \n & d \\vert k+1,\\; 2\\!\\! \\not\\vert\\, d\n \\\\\n p = \\pm d(k-1), \\; \n & d \\vert k-1,\\; 2\\!\\! \\not\\vert\\, d\n\\end{cases}\n \\\\\n\\text{3.}\\;\\;\n&\n\\begin{cases}\n p = \\pm d(k-1), \\;\n & d \\vert 2k+1,\\; 2\\!\\! \\not\\vert\\, \\textstyle{\\frac{2k+1}{d}}\n \\\\\n p = \\pm d(k+1), \\;\n & d \\vert 2k-1,\\; 2\\!\\! \\not\\vert\\, \\textstyle{\\frac{2k-1}{d}}\n\\end{cases}\n\\end{align*}\nfor any positive integer $d$ satisfying the above divisibility constraints.\nThe redundant oddness condition in case `3' is listed for\naesthetic reasons.\n\\end{prop}\n\n\\begin{proof}\nThe triple $(k^2, p^{-1},k)$ is genus-minimizing if and only if\n$p^{-1}=q \\in {{\\mathbb Z}}\/k^2$ (or equivalently, if and only if $p = q^{-1}$),\nfor one of the forms of $q$\nlisted in Proposition \\ref{prop: bookkeeping for q genus-minimizing}.\nIf $q = ik \\pm 1$, for some \n$i \\in {{\\mathbb Z}}\/k^2$ with $\\gcd(i,k) \\in \\{1,2\\}$,\nthen $q^{-1}=-ik \\pm 1 = (k-i)k \\pm 1$, with $\\gcd(k-i,k) \\in \\{1,2\\}$.\nIf, for some\n$m, c \\in \\{1,2\\}$, $\\alpha, \\gamma, \\mu, \\xi \\in \\{1,-1\\}$,\nand primitive $d \\in {{\\mathbb Z}}\/k^2$, we have\n\\begin{equation}\n\\xi dq = \\mu m k + \\gamma c k + \\alpha \\;\\in {{\\mathbb Z}}\/k^2,\n\\end{equation}\nthen\n\\begin{align}\n q^{-1}\n&= \\xi d \\left(\\xi d q\\right)^{-1}\n \\\\ \\nonumber\n&= -\\xi d ((\\mu m + \\gamma c)k - \\alpha).\n\\end{align}\nThese rules for inverting $q$ establish a bijection\nbetween the forms of $q$ listed in \nProposition \\ref{prop: bookkeeping for q genus-minimizing}\nand the correspondingly numbered forms of $p$ listed above.\n\\end{proof}\n\n\n\n\nThe observant reader might notice that the set of solutions listed in \nProposition \\ref{prop: classification of q^-1 for genus-minimizing q}\ncoincides with the set of solutions listed in\nProposition \\ref{prop: bookkeeping for q genus-minimizing}.\nThis is because\nthe set of genus-minimizing solutions for $q$ is closed under the\noperation of taking inverses in ${{\\mathbb Z}}\/k^2$.\n\n\n\n\n\n\n\n\n\\section{Case $q \\equiv k^{-2}(\\mod p)$}\n\\label{s:q=k-2}\n\nWe have now classified all \ngenus-minimizing triples of the form $(k^2, p^{-1}, k)$,\nbut in order to determine all simple knots in lens spaces\nthat have L-space homology sphere surgeries,\nwe need to classify all genus-minimizing triples of the form\n$(p, k^{-2}, k)$. The following proposition tells us\nthat these two classifications coincide when $p > k^2$.\n\n\n\\begin{prop}\n\\label{prop:section q=k^-2, (p,k^-2,k) is like (k^2, p^-1, k)}\nIf $p > k^2$, then the triple $(p, k^{-2}, k)$ is genus-minimizing if and only if\nthe triple $(k^2, p^{-1}, k)$ is genus-minimizing.\n\\end{prop}\n\\begin{proof}\nBy Proposition \\ref{prop:G properties}, we know that\n\\begin{equation}\n \\bar{G}\\!\\left(k^2, p^{-1}, k\\right)\n= \\bar{G}\\!\\left(k^2, p, p^{-1}k\\right)\n= \\bar{G}\\!\\left(k^2, -p, -p^{-1}k\\right).\n\\end{equation}\nLikewise, Proposition \\ref{prop:G properties} tells us that\n\\begin{equation}\n \\bar{G}\\!\\left(p, k^{-2}, k\\right)\n= \\bar{G}\\!\\left(p, k^2, k^{-2}\\cdot k\\right)\n= \\bar{G}\\!\\left(p, k^2, k^{-1}\\right).\n\\end{equation}\nThus, it is sufficient to show that\nthe triple $\\left(k^2, -p, -p^{-1}k\\right)$ is genus-minimizing\nif and only if the triple\n$\\left(p, k^2, k^{-1}\\right)$ is genus-minimizing.\n\n\nFor brevity, let $A$ and $B$ denote the triples\n\\begin{align}\n\\label{eq: section q=k^-2, def of triple A}\nA&:= \\left(k^2, -p, -p^{-1}k\\right) = \\left(k^2, \\epsilon, nk\\right),\n \\\\\n\\label{eq: section q=k^-2, def of triple B}\nB&:= \\left(p, k^2, k^{-1}\\right) = \\left(p, k^2, \\frac{np+1}{k}\\right),\n\\end{align}\nwhere $\\epsilon := \\left[-p\\right]_{k^2}$ and $n := \\left[-p^{-1}\\right]_k$.\nNote that the $n$ in\n(\\ref{eq: section q=k^-2, def of triple A})\nis the same as the $n$ in\n(\\ref{eq: section q=k^-2, def of triple B}),\nsince $\\left[k^{-1}\\right]_p = \\frac{\\left[\\!-\\!p^{-1}\\!\\right]_{\\!k} p \\;+ 1}{k}$.\n\n\nFurthermore, define\n\\begin{equation}\n{\\bar{v}}_A := \\frac{1}{k}v_A\n\\;\\;\\;\\;\\mathrm{and}\\;\\;\\;\\;\n{\\bar{v}}_B := \\frac{k^2}{p}\\cdot \\frac{1}{k}v_B,\n\\end{equation}\nso that for any $x, y \\in {{\\mathbb Z}}$ with $x \\leq y$, we have\n\\begin{align}\n {\\bar{v}}_A(x,y) \n&:= \\#\\left({{\\mathbb Z}}\\cap \\left\\langle x, y\\right]\\right) n\n \\;\\; - \\;\\;\n \\#\\left({\\tilde{Q}}_A \\cap \\left\\langle x, y\\right]\\right) k,\n \\\\\n {\\bar{v}}_B(x,y) \n&:= \\#\\left({{\\mathbb Z}}\\cap \\left\\langle x, y\\right]\\right) \\left(n + \n \\textstyle{\\frac{1}{p}}\\right)\n \\;\\; - \\;\\;\n \\#\\left({\\tilde{Q}}_B \\cap \\left\\langle x, y\\right]\\right) k,\n\\end{align}\nwhere\n\\begin{align}\nQ_A &:= \\left\\{0\\epsilon, \\ldots, (nk-1)\\epsilon\\right\\} \\subset {{\\mathbb Z}}\/k^2,\n \\\\ \nQ_B &:= \\left\\{0k^2, \\ldots, \\left(\\frac{np+1}{k}-1\\right)\\!k^2\\right\\}\n \\subset {{\\mathbb Z}}\/p,\n\\end{align}\nand ${\\tilde{Q}}_A := {\\pi}_A^{-1}(Q_A)$ and \n${\\tilde{Q}}_B := {\\pi}_B^{-1}(Q_B)$ are the preimages of\n$Q_A$ and $Q_B$ under the respective quotient maps\n${{\\mathbb Z}} \\stackrel{{\\pi}_A}{\\rightarrow} {{\\mathbb Z}}\/k^2$ and\n${{\\mathbb Z}} \\stackrel{{\\pi}_B}{\\rightarrow} {{\\mathbb Z}}\/p$.\nWhen $x > y$, one could take ${\\bar{v}}_A$ and ${\\bar{v}}_B$\nto be defined by the identities \n${\\bar{v}}_A(x,y) = -{\\bar{v}}_A(y,x)$ and\n${\\bar{v}}_B(x,y) = -{\\bar{v}}_B(y,x)$.\n\nRecall that according to Corollary \n\\ref{cor: intro defs, genus-minimizing if v < p + k},\nan arbitrary triple $(p_0, q_0, k_0)$ is genus-minimizing\nif and only if\n\\begin{equation}\nv_{(p_0, q_0, k_0)}(x,y) < p_0 + k_0\\;\\;\\;\\;\n\\text{for all}\\;x,y \\in Q_0,\n\\end{equation}\nwhere $Q_0 := \\{0q_0, \\ldots, (k_0-1)q_0)\\} \\subset {{\\mathbb Z}}\/{p_0}$.\nEquivalently, this condition can be phrased in terms of\nlifts of $x$ and $y$ to ${{\\mathbb Z}}$. That is, $(p_0, q_0, k_0)$\nis genus-minimizing if and only if\n$v_{(p_0, q_0, k_0)}(x,y) < p_0 + k_0$\nfor all $x, y \\in {\\tilde{Q}}_0$, where\n${\\tilde{Q}}_0 := {\\pi}_0^{-1}(Q_0)$ is the preimage of\n$Q_0$ under the quotient map\n${{\\mathbb Z}} \\stackrel{{\\pi}_0}{\\rightarrow} {{\\mathbb Z}}\/{p_0}$.\n\n\nWe can therefore phrase the respective conditions for\nthe triples $A$ and $B$ to be genus-minimizing as follows:\n\\begin{align}\n A\\;\\text{is genus-minimizing}\n \\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;\n {\\bar{v}}_A(x,y)\n&< k+n\n \\;\\;\\;\\;\\;\\;\\;\\;\\;\\:\\forall\\; x,y \\in {\\tilde{Q}}_A;\n \\\\\n B\\;\\text{is genus-minimizing}\n \\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;\n {\\bar{v}}_B(x,y)\n&< k+n + \\textstyle{\\frac{1}{p}}\n \\;\\;\\;\\forall\\; x,y \\in {\\tilde{Q}}_B.\n\\end{align}\nNow, ${\\bar{v}}_A \\in {{\\mathbb Z}}$, so ${\\bar{v}}_A < k + n$ if and only if\n${\\bar{v}}_A \\leq k + n - 1$. On the other hand,\n${\\bar{v}}_B \\in \\frac{1}{p}{{\\mathbb Z}}$, so\n${\\bar{v}}_B < k + n + \\frac{1}{p}$ if and only if\n${\\bar{v}}_B \\leq k + n$. However, ${\\bar{v}}_B(x,y) \\in {{\\mathbb Z}}$\nonly if $y-x$ is a multiple of $p$, which in turn implies that\n${\\bar{v}}_B(x,y) = 0$. Thus it is impossible to have\n${\\bar{v}}_B = k + n$. We therefore have the revised conditions,\n\\begin{align}\n A\\;\\text{is genus-minimizing}\n \\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;\n {\\bar{v}}_A(x,y)\n&\\leq k+n - 1\n \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\:\\forall\\; x,y \\in {\\tilde{Q}}_A;\n \\\\\n B\\;\\text{is genus-minimizing}\n \\;\\;\\;\\;\\Leftrightarrow\\;\\;\\;\\;\n {\\bar{v}}_B(x,y)\n&\\leq k+n - 1 + \\textstyle{\\frac{p-1}{p}}\n \\;\\;\\;\\forall\\; x,y \\in {\\tilde{Q}}_B.\n\\end{align}\n \\\\\n\nLet us next turn our attention to $Q_A$ and $Q_B$,\nrecalling that \n$Q_A = \\left\\{0\\epsilon, \\ldots, (nk-1)\\epsilon\\right\\} \\subset {{\\mathbb Z}}\/k^2$,\nand\n$Q_B = \\left\\{0k^2, \\ldots, (k^{-1} -1)k^2\\right\\} \\subset {{\\mathbb Z}}\/p$.\nIn a manner somewhat reminiscent of the construction\nof the tuples $\\{{\\bf{z}}^j\\}$ in Section \\ref{s:p=k2},\nwe arrange the elements of $\\left[ 0,p \\right\\rangle\\cap {\\tilde{Q}}_B$ into tuples\n$\\{{\\bf{w}}^j\\}$, defined by \n$w_i^j := \\left[j\\epsilon \\right]_{k^2} + ik^2$,\nwhere $\\epsilon = \\left[-p\\right]_{k^2}$,\nand for each $j$, we restrict $i$ to lie in \n$\\{ i^{\\prime} \\in {{\\mathbb Z}}_{\\geq 0}\\left\\vert\\; w_{i^{\\prime}}^j < p \\right.\\}$.\nThus, for some $j_* \\in {{\\mathbb Z}}$, we have\n\\begin{equation}\n \\left({\\bf{w}}^0, \\ldots, {\\bf{w}}^{j_*-1}\\right)\n= \\left(\\left[0k^2\\right]_p, \\ldots, \\left[(k^{-1}-1)k^2\\right]_p\\right) \n\\end{equation}\nOf course, this also requires that when $j = j_*-1$,\nwe restrict $i$ to lie in $\\{0, \\ldots, i_*-1\\}$,\nwhere $w_{i_*-1}^{j_*-1} = \\left[(k^{-1}-1)k^2\\right]_p$.\nLet us pause to determine $i_*$ and $j_*$.\n\\begin{align}\n \\nonumber\n w_{i_*-1}^{j_*-1}\n&= \\left[(k^{-1}-1)k^2\\right]_p\n \\\\ \\nonumber\n \\left[(j_*-1)\\epsilon \\right]_{k^2} + (i_*-1) k^2\n&= p + k - k^2\n \\\\ \\nonumber\n i_*\n&= \\frac{p + k - \\left[(j_*-1)\\epsilon \\right]_{k^2}}{k^2}\n \\\\\n i_*\n&= \\left\\lceil\\frac{p - \\left[(j_*-1)\\epsilon \\right]_{k^2}}{k^2}\\right\\rceil,\n\\end{align}\nwhere the second line uses the fact that $p > k^2$.\nTaking the second line modulo $k^2$, we next determine $j_*$.\nRecalling that $n = \\left[-p^{-1}\\right]_k$, we have\n\\begin{align}\n \\nonumber\n (j_*-1)\\epsilon + (i_* - 1) k^2\n&\\equiv p + k - k^2\\;\\;(\\mod k^2)\n \\\\ \\nonumber\n (j_*-1)(-p)\n&\\equiv p + k \\;\\;(\\mod k^2)\n \\\\ \\nonumber\n j_*\n&\\equiv -p^{-1}k \\;\\;(\\mod k^2)\n \\\\\n j_*\n&= nk.\n\\end{align}\nThis allows us to write out the tuples ${\\bf{w}}^j$ as follows:\n\\begin{align}\n {\\bf{w}}^0\n&= \\left(\\; \\left[0\\epsilon\\right]_{k^2} + 0k^2\\!,\\;\\;\n \\left[0\\epsilon\\right]_{k^2} + 1k^2\\!,\\;\\; \\ldots,\\;\\;\n \\left[0\\epsilon\\right]_{k^2}\n + \\left({\\left\\lceil\\!\\frac{p - \\left[0\\epsilon\\right]_{k^2}\\!}{k^2}\n \\right\\rceil} - 1 \\right)\\! k^2\n \\;\\right),\n \\\\ \\nonumber\n&\\;\\;\\vdots\n \\\\ \\nonumber\n {\\bf{w}}^j\n&= \\left(\\; \\left[j\\epsilon\\right]_{k^2} + 0k^2\\!,\\;\\;\n \\left[j\\epsilon\\right]_{k^2} + 1k^2\\!,\\;\\; \\ldots,\\;\\;\n \\left[j\\epsilon\\right]_{k^2}\n + \\left({\\left\\lceil\\!\\frac{p - \\left[j\\epsilon\\right]_{k^2}\\!}{k^2}\n \\right\\rceil} - 1 \\right)\\! k^2\n \\;\\right),\n \\\\ \\nonumber\n&\\;\\;\\vdots\n \\\\ \\nonumber\n {\\bf{w}}^{nk-1}\n&= \\left(\\; \\left[(nk\\!-\\!1)\\epsilon\\right]_{k^2} + 0k^2\\!,\\;\\;\n \\left[(nk\\!-\\!1)\\epsilon\\right]_{k^2} + 1k^2\\!,\\;\\; \\ldots,\n \\right.\n \\\\ \\nonumber\n &\\left.\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n \\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\n \\left[(nk\\!-\\!1)\\epsilon\\right]_{k^2}\n + \\left({\\left\\lceil\\!\\frac{p - \\left[(nk\\!-\\!1)\\epsilon\\right]_{k^2}\\!}{k^2}\n \\right\\rceil} - 1 \\right)\\! k^2\n \\;\\right).\n\\end{align}\nThus, for each\n$i \\in \\left\\{0, \\ldots, \\left\\lfloor \\frac{p}{k^2} \\right\\rfloor - 1\\right\\}$,\nwe have\n\\begin{align}\n \\left\\{w_i^j \\left\\vert\\; j\\in \\{0, \\ldots, nk-1 \\} \\right. \\right\\}\n&= ik^2 + \\left\\{ \\left[0\\epsilon\\right]_{k^2}, \\ldots, \n \\left[(nk-1)\\epsilon\\right]_{k^2} \\right\\}\n \\\\ \\nonumber\n&= \\left[ ik^2, (i+1)k^2 \\right\\rangle \\;\\cap\\; {\\tilde{Q}}_A.\n\\end{align}\nThis, in turn, implies that\n\\begin{align}\n \\left[ 0, \\left\\lfloor\\frac{p}{k^2}\\right\\rfloor\\! k^2 \\right\\rangle \\cap {\\tilde{Q}}_A\n&= \\left\\{w_i^j \\left\\vert\\;\n \\begin{array}{c} j\\in \\{0, \\ldots, nk-1 \\},\n \\\\\n i \\in \\left\\{0,\\ldots, \\left\\lfloor\\frac{p}{k^2}\\right\\rfloor - 1\\right\\}\n \\end{array}\n \\right.\\right\\}\n \\\\ \\nonumber\n&= \\left\\{\\left[ 0, p \\right\\rangle \\cap {\\tilde{Q}}_B\\right\\}\n \\setminus\n \\left\\{w_{\\left\\lfloor\\!\\frac{p}{k^2}\\!\\right\\rfloor}^j \\left\\vert\\;\n j \\in J \\right.\\right\\},\n\\end{align}\nwhere\n\\begin{equation}\nJ = \\left\\{ j \\in \\{0, \\ldots, nk-1\\} \\left\\vert\\;\n \\left[j\\epsilon\\right]_{k^2} < [p]_{k^2} \\right.\\right\\}.\n\\end{equation}\nBut this definition of $J$ implies that\n\\begin{equation}\n \\left\\{w_{\\left\\lfloor\\!\\frac{p}{k^2}\\!\\right\\rfloor}^j \\left\\vert\\;\n j \\in J \\right.\\right\\}\n= \\left[ \\left\\lfloor \\frac{p}{k^2}\\right\\rfloor\\! k^2, p \\right\\rangle \n \\cap {\\tilde{Q}}_A,\n\\end{equation}\nand thus, we have\n\\begin{equation}\n \\left[ 0 , p \\right\\rangle \\cap {\\tilde{Q}}_A\n= \\left[ 0 , p \\right\\rangle \\cap {\\tilde{Q}}_B.\n\\end{equation}\n \\\\\n\nWe now return to the question of genus-minimization.\nFirst, for brevity, set\n\\begin{equation}\n \\tilde{Q}\n:= \\left[ 0 , p \\right\\rangle \\cap {\\tilde{Q}}_A\n = \\left[ 0 , p \\right\\rangle \\cap {\\tilde{Q}}_B,\n\\end{equation}\nso that, for any $x, y \\in \\left[ 0 , p \\right\\rangle$ with $x \\leq y$,\nwe have\n\\begin{align}\n {\\bar{v}}_A(x,y) \n&:= \\#\\left({{\\mathbb Z}}\\cap \\left\\langle x, y\\right]\\right) n\n \\;\\; - \\;\\;\n \\#\\left(\\tilde{Q} \\cap \\left\\langle x, y\\right]\\right) k,\n \\\\\n {\\bar{v}}_B(x,y) \n&:= \\#\\left({{\\mathbb Z}}\\cap \\left\\langle x, y\\right]\\right) \\left(n + \n \\textstyle{\\frac{1}{p}}\\right)\n \\;\\; - \\;\\;\n \\#\\left(\\tilde{Q} \\cap \\left\\langle x, y\\right]\\right) k,\n\\end{align}\nwith ${\\bar{v}}_A(y,x) := -{\\bar{v}}_A(x,y)$ and ${\\bar{v}}_B(y,x) := -{\\bar{v}}_B(x,y)$.\nThus, for any $x, y \\in \\left[0, p\\right\\rangle$, we have\n\\begin{equation}\n\\label{eq: section p = k^-2, compare v_A to v_B}\n {\\bar{v}}_B(x,y) = {\\bar{v}}_A(x,y) + \\frac{y-x}{p}.\n\\end{equation}\n\n\nSuppose that $A$ is genus-minimizing.\nThen for any $x, y \\in \\tilde{Q}$, we have\n\\begin{align}\n {\\bar{v}}_B(x,y) \n&= {\\bar{v}}_A(x,y) + \\frac{y-x}{p}\n \\\\ \\nonumber\n&\\leq (n + k - 1) + \\frac{y-x}{p}\n \\\\ \\nonumber\n&\\leq n + k - 1 + \\frac{p-1}{p},\n\\end{align}\nso that $B$ is genus-minimizing.\n\n\n\nConversely, suppose that $B$ is genus-minimizing.\nThen for any $x, y \\in \\tilde{Q}$,\n(\\ref{eq: section p = k^-2, compare v_A to v_B}) implies\n\\begin{equation}\n{\\bar{v}}_B(x,y) \\equiv \\frac{y-x}{p}\\;(\\mod {{\\mathbb Z}}).\n\\end{equation}\nWe therefore have\n\\begin{align}\n {\\bar{v}}_A(x,y)\n&= {\\bar{v}}_B(x,y) - \\frac{y-x}{p}\n \\\\ \\nonumber\n&\\leq \\left(n + k - 1 + \\frac{y-x}{p}\\right) - \\frac{y-x}{p}\n \\\\ \\nonumber\n&= n + k - 1,\n\\end{align}\nso that $A$ is genus-minimizing.\n\n\n\\end{proof}\n\n\n\n\nCombining Propositions \n\\ref{prop: classification of q^-1 for genus-minimizing q} and\n\\ref{prop:section q=k^-2, (p,k^-2,k) is like (k^2, p^-1, k)}\nthen gives the result we have been seeking.\n\n\\begin{theorem}\nWhen $p > k^2$, Conjecture \\ref{conj: berge} is true.\n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzehkj b/data_all_eng_slimpj/shuffled/split2/finalzzehkj new file mode 100644 index 0000000000000000000000000000000000000000..d868da1da56acbb647a3fb0c93f54515eacf36e3 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzehkj @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nIn the limit of low-frequency magnetohydrodynamic (MHD) fluctuations,\ncharged relativistic particles \nare accelerated by mirror forces resulting from magnetic\ncompressions \\citep{Achterberg1981},\n\\begin{equation}\n \\label{eq:mirrorForce}\n \\frac{d p_\\parallel}{dt} = \\frac{p_\\perp v_\\perp}{2B}\n \\nabla_\\parallel \\mid\\boldsymbol{B} \\mid,\n\\end{equation}\nwhere $\\parallel$ and $\\perp$ denote directions relative to the local magnetic field. \nIn MHD, magnetic compressions are caused by slow modes and fast\nmodes, with slow modes containing most of the compressive energy in\nsubsonic turbulence. Because slow modes\npropagate approximately along the magnetic field in most regimes, a\npure linear resonance with relativistic particles requires\n$\\omega \\simeq k_\\parallel v_p = k_\\parallel v_\\parallel$, or\nequivalently $v_p \\simeq\nc$, where $v_p$ is the parallel phase velocity of slow modes. Thus\nlinear theory predicts no acceleration of \nhigh-energy particles by MHD-scale slow modes, because the resonance condition\ncannot be satisfied. As a result, fast modes have traditionally been\nbelieved to be the dominant source of relativistic particle\nacceleration by MHD-scale turbulent fluctuations\n\\citep{Achterberg1981,Miller1996}. However, subsonic\nturbulence does not contain significant fast mode energy (see, e.g., \\citealt{Yao2011a, Howes2012} for empirical constraints on the fast and slow mode energy in the solar wind). This appears\nto significantly limit the efficiency of relativistic particle\nacceleration by MHD turbulence in many astrophysical environments.\n\nIn strong MHD turbulence, the waves comprising MHD turbulence are not\nlong-lived but instead have a decay time comparable \nto their linear period. In this case,\nthe linear resonance is not the appropriate condition for\nwave-particle interaction. Instead, the \nresonance is nonlinearly broadened \n\\citep{Bieber1994a,Gruzinov1999b,Shalchi2004,Shalchi2004a,Qin2006,Yan2008a,Lynn2012}. \nResonance broadening allows waves to interact with\nrelativistic particles when $\\omega_{\\rm nl} \\gtrsim k_\\parallel\nc$, where $\\omega_{\\rm nl}^{-1}$ is the non-linear correlation time of\nthe turbulence. In this paper, we estimate the resulting\nparticle acceleration analytically (\\S\n\\ref{sec:transportProperties}) and numerically using simulations of\nrelativistic test particles interacting with MHD turbulence (\\S\n\\ref{sec:numericalMethods} \\& \\ref{sec:numericalResults}). Our\nresults are potentially relevant to a wide range of astrophysical\nplasmas; in \\S \\ref{sec:conclusions} we briefly assess the\nimplications of our results for non-thermal emission from accretion\ndisks around black holes. \n\n\\section{Relativistic momentum diffusion by low-frequency MHD\n turbulence} \n\\label{sec:transportProperties}\n\nWe first provide an order of magnitude estimate of the momentum\ndiffusion coefficient for relativistic particles interacting with magnetic field compressions associated with slow modes (the case of fast modes is considered separately). The diffusion coefficient for a particle with reduced momentum\n$\\overline{p} \\equiv p\/mc$ may be estimated as\n$D_{\\overline{p}, \\boldsymbol{k}} \\sim f^2 \n\\delta t \/ c^2$ where $f \\sim \\overline{p}\nc^2 k_\\parallel \\delta B_\\parallel(\\boldsymbol{k}) \/ B_0$ is the force felt by a particle\ninteracting with a given spatial scale labeled by $\\boldsymbol{k}$,\n$\\delta\nt \\sim \\omega_{\\rm nl}^{-1}$ refers to the timescale over which\nwave-particle interactions are correlated, and $\\delta B_\\parallel(\\boldsymbol{k})$ is the rms fluctuation in magnetic compressions on scale $\\boldsymbol{k}$. For a given $k_\\perp$,\nthe total acceleration will be determined by the \naverage of $k_\\parallel^2 \\delta B_\\parallel(\\boldsymbol{k})^2$ over $k_\\parallel$,\nlimited to those \n$k_\\parallel$ that satisfy the broadened resonance condition \n$k_\\parallel \\lesssim \\omega_{\\rm nl} \/c $. Provided that $\\delta \nB_\\parallel(\\boldsymbol{k})^2$ does not scale too steeply with $k_\\parallel$,\nparallel wavenumbers near $k_\\parallel \\sim \\omega_{\\rm nl} \/c$ will dominate, resulting in a \ndiffusion coefficient at fixed $k_\\perp$ of order\n\\begin{equation}\n \\label{eq:approxDiffusion}\n D_{\\overline{p}, k_\\perp} \\sim {\\overline{p}^2} \\, \\frac{v_A}{c} \\, \\frac{\\omega_{\\rm nl} \\, \\delta B_\\parallel(\\boldsymbol{k_\\perp})^2.}{B_0^2}\n\\end{equation}\nwhere we have used the fact that most of the turbulent energy in anisotropic MHD turbulence has $k_\\parallel \\lesssim \\omega_{\\rm nl}\/v_A$ and that only a fraction $v_A\/c$ of the energy in magnetic compressions satisfies the conditions required for efficient particle acceleration, namely $k_\\parallel \\lesssim \\omega_{\\rm nl} \/c$ {\\bf (this follows formally from the magnetic field power spectrum in eq. \\ref{eq:gsTurbulence} below, which implies a parallel energy spectrum of $dE\/d \\ln k_\\parallel \\propto k_\\parallel$).} Equation \\ref{eq:approxDiffusion} shows that the scaling of\n$\\omega_{\\rm nl} \\delta B_\\parallel(\\boldsymbol{k_\\perp})^2$ with $k_\\perp$\ndetermines which $k_\\perp$ dominates. For the strong MHD power\nspectrum of \\citet{Goldreich1995}, $\\omega_{\\rm nl} \\propto\nk_\\perp^{2\/3}$ and $\\delta B(\\boldsymbol{k_\\perp}) \\propto\nk_\\perp^{-1\/3}$, so that all scales contribute equally, provided that\nthey can satisfy $k_\\parallel \\lesssim \\omega_{\\rm nl} \/ c$ (which\nfavors long-wavelength fluctuations). \n\nMore formally, the\nresonance-broadened diffusion coefficient for the reduced momentum is\ngiven by \\citep{Dupree1966,Weinstock1969}\n\\begin{equation}\n \\label{eq:broadenedDiffusion}\n D_{\\overline{p}_\\parallel} = \\frac{\\overline{p}_\\perp^2 v_\\perp^2}{4 B_0^2} \\int\n d^3k k_\\parallel^2 I_B(\\boldsymbol{k}) R(\\boldsymbol{k}), \n\\end{equation}\nwhere $\\overline{p}_\\perp$ is the perpendicular component of the\ndimensionless momentum, $I_B(\\boldsymbol{k})$ is\nthe 3D power spectrum of magnetic field \nfluctuations, and $R(\\boldsymbol{k})$ is a resonance function that\ndescribes the time-averaged interaction of a test particle with waves\nat a given $\\boldsymbol{k}$. Quantitatively, $R(\\boldsymbol{k}) = \\Re \\int_0^\\infty dt \n\\exp{[\\imath (\\omega(\\boldsymbol{k}) - v_\\parallel k_\\parallel) t]} \\,\nf(t)$, where $f(t)$ is the time correlation function for wave-particle\ninteractions. $R(\\boldsymbol{k})$ is necessarily \nphenomenological, as it in principle depends on the momentum diffusion\ncoefficient itself. Standard models in the literature assume that\nwaves decay as an exponential or Gaussian in time due to non-linear\ninteractions; e.g, $f(t) = e^{-\\omega_{\\rm nl}^2 t^2}$ (a Gaussian\ndecay model), with a nonlinear decay frequency $\\omega_{\\rm nl}$. We\nfocus on a Gaussian decay model favored by our previous test particle\nsimulations \\citep{Lynn2012}.\nThese assumptions lead to \\begin{equation} \n \\label{eq:gaussianResonance}\n R(\\boldsymbol{k}) = \\frac{\\sqrt{\\pi}}{2 \\omega_{\\rm nl}}\n \\exp{\\left[-\\frac{k_\\parallel^2 (v_\\parallel-v_p)^2}{4 \\omega_{\\rm nl}^2}\\right]}.\n\\end{equation}\nwhere we have assumed for simplicity that the waves have a dispersion\nrelation $\\omega = k_\\parallel v_p$, a reasonable approximation for\nanisotropic slow modes with $k_\\perp \\gg k_\\parallel$. Physically, equation \\ref{eq:gaussianResonance} implies that for the particles to couple to the turbulent fluctuations, the frequency that the particles feel as they pass through a wave, $k_\\parallel (v_\\parallel - v_p)$, must be of order (or less than) the\nnonlinear frequency.\n\nTo perform the calculation in Equation \\ref{eq:broadenedDiffusion}, we\nassume that the magnetic power spectrum of slow modes is given by\nstrong anisotropic turbulence,\n\\begin{equation}\n \\label{eq:gsTurbulence}\n I(\\boldsymbol{k}) \\equiv \\frac{\\delta B_S^2 L^3}{12 \\pi} (k_\\perp L)^{-10\/3}\n g\\left(\\frac{k_\\parallel L^{1\/3}}{k_\\perp^{2\/3}}\\right),\n\\end{equation}\nwhere $L$ is the outer scale of the cascade, $\\delta B_S^2$ is the\ntotal energy in slow mode magnetic fluctuations, and $g(x) \\simeq 1$ for $x \\lesssim 1$ and falls off sharply to zero for $x \\gtrsim 1$. The cutoff $g(x)$ in equation \\ref{eq:gsTurbulence}\nrepresents the lack of power outside the\nanisotropic Goldreich-Sridhar cone \\citep{Goldreich1995} in weakly\ncompressible MHD turbulence,\\footnote{{\\bf \\citet{Cho2002a} show using numerical simulations that $g(x) \\propto \\exp[-x]$. Our results are insensitive to the precise functional form of g(x).}} and the power spectrum normalization is\nchosen so that \n$\\int d^3k I(\\boldsymbol{k}) \\equiv \\delta B_S^2 \/ 2$.\nThe nonlinear frequency in such turbulence is given by $\\omega_{\\rm nl} \\simeq (v_A\/L) (k_\\perp \nL)^{2\/3}$, which is of order the eddy turnover time on a given scale.\n\nFinally, to simplify the resulting estimate, we assume that the particles are \nrelativistic ($\\overline{p}_\\perp \\sim\n\\overline{p}_\\parallel \\gg 1$), and that wave speeds are\nnonrelativistic. To emphasize the net particle acceleration efficiency, we also present our results in terms of the total momentum diffusion coefficient, rather than the parallel momentum diffusion coefficient. This implicitly assumes that modest pitch angle scattering isotropizes the distribution function (see \\S \\ref{sec:numericalResults}). Under these approximations, the diffusion coefficient becomes \n\\begin{align}\n \\label{eq:slowModeIntermediateIntegral}\n & D_{\\overline{p}} = \\frac{\\sqrt{\\pi} \\overline{p}_\\perp^2 v_\\perp^2}{48 v_A}\n \\frac{\\delta B_S^2}{B_0^2} \\left(\\frac{v_\\parallel}{v}\\right)^2 \\times \\\\ & \\int \n dk_\\parallel dk_\\perp \\frac{k_\\parallel^2}{k_\\perp^3} g\\left(\\frac{k_\\parallel\n L^{1\/3}}{k_\\perp^{2\/3}}\\right) \\nonumber \\exp{\\left(-\\frac{k_\\parallel^2 L^2 v_\\parallel^2}{4 (k_\\perp L)^{4\/3} v_A^2}\\right)}. \n\\end{align}\nBoth the exponential and\nthe $g(x)$ term in equation \n\\ref{eq:slowModeIntermediateIntegral} \nhave the effect of cutting off the interaction at \nhigh $k_\\parallel$ for a given $k_\\perp$. When $v_A \\ll v_\\parallel$, the \nexponential cutoff will always be more constraining and requires\n$k_\\parallel \\lesssim L^{-1} v_A\/c (k_\\perp L)^{2\/3}$. Given this, we\nfind \n\\begin{equation}\n \\label{eq:slowModeDiffusion}\n {D_{\\overline{p}}}= {\\overline p^2} \\frac{\\pi}{24}\n \\frac{v_A^2}{c L} \\frac{\\delta B_S^2}{B_0^2} \n \\frac{\\left( 1 - \\alpha^2 \\right)^2}{\\alpha^3}\n \\ln{(k_{\\rm max} L)},\n\\end{equation}\nwhere we have rewritten the angular dependence in terms of $\\alpha =\nv_\\parallel \/ v$, the particle pitch-angle \ncosine. The result in equation \\ref{eq:slowModeDiffusion} is similar to that derived earlier by \\citet{Chandran2000}. The restriction to parallel velocities much larger than $v_A$\ncorresponds to $\\alpha \\gg v_A \/ v \\simeq v_A \/ c$.\nEquation \\ref{eq:slowModeDiffusion} corresponds to a particle acceleration time $t_{\\rm\n a} \\sim (\\delta B_S\/B_0)^{-2} (L\/v_A) (c\/v_A)$ . Note that this is independent of particle energy and is roughly the eddy turnover time divided by the fraction of the magnetic energy at low $k_\\parallel$ that satisfies $k_\\parallel \\lesssim \\omega_{\\rm nl}\/c$.\n\nEquation \\ref{eq:slowModeDiffusion} can be compared to the analogous result for acceleration of relativistic particles by fast modes. The latter is predominantly via a linear resonance with highly oblique waves. Versions of this\ncalculation have been performed in many other contexts\n\\citep{Miller1996, Yan2002}, so we restrict ourselves to briefly summarizing the salient features here.\n\nRelativistic test particles travelling along magnetic field lines can\nexperience a linear resonance with a highly \noblique fast mode. The linear resonance function is a delta-function,\n\\begin{equation}\n \\label{eq:linearFastModeResonance}\n R(\\boldsymbol{k}) = \\pi \\delta{(k v_p \\pm k_\\parallel v_\\parallel)},\n\\end{equation}\nwhere $v_p$ is the isotropic phase velocity of fast modes\n(approximately $v_A$ for $\\beta \\ll 1$ and $c_s$ for $\\beta \\gg 1$).\nThe power spectrum of fast modes in Alfv\\'enic turbulence is \nbelieved to be isotropic. The spectral index is uncertain but \nthe simulations of \\citet{Kowal2010a} suggest a 1D power spectrum\n$P(k) \\sim k^{-2}$ (though possibly shallower; see \\citealt{Cho2003, Chandran2005}).\n\nPerforming the integral in Equation \\ref{eq:broadenedDiffusion} with\nthe fast mode linear resonance and isotropic power spectrum $\\sim\n\\delta B_F^2 L^3 (k L)^{-\\alpha}$ leads to \\begin{equation}\n \\label{eq:fastModeDiffusion}\n D_{\\overline{p}} \\sim \\overline{p}^2 \\frac{v_p^2}{c L} \\frac{\\delta B_F^2}{B_0^2}\n \\int_{k_{\\rm min}}^{k_{\\rm max}} dk k^{1-\\alpha},\n\\end{equation}\nwhere $\\delta B_F^2$ is the energy in magnetic compressions associated with fast modes. For\n$\\alpha \\sim 2$, the diffusion coefficient due to fast modes\n(eq. \\ref{eq:fastModeDiffusion}) is of the same form as that due to\nslow modes (eq. \\ref{eq:slowModeDiffusion}). For $\\beta \\gg 1$, a comparison of these two expressions shows that ratio of the fast mode to slow mode diffusion coefficient is $\\sim (c_s\/v_A)^2 (\\delta B_F\/\\delta B_S)^2$. At high $\\beta$, fast modes lose their magnetic compressibility (becoming simply sound waves), so that the magnetic energy $\\delta B_F^2$ in fast modes decreases at fixed velocity amplitude, with $\\delta B_F^2 \\sim \\rho \\, \\delta v_F^2\/\\beta$. By contrast, $\\delta B_S^2 \\sim \\rho \\, \\delta v_S^2$. Thus the ratio of the fast to slow mode diffusion coefficients is in fact set by the relative turbulent energy in each mode. For subsonic turbulence, slow modes will in general dominate the particle acceleration because there is significantly more energy in slow modes. This depends, however, on the power spectrum of the fast modes. If $\\alpha \\sim 3\/2$ rather than $\\alpha \\sim 2$ (as in \\citealt{Chandran2005}) then the fast mode acceleration efficiency can be greater than that of slow modes even given the overall lower energy density in fast modes.\n\n\\section{Numerical methods}\n\\label{sec:numericalMethods}\n\nOur simulations consist of charged test particles evolving in\nthe macroscopic electric and magnetic fields of isothermal, subsonic\nMHD turbulence. Apart from modifying the particle pusher for\nrelativistic test \nparticles, which we describe below, our computational approach is \nidentical to that of \\cite{Lynn2012}. Dimensional quantities\nthroughout the paper are expressed in units of the sound speed $c_s$\nand the box scale \n$L$, when not explicitly stated.\n\n\\subsection{Turbulence simulations}\n\\label{sec:turbSims}\n\nWe simulate ideal MHD turbulence with the Athena code\n\\citep{Stone2008}. We drive an incompressible turbulent velocity field\nusing an Ornstein-Uhlenbeck process, and allow \ncompressible fluctuations to develop naturally. The OU process\nhas a characteristic autocorrelation time $t_{\\rm OU}$. Fiducial\nproperties for the MHD simulations used in this work \nare summarized in Table \\ref{table:fiducial}. We show results from higher resolution calculations in Figure \\ref{fig:Dp_vs_c} (discussed below) and find that the numerically determined diffusion coefficients are relatively independent of resolution. In our calculations,\nthe simulation box is extended along the\nmean magnetic field, because otherwise the particles (which undergo\nperiodic boundary conditions) would interact with the same eddies\nmultiple times before the eddies decorrelate.\n\n\\begin{table}\n \\begin{center}\n \\caption{Summary of fiducial simulation properties}\n \\begin{tabular}{ c c }\n \\\\\n \\hline \\hline\n Parameter & Value \\\\\n \\hline \\hline\n Resolution & $512\\times128^2$ \\\\\n Volume ($L^3$) & $8 \\times 2^2$ \\\\\n $\\dot{\\epsilon}$ ($c_s^3 \/ L$)\\footnote{The turbulent energy input\n rate, corresponding to a sonic Mach number of $\\simeq 0.35$. Calculations with $\\dot{\\epsilon} = 0.01$ yield similar results.} & 0.1 \\\\\n $\\beta$\\footnote{Ratio of thermal to magnetic pressure. Our calculations covered a range of $\\beta \\sim 0.1-10$.} & $1$ \\\\\n $t_{\\rm OU} \\, (L\/c_s)\\footnote{$t_{\\rm OU}$\n refers to the correlation time in the Ornstein-Uhlenbeck turbulence forcing.}$ & $1.5$ \\\\\n $l_D$ ($L$)\\footnote{Outer (driving) scale of the turbulence.} & $0.39$\\\\\n $\\delta B_{\\parallel}$ ($B_0$)\\footnote{RMS fluctuation in the parallel magnetic field.} & $0.12$\\\\\n $N_{\\mathrm{particles}}$ & $2^{11} \\times 10^3 \\simeq 2 \\times 10^6$ \\\\\n $\\Omega_0$ ($c_s\/L$)\\footnote{Test particle gyrofrequency.} & $2\n \\times 10^5$ \\\\\n \\end{tabular}\n \\label{table:fiducial}\n \\end{center}\n\\end{table}\n\n\\subsubsection{Measurement of turbulence properties}\n\\label{sec:turbulenceProperties}\n\nAn important property of our turbulence simulations for comparing test\nparticle results to analytical estimates is the rms deviation in\nparallel magnetic field, $\\delta B_{\\parallel}$, since this sets the\nmagnitude of the magnetic mirror forces. We define this quantity as\nthe spatial rms average of $\\mid \\boldsymbol{B}\\mid -\n\\mid\\boldsymbol{B_0}\\mid$, where \n$\\boldsymbol{B_0}$ is the initial mean magnetic field. This is equivalent\nto taking the local direction of the magnetic field as the\n``parallel'' direction. The magnitude of $\\delta B_\\parallel$ depends\non the \nmagnetic compressibility of the fast and slow modes at a given\n$\\beta$, in addition to their overall representation in the\nturbulence. For our fiducial\nsimulation summarized in Table \\ref{table:fiducial}, $\\delta B_\\parallel \/B_0 \\simeq 0.12$, while\nsimulations that have the same driving rate (and thus similar $v_{\\rm \n rms}$) but $\\beta=0.3$ \n($\\beta=3$) have $\\delta B_\\parallel \/B_0 \\simeq 0.05$ ($0.23$). Note\nthat for higher $\\beta$, the energy in magnetic compressions is\nlarger. \n\nWe also decompose the turbulent velocity field into the linear MHD\nAlfv\\'{e}n, slow, and fast modes, following the approximate Fourier\nspace method of \\citet{Cho2003}.\nFor our fiducial simulation, 50\\%, 45\\%, and 5\\% of the turbulent\nenergy is in the \nAlfv\\'{e}n, slow, and fast modes respectively. For both higher\nand lower $\\beta$, the proportion of energy in fast modes decreases\nsubstantially (to less than $1\\%$), while the slow mode energy remains\nat the same order of magnitude. Thus it is broadly appropriate to\nassume that all of the magnetic\nfield fluctuation energy is in the slow modes, and that\nthe particle acceleration is dominated by interactions with slow modes. \n\n\\subsection{Test particle integration}\n\\label{sec:testParticles}\n\nFor a given fluid simulation, we simulate a statistical ensemble of\ncharged test particles which are initialized randomly throughout the\nbox of fully saturated turbulence. These test particles are evolved\naccording to the Lorentz force \n\\begin{equation}\n \\label{eq:lorentzForce}\n \\frac{d \\boldsymbol{\\overline{p}}}{dt} = \\frac{q}{m c} \\boldsymbol{E} +\n \\frac{q}{m c} \\boldsymbol{\\beta} \\times \\boldsymbol{B},\n\\end{equation}\nwhere the dimensionless momentum $\\boldsymbol{\\overline{p}} \\equiv\n\\boldsymbol{p} \/ mc$, and $\\boldsymbol{\\beta} = \\boldsymbol{u}\/c$ is\nthe particle's physical velocity. The $E$ \nand $B$-fields are those on the MHD grid, interpolated to the\nparticle's location using the triangular-shaped cloud\n\\citep{Hockney1981} method in space and time. For each simulation, we\nalso choose a numerical value for the speed of light $c$, which\naffects the motion of the test particles. The choice of $c$ does not,\nhowever, affect the turbulence. Our\nchoice of non-relativistic turbulence and relativistic test particles\nis appropriate for studying high energy supra-thermal particle\nacceleration. \n\nParticles are integrated\nusing the \\citet{Vay2008} particle pusher, which is sympletic and\nsymmetric in time, and conserves energy and the magnetic moment\nadiabatic invariant to machine precision in tests with constant\nfields.\\footnote{The \\cite{Boris1970} pusher is not as accurate when fluid velocities are non-negligible fractions of the chosen value of $c$. In tests, these errors did not significantly affect our\n results, but we nevertheless prefer the Vay pusher for the relativistic\n case.} We initialize the test particles with sufficiently high\ngyrofrequencies that diffusion and heating is independent of\ngyrofrequency; i.e. $\\Omega \\gg \\omega$ where $\\omega$ is the\nfrequency of any turbulent motions and $\\Omega$ is the relativistic\ngyrofrequency. To calculate the momentum diffusion coefficients, we further initialize particles with a specific value of\n$\\overline{p}$, and generally take $\\overline p_\\perp = \\overline\np_\\parallel$. \nThe momentum is defined with respect to\nthe bulk rest-frame of the simulation, though for the\nrelativistic particles we focus on, this choice is\nunimportant because the particle velocities are much greater than the fluid velocities.\n\nThe velocity diffusion\ncoefficients are calculated according to \n\\begin{equation}\n \\label{eq:diffusionDefinition}\n D_{\\overline{p}} \\equiv \\frac{\\langle \\delta \\overline{p}^2 \\rangle} {2 \\, \\delta t},\n\\end{equation}\nwhere the average is over many particles with the same initial\nmomentum. \nOne subtlety is that because we initialize the particle momentum with\nrespect to the bulk rest frame of the simulation, they are not\ninitially moving with the local drift velocity. As they change their\nmotion to follow the drift velocity, the\nparticle momentum undergoes an initial transient ``jump'' which saturates \nat the rms velocity of the turbulence (times the Lorentz factor of the\ntest particle, for ultra-relativistic particles). We sidestep this\nsubtlety by fitting a linear function in $t$ to $\\langle \\delta\n\\overline{p}^2 \\rangle$ at later times using\nleast-squares.\n\n\\section{Numerical results}\n\\label{sec:numericalResults}\n\nIn the analytic calculations summarized in \\S \\ref{sec:transportProperties}, the particles are assumed to diffuse primarily in $p_\\parallel$, as expected for particles interacting with long wavelength, low frequency turbulent fluctuations. In our numerical calculations, we find that particles undergo diffusion in both $p_\\parallel$ and $p_\\perp$ (or, equivalently, in total momentum $p$ and magnetic moment $\\mu$). This is true for both the relativistic calculations presented here and our earlier non-relativistic test particle calculations \\citep{Lehe2009, Lynn2012}. For the non-relativistic test particle calculations, the diffusion time for $\\mu$ was somewhat longer than that for the total momentum $p$, while for the relativistic test particle results presented in this paper, the two timescales are comparable. This diffusion in $\\mu$ corresponds to an effective pitch angle scattering rate and may be due to violation of magnetic moment conservation by finite amplitude low frequency turbulent fluctuations \\citep{Chandran2010}. The theory for the latter has not been fully worked out for the $\\beta \\sim 1$ conditions we focus on here. In what follows, we defer a detailed analysis of the diffusion in $\\mu$ to future work and focus on the diffusion in total momentum $p$.\n\nFigure \\ref{fig:Dp_vs_p} demonstrates that for relativistic test\nparticles, the momentum diffusion coefficient is robustly of the form\n$D_{\\overline p} \\propto {\\overline p}^2$. \n\\begin{figure}\n \\includegraphics[width=3.25in]{Dp_vs_p.pdf}\n \\caption{Test particle momentum diffusion coefficients as a function of $\\overline p \\equiv p\/mc$ normalized by\n $\\overline{p}^2$ for several values of $\\beta$ (for fixed driving\n rate $\\dot{\\epsilon} = 0.01 c_s^3 \/ L$ and fixed $c=10 \\,\n c_s$). For each $\\overline p$, the diffusion coefficient is measured for an ensemble of particles with ${\\overline p}_{\\perp} = {\\overline\n p}_{\\parallel} = {\\overline p}\/\\sqrt{2}$ that are initially at random positions in the turbulent box. The numerical results demonstrate that $D_{\\overline p} \\propto {\\overline p}^2$ for ultrarelativistic\n particles (${\\overline p} \\gg 1$), consistent with the analytic\n expectations from equation \\ref{eq:slowModeDiffusion}.}\n \\label{fig:Dp_vs_p}\n \\\n\\end{figure}\nThis is in contrast to the case of non-relativistic particles where\nthe diffusion coefficient for particles interacting with subsonic\nturbulence is roughly $D \\propto p$ for supra-thermal particles\n\\citep{Lynn2012}. \n\nThe analytic results summarized in equation \\ref{eq:slowModeDiffusion} also predict that for relativistic particles, the magnitude of the diffusion coefficient depends on the magnetic compressibility of the turbulence and $v_A\/c$. We now test these expectations using our test particle simulations. \n\nFigure \\ref{fig:Dw_vs_dB2} shows how the measured diffusion coefficient (normalized by $\\overline{p}^2$) varies\nwith the rms magnetic field compression $\\delta B_\\parallel\/B_0$. The latter is directly measured in the simulations as described in \\S \\ref{sec:turbulenceProperties}. \nFigure \\ref{fig:Dw_vs_dB2} shows that the diffusion coefficient scales with the total turbulent energy in the magnetic field compressions, consistent with equation \\ref{eq:slowModeDiffusion}.\n\\begin{figure}\n \\includegraphics[width=3.25in]{Dw_vs_dB2.pdf}\n \\caption{Test particle momentum diffusion coefficient for relativistic particles as a function of the strength of the magnetic compressions $\\propto \\delta B_\\parallel$, for several values of $\\beta$. The simulations have \n $c = 10 \\, c_s$ and different turbulent driving rates $\\dot \\epsilon$. The numerically determined\n diffusion coefficients are $\\propto (\\delta B_\\parallel\/B_0)^2$\n (light dashed lines, with arbitrary normalization), consistent with the analytical predictions in \\S \\ref{sec:transportProperties}.}\n \\label{fig:Dw_vs_dB2}\n\\end{figure}\n\nFigure \\ref{fig:Dp_vs_c} shows the dependence of the measured diffusion\ncoefficient on the ratio $c\/v_A$, for three different values of\n$\\beta$. We reiterate that in each simulation, we\nchoose a value for the speed of light $c$ only for the purposes of evolving the test particles (the choice of $c$ has no impact on the properties of the turbulence). The diffusion coefficients in Figure \\ref{fig:Dp_vs_c} are normalized by the analytic\nprediction in equation\n\\ref{eq:slowModeDiffusion}.\\footnote{Specifically, we use\n $D_{\\overline{p}} \\simeq 0.4 \\overline{p}^2 (\\delta\n B_\\parallel\/B_0)^2 v_A^2\/cL$ for the analytic prediction from\n equation \\ref{eq:slowModeDiffusion}, with $\\delta B_\\parallel\/B_0$\n calculated for each simulation, where we have used\n$\\ln \\left( k_{\\rm max} L \\right) = \\ln 64 \\simeq 4.15$ for the fiducial simulation. For the\nhigher resolution simulations, the logarithmic factor is adjusted as\nappropriate.}\nFor each $\\beta$, $v_A$ and $c_s$\nare fixed, so the x-axis in Figure \\ref{fig:Dp_vs_c} corresponds to\ndifferent choices of $c$. \nFor $c \\gg v_A$, the results are reasonably consistent with the analytical\npredictions. In addition to the results for the fiducial simulation,\nFigure \\ref{fig:Dp_vs_c} also shows two other $\\beta = 1$ simulations:\n(1) one with the same resolution but a larger driving scale by a\nfactor of two, so that the inertial range is somewhat more extended\n(HR) (2) a second higher resolution 1024x256$^2$ simulation. The\nresults for both of these other simulations are very similar to the\nfiducial calculation, confirming that the large-scale fluctuations\nthat are well-resolved in a typical MHD simulation produce the\nmajority of the particle acceleration. \n\n\\begin{figure}\n \\includegraphics[width=3.25in]{Dw_vs_c.pdf}\n\\caption{Test particle momentum diffusion coefficients for relativistic particles normalized by\n the analytic prediction of equation \\ref{eq:slowModeDiffusion}\n for simulations with different $\\beta$ and $v_A\/c$. For $c \\gg\n v_A$, the results are well-described by the analytical predictions.\n The test particle calculations are for an ensemble of particles\n with ${\\overline p}_{\\perp} = {\\overline p}_{\\parallel} =\n {\\overline p}\/\\sqrt{2}$. The orange\n triangles (HR) are from a run with the fiducial number of grid\n cells but a larger turbulent driving scale, so that the\n turbulence has an inertial range that is 2 times larger. The blue triangles (HR2) are for a higher resolution $1024\\times256^2$ simulation. Both resolution tests yield very similar results indicating that large scale turbulent fluctuations dominate the particle acceleration.} \n \\label{fig:Dp_vs_c}\n\\end{figure}\n\n\n\\subsection{Long-time evolution of distribution function}\n\\label{sec:evolution}\n\nThe diffusion coefficients shown in Figure \\ref{fig:Dp_vs_c} correspond to particle acceleration times that are many eddy turnover times. As a result, it is computationally intensive to directly simulate the long timescale evolution of the distribution function. To study the latter, we instead separately solve the time dependent diffusion equation for the distribution function $f(p,t)$ using the momentum diffusion coefficients determined in our test particle calculations. In particular, we begin with a Maxwellian distribution function\nhaving $k_B T \\sim m c^2$, i.e., $\\langle \\overline p \\rangle \\sim 1$ and evolve it subject to a diffusion coefficient given by $D_{\\overline p} \\equiv \\overline p^2\/t_a$ where $t_a$ defines the acceleration time. Figure \\ref{fig:df_evol} ({\\em right panel}) shows the resulting distribution function at later times. For comparison, we also show a Maxwell-J{\\\"u}ttner distribution function (black dashed lines) that has the same total energy as the final distribution function in our diffusion calculations. Figure \\ref{fig:df_evol} shows that the distribution function quickly develops a significant non-thermal tail, on a timescale of $\\sim 0.25 \\, t_a$. As a consistency check, the {\\em left panel} in Figure \\ref{fig:df_evol} shows that over the timescale we can directly simulate the MHD turbulence with test particles, the evolution of the distribution function is indistinguishable from the solution of the momentum-space diffusion equation.\n\n\\begin{figure*}\n\\includegraphics[width=7in]{f_vs_t.pdf}\n \\caption{Evolution of the particle distribution function due to interaction with MHD turbulence. \\textit{Left panel:} Comparison of test particle simulations (dotted)\n with the numerical solution of the time dependent diffusion equation (solid blue) using\n a momentum diffusion coefficient given by $D_{\\overline p} \\equiv\n \\overline{p}^2\/t_a$, where $t_a$ defines the acceleration time and is derived from the test particle results. The particles are initially thermal and isotropic, and evolve over a\n long time baseline ($t = 20 \\, L\/c_s$ vs. $1 \\, L\/c_s$ for\n the calculations used to measure the diffusion coefficient). The numerical solution of the diffusion equation is shown at the same time and is in good agreement with the direct evolution of the test particles. \\textit{Right panel:} Longer-time evolution of the numerical\n solution of the diffusion equation. The particles gain energy exponentially, with an\n e-folding time of approximately $0.25 \\, t_a$. On a comparable timescale, the distribution function develops a significant non-thermal tail. For comparison, we also plot a thermal distribution with\n the same energy as the final distribution (black dashed curve), which highlights the substantial\n non-thermal tail at high energies.} \n \\vspace{0.15in}\n \\label{fig:df_evol}\n\\end{figure*}\n\\\n\\section{Conclusions \\& Implications}\n\\label{sec:conclusions}\n\nOur results demonstrate that subsonic MHD turbulence efficiently accelerates relativistic particles with a Fermi-like momentum diffusion coefficient $D_p \\propto p^2$. This is true for both $\\beta \\lesssim 1$ and $\\beta \\gtrsim 1$ and is thus a robust property of charged particles interacting with low frequency MHD turbulence. We have restricted our analysis to particles whose (relativistic) cyclotron frequencies are larger than the frequencies of the turbulent fluctuations. In practice this limits our analysis to particles that are not too relativistic. \n\nOur key analytic result is that nonlinear broadening of quasi-linear resonances implies that slow modes in strong MHD turbulence can interact efficiently with relativistic particles, despite being\nunable to satisfy the linear resonance condition (see \\citealt{Chandran2000} for a similar result). \n{\\bf In particular, resonance broadening allows long wavelength turbulent magnetic field compressions satisfying $k_\\parallel c \\lesssim \\omega_{\\rm nl}$ to accelerate particles, where $\\omega_{\\rm nl}$ is the non-linear decay rate of the turbulence at a given scale.}\n\nBecause slow modes tend to be energetically more important than fast modes in subsonic turbulence, this suggests that interactions with slow modes may dominate the overall\nparticle acceleration by low-frequency, weakly compressible MHD\nturbulence. This is contrary to the standard quasi-linear theory\nresults in the literature (e.g., \\citealt{Achterberg1981}). However, the particle acceleration efficiency by fast modes depends sensitively on their turbulent power spectrum, which is not fully understood. In particular, if the fast mode spectral index is $\\alpha \\sim 3\/2$ (which is not the case in our simulations, though it is suggested by some studies), fast modes may be more efficient than slow modes at accelerating particles even if their total energy density is smaller (see eq. \\ref{eq:fastModeDiffusion}). \n\nFor relativistic particles, momentum diffusion of the form $D_p \\propto p^2$ produces a power-law spectrum $dN\/dp \\propto p^{-1}$ so long as the acceleration time of particles (which is independent of particle energy) is shorter than the radiative loss timescale and the escape time from the acceleration region \\citep{Blandford1987}. The total energy in the accelerated particle population depends on the efficiency with which `seed' relativistic particles are created. Because suprathermal particle acceleration is inefficient for non-relativistic particles interacting with MHD turbulence \\citep{Lynn2012}, it is not clear if the net acceleration efficiency (by turbulent mechanisms alone) will be substantial for plasmas with non-relativistic temperatures, because the turbulence itself does not self-consistently seed relativistic particles. By contrast, for relativistically hot plasmas, the formation of a non-thermal tail of relativistic particles by the mechanism studied here is likely to be quite efficient. One particularly important application of our results is thus to accretion flows onto black holes, where the electrons can in some cases have $k T \\gtrsim m_e c^2$ even though the disk turbulence itself is non-relativistic. \n\n\\subsection{Implications for Black Hole Accretion Flows}\n\nWeakly compressible MHD turbulence is\ngeneric in black hole accretion flows as a consequence of the nonlinear\nevolution and \nsaturation of the magnetorotational instability \\citep{Balbus1998}.\nNon-thermal particle acceleration by such turbulence is of particular\nastrophysical interest in at least two circumstances. First, at low\naccretion rates onto a black hole or neutron star, the accretion flow\ncan adopt a low-collisionality state in which much of the emission can\nbe dominated by a non-thermal population of electrons, if such a\npopulation is present (e.g., \\citealt{Yuan2003}). Secondly, in luminous\nradiatively efficient accretion flows, non-thermal emission from the\ndisk surface layers (a ``corona\") can contribute significantly to the\nsynchrotron and high energy inverse Compton emission. We briefly\ndiscuss the implications of our results for these applications. \n\nThe momentum diffusion coefficient calculated in \\S\n\\ref{sec:transportProperties} and Figure \\ref{fig:Dp_vs_c}\ncorresponds to a rate of energy gain given by \\begin{equation} \n\\label{eq:Edotacc}\n\\dot E_{\\rm acc} \\sim \\frac{c \\, D_p}{p} \\equiv A \\, p \\,\n\\frac{v_A^2}{L}\\left(\\frac{\\delta B_\\parallel}{ B_0}\\right)^2 \n\\end{equation}\nwhere $A$ is a dimensionless coefficient that\nencapsulates the efficiency of the particle acceleration and can be\ncalibrated using our test particle simulations. In particular,\nFigure \\ref{fig:Dp_vs_c} corresponds to $A \\sim 1\/3$ for $c\/v_A\n\\sim 10-100$, the values expected in the inner regions of accretion\ndisks around black holes. \nThe exact value of $\\delta B_\\parallel\/B_0$ in accretion disk turbulence is somewhat uncertain. For the $\\beta \\sim 10-100$ conditions expected, $\\delta B_\\parallel \\sim 0.3 \\, B_0$ is plausible. However, the exact value depends in part on the effect of collisionless damping on the compressibility of accretion disk turbulence, which is not well understood. Moreover, small-scale fluctuations generated by the mirror instability may contribute significantly to the magnetic field compressions in collisionless disks \\citep{Kunz2014,Riquelme2014}.\n\nThe acceleration of particles by disk turbulence requires that the acceleration time is shorter than the viscous time. Given the acceleration rate in equation \\ref{eq:Edotacc} this is likely achieved in the inner regions close to the black hole. In addition, the acceleration of particles by disk turbulence is limited by\nradiative losses, in particular synchrotron and inverse Compton\nemission. Focusing on the former, \nwe find that the maximum Lorentz factor of accelerated electrons\nis given by \n\\begin{equation}\n\\label{eq:gmax}\n\\gamma_{\\rm max} \\sim A \\, \\left(\\frac{m_e}{m_p}\\right) \\tau_T^{-1} \\left(\\frac{\\delta B_\\parallel}{B_0}\\right)^{2}\n\\end{equation} \nwhere $\\tau_{T} \\equiv \\sigma_T n_e L$ is the Thompson optical depth\nacross the outer scale of the turbulent fluctuations $L$. Equation \\ref{eq:gmax} implies that non-thermal emission from accelerated\nelectrons is likely to be particularly important in low-luminosity\nsystems where $\\tau_T \\ll 1$. As a concrete example, in models of\nthe emission from Sgr A*, $\\tau_T \\sim 10^{-5}-10^{-6}$ (e.g.,\n\\citealt{Yuan2003,Gammie2009}) so that $\\gamma_{\\rm max} \\sim\n100$. This implies that the particle acceleration found here\nmay substantially modify the electron distribution function for\nelectrons that emit in the mm-infrared. This is particularly\nimportant to understand in the context of interpreting the variable\ninfrared emission and resolved mm images of Sgr A* (e.g.,\n\\citealt{Doeleman2008, Do2009}). In the near future, more\ndetailed calculations of test particle electron acceleration in\nshearing box simulations can be used to quantify the uncertain\ndimensionless coefficient A in the above acceleration efficiency.\n\nA second potential application of our results is to high energy emission from luminous accreting black holes, which can be produced by a combination of thermal and non-thermal processes. However, phenomenological models of this emission suggest that $\\tau_T \\sim\n0.1-1$ in the emission region \\citep{Haardt1991,Esin1997}. As a\nresult, it is unlikely that the particle acceleration found here is\nsufficiently rapid to compete with radiative losses by synchrotron and\ninverse Compton emission. \n\n\\begin{acknowledgements}\nThis material is based on work supported by the National Science\nFoundation Graduate Research Fellowship under Grant\nNo. DGE-1106400. This work was also supported in part by\nNASA HTP grant NNX11AJ37G, NSF grant AST-1333682, a Simons Investigator award from the Simons Foundation, the David and Lucile Packard Foundation,\nand the Thomas Alison Schneider Chair in Physics at UC Berkeley.\nComputing time was provided by the National Science Foundation\nTeraGrid\/XSEDE resource on the Trestles and Kraken\nsupercomputer.\n\\end{acknowledgements}\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nOur goal in this paper is to introduce a general framework---a parabolic blow up method---to study the asymptotic nature of a multiplicity 1 Brakke flow near certain generic singularities of the flow. The theorems we prove here using this framework, which we shall describe shortly, concern the simplest non-trivial situation, namely, the asymptotic behavior near a static triple junction of a Brakke flow of planar \\emph{networks}, i.e.\\ a 1-parameter family of \n1-dimensional sets (corresponding to integral 1-varifolds) moving by generalized curvature in a domain \n$U\\subset{\\mathbb R}^2$ over a time interval.\n\nIn what might be called the classical setting, such a 1-dimensional flow consists of a locally finite \nunion of smoothly embedded open curves moving smoothly \nwith velocity equal to the \ncurvature vector at each point and time, and such that at each of their boundary points (in $U$), three curves meet smoothly at 120 degree angles. Since globally in space the flow decreases the\ntotal length of the curves in time, the curvature flow of networks\nmodels the motion of grain boundaries driven by interfacial surface tension \\cite{Gurtin}.\nThe 120 degree angle condition in this context is called the \\emph{Herring condition}. \n\nWhile such classical solutions may stay classical time-globally in some special cases\n\\cite{Garcke,Ikota2005,Kinderlehrer,MMN,Mantegazza}, in general, various singularities may occur in finite time. For instance, physically, in the motion of grain boundaries, \none observes that \ntwo or more triple junctions collide with each other and small grains are eliminated. \nThis is a process of grain coarsening which should be an integral part of the mathematical modeling of the motion.\nMotivated partly by such phenomena, in the pioneering work \\cite{Brakke}, Brakke introduced \na generalized notion of mean curvature flow (abbreviated MCF hereafter) \nusing the notion of varifolds in Geometric Measure Theory, \nand studied existence and regularity of surfaces of any dimension and codimension moving by mean curvature.\nBrakke's MCF (which we also call ``Brakke flow'', see Sec. \\ref{defBF} for the definition)\nnaturally accommodates flows with\nsingularities, but allows the possibility of sudden loss of measure and non-uniqueness.\n\nWhile the study of regularity of various classes of stationary varifolds (generalized minimal submanifolds), which are the equilibrium solutions of the Brakke flow, \nhas seen several advances in the past 50 years or so, \nmuch less has been known concerning Brakke flows apart from Brakke's own original work \\cite{Brakke}.\nRecently, the work of Kasai and the first author \\cite{Kasai-Tonegawa} and \nof the first author \\cite{Tonegawa} gave a new, streamlined proof of a generalization of Brakke's local regularity theorem (\\cite{Brakke})\nwhich establishes a.e$.$ smoothness in time and space under the hypothesis that the moving surfaces have multiplicity 1 a.e. \nRoughly speaking, just like Allard's regularity theorem \\cite{Allard}\nfor stationary varifolds, \nthe results in \\cite{Brakke,Kasai-Tonegawa,Tonegawa} show that a nearly flat part of a unit density Brakke flow is necessarily a smooth MCF, i.e. locally $C^{\\infty}$ embedded submanifolds in space-time and moving smoothly\nby the mean curvature. \\cite{Brakke,Kasai-Tonegawa,Tonegawa} however do not give any structural information about the flow in the vicinity \nof singularities including triple junctions. \nWe note that \nthere have been important results on Brakke's regularity theorem\nwhen one is interested in special Brakke flows such as those arising as weak \nlimits of smooth MCFs (see for instance \\cite{Ecker1,Ecker,White1}) or those produced by Ilmanen's elliptic regularization method (\\cite{Ilmanen0}). \n\n\nFor both minimal submanifolds and mean curvature flows, as well as for numerous other problems in geometric analysis and non-linear PDE, describing the asymptotic behavior of the objects in question on approach to their singular sets, and understanding the structure of the singular sets themselves, remain largely open major challenges. For multiplicity 1 classes of minimal submanifolds,\nthe seminal work of Simon \\cite{Simon2.1, Simon2, Simon3} established asymptotics near certain singularities, and also the structure results for the singular sets in the full generality of varying tangent cone types and when there is no topological obstruction to perturbing singularities away. Earlier work of Allard--Almgren \\cite{AA1}, Taylor \\cite{Taylor2} and White \\cite{White4} proved similar results in situations where the tangent cones satisfy more restrictive conditions in addition to the multiplicity 1 condition. \nRecent work of \nthe second author \\cite{Wick, Wick1} and of Krummel and the second author \\cite{KW1, KW2} establish regularity results and asymptotics near singularities (branch points) for certain classes of minimal and related submanifolds for which the multiplicity 1 condition either fails or is not assumed a priori. The work \\cite{Taylor2, White4, Simon2, Simon3, Wick1, KW2} establish, for various classes of minimal submanifolds, fine properties of the singular sets themselves, such as smoothness or rectifiability. \n\n\nAmong the known results in this direction for MCF is the \nrecent deep work of Colding--Minicozzi \\cite{Colding, Colding1} (aided also by the work of Colding--Ilmanen--Minicozzi \\cite{Colding0}) which proves, for any flow of hypersurfaces, uniqueness of the tangent flow whenever a multiplicity 1 shrinking cylinder occurs as one tangent flow, and provides strong structural information on the singular sets of hypersurface flows whose tangent flows at singularities are all shrinking multiplicity 1 cylinders. In particular, these results apply to MCFs of mean-convex hypersurfaces, for which the earlier work of White \\cite{White3, White2, White6} had established that all tangent flows at singularities are shrinking multiplicity 1 cylinders. \nThe work of Schulze~\\cite{Schulze} established such asymptotics near compact singularities of multiplicity 1 flows. \n\nSimon's work \\cite{Simon2.1, Simon2, Simon3} mentioned above developed two far reaching methods---one based on an infinite dimensional Lojasiewicz inequality and the other based on the so called blow-up method---for studying the singular sets of minimal submanifolds. The work of Colding--Minicozzi \\cite{Colding, Colding1} and of Schulze \\cite{Schulze} referred to above study singularities of MCFs by establishing appropriate Lojasiewicz inequalities. In the present paper we introduce a parabolic version of the blow-up method for studying fine properties of Brakke flow singularities, and implement it fully in the simplest non-trivial case with moving singularities, namely, for 1-dimensional Brakke flows in the vicinity of a static multiplicity 1 triple junction. \n\n\nOur main result here (Theorem~\\ref{mainreg} below) is a precise version of the following:\n\n\\noindent\n{\\bf Theorem 1}. \\emph{If a 1-dimensional Brakke flow in a planar region is weakly close in a space-time neighborhood to a static multiplicity 1 triple junction $J$, then in a smaller space-time neighborhood, it is regular in the sense that it consists of three curves coming smoothly together at a single point, staying smoothly close to $J$ and moving by curvature.} \n\nHere, by \\emph{weakly close} we mean that the flow has small space-time $L^{2}$ distance from $J$ and satisfies suitable mass hypotheses at the initial and final times. (See the statement of Theorem~\\ref{mainreg}.) \nThis is a natural, easily verifiable criterion in the analysis of singularities. For example, if a $1$-dimensional Brakke flow at a space-time singular point has a tangent\nflow equal to a static multiplicity 1 triple junction, then our theorem is applicable near that point, and implies uniqueness of the tangent flow, and moreover, that in a space-time neighborhood of the point, the flow itself is a regular triple junction moving by curvature.\nUsing the above result and White's stratification theorem \\cite{White0}, we deduce the following partial regularity result (Theorem~\\ref{mainpa}) for 1-dimensional Brakke flows in a planar region:\n\n\\noindent\n{\\bf Theorem 2}. \\emph{If a 1-dimensional Brakke flow in a planar region has no static tangent flow consisting of more than 3 half-lines meeting at the origin (or, equivalently, if the density of every static tangent flow is $<2$), \nthen the flow is a classical network flow away from a closed singular set of parabolic Hausdorff dimension at most 1; in particular, such a flow is a classical network flow at each instance of time except for a closed set of times of ordinary Hausdorff dimension at most 1\/2}. \n\nThe hypothesis concerning tangent flows in Theorem 2 is motivated by the physics of motion of grain boundaries where the triple junction seems to be the unique stable junction; other types of junctions may form but seem to disappear instantly.\n(See more discussion after the statement of Theorem~\\ref{mainpa}.)\nIn this regard, \nTheorem 1 may also be understood as a result about local asymptotic stability of triple junction within a \nbroad class of weak varifold solutions and with respect to a weak topology of measure. There have been \nrelated works which prove global-in-time asymptotic stability of triple junction, i.e. classical \nnetwork curvature flow converges to a triple junction as $t\\rightarrow\\infty$ \\cite{Garcke,Ikota2005,Kinderlehrer,MMN}. \n\nThe above results are formulated and proved here for a class of 1-dimensional flows more general than Brakke flows, in which the ``velocity of motion'' is given by the curvature vector plus any given space-time dependent vector field satisfying an optimal integrability condition. \n\nThe simplicity of the spatial 1-dimensionality of the problem considered here allows us to essentially isolate the difficulties arising from the presence of the time variable. Although some of our arguments here take advantage of the spatial 1-dimensionality, the overall method introduced here appears to hold promise for much further development.\nIndeed, many of the estimates developed here either directly extend to or can easily be modified\nto work for Brakke flows of general dimension and codimension weakly close to certain types of multiplicity 1 tangent flows, including higher dimensional static triple junctions. However, there are also a few ingredients for which the arguments needed in higher dimensions seem to be much more complicated. We shall address such generalizations elsewhere. \n\nOne may also naturally wonder what could go wrong if the triple junction $J$ is replaced by a 1-dimensional multiplicity 1 stationary junction $J_{N}$ with $N(>3)$ half lines meeting at the origin. \nIn this case, the direct analogue of the conclusion of Theorem 1, namely that in the space-time interior the flow consists of $N$ embedded curves coming together smoothly at a single point, is false. For instance in case $N=4$, consider the static junction $J_{4}$ consisting of two intersecting lines at the origin with a 120 degree angle between them. We may construct a static configuration arbitrarily close to $J_{4}$ with precisely two triple junction singularities by splitting apart $J_{4}$ at the origin into two pairs of half-lines each making a 120 degree angle and connecting their vertices by a short line segment, and imagine non static flows that remain close to this configuration.\nFrom the point of view of our method here (see below for an outline), general uniform regularity estimates fail (as they must in view of the example just mentioned) without further hypotheses in case $N \\geq 4$ because the flow need not have the property that \nthe moving curve at time $t$ has a singular point of density $\\geq N\/2$ for a.e.\\ $t$. A further complicating issue in this case is that even when the curve does have singularities with the right density, its tangent cones may contain \\emph{higher multiplicity} lines or half-lines. Neither of these issues arises in the case $N = 3.$ \n\nSmoothly embedded 1-dimensional flows on the other hand cannot stay close to a singular static junction for too long. In higher dimensions, an analogue is the question of what one can say about minimal surfaces weakly close to a pair of transverse planes (say, in ${\\mathbb R}^{3}$). In that case, the difficulties are illustrated by Scherk's surfaces which show that no uniform estimates can hold without further hypotheses. \n\n\\noindent\n{\\bf An outline of the proof of Theorem 1:} Without loss of generality, let $B_{2} \\times [0, 4]$ be the space-time region in Theorem 1, where $B_{2}$ is the open ball in ${\\mathbb R}^{2}$ (space) with radius $2$ and center at the origin. Let $V_{t}$, $t \\in [0, 4]$ denote the moving 1-varifold at time $t$, and let $J$ denote a fixed stationary triple junction with vertex at the origin. Thus $J$ consists of three half-lines meeting at 120 degree angles at the origin. By assumption, the flow is weakly close to $J$, which in particular means that the space-time $L^{2}$ distance (height excess) $\\mu$ of the flow $\\{V_{t}\\}_{t \\in [0, 4]}$ relative to $J$, defined by $$\\mu = \\left(\\int_{0}^{4}\\int_{B_{2}} {\\rm dist}^{2} \\, (x, J) \\, d\\|V_{t}\\|(x) dt \\right)^{1\/2},$$ is small. \n\nAs mentioned before, our proof of Theorem 1 is based on a parabolic version of the blow-up method. We first use the full \nstrength of \\cite{Kasai-Tonegawa} to obtain (in Proposition~\\ref{smprop}) a graphical representation of the varifolds $V_{t}$, with an appropriate estimate, away from the center of $J$. We use this graphical representation to establish various a priori space-time and time uniform $L^{2}$-estimates that control the behavior of the flow in the region near the center of $J$. In particular, a key step is to show that $\\mu$ does not concentrate near the center of $J$. \n\nOur approach to establishing this a priori non-concentration estimate is inspired by the basic strategy developed by Simon \\cite{Simon2} for minimal submanifolds. A key ingredient in Simon's method is the monotonicity formula for minimal submanifolds, whose role here is played by (a certain local estimate inspired by) the Huisken monotonicity formula.\nIn the present parabolic setting, there are several interesting new aspects also. These stem from firstly the fact that all we have at our \ndisposal is Brakke's inequality defining the flow---which a priori only tells us something about the rate of change of mass (length) and not much about the velocity of motion---and secondly the fact that we need a number of nontrivial preliminary estimates \ninvolving curvature, which in the case of minimal submanifolds are not needed (regardless of the dimension). \nA key such estimate (established in Proposition~\\ref{hde}) gives an interior space-time $L^{2}$ bound for the generalized curvature $h = h(V_{t}, x)$ of $V_{t}$ (where $x \\in {\\rm spt} \\, \\|V_{t}||$) in terms of $\\mu$ whenever $\\mu$ is sufficiently small; said more precisely, \n\\begin{equation*}\n\\int_{1}^{3}\\int_{B_{3\/2}}|h|^2\\,\nd\\|V_t\\|dt\\leq c\\mu^2\n\\end{equation*}\nprovided $\\mu$ is sufficiently small, where $c$ is a fixed constant independent of the flow. We use this estimate and computations similar to those used in the derivation of the Huisken monotonicity formula \\cite{Huisken} (see also \\cite{Ilmanenp}) to \nestablish (in Proposition~\\ref{pade}), whenever $\\mu$ is sufficiently small, that \n\\begin{equation*}\n\\int_{5\/4}^{s}\\int_{B_{1}}\\left|h+\\frac{x^{\\perp}}{2(s-t)}\\right|^2 \\rho_{(0,s)}(x,t)\\, d{\\|V_{t}\\|}(x) dt \\leq c\\mu^{2}\n\\end{equation*}\nfor any $s \\in [3\/2, 3]$ such that $h(V_{s}, \\cdot) \\in L^{2}_{\\rm loc} \\, (\\|V_{s}\\|)$ and $\\Theta \\, (\\|V_{s}\\|, 0) \\geq \\Theta \\, (\\|J\\|, 0) = 3\/2$, where $c$ is a fixed constant independent of the flow, $\\rho_{(0,s)}(x,t) = (4\\pi(s-t))^{-1\/2}e^{\\frac{-|x|^{2}}{4(s-t)}}$ ($-\\infty < t < s < \\infty$) is the backwards heat kernel\nwith pole at $(0,s)$ and $\\Theta \\, (\\|V\\|, Z)$ denotes the density of $V$ at $Z$. \nThis bound is then used (in Proposition~\\ref{tildest}) to obtain, for any $s \\in [3\/2, 3]$ as above and any $\\kappa \\in (0, 1)$, the crucial estimate \n\\begin{equation*}\n\\sup_{t \\in [5\/4, s)} (s-t)^{-\\kappa}\\int_{B_{3\/4}}\\rho_{(0, s)}(\\cdot, t) {\\rm dist}^{2} \\, (\\cdot, J) \\, d\\|V_{t}\\|(x) \\leq c_{0}\\mu^{2},\n\\end{equation*}\nagain provided $\\mu$ is sufficiently small, where $c_{0}$ depends only on $\\kappa$. This says that the $L^2$ distance of $V_{t}$ from the triple junction $J$ weighted by the backwards heat kernel decays quickly in time; in particular, this estimate implies that the contribution to $\\mu^{2}$ coming from a small spatial neighborhood of the origin and a slightly smaller time interval is a small proportion of $\\mu^{2}$. \n\n\nUsing these estimates, we carry out a careful blow-up analysis in Sec.~\\ref{blowup-analysis}. We emphasize that the term $(s-t)^{-\\kappa}$ appearing in the preceding estimate, though not needed for the non-concentration conclusion just pointed out, plays an important role in the blow up analysis. Once the appropriate asymptotic decay for the blow-ups are established, we obtain a space-time excess improvement lemma (Lemma~\\ref{blowprop7}) for the flow, the iteration of which leads to Theorem 1 in a fairly standard way. \n\n\n\\noindent\n{\\bf Organization of the paper:} In Sec.\\,2 we fix notation and state our main results. In Sec.\\,3\nwe use results of \\cite{Kasai-Tonegawa} to give a graph representation of the moving curves away from the center of the triple junction. The \nmain result in Sec.\\,4 is Proposition~\\ref{hde}, which gives a time-uniform estimate on the difference of length\nbetween the moving curve and the triple junction in terms of the space-time $L^2$ distance of the flow to the triple junction. The \nsame estimate gives an $L^2$ curvature estimate in terms of the $L^2$ distance. Sec.\\,5 contains\nthe main non-concentration estimate, Proposition~\\ref{tildest}, which shows that the $L^2$ distance does not concentrate\naround the junction point. This is used in Sec.\\,6 to estimate the location of and the H\\\"{o}lder norm (in time) for the junction points in term of the $L^2$ distance. All of these estimates are used to carry out a blow-up argument in Sec.\\,7 on each of the three rays of the triple \njunction and to show that the three pieces of the blow-up come together at a single point in a regular fashion. Sec.\\,8 describes the iteration procedure giving a H\\\"{o}lder estimate\nof the gradient up to the (moving) junction points, proving the main local regularity theorem \n(Theorem~\\ref{mainreg}).\nSec.\\,9 contains the proof of the partial regularity theorem (Theorem~\\ref{mainpa}). \nSec.\\,10 contains a further result concerning the nature of the tangent flows at singular points of a flow satisfying the hypotheses of Theorem~\\ref{mainpa}. \n\\section{Notation, background and the main theorems}\n\\subsection{Basic notation}\n Let ${\\mathbb N}$ be the the set of natural numbers and let ${\\mathbb R}^+=\\{x\\geq 0\\}$. For $r\\in (0,\\infty)$\n and $a\\in {\\mathbb R}^2$, define $B_r(a)=\\{x\\in {\\mathbb R}^2 : |x-a| 0$, let $Q_{R} =\\{(s,t)\\in (-R,R)\\times(-R^2,R^2)\\}$. We shall use the following norm for functions $f \\,: \\, Q_{R} \\to {\\mathbb R}$:\n\\begin{equation*}\n\\begin{split}\n\\|f\\|_{C^{1,\\zeta}(Q_{R})}=&\\sup_{(s,t)\\in Q_{R}}(R^{-1}|f(s,t)| +|\\nabla f(s,t)|)\\\\\n&+\\sup_{(s_1,t_2),(s_2,t_2)\\in Q_{R}, (s_1,t_2) \\neq (s_2,t_2)}\n\\frac{R^{\\zeta}|\\nabla f (s_1,t_1)-\\nabla f (s_2,t_2)|}{\\max\\{|s_1-s_2|,|t_1-t_2|^{\\frac{1}{2}}\\}^{\\zeta}} \\\\\n&+\\sup_{(s,t_1),(s,t_2)\\in Q_{R}, t_{1} \\neq t_{2}} \\frac{R^{\\zeta}|f(s,t_1)-f(s,t_2)|}{|t_1-t_2|^{\\frac{1+\\zeta}{2}}}.\n\\end{split}\n\\end{equation*}\n Note that this norm is \ninvariant under the parabolic change of variables in the sense that if $\\tilde f(\\tilde s, \\tilde t)=R^{-1}f(s,t)$ where $\\tilde s=R^{-1} s$ and $\\tilde t=R^{-2}t$, then $\\|\\tilde f\\|_{C^{1, \\zeta}(Q_{1})} = \\|f\\|_{C^{1, \\zeta}(Q_{R})}$. We shall denote by $C^{1,\\zeta}(Q_{R})$ the space of functions $f \\, : \\, Q_{R} \\to {\\mathbb R}$ with $\\|f\\|_{C^{1, \\zeta}} (Q_{R}) < \\infty$. \n\\subsection{Hypotheses and the main theorems}\\label{hypotheses}\nLet $U\\subseteq {\\mathbb R}^2$ be open and let $I \\subseteq {\\mathbb R}$ be an interval (i.e.\\ a connected open subset of ${\\mathbb R}$). Assume\n\n\\begin{itemize}\n\\item[(A0)] $p \\in [2, \\infty)$ and $q \\in (2, \\infty)$ are fixed numbers such that \n\\begin{equation}\n\\zeta \\equiv 1-\\frac{1}{p}-\\frac{2}{q}>0.\n\\label{power}\n\\end{equation}\n\\end{itemize}\nFor each $t \\in I$, let $V_{t}$ be a 1-varifold in $U$ and \n$u(\\cdot, t) \\, : \\, U \\to {\\mathbb R}^{2}$ a $\\|V_{t}\\|$ measurable vector field such that: \n\n\\begin{itemize}\n\\item[(A1)] $V_t\\in {\\bf IV}_1(U)$ for a.e.\\ $t\\in I$;\n\\item[(A2)] there exists $E_{1} \\in [1, \\infty)$ such that for each $B_r(x)\\subset U$ and each $t\\in I$,\n\\begin{equation}\n\\|V_t\\|(B_r(x))\\leq 2r E_1;\n\\label{ddd}\n\\end{equation}\n\n\\item[(A3)] $u$ satisfies\n\\begin{equation}\n\\left(\\int_{I}\\left(\\int_U |u(x,t)|^p\\, d\\|V_t\\|(x)\\right)^{\\frac{q}{p}}\\, dt\\right)^{\\frac{1}{q}}<\\infty;\n\\label{powerfin}\n\\end{equation}\n\\item[(A4)] for each $\\phi\\in C^1(U\\times I;{\\mathbb R}^+)$ with $\\phi(\\cdot,t)\\in C_c^1(U)$ for $t \\in I,$ and for any $t_{1}, t_{2} \\in I$ with $t_1-\\infty$ for a.e.\\ $t\\in I$ hence $h(V_t,\\cdot)$ exists and in $L^2_{loc}(\\|V_t\\|)$. Thus in this case, (A1), (A3) and (A4)\nare equivalent to $\\{V_t\\}$ being a Brakke flow.\nMoreover, Huisken's monotonicity formula which can be derived from \\eqref{firstlen3} \nshows (see e.g.\\ \\cite{Ilmanenp}) that\n(A2) is locally satisfied for some $E_1$ so that Brakke flows always satisfy (A1)-(A4) locally. \n(A4) is a weak formulation of the condition $v=h+u^{\\perp}$, \nas discussed in Sec. \\ref{defBFG}.\n\nThe following is our $\\varepsilon$-regularity theorem, whose proof takes up a \nmajor part of the paper:\n\n\\begin{thm}\\label{mainreg}\nCorresponding to $p$, $q$ as in $({\\rm A}0)$, $E_{1} \\in [1, \\infty)$ and $\\nu\\in (0,1)$, there exist $\\Cl[eps]{e-8}\\in (0,1)$ and $\\Cl[c]{c-14}\\in (1,\\infty)$ such that\nthe following holds: For $R\\in (0,\\infty)$ and $U=B_{4R}$, let $\\{V_t\\}_{t\\in [-2R^2,2R^2]}$ and\n$\\{u(\\cdot,t)\\}_{t\\in [-2R^2,2R^2]}$ satisfy (A1)-(A4) with $I = [-2R^{2}, 2R^{2}]$. Suppose \n\\begin{equation}\n\\mu \\equiv \\Big( R^{-5}\\int_{-2R^2}^{2R^2} \\int_{B_{4R}} {\\rm dist}\\,(\\cdot,J)^2 \\, d\\|V_t\\|dt\\Big)^{\\frac12}<\\Cr{e-8};\n\\label{mr1}\n\\end{equation}\nthere exist $j_{1}, j_{2} \\in \\{1, 2, 3\\}$ such that \n\\begin{equation}\nR^{-1}\\|V_{-2R^2}\\|(\\phi_{j_1,J,R})\\leq (2-\\nu)\\Cr{c-p}, \\ \\ R^{-1}\\|V_{2R^2}\\|(\\phi_{j_2,J,R})\\geq \\nu\\Cr{c-p}\n\\label{mr2}\n\\end{equation}\nwhere $J$, $\\Cr{c-p}$ and $\\phi_{j,J,R}$ are as defined in \\eqref{regtridef}, \\eqref{c-pdef} and \\eqref{hje2} respectively, and \n\\begin{equation}\n\\|u\\| \\equiv R^{\\zeta} \\Big(\\int_{-2R^2}^{2R^2}\\big(\\int_{B_{4R}} |u|^p\\, d\\|V_t\\|\\big)^{\\frac{q}{p}}\\Big)^{\\frac{1}{q}}\n<\\Cr{e-8}.\n\\label{mr3}\n\\end{equation}\nThen there exists $\\hat a\\in C^{\\frac{1+\\zeta}{2}}([-R^2,R^2];B_R)$\nand, letting \n\\begin{equation}\nl_j(t)=\\mbox{first coordinate of }{\\bf R}_{-\\frac{2\\pi(j-1)}{3}}\n\\left(\\hat a(t)\\right)\\ \\ and \\ \\ \nD_j =\\cup_{t\\in [-R^2,R^2]} \\big([l_j(t),R]\\times\\{t\\}\\big),\n\\label{mr4}\n\\end{equation}\nthere exist functions $f_j\\in C^{1,\\zeta}(D_j),$ $j=1,2,3,$\nsuch that for all $t\\in [-R^2,R^2]$, we have\n\\begin{equation}\n\\label{mr6}\n{\\bf R}_{-\\frac{2\\pi(j-1)}{3}}({\\rm spt}\\,\\|V_t\\|)\\cap \\big([l_j(t),R]\\times\n[-R,R]\\big)=\\{(x,f_j(x,t)): x\\in [l_j(t),R]\\}\n\\end{equation}\nfor $j=1,2,3$ and \n\\begin{equation}\n\\frac{\\partial f_1}{\\partial x}(l_1(t),t)=\\frac{\\partial f_2}{\\partial x}(l_2(t),t)\n=\\frac{\\partial f_3}{\\partial x}(l_3(t),t).\n\\label{mr6.5}\n\\end{equation}\nFurthermore, we have\n\\begin{equation}\n\\label{mr7}\n\\|\\hat a\\|_{C^{\\frac{1+\\zeta}{2}}([-R^2,R^2];B_R)}+\\sum_{j=1}^3\n\\|f_j\\|_{C^{1,\\zeta}(D_j)}\\leq \\Cr{c-14}\\max\\{\\mu,\\|u\\|\\}.\n\\end{equation}\n\\end{thm}\n\n\\noindent\n{\\bf Remark:} In case $u\\in C^{\\beta}(B_{4R}\\times [-2R^2,2R^2];{\\mathbb R}^2)$ \n(where H\\\"{o}lder \ncontinuity is in the usual parabolic sense), the result of\n\\cite{Tonegawa} shows that $f_j$ are $C^{2,\\beta}$ away from \nthe junction point and that the flow satisfies $v=h+u^{\\perp}$ in the classical\nsense. In fact in this case, up-to-the-junction-point $C^{2,\\beta}$ regularity of $f_j$ as well as \n$C^{1,\\frac{\\beta}{2}}$ regularity of $\\hat a$ also hold, and can be proved \nby the well-know reflection technique of \\cite{KNS} combined with the\nregularity theory of linear parabolic systems \\cite{Solo}. If $u$ is smooth, \nor zero in particular, then $\\hat a$ is smooth, and $f_j$ are smooth up to the junction point. Note that\nfor the reflection technique of \\cite{KNS}, having $C^{1,\\zeta}$ regularity given by our theorem \nprovides the crucial starting hypothesis. \n\n\\vspace{.2in}\n\nWe note that all quantities are scale invariant under parabolic change of variables\nso we may and we shall, without loss of generality, set $R=1$ in the proof of Theorem~\\ref{mainreg}. The inequality \\eqref{mr1} provides a closeness\nto $J$ of $\\|V_t\\|$ in the $L^2$ distance, and \\eqref{mr2} requires some closeness to $J$\nin terms of measure at the initial and final times. The latter also prevents complete loss\nof measure $\\|V_t\\|$ during this time interval. Assumption \\eqref{mr3} ensures smallness\nof the perturbation from the $u=0$ case, and obviously is not needed if $u=0$. \nThe conclusion is that each time slice of the flow in a smaller space-time domain consists of three embedded curves meeting precisely at one common junction point, that this junction point $\\hat{a}(t)$ at time $t$ is a\nH\\\"{o}lder continuous function of $t$, and that the three curves are represented as\n$C^{1,\\zeta}$ graphs up to the junction point as in \\eqref{mr4} and \\eqref{mr6}, satisfying the estimate \\eqref{mr7}. Moreover, \nat each time, the three curves meet at 120 degree angles, as expressed by \\eqref{mr6.5}. \nIf more regularity is assumed on $u$, then we have, as stated in\nthe Remark above, better regularity up to the junction point.\n\nBy combining Theorem \\ref{mainreg} with a stratification theorem of White \\cite{White0}, we obtain \nthe following partial regularity theorem. We make an assumption on the density of static\ntangent flows, which is equivalent to assuming that any static tangent flow is either a unit\ndensity line or a unit density triple junction. For \nthe precise definition of tangent flow, \nsee Sec.\\,\\ref{secpart}. Tangent flows are analogous to tangent cones for minimal submanifolds:\nat each point in space-time, at least one tangent flow is obtained by passing to a subsequential varifold limit of parabolic rescalings of the flow, and tangent flows enjoy a nice\nhomogeneity property called backwards-cone-like (see Sec.\\,\\ref{secpart} (b)). \n\\begin{thm}\nIn addition to the assumptions ($A1$)-($A4$), assume that \n\\begin{itemize}\n\\item[($A5$)] at each point in space-time, whenever a tangent flow to $\\{V_{t}\\}_{t \\in I}$ is static, the density at the\norigin of the tangent flow is strictly less than 2. \n\\end{itemize}\nThen there exists a closed set $\\Sigma_1\\subset U\\times I$ with the parabolic Hausdorff \ndimension at most 1 such that the flow in $U \\times I \\setminus \\Sigma_{1}$ is classical in the sense that for any $(x,t)\\in U\\times I \\setminus \\Sigma_1$, there exists \na space-time neighborhood $U_{x,t}$ containing $(x,t)$ such that $U_{x,t}\\cap \\cup_{t'}({\\rm spt}\\,\\|V_{t'}\\|\\times\\{t'\\})$ is\neither empty, a $C^{1,\\zeta}$ graph over a line segment or a $C^{1,\\zeta}$ triple junction as described in Theorem~\\ref{mainreg}.\n\\label{mainpa}\n\\end{thm}\n\n\\noindent\n{\\bf Remarks:} {\\bf (1)} Under hypotheses (A1)-(A4), we may completely classify all non-trivial static tangent flows. \nThey are time-independent stationary integral\nvarifolds whose supports are unions of half-lines emanating from the origin.\nThus hypothesis (A5) requires that any static tangent flow is a single line or\na triple junction, either one with unit density, and nothing else. \nThis hypothesis is motivated by the fact that any static tangent flow with density greater than or equal to 2 should be unstable for various physical models. For the motion of grain boundaries, one\nobserves that junctions with more than 3 edges appear and break up instantaneously. \nMathematically, any junction (including lines) with multiplicity strictly greater than 1 is not\nmass minimizing in the sense that one can always set the multiplicity equal to 1 and reduce\nthe mass. Any unit density junction with more than 3 edges may be mapped by a \nsuitable Lipschitz function so that the image of the map has locally \nless ${\\mathcal H}^1$ measure {\\it as a set}.\n(Note that the usual varifold push-forward counts multiplicities of the image and the mapping\nhere is different from it.) It is called {\\it reduced mass model} according to Brakke \\cite[p.57]{Brakke}.\n\n\\noindent\n{\\bf (2)} Other than the singularities coming from collisions of triple junctions, \nwe may also have some curve disappearing suddenly. Such singularities are included\nin the closed set $\\Sigma_1$. \n\n\\noindent\n{\\bf (3)} The parabolic Hausdorff dimension counts the time variable as 2. \nThus, $\\Sigma_{1}$ having parabolic dimension at most 1 implies that the times at which singularities can occur form a closed subset of $I$ of usual \nHausdorff dimension at most $\\frac{1}{2}.$ \n\n\\noindent\n{\\bf (4)} The short-time existence of classical network flows (i.e.\\ those consisting of curves meeting smoothly and only at a locally finite number of triple junctions) was established by Bronsard--Reitich \\cite{Bronsard} when the initial network itself is classical, and has recently been extended to more general initial networks satisfying certain regularity and non-degeneracy assumptions by Ilmanen--Neves--Schulze \\cite{INS}. (The work \\cite{INS} also gives a result that says that a flow weakly close to a triple junction $J$ is $C^{1, \\alpha}$ close to $J$ in the interior (see \\cite{INS}, Theorem~1.3 and remark (iii)) in the special case when the flow is a priori assumed to be regular, using methods limited to such an priori regularity hypothesis.) It remains an interesting open problem to prove a general existence theorem for curvature flows\nsatisfying (A1)-(A5). \n\n\\noindent\n{\\bf (5)} See Sec.\\,\\ref{difpat} for a more detailed characterization of $\\Sigma_1$\nin terms of tangent flows. \n\n\\section{A graph representation away from the singularity of $J$}\n We apply results from \\cite{Kasai-Tonegawa}\nto show that the supports of the moving varifolds, in the region outside a small neighborhood of the singularity of $J$, are represented as a $C^{1,\\zeta}$ graphs, with the $C^{1,\\zeta}$\nnorm bounded in terms of the $L^2$ distance of the flow to $J$. This result will be used frequently in the rest of the paper. \n\\begin{prop}\nCorresponding to $\\tau\\in (0,\\frac12),\\nu\\in (0,1),E_1 \\in [1, \\infty)$ and $p \\in [2, \\infty)$, $q \\in (2, \\infty)$ satisfying (\\ref{power}), there exist $\\Cl[eps]{e-p}\\in (0,1)$ and \n$\\Cl[c]{c-p-1}\\in (1,\\infty)$ such that the following holds: Suppose that $\\{V_t\\}_{t\\in [0,4]}$ and $\\{u(\\cdot,t)\\}_{t\\in [0,4]}$ satisfy (A1)-(A4) with $U = B_{2}$ and $I = [0,4],$ and that\n\\begin{equation}\n\\mu \\equiv \\left(\\int_0^{4}\\int_{B_{2}}{\\rm dist}(\\cdot,J)^2\\, d\\|V_t\\|dt\\right)^{\\frac12}\\leq \\Cr{e-p},\n\\label{smep1}\n\\end{equation}\n\\begin{equation}\n\\|u\\| \\equiv \\left(\\int_0^{4}\\left(\\int_{B_{2}} |u|^p\\, d\\|V_t\\|\\right)^{\\frac{q}{p}}dt\\right)^{\\frac{1}{q}}\\leq \\Cr{e-p},\n\\label{smep2}\n\\end{equation}\n\\begin{equation}\n\\|V_0\\|(\\phi_{j_1})\\leq (2-\\nu)\\Cr{c-p} \\ \\ \\mbox{for some} \\ \\ j_{1} \\in \\{1, 2, 3\\} \\ \\ \\mbox{and}, \n\\label{smep3}\n\\end{equation}\n\\begin{equation}\n \\|V_{4}\\|(\\phi_{j_2})\\geq \\nu \\Cr{c-p} \\ \\ \\mbox{for some} \\ \\ j_2\\in \\{1,2,3\\}.\n\\label{smep4}\n\\end{equation}\nThen \n\\begin{equation}\nB_{2-\\frac{\\tau}{5}}\\cap\\left\\{{\\rm dist}(\\cdot,J)>\\frac{\\tau}{5}\\right\\}\\cap {\\rm spt}\\,\\|V_t\\|=\\emptyset \\ \\ \\ \\mbox{for each} \\ \\ t\\in [\\tau,4]\n\\label{smep5}\n\\end{equation}\nand there exist $f_j: [\\tau ,2-\\tau]\\times [\\tau,4-\\tau]\\rightarrow {\\mathbb R}$, $j=1,2,3$, such that\n$f_1,f_2,f_3$ are $C^{1,\\zeta}$ in space and $C^{\\frac{1+\\zeta}{2}}$ in time with \n\\begin{equation}\n\\begin{split}\n\\|f_j\\|_{C^{1,\\zeta}(Q)}\n\\leq \\Cr{c-p-1}\\max\\{\\mu,\\|u\\|\\}\n\\end{split}\n\\label{smep6}\n\\end{equation}\nwhere $Q=[\\tau,2-\\tau]\\times [\\tau,4-\\tau]$, \nand\n\\begin{equation}\n([\\tau ,2-\\tau]\\times [-\\tau,\\tau])\\cap \\big({\\bf R}_{-\\frac{2(j-1)\\pi}{3}}({\\rm spt}\\,\\|V_t\\|)\\big)=\n\\{(s,f_j(s,t))\\,:\\, s\\in [\\tau ,2-\\tau]\\},\n\\label{smep7}\n\\end{equation}\nfor $j=1,2,3$ and for all $t\\in [\\tau,4-\\tau]$. \nFurthermore, for a.e$.$ $t\\in [\\tau,4-\\tau]$, we have that\n\\begin{equation}\n\\label{exden}\n\\Theta(\\|V_t\\|,x) \\in \\{1, 3\/2\\} \\;\\; \\mbox{for each} \\;\\; x\\in B_{2-\\tau}\\cap {\\rm spt}\\,\\|V_t\\|.\n\\end{equation}\n\\label{smprop}\n\\end{prop}\n{\\it Proof}. \nThe claim \\eqref{smep5} may be deduced by applying \\cite[Prop.\\,6.4 \\& Cor.\\,6.3]{Kasai-Tonegawa}.\nWe also see that for fixed $\\tau \\in (0, 1\/2)$, ${\\rm spt}\\|V_t\\|\\cap B_{2-\\tau}$ approaches $J$ as $\\mu,\\,\\|u\\|\\rightarrow 0$ for all $t\\in [\\tau,4]$. \n\nTo prove the existence of a graph representation\nand the $C^{1,\\zeta}$ estimate as asserted, assume for fixed $E_{1}$, $\\nu$, $\\tau$ that\nthe claim is false. Then for each $m\\in {\\mathbb N}$\nthere exist $\\{V_t^{(m)}\\}_{t\\in [0,4]}$ and $\\{u^{(m)}(\\cdot,t)\\}_{t\\in [0,4]}$ satisfying (A1)-(A4) and \\eqref{smep1}-\\eqref{smep4} with $U = B_{2}$, $I=[0,4]$, $\\Cr{e-p} = m^{-1}$ and with $V_{t}^{(m)}, u^{(m)}$ in place of $V_{t}, u$, but there are no \nfunctions $f_{1}$, $f_{2}$, $f_{3}$ with the stated regularity satisfying \\eqref{smep6} and \\eqref{smep7} with $\\Cr{c-p-1}=m,$ $u^{(m)}$ in place of $u$ and $\\mu^{(m)} = \\left(\\int_{0}^{4}\\int_{B_{2}} {\\rm dist}\\, (\\cdot, J)^{2} \\, d\\|V_{t}^{(m)}\\| dt \\right)^{\\frac{1}{2}}$ in place of $\\mu$. \nTo obtain a contradiction, we will use \\cite[Th.\\,8.7]{Kasai-Tonegawa} which shows the existence of such a graph \nrepresentation as in the asserted conclusion under a set of hypotheses. In order to check that the hypotheses of \\cite[Th.\\,8.7]{Kasai-Tonegawa}, with $V^{(m)}_{t},$ $u^{(m)}$ in place of $V_{t}$, $u$, are satisfied for sufficiently large $m$, we first prove that\n\\begin{equation}\n\\|V_t^{(m)}\\|\\rightarrow {\\mathcal H}^1\\hbox{ {\\vrule height .3cm}{\\leaders\\hrule\\hskip.3cm}}\\hskip5.0\\mu_{J}\n\\label{smep8}\n\\end{equation}\nas $m\\rightarrow\\infty$ on $B_{2}$ for all $t\\in (0,4)$. To see this, take any $\\varphi\\in C_c^2(B_{2};{\\mathbb R}^+)$ and use\n(A4) to obtain for any $t_1,t_2\\in [0,4]$ with $t_10,\\, |v|=1,\\, \\cos\\frac{\\pi}{3}>v\\cdot (1,0)>\\cos\\frac{5\\pi}{6}\\}$, where\nwe have $|f_j(s)|>{\\rm dist}\\, ((s,f_j(s)),J)$. Denote $\\tilde{\\delta}=\n\\tilde{s}_2-\\tilde{s}_1$ and note that $\\tilde{\\delta}$ is small when $\\Cr{alpha-1}$ and $\\Cr{beta-1}$ are \nsmall. Considering the geometry of graph, for sufficiently small $\\sup |f'_j|$, we have\n\\begin{equation}\n\\tilde{\\delta}\\sup_{s\\in (\\tilde{s}_1,\\tilde{s}_2)}|f_j(s)|^2\\leq 2\\tilde{\\delta}\\inf_{s\\in (\\tilde{s}_2,\\tilde{s}_2+\\tilde{\\delta})}|f_j(s)|^2\n\\leq 2\\int_{l_j\\cap B_1}{\\rm dist}\\, (x,J)^2\\, d\\|V\\|(x).\n\\label{fir13.1}\n\\end{equation}\nOutside of $(\\tilde{s}_1,\\tilde{s}_2)$, as stated, we have $|f_j(s)|\\leq {\\rm dist}\\,((s,f_j(s)),J)$, thus we have\nby \\eqref{fir13.1}\n\\begin{equation}\n\\int_{s_j}^1|f_j|^2\\leq 3\\int_{l_j\\cap B_1}{\\rm dist}\\, (x,J)^2\\, d\\|V\\|(x).\n\\label{fir14}\n\\end{equation}\nCombining \\eqref{fir12}, \\eqref{fir13} and \\eqref{fir14}, we obtain \\eqref{fir5}. To obtain \\eqref{fir5.5}, \nnote that $\\phi_{\\rm rad}=1$ on $B_1$ and thus we need to be concerned with region of integration \nover $B_{\\frac32}\\setminus B_1$ of $\\|V\\|$. But we have \\eqref{fir2}, thus the difference of\nintegrations in this region can be estimated by $c\\hat\\beta^2$. In the estimate one uses the radial \nsymmetry of $\\phi_{\\rm rad}$ to obtain the quadratic estimate. Thus we obtain \\eqref{fir5.5} with some \nsuitable constant $\\Cr{c-1}$. \n\\hfill{$\\Box$}\n\\begin{cor}\nFor a given $E_1\\in [1,\\infty)$, let $\\Cr{c-1},\\Cr{alpha-1},\\Cr{beta-1}$ be the\ncorresponding constants obtained in Proposition \n\\ref{firprop}. For $V\\in {\\bf IV}_1(B_{2})$ with $h(V,\\cdot)\\in L^2(\\|V\\|)$, define\n$\\hat\\alpha,\\hat\\beta, \\hat\\mu$ as \\eqref{fir3.2}, \\eqref{fir2} and \\eqref{fir4}. \nAssume \\eqref{firm1}-\\eqref{fir6}.\nDefine\n\\begin{equation}\n\\hat{E}=\\|V\\|(\\phi_{\\rm rad}^2)-{\\bf c},\n\\label{hatE}\n\\end{equation}\nand assume that\n\\begin{equation}\n3\\Cr{c-1}\\hat\\beta^2\\leq |\\hat{E}|.\n\\label{cor1}\n\\end{equation}\nThen we have\n\\begin{equation}\n\\hat\\alpha^2\\geq \\min\\{\\Cr{alpha-1}^2,(2\\Cr{c-1}\\hat \\mu)^{-2}|\\hat{E}|^2\\}.\n\\label{cor2}\n\\end{equation}\n\\label{fircor}\n\\end{cor}\n{\\it Proof}. If $\\hat\\alpha\\geq \\Cr{alpha-1}$ holds, then \\eqref{cor2} holds and there is nothing further to prove. \nThus consider the case $\\hat\\alpha< \\Cr{alpha-1}$. Since \\eqref{firm1}-\\eqref{fir6} are assumed, \nwe fulfill all the assumptions of Proposition \\ref{firprop}, thus we have \\eqref{fir5.5}. Using the\nnotation of \\eqref{hatE}, this implies that we have either $|\\hat{E}|\\leq 2\\Cr{c-1} \\hat\\alpha\\hat\\mu$, \nor $|\\hat{E}|\\leq 2\\Cr{c-1}\\hat\\beta^2$. The last possibility is excluded by \\eqref{cor1}.\nThus we have $\\hat\\alpha^2\\geq (2\\Cr{c-1}\\hat\\mu)^{-2}|\\hat{E}|^2$. \nThus we have either $\\hat\\alpha^2\\geq \\Cr{alpha-1}^2$ or the last possibility. \nThis proves \\eqref{cor2}.\n\\hfill{$\\Box$}\n\nThe next ODE lemma connects Corollary \\ref{fircor} to (A4). \n\\begin{lemma}\nCorresponding to $P,T\\in (0,\\infty)$ there exist $\\Cl[c]{ha},\\Cl[c]{ha2}\\in (0,\\infty)$ \nsuch that the following holds: Given a non-negative function $g \\in L^2([0,T])$ and a monotone\ndecreasing function $\\Phi\\,:\\, [0,T]\\rightarrow{\\mathbb R}$, define\n$f\\,:\\, [0,T]\\rightarrow {\\mathbb R}^+$ by\n\\begin{equation}\nf(t)=P\\min\\left\\{1,g(t)^{-2} |\\Phi(t)|^2\\right\\}\n\\label{le1}\n\\end{equation}\nwhen $g(t)>0$ and $f(t)=P$ when $g(t)=0,$ \nand suppose that \n\\begin{equation}\n\\Phi(t_2)-\\Phi(t_1)\\leq -\\int_{t_1}^{t_2}f(t)\\, dt,\\hspace{.2cm}\n0\\leq \\forall t_1<\\forall t_2\\leq T.\n\\label{le2}\n\\end{equation}\nThen \n\\begin{itemize}\n\\item[(1)] if $\\Phi(0)\\leq \\Cr{ha}$, then \n\\begin{equation}\n\\Phi(T)\\leq \\Cr{ha2} \\|g\\|_{L^{2}([0, T])}^{2}.\n\\label{le3}\n\\end{equation}\n\\item[(2)] if $\\Phi(T)\\geq -\\Cr{ha}$, then\n\\begin{equation}\n\\Phi(0)\\geq - \\Cr{ha2}\\|g\\|_{L^{2}([0, T])}^{2}.\n\\label{le4}\n\\end{equation}\n\\end{itemize}\n\\label{ode}\n\\end{lemma}\n{\\it Proof}.\nWe prove (1) first. Set\n\\begin{equation}\n\\Cr{ha}=\\frac{PT}{8},\\ \\ \\ \\Cr{ha2}=\\frac{8}{PT^2}.\n\\label{kys}\n\\end{equation}\nWe may assume $\\Phi(t)>0$ for \nall $t\\in [0,T]$ since $\\Phi(T)\\leq 0$ otherwise and \\eqref{le3} is trivially true. \nAssume for a contradiction that \\eqref{le3} were false. \nSet \n\\begin{equation}\nc=\\|g\\|_{L^2}\\sqrt{2\/T}\n\\label{ky0}\n\\end{equation}\nand define $A_1=\\{t\\in [0,T]\\, :\\, g(t)\\geq c\\}$\nand $A_2=[0,T]\\setminus A_1$. It is easy to check that ${\\mathcal L}^1(A_1)\n\\leq T\/2$, and thus \n\\begin{equation}\n{\\mathcal L}^1 (A_2)\\geq T\/2.\n\\label{ky1}\n\\end{equation}\nWe next define \n$A_{2,a}=\\{t\\in A_2\\, :\\, g(t)^{-2}|\\Phi(t)|^2\\geq 1\\}$ and $A_{2,b}=A_2\\setminus A_{2,a}$.\nBy \\eqref{ky1}, we have either ${\\mathcal L}^1(A_{2,a})\\geq T\/4$ or ${\\mathcal L}^1(A_{2,b})\\geq T\/4$.\nIn the first case, since $f(t)=P$ on $A_{2,a}$, we have \n\\begin{equation}\n\\Phi(T)-\\Phi(0)\\leq -\\int_{A_{2,a}} f(t)\\, dt=-P{\\mathcal L}^1(A_{2,a})\\leq -\\frac{PT}{4}.\n\\label{ky2}\n\\end{equation}\nWe have $\\Phi(T)-\\Phi(0)\\geq -\\Cr{ha}$ and \\eqref{ky2} gives a contradiction\nto \\eqref{kys}. \nIn the second case, using $f(t)=P g(t)^{-2}\\Phi(t)^2$ on $A_{2,b}$ and integrating $(-\\Phi(t)^{-1})'\n\\leq -Pg(t)^{-2}$ we have\n\\begin{equation}\n-\\Phi(T)^{-1}+\\Phi(0)^{-1}\\leq -P\\int_{A_{2,b}}\\frac{dt}{g(t)^2}\n\\leq -P c^{-2} {\\mathcal L}^1 (A_{2,b})\\leq -\\frac{PT^2}{8} \\|g\\|_{L^2}^{-2}.\n\\label{ky2.5}\n\\end{equation}\nWe used $g(t)\\leq c$ on $A_{2,b}\\subset A_2$ and \\eqref{ky0}. \nSince we are assuming \\eqref{le3} is false, by \\eqref{kys}, we have\n\\begin{equation}\n-\\Phi(T)^{-1}> -\\Cr{ha2}^{-1} \\|g\\|_{L^2}^{-2}=-\\frac{PT^2}{8} \\|g\\|_{L^2}^{-2}.\n\\label{ky3}\n\\end{equation}\nTwo inequalities \\eqref{ky2.5} and \\eqref{ky3} give a contradiction. This \nproves (1). \nFor (2), replace $\\Phi(\\cdot)$ by $-\\Phi(T-\\cdot)$ and $f(\\cdot)$ by $f(T-\\cdot)$,\nand then apply the previous argument. If $-\\Phi(T)\\leq \\Cr{ha}$, then one concludes that\n$-\\Phi(0)\\leq \\Cr{ha2}\\|g\\|_{L^2}^2$. Thus we obtain the result of (2).\n\\hfill{$\\Box$}\n\\begin{prop}\nCorresponding to $\\nu,E_1,p,q$ there exist $\\Cl[eps]{e-1}\\in (0,1)$\nand $\\Cl[c]{c-2}\\in (1,\\infty)$ with the following property. \nSuppose $\\{V_t\\}_{t\\in [0,4]}$ and $\\{u(\\cdot, t)\\}_{t\\in [0,4]}$\nsatisfy (A1)-(A4) on $B_{2}\\times [0,4]$. Assume \\eqref{smep1} and \\eqref{smep2}\nwith $\\Cr{e-p}$ there replaced by $\\Cr{e-1}$, \\eqref{smep3} and \\eqref{smep4}.\nThen we have\n\\begin{equation}\n\\sup_{t\\in [1,3]}\\big|\\|V_t\\|(\\phi_{\\rm rad}^2)-{\\bf c}\\big|\n\\leq \\Cr{c-2}\\max\\{\\mu,\\|u\\|\\}^2\n\\label{thap9}\n\\end{equation}\nand \n\\begin{equation}\n\\int_{1}^{3}\\int_{B_2}|h(V_t,\\cdot)|^2\\phi_{\\rm rad}^2\\,\nd\\|V_t\\|dt\\leq \\Cr{c-2}\\max\\{\\mu,\\|u\\|\\}^2.\n\\label{thap10}\n\\end{equation}\n\\label{hde}\n\\end{prop}\n{\\it Proof}. \nWe first use Proposition \\ref{smprop} with $\\tau=\\frac14$ to obtain $\\Cr{e-p}$ and $\\Cr{c-p-1}$ so that we have\n\\eqref{smep5}-\\eqref{exden} with $\\tau=\\frac14$ there. We also use Proposition \\ref{firprop} to obtain $\\Cr{c-1},\n\\Cr{alpha-1},\\Cr{beta-1}$ corresponding to $E_1$. Then, for a.e$.$ $t\\in [\\frac{1}{2},\\frac{7}{2}]$, \nwe have conditions \\eqref{firm1}-\\eqref{fir1} and \\eqref{fir6} satisfied for $V=V_t$ there. If we further assume that \n\\begin{equation}\n\\Cr{c-p-1}\\max\\{\\mu,\\|u\\|\\}\\leq \\Cr{beta-1},\n\\label{bb1}\n\\end{equation}\nthen \\eqref{fir2} is also satisfied due to \\eqref{smep6}. \nWe restrict $\\Cr{e-1}$ so that\n\\eqref{bb1} holds by the following: \n\\begin{equation}\n\\Cr{e-1}\\leq \\min\\{\\Cr{e-p}, \\Cr{beta-1} \\Cr{c-p-1}^{-1}\\}.\n\\label{asu1}\n\\end{equation}\nNext fix $P$ and $T$ as \n\\begin{equation}\nP=\\frac{1}{16}\\min\\{\\Cr{alpha-1}^2,(2\\Cr{c-1})^{-2}\\}, \\ \\ T=\\frac12.\n\\label{th11.1}\n\\end{equation}\nWith these choices of $P$ and $T$, we obtain $\\Cr{ha}$ and\n$\\Cr{ha2}$ by Lemma \\ref{ode}.\nWith $\\Cr{ha}$ fixed, we choose a small $\\tau$ and then restrict $\\Cr{e-1}$ so that, by using Proposition \\ref{smprop}, we have\n\\begin{equation}\n\\|V_{\\frac12}\\|(\\phi_{\\rm rad}^2)\\leq {\\bf c}+\\frac{\\Cr{ha}}{2}\n\\label{va1}\n\\end{equation}\nand\n\\begin{equation}\n\\|V_{\\frac72}\\|(\\phi_{\\rm rad}^2)\\geq {\\bf c}-\\frac{\\Cr{ha}}{2}.\n\\label{va2}\n\\end{equation}\nWe will fix $\\Cr{c-2}$ later.\nWe also set \n\\begin{equation}\n\\beta_*=\\max_{j=1,2,3} \\sup_{t\\in [\\frac{1}{2},\\frac{7}{2}]} \\|f_j(\\cdot, t)\\|_{C^1(\\{|s-1|\\leq \\frac12\\})}(\\leq \\Cr{c-p-1}\\max\\{\\mu,\\|u\\|\\}\\leq \\Cr{beta-1}),\n\\label{thap4}\n\\end{equation}\nand define $C(u)$ and estimate it by H\\\"{o}lder's inequality as follows: \n\\begin{equation}\nC(u)=\\int_0^4\\int_{B_2}|u|^2\\, d\\|V_t\\|dt\\leq \\Cl[c]{cuc}(p,q,E_1)\\|u\\|^2.\n\\label{bb2}\n\\end{equation}\nDefine for $t\\in [\\frac{1}{2}, \\frac{7}{2}]$\n\\begin{equation}\nE(t)=\\|V_t\\|(\\phi_{\\rm rad}^2)-{\\bf c}-\\int_{\\frac{1}{2}}^t\\int_{B_2} |u|^2\\phi_{\\rm rad}^2\\, d\\|V_s\\|ds-\\Cl[c]{c-3} \\beta_*^2 (t-\\frac{1}{2}),\n\\label{th1}\n\\end{equation}\nwhere $\\Cr{c-3}$ will be fixed later.\nWe first prove that \n\\begin{equation}\nE(t_2)-E(t_1)\\leq -\\frac14 \\int_{t_1}^{t_2}\\int_{B_2} |h(V_t,\\cdot)|^2\\phi_{\\rm rad}^2\\, d\\|V_t\\|dt,\n\\hspace{.2cm}\\frac{1}{2}\\leq \\forall t_1<\\forall t_2\\leq \\frac{7}{2}.\n\\label{th2}\n\\end{equation}\nBy (A1), (A2) and (A4), for a.e$.$ $t$, we have $V_t\\in {\\bf IV}_1(B_2)$, $h(V_t,\\cdot)\\in L^2(\\|V_t\\|)$, $u(\\cdot,t)\\in L^2(\\|V_t\\|)$. At such time $t$, using the perpendicularity of \nmean curvature \\eqref{perpthm}, (omitting $t$ dependence for simplicity)\n\\begin{equation}\n{\\mathcal B}(V,u,\\phi_{\\rm rad}^2)\\leq \\int_{B_2}-|h|^2\\phi_{\\rm rad}^2+\\phi_{\\rm rad}^2|h||u|\n+|u^{\\perp}\\cdot\\nabla\\phi_{\\rm rad}^2|+(\\nabla\\phi_{\\rm rad}^2)^{\\perp}\\cdot h\\, d\\|V\\|.\n\\label{th3}\n\\end{equation}\nThe last term of \\eqref{th3} may be computed as \n\\begin{equation}\n\\int_{G_1(B_2)} 2\\phi_{\\rm rad}S^{\\perp}(\\nabla\\phi_{\\rm rad})\\cdot h\\, dV\n\\leq \\frac14 \\int_{B_2}|h|^2\\phi_{\\rm rad}^2\\,d\\|V\\|+4\\int_{G_1(B_2)}|S^{\\perp}(\\nabla\\phi_{\\rm rad})|^2\\, dV.\n\\label{th4}\n\\end{equation}\nNote that $\\nabla\\phi_{\\rm rad}$ is $0$ outside of $B_{\\frac32}\\setminus B_1$, and ${\\rm spt}\\, \\|V\\|$\nis represented as the union of three graphs of $C^1$ functions by \\eqref{smep5} and \\eqref{smep7} in $B_{\\frac32}\n\\setminus B_1$. \nConsider the neighborhood of $(1,0)$ in which ${\\rm spt}\\, \\|V\\|$ is represented by $f_1$. \nSince $\\phi_{\\rm rad}$ is radially symmetric function, $\\nabla\\phi_{\\rm rad}$ at\n$(s,f_1(s))$ is parallel to $(s,f_1(s))$ and $|\\nabla\\phi_{\\rm rad}|\\leq 4$. \nOn the other hand, the projection matrix $S^{\\perp}$ at the same point is easily seen to be\n\\begin{equation}\nS^{\\perp}=(1+(f_1')^2)^{-1}\\left(\\begin{array}{ll} (f_1')^2 & -f_1' \\\\ -f_1' & 1\\end{array}\\right)\n\\label{th5}\n\\end{equation}\nwhich is obtained by computing $I-\\hat\\nu\\otimes\\hat\\nu$ with $\\hat\\nu=(1+(f_1')^2)^{-\\frac12}(1,f_1')$, $I$ being the\nidentity $2\\times 2$ matrix. Thus, we have\n\\begin{equation}\n|S^{\\perp}(\\nabla\\phi_{\\rm rad})|\\leq \\frac{4}{\n\\sqrt{s^2+f_1(s)^2}}\\big|S^{\\perp}\\left(\\begin{array}{l} s\\\\ f_1(s)\\end{array}\n\\right)\\big|\\leq c\\sqrt{(f_1')^2+f_1^2}\\leq c\\beta_*\n\\label{th6}\n\\end{equation}\nby \\eqref{thap4}, where $c$ is an absolute constant. \nWe have similar computations for $f_2$ and $f_3$. \nThus by \\eqref{th4} and \\eqref{th6}, the last term of \\eqref{th3} may be\nestimate by\n\\begin{equation}\n\\int_{B_2}(\\nabla\\phi_{\\rm rad}^2)^{\\perp}\\cdot h\\, d\\|V\\|\n\\leq \\frac14\\int_{B_2}|h|\\phi_{\\rm rad}^2\\, d\\|V\\|+c\\beta_*^2.\n\\label{th7}\n\\end{equation}\nThe same computations show that the third term of \\eqref{th3} may be estimated by\n\\begin{equation}\n\\int_{B_2}|u^{\\perp}\\cdot\\nabla\\phi_{\\rm rad}^2|\\, d\\|V\\|\n\\leq \\frac12 \\int_{B_2}|u|^2\\phi_{\\rm rad}^2\\, d\\|V\\|+c\\beta_*^2.\n\\label{th8}\n\\end{equation}\nThe second term of \\eqref{th3} may be estimated by \n\\begin{equation}\n\\int_{B_2}\\phi_{\\rm rad}^2|h||u|\\, d\\|V\\|\\leq \\frac12\\int_{B_2}\\phi_{\\rm rad}^2|h|^2\\,d\\|V\\|\n+\\frac12\\int_{B_2}\\phi_{\\rm rad}^2|u|^2\\, d\\|V\\|.\n\\label{th9}\n\\end{equation}\nCombining \\eqref{th3}, \\eqref{th7}-\\eqref{th9}, we obtain (by recovering the notation for $t$ dependence)\n\\begin{equation}\n{\\mathcal B}(V_t,u(\\cdot,t),\\phi_{\\rm rad}^2)\\leq -\\frac14\\int_{B_2}|h(V_t,\\cdot)|^2\\phi_{\\rm rad}^2\n\\, d\\|V_t\\|+\\int_{B_2}|u(\\cdot,t)|^2\\phi_{\\rm rad}^2\\, d\\|V_t\\|+\\Cr{c-3}\\beta_*^2,\n\\label{th10}\n\\end{equation}\nwhere $\\Cr{c-3}$ is an absolute constant, and this holds for a.e$.$ $t\n\\in [\\frac{1}{2},\\frac{7}{2}]$. Due to (A4) and \\eqref{th10}, now it is \nclear that the inequality \\eqref{th2} holds if we define $E(t)$ as in\n\\eqref{th1}. \nWe restrict $\\Cr{e-1}$ further by\n\\begin{equation}\n\\Cr{e-1}^2\\leq \\min\\left\\{\\frac{\\Cr{ha}}{4\\Cr{cuc}}, \\frac{\\Cr{ha}}{16\\Cr{c-3}\\Cr{c-p-1}^{2}}\\right\\}.\n\\label{th12.1}\n\\end{equation}\nWe proceed to prove \\eqref{thap9}. \nFor a.e$.$ $t\\in [\\frac12,\\frac72]$, we have \\eqref{firm1}-\\eqref{fir6}. \nDenoting\n\\begin{equation}\n\\hat{E}(t)=\\|V_t\\|(\\phi_{\\rm rad}^2)-{\\bf c},\n\\label{th13}\n\\end{equation}\n\\begin{equation}\n\\alpha(t)=\\left(\\int_{B_2}|h(V_t,\\cdot)|^2\\phi_{\\rm rad}^2\\, d\\|V_t\\|\\right)^{\\frac12},\\,\\,\n\\mu(t)=\\left(\\int_{B_2}{\\rm dist}\\, (\\cdot, J)^2\\, d\\|V_t\\|\\right)^{\\frac12},\n\\label{th14}\n\\end{equation}\nCorollary \\ref{fircor} and the definition of $\\beta_*$ in \\eqref{thap4} show that\n\\begin{equation}\n3\\Cr{c-1}\\beta_*^2\\leq |\\hat{E}(t)|\\, \\Longrightarrow\\, \\alpha(t)^2 \n\\geq \\min\\{\\Cr{alpha-1}^2,\n(2\\Cr{c-1}\\mu(t))^{-2}|\\hat{E}(t)|^2\\}.\n\\label{th15}\n\\end{equation}\nFix any ${\\hat s}\\in [1,3]$ and we first prove the following upper bound,\n\\begin{equation}\n{\\hat E}({\\hat s})\\leq \\Cr{ha2}\\mu^2+ (3\\Cr{c-1}+4\\Cr{c-3})\\beta_*^2+2C(u).\n\\label{th16}\n\\end{equation}\n{\\it Proof of \\eqref{th16}}.\n\\newline\n{\\bf (i)} Suppose that there exists some $t_0\\in [\\frac12, 1]$\nsuch that \n\\begin{equation}\n{\\hat E}(t_0)<(3\\Cr{c-1}+\\Cr{c-3})\\beta_*^2+C(u).\n\\label{th17}\n\\end{equation}\nBy the monotone decreasing property of $E(\\cdot)$, \\eqref{th1} and \\eqref{th17}, we then have\n\\begin{equation}\nE({\\hat s})\\leq E(t_0)\\leq {\\hat E}(t_0)<(3\\Cr{c-1}+\\Cr{c-3})\\beta_*^2+C(u).\n\\label{th18}\n\\end{equation}\nBut then again by \\eqref{th1}, \\eqref{th18} and \\eqref{thap4}, we have\n\\begin{equation}\n{\\hat E}({\\hat s})\\leq E({\\hat s})+C(u)+3\\Cr{c-3}\\beta_*^2\n<(3\\Cr{c-1}+4\\Cr{c-3})\\beta_*^2+2C(u).\n\\label{th19}\n\\end{equation}\nWith \\eqref{th19} we proved \\eqref{th16} under the assumption of\n(i). Now consider the complementary situation.\n\\newline\n{\\bf (ii)} Suppose that for all $t\\in [\\frac12,1]$, we have\n\\begin{equation}\n{\\hat E}(t)\\geq (3\\Cr{c-1}+\\Cr{c-3})\\beta_*^2+C(u).\n\\label{th20}\n\\end{equation}\nThis in particular means $|{\\hat E}(t)|\\geq 3\\Cr{c-1}\\beta_*^2$, \nthus \\eqref{th20} and \\eqref{th15} show\n\\begin{equation}\n\\alpha(t)^2\\geq \\min\\{\\Cr{alpha-1}^2,(2\\Cr{c-1}\\mu(t))^{-2}|{\\hat E}(t)|^2\\}\n\\label{th21}\n\\end{equation}\nfor a.e$.$ $t\\in [\\frac12,1]$. \nBy \\eqref{th1} and \\eqref{th20},\n\\begin{equation}\n{\\hat E}(t)\\geq E(t)\\geq {\\hat E}(t)-\\Cr{c-3}\\beta_*^2-C(u)\n\\geq 0.\n\\label{th22}\n\\end{equation}\n\\eqref{th22} shows $|{\\hat E}(t)|\\geq |E(t)|$ in particular and by \\eqref{th21} \nand \\eqref{th11.1} we have\n\\begin{equation}\n\\alpha(t)^2 \n\\geq 4 P\\min\\{1,\\mu(t)^{-2}|E(t)|^2\\}\n\\label{th23}\n\\end{equation}\nfor a.e$.$ $t\\in [\\frac12,1]$.\nNow we are in the position to apply Lemma \\ref{ode}. \nDefine\n\\begin{equation}\n\\Phi(t)=E(t+\\frac12),\\,\\, f(t)=P\\min\\{1,\\mu(t+\\frac12)^{-2}\n|\\Phi(t)|^2\\},\\,\\,\\,t\\in [0,\\frac12]. \n\\label{th24}\n\\end{equation}\nBy \\eqref{th2} \\eqref{th23} and \\eqref{th24}, we have for $0\\leq \\forall t_1<\\forall t_2\\leq \\frac12$\n\\begin{equation}\n\\Phi(t_2)-\\Phi(t_1)\\leq -\\frac14 \\int_{t_1+\\frac12}^{t_2+\\frac12}\n\\alpha(t)^2\\, dt\\leq -\\int_{t_1}^{t_2}f(t)\\, dt.\n\\label{th25}\n\\end{equation}\nWe also have \n\\begin{equation}\n\\Phi(0)=E(\\frac12)= \\|V_{\\frac12}\\|(\\phi_{\\rm rad}^2)-{\\bf c}\\leq \\frac{\\Cr{ha}}{2}\n\\label{th26}\n\\end{equation}\nby \\eqref{th1} and \\eqref{va1}. Hence the assumptions of Lemma \\ref{ode}\nare all satisfied with $g(t)=\\mu(t+\\frac12)$, and noticing that $\\|g\\|_{L^2}^2\\leq \\mu^2$, \nwe conclude\n\\begin{equation}\n (E(1)=)\\,\\Phi(\\frac12)\\leq \\Cr{ha2}\\mu^2.\n\\label{th27}\n\\end{equation}\nFor any $\\hat s \\in [1,3]$, by \\eqref{th1}, \\eqref{th27} and the monotone decreasing property of $E$, we have\n\\begin{equation}\n{\\hat E}({\\hat s})\\leq E({\\hat s})+C(u)+3\\Cr{c-3}\\beta_*^2 \n\\leq \\Cr{ha2}\\mu^2+C(u)+3\\Cr{c-3}\\beta_*^2.\n\\label{th28}\n\\end{equation}\nThus \\eqref{th28} shows that \n\\eqref{th16} holds under the assumption of (ii). This concludes the proof of \\eqref{th16}. \n\\newline\nFix any ${\\hat s}\\in [1,3]$ and we next prove the following lower\nbound,\n\\begin{equation}\n-\\Cr{ha2}\\mu^2-(3\\Cr{c-1}+8\\Cr{c-3})\\beta_*^2-2C(u)\\leq {\\hat E}({\\hat s}).\n\\label{th29}\n\\end{equation}\nThe idea is similar to the upper bound estimate with a few differences, \nbut we present the proof for the completeness. \n\\newline\n{\\it Proof of \\eqref{th29}}.\n\\newline\n{\\bf (i)} Suppose that there exists some $t_0\\in [3,\\frac72]$\nsuch that\n\\begin{equation}\n{\\hat E}(t_0)>-(3\\Cr{c-1}+4\\Cr{c-3})\\beta_*^2 -C(u).\n\\label{th30}\n\\end{equation}\nBy \\eqref{th1} and \\eqref{th30},\n\\begin{equation}\nE(t_0)\\geq {\\hat E}(t_0)-C(u)-4\\Cr{c-3}\\beta_*^2\n>-(3\\Cr{c-1}+8\\Cr{c-3})\\beta_*^2-2C(u).\n\\label{th31}\n\\end{equation}\nBy the monotone decreasing property of $E$, we have\n$E({\\hat s})\\geq E(t_0)$ while ${\\hat E}({\\hat s})\\geq E({\\hat s})$ by\n\\eqref{th1}. Thus \\eqref{th31} proves \\eqref{th29} in case of (i).\n\\newline\n{\\bf (ii)} Suppose that for all $t\\in [3,\\frac72]$, we have\n\\begin{equation}\n{\\hat E}(t)\\leq -(3\\Cr{c-1}+4\\Cr{c-3})\\beta_*^2-C(u).\n\\label{th32}\n\\end{equation}\nThis means $|{\\hat E}(t)|\\geq 3\\Cr{c-1}\\beta_*^2$, thus by \\eqref{th15}, we have\n\\eqref{th21} for a.e$.$ $t\\in [3,\\frac72]$. \nWe need to change ${\\hat E}$ in \\eqref{th21} to $E$. To do so, observe that\n\\begin{equation}\n|E(t)|\\leq |{\\hat E}(t)|+C(u)+4\\Cr{c-3}\\beta_*^2\n\\leq 2|{\\hat E}(t)|,\n\\label{th33}\n\\end{equation}\nthe last inequality of \\eqref{th33} coming from \\eqref{th32}. \nThus, \\eqref{th21} with \\eqref{th33} (as well as recalling \\eqref{th11.1}) shows\n\\eqref{th23} for a.e$.$ $t\\in[3,\\frac72]$. Again we apply\nLemma \\ref{ode}. Set\n\\begin{equation}\n\\Phi(t)=E(t+3),\\,\\, f(t)=P\\min\\{1,\\mu(t+3)^{-2}|\\Phi(t)|^2\\},\\,\\, t\\in [0,\\frac12].\n\\label{th34}\n\\end{equation}\nBy having \\eqref{th23}, we have \\eqref{th25} and \n\\begin{equation}\n\\Phi(\\frac12)=E(\\frac72)\\geq \\|V_{\\frac72}\\|(\\phi_{\\rm rad}^2)-{\\bf c}-C(u)-4\\Cr{c-3}\\beta_*^2\n\\geq -\\frac{\\Cr{ha}}{2}-\\Cr{cuc}\\Cr{e-1}^2-4\\Cr{c-3}\\Cr{c-p-1}^2\\Cr{e-1}^2\\geq -\\Cr{ha}\n\\label{th35}\n\\end{equation}\nby \\eqref{th34}, \\eqref{th1}, \\eqref{va2}, \\eqref{bb2}, \\eqref{thap4} and the last inequality due to \\eqref{th12.1}. \nThe assumptions of Lemma \\ref{ode} (for case (2))\nare thus satisfied, and we obtain\n\\begin{equation}\n-\\Cr{ha2}\\mu^2\\leq \\Phi(0)(=E(3)).\n\\label{th36}\n\\end{equation}\nSince $E$ is decreasing, for any $\\hat s\\in [1,3]$, we have\n\\begin{equation}\n\\hat E (\\hat s)\\geq E(\\hat s) \\geq E(3).\n\\label{th36s}\n\\end{equation}\nHence under the assumption of (ii), \\eqref{th36} and \\eqref{th36s} show \\eqref{th29}.\n\nSince ${\\hat s}\\in [1,3]$ is arbitrary, \\eqref{th16} and\n\\eqref{th29} combined with \\eqref{thap4} and \\eqref{bb2} prove the first claim \\eqref{thap9}\nwith a suitable constant $\\Cr{c-2}$. \n\nTo prove \\eqref{thap10}, observe that \\eqref{th2} with $t_2=3$ and\n$t_1=1$ shows (recalling \\eqref{th1})\n\\begin{equation}\n\\int_{1}^{3}\\int_{B_2}|h|^2\\phi_{\\rm rad}^2\\, d\\|V_t\\|dt\n\\leq 4(E(1)-E(3)) \n\\leq 4(|{\\hat E}(1)|+\n|{\\hat E}(3)|+C(u)+2\\Cr{c-3}\\beta_*^2).\n\\label{th37}\n\\end{equation}\nThen using \\eqref{thap9} to \\eqref{th37}, we obtain \\eqref{thap10}, again with a suitable choice of $\\Cr{c-2}$.\n\\hfill{$\\Box$}\n\\vspace{.2in}\n\nIn the next two sections we derive further a priori estimates for the flow $\\{V_{t}\\}$ whenever it is weakly close, in space-time at scale one, to the static triple junction $J$. These estimates provide enough control of the behavior of the moving curves near the singularity of $J$ for us to establish (in Section~\\ref{blowupsection}) decay, by a fixed factor at a fixed smaller scale, of the space-time $L^{2}$ distance of the flow to $J$, and consequently (by iterating this decay result) Theorem~\\ref{mainreg}. These estimates are in the spirit of those proved first by L.~Simon (\\cite{Simon2}), for a similar purpose, for the case of multiplicity 1 minimal submanifolds weakly close to certain cylindrical minimal cones (in arbitrary dimension and codimension). However, in the present parabolic setting, their statements are often different and proofs require new ideas. \n\n\n\\section{A priori estimates II: non-concentration of the $L^2$-distance near the singularity of $J$}\nThe main result in this section is the estimate (\\ref{pa21}) of Proposition~\\ref{tildest}. This estimate in full strength plays an important role in Section~\\ref{blowupsection} where we establish asymptotics for the blow-ups of sequences of flows converging weakly to the static triple junction $J$. It also implies that the space-time $L^{2}$ distance $\\mu$ of the flow from $J$ does not concentrate near the singularity of $J$, a fact that is indispensable in the proof of the key decay result (Proposition~\\ref{blowprop7}) for $\\mu$. \n\nAn essential ingredient in the proof of Proposition~\\ref{tildest} is Proposition~\\ref{pade} below, which is based on the results of Section~\\ref{curv-est} and (the main idea behind) Huisken's monotonicity formula. \n\\begin{prop} Corresponding to $\\nu \\in (0, 1)$, $E_1 \\in [1, \\infty)$ and $p$, $q$ as in $({\\rm A}0),$\n there exist $\\Cl[eps]{e-2}\\in (0,\\Cr{e-1}]$ (where $\\Cr{e-1} = \\Cr{e-1}(\\nu, E_{1}, p, q)$ is as in Proposition~\\ref{hde}) and $\\Cl[c]{c-4}\\in (1,\\infty)$ with the following property: \nIf $\\{V_t\\}_{t\\in [0,4]}$ and $\\{u(\\cdot, t)\\}_{t\\in [0,4]}$\nsatisfy (A1)-(A4) with $U = B_{2}$ and $I =[0,4]$, if $\\eqref{smep1}, \\eqref{smep2}, \\eqref{smep3}, \\eqref{smep4}$ hold with $\\Cr{e-2}$ in place of $\\Cr{e-1}$ and if\n\\begin{equation}\nV_{t_0}\\in {\\bf IV}_1(B_2),\\ \\ h(V_{t_0},\\cdot)\\in L^2(\\|V_{t_0}\\|)\\ \\mbox{ and }\\ \n\\Theta(\\|V_{t_0}\\|,0)=\\frac32\n\\label{pa1}\n\\end{equation}\nfor some $t_0\\in [\\frac32, 3]$, then \n\\begin{equation}\n\\int_{5\/4}^{t_0}\\int_{B_1} \\left|h+\\frac{x^{\\perp}}{2(t_0-t)}\\right|^2\\rho_{(0,t_0)}(x,t)\n\\, d\\|V_t\\|dt\\leq \\Cr{c-4}\\max\\{\\mu,\\|u\\|\\}^2.\n\\label{pa2}\n\\end{equation}\n \\label{pade}\n\\end{prop}\n{\\it Proof}. By \\eqref{pa1}, there exists a tangent cone to $V_{t_0}$ at $x=0$ which is \n$|{\\bf R}_{\\theta}(J)|$ for some $\\theta\\in [0,2\\pi)$. From this fact, it follows that \n\\begin{equation}\n\\lim_{\\epsilon\\searrow 0} \\int_{B_2}\\phi_{\\rm rad}^2(x)\\rho_{(0,t_0+\\epsilon)}(x,t_0)\\, d\\|V_{t_0}\\|(x)=\\frac32.\n\\label{pa3}\n\\end{equation}\nIn the following, we fix $\\epsilon>0$ arbitrarily close to 0. We choose\n$t_1\\in [1,\\frac54]$ so that\n\\begin{equation}\n\\int_{B_2}|h(V_{t_1},\\cdot)|^2\\phi_{\\rm rad}^2\\, d\\|V_{t_1}\\|\\leq\n8\\Cr{c-2}\\max\\{\\mu,\\|u\\|\\}^2\n\\label{pa4}\n\\end{equation}\nwhere $\\Cr{c-2} = \\Cr{c-2}(\\nu, E_{1}, p, q)$ is as in Proposition~\\ref{hde}. This is possible in view of the estimate \\eqref{thap10}.\nArguing as in Proposition~\\ref{firprop}, for $\\Cr{e-2}$ suitably small (so that\n$8\\Cr{c-2}\\max\\{\\mu,\\|u\\|\\}^2\\leq \\Cr{alpha-1}^2$ where $\\alpha_{1} = \\alpha_{1}(E_{1})$ is as in Proposition~\\ref{firprop}), we may \nconclude that ${\\rm spt}\\,\\|V_{t_1}\\| \\cap B_{1}$ consists of three $C^{1,\\frac12}$ curves $l_1,l_2,l_3$ meeting\nat a common point $p$ near the origin, with associated numbers $s_{1}, s_{2}, s_{3} \\in (-1\/2, 1\/2)$ and functions $f_{j} \\in C^{1, 1\/2}([s_{j}, 1])$, $j=1, 2, 3,$ such that \n\\begin{equation}\n{\\bf R}_{-\\frac{2\\pi (j-1)}{3}} l_{j} = \\{(s,f_j(s))\\in {\\mathbb R}^2\\,:\\, s\\in [s_j,1]\\}\n\\end{equation}\nfor $j=1, 2, 3;$ furthermore, using the estimate \\eqref{fir10.5}, we see that $\\sup \\, \\{{\\rm dist} \\, (x, J) \\, : \\, x \\in l_{j}\\}$ for $j=1,2,3,$ and hence also $|s_j|$, are all $\\leq 2\\beta_*+2\\sqrt{8\\Cr{c-2}}\\max\\{\\mu,\\|u\\|\\}$, where $\\beta_*$ is as in \\eqref{thap4}. \nThese estimates and radial symmetry of $\\phi_{\\rm rad}$ and $\\rho_{(0,t_0)}(\\cdot,t_1)$ imply, \nfor a suitable choice of $\\Cr{c-4}$ depending only on $p,q,\\nu,E_1,$ that\n\\begin{equation}\n\\left|\\int_J \\phi_{\\rm rad}^2\\rho_{(0,t_0)}(\\cdot,t_1)\\, d{\\mathcal H}^1\n-\\int_{B_2} \\phi_{\\rm rad}^2\\rho_{(0,t_0)}(\\cdot,t_1)\\, d\\|V_{t_1}\\|\\right|\\leq\\Cr{c-4}\n\\max\\{\\mu,\\|u\\|\\}^2.\n\\label{pa5}\n\\end{equation}\nHere it is important that $t_1\\in [1,\\frac54]$ so that $t_0-t_1\\geq \\frac14$, allowing the choice\nof $\\Cr{c-4}$ to be independent of $t_0$ and $t_1$. \n\nWe next use $\\rho_{(0,t_0+\\epsilon)}(\\cdot,t)\\phi_{\\rm rad}^2$ as a \ntest function in \\eqref{meq} with $t\\in [t_1, t_0]$. For simplicity of notation write $\\rho$ for \n$\\rho_{(0, t_{0} + \\epsilon)}$ and define ${\\hat \\rho}(x, t) =\\rho_{(0,\nt_0+\\epsilon)}(x,t)\\phi_{\\rm rad}(x)^2$. By direct computation, \n\\begin{equation}\n\\begin{split}\n{\\mathcal B}(V_t,u(\\cdot,t),{\\hat \\rho}(\\cdot,t))&=\\int_{B_2}(-h{\\hat \\rho}+\\nabla{\\hat\\rho})\\cdot(h+u^{\\perp})\\, d\\|V_t\\|\n\\\\ & =\\int_{B_2}-|h|^2{\\hat\\rho}+2\\nabla{\\hat\\rho}\\cdot h+u^{\\perp}\\cdot(-h{\\hat\\rho}+\\nabla{\\hat\\rho})\n-\\nabla{\\hat\\rho}\\cdot h\\, d\\|V_t\\|\\\\\n&=\\int_{G_1(B_2)}-{\\hat\\rho}\\big|h-\\frac{(\\nabla{\\hat\\rho})^{\\perp}}{\\hat\\rho}\\big|^2\n+\\frac{|(\\nabla{\\hat\\rho})^{\\perp}|^2}{\\hat\\rho}+u\\cdot(-h{\\hat\\rho}+(\\nabla{\\hat\\rho})^{\\perp})+S\\cdot\\nabla^2\n{\\hat\\rho}\\, d V_t \\\\\n&\\leq \\int_{G_1(B_2)}-\\frac{\\hat\\rho}{2}\\big|h-\\frac{(\\nabla{\\hat\\rho})^{\\perp}}{\\hat\\rho}\\big|^2\n+\\frac{|(\\nabla{\\hat\\rho})^{\\perp}|^2}{\\hat\\rho}+\\frac{|u|^2{\\hat\\rho}}{2}+S\\cdot\\nabla^2{\\hat\\rho}\\,\ndV_t,\n\\end{split}\n\\label{pa7}\n\\end{equation}\nwhere we have used the fact that by \\eqref{perpthm}, for $\\|V_t\\|$ a.e$.$, $h\\cdot \\nabla{\\hat\\rho}=\nh\\cdot(\\nabla{\\hat\\rho})^{\\perp}$. \nWe now need to carefully evaluate the terms involving $\\nabla\\phi_{\\rm rad}^2$ in the above. For the second term on the right hand side of \\eqref{pa7}, we have\n\\begin{equation}\n\\frac{|(\\nabla{\\hat\\rho})^{\\perp}|^2}{\\hat\\rho}=\\phi_{\\rm rad}^2\\frac{|(\\nabla\\rho)^{\\perp}|^2}{\\rho}\n+2(\\nabla\\rho)^{\\perp}\\cdot(\\nabla\\phi_{\\rm rad}^2)^{\\perp}\n+\\rho\\frac{|(\\nabla\\phi_{\\rm rad}^2)^{\\perp}|^2}{\\phi_{\\rm rad}^2}\n\\leq \\phi_{\\rm rad}^2\\frac{|(\\nabla\\rho)^{\\perp}|^2}{\\rho}+c\\beta_*^2\n\\label{pa8}\n\\end{equation}\nwhere the last inequality follows from the estimate \\eqref{th6} which holds also with $\\rho$ in place of $\\phi_{\\rm rad}$.\nThe constant $c$ is an absolute constant which may differ from line to line.\nBy combining \\eqref{pa7} and \\eqref{pa8}, we obtain\n\\begin{equation}\n\\begin{split}\n{\\mathcal B}(V_t,u(\\cdot,t),{\\hat\\rho}(\\cdot,t))\\leq& \\int_{G_1(B_2)}\n-\\frac{\\hat\\rho}{2}\\big|h-\\frac{(\\nabla{\\hat\\rho})^{\\perp}}{\\hat\\rho}\\big|^2\n+\\phi_{\\rm rad}^2\\big(\\frac{|(\\nabla\\rho)^{\\perp}|^2}{\\rho}+S\\cdot\\nabla^2\\rho\\big)\\\\\n&\\hspace{.5in}+\\frac{|u|^2{\\hat\\rho}}{2}+c\\beta_*^2 +2S\\cdot(\\nabla\\rho\\otimes\n\\nabla\\phi_{\\rm rad}^2)+\\rho S\\cdot\\nabla^2\\phi_{\\rm rad}^2\\, dV_t.\n\\end{split}\n\\label{pa9}\n\\end{equation}\nBy \\eqref{pa9}, (A4) and the identity \n\\begin{equation}\\label{idenbh}\n\\frac{\\partial \\rho}{\\partial t}+S\\cdot \\nabla^2\\rho+\\frac{|S^{\\perp}\n(\\nabla\\rho)|^2}{\\rho}=0,\n\\end{equation}\nwe conclude that\n\\begin{equation}\n\\begin{split}\n\\left.\\int_{B_2}{\\hat\\rho}(\\cdot,t)\\, d\\|V_t\\|\\right|_{t=t_1}^{t_0}\\leq& \\int_{t_1}^{t_0}\\int_{G_1(B_2)}-\\frac{\\hat\\rho}{2}\n\\big|h-\\frac{(\\nabla{\\hat\\rho})^{\\perp}}{\\hat\\rho}\\big|^2 \n+\\frac{|u|^2{\\hat\\rho}}{2}+c\\beta_*^2 \\\\ & \\hspace{1in}+2S\\cdot(\\nabla\\rho\\otimes\n\\nabla\\phi_{\\rm rad}^2)+\\rho S\\cdot\\nabla^2\\phi_{\\rm rad}^2\\, dV_t dt.\n\\end{split}\n\\label{pa10}\n\\end{equation}\nWe next proceed to estimate the last two terms on the right hand side of \\eqref{pa10}. \nNote that the integrands in these two terms are zero outside $B_{\\frac32}\\setminus\nB_{\\frac12}$. (So in particular the derivatives of $\\rho$ appearing there are bounded uniformly.) \nSince by Proposition~\\ref{smprop} integration with respect to $\\|V_{t}\\|$ in $B_{\\frac 32} \\setminus B_{\\frac 12}$ is along three $C^{1,\\frac12}$ curves $l_{1}^{(t)}, l_{2}^{(t)}, l_{3}^{(t)}$ represented as graphs of functions $f_1(\\cdot, t),f_2(\\cdot, t),f_3(\\cdot, t)$ respectively as in \\eqref{smep7}, one can compute them explicitly. For instance for the curve $l_{1}^{(t)}$, \nby explicit calculation (suppressing the $t$ dependence of the functions involved), \n\\begin{equation}\n\\begin{split}\n\\int_{l_{1}^{(t)}} & 2S\\cdot(\\nabla\\rho\\otimes\n\\nabla\\phi_{\\rm rad}^2)+\\rho S\\cdot\\nabla^2\\phi_{\\rm rad}^2\\\\\n & \\begin{split}= \\int_{\\frac12}^{\\frac32}&\n(1+(f_1')^2)^{-\\frac12}\\left(\\begin{array}{ll} 1 & f_1' \\\\ f_1' & (f_1')^2\\end{array}\\right)\\cdot\n\\left\\{2r^{-2}(x\\otimes x)\\frac{d\\rho}{dr}\\frac{d\\phi_{\\rm rad}^2}{dr}\\right. \\\\ &+\\left.\\rho r^{-2}\n(x\\otimes x)(\\frac{d^2}{dr^2}-r^{-1}\\frac{d}{dr} )\\phi_{\\rm rad}^2+r^{-1}\\rho I\\frac{d\\phi_{\\rm rad}^2}{dr}\\right\\}ds,\n\\end{split}\n\\end{split}\n\\label{pa11}\n\\end{equation}\nwhere $f_1=f_1(s,t)$, $r=\\sqrt{s^2+(f_1)^2}$, $x=(s,f_1(s,t))$, $I$ is the identity $2\\times 2$\nmatrix and $d\/dr$ is the differentiation with respect to the radial direction. Since \n$|f_1|,\\,|f_1'|\\leq \\beta_*$ by \\eqref{thap4}, we may estimate terms\non the right hand side of \\eqref{pa11} up to errors of order $\\beta_*^2$ to obtain \n\\begin{equation}\n\\begin{split}\n\\int_{l_{1}^{(t)}} & 2S\\cdot(\\nabla\\rho\\otimes\n\\nabla\\phi_{\\rm rad}^2)+\\rho S\\cdot\\nabla^2\\phi_{\\rm rad}^2\\\\ &\\leq \\int_{\\frac12}^{\\frac32}\\left(2\\frac{d\\rho}{dr}\\frac{d\\phi_{\\rm rad}^2}{dr}+\n\\rho\\left(\\frac{d^2}{dr^2}-s^{-1}\\frac{d}{dr}\\right)\\phi_{\\rm rad}^2+s^{-1}\\rho\\frac{d\\phi_{\\rm rad}^2}{dr}\\right)\\, ds\n+c\\beta_*^2\\\\\n&=\\int_{\\frac12}^{\\frac32}\\left(2\\frac{d\\rho}{dr}\\frac{d\\phi_{\\rm rad}^2}{dr}+\n\\rho\\frac{d^2\\phi_{\\rm rad}^2}{dr^2}\\right)\\, ds+c\\beta_*^2.\n\\end{split}\n\\label{pa12}\n\\end{equation}\nSince the functions appearing in the integrand on the right hand side of the above are radially symmetric, their values at \n$(s,f_1(s,t))$ and those at $(s,0)$ differ by at most $c\\beta_*^2$. Thus we have \n\\begin{equation}\n\\begin{split}\n\\int_{l_{1}^{(t)}} & 2S\\cdot(\\nabla\\rho\\otimes\n\\nabla\\phi_{\\rm rad}^2)+\\rho S\\cdot\\nabla^2\\phi_{\\rm rad}^2\\\\ & \\leq \\int_{\\frac12}^{\\frac32}\\left(2\\frac{\\partial \\rho}{\\partial x}\\frac{\\partial \\phi_{\\rm rad}^2}{\\partial x}\n+\\rho\\frac{\\partial^2 \\phi_{\\rm rad}^2}{\\partial x^2}\\right)(s,0)\\, ds+c\\beta_*^2\n=\\int_{\\frac12}^{\\frac32} \\frac{\\partial \\rho}{\\partial x}\\frac{\\partial \\phi_{\\rm rad}^2}{\\partial x}(s,0)\\, ds\n+c\\beta_*^2\n\\end{split}\n\\label{pa13}\n\\end{equation}\nwhere we integrated by parts and used the property that $\\frac{\\partial\\phi_{\\rm rad}^2}{\\partial x}(s,0)=0$\nat $s=1\\pm \\frac12$. The same computation holds after rotation for the other two curves $l_{2}^{(t)}, l_{3}^{(t)}$, so by \\eqref{pa10} we deduce \n\\begin{equation}\n\\begin{split}\n\\left.\\int_{B_2}{\\hat\\rho}(\\cdot,t)\\, d\\|V_t\\|\\right|_{t=t_1}^{t_0}\\leq &\n\\int_{t_1}^{t_0}\\int_{B_2}-\\frac{\\hat\\rho}{2}\n\\big|h-\\frac{(\\nabla{\\hat\\rho})^{\\perp}}{\\hat\\rho}\\big|^2 \n+\\frac{|u|^2{\\hat\\rho}}{2}\\, d\\|V_t\\|dt \\\\\n& +3 \\int_{t_1}^{t_0}\\int_{\\{(x,0)\\in{\\mathbb R}^2\\,:\\, |x-1|<\\frac12\\}} \\frac{\\partial \\rho}{\\partial x}\\frac{\\partial \\phi_{\\rm rad}^2}{\\partial x}\\, d{\\mathcal H}^1 dt+c\\beta_*^2.\n\\end{split}\n\\label{pa14}\n\\end{equation}\nWe note that \n\\begin{equation*}\n\\left.\\int_{J}{\\hat\\rho}(\\cdot,t)\\, d{\\mathcal H}^1\\right|_{t=t_1}^{t_0}=\\int_J\\int_{t_1}^{t_0}\n\\frac{\\partial{\\hat\\rho}}{\\partial t}(\\cdot,t)\\, dtd{\\mathcal H}^1=3\\int_{t_1}^{t_0}\\int_{\\{(x,0)\\in\n{\\mathbb R}^2\\,:\\, x\\geq 0\\}}\\frac{\\partial{\\hat\\rho}}{\\partial t}(\\cdot,t)d{\\mathcal H}^1 dt\n\\label{pa15}\n\\end{equation*}\nby radial symmetry of $\\hat\\rho$, and thus, since $\\frac{\\partial{\\hat\\rho}}{\\partial\nt}=-\\phi_{\\rm rad}^2 \\frac{\\partial^2 \\rho}{\\partial x^2}$ on the $x$-axis, \n\\begin{equation}\n\\begin{split}\n\\left.\\int_{J}{\\hat\\rho}(\\cdot,t)\\, d{\\mathcal H}^1\\right|_{t=t_1}^{t_0}&=3\\int_{t_1}^{t_0}\\int_{\\{(x,0)\\in\n{\\mathbb R}^2\\,:\\, x\\geq 0\\}} \\frac{\\partial \\rho}{\\partial x}\\frac{\\partial \\phi_{\\rm rad}^2}{\\partial x}d{\\mathcal H}^1 dt\\\\\n&=3\\int_{t_1}^{t_0}\\int_{\\{(x,0)\\in\n{\\mathbb R}^2\\,:\\, |x-1|<\\frac12\\}} \\frac{\\partial \\rho}{\\partial x}\\frac{\\partial \\phi_{\\rm rad}^2}{\\partial x}d{\\mathcal H}^1 dt\n\\end{split}\n\\label{pa16}\n\\end{equation}\nwhere we used integration by parts and the fact that $\\frac{\\partial\\rho}{\\partial x}=0$ at $x=0$ and $\\phi_{\\rm rad}^2=0$ at $x=\\frac32$.\nSubstituting \\eqref{pa16} into \\eqref{pa14}, we obtain\n\\begin{equation}\n\\left.\\big(\\int_{B_2}{\\hat\\rho}(\\cdot,t)\\, d\\|V_t\\|-\\int_J{\\hat\\rho}(\\cdot,t)\\, d{\\mathcal H}^1\\big)\\right|_{t=t_1}^{t_0}\n\\leq \\int_{t_1}^{t_0}\\int_{B_2}-\\frac{\\hat\\rho}{2}\\big|h-\\frac{(\\nabla{\\hat\\rho})^{\\perp}}{\\hat\\rho}\\big|^2+{\\hat\\rho}\n\\frac{|u|^2}{2}\\, d\\|V_t\\|dt+c\\beta_*^2.\n\\label{pa17}\n\\end{equation}\nWe now let $\\epsilon\\rightarrow 0$ in \\eqref{pa17}. Since $\\int_J{\\hat\\rho}_{(0,t_0+\\epsilon)}(x,t_0)\\, d{\\mathcal H}^1\n\\rightarrow \\frac32$, in view of \\eqref{pa3} and \\eqref{pa5}, we obtain from \\eqref{pa17} (using also \nthe fact that $\\phi_{\\rm rad}=1$ on $B_1$ and $=0$ on ${\\mathbb R}^{2} \\setminus B_{\\frac32}$) that\n\\begin{equation}\n\\int_{t_1}^{t_0}\\int_{B_1}\\frac{\\rho}{2}\\big|h-\\frac{(\\nabla\\rho)^{\\perp}}{\\rho}\\big|^2\\, d\\|V_t\\|dt\\leq \\int_{t_1}^{t_0}\n\\int_{B_{\\frac32}}{\\rho}\\frac{|u|^2}{2}\\, d\\|V_t\\|dt+c\\beta_*^2+\\Cr{c-4}\\max\\{\\mu,\\|u\\|\\}^2\n\\label{pa18}\n\\end{equation}\nwhere $\\rho=\\rho_{(0,t_0)}(x,t)$. Lastly, the term above involving $u$ may be estimated as in \\cite[(6.7)-(6.8)]{Kasai-Tonegawa} to get \n\\begin{equation}\n\\int_{t_1}^{t_0}\\int_{B_{\\frac32}}\\rho|u|^2\\, d\\|V_t\\|dt\\leq c(p,q)E_1 ^{1-\\frac{2}{p}}\\|u\\|^2.\n\\label{pa19}\n\\end{equation}\nSince $\\frac{\\nabla\\rho}{\\rho}=-\\frac{x}{2(t_0-t)}$, the desired estimate follows after redefining $\\Cr{c-4}$ depending only on $p,q,\\nu,E_1$. \n\\hfill{$\\Box$}\n\n\\begin{prop} Fix $\\kappa\\in [0,1)$. Under the same assumptions as in Proposition \\ref{pade}, we have\n\\begin{equation}\n\\sup_{t\\in [\\frac54,t_0)}(t_0-t)^{-\\kappa}\\int_{B_{\\frac34}} \\rho_{(0,t_0)}(\\cdot,t){\\rm dist}\\, (\\cdot,J)^2\\, \nd\\|V_t\\|\\leq \\Cl[c]{c-5}\\max\\{\\mu,\\|u\\|\\}^2\n\\label{pa21}\n\\end{equation}\nwhere $\\Cr{c-5}$ depends only on $\\kappa,p,q,\\nu,E_1$. \n\\label{tildest}\n\\end{prop}\n{\\it Proof}. \nDefine ${\\tilde d}:{\\mathbb R}^2\\rightarrow{\\mathbb R}$ such that ${\\tilde d}$ \nis positively homogeneous of degree one (i.e.\\ ${\\tilde d}(\\lambda x)=\\lambda {\\tilde d}(x)$ $\\forall \\lambda\\geq 0$\nand $\\forall x\\in{\\mathbb R}^2$), smooth away from $J$, and\n\\begin{equation}\n\\begin{array}{ll}\n {\\tilde d}(x)={\\rm dist}\\,(x,J)&\\mbox{ $\\forall x$ with }{\\rm dist}\\,(x,J)<\\frac{|x|}{5},\\\\\n \\frac12 {\\rm dist}\\, (x,J)\\leq {\\tilde d}(x)\\leq 2\\,{\\rm dist}\\, (x,J) & \\forall x\\in {\\mathbb R}^2,\\\\\n |\\nabla {\\tilde d}(x)|\\leq 1 & \\forall x\\notin J.\n\\end{array}\n\\label{pa20}\n\\end{equation}\nBy homogeneity, we have $x\\cdot \\nabla ({\\tilde d}^2\/|x|^2)=0$, which gives after a little computation that\n\\begin{equation}\nx\\cdot\\nabla{\\tilde d}^2=2{\\tilde d}^2.\n\\label{pa20.5}\n\\end{equation} \nLet $0\\leq \\eta\\leq 1$ be a non-negative smooth radially symmetric function such that \n\\begin{equation}\n\\eta=0 \\quad \\forall x\\notin B_1 \\quad \\mbox {and} \\quad \\eta=1 \\quad \\forall x\\in B_{\\frac34}.\n\\label{pa22}\n\\end{equation}\nSince ${\\rm spt} \\, \\nabla \\eta \\subset B_{1} \\setminus B_{\\frac 34}$, we may assume that\n${\\rm spt}\\, |\\nabla\\eta| \\, \\cap \\, {\\rm spt}\\,\\|V_t\\|$ is contained in $\\cup _{j=1}^{3}{\\rm graph}\\,f_j(\\cdot,t)$ for all $t\\in [1,3]$, where $f_{j}$ are as in Proposition~\\ref{smprop}. Fix $t_1\\in [1, \\frac54]$ such that\n\\begin{equation}\n\\int_{B_2}{\\rm dist}\\,(\\cdot,J)^2\\, d\\|V_{t_1}\\|\\leq 4 \\mu^2.\n\\label{pa23}\n\\end{equation}\nSuch $t_1$ exists by \\eqref{smep1}, the definition of $\\mu$.\nWe next use $(t_0-t)^{-\\kappa} g(t)\\rho_{(0,t_0)}(x,t){\\tilde d}(x)^2\\eta(x)$ as a test function in \n\\eqref{meq} \nover the time interval $[t_1,t_2]$ with arbitrary $t_2\\in [\\frac54,t_0)$, where $g$ is a fixed smooth non-negative function with\n\\begin{equation}\n00.\n\\label{pa37.5}\n\\end{equation}\nWe emphasize that $c(\\kappa)$ here may be chosen independently of $t_1\\in [1, \\frac54]$ and $t_0\\in[\\frac32,3]$. \nWith this choice of $g$, \n\\begin{equation}\n\\rho\\frac{d}{dt}\\left((t_0-t)^{-\\kappa}g\\right)=\\frac{\\kappa{\\hat\\rho}}{t_0-t}\n-8(t_0-t)^{-\\kappa}{\\hat\\rho}\n\\label{pa38}\n\\end{equation}\nso we have\n\\begin{equation}\nI_5= \\int_{t_1}^{t_2}\\int_{B_1}\n \\frac{\\kappa{\\hat\\rho} \\eta{\\tilde d}^2}{t_0-t}\n-8(t_0-t)^{-\\kappa}{\\hat\\rho} \\eta{\\tilde d}^2\\, d\\|V_t\\|dt.\n\\label{pa39}\n\\end{equation}\n{\\bf Conclusion}.\n\\newline\nBy combining \\eqref{pa31}, \\eqref{pa34}, \\eqref{pa36} and \n\\eqref{pa39}, and using the estimates \\eqref{pa2},\n\\eqref{pa23}, \\eqref{pa37.5} and \\eqref{pa20}, we deduce the desired estimate \\eqref{pa21}.\n\\hfill{$\\Box$}\n\\section{A priori estimates III: Distance and approximate continuity estimates for the junction point}\n\nFix $p, q$ as in (A0), $\\nu \\in (0, 1), E_1 \\in [1, \\infty)$ and $\\kappa\\in [0,1).$ Let $\\Cr{c-2}$ be as in Proposition~\\ref{hde}, $\\Cr{e-2},\\Cr{c-4}$ be as in Proposition~\\ref{pade}, and \n$\\Cr{c-5}$ be as in Proposition~\\ref{tildest}. With $U=B_{3}$ and $I = [0, 4]$, suppose $\\{V_t\\}_{t\\in[0,4]}$ and $\\{u(\\cdot, t)\\}_{t\\in[0,4]}$ \nsatisfy (A1)-(A4). Assume further, with the following new definitions of $\\mu$ and $\\|u\\|$ (in which spatial integration is over $B_3$ rather than over $B_2$), that \n\\begin{equation}\n\\mu=\\left(\\int_0^4 \\int_{B_3}{\\rm dist}\\,(\\cdot, J)^2\\, d\\|V_t\\|dt\\right)^{\\frac12}\\leq \\frac{\\Cr{e-2}}{2},\n\\label{smep1c}\n\\end{equation}\n\\begin{equation}\n\\|u\\|=\\left(\\int_0^4 \\big(\\int_{B_3}|u|^p\\, d\\|V_t\\|\\big)^{\\frac{q}{p}}dt\\right)^{\\frac{1}{q}}\\leq \\frac{\\Cr{e-2}}{2},\n\\label{smep2c}\n\\end{equation}\n\\begin{equation}\n\\|V_0\\|(\\phi_{j_1})\\leq \\frac{3-\\nu}{2} \\Cr{c-p}, \\ \\ \\|V_4\\|(\\phi_{j_2})\\geq \\frac{1+\\nu}{2} \\Cr{c-p} \\quad \\mbox {for some} \\quad j_{1}, j_{2} \\in \\{1, 2, 3\\} \n\\label{smep34c}\n\\end{equation}\nwhere $\\phi_j$ are as in \\eqref{phij}.\nNote that \\eqref{smep1c}-\\eqref{smep34c} are more restrictive conditions than hypotheses (\\ref{smep1})-(\\ref{smep4}) of Proposition \\ref{pade}. \nFor $\\xi\\in {\\mathbb R}^2$ with $|\\xi|<1$, let $V_t^{(\\xi)}$ be the translation of $V_t$ by $-\\xi$; thus, \n\\begin{equation}\nV_t^{(\\xi)}(\\phi)=\\int_{G_1(B_3)}\\phi(x-\\xi,S)\\, dV_t(x,S) \\quad \\mbox{for each} \\quad \\phi\\in C_c(G_1(B_2)).\n\\label{ya7}\n\\end{equation} \nNote that if ${\\rm spt}\\, \\|V_t\\|$ has a junction point at $\\xi$, then \n${\\rm spt}\\, \\|V_t^{(\\xi)}\\|$ has a junction point at the origin. Now, depending only on\n$\\nu,E_1,\\Cr{e-2}$ (hence ultimately only on $p,q,\\nu,E_1$), \nthere exists a small $\\Cl[de]{delta-1} \\in (0, 1)$ such that, if $|\\xi|\\leq\\Cr{delta-1}$, then $\\{V_t^{(\\xi)}\\}_{t\\in[0,4]}$\nand $\\{u(\\cdot+\\xi,t)\\}_{t\\in [0,4]}$ satisfy \\eqref{smep1}-\\eqref{smep4} with $\\Cr{e-2}$ in place of \n$\\Cr{e-1}$. In particular, if $|\\xi|\\leq \\Cr{delta-1}$,\n$\\{V_t^{(\\xi)}\\}$ and $\\{u(\\cdot+\\xi,t)\\}$ satisfy the hypotheses of Proposition \\ref{pade} on $B_2\\times [0,4].$ Let us fix such $\\Cr{delta-1}$. \n\\begin{lemma}\nFor $\\xi\\in {\\mathbb R}^2\\setminus\\{0\\}$ define $J_{\\xi}=\\{x\\in {\\mathbb R}^2 : x-\\xi\\in J\\}$. \nThen on one of the three connected components of $\\{x\\in{\\mathbb R}^2 : {\\rm dist}\\,(x,{\\bf R}_{\\frac{\\pi}{3}}(J))>|\\xi|\\}$, we\nhave \n\\begin{equation}\n\\frac{\\sqrt{3}}{2}|\\xi|\\leq {\\rm dist}\\,(x,J)+{\\rm dist}\\,(x,J_{\\xi}).\n\\label{xie2}\n\\end{equation}\n\\label{little}\n\\end{lemma}\n{\\it Proof}. First note that ${\\bf R}_{\\frac{\\pi}{3}}(J)\\setminus\\{0\\}$ is the set of points $x$ such that the closest point to $J$ from $x$ is not unique.\nGiven $\\xi\\in {\\mathbb R}^2\\setminus \\{0\\}$, let $A=\\{x\\in{\\mathbb R}^2 : {\\rm dist}\\,(x,{\\bf R}_{\\frac{\\pi}{3}}(J))>|\\xi|\\}$. \nSince $x\\in A$ is away from ${\\bf R}_{\\frac{\\pi}{3}}(J)$ by at least $|\\xi|$, the closest points in $J$ and $J_{\\xi}$ to $x$ are\nboth unique. Let $x_{J}\\in J$ be the closest point to $x$ in $J$. \nOne checks easily that the closest point in $J_{\\xi}$ from $x$ is $x_{J}+\\xi^{\\perp}$, where $\\xi^{\\perp}$ is $(T_{x_J} J)^{\\perp}(\\xi)$.\nThis implies ${\\rm dist}\\, (x,J)=|x-x_J|$ and ${\\rm dist}\\, (x,J_{\\xi})=|x-x_J-\\xi^{\\perp}|$. Then the triangle inequality gives\n$|\\xi^{\\perp}|\\leq {\\rm dist}\\, (x,J)+{\\rm dist}\\,(x,J_{\\xi})$. For $\\xi$, there is at least one component of $A$ on which \n$|\\xi^{\\perp}|\\geq \\frac{\\sqrt{3}}{2}|\\xi|$ holds. On this component, we have \\eqref{xie2}.\n\\hfill{$\\Box$}\n\n\\begin{prop}\nThere exist $\\Cl[eps]{e-4}\\in (0,\\frac{\\Cr{e-2}}{2}]$ and $\\Cl[c]{c-6}\\in (1,\\infty)$ depending only on \n$p,q,\\nu,E_1$ such that if $\\{V_t\\}_{t\\in [0,4]}$, $\\{u(\\cdot,t)\\}_{t\\in [0,4]}$\nsatisfy (A1)-(A4) with $U=B_3$, $I = [0, 4]$ and \\eqref{smep1c}, \\eqref{smep2c}, \\eqref{smep34c}\n hold with $\\Cr{e-4}$ in place of $\\frac{\\Cr{e-2}}{2}$\nthen for any $\\xi\\in B_1$ and $t_0\\in [\\frac32,3]$ with\n$h(V_{t_0},\\cdot)\\in L^2(\\|V_{t_0}\\|)$ and $\\Theta(\\|V_{t_0}\\|,\\xi)=\\frac32$,\nwe have\n\\begin{equation}\n|\\xi|\\leq \\Cr{c-6}\\max\\{\\mu,\\|u\\|\\}.\n\\label{xie4}\n\\end{equation}\nIn addition, given $\\kappa\\in [0,1)$, there exists $\\Cl[c]{c-7}\\in (1,\\infty)$ depending only on $\\kappa,p,q,\\nu,E_1$ \nsuch that\n\\begin{equation}\n\\sup_{t\\in [\\frac54,t_0)}(t_0-t)^{-\\kappa}\\int_{B_{\\frac34}(\\xi)}\\rho_{(\\xi,t_0)}(\\cdot,t)\n{\\rm dist}(\\cdot,J_{\\xi})^2\\, d\\|V_t\\|\\leq \\Cr{c-7}\\max\\{\\mu,\\|u\\|\\}^2.\n\\label{xie4.5}\n\\end{equation}\n\\label{distrip}\n\\end{prop}\n{\\it Proof}. Recall $\\Cr{delta-1}$ which is fixed before Lemma \\ref{little}.\nCorresponding to $\\kappa=1\/2$, let $\\Cr{c-5}$ be chosen using\nProposition \\ref{tildest}. Fix $r_0\\in (0,\\frac14]$ by\n\\begin{equation}\nr_0 = \\min\\left\\{\\frac14, \\frac12\\left(\\frac{16}{3}32eE_1 \\sqrt{16\\pi}\\Cr{c-5}\\right)^{-1}\\right\\}.\n\\label{defr}\n\\end{equation}\nCorresponding to $\\tau=\\min\\{\\Cr{delta-1}\/2,\\, r_0\/8\\}$, fix $\\Cr{e-p}$ \nusing Proposition \\ref{smprop}. We will choose $\\Cr{e-4}\\in (0, \\Cr{e-p}]$ by restricting further in the following.\nFor any $\\xi\\in B_1$ and $t_0$ satisfying the assumptions, due to the\nchoice of $\\Cr{e-p}$, the claim of Proposition \\ref{smprop} shows that we have $|\\xi|\\leq 2\\tau\\leq \\Cr{delta-1}$. Then due to the\nchoice of $\\Cr{delta-1}$ (see the discussion before Lemma \\ref{little}),\n$\\{V_t^{(\\xi)}\\}_{t\\in [0,4]}$ and $\\{u(\\cdot+\\xi,t)\\}_{t\\in [0,4]}$\nsatisfy assumptions of Proposition \\ref{pade} and \\ref{tildest}.\nThus we have\n\\begin{equation}\n\\sup_{t\\in [\\frac54,t_0)}(t_0-t)^{-1\/2}\\int_{B_{\\frac34}}\n\\rho_{(0,t_0)}(\\cdot,t)\\, {\\rm dist}\\,(\\cdot, J)^2\\, d\\|V_t^{(\\xi)}\\|\n\\leq \\Cr{c-5}\\max\\{\\mu_{\\xi},\\|u(\\cdot+\\xi)\\|\\}^2,\n\\label{xie5}\n\\end{equation}\nwhere $\\mu_{\\xi}$ is the corresponding quantities for $V_{t}^{(\\xi)}$ with integration over $B_2$ and $\\|u(\\cdot+\\xi)\\|$\nis integration over $B_2$.\nBy the definition of $V_t^{(\\xi)}$ and ${\\rm dist}\\,(x,J_{\\xi})^2\n\\leq 2\\, {\\rm dist}\\,(x,J)^2+2|\\xi|^2$, we have\n\\begin{equation}\n\\mu_{\\xi}^2\\leq 2\\mu^2+ 32E_1|\\xi|^2\n\\label{xie6}\n\\end{equation}\nand \n\\begin{equation}\n\\|u(\\cdot+\\xi)\\|\\leq \\|u\\|,\n\\label{xie7}\n\\end{equation}\nwhere $\\|u\\|$ is integration over $B_3$.\nIn the time interval $[t_0-2r_0^2,t_0-r_0^2]\\subset[\\frac54,t_0)$, we choose $t_1$ so that\n\\begin{equation}\n\\int_{B_3}{\\rm dist}\\,(\\cdot, J)^2\\, d\\|V_{t_1}\\|\\leq \\frac{3}{r_0^2} \\mu^2,\n\\label{xie8}\n\\end{equation}\n\\begin{equation}\n\\int_{B_1}|h(V_{t_1},\\cdot)|^2\\, d\\|V_{t_1}\\|\\leq \\frac{3\\Cr{c-2}}{r_0^2}\\max\\{\\mu,\\|u\\|\\}^2.\n\\label{xie8.5}\n\\end{equation}\nSuch $t_1$ exists by the definition of $\\mu$ and by \\eqref{thap10}.\nUsing $t_1$ in \\eqref{xie5} as well as \\eqref{xie6}, \\eqref{xie7} and recalling the definition of $V_t^{(\\xi)}$, we then obtain\n\\begin{equation}\n\\int_{B_{\\frac34}(\\xi)}\\rho_{(\\xi,t_0)}(\\cdot,t_1){\\rm dist}(\\cdot,J_{\\xi})^2\\, d\\|V_{t_1}\\|\n\\leq \\Cr{c-5} (t_0-t_1)^{1\/2} \\max\\{2\\mu^2+32E_1|\\xi|^2,\\|u\\|^2\\}.\n\\label{xie9}\n\\end{equation}\nOn $B_{r_0}$, we have (since $|\\xi|\\leq 2\\tau\\leq \\frac{r_0}{4}$ and $2r_0^2\\geq t_0-t_1\\geq r_0^2$)\n\\begin{equation}\n\\rho_{(\\xi,t_0)}(x,t_1)\\geq \\frac{\\exp\\left(-\\frac{|x|^2+|\\xi|^2}{2(t_0-t_1)}\\right)}{\\sqrt{8\\pi r_0^2}}\n\\geq \\frac{e^{-1}}{\\sqrt{8\\pi}r_0}.\n\\label{xie10}\n\\end{equation}\nUsing $t_0-t_1\\leq 2r_0^2$ and noting that $r_0\\leq \\frac14$ and $|\\xi|\\leq \n\\frac{r_0}{4}$ (implying $B_{r_0}\\subset B_{\\frac34}(\\xi)$), we obtain from \\eqref{xie9} and\n\\eqref{xie10}\n\\begin{equation}\n\\frac{1}{r_0}\\int_{B_{r_0}} {\\rm dist}(\\cdot,J_{\\xi})^2\\, d\\|V_{t_1}\\|\\leq e\\sqrt{16\\pi}\\Cr{c-5} r_0\n \\max\\{2\\mu^2+32E_1 |\\xi|^2,\\|u\\|^2\\}.\n \\label{xie11}\n \\end{equation}\nNext, we consider the set $\\{x\\in {\\mathbb R}^2\\,:\\, {\\rm dist}(x,{\\bf R}_{\\frac{\\pi}{3}}(J))>\n\\frac{r_0}{4}\\}$. Denote the three connected components of this set by $W_1,W_2,W_3$. \nBy the argument in the proof of Proposition \\ref{firprop}, by choosing $\\Cr{e-4}$ \ndepending only on $r_0$ (as in \\eqref{xie8}) and $\\Cr{c-2}$ (as in \\eqref{xie8.5}) \n(which ultimately depend only on $p,q,\\nu,E_1$), \nwe can ensure that\n\\begin{equation}\n\\|V_{t_1}\\|(W_j\\cap B_{r_0})\\geq \\frac{r_0}{2},\\,\\,j=1,2,3.\n\\label{xie12}\n\\end{equation}\nBy Lemma \\ref{little} and the fact that $|\\xi|\\leq r_0\/4$, on one of the components, we have\n\\eqref{xie2}. Without loss of generality, let this component be $W_1$. Then \\eqref{xie12} implies\n\\begin{equation}\n|\\xi|^2\\leq \\frac{2}{r_0}\\int_{W_1\\cap B_{r_0}}|\\xi|^2\\,\nd\\|V_{t_1}\\|\\leq \\frac{16}{3r_0}\\int_{W_1\\cap B_{r_0}} \n\\left({\\rm dist}(\\cdot,J)^2 +{\\rm dist}(\\cdot,J_{\\xi})^2\\right)\\, d\\|V_{t_1}\\|.\n\\label{xie13}\n\\end{equation}\nThe first term of the right-hand side of \\eqref{xie13} may be estimated by \nan appropriate constant times $\\mu^2$ due to \\eqref{xie8}. \nFor the second term, we use \\eqref{xie11} and \\eqref{defr} to deduce\n\\begin{equation}\n\\frac{16}{3r_0}\\int_{B_{r_0}}{\\rm dist}(\\cdot,J_{\\xi})^2\\, d\\|V_{t_1}\\|\n\\leq \\frac{16}{3}2e\\sqrt{16\\pi}\\Cr{c-5} r_0\\max \\{\\mu,\\|u\\|\\}^2\n+\\frac{|\\xi|^2}{2}.\n\\label{xie14}\n\\end{equation}\nBy relegating the last term of \\eqref{xie14} to the left-hand side in \\eqref{xie13}\nand setting an appropriate $\\Cr{c-6}$, we obtain the desired estimate \\eqref{xie4}.\nBy applying Proposition \\ref{tildest} to \n$V_{t}^{(\\xi)}$ and using \\eqref{xie6} and \\eqref{xie4}, we obtain \\eqref{xie4.5}.\n\\hfill{$\\Box$}\n\\begin{prop} Corresponding to $\\gamma\\in (0,\\frac12),\\ \\kappa\\in (0,1),\\ p,q,\\nu,E_1$, there exist $\\Cl[eps]{e-5}\\in (0, \\Cr{e-4}]$ \ndepending only on $p,q,\\nu,E_1,\\gamma$ and $\\Cl[c]{c-8}\\in (1,\\infty)$ depending only on $p,q,\\nu,\nE_1,\\gamma,\\kappa$ such that \nthe following holds: Suppose $\\{V_t\\}_{t\\in [0,4]}$ and $\\{u(\\cdot,t)\\}_{t\\in [0,4]}$\nsatisfy (A1)-(A4) with $U=B_3$ and $I = [0, 4]$. Assume that we have \\eqref{smep1c}-\\eqref{smep2c}\nwith $\\Cr{e-5}$ in place of $\\frac{\\Cr{e-2}}{2}$ and \\eqref{smep34c}.\nSuppose that we have two points $\\xi_1,\\xi_2\\in B_1$ and two \ntimes $t_1,t_2\\in [\\frac32,3]$ such that $h(V_{t_j},\\cdot)\\in L^2(\\|V_{t_j}\\|)$\nand $\\Theta(\\|V_{t_j}\\|,\\xi_j)=\\frac32$ for $j=1,2$. \nThen we have\n\\begin{equation}\n|\\xi_1-\\xi_2|\\leq \\Cr{c-8} \\big(\\max\\{\\mu,\\|u\\|\\}^{\\gamma}+\\sqrt{|t_1-t_2|}\\big)^{\\kappa} \\max\\{\\mu,\\|u\\|\\}.\n\\label{hoe0.5}\n\\end{equation}\n\\label{hoeld}\n\\end{prop}\n{\\it Proof}. Assume $t_1\\leq t_2$ without loss of generality. Write for simplicity \n\\begin{equation}\n{\\hat \\mu}=\\max\\{\\mu,\\|u\\|\\},\\,\\, {\\hat s}={\\hat \\mu}^{\\gamma}+\\sqrt{|t_2-t_1|},\\,\\, {\\hat \\xi}=\\xi_2-\\xi_1.\n\\label{hoe0}\n\\end{equation}\nFrom Proposition \\ref{distrip}, we already know that $|{\\hat \\xi}|\\leq 2\\Cr{c-6} {\\hat \\mu}$. \nWe choose \n${\\hat t}\\in [t_1-2{\\hat s}^2,t_1-{\\hat s}^2]$ \nso that \n\\begin{equation}\n\\int_{B_1}|h(V_{\\hat t},\\cdot)|^2\\, d\\|V_{\\hat t}\\|\\leq \\frac{2\\Cr{c-2}}{{\\hat s}^2}\\hat\\mu^2.\n\\label{hoe1}\n\\end{equation}\nThis is possible by \\eqref{thap10}. Since ${\\hat s}\\geq {\\hat \\mu}^{\\gamma}$ with\n$\\gamma<\\frac12$, ${\\hat s}$ is relatively larger than ${\\hat \\mu}$ for all sufficiently small ${\\hat \\mu}$. \nWe utilize this in the following. We restrict ${\\hat \\mu}$ depending only on $\\Cr{c-6}$ and $\\gamma$ so that \n\\begin{equation}\n2\\Cr{c-6}{\\hat \\mu}\\leq \\frac{\\hat s}{10} \n\\label{hoe2}\n\\end{equation}\nE.g. $\\hat\\mu \\leq (20\\Cr{c-6})^{-\\frac{1}{1-\\gamma}}$ is sufficient.\nConsider the set $B_{\\hat s}\\cap \\{x\\in {\\mathbb R}^2\\,:\\,\n{\\rm dist}(x,{\\bf R}_{\\frac{\\pi}{3}}(J))>2\\Cr{c-6} {\\hat \\mu}\\}$. Due to \\eqref{hoe2}, this set consists of \nthree non-empty connected components, denoted by $W_1,W_2,W_3$. We have \n\\begin{equation}\n\\frac{{\\hat \\mu}^2}{{\\hat s}^2}\\leq{\\hat \\mu}^{2-2\\gamma}={\\hat\\mu}^{2-4\\gamma}\\cdot{\\hat\\mu}^{2\\gamma}\n\\label{hoe3}\n\\end{equation}\nwith ${\\hat \\mu}^{2-4\\gamma}$ chosen as small as one likes (note $\\gamma<1\/2$). \nThus, restricting ${\\hat \\mu}$ depending only on $\\gamma$ and $\\Cr{c-2}$, we can make sure\nusing \\eqref{hoe1} and \\eqref{hoe3} that ${\\rm spt}\\,\\|V_{\\hat t}\\|$ \nlies $o({\\hat \\mu}^{\\gamma})$-neighborhood of $J$ (using the argument\nin Proposition \\ref{firprop}) in $B_1$. The same can be said about\n${\\rm spt}\\,\\|V_{\\hat t}^{(\\xi_1)}\\|$, since $|\\xi_1|\\leq\\Cr{c-6}{\\hat \\mu}$. \nIn particular, since $\\hat s\\geq \\hat \\mu^{\\gamma}$, we have\n\\begin{equation}\n\\|V_{\\hat t}^{(\\xi_1)}\\|(W_j)\\geq \\frac{\\hat s}{4}\n\\label{hoe4}\n\\end{equation}\nfor $j=1,2,3$ under this restriction on ${\\hat \\mu}$. Since\n$|{\\hat \\xi}|\\leq 2\\Cr{c-6}{\\hat \\mu}$, by Lemma \\ref{little}, on one of \n$W_j$, say on $W_1$, we have \\eqref{xie2} with ${\\hat \\xi}$ in place of $\\xi$. Thus \nby \\eqref{hoe4} and \\eqref{xie2} we obtain\n\\begin{equation}\n\\begin{split}\n|{\\hat \\xi}|^2 \\leq \\frac{4}{\\hat s}\\int_{W_1}|{\\hat\\xi}|^2\\, d\\|V_{\\hat t}^{(\\xi_1)}\\|\n&\\leq \\frac{32}{3\\hat s}\\int_{W_1}{\\rm dist}(\\cdot,J)^2+{\\rm dist}(\\cdot,J_{\\hat\\xi})^2\\,\nd\\|V_{\\hat t}^{(\\xi_1)}\\| \\\\\n&\\leq \\frac{32}{3\\hat s}\\sum_{j=1}^2\\int_{B_{2\\hat s}(\\xi_j)}\n{\\rm dist}(\\cdot, J_{\\xi_j})^2\\, d\\|V_{\\hat t}\\|.\n\\end{split}\n\\label{hoe5}\n\\end{equation}\nFor each $j=1,2$ and $x\\in B_{2\\hat s}(\\xi_j)$, we have\n\\begin{equation}\n\\rho_{(\\xi_j,t_j)}(x,{\\hat t})=\\frac{\\exp\\big(-\\frac{|x-\\xi_j|^2}{4(t_j-{\\hat t})}\\big)\n}{\\sqrt{4\\pi (t_j-{\\hat t})}}\\geq \\frac{\\exp(-1)}{\\sqrt{12\\pi} {\\hat s}},\n\\label{hoe6}\n\\end{equation}\nwhere we used ${\\hat s}^2\\leq t_j-{\\hat t}\\leq 3{\\hat s}^2$ which follows\neasily from the definition of ${\\hat s}$ and ${\\hat t}$. By \\eqref{hoe5}\nand \\eqref{hoe6}, we obtain\n\\begin{equation}\n|{\\hat \\xi}|^2\\leq \\frac{32 e \\sqrt{12\\pi}}{3}\n\\sum_{j=1}^2\\int_{B_{2\\hat s}(\\xi_j)}\\rho_{(\\xi_j,t_j)}(\\cdot,{\\hat t})\n{\\rm dist}(\\cdot,J_{\\xi_j})^2\\, d\\|V_{\\hat t}\\|.\n\\label{hoe7}\n\\end{equation}\nFor $\\kappa\\in [0,1)$, by Proposition \\ref{distrip}, each of the last two integrals\nis bounded by $\\Cr{c-7}(t_j-{\\hat t})^{\\kappa}{\\hat\\mu}^2$. Since $t_j-{\\hat t}\\leq 3{\\hat s}^2$,\nby defining $\\Cr{c-8}$ appropriately, we obtain the desired estimate \\eqref{hoe0.5}.\n\\hfill{$\\Box$}\n\\section{Blow-up analysis and improvement of the space-time $L^{2}$-distance}\\label{blowup-analysis}\n\\label{blowupsection}\nThroughout this section, we prove a sequence of propositions under the following assumptions. \nSuppose that $\\{V_t^{(m)}\\}_{t\\in[0,4]}$ and $\\{u^{(m)}(\\cdot,t)\\}_{t\\in [0,4]}$ ($m\\in {\\mathbb N}$) are arbitrary sequences\nsatisfying (A1)-(A4) and \\eqref{smep34c} with $U=B_3$, $I = [0, 4]$ and with $V_{t}^{(m)}$, $u^{(m)}$ in place of $V_{t}$, $u$. Suppose that we have sequences $\\{\\mu^{(m)}\\}_{m\\in {\\mathbb N}}$\nand $\\{\\|u^{(m)}\\|\\}_{m\\in {\\mathbb N}}$ which converge to 0 (and which will be defined in the next section) with the property\n\\begin{equation}\n\\big(\\int_0^4\\int_{B_3}{\\rm dist}\\,(\\cdot,J)^2\\, d\\|V_t^{(m)}\\|dt\\big)^{\\frac12}\\leq \\mu^{(m)},\n\\label{blow1}\n\\end{equation}\n\\begin{equation}\n\\big(\\int_0^4\\big(\\int_{B_3}|u^{(m)}|^p\\, d\\|V^{(m)}_t\\|\\big)^{\\frac{q}{p}}dt\\big)^{\\frac{1}{q}}\\leq \\|u^{(m)}\\|\n\\label{blow2}\n\\end{equation}\nand\n\\begin{equation}\n\\lim_{m\\rightarrow\\infty}(\\mu^{(m)})^{-1}\\|u^{(m)}\\|=0.\n\\label{blow3}\n\\end{equation}\nFix a decreasing sequence $\\{\\tau_m\\}_{m\\in {\\mathbb N}}\\subset (0,\\frac12)$ with $\\lim_{m\\rightarrow\\infty}\\tau_m=0$ and use\nProposition \\ref{smprop}\nwith $\\tau=\\tau_m$ to obtain $\\Cr{e-p}(m)$ and $\\Cr{c-p-1}(m)$ corresponding to $\\tau_m$. We choose a subsequence \nso that, after relabelling, $\\max\\{\\mu^{(m)},\\|u^{(m)}\\|\\}\\leq \\Cr{e-p}(m)$ for all $m\\in {\\mathbb N}$. Then we can apply Proposition\n\\ref{smprop} to $\\{V_t^{(m)}\\}_{t\\in[0,4]}$ and $\\{u^{(m)}(\\cdot,t)\\}_{t\\in [0,4]}$ with $\\tau=\\tau_m$. Let $f_j^{(m)}:[\\tau_m,2-\\tau_m]\n\\times [\\tau_m,4-\\tau_m]\\rightarrow{\\mathbb R}$, $j=1,2,3$, be the resulting functions, satisfying \\eqref{smep6} and \\eqref{smep7}. For each fixed $\\tau\\in (0,\\frac12)$ and for all $m\\in {\\mathbb N}$ with\n$\\tau_m\\leq \\tau$, note that $f_j^{(m)}$ satisfies \\eqref{smep6} with $Q = Q_{\\tau}=[\\tau,2-\\tau]\\times[\\tau,4-\\tau]$ \nand $\\Cr{c-p-1} = \\Cr{c-p-1}(\\tau)$, i.e., \n\\begin{equation}\n\\|f_j^{(m)}\\|_{C^{1,\\zeta}(Q_{\\tau})}\\leq \\Cr{c-p-1}(\\tau) \n\\max\\{\\mu^{(m)},\\|u^{(m)}\\|\\}.\n\\label{blow4}\n\\end{equation}\nFor each $m\\in {\\mathbb N}$ and $j=1,2,3$, define\n\\begin{equation}\n\\tilde f_j^{(m)}=(\\mu^{(m)})^{-1}f_j^{(m)}.\n\\label{blow5}\n\\end{equation}\nBy \\eqref{blow4}, \\eqref{blow5}, \\eqref{blow3} and the Ascoli-Arzel\\`{a} compactness theorem, $\\{\\tilde f_j^{(m)}\\}_{m\\in {\\mathbb N}}$ has a \nsubsequence which converges locally uniformly on $(0,2)\\times(0,4)$ to some limit function $\\tilde f_j$, $j=1,2,3$.\nWe also have the estimate\n\\begin{equation}\n\\|\\tilde f_j\\|_{C^{1,\\zeta}(Q_{\\tau})} \\leq \\Cr{c-p-1}(\\tau).\n\\label{blow6}\n\\end{equation}\nIn the following, we denote subsequences by the same index. \n\\begin{prop}\nThe function $\\tilde f_j$ belongs to $C^{\\infty}((0,1)\\times(1,3))$ and satisfies \nthe heat equation $\\frac{\\partial \\tilde f_j}{\\partial t}=\\frac{\\partial^2 \\tilde f_j}{\\partial x^2}$ on \n$(0,1)\\times (1,3)$ for $j=1,2,3$. \n\\label{blowprop1}\n\\end{prop}\n{\\it Proof}. It is enough to prove the claim for $\\tilde f_1$ since the proof for the other two is\nsimilarly carried out after suitable rotations. \nFix $\\phi\\in C^{\\infty}_c((0,1)\\times(1,3);{\\mathbb R}^+)$, and fix $\\tau\\in (0,\\frac12)$ so that \n${\\rm spt}\\, \\phi\\subset Q_{\\tau}$. For all sufficiently large $m$, we have $\\tau_m<\\tau$ and we \nonly consider such $m$. Let $\\Cr{c-p-1}=\\Cr{c-p-1}(\\tau)$ be a constant to be fixed depending only on $\\tau$. We take \nin \\eqref{meq} $\\phi^{(m)}(x,t) \\equiv (x_2+2\\Cr{c-p-1} \\mu^{(m)})\\phi(x_1,t)\\eta^{(m)}(x_2)$ as a test function,\nwhere, $x=(x_1,x_2)$ and $\\eta^{(m)}$ is a $C^{\\infty}$ function such that $\\eta=1$ for \n$x_2\\in [-\\Cr{c-p-1} \\mu^{(m)},\\Cr{c-p-1} \\mu^{(m)}]$,\n$\\eta^{(m)}=0$ for $x_2\\notin [-2\\Cr{c-p-1} \\mu^{(m)},2\\Cr{c-p-1} \\mu^{(m)}]$ \nand $0\\leq \\eta^{(m)}\\leq 1$. \nNote that $\\phi^{(m)}(\\cdot,t)$ has compact support in $B_3$ and is non-negative, so is a valid choice as a test function. Since \n$x_2=f_1^{(m)}(x_1,t)$ for $(x_1,x_2)\\in {\\rm spt}\\,\\|V_t^{(m)}\\|$, and since $|f^{(m)}|\n\\leq \\Cr{c-p-1} \\mu^{(m)}$ by \\eqref{smep6} and \\eqref{blow3}, for all sufficiently large $m$, \nwe have $\\eta^{(m)}=1$ on ${\\rm spt}\\, \\|V_t^{(m)}\\|$. Thus in the following computation, even though we need \n$\\eta^{(m)}$ for $\\phi^{(m)}$ to have non-negativity, we ignore $\\eta^{(m)}$. We then have\n\\begin{equation}\n0\\leq \\int_1^3\\int_{B_2} (-h(V_t^{(m)},\\cdot)\\phi^{(m)}+\\nabla\\phi^{(m)})\\cdot(h(V_t^{(m)},\\cdot)+(u^{(m)})^{\\perp})\n+\\frac{\\partial\\phi^{(m)}}{\\partial t}\\, d\\|V_t^{(m)}\\|dt.\n\\label{blow7}\n\\end{equation}\nBy the Cauchy-Schwarz inequality and by dropping a negative $|h|^2$ term, we obtain from \\eqref{blow7}\n\\begin{equation}\n\\begin{split}\n0&\\leq \\int_1^3\\int_{B_2}|u^{(m)}|^2\\phi^{(m)}+|u^{(m)}||\\nabla\\phi^{(m)}|+\\frac{\\partial\\phi^{(m)}}{\\partial t}\n+\\nabla\\phi^{(m)}\\cdot h(V_t^{(m)},\\cdot)\\, d\\|V_t^{(m)}\\|dt\\\\\n&=I_1^{(m)}+I_2^{(m)}+I_3^{(m)}+I_4^{(m)}, \\quad {\\rm say}.\n\\end{split}\n\\label{blow8}\n\\end{equation}\nWe next estimate $\\lim_{m\\rightarrow\\infty}(\\mu^{(m)})^{-1}I_j^{(m)}$. By the H\\\"{o}lder inequality, we have\n\\begin{equation}\n\\lim_{m\\rightarrow\\infty}(\\mu^{(m)})^{-1}I_1^{(m)}\\leq \\lim_{m\\rightarrow\\infty}c(p,q)(\\mu^{(m)})^{-1}\\|u^{(m)}\\|^2=0,\n\\label{blow9}\n\\end{equation}\nwhere we used \\eqref{blow3}. Similarly, since $|\\nabla\\phi^{(m)}|\\leq c(\\phi)$ on ${\\rm spt}\\, \\|V_t^{(m)}\\|$\nand by \\eqref{blow3},\n\\begin{equation}\n\\lim_{m\\rightarrow\\infty} (\\mu^{(m)})^{-1}I_2^{(m)}\\leq \\lim_{m\\rightarrow\\infty}c(\\phi)(\\mu^{(m)})^{-1}\\|u^{(m)}\\|=0.\n\\label{blow10}\n\\end{equation}\nFor $I_3^{(m)}$, we have\n\\begin{equation}\n\\begin{split}\n\\lim_{m\\rightarrow\\infty}(\\mu^{(m)})^{-1}I_3^{(m)}&=\\lim_{m\\rightarrow\\infty}\\int_1^3\\int_0^1(\\tilde f_1^{(m)}+2\\Cr{c-p-1})\n\\frac{\\partial\\phi}{\\partial t}\\, \\sqrt{1+|\\partial_{x_1} f_1^{(m)}|^2}\\, dx_1dt\\\\\n&=\\int_1^3\\int_0^1 (\\tilde f_1+2\\Cr{c-p-1})\n\\frac{\\partial\\phi}{\\partial t}\\, dx_1dt,\n\\end{split}\n\\label{blow11}\n\\end{equation}\nwhere we used the uniform convergence $\\tilde f_1^{(m)}\\rightarrow\\tilde f_1$ and \\eqref{blow4}.\nFor $I_4^{(m)}$, since $\\nabla\\phi^{(m)}=(0,1)\\phi+(x_2+2\\Cr{c-p-1} \\mu^{(m)})\\nabla\\phi$, writing\n$h(V_t^{(m)},x)=h=(h_1,h_2)$, we have\n\\begin{equation}\n(\\mu^{(m)})^{-1}I_4^{(m)}=(\\mu^{(m)})^{-1}\\int_1^3\\int_{B_2} \\{\\phi h_2+(x_2+2\\Cr{c-p-1} \\mu^{(m)})\\nabla\\phi\\cdot h\\}\\, d\\|V_t^{(m)}\\|dt.\n\\label{blow12}\n\\end{equation}\nSince $|x_2+2\\Cr{c-p-1} \\mu^{(m)}|\\leq 3\\Cr{c-p-1} \\mu^{(m)}$ and using the estimate \\eqref{thap10}\nwhich is valid here, the second term of the integral converges to $0$. For the first term, by the first variation formula,\n\\begin{equation}\n\\int_{B_2}\\phi h_2\\, d\\|V_t^{(m)}\\|=-\\int_{B_2} S\\cdot (\\nabla\\phi\\otimes (0,1))\\, dV_t^{(m)}(\\cdot,S).\n\\label{blow13}\n\\end{equation}\nSince $S=(1+|\\partial_{x_1} f^{(m)}_1|^2)^{-1}(1,\\partial_{x_1} f^{(m)}_1)\\otimes(1,\\partial_{x_1} f^{(m)}_1)$ and $\\nabla\\phi\n=(\\partial_{x_1}\\phi,0)$, we have\n\\begin{equation}\n-\\int_{B_2} S\\cdot (\\nabla\\phi\\otimes (0,1))\\, dV_t^{(m)}(\\cdot,S)=\n-\\int_0^1 (1+|\\partial_{x_1}f_1^{(m)}|^2)^{-\\frac12}\\partial_{x_1}f_1^{(m)}\\partial_{x_1}\\phi\\,dx_1.\n\\label{blow14}\n\\end{equation}\nSince $\\nabla \\tilde f_1^{(m)}\\rightarrow \\nabla\\tilde f_1$ uniformly, \\eqref{blow12}-\\eqref{blow14} show that\n\\begin{equation}\n\\lim_{m\\rightarrow\\infty} (\\mu^{(m)})^{-1}I_4^{(m)}=-\\int_1^3 \\int_0^1 \\partial_{x_1}\\tilde f_1\\partial_{x_1}\\phi\\, dx_1.\n\\label{blow15}\n\\end{equation}\nCombining \\eqref{blow8}-\\eqref{blow11} and \\eqref{blow15}, we obtain (writing $x_1$ as $x$)\n\\begin{equation}\n0\\leq \\int_1^3\\int_0^1 (\\tilde f_1+2\\Cr{c-p-1})\\frac{\\partial\\phi}{\\partial t}-\n\\frac{\\partial \\tilde f_1}{\\partial x}\\frac{\\partial\\phi}{\\partial x}\\, dxdt.\n\\label{blow16}\n\\end{equation}\nWe carry out the same argument with $\\phi^{(m)}=(2\\Cr{c-p-1}\\mu^{(m)}-x_2)\\phi(x_1,t)\\eta^{(m)}(x_2)$, which\nis again non-negative with compact support. The limit in this case produces\n\\begin{equation}\n0\\leq \\int_1^3\\int_0^1 (2\\Cr{c-p-1}-\\tilde f_1)\\frac{\\partial\\phi}{\\partial t}+\\frac{\\partial \\tilde f_1}{\\partial x}\\frac{\\partial \\phi}{\\partial x}\\,\ndxdt.\n\\label{blow17}\n\\end{equation}\nSince $\\phi$ has a compact support in $(0,1)\\times (1,3)$, the term involving \n$\\Cr{c-p-1}$ is $0$. Thus \\eqref{blow16} and \\eqref{blow17} give\n\\begin{equation}\n0=\\int_1^3\\int_0^1 \\tilde f_1\\frac{\\partial\\phi}{\\partial t}-\\frac{\\partial \\tilde f_1}{\\partial x}\\frac{\\partial \\phi}{\\partial x}\\,\ndxdt.\n\\label{blow18}\n\\end{equation}\nWe have proved that \\eqref{blow18} holds for arbitrary $\\phi\\in C^{\\infty}_c((0,1)\\times(1,3);{\\mathbb R}^+)$. One can \nthen prove that \\eqref{blow18} holds for $\\phi\\in C^{\\infty}_c((0,1)\\times(1,3))$ which is not necessarily non-negative.\nBy the standard regularity theory of parabolic equation, $\\tilde f_1$ is smooth and satisfies the heat equation.\n\\hfill{$\\Box$}\n\nFor the sequence $\\{V_t^{(m)}\\}_{t\\in [0,4]}$ ($m\\in{\\mathbb N}$) under consideration, we define the following.\n\\begin{define}\n\\begin{equation}\n\\begin{split}\nT_g=\\cap_{m\\in{\\mathbb N}}\\Big\\{t\\in [\\frac32,3]\\, :&\\, V_t^{(m)} \\mbox{is a unit density 1-varifold},\\ \\ \nh(V_t^{(m)},\\cdot)\\in L_{loc}^2(\\|V_t^{(m)}\\|)\\\\\n&\\mbox{ and } \\Theta(\\|V_t^{(m)}\\|,x)=1\\mbox{ or }\\frac32,\\, \\ \\forall x\\in {\\rm spt}\\,\\|V_t^{(m)}\\|\\Big\\},\n\\end{split}\n\\label{trisin1}\n\\end{equation}\nwhere all the conditions are required to be satisfied in $B_3$.\n\\end{define}\nSince above conditions are satisfied for a.e$.$ $t\\in [\\frac12,4]$ for each $\\{V_t^{(m)}\\}_{t\\in [0,4]}$, $T_g$ is a \nfull measure set in $[\\frac32,3]$, i.e., ${\\mathcal L}^1([\\frac32,3]\\setminus T_g)=0$. \nWe next define the following sets.\n\\begin{define}\nFor $m\\in{\\mathbb N}$ and $t\\in T_g$, define \n\\begin{equation}\n\\xi^{(m)}(t)=\\Big\\{x\\in B_1\\, :\\, \\Theta(\\|V_t^{(m)}\\|,x)=\\frac32\\Big\\},\\ \\ \\tilde\\xi^{(m)}(t)=\\Big\\{\\frac{x}{\\mu^{(m)}}\\ : \\ x\\in \\xi^{(m)}(t)\n\\Big\\}.\n\\label{trisin2}\n\\end{equation}\n\\end{define}\nSince ${\\rm spt}\\, \\|V_t^{(m)}\\|$ away from the origin consists of three $C^{1,\\zeta}$ curves, there has to be at least one point $x$\nin $B_1$ with $\\Theta(\\|V_t^{(m)}\\|,x)=\\frac32$. Otherwise $\\Theta(\\|V_t\\|.x)=1$ $\\forall x\\in {\\rm spt}\\, \\|V_t^{(m)}\\|\\cap B_1$\nand ${\\rm spt}\\, \\|V_t^{(m)}\\|$ has to be a union of regular embedded curves, a contradiction. Thus $\\xi^{(m)}(t)\\neq \\emptyset$\nfor all $t\\in T_g$.\nWe now apply Proposition \\ref{distrip} and \\ref{hoeld}. Fix $\\gamma\\in (0,\\frac12)$ and $\\kappa\\in (0,1)$. Then for all sufficiently\nlarge $m$, \\eqref{xie4} and \\eqref{hoe0.5} combined with \\eqref{blow3} show that for any $a\\in \\tilde\\xi^{(m)}(t)$, \n\\begin{equation}\n|a|\\leq \\Cr{c-6},\n\\label{trisin3}\n\\end{equation}\nand for any $a\\in \\tilde\\xi^{(m)}(t_1)$ and $b\\in \\tilde\\xi^{(m)}(t_2)$, \n\\begin{equation}\n|a-b|\\leq \\Cr{c-8}((\\mu^{(m)})^{\\gamma}+\\sqrt{|t_1-t_2|})^{\\kappa}.\n\\label{trisin4}\n\\end{equation}\n\\begin{prop}\nThere exists a $\\frac{\\kappa}{2}$-H\\\"{o}lder continuous \nfunction $\\tilde\\xi:[\\frac32,3]\\rightarrow{\\mathbb R}^2$ such that \n\\begin{equation}\n\\sup_{t\\in [\\frac32,3]}|\\tilde\\xi(t)|\\leq \\Cr{c-6},\n\\label{trisin5}\n\\end{equation}\n\\begin{equation}\n\\sup_{t_1,t_2\\in [\\frac32,3],\\, t_1\\neq t_2}\\frac{ |\\tilde\\xi(t_1)-\\tilde\\xi(t_2)|}{|t_1-t_2|^{\\frac{\\kappa}{2}}}\\leq \\Cr{c-8}\n\\label{trisin6}\n\\end{equation}\nand $\\tilde\\xi^{(m)}(t)$ converges uniformly on $T_g$ to $\\tilde\\xi(t)$ in the Hausdorff distance.\n\\label{blowprop2}\n\\end{prop}\n{\\it Proof}. Choose a countable dense set $\\{t_i\\}_{i\\in {\\mathbb N}}\\subset T_g$. For all sufficiently large $m$, \n$\\tilde\\xi^{(m)}(t)$ is bounded uniformly by \\eqref{trisin3}. Also by \\eqref{trisin4}, one notes\nthat the diameter of $\\tilde\\xi^{(m)}(t)$ is $\\leq \\Cr{c-8}(\\mu^{(m)})^{\\gamma}$ and \nconverges to $0$ as $m\\rightarrow\\infty$. Thus for each fixed $i\\in {\\mathbb N}$,\n$\\{\\tilde\\xi^{(m)}(t_i)\\}_{m\\in {\\mathbb N}}$ has a converging subsequence whose limit is a single point. \nBy the diagonal argument, we can extract\na subsequence (denoted by the same index) so that $\\{\\tilde\\xi^{(m)}(t_i)\\}_{m\\in {\\mathbb N}}$ converges to a limit point\ndenoted by $\\tilde\\xi(t_i)$. By \\eqref{trisin4}, $\\tilde\\xi$ is $\\frac{\\kappa}{2}$-H\\\"{o}lder continuous on this countable set,\nand one can extend the definition of $\\tilde\\xi$ uniquely to the whole $[\\frac32,3]$ with the same H\\\"{o}lder constant.\nFor $t\\in T_g\\setminus \\{t_i\\}_{i\\in \\mathbb N}$, by using \\eqref{trisin4}, one\ncan prove that $\\{\\tilde\\xi^{(m)}(t)\\}_{m\\in{\\mathbb N}}$ also converges to $\\tilde\\xi(t)$ and that the convergence is uniform. \n\\hfill{$\\Box$}\n\\begin{define}\nFor each $t\\in [\\frac32,3]$ and $j=1,2,3$, let $\\tilde\\xi_j^{\\perp}(t)\\in {\\mathbb R}$ be obtained as follows. \nFor $j=1$, set $\\tilde\\xi_1^{\\perp}(t)$ be the second coordinate of \n$\\tilde\\xi(t)$. For $j=2,3$, rotate $\\tilde\\xi(t)$ by $\\frac{2\\pi(j-1)}{3}$ \nclockwise, and take its second coordinate to be $\\tilde\\xi_j^{\\perp}(t)$. \n\\label{defcp}\n\\end{define}\nSince ${\\bf R}_0+{\\bf R}_{-\\frac{2\\pi}{3}}+{\\bf R}_{-\\frac{4\\pi}{3}}={\\bf 0}$, we have\n\\begin{equation}\n\\tilde\\xi_1^{\\perp}(t)+\\tilde\\xi_2^{\\perp}(t)+\\tilde\\xi_3^{\\perp}(t)=0\n\\label{trisin7}\n\\end{equation}\nfor all $t\\in [\\frac32,3]$. \n\\begin{prop}\n\\label{blowprop3}\nFor any $t_0\\in [\\frac32,3]$, we have\n\\begin{equation}\n\\sup_{t\\in [\\frac54,t_0)} (t_0-t)^{-\\kappa-\\frac12}\\int_{0}^{\\frac12} e^{-\\frac{x^2}{4(t_0-t)}}\n\\sum_{j=1}^3 |\\tilde f_j(x,t)-\\tilde\\xi_j^{\\perp}(t_0)|^2\\, dx\\leq \\sqrt{4\\pi}\\,\\Cr{c-7}.\n\\label{trisin8}\n\\end{equation}\n\\end{prop}\n{\\it Proof}. If we prove \n\\begin{equation}\n(t_0-t)^{-\\kappa-\\frac12}\\int_{\\tau}^{\\frac12}e^{-\\frac{x^2}{4(t_0-t)}} \\sum_{j=1}^3 |\\tilde f_j(x,t)-\\tilde\\xi_j^{\\perp}(t_0)|^2\\, dx\n\\leq \\sqrt{4\\pi}\\, \\Cr{c-7},\n\\label{trisin9}\n\\end{equation}\nfor arbitrary $t_0\\in T_g$, $t\\in [\\frac54,t_0)$ and $\\tau\\in (0,\\frac12)$,\nthen by the continuity of $\\tilde\\xi_j^{\\perp}$, \\eqref{trisin9} is true for all \n$t_0\\in [\\frac32,3]$ and we will end the proof of \\eqref{trisin8}. Thus we \nfix arbitrary such $t_0,t,\\tau$. By \\eqref{trisin1}, there exists a sequence of non-empty sets\n$\\{\\xi^{(m)}(t_0)\\}_{m\\in {\\mathbb N}}$ as in \\eqref{trisin2}. From each $\\xi^{(m)}(t_0)$,\nchoose one point $\\xi_*^{(m)}(t_0)\\in \\xi^{(m)}(t_0)$. Now, for all sufficiently large $m$,\nwe may apply Proposition \\ref{distrip} with $\\xi$ there replaced by $\\xi_*^{(m)}(t_0)$. \nThus for all sufficiently large $m$, we have\n\\begin{equation}\n(t_0-t)^{-\\kappa}\\int_{B_{\\frac58}\\setminus B_{\\frac{\\tau}{2}}} \\rho_{(\\xi_*^{(m)}(t_0),t_0)}(\\cdot,t){\\rm dist}\\,(\\cdot, J_{\\xi_*^{(m)}(t_0)})^2\n\\, d\\|V_t^{(m)}\\|\\leq \\Cr{c-7}(\\mu^{(m)})^2.\n\\label{trisin10}\n\\end{equation}\nAs we have seen already, we may represent ${\\rm spt}\\, \\|V_t^{(m)}\\|$ by $f_j^{(m)}$ in the relevant domain\nafter suitable rotations. At a point in $x\\in {\\rm spt}\\, \\|V_t^{(m)}\\|$ represented by $(s,f_j^{(m)}(s,t))$ after a rotation, \n\\begin{equation}\n{\\rm dist}\\, (x,J_{\\xi_*^{(m)}(t_0)})=|f_j^{(m)}(s,t) - \\xi_j^{(m)\\perp}(t_0)|,\n\\label{trisin11}\n\\end{equation}\nwhere $\\xi_j^{(m)\\perp}(t_0)=\\mbox{second coordinate of }{\\bf R}_{-\\frac{2(j-1)\\pi}{3}}(\\xi_*^{(m)}(t_0))$. Since\n$(\\mu^{(m)})^{-1}\\xi^{(m)}_*(t_0)\\rightarrow\\tilde \\xi(t_0)$, we have $(\\mu^{(m)})^{-1} \\xi_j^{(m)\\perp}(t_0)\n\\rightarrow \\tilde\\xi_j^{\\perp}(t_0)$. Since $(\\mu^{(m)})^{-1}f_j^{(m)}\\rightarrow \\tilde f_j$ uniformly away from\nthe origin, with \\eqref{trisin11}, we have\n\\begin{equation}\n\\lim_{m\\rightarrow\\infty}(\\mu^{(m)})^{-2} \\int_{B_{\\frac58}\\setminus B_{\\frac{\\tau}{2}}}\\rho_{(\\xi_*^{(m)}(t_0),t_0)}{\\rm dist}\\,(\\cdot,J_{\\xi_*^{(m)}(t_0)})^2\\, d\\|V_t^{(m)}\\|= \\sum_{j=1}^3 \\int_{\\frac{\\tau}{2}}^{\\frac58} \\rho_{(0,t_0)}|\\tilde f_j -\\tilde\\xi_j^{\\perp}(t_0)|^2\\, dx.\n\\label{trisin12}\n\\end{equation}\nRecalling the definition of $\\rho_{(0,t_0)}$, \\eqref{trisin10} and \\eqref{trisin12} prove \\eqref{trisin9} and we end the proof.\n\\hfill{$\\Box$}\n\\begin{lemma}\n\\label{blowlem1}\nThere exists $\\Cl[c]{c-9}\\in (1,\\infty)$ depending only on $\\kappa,p,q,\\nu,E_1$ with the property that\n\\begin{equation}\n\\sup_{t\\in [\\frac54,\\frac52]}\\left( \\int_0^{\\frac12}|\\tilde f_j(x,t)|^2\\, dx\\right)^{\\frac12}\\leq \\Cr{c-9}\n\\label{trisin13}\n\\end{equation}\nfor $j=1,2,3$.\n\\end{lemma}\n{\\it Proof}. We simply choose $t_0=3$ in Proposition \\ref{blowprop3}. \nThen, for any $t\\in[\\frac54,\\frac52]$ and $x\\in(0,\\frac12)$, $t_0-t\\leq \\frac74$ and\n$\\frac{x^2}{4(t_0-t)}\\leq \\frac18$. Moreover, $|\\tilde\\xi_j^{\\perp}(3)|\\leq\\Cr{c-6}$ by \\eqref{trisin5}. Combining these \nfacts and with a suitable constant depending only on $\\Cr{c-6}$ and $\\Cr{c-7}$, and thus ultimately depending\nonly on $\\kappa,p,q,\\nu,E_1$, we obtain \\eqref{trisin13} from \\eqref{trisin8}.\n\\hfill{$\\Box$}\n\\begin{define}\nWe define $\\tilde f_{AV}\\in C^{\\infty}((0,1)\\times(\\frac54,\\frac52))$ by\n\\begin{equation}\n\\label{defavef}\n\\tilde f_{AV}(x,t)=\\frac13 (\\tilde f_1(x,t)+\\tilde f_2(x,t)+\\tilde f_3(x,t)).\n\\end{equation}\n\\end{define}\n\\begin{prop}\n\\label{blowprop4}\nThe odd extension of $\\tilde f_{AV}$ with respect to $x$ satisfies the heat equation on $(-1,1)\\times(\\frac54,\\frac52)$\nand is $C^{\\infty}$ there. In particular, $\\tilde f_{AV}(0,t)=0$ for $t\\in (\\frac54,\\frac52)$. \n\\end{prop}\n{\\it Proof}. Fix $\\tau\\in (0,\\frac14)$. In \\eqref{trisin8}, we use $t\\in (\\frac54,\\frac52)$ and $t_0=\\frac{\\tau^2}{4}+t$. Then\nwe obtain\n\\begin{equation}\n\\int_0^{\\tau} e^{-1} \\sum_{j=1}^3 |\\tilde f_j(x,t)-\\tilde\\xi_j^{\\perp}(t_0)|^2\\, dx\\leq \\sqrt{4\\pi}\\, \\Cr{c-7}\n\\left(\\frac{\\tau^2}{4}\\right)^{\\kappa+\\frac12}.\n\\label{trisin14}\n\\end{equation}\nUsing \\eqref{trisin7}, \\eqref{defavef} and \\eqref{trisin14}, we obtain\n\\begin{equation}\n\\int_0^{\\tau}|\\tilde f_{AV}(x,t)|^2\\, dx\\leq e\\sqrt{4\\pi} 4^{-\\kappa-\\frac12}\\Cr{c-7} \\tau^{2\\kappa+1}.\n\\label{trisin15}\n\\end{equation}\nWe continue to denote the odd extension of $\\tilde f_{AV}(x,t)$ for $x\\in (-1,0)$ by the same notation. \nFor $\\phi\\in C^{\\infty}_c((-1,1)\\times(\\frac54,\\frac52))$, we need to prove\n\\begin{equation}\n\\int_{Q} \\tilde f_{AV}\\frac{\\partial \\phi}{\\partial t}+\\tilde f_{AV}\\frac{\\partial^2 \\phi}{\\partial x^2}\\, dxdt=0\n\\label{trisin16}\n\\end{equation}\nwith $Q=(-1,1)\\times(\\frac54,\\frac52)$. Since $\\tilde f_{AV}$ is odd with respect to $x$, we only \nneed to prove \\eqref{trisin16} for odd $\\phi$. Let $\\eta_{\\tau}\\in C^{\\infty}({\\mathbb R})$ be a function such \nthat $\\eta_{\\tau}(x)=1$ for $|x|\\geq \\tau$, $\\eta_{\\tau}(x)=0$ for $|x|\\leq\\frac{\\tau}{2}$, $0\\leq \\eta_{\\tau}\\leq 1$\nand $|\\eta_{\\tau}'|\\leq 4\\tau^{-1}$ and $|\\eta_{\\tau}''|\\leq 16\\tau^{-2}$. By integration by parts and the fact that $\\tilde f_{AV}$\nsatisfies the heat equation away from $\\{x=0\\}$, we have\n\\begin{equation}\n\\begin{split}\n\\int_{Q} \\tilde f_{AV}\\frac{\\partial \\phi}{\\partial t}+\\tilde f_{AV}\\frac{\\partial^2 \\phi}{\\partial x^2}\\, dxdt&=\\lim_{\\tau\\rightarrow 0+}\n\\int_{Q}\\tilde f_{AV}\\eta_{\\tau}\\frac{\\partial \\phi}{\\partial t}+\\tilde f_{AV}\\eta_{\\tau}\\frac{\\partial^2 \\phi}{\\partial x^2}\\, dxdt\\\\\n&=-\\lim_{\\tau\\rightarrow 0+} \\int_{Q\\cap\\{|x|\\leq \\tau\\}} \\big(2\\eta_{\\tau}'\\frac{\\partial \\phi}{\\partial x}+\\phi\\eta_{\\tau}''\\big)\n\\tilde f_{AV}\\, dxdt.\n\\end{split}\n\\label{trisin17}\n\\end{equation}\nSince $|\\phi(x,t)|\\leq c(|\\nabla\\phi|)|x|$ by the oddness of $\\phi$, we have $|\\phi \\eta_{\\tau}''|\\leq c\\tau$. Then by using \n\\eqref{trisin15}, we may prove that \\eqref{trisin17} is $=0$, which proves \\eqref{trisin16}. The standard regularity theory\nshows that $\\tilde f_{AV}$ is $C^{\\infty}$ on $Q$.\n\\hfill{$\\Box$}\n\\begin{prop}\n\\label{blowprop5}\nFor $j,j'\\in \\{1,2,3\\}$, $t\\in (\\frac54,\\frac52)$ and $x\\in (-1,0)$, define $(\\tilde f_j-\\tilde f_{j'})(x,t)=(\\tilde f_j-\\tilde f_{j'})(-x,t)$.\nThen $\\tilde f_j-\\tilde f_{j'}$ satisfies the heat equation on $(-1,1)\\times\\left(\\frac54,\\frac52\\right)$ and is $C^{\\infty}$ there. \nIn particular, we have\n$\\frac{\\partial (\\tilde f_j-\\tilde f_{j'})}{\\partial x}(0,t)=0$ for $t\\in \\left(\\frac54,\\frac52\\right)$. \n\\end{prop}\n{\\it Proof}. We consider $\\tilde f_1-\\tilde f_2$ since others can be similarly proved. \nGiven $\\tau\\in (0,\\frac14)$, fix an arbitrary $\\tau'\\in (0,\\tau)$. With respect to $\\tau'$, we obtain\n$\\Cr{c-p-1}(\\tau')$ by Proposition \\ref{smprop}. For all sufficiently large $m$, define\n\\begin{equation*}\nT_g^{(m)}=\\left\\{t\\in T_g\\cap \\left[\\frac54,\\frac52\\right]\\,:\\, \\int_{B_2}|h(V_t^{(m)},\\cdot)|^2\\phi_{\\rm rad}^2\\, d\\|V_t^{(m)}\\|\\leq \\Cr{alpha-1}^2\\right\\},\n\\end{equation*}\nwhere $\\Cr{alpha-1}$ is from Proposition \\ref{firprop}. By \\eqref{thap10}, note that we have\n\\begin{equation}\n{\\mathcal L}^1\\left([\\frac54,\\frac52]\\setminus T_g^{(m)}\\right)\\leq \\Cr{alpha-1}^{-2}(\\mu^{(m)})^2 \\Cr{c-2} .\n\\label{trisin18}\n\\end{equation}\nFor any $t\\in T_g^{(m)}$, by Proposition \\ref{firprop}, ${\\rm spt}\\,\\|V_t^{(m)}\\|$ is represented by \n$f^{(m)}_j$ ($j=1,2,3$) inside $B_1$, and there is only one junction point where\nthree curves are joined with angle $\\frac{2\\pi}{3}$. Using the similar notation in the proof of Proposition \\ref{firprop}, \neach curves are represented as in \\eqref{fir7} with $f_j^{(m)}(\\cdot,t)$ and $s_j^{(m)}$ in place of $f_j$ and $s_j$ there. \nWe have \\eqref{fir8}-\\eqref{fir10} satisfied similarly. \nSuppose without loss of generality that $s_1^{(m)}\\leq s_2^{(m)}$. For all sufficiently large $m$, we have \n$|s_2^{(m)}|<\\tau'$ since $|s_j^{(m)}|\\leq \\Cr{c-6}\\mu^{(m)}$ by \\eqref{xie4}. For each $s\\in (\\tau',\\tau)$, \nwe have (omitting $t$ dependence)\n\\begin{equation}\n\\begin{split}\n\\big|\\frac{\\partial( f_1^{(m)}-f_2^{(m)})}{\\partial x}(s)\\big|&\\leq \n\\big|\\frac{\\partial( f_1^{(m)}-f_2^{(m)})}{\\partial x}(s_2^{(m)})\\big|\n+\\int_{s_2^{(m)}}^s\\sum_{j=1,2} \\big|\\frac{\\partial^2 f_j^{(m)}}{\\partial x^2}\\big|\\, dx \\\\\n&\\leq \\big|\\frac{\\partial f_1^{(m)}}{\\partial x}(s_2^{(m)})-\\frac{\\partial f_1^{(m)}}{\\partial x}(s_1^{(m)})\\big|\n+2\\int_{B_{2\\tau}}|h(V_t^{(m)},\\cdot)|\\, d\\|V_t^{(m)}\\|,\n\\end{split}\n\\label{trisin19}\n\\end{equation}\nwhere we have used $\\frac{\\partial f_1^{(m)}}{\\partial x}(s_1^{(m)})=\\frac{\\partial f_2^{(m)}}{\\partial x}(s_2^{(m)})$\nwhich follows from \\eqref{fir10}. The first term of \\eqref{trisin19} may be bounded by the second term, so we\nobtain from \\eqref{trisin19}\n\\begin{equation}\n\\sup_{s\\in (\\tau',\\tau)}\\big|\\frac{\\partial (f_1^{(m)}-f_2^{(m)})}{\\partial x}(s,t)\\big|\n\\leq 4\\int_{B_{2\\tau}}|h(V_t^{(m)},\\cdot)|\\, d\\|V_t^{(m)}\\|.\n\\label{trisin20}\n\\end{equation}\nFor any $t\\in [\\frac54,\\frac52]\\setminus T_g^{(m)}$, we have\n\\begin{equation}\n\\sup_{s\\in (\\tau',\\tau)}\\big|\\frac{\\partial( f_1^{(m)}-f_2^{(m)})}{\\partial x}(s,t)\\big|\n\\leq 2 \\Cr{c-p-1}(\\tau') \\mu^{(m)}.\n\\label{trisin21}\n\\end{equation}\nCombining \\eqref{trisin18}, \\eqref{trisin20} and \\eqref{trisin21}, we obtain\n\\begin{equation}\n\\int_{\\frac54}^{\\frac52}\\sup_{s\\in (\\tau',\\tau)}\\big|\\frac{\\partial (f_1^{(m)}-f_2^{(m)})}{\\partial x}(s,t)\\big|\n\\, dt\\leq 4\\int_{\\frac54}^{\\frac52}\\int_{B_{2\\tau}}|h(V_t^{(m)},\\cdot)|\\, d\\|V_t^{(m)}\\|+2\\Cr{c-p-1}\\Cr{alpha-1}^{-2}\\Cr{c-2}(\\mu^{(m)})^3.\n\\label{trisin22}\n\\end{equation}\nWe may estimate\n\\begin{equation}\n\\begin{split}\n\\int_{\\frac54}^{\\frac52}\\int_{B_{2\\tau}}|h(V_t^{(m)},\\cdot)|\\, d\\|V_t^{(m)}\\|&\\leq (5 \\tau E_1)^{\\frac12} \\big(\\int_1^3\\int_{B_2}\n|h(V_t^{(m)},\\cdot)|^2\\phi_{\\rm rad}^2\\, d\\|V_t^{(m)}\\|\\big)^{\\frac12}\\\\\n&\\leq (5\\tau E_1)^{\\frac12}\\Cr{c-2}^{\\frac12}\\mu^{(m)},\n\\end{split}\n\\label{trisin23}\n\\end{equation}\nby \\eqref{thap10}. Thus dividing \\eqref{trisin22} by $\\mu^{(m)}$ and letting $m\\rightarrow\\infty$, \\eqref{trisin23} and \nthe uniform convergence of $\\frac{\\partial \\tilde f_j^{(m)}}{\\partial x}$ to $\\frac{\\partial \\tilde f_j}{\\partial x}$ show that\n\\begin{equation}\n\\int_{\\frac54}^{\\frac52}\\sup_{s\\in (\\tau',\\tau)}\\big|\\frac{\\partial(\\tilde f_1-\\tilde f_2)}{\\partial x}(s,t)\\big|\\, dt\n\\leq (5\\tau E_1)^{\\frac12}\\Cr{c-2}^{\\frac12}.\n\\label{trisin24}\n\\end{equation}\nSince the right-hand side of \\eqref{trisin24} does not depend on $\\tau'$, the same inequality holds with $\\tau'=0$. \nArguing as in the proof for $\\tilde f_{AV}$, we may prove that the even extension of $\\tilde f_1-\\tilde f_2$ with\nrespect to $x$ now satisfies the heat equation weakly by using \\eqref{trisin24} (with $\\tau'=0$). \nThis time, it is sufficient to use even test functions\n$\\phi$, and use also $|\\frac{\\partial \\phi}{\\partial x}|\\leq c(|\\nabla^2\\phi|)|x|$ in the proof to estimate the truncation error due to $\\eta_{\\tau}$. \nWe omit the detail since it is \nsimilar to the previous case. In particular, we have $\\tilde f_1-\\tilde f_2$ is $C^{\\infty}$ on $(-1,1)\\times(\\frac54,\\frac52)$.\nSince it is evenly extended, the $x$-derivative vanishes on $x=0$. \n\\hfill{$\\Box$}\n\\begin{prop}\nWe have $\\tilde f_j\\in C^{\\infty}([0,1)\\times(\\frac54,\\frac52))$ and $\\tilde f_j(0,t)=\\tilde\\xi_j^{\\perp}(t)$\nfor $t\\in (\\frac54,\\frac52)$ and for $j=1,2,3$. Moreover, for any $k,l\\in {\\mathbb N}\\cup\\{0\\}$, there exists\n$\\Cl[c]{c-10}\\in (1,\\infty)$ depending only on $\\kappa,p,q,\\nu,E_1,k,l$ such that, for $j=1,2,3$, \n\\begin{equation}\n\\sup_{[0,\\frac14)\\times(\\frac32,\\frac52)}\\Big|\\frac{\\partial^{k+l}\\tilde f_j}{\\partial x^{k}\\partial t^{l}}\\Big|\n\\leq \\Cr{c-10}.\n\\label{trisin27}\n\\end{equation}\n\\label{blowprop6}\n\\end{prop}\n{\\it Proof}. We prove this for the case $j=1$; the other cases follow by similar reasoning.\nSince we may write $\\tilde f_1=\\tilde f_{AV}+\\frac13(\\tilde f_1-\\tilde f_2)+\\frac13(\\tilde f_1-\\tilde f_3)$,\nProposition \\ref{blowprop4} and \\ref{blowprop5} show that $\\tilde f_1$ may be smoothly extended for $\\{x\\leq 0\\}$.\nMore precisely, for $x\\in (-1,0)$ and $t\\in (\\frac54,\\frac52)$, define \n\\begin{equation}\n\\tilde f_1 (x,t)=\\big\\{-\\tilde f_{AV}+\n\\frac13(\\tilde f_1-\\tilde f_2)+\\frac13(\\tilde f_1-\\tilde f_3)\\big\\}\\Big|_{(-x,t)}=\\frac13(\\tilde f_1-2\\tilde f_2-2\\tilde f_3)(-x,t).\n\\label{trisin25}\n\\end{equation}\nThen $\\tilde f_1$ is in $C^{\\infty}((-1,1)\\times(\\frac54,\\frac42))$ and satisfies the heat equation on its domain. Moreover, by\n\\eqref{trisin13} and \\eqref{trisin25}, we have\n\\begin{equation}\n\\sup_{t\\in [\\frac54,\\frac52]}\\left(\\int_{-\\frac12}^{\\frac12}|\\tilde f_1(x,t)|^2\\, dx\\right)^{\\frac12}\\leq \\frac83\\Cr{c-9},\n\\label{trisin26}\n\\end{equation}\nand by \\eqref{trisin8}, $\\tilde f_1(0,t)=\\tilde \\xi_1^{\\perp}(t)$ holds for $t\\in (\\frac54,\\frac52)$. \nSince $\\tilde f_1$ satisfies the heat equation with the estimate \\eqref{trisin26}, the standard regularity theory \n(\\cite{ladyzhenskaja}) shows that any partial derivatives of $\\tilde f_1$ on $(-\\frac14,\\frac14)\\times (\\frac32,\\frac52)$\ncan be bounded by a constant depending only on $\\Cr{c-9}$ and the order of differentiation. Since $\\Cr{c-9}$ depends only on $\\kappa,p,q,\\nu,E_1$,\nwe have \\eqref{trisin27} with a suitable constant $\\Cr{c-10}$.\n\\hfill{$\\Box$}\n\\begin{define}\nRecall the definition of $\\tilde\\xi(t)$ (in Definition \\ref{defcp}). For each $m\\in {\\mathbb N}$, define $\\tilde J^{(m)}\\subset {\\mathbb R}^2$ to be the set obtained by first rotating $J$ counterclockwise by $\\arctan(\\mu^{(m)}\\frac{\\partial\\tilde f_1}{\\partial x}(0,2))$ and then translating by\n$\\mu^{(m)}\\tilde \\xi(2)$. In the similarity class of $J$, $\\tilde J^{(m)}$ is the element characterized by the properties that it has junction point at $\\mu^{(m)}\\tilde \\xi(2)$, and the slope of its ray close to the positive $x$-axis is equal to $\\mu^{(m)}\\frac{\\partial\\tilde f_1}{\\partial x}(0,2)$.\nDenote the junction point of ${\\tilde J}^{(m)}$ by $a^{(m)}$ (thus $a^{(m)} =\\mu^{(m)}{\\tilde \\xi}(2)$). \n\\label{deftp}\n\\end{define}\nTo clarify the property of $\\tilde J^{(m)}$ concerning the slope of its ray close the the $x$-axis, we recall that the second coordinate of ${\\bf R}_{-\\frac{(j-1)2\\pi}{3}}(\\tilde \\xi(t))$ has been denoted by \n$\\tilde\\xi_j^{\\perp}(t)$ and is equal to $\\tilde f_j(0,t)$ for $j=1,2,3$. The ray of $\\tilde J^{(m)}$ close to the\n$x$-axis can be expressed as\n\\begin{equation}\n\\Big\\{\\Big(x,\\mu^{(m)}\\tilde f_1(0,2)+\\mu^{(m)}(x-\\mu^{(m)}\\tilde \\xi_1(2))\\frac{\\partial \\tilde f_1}{\\partial x}(0,2)\\Big)\\in {\\mathbb R}^2\\, :\\,\nx\\in (\\mu^{(m)}\\tilde \\xi_1(2),\\infty)\\Big\\}.\n\\label{trisin28}\n\\end{equation}\nMore generally, for $j=1,2,3$, the half line of ${\\bf R}_{-\\frac{(j-1)2\\pi}{3}}(\\tilde J^{(m)})$ close to the $x$-axis is\n\\begin{equation}\n\\Big\\{\\Big(x,\\mu^{(m)}\\tilde f_j(0,2)+\\mu^{(m)}(x-\\mu^{(m)}v_j)\\frac{\\partial \\tilde f_j}{\\partial x}(0,2)\\Big)\\in {\\mathbb R}^2\\, :\\,\nx\\in (\\mu^{(m)}v_j,\\infty)\\Big\\},\n\\label{trisin29}\n\\end{equation}\nwhere $v_j$ is the first coordinate \nof ${\\bf R}_{-\\frac{(j-1)2\\pi}{3}}(\\tilde\\xi(2))$. It is important to note that we used\n$\\frac{\\partial \\tilde f_1}{\\partial x}(0,2)=\\frac{\\partial \\tilde f_j}{\\partial x}(0,2)$ ($j=2,3$) which follows from Proposition \\ref{blowprop5}\nand \\ref{blowprop6}. \n\\begin{prop}\n\\label{blowprop7}\nThere exists $\\Cl[c]{c-11}$ depending only on $p,q,\\nu,E_1$ such that, for all $\\theta\\in (0,\\frac14)$, we have\n\\begin{equation}\n\\limsup_{m\\rightarrow\\infty} \\frac{1}{(\\mu^{(m)})^2\\theta^5}\\int_{2-\\theta^2}^{2+\\theta^2}\\int_{B_{\\theta}\n(a^{(m)})} {\\rm dist}\\,(\\cdot,\\tilde J^{(m)})^2\\, d\\|V_t^{(m)}\\|\ndt\\leq \\Cr{c-11}\\theta^{2}\n\\label{trisin30}\n\\end{equation}\nand\n\\begin{equation}\nd(\\tilde J^{(m)},J)\\leq \\Cr{c-11}\\mu^{(m)}.\n\\label{trisin30.5}\n\\end{equation}\n\\end{prop}\n{\\it Proof}. Fix $\\theta\\in (0,\\frac14)$ and $\\tau\\in (0,\\theta)$. For any $t\\in (2-\\theta^2,2+\\theta^2)$ with $t+\\tau^2\\in T_g$, choose a point\n$\\xi_*^{(m)}\\in \\xi^{(m)}(t+\\tau^2)$ (recall \\eqref{trisin2}). Then by Proposition \\ref{distrip} with\n$\\kappa=\\frac12$ fixed, for all sufficiently large $m$, we have\n\\begin{equation}\n\\tau^{-2\\kappa}\\int_{B_{\\frac34}(\\xi_*^{(m)})} \\rho_{(\\xi_*^{(m)},t+\\tau^2)}(\\cdot,t){\\rm dist}\\,(\\cdot,J_{\\xi_*^{(m)}})^2\\, d\\|V_t^{(m)}\\|\n\\leq \\Cr{c-7}(\\mu^{(m)})^2.\n\\label{trisin31}\n\\end{equation}\nSince $\\rho_{(\\xi_*^{(m)},t+\\tau^2)}(x,t)\\geq (4\\pi\\tau^2)^{-\\frac12}e^{-1}$ for $|x-\\xi_*^{(m)}|\\leq 2\\tau$, we have from \\eqref{trisin31}\n\\begin{equation}\n\\int_{B_{\\tau}} {\\rm dist}(\\cdot,J_{\\xi_*^{(m)}})^2\\, d\\|V_t^{(m)}\\|\\leq \\Cr{c-7}(4\\pi)^\\frac12 e \\tau^{1+2\\kappa}\n(\\mu^{(m)})^2.\n\\label{trisin32}\n\\end{equation}\nHere, the integration should be over $B_{2\\tau}(\\xi_*^{(m)})$, but since $\\xi_*^{(m)}\\rightarrow 0$ (uniformly in $t$) as $m\\rightarrow \\infty$, \nwe have $B_{\\tau}\\subset B_{2\\tau}(\\xi_*^{(m)})$ for sufficiently large $m$ and we obtain \\eqref{trisin32}. We next wish to replace\n$J_{\\xi_*^{(m)}}$ by $\\tilde J^{(m)}$ in \\eqref{trisin32}. The (Hausdorff) distance between $J_{\\xi_*^{(m)}}$ and $J_{\\mu^{(m)}\\tilde\\xi(2)}$ \nin $B_1$ may be estimated by $3\\Cr{c-6}\\mu^{(m)}$ for all sufficiently large $m$ due to\n$|(\\mu^{(m)})^{-1}\\xi_*^{(m)}-\\tilde \\xi(t+\\tau^2)|\\leq o(1)$ and $|\\tilde \\xi(t+\\tau^2)-\\tilde \\xi(2)|\\leq\n2\\Cr{c-6}$ by Proposition \\ref{blowprop2}. The distance between $J_{\\mu^{(m)}\\tilde\\xi(2)}$ and $\\tilde J^{(m)}$ is \nbounded by $\\Cr{c-10}\\mu^{(m)}$ due to the estimate \\eqref{trisin27} (with $k=1$ and $l=0$) \nfor the angle of rotation. Thus we have for any $x\\in B_{\\tau}$\n\\begin{equation}\n{\\rm dist}(x,\\tilde J^{(m)})^2\\leq \n2 \\,{\\rm dist}(x,J_{\\xi_*^{(m)}})^2\n+2(\\mu^{(m)})^2 (3\\Cr{c-6}+\\Cr{c-10})^2.\n\\label{trisin33}\n\\end{equation}\nNow combining \\eqref{trisin32} and \\eqref{trisin33}, we obtain\n\\begin{equation}\n\\limsup_{m\\rightarrow\\infty}\\frac{1}{(\\mu^{(m)})^{2}}\\int_{B_{\\tau}}{\\rm dist}\\, (\\cdot,\\tilde J^{(m)})^2\\, d\\|V_t^{(m)}\\|\n\\leq 2\\Cr{c-7}(4\\pi)^\\frac12 e \\tau^{1+2\\kappa}+6\\tau(3\\Cr{c-6}+\\Cr{c-10})^2.\n\\label{trisin34}\n\\end{equation}\nWe next estimate the integration over $B_{2\\theta}\\setminus B_{\\tau}$. Fix any $t\\in (2-\\theta^2,2+\\theta^2)$. For all \nsufficiently large $m$ depending on $\\tau$, ${\\rm spt}\\, \\|V_t^{(m)}\\|\\cap B_{2\\theta}\n\\setminus B_{\\tau}$ is represented as a union of graphs using $f_j^{(m)}(\\cdot,t)=\\mu^{(m)}\\tilde f_j^{(m)}(\\cdot,t)$. \nRecalling \\eqref{trisin29}, we have (denoting $\\frac{\\partial}{\\partial x}$ by sub-index $x$ for simplicity)\n\\begin{equation}\n\\begin{split}\n&\\frac{1}{(\\mu^{(m)})^2}\\int_{B_{2\\theta}\\setminus B_{\\tau}}{\\rm dist}(\\cdot,\\tilde J^{(m)})^2\\, d\\|V_t^{(m)}\\| \\\\\n&\\leq \\sum_{j=1}^3 \\int_{\\frac{\\tau}{2}} ^{2\\theta}\n\\big|\\tilde f^{(m)}_j(x,t)-\\tilde f_j(0,2)-(x-\\mu^{(m)}v_j) (\\tilde f_j)_x(0,2)\\big|^2\\,\n \\sqrt{1+(\\mu^{(m)})^2 (\\tilde f_j^{(m)})_x^2}\\, dx \\\\\n &\\leq \\sum_{j=1}^3 \\int_{\\frac{\\tau}{2}} ^{2\\theta}\n2\\big|\\tilde f^{(m)}_j(x,t)-\\tilde f_j(0,2)-x (\\tilde f_j)_x(0,2)\\big|^2\\, dx+ c(\\Cr{c-p-1}(\\tau),\\Cr{c-10})(\\mu^{(m)})^2.\n \\end{split}\n \\label{trisin35}\n \\end{equation}\n We know already that $\\tilde f_j^{(m)}$ converges to $\\tilde f_j$ on $[\\frac{\\tau}{2},2\\theta]$, and\n \\begin{equation}\n |\\tilde f_j(x,t)-\\tilde f_j(0,2)-x(\\tilde f_j)_x (0,2)|\\leq \\Cr{c-10}(|x|^2+|t-2|)\n \\label{trisin36}\n \\end{equation}\n by Taylor's theorem and \\eqref{trisin27}. Since $|x|\\leq2 \\theta$ and $|t-2|\\leq \\theta^2$, \\eqref{trisin35}\n and \\eqref{trisin36} prove\n \\begin{equation}\n \\limsup_{m\\rightarrow\\infty}\\frac{1}{(\\mu^{(m)})^2} \\int_{B_{2\\theta}\\setminus B_{\\tau}}{\\rm dist}\\,(\\cdot, \\tilde J^{(m)})^2\\, d\\|V_t^{(m)}\\|\n \\leq 24\\Cr{c-10}^2 \\theta^5.\n \\label{trisin37}\n \\end{equation}\n Since $\\tau$ is arbitrary, combining \\eqref{trisin34} and \\eqref{trisin37} and setting $\\Cr{c-11}\\geq 48\\Cr{c-10}^2$, we obtain the\n desired estimate \\eqref{trisin30}, also by observing $B_{\\theta}(a^{(m)})\\subset\n B_{2\\theta}$ for all sufficiently large $m$. \n By the definition of $\\tilde J^{(m)}$, \\eqref{trisin30.5} follows as well with a suitable choice of \n constant.\n \\hfill{$\\Box$}\n\\section{Pointwise estimates: Proof of Theorem~\\ref{mainreg}}\nWith Proposition \\ref{blowprop7} established, a standard iteration argument establishes the desired \nestimates as well as the expected geometry of the flow as a regular triple junction moving by curvature. For completeness, we present the detailed argument. \n\\begin{prop}\nCorresponding to $p,q,\\nu,E_1$, there exist $\\Cl[eps]{e-6}\\in (0,1)$, $\\theta_*\\in (0,\\frac14)$ and $\\Cl[c]{c-12}\\in (1,\\infty)$ such that the following holds: For $R\\in (0,\\infty)$ and $U=B_{4R}$, suppose $\\{V_t\\}_{t\\in [-2R^2,2R^2]}$ and\n$\\{u(\\cdot,t)\\}_{t\\in [-2R^2,2R^2]}$ satisfy (A1)-(A4). Assume \n\\begin{equation}\n\\mu=\\left( R^{-5}\\int_{-2R^2}^{2R^2} \\int_{B_{4R}} {\\rm dist}\\,(\\cdot,J)^2 \\, d\\|V_t\\|dt\\right)^{\\frac12}<\\Cr{e-6},\n\\label{dec3}\n\\end{equation}\n\\begin{equation}\n\\exists j_1,j_2\\in \\{1,2,3\\} : \nR^{-1}\\|V_{-2R^2}\\|(\\phi_{j_1,J,R})\\leq \\frac{3-\\nu}{2}\\Cr{c-p}, \\ \\ R^{-1}\\|V_{2R^2}\\|(\\phi_{j_2,J,R})\\geq \\frac{1+\\nu}{2}\\Cr{c-p}, \n\\label{dec5}\n\\end{equation}\nand denote\n\\begin{equation}\n\\|u\\|=R^{\\zeta} \\left(\\int_{-2R^2}^{2R^2}\\left(\\int_{B_{4R}} |u|^p\\, d\\|V_t\\|\\right)^{\\frac{q}{p}}\\right)^{\\frac{1}{q}}.\n\\label{dec4}\n\\end{equation}\nThen there exists $ J'={\\bf R}_{\\theta}(J)+\\xi\\in {\\mathcal J}$ such that\n\\begin{equation}\nd_R(J',J)\\leq \\Cr{c-12}\\mu \\quad and\n\\label{dec7}\n\\end{equation}\n\\begin{equation}\n\\left((\\theta_* R)^{-5}\\int_{-2(\\theta_*R)^2}^{2(\\theta_*R)^2} \\int_{B_{4\\theta_* R}(\\xi)} {\\rm dist}\\, (\\cdot, J')^2\\, d\\|V_t\\|dt\\right)^{\\frac12}\n\\leq \\theta_*^{\\zeta} \\max\\{\\mu,\\Cr{c-12}\\|u\\|\\}.\n\\label{dec8}\n\\end{equation}\nMoreover, if we additionally assume that $\\|u\\|\\leq \\Cr{e-6}$, then we have\n\\begin{equation}\n(\\theta_* R)^{-1}\\|V_{-2(\\theta_*R)^2}\\|(\\phi_{j, J',\\theta_*R})\\leq \\frac{3-\\nu}{2}\\Cr{c-p}, \\ \\ (\\theta_* R)^{-1}\\|V_{2(\\theta_* R)^2}\\|\n(\\phi_{j,J',\\theta_*R})\\geq \\frac{1+\\nu}{2}\\Cr{c-p}, \\ \\ j=1,2,3.\n\\label{dec8.5}\n\\end{equation}\n\\label{disjun2}\n\\end{prop}\n{\\it Proof}. We may assume that $R=1$ after a parabolic change of variables. \nWe prove the claim by contradiction. For all $m\\in {\\mathbb N}$, consider a set of sequences\n$\\{V_t^{(m)}\\}_{t\\in [-2,2]}$ and $\\{u^{(m)}(\\cdot, t)\\}_{t\\in [-2,2]}$ satisfying (A1)-(A4) with $U=B_4$\nsuch that \\eqref{dec3} and \\eqref{dec5} are satisfied with $V_t^{(m)},\\frac{1}{m}$, that is, for all $m\\in {\\mathbb N}$,\n\\begin{equation}\n\\mu^{(m)} \\equiv \\left(\\int_{-2}^{2}\\int_{B_4}{\\rm dist}\\,(\\cdot, J)^2\\, d\\|V_t^{(m)}\\|dt\\right)^{\\frac12}<\\frac{1}{m},\n\\label{dec10}\n\\end{equation}\n\\begin{equation}\n\\|V_{-2}^{(m)}\\|(\\phi_{j})\\leq \\frac{3-\\nu}{2} \\Cr{c-p}, \\ \\ \\|V_{2}^{(m)}\\|(\\phi_{j})\\geq \\frac{1+\\nu}{2}\\Cr{c-p}, \\ \\ j=1,2,3.\n\\label{dec11}\n\\end{equation}\nDefine $\\|u^{(m)}\\|$ as in \\eqref{dec4} with $R=1$ and $u^{(m)}$ in place of $u$. \nThe negation then implies that for any $J'={\\bf R}_{\\theta}(J)+\\xi\\in {\\mathcal J}$ with\n\\begin{equation}\nd(J',J)\\leq m\\mu^{(m)},\n\\label{dec13}\n\\end{equation}\nwe have\n\\begin{equation}\n\\left(\\theta_*^{-5}\\int_{-2\\theta_*^2}^{2\\theta_*^2} \\int_{B_{4\\theta_* }(\\xi)} {\\rm dist}\\, (\\cdot,J')^2\\, d\\|V^{(m)}_t\\|dt\\right)^{\\frac12}\n> \\theta_*^{\\zeta} \\max\\{\\mu^{(m)}, m\\|u^{(m)}\\|\\}.\n\\label{dec14}\n\\end{equation}\nHere $\\theta_*\\in (0,\\frac14)$ will be chosen depending only on $p,q,E_1$. \n\nWe next proceed to use the argument in the previous section. First,\nuse $J'=J$ in \\eqref{dec14}. \nThen we have\n\\begin{equation}\n\\theta_*^{\\zeta}\\max\\{\\mu^{(m)},m\\|u^{(m)}\\|\\}<\\left(\\theta_*^{-5}\\int_{-2}^2\\int_{B_4}{\\rm dist}\\, (\\cdot,J)^2\\, d\\|V_t^{(m)}\\|dt\\right)^{\\frac12}\n\\leq \\theta_*^{-\\frac52} \\mu^{(m)}.\n\\label{dec15}\n\\end{equation}\nThus, \\eqref{dec15} shows \\eqref{blow3} is satisfied. We also have $\\lim_{m\\rightarrow\\infty}\\mu^{(m)}=0$ by \\eqref{dec10}. \nWe shift $t$ by $-2$ so that the time interval will be $[0,4]$ from $[-2,2]$. \nWe have \\eqref{blow1}, \\eqref{blow2} and \\eqref{smep34c} satisfied\nthus all the assumptions in the previous section are satisfied. In the argument, we may fix $\\kappa=\\frac12$ from the beginning. The \nconclusion of Proposition \\ref{blowprop7} shows that for all $m>\\Cr{c-11}$, $\\tilde J^{(m)}$ satisfies \\eqref{dec13} due to \\eqref{trisin30.5}\nwhile we have \\eqref{trisin30}. On the other hand, \\eqref{dec14} shows\n\\begin{equation}\n\\theta_*^{2\\zeta}\\leq \\limsup_{m\\rightarrow\\infty} \\frac{1}{(\\mu^{(m)})^2\\theta_*^5} \\int_{2-2\\theta_*^2}^{2+2\\theta_*^2}\n\\int_{B_{4\\theta_*}(a^{(m)})}{\\rm dist}\\,(\\cdot,\\tilde J^{(m)})^2\\, d\\|V_{t-2}^{(m)}\\|dt.\n\\label{dec16}\n\\end{equation}\nIf we let $\\theta=4\\theta_*$ in \\eqref{trisin30} and compare \\eqref{dec16}, we obtain\n\\begin{equation}\n\\theta_*^{2\\zeta}\\leq 4^7 \\Cr{c-11}\\theta_*^2.\n\\label{dec17}\n\\end{equation}\nNote that $\\Cr{c-11}$ and $\\zeta$ depend only on $p,q,\\nu,E_1$. Since $\\zeta\\in (0,1)$, we obtain a contradiction for \nsuitably small $\\theta_*$ depending only on $\\Cr{c-11},\\zeta$ and thus ultimately only on $p,q,\\nu,E_1$. Once $\\theta_*$\nis fixed, then we may use Proposition \\ref{smprop} with $\\tau=\\frac{\\theta_*}{2}$. For suitably small $\\Cr{e-6}$, \nwe can make sure that ${\\rm spt}\\|V_t\\|$ on $B_{4\\theta_*}(\\xi)\\setminus B_{\\frac{\\theta_*}{2}}(\\xi)$ is close to $J$ (and thus to $J'$)\nin $C^{1,\\zeta}$ and \\eqref{dec8.5} can be guaranteed. \n\\hfill{$\\Box$}\n\\begin{prop}\nCorresponding to $\\nu,E_1,p,q$ there exist $\\Cl[eps]{e-7}\\in (0,1)$ and $\\Cl[c]{c-13}\\in (1,\\infty)$\nwith the following property. Under the assumptions of Proposition \\ref{disjun2} where\n$\\Cr{e-6}$ is replaced by $\\Cr{e-7}$ and with $\\|u\\|\\leq \\Cr{e-7}$, \n\\newline\n(1) there exists $J_0\\in {\\mathcal J}$ with the junction point at $\\hat a$ such that \n\\begin{equation}\nd_R(J_{0},J)\\leq \\Cr{c-13}\\max\\{\\mu,\\Cr{c-12}\\|u\\|\\},\n\\label{del9}\n\\end{equation}\n(2) for $0 \\lambda$, and assuming\n$x_1-l_j(s_1)>\\lambda$ without loss of generality, we may \nuse Proposition \\ref{disjun3} with $\\lambda=x_1-l_j(s_1)$ and \nProposition \\ref{smprop} to obtain a $C^{1,\\zeta}$ estimate.\nWith an appropriate choice of $\\Cr{c-14}$, we may finish the proof of\n\\eqref{mr7}. \n\\hfill{$\\Box$}\n\\section{Partial regularity: Proof of Theorem~\\ref{mainpa}} \\label{secpart}\nIn this section, we combine Theorem \\ref{mainreg} with a \nstratification theorem of singular sets. The general idea of\nstratification using tangent cone goes back to Federer \\cite{Fed}\nand it has been adapted in a number of variational problems. \nHere we use a non-trivial adaptation to Brakke flows due to White \\cite{White0}. \nFor the proof of Theorem \\ref{mainpa}, we first recall some definitions and results from \n\\cite{Ilmanenp,White0}. \n\n(a) {\\it Existence of tangent flow}.\nFor any fixed $(y,s)\\in U\\times (0,\\Lambda)$ and $\\lambda>0$, define\n\\begin{equation}\nV^{(y,s),\\lambda}_t(\\phi)=\\lambda^{-1}\\int_{G_1(\\lambda^{-1}(U-y))} \\phi(y+\\lambda x, S)\\, d V_{s+\\lambda^2 t}(x,S)\\label{pat1}\n\\end{equation}\n for $\\phi\\in C_c(G_1(\\lambda^{-1}(U-y)))$ and \n $t\\in (-\\lambda^{-2} s,\\lambda^{-2}(\\Lambda-s))$.\n$V^{(y,s),\\lambda}_t$ is a parabolically rescaled flow at $(y,s)$. For any\npositive sequence $\\{\\lambda_i\\}_{i\\in {\\mathbb N}}$ converging to 0, \nthere exist a subsequence $\\{\\lambda_{i_j}\\}_{j\\in {\\mathbb N}}$\nand a family of varifolds $\\{\\tilde V_t\\}_{t\\in {\\mathbb R}}$ with the following \nproperty: we have\n$\\tilde V_t\\in {\\bf IV}_1\n({\\mathbb R}^2)$ for a.e$.$ $t\\in {\\mathbb R}$, $\\{\\tilde V_t\\}_{t\\in\n{\\mathbb R}}$ satisfies \\eqref{meq} with $u=0$ on ${\\mathbb R}^2\\times{\\mathbb R}$,\n$\\tilde V_t=\\tilde V_{\\lambda^2 t}^{(0,0),\\lambda}$ for all $t<0$ and $\\lambda>0$, and\n$\\lim_{j\\rightarrow\\infty}\\|V^{(y,s),\\lambda_{i_j}}_t\\|=\\|\\tilde V_t\\|$ for all $t\\in \n{\\mathbb R}$. The proof of the existence of such flow is in \n\\cite[Lem. 8]{Ilmanenp} and also see \\cite[Sec. 7]{White0}. \nIt is for $u=0$, \nbut the proof goes through\neven with non-zero $u$ since Huisken's monotonicity formula \\cite{Huisken} \nholds with\na minor error term due to $u$ (see \\cite[Sec. 6]{Kasai-Tonegawa} for the\ndetail) and it\nvanishes as $\\lambda\\rightarrow 0$. We call $\\{\\tilde V_t\\}_{t\\in {\\mathbb R}}$\na {\\it tangent flow} at $(y,s)$. Note that $\\{\\tilde V_t\\}_{t\\in {\\mathbb R}}$ inherits the property of \n(A2) with the same constant $E_1$.\n\n(b) {\\it backwards-cone-like functions}.\nFor any \ntangent flow $\\{\\tilde V_t\\}_{t\\in \n{\\mathbb R}}$ at $(y,s)\\in U\\times(0,\\Lambda)$ and for $(x,t)\\in {\\mathbb R}^2\\times {\\mathbb R}$, define\n\\begin{equation}\ng(x,t)=\\lim_{\\tau\\rightarrow 0+}\n\\int_{{\\mathbb R}^2}\\rho_{(x,t)}(x',t-\\tau)\\, d\\|\\tilde V_{t-\\tau}\\|(x').\n\\label{pat2}\n\\end{equation}\nBy Huisken's monotonicity formula, the limit in \\eqref{pat2} always exists. \nThe set of all such $g$ obtained from a tangent flow at $(y,s)$ is denoted\nby ${\\mathcal G}(y,s)$.\nThe function $g$ has the following property which is called {\\it backwards-cone-like} \\cite[Sec. 8]{White0}: \n\\begin{equation}\n\\label{pat2.1}\ng(x,t)\\leq g(0,0) \\ \\ \\forall (x,t)\\in {\\mathbb R}^2\\times {\\mathbb R},\n\\end{equation}\n\\begin{equation}\ng(x,t)=g(0,0) \\Longrightarrow g(x+x', t+t')=g(x+\\lambda x',t+\\lambda^2 t')\n\\ \\forall t'\\leq 0, \\ \\forall x'\\in {\\mathbb R}^2, \\ \\forall \\lambda>0.\n\\label{pat2.2}\n\\end{equation}\nDefine\n\\begin{equation}\n{\\mathcal V}(g)=\\{x\\in {\\mathbb R}^2 : g(x,0)=g(0,0)\\}, \\ \\\n{\\mathcal S}(g)=\\{(x,t)\\in {\\mathbb R}^2\\times {\\mathbb R} : g(x,t)=g(0,0)\\}.\n\\label{pat3}\n\\end{equation}\nWe note that ${\\mathcal V}(g)$ and ${\\mathcal S}(g)$ are\ndenoted by $V(g)$ and $S(g)$ in \\cite{White0}, respectively, but we changed the \nnotation here to avoid possible confusion. \nThen \\cite[Th. 8.1]{White0} proves that \n\\begin{equation}\n\\label{pat3.5}\ng(x,t)=g(x+x',t) \\ \\ \\forall x\\in {\\mathbb R}^2, \\, \\forall x' \\in {\\mathcal V}(g),\\,\n\\forall t\\leq 0,\n\\end{equation}\n${\\mathcal V}(g)$ is a vector subspace of ${\\mathbb R}^2$, and\n${\\mathcal S}(g)$ is either ${\\mathcal V}(g)\\times \\{0\\}$ or ${\\mathcal V}(g)\n\\times (-\\infty,a]$ for some $a\\in [0,\\infty]$. In the latter case, $g$ is\ntime-independent up to time $t=a$: that is, $g(x,t)=g(x,t')$ for all $t\\leq t'0$ depending only on $E_1$ \nsuch that either\n$g(x,t)\\geq g_0$ or $g(x,t)=0$. In the latter case, there exists a \nspace-time neighborhood $U_{x,t}$ of $(x,t)$ with $U_{x,t}\\cap\n\\cup_{t'}({\\rm spt}\\,\\|\\tilde V_{t'}\\|\\times\\{t'\\})=\\emptyset$. \n\\newline\n{\\it Proof of claim 1}. This is a well-known fact but we include the proof\nfor the convenience of the reader. By Brakke's clearing out lemma \\cite{Brakke} or \n\\cite[Cor.\\,6.3]{Kasai-Tonegawa}, there exist constants $\\tilde g_0>0$ \nand $L>1$ depending on $E_1$\nsuch that for any $\\tau>0$, $\\|\\tilde V_{t-2\\tau}\\|(B_{\\sqrt{\\tau}L}(x))<\\tilde g_0 \\sqrt{\\tau}$ implies $\\|\\tilde V_{t'}\\|\n(B_{\\sqrt{\\tau}}(x))=0$ for all $t'\\in [t-\\tau,t+\\tau]$. Assume \n$g(x,t)>0$. By the monotone property of \\eqref{pat2}, for sufficiently small $\\tau$, we have\n\\begin{equation*}\n2 g(x,t)>\\int_{B_{\\sqrt{\\tau} L}(x)} \\rho_{(x,t)}(x',t-2\\tau)\\, d\\|\\tilde V_{t-2\\tau}\\|(x')\n\\geq \\frac{e^{-\\frac{L^2}{8}}}{\\sqrt{8\\pi\\tau}} \\|\\tilde V_{t-2\\tau}\\|(B_{\\sqrt{\\tau}\nL}(x)). \\label{par5.1}\n\\end{equation*}\nThus, for sufficiently small $g_0$ depending on $\\tilde g_0$ and $L$, \nif $g(x,t)0$. If $g(x,t)=0$, then the same argument shows\nthe last statement, concluding the proof of claim 1. \n\\newline\n{\\it Claim 2}. ${\\rm dim}{\\mathcal V}(g)=2$ implies that there exists\na space-time neighborhood $U_{y,s}\\subset U\\times(0,\\Lambda)$ such that\n$U_{y,s}\\cap \\cup_{t}({\\rm spt}\\,\\|V_t\\|\\times\\{t\\})=\\emptyset$. \n\\newline\n{\\it Proof of claim 2}. By \\eqref{pat3}, $g(\\cdot,0)$ is a constant function on \n${\\mathbb R}^2$. Suppose $g(0,0)\\geq g_0$. By the monotone property of\n\\eqref{pat2} and using (A2), we have for any $x\\in {\\mathbb R}^2$,\n$\\tau>0$ and $R>0$\n\\begin{equation}\ng_0\\leq g(x,0)\\leq \\int_{B_{\\sqrt{\\tau} R}(x)}\\rho_{(x,0)}(x',-\\tau)\\, d\\|\\tilde V_{-\\tau}\\|(x')\n+E_1 o(1)\n\\label{pat6}\n\\end{equation}\nwhere $o(1)$ here means $\\lim_{R\\rightarrow\\infty}o(1)=0$. Hence, \nfixing large $R$ depending only on $E_1$ so the last term is less than \n$\\frac{g_0}{2}$, and then set $\\delta=\\frac{g_0}{2} \\sqrt{4\\pi}$. \nWith this choice and \\eqref{pat6} show\n\\begin{equation}\n\\delta\\sqrt{\\tau}\\leq \\|\\tilde V_{-\\tau}\\|(B_{\\sqrt{\\tau} R}(x))\n\\label{pat7}\n\\end{equation}\nfor all $\\tau>0$ and $x\\in {\\mathbb R}^2$. But then, \nsince $B_R$ may contain $O(\\tau^{-1})$ number of disjoint balls\nof radius $\\sqrt{\\tau}R$, we may prove that\n$\\|\\tilde V_{-\\tau}\\|(B_R)\\geq C \\delta\/\\sqrt{\\tau}$\nwhich goes to infinity as $\\tau\\rightarrow 0$. This is a contradiction\nto (A2). Thus $g(0,0)=0$, and by \\eqref{pat2.1}, $g$ is identically 0 on ${\\mathbb R}^2\n\\times {\\mathbb R}$. Then claim 1 shows \n$\\|\\tilde V_t\\|=0$ for all $t\\in {\\mathbb R}$. \nLet $\\{\\lambda_j\\}_{j=1}^{\\infty}$ be a sequence such that \n$\\|V_t^{(y,s),\\lambda_j}\\|\\rightarrow \\|\\tilde V_t\\|(=0)$ as was\ndescribed in (a). Then one has $\\lim_{j\\rightarrow \\infty}\n\\|V_{t'-2}^{(y,s),\\lambda_j}\\|(B_{L})=0$ for $t'\\in [-1,1]$, $L$ as in claim 1.\nThen by \\cite[Cor.6.3]{Kasai-Tonegawa}, \nfor sufficiently large $j$, we have\n$\\|V_{t'}^{(y,s),\\lambda_j}\\|(B_{1})=0$ for all $t'\\in [-1,1]$.\nThis shows that there exists a space-time neighborhood of $(y,s)$ \non which $\\|V_t\\|$ has measure zero, completing the proof of claim 2. \n\\newline\n{\\it Claim 3}. ${\\rm dim}{\\mathcal V}(g)=1$ implies that there exists a space-time\nneighborhood $U_{y,s}\\subset U\\times(0,\\Lambda)$ such that\n$U_{y,s}\\cap \\cup_{t} ({\\rm spt}\\,\\|V_t\\|\\times \\{t\\})$ is represented \nas a $C^{1,\\zeta}$ graph. \n\\newline\n{\\it Proof of claim 3}. Since ${\\mathcal D}(g)\\geq 2$, \\eqref{pat4} shows\nthat $g$ is static, i.e., ${\\mathcal S}(g)={\\mathcal V}(g)\\times{\\mathbb R}$.\nWe have $g(0,0)\\geq g_0$, or else, $g$ is identically 0 and ${\\rm dim}{\\mathcal V}(g)=2$.\nAs described in (b), $g$ is independent of $t$, and by \\eqref{pat3.5}, \ninvariant in ${\\mathcal V}(g)$ direction. Thus, if $g(x,t)(=g(x,0))>0$ for some\n$x\\in {\\mathbb R}^2\\setminus {\\mathcal V}(g)$, then $g(\\lambda x+x',t)\n=g(x,t)$ for all $x'\\in {\\mathcal V}(g)$ and $\\lambda>0$, letting\n$g$ having a positive constant on the half-space of ${\\mathbb R}^2$\nwith the boundary ${\\mathcal V}(g)$.\nThis leads to a contradiction by the similar argument in the proof of claim 2.\nThus we have $g=0$ outside of ${\\mathcal S}(g)$ and positive constant\non ${\\mathcal S}(g)$. Similarly, we may also prove that $\\|\\tilde V_t\\|(\n{\\mathbb R}^2\\setminus {\\mathcal V}(g))=0$ for all $t\\in {\\mathbb R}$.\nFor a.e$.$ $t\\in {\\mathbb R}$, we have \n$\\tilde V_t\\in {\\bf IV}_1({\\mathbb R}^2)$, thus \n$\\tilde V_t=\\theta(x,t)|{\\mathcal V}(g)|$ for some ${\\mathcal H}^1$ a.e$.$ \ninteger-valued function $\\theta(\\cdot,t)$. Since \n$h({\\tilde V_t},\\cdot)\\in L^2_{loc}$ for a.e$.$ $t$, going back to the\ndefinition of the first variation, one can check that $\\theta(\\cdot,t)$ has to \nbe a constant function. \nSince $g$ is constant on ${\\mathcal S}(g)$,\none easily sees that $\\theta$ is independent of $t$. \nThus with some integer $\\theta_0$, $\\tilde V_t=\\theta_0 |{\\mathcal V}(g)|$ \nfor all $t$. Now by (A5), we necessarily have $\\theta_0=1$. Let $V_t^{(y,s),\\lambda_j}$ be a \nsequence converging to $\\tilde V_t$. \nThen by \\cite[Th.\\,8.7 or Prop.\\,9.1]{Kasai-Tonegawa}, \nfor sufficiently large $j$, we may conclude that $\\cup_t ({\\rm spt}\\,\\|V_t^{(y,s),\n\\lambda_j}\\|\\times \\{t\\})$ is represented as a $C^{1,\\zeta}$ graph in \n$B_1\\times (-1,1)$. This concludes the proof of claim 3. \n\\newline\n{\\it Claim 4}. ${\\rm dim}{\\mathcal V}(g)=0$ implies that there exists a space-time\nneighborhood $U_{y,s}\\subset U\\times(0,\\Lambda)$ such that\n$U_{y,s}\\cap \\cup_{t} ({\\rm spt}\\,\\|V_t\\|\\times \\{t\\})$ is represented \nas a $C^{1,\\zeta}$ triple junction as in Theorem \\ref{mainreg}. \n\\newline\n{\\it Proof of claim 4}. Since ${\\mathcal D}(g)\\geq 2$, by \\eqref{pat4}, \n$g$ is static. $g$ is independent of $t$ and \n$g(x,t)=g(\\lambda x,0)$ for all $x\\in{\\mathbb R}^2$ and $\\lambda>0$. \nWe shall write $g(x)$ instead of $g(x,t)$. \nDefine $W=\\{x\\in {\\mathbb R}^2 : |x|=1,\\, g(x)\\geq g_0\\}$. Then \nfollowing the similar argument as in the proof of claim 2, we may prove\nthat the number of element of $W$ is finite. More precisely, if we pick\n$W'\\subset W$ consisted of $N$ elements, \nwe may choose $O(N\/\\sqrt{\\tau})$ number of\ndisjoint balls of radius $\\sqrt{\\tau} R$ centered at $\\cup_{\\lambda>0}\n\\lambda W'$ inside of $B_{R}$, each satisfying \\eqref{pat7}. \nThen we would have $\\|\\tilde V_{-\\tau}(B_R)\\|\\geq O(N)$, thus \n$N$ cannot go to $\\infty$. Thus $\\{g\\geq g_0\\}$ consists of \na finite number of half rays denoted by $\\{l_j \\subset {\\mathbb R}^2: j=1,\\cdot, N\\}$\nemanating from 0, and $g$ is constant\non each half ray. As in the proof of claim 3, one can argue that \nthere exist some positive integers $\\theta_j$ such that $\\tilde V_t\n=\\sum_{j=1}^{N} \\theta_j |l_j|$. Then again using (A5) which says $\\Theta(\\|V_s\\|,y)<2$\nand arguing as before, we have $N\\leq 3$ and $\\sum_{j=1}^N\\theta_j\n\\leq 3$. The conditions $h(\\tilde V_t,\\cdot)\n\\in L^2_{loc}$ and ${\\rm dim}{\\mathcal V}(g)=0$ limit the possibility to \n$\\tilde V_t=|{\\bf R}_{\\theta}(J)|$ with some $\\theta\\in [0,2\\pi)$. \nWe are now ready to apply Theorem \\ref{mainreg} to \n$V_t^{(y,s),\\lambda_j}$. Note that (after a rotation by $\\theta$) \nthat \\eqref{mr1}-\\eqref{mr3} are satisfied for all sufficiently large $j$.\nThus this concludes the proof of claim 4. Since $U_{y,s}$ in all three\ncases do not intersect with $\\Sigma_1$, $\\Sigma_1$ is a closed set \nand this ends the proof of Theorem \\ref{mainpa}.\n\\hfill{$\\Box$}\n\n\\section{The top dimensional part of the genuine singular set}\n\n\n \n\n\n\\label{difpat}\nUnder the hypotheses (A1)-(A5) of Theorem \\ref{mainpa}, let $\\Sigma_1$ and $\\Sigma_0$\nbe defined as in \\eqref{pat5}. We know that $\\Sigma_1$ is closed (by Theorem \\ref{mainpa}), \nand that ${\\rm dim} \\, \\Sigma_{1} \\leq 1$ where ${\\rm dim}$ is the parabolic Hausdorff dimension. Moreover, \n$\\Sigma_0$ is discrete by \\cite[Th.\\,8.2]{White0}. \n\nWe may further characterize the ``top dimensional part'' of the singular set, i.e.\\ $\\Sigma_1\n\\setminus \\Sigma_0,$ in terms of tangent flows as follows: \n\\begin{thm}\nFor any $(y,s)\\in \\Sigma_1\\setminus \\Sigma_0$, there exists a \nquasi-static tangent flow $\\{\\tilde V_t\\}_{t\\in {\\mathbb R}}$ such that, ${\\rm spt} \\, \\|\\tilde V_{t}\\| = \\emptyset$ or \n${\\rm spt}\\,\\|\\tilde V_t\\|=S$ for some $S\\in {\\bf G}(2,1)$. Moreover, there exists a set of integers $\\theta_1>\\cdots\n>\\theta_N\\geq 0$ ($N\\geq 2$) and real numbers $0\\leq a_1<\\cdots0\\}$ then consists of a finite number of lines \nparallel to ${\\mathcal V}(g)$ and by the argument in the proof of Theorem \\ref{mainpa}, \none can prove that $\\tilde V_{-1}$ is a sum of varifolds with integer multiples supported on such lines. Due to the\nbackwards-cone-like property and also the fact that $\\tilde V_t$ is a curvature flow, one\ncan prove that $\\tilde V_t=\\theta_1 |{\\mathcal V}(g)|$ ($\\theta_1\\in {\\mathbb N}$) for $t<0$ (otherwise it has to move\nat non-zero speed even if it is a line). This shows that $g$ has to be quasi-static. \nBy \\cite{White0}, we know that ${\\mathcal S}(g)={\\mathcal V}(g)\\times(-\\infty,a]$ for some $a\\in [0,\\infty)$.\nThis in particular shows $\\tilde V_t=\\theta_1|{\\mathcal V}(g)|$ for $t\\in (-\\infty,a)$. \nOne can then prove, for example using the clearing-out lemma \\cite{Brakke}, that \n${\\rm spt}\\,\\|\\tilde V_t\\|\\subset {\\mathcal V}(g)$ for all $t>0$ and by $h(\\tilde V_t,\\cdot)\\in L^2_{loc}$, \nthat $\\tilde V_t=\\theta(t)|{\\mathcal V}(g)|$ for some $\\theta(t)\\in {\\mathbb N}$. The fact that\nthey are time-discretely decreasing can be easily seen from the curvature flow inequality.\nIf $\\theta_1=1$, and if $a_1>0$, then this would mean that ${\\tilde V}_t=|{\\mathcal V}(g)|$\nin a neighborhood of $(0,0)$. Since $V_t^{(y,s),\\lambda_j}$ is approaching to ${\\tilde V}_t$,\nfor sufficiently large $j$, we may apply \\cite[Th.\\,8.7]{Kasai-Tonegawa} to $V_t^{(y,s),\\lambda_j}$ \nin some small neighborhood of $(0,0)$ and conclude that $(y,s)$ is a $C^{1,\\zeta}$ regular point of $V_t$\n(see the definition in \\cite{Kasai-Tonegawa}). But then \nthe tangent flow at $(y,s)$ should be static, a contradiction. Thus if $\\theta_1=1$, then \n$a_1=0$. This completes the proof.\n\\hfill{$\\Box$}\n\n\\noindent\n{\\bf Remark:} If we assume further that there exists no quasi-static tangent flow with ${\\rm dim}\\,{\\mathcal V}(g)=1$, \nTheorem \\ref{difpatreg} \nshows that $\\Sigma_1\\setminus\\Sigma_0=\\emptyset$. If this is satisfied,\nthe picture is akin to that of the motion of grain boundaries\nwhere networks of curves joined by triple junctions move continuously with occasional\ncollisions of junctions only at discrete points in $\\Sigma_0$. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Conclusion}\n\\label{sec:concl}\n\nIn this paper, we proposed a method for verifying CTL properties with\nrespect to a (possibly infinite-space) program. \nThe method takes a transition system that models the input program and\na CTL formula specifying the property to prove as inputs.\nIt first applies proof rules from its proof system to generate a set\nof forall-exists quantified Horn constraints and well-foundedness\nconstraints. \nThen, it applies the solving algorithms \\textsc{E-HSF}\\xspace to solve the\ngenerated set of Horn constraints.\nThe defining feature of this approach is the separation of concerns\nbetween the encoding and the solving of the verification problem.\nAlthough our method is based on generic Horn constraint solving\nengine, it is able to outperform state-of-art methods \nspecialised for CTL verification.\nWe also demonstrate the practical applicability of the approach by\npresenting an experimental evaluation using examples from the\nPostgreSQL database server, the SoftUpdates patch system, the Windows\nOS kernel.\n\n\n\n\\section{Constraint generation}\n\\label{sec:cons-gen}\n\nThe contraint generation procedure performs a top-down, recursive\ndescent through the syntax tree of the given CTL formula.\nAt each level of recursion, the procedure takes as input a CTL\nsatisfaction $(p(v),\\mathit{next}(v,v')) \\models_{\\mathit{CTL}} \\varphi$, where $\\varphi$\nis a CTL formula and assertions $p(v)$ and\n$\\mathit{next}(v, v')$ describe a set of states and a transition\nrelation, respectively.\nThe constraint generation procedure applies proof rules from the proof\nsystem presented in the previous section to recursively decompose\ncomplex satisfactions and eventually generate forall-exists quantified\nHorn constraints with well-foundedness conditions.\nBefore starting the actual constraint generation, the procedure\nrecursively rewrites the input satisfaction of a given CTL formula with arbitrary\nstructure into a set of satisfactions of simple CTL formulas where\neach simple formula is either a basic CTL state formula or an\nassertion over the background theory.\nThe procedure then takes each satisfaction involving simple formula,\nintroduces auxiliary predicates and generates a sequence of\nforall-exists quantified Horn constraints and well-foundedness\nconstraints (when needed) over these predicates.\n\n \n\\paragraph*{Complexity and Correctness}\nThe procedure performs a single top-down descent through the syntax tree\nof the given CTL formula $\\varphi$.\nThe run time for constraints generation, and hence the size of the\ngenerated constraints, is linear in the size of $\\varphi$.\nFinding a solution for the generated Horn constraints is undecidable in\ngeneral.\nIn practice however, our solving algorithm \\textsc{E-HSF}\\xspace often succeeds in\nfinding a solution (see Section~\\ref{sec:eval}).\nWe formalize the correctness of the constraint generation procedure in\nthe following theorem.\n\\begin{theorem}\nFor a given program $\\ensuremath{P}$ with $\\mathit{init}(v)$ and $\\mathit{next}(v, v')$ over $v$ and\na CTL formula $\\varphi$ the Horn constraints generated from\n$(p(v),\\mathit{next}(v,v')) \\models_{\\mathit{CTL}} \\varphi$ are satisfiable if and only\nif $\\ensuremath{P} \\models \\varphi$.\n\\end{theorem}\nThe proof can be found in~\\cite{KestenTCS95}.\n\n\\paragraph*{Example}\nLet us consider the program given in\nFigure~\\ref{fig-ctl-example-code}. \nIt contains the variable \\emph{rho} which is assigned a non-deterministic\nvalue at Line~2.\nThis assignment results in the program control to move\nnon-deterministically following the evaluation of the condition at\nLine~4.\nIt is common to verify such programs with respect to various CTL\nproperties as the non-determinism results in different computation\npaths of the program. \nNow, we would like to verify the program with respect to the CTL property\n$\\mathit{AG(EF~(WItemsNum \\geq 1))}$, i.e., from every reachable \nstate of the program, there exists a path to a state where\n\\emph{WItemsNum} has a positive integer value. \n\\begin{figure}[h]\n \\begin{lstlisting}[language=C]\n int main () {\n1: while(1) {\n2: while(1) { \n rho = nondet();\n3: if (WItemsNum<=5) { \n4: if (rho>0) break; }\n5: WItemsNum++;\n6: } \n7: while(1) { \n8: if (!(WItemsNum>2)) break;\n9: WItemsNum--;\n10: }\n11: }\n12: }\n \\end{lstlisting}\n \\caption{An example program}\n \\label{fig-ctl-example-code}\n\\end{figure}\n\nWe can make the following observations about the program.\nThe value of the variable \\emph{WItemsNum} is not set initially. \nTherefore, the property is checked for any arbitrary initial value of\n\\emph{WItemsNum}.\nThe verification problem is more interesting for the case when\n\\emph{WItemsNum} has a non-positive integer value.\n\\begin{figure}[h]\n\\begin{equation*}\n \\begin{array}[t]{@{}rl@{\\;}l@{\\;}}\n v & =& (w,pc) \\\\[\\jot]\n \\mathit{init}(v) & = &(pc=1) \\\\[\\jot]\n \\mathit{next}(v,v') & = &(pc=\\ell_1 \\land \\mathit{pc}'=\\ell_2 \\land w'=w ~\\lor pc=\\ell_2 \\land \\mathit{pc}'=\\ell_3 \\land w'=w ~\\lor \\\\[\\jot]\n & & pc=\\ell_3 \\land w \\leq 5 \\land \\mathit{pc}'=\\ell_4 \\land w'=w ~\\lor pc=\\ell_3 \\land w >5 \\land \\mathit{pc}'=\\ell_5 \\land w'=w ~\\lor \\\\[\\jot]\n & & pc=\\ell_4 \\land \\mathit{pc}'=\\ell_5 \\land w'=w ~\\lor pc=\\ell_4 \\land \\mathit{pc}'=\\ell_7 \\land w'=w ~\\lor \\\\[\\jot]\n & & pc=\\ell_5 \\land \\mathit{pc}'=\\ell_6 \\land w'=w+1 ~\\lor pc=\\ell_6 \\land \\mathit{pc}'=\\ell_3 \\land w'=w ~\\lor \\\\[\\jot]\n & & pc=\\ell_7 \\land \\mathit{pc}'=\\ell_8 \\land w'=w ~\\lor pc=\\ell_8 \\land w \\leq 2 \\land \\mathit{pc}'=\\ell_{11} \\land w'=w ~\\lor \\\\[\\jot]\n & & pc=\\ell_8 \\land w > 2 \\land \\mathit{pc}'=\\ell_9 \\land w'=w ~\\lor pc=\\ell_9 \\land \\mathit{pc}'=\\ell_{10} \\land w'=w-1 ~\\lor \\\\[\\jot]\n & & pc=\\ell_{10} \\land \\mathit{pc}'=\\ell_8 \\land w'=w ~\\lor pc=\\ell_{11} \\land \\mathit{pc}'=\\ell_3 \\land w'=w)\n \\end{array}\n\\end{equation*}\n \\caption{Transition system for the example program}\n \\label{fig-TS-example}\n\\end{figure}\nThis is because depending on how the variable \\emph{rho} is\ninstantiated at Line~2, we may get a path that will not reach a state where\n\\emph{WItemsNum} gets a positive integer value.\nFor example, if we assume \\emph{WItemsNum} has the value 0 initially\nand \\emph{WItemsNum} is instantiated to the value 1, the program\ncontrol swings between the two internal loops by keeping the value of\n\\emph{WItemsNum} the same.\nThis resulting path will not reach the state with \\emph{WItemsNum\n $\\geq$ 1}. \nHowever, if \\emph{rho} is assigned a non-positive value, no matter\nwhat the value of \\emph{rho} is initially, it will eventually reach a\nvalue greater than 5 before exiting the first nested loop.\nSuch a path will eventually reach the state with \\emph{WItemsNum\n $\\geq$ 1} and hence the program satisfies the CTL property\n$AG(EF~(WItemsNum \\geq 1))$.\n\nOur method abstracts away from the concrete syntax of a programming\nlanguage by modeling a program as a transition system.\nThe transition system for the program is given in Figure~\\ref{fig-TS-example}.\n \n \n \n \n \n \n \n \n \n \n \n \n \n \nIn the tuple of variables $v$, the variable $w$ corresponds to the\nprogram variable \\emph{WItemsNum} and $pc$ is the program counter\nvariable.\nThe problem of verifying the program with respect to the given property\namounts to checking if $(\\mathit{init}(v), \\mathit{next}(v,v'))$ satisfies\n$AG(EF(w \\geq 1))$, i.e., if the satisfaction $(\\mathit{init}(v), \\mathit{next}(v,\nv')) \\models_{\\mathit{CTL}} AG(EF(w \\geq 1))$ holds.\nOur method first generates a set of Horn constraint corresponding to\nthe verification problem by applying the proof system.\n\nWe start constraint generation by considering the nesting structure of\n$AG(EF(w \\geq 1))$.\nDue to the fact that $AG(EF(w \\geq 1))$ has $AG$ as the outermost operator, we apply $\\textsc{RuleCtlDecompUni}\\xspace$\nfrom Figure~\\ref{fig-ctl-proof-rule-decompUni} to split the original\nsatisfaction $(\\mathit{init}(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}} AG(EF(w \\geq 1))$ into\na reduced satisfaction $(\\mathit{init}(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}} AG(p_1(v))$ and\na new satisfaction $(p_1(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}} EF(w \\geq 1)$. \nWe need to solve for the auxiliary assertion $p_1(v)$ satisfying both of\nthe satisfactions.\n\nThe assertion $p_1(v)$ corresponds to a set of program states that\nneeds to be discovered from the initial state. \nThis is represented by the new satisfaction $(\\mathit{init}(v), \\mathit{next}(v, v'))\n\\models_{\\mathit{CTL}} AG(p_1(v))$ which is reduced directly to a set of Horn\nconstraints by applying $\\textsc{RuleCtlAG}\\xspace$\nfrom Figure~\\ref{fig-ctl-proof-rule-AG}.\nThis set of Horn constraints is over an auxiliary\npredicate~$\\mathit{inv}_1(v)$ and given below.\n\\begin{equation*}\n \\begin{array}[t]{@{}l@{}}\n \\mathit{init}(v) \\rightarrow \\mathit{inv}_1(v),\\\\[\\jot]\n \\mathit{inv}_1(v) \\land \\mathit{next}(v, v') \\rightarrow \\mathit{inv}_1(v'),\\\\[\\jot]\n \\mathit{inv}_1(v) \\rightarrow p_1(v).\n \\end{array}\n\\end{equation*}\n\nThe formula $EF(w \\geq 1)$, which was nested in the original formula\n$AG(EF(w \\geq 1))$, must also be satisfied from the set of\nstates represented by $p_1(v)$.\nThis is represented by the new satisfaction $(p_1(v), \\mathit{next}(v,\nv')) \\models_{\\mathit{CTL}} EF(w \\geq 1)$.\nSuch new satisfactions may not lead directly to Horn constraints\ngeneration and may require further reduction into simpler\nsatisfactions. \nSince $EF(w \\geq 1)$ has $EF$ as the outermost operator, we apply\nagain $\\textsc{RuleCtlDecompUni}\\xspace$ from\nFigure~\\ref{fig-ctl-proof-rule-decompUni} to split the satisfaction\n$(\\mathtt{p}_1(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}} EF(w \\geq 1)$ into a reduced satisfaction\n$(\\mathtt{p}_1(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}} EF(p_2(v))$ and a new\nsatisfaction $(p_2(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}} w \\geq 1$. \nHere also, we need to solve for the auxiliary assertion $p_2(v)$ satisfying both of\nthe satisfactions.\n\nThe equivalence between the formulas $EF(p_2 (v))$ and $EU(\\mathit{true}, p_2\n(v))$ is used to reduce the CTL satisfaction problem $(\\mathtt{p}_1(v),\n\\mathit{next}(v, v')) \\models_{\\mathit{CTL}} EF(p_2(v))$ into $(p_1(v), \\mathit{next}(v,v'))\n\\models_{\\mathit{CTL}} EU(true, p_2(v))$. \nThe corresponding set of Horn constraints are generated by applying\n$\\textsc{RuleCtlEU}\\xspace$ from Figure~\\ref{fig-ctl-proof-rule-EU}. \nDue to the existential path quantifier in $EU(true, p_2(v))$, we\nobtain clauses that contain existential quantification.\nWe deal with the eventuality by imposing a well-foundedness condition.\nThis set of Horn constraints is over the auxiliary\nassertions $\\mathit{inv}_2(v)$ and $\\mathit{rank}(v, v')$, and it is given below.\n\\begin{equation*}\n \\begin{array}[t]{@{}l@{}}\n p_1(v) \\rightarrow \\mathit{inv}_2(v),\\\\[\\jot]\n \\mathit{inv}_2(v) \\land \\neg p_2(v) \\rightarrow \n \\exists v': \\mathit{next}(v, v') \\land \\mathit{inv}_2(v') \\land \\mathit{rank}(v, v'),\n \\\\[\\jot]\n \\mathit{wf}(\\mathit{rank})\n \\end{array}\n\\end{equation*}\n\nComing to the new satisfaction $(p_2(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}}\nw \\geq 1$, we see that its formula $w \\geq 1$ is an assertion with no\ntemporal operators. \nSince no further decomposition is possible, we apply $\\textsc{RuleCtlInit}\\xspace$ from\nFigure~\\ref{fig-ctl-proof-rule-init} to generate directly the\nclause:\n\\begin{equation*}\n p_2(v) \\rightarrow w \\geq 1\n\\end{equation*}\n\nAs the original satisfaction $(\\mathit{init}(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}}\nAG(EF(w \\geq 1))$ is reduced into the satisfactions \n$(\\mathit{init}(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}} AG(p_1(v))$, \n$(p_1(v), \\mathit{next}(v,v')) \\models_{\\mathit{CTL}} EF(p_2(v))$ and \n$(p_2(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}} w \\geq 1$, \nthe constraints for the original satisfaction will be the union of the\nconstraints for each of the decomposed satisfactions.\nThe Horn constraints are over the auxiliary assertions $p_1(v)$,\n$\\mathit{inv}_1(v)$, $p_2(v)$, $\\mathit{inv}_2(v)$ and $\\mathit{rank}(v, v')$, and they are\ngiven below.\n\\begin{equation*}\n \\begin{array}[t]{@{}l@{}}\n \\mathit{init}(v) \\rightarrow \\mathit{inv}_1(v),\\\\[\\jot]\n \\mathit{inv}_1(v) \\land \\mathit{next}(v, v') \\rightarrow \\mathit{inv}_1(v'),\\\\[\\jot]\n \\mathit{inv}_1(v) \\rightarrow p_1(v),\\\\[\\jot]\n p_1(v) \\rightarrow \\mathit{inv}_2(v),\\\\[\\jot]\n \\mathit{inv}_2(v) \\land \\neg p_2(v) \\rightarrow \n \\exists v': \\mathit{next}(v, v') \\land \\mathit{inv}_2(v') \\land \\mathit{rank}(v, v'),\n \\\\[\\jot]\n \\mathit{wf}(\\mathit{rank}) \\\\[\\jot]\n p_2(v) \\rightarrow w \\geq 1\n \\end{array}\n\\end{equation*}\nThis will be the final output of our Horn constraint generation\nprocedure.\n\n\n\n\\section{Evaluation}\n\\label{sec:eval}\n\nWe evaluate our method of CTL verification by applying the\nimplementation of the \\textsc{E-HSF}\\xspace solver on a set of industrial benchmarks\nfrom~\\cite[Figure~7]{CookPLDI13}. \nThese benchmarks consists of seven programs:\n\\texttt{Windows OS fragment 1}, \\texttt{Windows OS fragment 2},\n\\texttt{Windows OS fragment 3}, \\texttt{Windows OS fragment 4},\n\\texttt{Windows OS fragment 5}, \\texttt{PostgreSQL pgarch} and\n\\texttt{Software Updates}. \nFor each of these programs, four slightly different versions are\nconsidered for evaluation.\nIn general, the four versions of a given program are the same\nin terms of the main logic of the program and what the program does,\nbut they may differ on the value assigned to a particular variable or\nthe condition for exiting a loop, etc.\nThis gives us a set of 28 programs.\nEach such program $P$ is provided with a CTL property $\\varphi$, and\nthere are two verification tasks associated with it: $P \\models_{\\mathit{CTL}}\n\\varphi$ and $P \\models_{\\mathit{CTL}} \\neg\\varphi$.\nThe existence of a proof for a property $\\varphi$ for $P$ implies that $\\neg\\varphi$\nis violated by the same program $P$, and similarly, a proof for\n$\\neg\\varphi$ for $P$ implies that $\\varphi$ is violated by $P$.\nHowever, it may also be the case that neither $P \\not\\models_{\\mathit{CTL}} \\varphi$ nor $P \\not\\models_{\\mathit{CTL}}\n\\neg\\varphi$ hold.\n\n\\paragraph{Templates: }\nAs discussed in Section~\\ref{subsec-ehsf}, \\textsc{E-HSF}\\xspace requires the\ntemplate functions to be provided by the user for relations with\nexistentially quantified variables. \nFor the application of CTL verification, which is the main topic of\ninterest in the paper, we claim that the transition relation\n$\\mathit{next}(v,v')$ can be used as a template by adding constraints at\neach location of non-determinism.\nThere are two kinds of constraints that can be added depending on the\ntwo types of possible non-determinism in $\\mathit{next}(v,v')$.\n\\begin{itemize} \n\\item \\textbf{non-deterministic guards: } this is the case when\n $\\mathit{next}(v,v')$ has a set of more than one disjuncts with the\n same guard, i.e., there can be more than one enabled moves from a\n certain state of the program. \n For each such set, we introduce a fresh case-splitting variable and\n we strengthen the guard of each disjunct by adding a distinct\n constraint on the fresh variable. \n For example, if the set has $n$ disjuncts and $B$ is a fresh\n variable, we add the constraint $B=i$ for each disjunct $i$ where\n $1 \\leq i \\leq n$.\n To reason about existentially quantified queries, then it will\n suffices to instantiate $B$ to one of the values in the range\n $1\\dots n$.\n Such reasoning is done by the \\textsc{E-HSF}\\xspace solver. \n\\item \\textbf{non-deterministic assignments:} this is the case when\n $\\mathit{next}(v,v')$ has a disjunct in which some $w'$, which is a\n subset of $v'$, is left unconstrained in the disjunct.\n In such case, we strengthen the disjunct by adding the constraint\n $x'=T_x*v+t_x $ as conjunct for each variable $x'$ in $w'$.\n Solving for $T_x$ and $t_x$ is done by the \\textsc{E-HSF}\\xspace solver. \n\\end{itemize}\nIn our CTL verification examples, both non-deterministic guards and\nassignments are explicitly marked in the original\nbenchmark programs using names~$\\mathtt{rho1}, \\mathtt{rho2},$ etc.\nWe apply the techniques discussed above to generate templates from the\ntransition relation of each program. \nIn these examples, linear templates are sufficiently expressive. \nFor dealing with well-foundedness we use linear ranking functions. \n\\input{table-ehsf-eval}\nWe report the results in Table~\\ref{table-ehsf-eval}.\nFor each program in Column~1, we report the shape of the property\n$\\varphi$ in Column~2. \nThe variables $p$ and $q$ in Column~2 range over the theory of\nquantifier-free linear integer arithmetic. \nThe result as well as the time it took the \\textsc{E-HSF}\\xspace engine to prove the property\n$\\varphi$ is given in Columns~3~and~4, and similarly, the result as well\nas the time it took the engine to discover a counterexample for the\nnegated property $\\neg\\varphi$ is given in Columns~5~and~6. \nThe symbol \\checkmark\\xspace marks the cases where \\textsc{E-HSF}\\xspace was able to find a\nsolution, i.e., a proof that the CTL property $\\varphi$ is valid, and \nthe symbol $\\times$~marks the cases where \\textsc{E-HSF}\\xspace was able to find a\ncounter-example, i.e., a proof that the negated CTL property\n$\\neg\\varphi$ is not valid.\nThe number of LOC of each program is also given in Column~1.\n\nThe \\textsc{E-HSF}\\xspace engine is able to find proofs that the CTL property $\\varphi$ is\nvalid (and the negated CTL property $\\neg\\varphi$ is not\nvalid) for all of the programs except the last three programs.\nFor the last three versions of \\texttt{Software Updates}, not only the\nnegated CTL property $\\neg\\varphi$ but also the CTL property $\\varphi$\nis not valid.\nThis was because $\\varphi$ was satisfied only for some initial states.\nThe method takes a total time of 52 seconds to complete the\nverifications tasks. \n\\input{table-ehsf-comp}\nOur method also compares favourably with state-of-art automated CTL \nverification methods. \nWe present in Table~\\ref{table-ehsf-comp} the comparison between the\nour solving algorithm \\textsc{E-HSF}\\xspace and a CTL verification method from\nCook~\\cite{Cook2014FTR}.\nHere also, we use the programs from Table~\\ref{table-ehsf-eval},\nhowever, for the sake of focusing on the comparison, we exclude\nprograms for which the two methods have different outcomes.\nFor each program in Column~1, we report the shape of the property in\nColumn~2.\nThe time it takes \\textsc{E-HSF}\\xspace to prove the property $\\varphi$ is given in\nColumn~3, and the corresponding time for Cook is given in Column~4.\nSimilarly, the time it takes \\textsc{E-HSF}\\xspace to discover a counterexample for\nthe negated property $\\neg \\varphi$ is given in\nColumn~5, and the corresponding time for Cook is given in Column~6.\n\nFrom the result, we can see that while \\textsc{E-HSF}\\xspace takes a total of 48\nseconds to finish the task, Cook takes a total of\n149 seconds. \nThis amounts to an approximate reduction of 70\\%. \nThere are a few cases where \\textsc{E-HSF}\\xspace takes longer than\nCook.\nWe suspect that a more efficient modeling of the original c program as\na transition system can help our method a lot. \nThe presence of many temporary program variables in the transition\nrelation which are not involved in any computation of the program can\naffect the performance of our method.\n\n\n\n\\section{Introduction}\n\\label{sec-ctl-intro}\n\nSince Pnueli's pioneering work~\\cite{Pnueli1977}, the use of temporal logics has\nlong been recognised as a fundamental approach to the formal\nspecification and verification of reactive systems~\\cite{Manna1992,\n Emerson1991TM}. \nTemporal logics allow precise specification of complex properties.\nThere have been decades of effort on temporal verification of finite state\nsystems~\\cite{Kupferman2000, Burch1990, Clarke02treelikeCEX,\n Clarke1983AVF}. \nFor CTL and other state-based properties, the standard procedure is to adapt\n\"bottom-up\" (or \"tableaux\") techniques for reasoning on finite-state\nsystems.\nIn addition, various classes of temporal logics support\nmodel-checking whose success over the last twenty years allows\nlarge and complex (finite) systems to be verified automatically~\\cite{Burch1990,\n Clarke1990TLM, McMillan1993SMC, Hassan2012IIC}.\nIn recent decades, however, the research focus has shifted to\ninfinite-state systems in general and to software systems in\nparticular as ensuring correctness for software is\nin high demand. \nMost algorithms for verifying CTL properties on infinite-state systems\ntypically involve first abstracting the state space into a finite-state\nmodel, and then applying finite reasoning strategies on the abstract\nmodel. \nThere is also a lot of effort on algorithms that are focused on a\nparticular fragment of CTL, such as the universal\nfragment~\\cite{Penczek2002BMC} and the existential\nfragment~\\cite{GurfinkelWC06}, or some\nparticular classes of infinite-state systems such as pushdown\nprocesses \\cite{Song2011, Song2013, Walukiewicz2000,\n walukiewicz2001pushdown} or parameterised\nsystems~\\cite{Emerson1996, Demri2010checkingctl}.\n\nIn this paper, we take on the problem of automatically verifying CTL\nproperties for a given (possibly infinite-state) program. \nWe propose a method based on solving a set of forall-exists quantified\nHorn constraints.\nOur method takes a program $\\ensuremath{P}$ modeled by a transition system\n$(\\mathit{init}(v), \\mathit{next}(v,v'))$ and a property given by a CTL formula\n$\\varphi(v)$, and then it checks if $\\ensuremath{P}$ satisfies $\\varphi(v)$,\ni.e., if $(\\mathit{init}(v), \\mathit{next}(v, v'))\\models_{\\mathit{CTL}} \\varphi(v)$. \nThe method first generates a set of forall-exists quantified Horn constraints\nwith well-foundedness conditions by exploiting the syntactic structure\nof the CTL formula $\\varphi(v)$.\nIt then solves the generated set of Horn constraints by applying an\noff-the-shelf solving engine \\textsc{E-HSF}\\xspace~\\cite{ehsf} for such\nconstraints.\nWe claim that $\\ensuremath{P}$ satisfies $\\varphi(v)$ if and only if\nthe generated set of Horn constraints has a solution. \nWe demonstrate the practical applicability of the method by presenting\nexperimental evaluation using examples from the PostgreSQL database\nserver, the SoftUpdates patch system, the Windows OS kernel.\n\nThe rest of the paper is organised as follows. \nWe start by summarising the syntax and semantics of CTL and by giving a brief\nintroduction to forall-exists quantified Horn constraints and their\nsolver \\textsc{E-HSF}\\xspace in Section~\\ref{sec:prelims}.\nIn Section~\\ref{sec:proof-system}, we present our CTL proof system\nthat generates a set of forall-exists quantified Horn constraints for\na given verification problem.\nWe illustrate application of the proof rules on an example in\nSection~\\ref{sec:cons-gen}.\nThe experimental evaluation of our method is given in\nSection~\\ref{sec:eval}.\nFinally, we present a brief discussion on related work in\nSection~\\ref{sec:rel-work} and concluding remarks in\nSection~\\ref{sec:concl}. \n\n\n\n\n\n\\section{Preliminaries} \n\\label{sec:prelims}\n\\subsection{CTL basics} \n\nIn this section, we review the syntax and the semantics of the logic\nCTL following~\\cite{KestenTCS95}. \nLet $\\mathcal{T}$ be a first order theory and $\\models_{\\mathcal{T}}$ denote its\nsatisfaction relation that we use to describe sets and relations\nover program states.\nLet $c$ range over assertions in $\\mathcal{T}$.\nA CTL formula $\\varphi$ is defined by the following grammar using\nthe notion of a path formula~$\\phi$.\n\\begin{equation*}\n \\begin{array}[t]{@{}r@{\\;::=\\;}l@{}}\n \\varphi &\n c \\mid\n \\varphi \\land \\varphi \\mid\n \\varphi \\lor \\varphi \\mid\n \\pathA \\, \\phi \\mid\n \\pathE \\, \\phi \\\\ [\\jot]\n \\phi & \\ltlNext \\varphi \\mid \\ltlG \\varphi \\mid \\varphi \\ltlU \\varphi\n \\end{array}\n\\end{equation*}\n$\\ltlNext$, $\\ltlG$, and $\\ltlU$ are called temporal operators, and\n$\\pathA$ and $\\pathE$ are called path quantifiers. \nA CTL formula whose principal operators are a pair QT, where Q is a\npath quantifier and T is a temporal operator, and which does not \ncontain any additional temporal operators or path quantifiers is\ncalled a basic CTL formula.\nAs usual, we define $\\ltlF \\varphi = (\\mathit{true}~ \\ltlU \\varphi)$.\nThe satisfaction relation $\\ensuremath{P}\\models \\varphi$ holds if and only\nif for each $s$ such that $\\mathit{init}(s)$ we\nhave~$\\sat{\\ensuremath{P}}{s}{\\varphi}$.\nWe define $\\sat{\\ensuremath{P}}{s}{\\varphi}$ as follows using an auxiliary\nsatisfaction relation $\\sat{\\ensuremath{P}}{\\pi}{\\phi}$.\n\n\\begin{equation*}\n \\begin{array}[t]{@{}l@{\\text{ iff }}l@{}}\n \\sat{\\ensuremath{P}}{s}{c} \n &\n s \\models_{\\mathcal{T}} c \\\\[\\jot]\n \\sat{\\ensuremath{P}}{s}{\\varphi_1 \\land \\varphi_2} \n &\n \\sat{\\ensuremath{P}}{s}{\\varphi_1} \\text{ and }\n \\sat{\\ensuremath{P}}{s}{\\varphi_2}\\\\[\\jot]\n \\sat{\\ensuremath{P}}{s}{\\varphi_1 \\lor \\varphi_2} \n &\n \\sat{\\ensuremath{P}}{s}{\\varphi_1} \\text{ or }\n \\sat{\\ensuremath{P}}{s}{\\varphi_2}\\\\[\\jot]\n \\sat{\\ensuremath{P}}{s}{\\pathA\\, \\phi} \n &\n \\text{for all $\\pi \\in \\computations{\\ensuremath{P}}{s}$ holds } \n \\sat{\\ensuremath{P}}{\\pi}{\\phi} \\\\[\\jot]\n \\sat{\\ensuremath{P}}{s}{\\pathE\\, \\phi}\n &\n \\text{exists $\\pi \\in \\computations{\\ensuremath{P}}{s}$ such that }\n \\sat{\\ensuremath{P}}{\\pi}{\\phi} \\\\[\\jot]\n \\sat{\\ensuremath{P}}{\\pi}{\\ltlNext \\varphi}\n &\n \\pi = s_1, s_2, \\ldots \\text{ and }\n \\sat{\\ensuremath{P}}{s_2}{\\varphi} \\\\[\\jot]\n \\sat{\\ensuremath{P}}{\\pi}{\\ltlG \\varphi}\n &\n \\pi = s_1, s_2, \\ldots \\text{for all $i\\geq 1$ holds }\n \\sat{\\ensuremath{P}}{s_i}{\\varphi} \\\\[\\jot]\n \\sat{\\ensuremath{P}}{\\pi}{\\varphi_1 \\ltlU \\varphi_2}\n &\n \\begin{array}[t]{@{}l@{}}\n \\pi = s_1, s_2, \\ldots \n \\text{ and exists $j\\geq 1$ such that }\\\\[\\jot]\n \\sat{\\ensuremath{P}}{s_j}{\\varphi_2} \\text{ and }\n \\sat{\\ensuremath{P}}{s_i}{\\varphi_1} \\text{ for } 1 \\leq i < j\n \\end{array}\n \\end{array}\n\\end{equation*}\nIn this paper, we represent a satisfaction relation $\\ensuremath{P}\\models\n\\varphi$ by the relation $\\ensuremath{P}\\models_{\\mathit{CTL}} \\varphi$ to\nexplicitly indicate that $\\varphi$ is a CTL formula.\nWe call such relation a CTL satisfaction, and $\\varphi$ is said to be\nits formula.\n\n\\subsection{The solving algorithm \\textsc{E-HSF}\\xspace} \n\\label{subsec-ehsf}\n\nOur proof rules are automated using the \\textsc{E-HSF}\\xspace engine for resolving\nforall-exists Horn-like clauses extended with well-foundedness\ncriteria.\n\nWe skip the syntax and semantics of the clauses targeted by this\nsystem --- see \\cite{ehsf} for more details. \nInstead, we illustrate these clauses with the following example:\n\\begin{equation*}\n \\begin{array}[t]{@{}l@{\\qquad}l@{}}\n x \\geq 0 \\rightarrow \\exists y: x \\geq y \\land \\mathit{rank}(x, y), &\n \\mathit{rank}(x, y) \\rightarrow \\mathit{ti}(x, y),\\\\[\\jot]\n \\mathit{ti}(x, y) \\land \\mathit{rank}(y, z) \\rightarrow \\mathit{ti}(x,\n z), &\n \\mathit{dwf}(ti).\n\\end{array}\n\\end{equation*}\n\nIntuitively, these clauses represent an assertion over the interpretation of\n``query symbols'' $\\mathit{rank}$ and~$\\mathit{ti}$ (the predicate\n$\\mathit{dwf}$ represents disjunctive well-foundedness, and is not a\nquery symbol).\nThe semantics of these clauses maps each predicate symbol \noccurring in them into a constraint over~$v$.\nSpecifically, the above set of clauses has a solution that maps both\n$\\mathit{rank}(x, y)$ and $\\mathit{ti}(x, y)$ to the constraint $(x\n\\geq 0 \\land y \\geq x-1)$.\n\n\\textsc{E-HSF}\\xspace resolves clauses like the above using a CEGAR scheme to\ndiscover witnesses for existentially quantified variables. \nThe refinement loop collects a global constraint that declaratively\ndetermines which witnesses can be chosen. \nThe chosen witnesses are used to replace existential quantification,\nand then the resulting universally quantified clauses are passed to a\nsolver for such clauses. \nAt this step, we can benefit from emergent tools in the area of\nsolving Horn clauses over decidable theories, e.g.,\nHSF~\\cite{GrebenshchikovTACAS12} or $\\mu$Z~\\cite{muz}.\nSuch a solver either finds a solution, i.e., a model for uninterpreted\nrelations constrained by the clauses, or returns a counterexample,\nwhich is a resolution tree (or DAG) representing a contradiction. \n\\textsc{E-HSF}\\xspace turns the counterexample into an additional constraint on the\nset of witness candidates, and continues with the next iteration of\nthe refinement loop. \nNotably, this refinement loop conjoins constraints that are obtained\nfor all discovered counterexamples. \nThis way \\textsc{E-HSF}\\xspace guarantees that previously handled counterexamples\nare not rediscovered and that a wrong choice of witnesses can be mended.\n\nFor the existential clause above, \\textsc{E-HSF}\\xspace introduces a\nwitness\/Skolem relation $\\mathit{rel}$ over variables $x$ and\n$y$, i.e., $x\\geq0 \\land \\mathit{rel}(x,y) \\rightarrow x \\geq y \\land\n\\mathit{rank}(x, y)$.\nFor each $x$ such that $x\\geq0$, we require the skolem relation to\nprovide the corresponding value for $y$, i.e., we require all such\n$x$ is in the domain of the Skolem relation. \nThis is encoded by an additional clause $x\\geq0 \\rightarrow \\exists y:\n\\mathit{rel}(x,y)$. \nIn the \\textsc{E-HSF}\\xspace approach, the search space of a skolem relation \n$\\mathit{rel}(x,y)$ is restricted by a template function\n$\\funTemplateOf{\\mathit{rel}}{x,y}$.\nIn general, \\textsc{E-HSF}\\xspace requires such template functions to be given by\nthe user. \n\n\n\\section{Proof system}\n\\label{sec:proof-system}\n\nOur CTL verification method encodes the verification problem as a problem of\nsolving forall-exists quantified Horn constraints with well-foundedness\nconditions. \nThis is done by applying a proof system that consists of various proof\nrules for handling different kinds of CTL formulas.\nThis proof system is based on a deductive proof system for CTL*\nfrom~\\cite{KestenTCS95} which is adapted in this work to be suitable\nfrom the perspective of constraint generation for a CTL satisfaction.\n\n\nGiven a transition system $(\\mathit{init}(v),\\mathit{next}(v,v'))$ and a CTL\nformula $\\varphi(v)$, the appropriate proof rules are used from the\nproof system to generate the corresponding set of Horn constraints for\nthe CTL satisfaction $(\\mathit{init}(v), \\mathit{next}(v, v'))\\models_{\\mathit{CTL}} \\varphi(v)$.\nThere are two sets of proof rules in the proof system. \n\n\\subsection{Proof rules for decomposition}\n\nThese proof rules are applied recursively to a CTL satisfaction whose\nformula is neither an assertion nor a basic CTL formula. \nThe proof rules decompose the given CTL formula into new\nsub-formulas by following the nesting structure of the formula.\nThen, the original satisfaction is reduced to new satisfactions over the\nnew sub-formulas and a Horn constraint relating the new\nsatisfactions.\n\nThere are different proof rules depending on the outermost operator\nof the formula. \nOne case is when the given formula $f(\\psi(v))$ nests another\nformula $\\psi(v)$ such that the outermost operator $f$ is a pair of a\ntemporal path operator and a unary temporal state operator, i.e., $f\n\\in \\{AX, AG, EX, EG\\}$.\nThe corresponding proof rule $\\textsc{RuleCtlDecompUni}\\xspace$ is given in\nFigure~\\ref{fig-ctl-proof-rule-decompUni} that shows how such satisfactions\nare decomposed.\n\\begin{figure}[h]\n \\makeFramedRule{Find an assertion $q(v)$ such that:}{ \n (p(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}} f(q(v)) \\quad (q(v), \\mathit{next}(v,\n v')) \\models_{\\mathit{CTL}} \\psi(v)}{ \n (p(v), \\mathit{next}(v, v'))\\models_{\\mathit{CTL}} f(\\psi(v))} \n \\caption{Proof rule \\textsc{RuleCtlDecompUni}\\xspace}\n \\label{fig-ctl-proof-rule-decompUni}\n\\end{figure}\nAnother case is when the given formula has a structure $f(\\psi_1(v),\\\n\\psi_2(v))$ nesting the formulas $\\psi_1(v)$ and $\\psi_2(v)$ such that\nthe outermost operator $f$ is either a pair of a temporal path\noperator and the state operator until or a disjunction\/conjunction, i.e., $f\n\\in \\{AU, EU, \\land, \\lor\\}$.\nNote that when $f$ is $\\land$ (resp. $\\lor$), the given formula\n$f(\\psi_1(v),\\ \\psi_2(v))$ corresponds to $\\psi_1(v)\\land\n\\psi_2(v)$ (resp $\\psi_1(v) \\lor \\psi_2(v)$).\nThe corresponding proof rule $\\textsc{RuleCtlDecompBin}\\xspace$ is given in\nFigure~\\ref{fig-ctl-proof-rule-decompBin} that shows how such satisfactions\nare decomposed.\n\\begin{figure}[h]\n \\makeFramedRule{Find assertions $q_1(v)$ and $q_2(v)$\n such that:}{\n p(v) \\rightarrow f(q_1(v), q_2(v)), \\\\\n (q_1(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}} \\psi_1(v) \\quad (q_2(v),\n \\mathit{next}(v, v')) \\models_{\\mathit{CTL}} \\psi_2(v)}{\n (p(v), \\mathit{next}(v, v'))\\models_{\\mathit{CTL}} f(\\psi_1(v),\n \\psi_2(v))\n }\n \\caption {Proof rule \\textsc{RuleCtlDecompBin}\\xspace}\n \\label{fig-ctl-proof-rule-decompBin}\n\\end{figure}\n\n\\subsection{Proof rules for constraints generation}\n\nThis set of proof rules is applied to a CTL satisfaction whose formula\nis either an assertion or a basic CTL formula.\nAny CTL satisfaction can be decomposed into a set of such simple CTL\nsatisfactions by applying the proof rules from the previous section. \nThe next step will be to generate forall-exists quantified Horn constraints\n(possibly with well-foundedness condition) that constrain a set of\nauxiliary assertions over program states. \n\nThe simplest of all is the proof rule \\textsc{RuleCtlInit}\\xspace, see\nFigure~\\ref{fig-ctl-proof-rule-init}, which is applied when the CTL\nformula is an assertion. \\\\\n\\begin{minipage}{.48\\textwidth} \n\\begin{figure}[H]\n \\makeFramedRule{}{\n p(v) \\rightarrow \\psi(v)\n }{\n (p(v), \\mathit{next}(v, v')) \\models_{\\mathit{CTL}} \\psi(v)\n }\n \\caption{Proof rule \\textsc{RuleCtlInit}\\xspace}\n \\label{fig-ctl-proof-rule-init}\n\\end{figure}\n\\end{minipage} %\n\\hspace{1 em}\n\\begin{minipage}{.48\\textwidth} %\n\\begin{figure}[H]\n \\makeFramedRule{}{\n p(v) \\rightarrow \\exists v': \\mathit{next}(v,v') \\land q(v')\n }{\n (p(v), \\mathit{next}(v,v')) \\models_{\\mathit{CTL}} EX~q(v)\n }\n \\caption{Proof rule \\textsc{RuleCtlEX}\\xspace}\n \\label{fig-ctl-proof-rule-EX}\n\\end{figure}\n\\end{minipage}\n\\vspace{2 em}\\\\\nThe proof rules \\textsc{RuleCtlEX}\\xspace (see Figure~\\ref{fig-ctl-proof-rule-EX}),\n\\textsc{RuleCtlEG}\\xspace (see Figure~\\ref{fig-ctl-proof-rule-EG}), and \\textsc{RuleCtlEU}\\xspace\n(see Figure~\\ref{fig-ctl-proof-rule-EU}) are applied for generating\nHorn constraints when the CTL satisfaction problem has a basic CTL\nformula with existential path operator. \\\\\n\\begin{minipage}{.4\\textwidth} \n\\begin{figure}[H]\n \\makeFramedRuleSS{Find an assertion $\\mathit{inv}(v)$ such that:}{\n p(v) & \\rightarrow & \\mathit{inv}(v)\\\\\n \\mathit{inv}(v) & \\rightarrow & \\exists v': \\mathit{next}(v,v') \\land \\mathit{inv}(v')\\\\\n \\mathit{inv}(v) & \\rightarrow & q(v)\n }{\n (p(v), \\mathit{next}(v,v')) \\models_{\\mathit{CTL}} EG~q(v)\n }\n \\caption{Proof rule \\textsc{RuleCtlEG}\\xspace }\n \\label{fig-ctl-proof-rule-EG}\n\\end{figure}\n\\end{minipage} %\n\\hspace{1 em}\n\\begin{minipage}{.56\\textwidth} %\n\\begin{figure}[H]\n \\makeFramedRuleSS{Find assertions $\\mathit{inv}(v)$ and $\\mathit{rank}(v,v')$ such that: }{\n p(v) & \\rightarrow & \\mathit{inv}(v)\\\\\n \\mathit{inv}(v) \\land \\neg r(v) & \\rightarrow & q(v) \\land \\exists v': \\mathit{next}(v,v') ~\\land \\\\[\\jot]\n & & \\mathit{inv}(v') \\land \\mathit{rank}(v,v') \\\\[\\jot]\n & \\mathit{wf}(\\mathit{rank})\n }{\n (p(v), \\mathit{next}(v,v')) \\models_{\\mathit{CTL}} EU(q(v),r(v))\n }\n \\caption{Proof rule \\textsc{RuleCtlEU}\\xspace}\n \\label{fig-ctl-proof-rule-EU}\n\\end{figure}\n\\end{minipage}\n\\vspace{2 em}\\\\\n\nSimilarly, the proof rules \\textsc{RuleCtlAX}\\xspace (see Figure~\\ref{fig-ctl-proof-rule-AX}),\n\\textsc{RuleCtlAG}\\xspace (see Figure~\\ref{fig-ctl-proof-rule-AG}), and \\textsc{RuleCtlAU}\\xspace\n(see Figure~\\ref{fig-ctl-proof-rule-AU}) are applied for generating\nHorn constraints when the CTL satisfaction has a basic CTL formula with\nuniversal path operator.\n\n\\begin{minipage}{.48\\textwidth} \n\\begin{figure}[H]\n \\makeFramedRuleSS{}{\n p(v) \\land \\mathit{next}(v,v') & \\rightarrow & q(v')\n }{\n (p(v), \\mathit{next}(v,v')) \\models_{\\mathit{CTL}} AX~q(v)\n }\n \\caption{Proof rule \\textsc{RuleCtlAX}\\xspace}\n \\label{fig-ctl-proof-rule-AX}\n\\end{figure}\n\\end{minipage} %\n\\hspace{1 em}\n\\begin{minipage}{.48\\textwidth} %\n\\begin{figure}[H]\n \\makeFramedRuleSS{Find an assertion $\\mathit{inv}(v)$ such that:}{\n p(v) & \\rightarrow & \\mathit{inv}(v)\\\\\n \\mathit{inv}(v) \\land \\mathit{next}(v,v') & \\rightarrow & \\mathit{inv}(v')\\\\\n \\mathit{inv}(v) & \\rightarrow & q(v)\n }{\n (p(v), \\mathit{next}(v,v')) \\models_{\\mathit{CTL}} AG~q(v)\n }\n \\caption{Proof rule \\textsc{RuleCtlAG}\\xspace}\n \\label{fig-ctl-proof-rule-AG}\n\\end{figure}\n\\end{minipage}\n\\vspace{2 em}\\\\\n\\begin{figure}[h]\n \\makeFramedRule{Find assertions $\\mathit{inv}(v)$ and $\\mathit{rank}(v,v')$ such that: }{\n p(v) \\rightarrow \\mathit{inv}(v)\\\\\n \\mathit{inv}(v) \\land \\neg r(v) \\land \\mathit{next}(v,v') \\rightarrow\n \\begin{array}[t]{l} \n q(v) \\land \\mathit{inv}(v') \\land \\mathit{rank}(v,v')\n \\end{array}\\\\\n \\mathit{wf}(\\mathit{rank}).\n }{\n (p(v), \\mathit{next}(v,v')) \\models_{\\mathit{CTL}} AU(q(v), r(v))\n }\n \\caption{Proof rule \\textsc{RuleCtlAU}\\xspace}\n \\label{fig-ctl-proof-rule-AU}\n\\end{figure}\n\nOur proof system is not exhaustive in terms of having proof rules for\nall kinds of basic CTL formulas.\nHowever, we utilize equivalence between CTL formulas to generate\nHorn constraints for basic CTL formulas whose proof rules are not\ngiven in the proof system. \nFor example, the equivalence between the formulas\n$EU(\\mathit{true},q(v))$ and $EF(q(v))$ can be used to reduce the CTL\nsatisfaction problem $(p(v), \\mathit{next}(v,v')) \\models_{\\mathit{CTL}} EF(q(v)$ into\n$(p(v), \\mathit{next}(v,v')) \\models_{\\mathit{CTL}} EU(true, q(v))$.\n\n\n\\section{Related work}\n\\label{sec:rel-work}\n\nVerification of properties specified in temporal logics such as CTL\nhas been extensively explored for finite-state\nsystems\\cite{Kupferman2000, Burch1990, Clarke02treelikeCEX,\n Clarke1983AVF}. \nThere has also been studies on the verification of CTL properties for\nsome restricted types of infinite-state systems.\nSome examples are pushdown processes \\cite{Song2013, Song2011, Walukiewicz2000},\npushdown games \\cite{walukiewicz2001pushdown},\nand parameterised systems \\cite{Emerson1996}.\nFor such restricted systems, the standard procedure is to abstract the\ninfinite-state system model into finite-state model and\napply the known methods for finite-state systems. \nBut existing abstraction methods usually do not allow reliable\nverification of CTL properties where alternation between universal and\nexistential modal operators is common. \nMany methods of proving CTL properties with only universal path\nquantifiers are known\\cite{Chaki2005, Cook2012TPV}.\n There also a few methods mainly focused on proving branching-time\nproperties with only existential path quantifiers.\nOne example is the tool Yasm \\cite{GurfinkelWC06} which implements a\nproof procedure aimed primarily at the non-nested existential subset of\nCTL.\nThere are also known techniques for proving program\ntermination~\\cite{Bradley05polyrankingfor, TerminatorPLDI06}\n(resp. non-termination~\\cite{TNTPOPL08}) which is equivalent with\nproving the CTL formula $AF~false$ (resp. $EG~true$).\n\nBanda et al.~\\cite{Banda2010CAS} proposed a CTL verification approach\nfor infinite state reactive systems based on CLP and abstraction of a\nCTL semantic function. \nAn automatic proof method that supports both universal and existential\nbranching-time modal operators for (possibly infinite-state) programs\nis proposed in by Cook et al.~\\cite{CookPLDI13}. \nCook's approach is based on reducing existential reasoning to universal\nreasoning when an appropriate restriction is placed on the the\nstate-space of the system. \nWhile this approach comes close to our approach, the refinement\nprocedure for state-space restrictions may make\nincorrect choices early during the iterative proof search.\nThese choices may limit the choices available later in the search\nleading to failed proof attempts in some cases.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn cosmological models of galaxy formation the formation history of galaxies, and hence their present day properties, are intimately linked with the formation history and properties of the dark matter (DM) haloes in which they reside. Therefore, if we wish to understand the process of galaxy formation it is important to observationally establish a link between the properties of galaxies and those of the halos in which they reside.\n\nIn recent years many studies have linked galaxy properties to halo mass using clustering, gravitational lensing, satellite kinematics, group catalogues or subhalo abundance matching, with the main focus being on linking the stellar properties (stellar luminosity, stellar mass, color, SFR) or those of the central black hole (luminosity, black hole mass) to the galaxy host dark matter halo properties. This work has shown strong correlations between halo mass and stellar mass, color, star formation rate, or morphology, such that more massive (luminous), redder (less star-forming), or more spheroidal galaxies live in more massive halos \\citep[e.g.][]{Li06a,Yang08,Yang09,Zehavi11,More11,Mandelbaum06,Leauthaud11}. For AGN the relationships are less clear although correlations do exist between narrow line AGN luminosity or radio luminosity \\citep{Wake04,Li06b,Wake08b,Mandelbaum09} and halo mass.\nPerhaps unsurprisingly the strongest relationship occurs with stellar mass, which for central galaxies in halos show a tight correlation with a relatively small scatter of ~0.17 dex \\citep{Yang11}. At fixed stellar mass properties that indicate the star formation history of a galaxy then show little dependence on halo mass \\citep{More11}, when only considering central galaxies. \n\nIn this work we focus on investigating how the dynamical properties of galaxies, such as velocity dispersion (\\Vdisp) or surface mass density (\\Mden), in addition to stellar mass are related to the clustering amplitude of galaxies and hence their host dark matter halo properties. This is a particularly relevant question as it is becoming increasingly clear that many galaxy properties such as their current star-formation rates, star-formation histories, metalicities, and black hole masses are more closely related to these dynamical properties than their total stellar mass \\citep[e.g.][]{Bender93,Ferrarese00,Gebhardt00,Kauffmann03b,Franx08}. If the formation history and properties of the host dark matter halo are the major underlying cause of these galaxy properties, we might expect that \\Vdisp~or \\Mden~are better indicators than stellar mass of these halo properties.\n \n\nIn this paper we make use of the very large sample of galaxies with velocity dispersion, stellar mass, size, color and morphology measurements from the seventh data release of the Sloan Digital Sky Survey \\citep[SDSS DR7;][]{Abazajian09}. We split these galaxies into narrow bins of one parameter, in particular mass or \\Vdisp, and then see if there is any dependence of the clustering amplitude on another galaxy parameter within that narrow bin. In this way we can investigate if there are any residual dependencies on halo properties manifested by higher or lower clustering amplitudes as one parameter is varied whilst another remains fixed. \n\nThroughout this paper, we assume a flat $\\Lambda$--dominated CDM cosmology with $\\Omega_m=0.27$, $H_0=73 $km s$^{-1} $Mpc$^{-1}$, and $\\sigma_8=0.8$ unless otherwise stated.\n\n\\section{Data}\n\\label{sec:data}\n\nThe galaxy data used in this analysis are gathered from the seventh data release of the SDSS \\citep{Abazajian09}. We begin with the Large Scale Structure samples of the NYU Value Added Galaxy catalog \\citep[VAGC;][]{Blanton05}. These samples have been carefully constructed for measurements of large scale structure and include corrections for fiber collisions, the tracking of spatially dependent completeness and appropriate random catalogs all of which are required to accurately measure clustering. The sample we use has an $r$ band magnitude range of $14.5 < r < 17.6$. In addition the NYU VAGC gives k-corrected (to z=0.1) absolute magnitudes \\citep{Blanton03}, velocity dispersion measurements from the Princeton Spectroscopic pipeline, and circularised sersic fits for each galaxy, all of which we make use of in this analysis.\n\nFor estimates of the stellar mass we make use of the MPA-JHU value added catalog which provides stellar mass estimates based on stellar population fits to the SDSS photometry \\citep{Kauffmann03, Salim07}. The overlap between the MPA-JHU and NYU VAGCs is close to but to quite 100\\% and so we remove the regions where they do not match from the analysis and adjust the completeness corrections appropriately (see section \\ref{sec:clus}). We also remove any region of the survey that has a spectroscopic completeness less than 70\\%. This leaves an area of 7640 deg$^2$ and a total sample of 521,313 galaxies.\n\nThe SDSS velocity dispersions are measured within the 3\" diameter SDSS fiber. We correct to a common aperture of one eighth of an effective radius ($r_e$), the central velocity dispersion, using the relation $\\sigma_0 = \\sigma_{ap}(8r_{ap}\/r_e)^{0.066}$ where $r_{ap}$ = 1.\"5 \\citep{Cappellari06}. $r_e$ is taken from the best fitting circularised sersic profile fit.\n\nIn addition to investigating how clustering depends on stellar mass and velocity dispersion, we also include the dynamical mass (\\Mdyn) as an addition galaxy mass indicator. Estimates of galaxy stellar masses are uncertain due to the complex nature of the star formation history, stellar population synthesis modeling, extinction law and initial mass function \\citep[e.g. see][]{Marchesini09, Muzzin09, Conroy09}. The dynamical mass, which is estimated purely from the velocity dispersion and the size of the galaxy, will not be affected by these same systematic uncertainties and will likely have errors more similar to those of the velocity dispersion. In addition there is a strong and fairly tight correlation between stellar mass and dynamical mass in the SDSS \\citep[][see also Figure \\ref{fig:MstarMdyn}]{Taylor10} and so both masses are largely tracing the same physical property of a galaxy.\n\nTo estimate dynamical mass we follow \\citet{Taylor10} where\n\\begin{equation}\n\t M_{dyn} = K_V \\sigma_0^2 r_e\/G \n\\end{equation}\nwith\n\\begin{equation}\n\tK_V = \\frac{73.32}{10.465 + (n-0.95)^2} + 0.954\t\n\\end{equation}\nwhere $n$ is the sersic $n$ parameter and the gravitational constant $G = 4.3 \\times 10^{-3}$ pc $M_\\odot^{-1}$ km$^2$ s$^{-2}$. \n\n \nFinally we will make use of both galaxy color and morphology to further parameterize our galaxy samples. Throughout we use $g-r$ colors from the K-corrected NYU VAGC absolute magnitudes. As already stated these are corrected to $z=0.1$; although we will refer to them as $g-r$ colors they are not quite $g-r$ at rest. The morphological classifications we use come from Galaxy Zoo \\citep{Lintott11} which provides multiple visual classifications for each galaxy in the SDSS spectroscopic sample. The parameter we use is the probability that a galaxy is an elliptical ($P_{el}$) corrected for redshift bias whereby higher redshift galaxies are more likely to be classified as ellipticals due to their smaller apparent sizes \\citep[see ][ for details]{Lintott11}.\n \n\\begin{figure}\n\n\\vspace{11cm}\n\\special{psfile=vdispSM_paper.ps hoffset=0 voffset=0 hscale=43 vscale=43 angle=0}\n\\caption{\\small The distribution of galaxies from the parent sample with \\Mstar~$> 6\\times10^{10}M_\\odot$ in the \\Mstar~-- \\Vdisp~plane. The colored regions in the top panel show the high and low \\Vdisp~samples at fixed \\Mstar~and in the bottom panel they show the high and low \\Mstar~samples at fixed \\Vdisp.\n\\label{fig:VdSM}}\n\\end{figure}\n\n\\begin{figure*}\n\n\\vspace{11cm}\n\\special{psfile=2ptcross_wp_VdatSM.ps hoffset=-10 voffset=0 hscale=40 vscale=40 angle=0}\n\\special{psfile=2ptcross_wp_SMatvd.ps hoffset=250 voffset=0 hscale=40 vscale=40 angle=0}\n\\caption{\\small The projected correlation functions of high and low \\Vdisp~samples at fixed \\Mstar~(left). The projected correlation functions of high and low \\Mstar~at fixed \\Vdisp~(right). The correlation function measurements are colored to match the selection regions shown in Figure \\ref{fig:VdSM}.\n\\label{fig:2ptVdSM1}}\n\\end{figure*}\n\n\\begin{figure*}\n\n\\vspace{11cm}\n\\special{psfile=2ptcross_wp_ratio_VdatSM.ps hoffset=-10 voffset=0 hscale=40 vscale=40 angle=0}\n\\special{psfile=2ptcross_wp_ratio_SMatvd.ps hoffset=250 voffset=0 hscale=40 vscale=40 angle=0}\n\\caption{\\small The ratio of the projected correlation functions between high and low velocity dispersion samples at fixed stellar mass (left). The ratio of the projected correlation functions between high and low stellar mass samples at fixed velocity dispersion (right). The best fit ratio on scales $1.6 < r_p < 25.1$ h$^{-1}$Mpc is shown as the dashed line.\n\\label{fig:2ptVdSM}}\n\\end{figure*}\n\n\n\\begin{figure*}\n\n\\vspace{11cm}\n\\special{psfile=2ptcross_wp_ratio_VdatdM.ps hoffset=-10 voffset=0 hscale=40 vscale=40 angle=0}\n\\special{psfile=2ptcross_wp_ratio_dMatvd.ps hoffset=250 voffset=0 hscale=40 vscale=40 angle=0}\n\\caption{\\small The ratio of the projected correlation functions between high and low velocity dispersion samples at fixed dynamical mass (left). The ratio of the projected correlation functions between high and low dynamical mass samples at fixed velocity dispersion (right). The best fit ratio on scales $1.6 < r_p < 25.1$ h$^{-1}$Mpc is shown as the dashed line.\n\\label{fig:2ptVddM}}\n\\end{figure*}\n\n\n\\begin{figure*}\n\n\\vspace{14.7cm}\n\\special{psfile=2ptcross_wp_ratio_MdenatSM.ps hoffset=-10 voffset=0 hscale=53 vscale=53 angle=0}\n\\special{psfile=2ptcross_wp_ratio_SMatMden.ps hoffset=250 voffset=0 hscale=53 vscale=53 angle=0}\n\\caption{\\small The ratio of the projected correlation functions between high and low surface mass density samples at fixed stellar mass (left) and high and low stellar mass samples at fixed surface mass density (right). The best fit ratio on scales $1.6 < r_p < 25.1$ h$^{-1}$Mpc is shown as the dashed line.\n\\label{fig:2ptMdenSM}}\n\\end{figure*}\n\n\\begin{figure*}\n\n\\vspace{18cm}\n\\special{psfile=wpratiofit_SMVd.ps hoffset=60 voffset=560 hscale=45 vscale=45 angle=-90}\n\\special{psfile=wpratiofit_dMVd.ps hoffset=60 voffset=390 hscale=45 vscale=45 angle=-90}\n\\special{psfile=wpratiofit_MdenSM.ps hoffset=60 voffset=220 hscale=45 vscale=45 angle=-90}\n\\caption{\\small The best fit large scale bias ratios for the samples plotted in Figures \\ref{fig:2ptVdSM}, \\ref{fig:2ptVddM} and \\ref{fig:2ptMdenSM}. The top panels show the bias ratio for high and low \\Mstar~at fixed \\Vdisp~and high and low \\Vdisp~at fixed \\Mstar. The middle panels show the bias ratio for high and low \\Mdyn~at fixed \\Vdisp~and high and low \\Vdisp~at fixed \\Mdyn. The bottom panels show the bias ratio for high and low \\Mden~at fixed \\Vdisp~and high and low \\Vdisp~at fixed \\Mden. There is a significant dependence of the bias on \\Vdisp~or \\Mden~at fixed mass, whereas there is little or no dependence on mass at fixed \\Vdisp~or \\Mden.\n\\label{fig:biasrat}}\n\\end{figure*}\n\n\\begin{table*}\n \\begin{center}\n \\caption{\\label{tab:ratfit} Fits to correlation funtion ratios}\n\t\\begin{tabular}{lcccc}\n\t\\tableline\n \t\\multicolumn{1}{l}{Sample} &\n \t\\multicolumn{1}{c}{Best fit ratio} &\n \t\\multicolumn{1}{c}{Best fit $\\chi^2$} &\n \t\\multicolumn{1}{c}{Ratio = 1 $\\chi^2$} &\n \t\\multicolumn{1}{c}{Ratio = 1 prob} \\\\\n\t\\tableline\n\\\\\n\\Mstar~at \\Vdisp~1 & 0.92 $\\pm$ 0.06 & 1.69 & 3.26 & 0.661\\\\\n\\Mstar~at \\Vdisp~2 & 1.01 $\\pm$ 0.07 & 2.07 & 2.07 & 0.839\\\\\n\\Mstar~at \\Vdisp~3 & 1.15 $\\pm$ 0.12 & 5.52 & 6.97 & 0.223\\\\\n\\Mstar~at \\Vdisp~All & - & - & 12.30 & 0.66\\\\\\\\\n\\Vdisp~at \\Mstar~1 & 1.17 $\\pm$ 0.07 & 7.51 & 12.10 & 0.033\\\\\n\\Vdisp~at \\Mstar~2 & 1.23 $\\pm$ 0.10 & 2.24 & 7.31 & 0.198\\\\\n\\Vdisp~at \\Mstar~3 & 1.32 $\\pm$ 0.12 & 4.88 & 11.54 & 0.042\\\\\n\\Vdisp~at \\Mstar~All & - & - & 30.95 & 0.0089\\\\\\\\\n\\Mdyn~at \\Vdisp~1 & 0.97 $\\pm$ 0.05 & 0.40 & 0.74 & 0.981\\\\\n\\Mdyn~at \\Vdisp~2 & 1.11 $\\pm$ 0.07 & 3.15 & 5.27 & 0.384\\\\\n\\Mdyn~at \\Vdisp~3 & 1.13 $\\pm$ 0.12 & 2.13 & 3.18 & 0.673\\\\\n\\Mdyn~at \\Vdisp~All & - & - & 9.18 & 0.87\\\\\\\\\n\\Vdisp~at \\Mdyn~1 & 1.18 $\\pm$ 0.05 & 6.81 & 14.87 & 0.011\\\\\n\\Vdisp~at \\Mdyn~2 & 1.26 $\\pm$ 0.09 & 4.62 & 12.80 & 0.025\\\\\n\\Vdisp~at \\Mdyn~3 & 1.56 $\\pm$ 0.16 & 3.06 & 13.89 & 0.016\\\\\n\\Vdisp~at \\Mdyn~All & - & - & 41.57 & 0.00026\\\\\\\\\n\\Mstar~at \\Mden~1 & 1.09 $\\pm$ 0.05 & 0.22 & 2.60 & 0.762\\\\\n\\Mstar~at \\Mden~2 & 1.04 $\\pm$ 0.16 & 2.98 & 3.05 & 0.692\\\\\n\\Mstar~at \\Mden~3 & 1.00 $\\pm$ 0.14 & 0.46 & 0.46 & 0.994\\\\\n\\Mstar~at \\Mden~4 & 1.17 $\\pm$ 0.07 & 5.88 & 10.73 & 0.057\\\\\n\\Mstar~at \\Mden~All & - & - & 16.84 & 0.66\\\\\\\\\n\\Mden~at \\Mstar~1 & 1.23 $\\pm$ 0.09 & 4.24 & 10.31 & 0.067\\\\\n\\Mden~at \\Mstar~2 & 1.12 $\\pm$ 0.05 & 6.55 & 10.69 & 0.058\\\\\n\\Mden~at \\Mstar~3 & 1.22 $\\pm$ 0.09 & 1.96 & 6.54 & 0.257\\\\\n\\Mden~at \\Mstar~4 & 1.11 $\\pm$ 0.08 & 9.47 & 11.23 & 0.047\\\\\n\\Mden~at \\Mstar~All & - & - & 38.77 & 0.0071\\\\\\\\\n\n\t\\tableline\n \\end{tabular}\n\\end{center}\n\\end{table*}\n\n\\section{Samples}\n\\label{sec:samples}\n\nWe define a parent sample with stellar mass $> 6 \\times 10^{10}M_\\odot$ and 0.04 $< z <$ 0.113. The stellar mass limit and redshift cuts are chosen to yield the largest stellar mass limited sample of galaxies that can be defined from the NYU SDSS VAGC galaxy catalog, thus maximizing the number of galaxies available for clustering measurements. This relatively high stellar mass cut is also helpful as the SDSS velocity dispersions are only reliable at $>$ 75 km\/s and thus become incomplete in \\Vdisp~at low stellar masses. By inspecting the distribution of velocity dispersion in narrow stellar mass bins we estimate that 98\\% of galaxies with stellar mass $> 6 \\times 10^{10}M_\\odot$ have velocity dispersion in excess of 75 km\/s. Finally we remove galaxies with unreliable \\Vdisp~measurements by requiring that the error in \\Vdisp~be less than 10\\%.\n\nSince there is a significant scatter in the relation between stellar mass and velocity dispersion we must make further cuts if we wish to define samples that are complete in both. To define the velocity dispersion completeness limit \nwe investigate the distribution of stellar mass in narrow velocity dispersion bins for the full DR7 sample. Using these measurements we find the velocity dispersion that defines a complete sample of galaxies at the stellar mass of $6 \\times 10^{10} M_\\odot$. We find that for galaxies with a velocity dispersion of 210 km\/s 85\\% have stellar masses greater than $6 \\times 10^{10}M_\\odot$ and thus we restrict our sample to have \\Vdisp~grater than this limit when defining \\Vdisp~limited samples. \n\nThe aim of this work is to investigate which is more fundamental in determining the clustering amplitude, a galaxy's mass, either stellar or dynamical, or its central velocity dispersion. We approach this question by measuring the clustering where we have fixed the mass and allowed \\Vdisp~to vary and visa-versa.\nWe define a series of samples with a narrow range in one parameter and then take the upper and lower quartile of the second parameter. We begin by defining the first velocity dispersion range, starting at the velocity dispersion completeness limit (210 km\/s) and cutting at a dispersion of 242.3 km\/s, the limit that contains 20\\% of the galaxies with \\Vdisp~larger than 210 km\/s. We then split that sample by either stellar mass or dynamical mass into upper and lower quartiles. In a similar manner we construct two more samples with the same interval in \\Vdisp~with successively higher \\Vdisp~limits again splitting into highest and lowest quartiles in either stellar mass or dynamical mass. Details of these samples are given in Tables \\ref{tab:SMatVd} and \\ref{tab:dMatVd} in Appendix \\ref{sec:appsamp}. To make the reciprocal samples, i.e. highest and lowest quartiles in \\Vdisp~at fixed stellar mass or dynamical mass, we find the appropriate starting minimum mass and interval that produces three samples with the same number of galaxies as in the \\Vdisp~ range samples. We then spit these into the highest and lowest quartiles of \\Vdisp. Details of these samples are given in Tables \\ref{tab:VdatSM} and \\ref{tab:VdatdM}. Figure 1 shows the distribution of galaxies in the \\Vdisp~-\\Mstar~plane for the parent sample as well as the samples at fixed \\Vdisp~and \\Mstar. \n\nWe also define a series of samples in narrow stellar mass ranges where we take the highest and lowest quartiles in stellar surface mass density (\\Mden) and their reciprocals (Tables \\ref{tab:MdenatSM} and \\ref{tab:SMatMden}) again ensuring that they are complete in both stellar mass and surface mass density. This is an important consistency check for our measurements. It is possible that the scatter introduced by larger errors on \\Mstar~could reduce any differences in relative clustering amplitude compared to \\Vdisp, something which should be reduced by using \\Mdyn. Whilst any errors on \\Mdyn~will be different and likely to be smaller than those on \\Mstar~they will still be larger than the error on \\Vdisp~due to the additional uncertainties in fitting the profile to determine \\Re~and the sersic-n parameter. However, \\Mden, which should show similar trends to \\Vdisp, contains both the errors on \\Mstar~and on \\Re. Therefore, any reduction in clustering trends due to scatter in the measurement of the physical parameters should be largest for this sample. \n\nFor each pair of samples (e.g. high and low stellar mass at fixed dispersion) we find that the mean of the fixed parameter changes very little between the samples split into high and low quartiles. For a fair comparison we compare the mass ratio to the square of the \\Vdisp~ratio since mass is proportional to $\\sigma^2$. For all samples the ratio of the mean of the fixed parameter varies by less than 5\\% and so should have a negligible effect on the clustering amplitude. Conversely the ratio of the means of the varying parameter covers a factor of 2 to 3.5.\n\nIn the Appendix \\ref{sec:appsamp} we plot redshift, velocity, mass, mass density and morphology distributions for each sample where appropriate. As expected, since we have defined a volume limited parent sample and have only made cuts where we are complete in \\Vdisp, $M_{den}$ and mass, the redshift distributions for all samples are the same. \n\\\\\n\n\\section{Clustering measurement techniques}\n\\label{sec:clus}\n\nThe two-point correlation function, $\\xi(r)$, is defined as the excess probability above Poisson of finding an object at a physical separation $r$ from another object. This is calculated by comparing the number of pairs as a function of $r$ in our galaxy catalogs with the number in a random catalog that covers the same volume as our data.\n\nSince the space density of galaxies in our subsamples is very low shot noise would be a significant source of error for measurements of their auto-correlation functions. However, we can overcome this problem by using a much denser sample of galaxies to trace the underlying density field and measure the clustering of our sub-samples relative to this larger sample using the cross-correlation function.\nThe obvious choice for the larger sample is the parent sample we defined above containing all galaxies passing our basic selection criteria. This sample contains almost 65,000 galaxies and so has a space density almost 20 times higher than our largest sub-sample. \nWhilst the individual cross-correlation functions will reflect the intrinsic clustering amplitude of both the sub- and parent samples the ratio between any two of these cross correlations will just reflect the ratio between the clustering of the two sub-samples in question, with the clustering of the parent sample in effect being 'canceled out'. \n\nMeasuring the line-of-sight distance using redshifts introduces distortions into the correlation function. The effect of these distortions can be overcome by separating the clustering signal into contributions perpendicular ($r_p$) and parallel ($\\pi$) to the line-of-sight ($\\xi(r_p,\\pi)$). One can then integrate over the line of sight direction to estimate the projected correlation function \n\\begin{equation}\n\t\\label{eq:wprp}\n\tw_p(r_p) = 2\\int^{\\infty}_0 d\\pi\\,\\xi(r_p,\\pi)\n\t = 2\\int^{\\infty}_{r_p} \\frac{r\\,dr\\,\\xi(r)}{(r^2-r_p^2)^{1\/2}}.\n\\end{equation}\nThe final expression only involves the real-space correlation function $\\xi(r)$ showing that $w_p(r_p)$ is not compromised by redshift space distortions \\citep{Davis83}. In practice it is only possible to integrate out to some maximum $\\pi$ because $\\xi(r_p,\\pi)$ is poorly known on very large scales resulting in additional noise being introduced to the measurement. We integrate to 60$h^{-1}$Mpc which is sufficiently large to include most correlated pairs and gives stable results. \n\nWe make this measurement using the \\citet{Landy93} estimator in the cross-correlation form, with\n\\begin{displaymath}\n \\xi(r_p,\\pi) = \\hspace{0.77\\columnwidth}\n\\end{displaymath}\n\\begin{equation}\n \\frac{DD1}{RR}\\left(\\frac{n_R^2}{n_D n_{D1}}\\right) - \\frac{DR}{RR}\\left(\\frac{n_R}{n_D}\\right) - \\frac{D1R}{RR}\\left(\\frac{n_R}{n_{D1}}\\right) + 1\n\\end{equation}\n where DD1 are the pair counts between the parent and sub-sample, DR are the pair counts between parent and random sample, D1R are the pair counts between sub-sample and random sample and RR are the pair counts between the random sample, all split into the $r_p$ and $\\pi$ directions. We are able to use the same random sample in each term as all samples are complete throughout the same volume, i.e. they cover the same spatial extent and have the same redshift distributions. We calculate the pair counts in a grid which is logarithmically spaced in $r_p$ of width 0.2 log($h^{-1}{\\rm Mpc}$) and linearly spaced in the $\\pi$ direction of width 2$h^{-1}{\\rm Mpc}$~with all separations in co-moving coordinates.\nThe random catalogue is based on the one provided by the NYU VAGC with additional cuts to match the exact spatial coverage of the MPA\/JHU stellar masses and redshifts matching the redshift distribution of our samples. The resulting random catalogue contains almost 1.9 million randoms, close to 29 times as many as in our parent galaxy catalog, which is a sufficient number such that our errors are never dominated by shot noise in the random pair counts.\n\nSince the individual bins in a correlation function measurement are highly correlated we need to generate accurate covariance matrices if we wish to compare our clustering measurements in a meaningful way. The simplest approach and the one we chose to follow is to use jackknife resampling \\citep{Scranton02}. There is some debate about the accuracy of covariance matrices generated in this way; for instance \\citet{Norberg09} suggest that the structure of jackknife covariance matrices may not be entirely accurate for $w_p(r_p)$~ measurements, although the variance is reliable. Conversely \\citet{Zehavi02, Zehavi04, Zehavi05} find that they can closely reproduce the structure and amplitude of covariance matrices generated by mock catalogs using jackknife resampling. Because of this uncertainty, it might be preferable to generate covariance matrices from large numbers of mock galaxy catalogues generated by populating dark matter halo catalogs so as to reproduce the clustering and density of each sample. However, since we have so many sub-samples, each with a complex selection, it would be enormously challenging to undertake such a scheme. We have therefore chosen to use jackknife resampling which should be sufficient for our purposes. We split the SDSS area into 146 equal area regions and then repeatedly calculate $w_p(r_p)$~removing one area at a time. These 146 $w_p(r_p)$~measurements are then used to generate a full covariance matrix.\n\n\\section{Results}\n\n\\begin{figure}\n\n\\vspace{8.1cm}\n\\special{psfile=MstarMdynMorph_paper.ps hoffset=-10 voffset=0 hscale=57 vscale=57 angle=0}\n\\caption{\\small The relationship between dynamical and stellar mass for all DR7 galaxies with 0.04 $< z <$ 0.113 and \\Vdisp~$>$ 75 km\/s (black contours) and those galaxies classified as either ellipticals (red contours) or spirals (blue contours). The points show the median \\Mstar~in bins of \\Mdyn~and the solid lines the biweight best fit. The dotted line shows the one-to-one relation. This relationship is almost identical independent of morphology over the whole mass range covered by this plot, and becomes indistinguishable for the mass range of interested in this paper. \n\\label{fig:MstarMdyn}}\n\\end{figure}\n\n\\subsection{Is galaxy mass or \\Vdisp~more closely related to clustering amplitude?}\n\nFigure \\ref{fig:2ptVdSM1} shows the $w_p(r_p)$~measurements for the high and low \\Vdisp~samples at fixed \\Mstar~(left) and the high and low \\Mstar~samples at fixed \\Vdisp~(right). We find that when the mass is held fixed the high \\Vdisp~samples have a higher clustering amplitude than the low \\Vdisp~samples. This is true on all scales and for all mass ranges. However, when \\Vdisp~is held fixed there appears to be little or no dependence of the clustering amplitude on \\Mstar. This is the central result of this paper: clustering amplitude depends on \\Vdisp~at fixed mass, but does not depend on mass at fixed \\Vdisp. \n\nThis result is illustrated more clearly in Figure \\ref{fig:2ptVdSM} which shows the ratio of $w_p(r_p)$~between the high and low \\Vdisp~samples at fixed \\Mstar~(left) and the high and low \\Mstar~samples at fixed \\Vdisp~(right). Figure \\ref{fig:2ptVddM} is similar to Figure \\ref{fig:2ptVdSM}, except \\Mstar~is replaced by \\Mdyn~and shows exactly the same trends as with \\Mstar~. We also show in Figure \\ref{fig:2ptMdenSM} the ratio of $w_p(r_p)$~ between high and low \\Mden~samples at fixed \\Mstar~and high and low \\Mstar~samples at fixed \\Mden. This is, in effect, the same as splitting by size at fixed mass or mass at fixed size. Since \\Mdyn~and \\Mstar~are highly correlated one would expect galaxies with smaller \\Re~at a fixed \\Mstar, thus higher \\Mden, to have a higher \\Vdisp. These samples should then be analogous to those with varying \\Vdisp~at fixed \\Mstar~and indeed they show the same trend of higher clustering amplitude at higher \\Mden~when \\Mstar~is fixed and very little dependence with \\Mstar~when \\Mden~is fixed. As discussed in Section \\ref{sec:samples} this also implies that these observed trends are not the result of larger measurement errors on \\Mstar~or \\Mdyn~compared to \\Vdisp, which could have reduced the clustering dependence on mass compared to \\Vdisp.\n\nTo quantify these ratios in $w_p(r_p)$, which constitutes a ratio in galaxy bias, we find the best fitting scale independent amplitude on scales $1.6 < r_p < 25.1$ h$^{-1}$Mpc. These scales are sufficiently large as to be dominated by pair counts between DM halos, rather than galaxies within halos, and so should show a scale independence with respect to the dark matter clustering and give an indication of the relative linear bias of the two samples. For the $w_p(r_p)$~ratios shown in Figures \\ref{fig:2ptVdSM}, \\ref{fig:2ptVddM} and \\ref{fig:2ptMdenSM} this appears to be the case and indeed a scale independent ratio is an acceptable fit for all but one of the samples. These fits are made using the full covariance matrices for the $w_p(r_p)$~ratio, and are shown as the dashed lines in Figures \\ref{fig:2ptVdSM}, \\ref{fig:2ptVddM} and \\ref{fig:2ptMdenSM}. Details of the fits are given in Table \\ref{tab:ratfit} and the best fit bias ratios and errors are plotted for all these samples in Figure \\ref{fig:biasrat}. In addition to the best fit we also calculate the $\\chi^2$ for a $w_p(r_p)$~ratio of one over the same scales, and give the $\\chi^2$ value as well as the probability of an acceptable fit in Table \\ref{tab:ratfit}. In all cases the best fit ratio is higher when \\Vdisp~or \\Mden~is varied at fixed mass than when \\Vdisp~or \\Mden~are fixed and the mass allowed to vary. Similarly when \\Vdisp~or \\Mden~are fixed the $w_p(r_p)$~ratios between the high and low mass samples are consistent with one in all but one case whereas the varying \\Vdisp~or \\Mden~samples at fixed mass are nearly all inconsistent with a $w_p(r_p)$~ratio of one. For each parameter pair we can combine the samples and determine the probability that all are consistent with a $w_p(r_p)$~ratio of 1. Again we find that at fixed \\Vdisp~or \\Mden~the samples are indeed consistent with showing no dependence of the clustering amplitude on mass at the 66\\%, 87\\% and 66\\% levels, whereas there is only a 0.9\\%, 0.03\\% and 0.7\\% chance that all three of the varying \\Vdisp~or \\Mden~samples at fixed mass are consistent with no variation in clustering amplitude on large scales. \n\nOn small scales the differences are just as clear with the varying \\Vdisp~or \\Mden~samples typically showing even larger $w_p(r_p)$~ratios than at large scales. The samples with varying mass at fixed \\Vdisp~or \\Mden~ occasionally show moderately larger $w_p(r_p)$~ratios on small scales than on large scales, but often there is no difference with the $w_p(r_p)$~ratio remaining consistent with one on all scales. \n\nOverall these measurements make it clear that velocity dispersion or stellar mass surface density are more closely related to galaxy bias than either stellar mass or dynamical mass.\n\\\\\n\\\\\n\n\\subsection{The Importance of Morphology and Color}\n\n\\begin{table*}\n \\begin{center}\n \\caption{\\label{tab:ratfitcolel} Fits to correlation funtion ratios for sample split by morphology and color}\n\t\\begin{tabular}{lcccc}\n\t\\tableline\n \t\\multicolumn{1}{l}{Sample} &\n \t\\multicolumn{1}{c}{Best fit ratio} &\n \t\\multicolumn{1}{c}{Best fit $\\chi^2$} &\n \t\\multicolumn{1}{c}{Ratio = 1 $\\chi^2$} &\n \t\\multicolumn{1}{c}{Ratio = 1 prob} \\\\\n\t\\tableline\n\\\\\n\\Pel~at \\Vdisp~1 & 0.92 $\\pm$ 0.05 & 2.44 & 4.55 & 0.473\\\\\n\\Pel~at \\Vdisp~2 & 1.06 $\\pm$ 0.09 & 4.29 & 4.71 & 0.452\\\\\n\\Pel~at \\Vdisp~3 & 1.07 $\\pm$ 0.10 & 2.95 & 3.41 & 0.638\\\\\n\\Pel~at \\Vdisp~All & - & - & 12.67 & 0.63\\\\\n\\\\\n\\Pel~at \\Mdyn~1 & 1.12 $\\pm$ 0.05 & 2.60 & 6.16 & 0.291\\\\\n\\Pel~at \\Mdyn~2 & 1.20 $\\pm$ 0.08 & 1.05 & 5.77 & 0.329\\\\\n\\Pel~at \\Mdyn~3 & 1.10 $\\pm$ 0.12 & 1.43 & 2.02 & 0.847\\\\\n\\Pel~at \\Mdyn~All & - & - & 13.95 & 0.53\\\\\n\\\\\n$g-r$ at \\Vdisp~1 & 1.35 $\\pm$ 0.11 & 3.35 & 11.81 & 0.037\\\\\n$g-r$ at \\Vdisp~2 & 1.36 $\\pm$ 0.14 & 4.55 & 11.34 & 0.045\\\\\n$g-r$ at \\Vdisp~3 & 1.17 $\\pm$ 0.17 & 5.69 & 6.64 & 0.248\\\\\n$g-r$ at \\Vdisp~All & - & - & 29.80 & 0.013\\\\\n\\\\\n$g-r$ at \\Mdyn~1 & 1.34 $\\pm$ 0.10 & 6.66 & 16.34 & 0.006\\\\\n$g-r$ at \\Mdyn~2 & 1.50 $\\pm$ 0.16 & 2.60 & 12.76 & 0.026\\\\\n$g-r$ at \\Mdyn~3 & 1.13 $\\pm$ 0.11 & 3.83 & 5.19 & 0.394\\\\\n$g-r$ at \\Mdyn~All & - & - & 34.29 & 0.0031\\\\\n\\\\\n\\Vdisp~at \\Mdyn~El 1 & 1.29 $\\pm$ 0.08 & 1.51 & 12.27 & 0.031\\\\\n\\Vdisp~at \\Mdyn~El 2 & 1.32 $\\pm$ 0.10 & 4.73 & 13.46 & 0.019\\\\\n\\Vdisp~at \\Mdyn~El 3 & 1.29 $\\pm$ 0.12 & 2.82 & 7.57 & 0.181\\\\\n\\Vdisp~at \\Mdyn~El All & - & - & 33.30 & 0.0043\\\\\n\\\\\n\\Vdisp~at \\Mdyn~red 1 & 1.21 $\\pm$ 0.06 & 2.32 & 12.08 & 0.034\\\\\n\\Vdisp~at \\Mdyn~red 2 & 1.15 $\\pm$ 0.08 & 1.08 & 4.49 & 0.482\\\\\n\\Vdisp~at \\Mdyn~red 3 & 1.48 $\\pm$ 0.15 & 1.34 & 10.71 & 0.058\\\\\n\\Vdisp~at \\Mdyn~red All & - & - & 27.28 & 0.027\\\\\n\t\\tableline\n \\end{tabular}\n\\end{center}\n\\end{table*}\n \n\n\\begin{figure*}\n\n\\vspace{12cm}\n\\special{psfile=wpratiofit_Pel.ps hoffset=60 voffset=390 hscale=45 vscale=45 angle=-90}\n\\special{psfile=wpratiofit_col.ps hoffset=60 voffset=220 hscale=45 vscale=45 angle=-90}\n\\caption{\\small The dependence of large scale bias on elliptical probability (top) and $g-r$ color (bottom) at fixed \\Mdyn~(left)and \\Vdisp~(right). The bias does depended on elliptical probability at fixed mass but not at fixed \\Vdisp. However, there is a significant dependence of the bias on color at both fixed mass and \\Vdisp.\n\\label{fig:biasrat2}}\n\\end{figure*}\n\n\\begin{figure*}\n\n\\vspace{12cm}\n\\special{psfile=wpratiofit_dMVd_El.ps hoffset=60 voffset=390 hscale=45 vscale=45 angle=-90}\n\\special{psfile=wpratiofit_dMVd_red.ps hoffset=60 voffset=220 hscale=45 vscale=45 angle=-90}\n\\caption{\\small The dependence of large scale bias on \\Vdisp~at fixed \\Mdyn~(left) and \\Mdyn~at fixed \\Vdisp~(right) for elliptical galaxies (P$_{el} >$ 0.6; top) and red galaxies ($g-r >$ 0.9; bottom). The same result as for the whole sample remains, there is significant dependence of the bias on \\Vdisp~at fixed \\Mdyn~but none on \\Mdyn~at fixed \\Vdisp.\n\\label{fig:biasrat3}}\n\\end{figure*}\n\n\n\nWe have included the distribution in elliptical probability for each of the samples in the plots in Appendix \\ref{sec:appsamp} as it demonstrates one area that is both potentially a concern but also scientifically important for this analysis. \\Vdisp~is much better correlated with morphology or color than stellar mass is. \\Mden~is also a better discriminator than mass, but not as good as \\Vdisp~(see Wake et al. 2012 for a detailed study of these trends). Because of this when we fix \\Vdisp~we come pretty close to fixing morphology regardless of the mass and when we split into high and low \\Vdisp~samples at fixed mass we are pretty close to splitting into disks and spheriods (we also see a splitting by color in the same way). This is of course the heart of our investigation: is a tighter correlation between \\Vdisp~and halo properties (compared to the correlation between mass and halo properties) the explanation as to why \\Vdisp~is a better indicator than stellar mass of a galaxy's stellar population? This does however raise a potential worry regarding systematics: are we measuring the same thing when we measure \\Vdisp~for a disk galaxy as compared to a spheroidal galaxy? \n\nWe tackle this question in several ways. Firstly we show in Figure \\ref{fig:MstarMdyn} the observed relationship between \\Mstar~and \\Mdyn~for all the galaxies in SDSS with $0.04 < z < 0.113$ and \\Vdisp~$>$ 75 km\/s. We further split this sample into spheriods or disks based on the Galaxy Zoo classifications. The relationship is essentially identical regardless of morphology over practically the entire mass range. There is a small deviation at lower masses but this is below the mass range considered in this work. This gives us confidence that the velocity dispersion is measuring the same physical quantity for both disks and ellipticals within the SDSS fiber radius.\n\nThe second approach is a more direct one: we can see if morphology itself has any effect on the clustering amplitude of our galaxies at fixed mass or \\Vdisp. We thus define samples with high and low elliptical probabilities in narrow ranges of \\Mdyn~and \\Vdisp. We note that we are unable to just take the quartiles of the distribution in elliptical probability as we have done for parameters previously as there is a redshift dependence to the probabilities assigned. This results from galaxies being harder to classify at higher redshift due to their smaller angular size and surface brightness dimming. Whilst there has been some correction for this effect applied to the Galaxy Zoo probabilities we find that it does not entirely remove this bias resulting in the samples split into upper and lower quartiles in \\Pel~having different redshift distributions. We thus define our samples making use of the expectation that the morphological mix does not vary significantly over our narrow redshift range. We find the best fit linear relations between elliptical probability and redshift that produce the 25\\% most and least likely ellipticals at each redshift and use these relations to define our samples. Details of these samples are given in Tables \\ref{tab:PelatdM} and \\ref{tab:PelatVd} and their distributions are plotted in Appendix \\ref{sec:appsamp}. \n\nFigure \\ref{fig:biasrat2} shows the large scale $w_p(r_p)$~ratios of these high and low \\Pel~samples at fixed mass and \\Vdisp. As before we fit the $w_p(r_p)$~ratio (shown in Figure \\ref{fig:2ptPel} in Appendix \\ref{sec:appsamp}) between $1.6 < r_p < 25.1$ h$^{-1}$Mpc and determine the probability that the ratio is consistent with one. Details of these fits are give in Table \\ref{tab:ratfitcolel}. At fixed \\Vdisp~there is no evidence for any dependence of the clustering amplitude on morphology at any scale. At fixed mass the ratios are all non-zero with galaxies with a higher \\Pel~showing a higher clustering amplitude than those with a low \\Pel. However, the significance of these positive $w_p(r_p)$~ratios is low with all being consistent with no difference in the clustering amplitude. This implies that the correlation of morphology with dispersion is not the determining factor in the strong correlation of clustering amplitude with \\Vdisp~at fixed mass. \n\nWhilst the strong correlation of color with \\Vdisp~is unlikely to introduce any systematics in our measurements it is still informative to investigate whether there is any residual clustering dependence on color at either fixed mass or \\Vdisp. Since color is better correlated with \\Vdisp~than mass one might imagine that any residual clustering dependence on color would be lower at fixed \\Vdisp~than at fixed mass if the halo properties are key to determining the color, much as is seen with morphology above. To test this we show in Figure \\ref{fig:biasrat2} the best fit large scale correlation function ratios of high and low $g-r$ color samples at fixed mass and dispersion and list the fitting results in Table \\ref{tab:ratfitcolel} (the full $w_p(r_p)$~ratios are shown in Figure \\ref{fig:2ptcol} in Appendix \\ref{sec:appsamp}). Since the color evolves with redshift we have again had to split the samples in a similar way to the elliptical probability samples such that we have the reddest and bluest quartiles at any redshift. \nNow, unlike with the elliptical probability, we see a residual clustering dependence on color at both fixed mass and \\Vdisp~such that red galaxies cluster more strongly than blue. \nThis is most likely the result of the truncation of star formation in satellite galaxies in massive dark halos, resulting in satellite galaxies at fixed mass or \\Vdisp~having redder colors than central galaxies with the same mass or \\Vdisp~\\citep[e.g.][]{Yang09}. The fact that we see a significant residual color dependence and not much of a residual morphology dependence suggests that the environmental mechanism for transforming the color of a galaxy from red to blue does not simultaneously change the morphology. A similar result was observed by \\citet{Skibba09} using the marked correlation function applied to SDSS galaxies.\n\nSince we do see some residual color dependence and perhaps a hint of morphology dependence in the clustering amplitude at fixed mass it is worth asking if the \\Vdisp~dependence remains when morphology or color is fixed. We show in Figure \\ref{fig:biasrat3} the large scale $w_p(r_p)$~ratio of high and low \\Vdisp~samples at fixed \\Mdyn~and high and low \\Mdyn~samples at fixed \\Vdisp~where we have restricted the galaxies to be either red ($g-r > 0.9$) or have high elliptical probabilities ($P_{el} >$ 0.6). The fit details given in Table \\ref{tab:ratfitcolel} and the full $w_p(r_p)$~ratios shown in Figures \\ref{fig:2ptVddMred} and \\ref{fig:2ptVddMel} in Appendix \\ref{sec:appsamp}. Despite the sample sizes being reduced and hence the errors increasing, the original trend of high \\Vdisp~galaxies having a higher clustering amplitude at fixed mass remains significant and the large scale ratios stay essentially the same within the errors. There is a hint that the $w_p(r_p)$~ratios are slightly reduced, but by removing the blue or spiral galaxies we have removed many lower \\Vdisp~galaxies and reduced the difference between the high and low \\Vdisp~samples.\nThis is a clear demonstration that galaxies with higher \\Vdisp~cluster more strongly than galaxies with lower \\Vdisp~ regardless of their mass, morphology or color. \n\n\n\\section{Discussion}\n\\label{sec:dis}\n\n\\begin{figure}\n\n\\vspace{11cm}\n\\special{psfile=HODmassbin_modcenmt.ps hoffset=0 voffset=0 hscale=40 vscale=40 angle=0}\n\\caption{\\small Top: An example halo occupation distribution representing the case where correlation between \\Vdisp~and dark matter halo mass is tighter than the correlation between \\Mstar~and halo mass. The black line represents an HOD for a galaxy population with a narrow slice in \\Mstar. The red and blue lines each contain 25\\% of the \\Mstar slice sample and have had their central halo mass thresholds adjusted to sample slightly higher and lower halo masses from within that sample, to represent high and low \\Vdisp~samples. In each case the dashed lines show the central galaxies and the dotted the satellites. Bottom: The auto- (solid) and cross- (dotted) correlation function ratios between the example high and low \\Vdisp HODs. They show the very similar characteristics to the measured correlation function ratios on both small and large scales. \n\\label{fig:HOD}}\n\\end{figure}\n\nWe have shown that at fixed mass (stellar or dynamical) galaxies with high \\Vdisp~(or \\Mden) cluster more strongly than galaxies with low \\Vdisp. Conversely, at fixed \\Vdisp~(or \\Mden) our measurements are consistent with high and low mass galaxies clustering the same. If the clustering of galaxies is dominated by the clustering of the DM halos in which they reside, as expected in the CDM paradigm, then this implies that even at fixed galaxy mass higher \\Vdisp~galaxies live in more clustered halos than low \\Vdisp~galaxies. The main determinant of a halo's clustering amplitude is its mass, but it has also been shown in simulations that halos with different formation histories also cluster differently, with older more centrally concentrated halos clustering more strongly \\citep{Gao05,Wechsler06,Wetzel07,Gao07,Jing07,Li08}. This means that galaxies with higher \\Vdisp~at fixed mass could be preferentially be living in either more massive or older more centrally concentrated halos. \n\nWhen considering how galaxies occupy halos it is also important to make the distinction between the galaxy that has been formed at the center of a given DM halo, the central galaxy, and those that initially formed at the centers of other halos and have later been accreted into this halo, satellite galaxies. The properties of a central galaxy will be largely determined by the formation history and properties of its host halo, whereas any effect the current host halo has on satellite galaxies is limited to the time after it has been accreted. \n\nIn the following sections we address whether our result can be explained by a relationship between \\Vdisp~and halo mass for central galaxies (Section \\ref{sec:cenmass}) or for satellite galaxies (Section \\ref{sec:satmass}), or by a relationship between \\Vdisp~and halo formation history (Section \\ref{sec:age}) \n\n\n\\subsection{Central galaxy halo mass correlations}\n\\label{sec:cenmass}\nSince DM halo mass has the largest effect on clustering amplitude, the simplest explanation for our results is that \\Vdisp~is better correlated with its host halo mass than stellar or dynamical mass are. For galaxies living at the centers of halos, central galaxies, there is a strong correlation between stellar mass and halo mass \\citep[e.g.][]{Yang09,Wake11}, but with a significant scatter \\citep{Yang09,Yang11}.\nIf the scatter between \\Vdisp~and halo mass for centrals is substantially lower than the scatter with \\Mstar~this could result in clustering observations much as we have observed. \n\nIt is striking in Figures \\ref{fig:2ptVdSM} and \\ref{fig:2ptVddM} that the clustering ratio between high and low \\Vdisp~samples becomes even larger as the scale decreases. The clustering amplitude on such scales is determined by the separations of galaxies within individual halos, so between central and satellite galaxies. Thus these smaller scale clustering measurements can provide an additional constraint as to whether a tighter correlation between \\Vdisp~and halo mass (\\Mhalo) for central galaxies could produce the observed clustering results. \n\nWe show in Figure \\ref{fig:HOD} an illustrative attempt at modeling the effect of a smaller scatter between \\Vdisp~and \\Mhalo~than between \\Mstar~and \\Mhalo~using the halo model \\citep[see][for details of our model]{Wake11}. We have fitted halo occupation distributions (HOD), the mean number of central and satellite galaxies as a function of halo mass, to the clustering measurements for a series of samples of SDSS galaxies limited in stellar mass (i.e. \\Mstar $>$M$_{\\rm low}$) \\citep[see][for details of these measurements]{Wake12}. Subtracting two of the resulting HODs with slightly different M$_{\\rm low}$ gives the HOD for a sample of galaxies with a narrow range in \\Mstar. We show an HOD for such a sample as the black line in the top panel of Figure \\ref{fig:HOD}, which has been chosen as it matches the lowest \\Mstar~ sample we use ($11.04 <$ log(\\Mstar) $< 11.19$). We can then attempt to mimic the effect of splitting into high and low \\Vdisp~samples under the assumption that \\Vdisp~for central galaxies is better correlated with halo mass than \\Mstar~is, so that the high \\Vdisp~sample will have higher average halo masses than the low \\Vdisp~sample. We do this by slightly adjusting the central galaxy halo mass thresholds that define the high and low halo mass cut offs in our HODs. The HODs of these two samples, representing high and low \\Vdisp~quartiles at fixed \\Mstar, are shown as the red and blue lines in the top panel of Figure \\ref{fig:HOD}. These mock high and low \\Vdisp~samples each contain 25\\% of the central and satellite galaxies of the full sample, with the size of the central HOD controlled by the mass threshold adjustments and the satellite just random sampled to 25\\% of the number density, i.e. the form of the satellite HOD is left unchanged. The resulting $w_p(r_p)$ ratio is shown in the bottom panel of Figure \\ref{fig:HOD} for both the auto- and cross-correlations. The cross-correlation is calculated with an HOD suitable for the parent sample we use throughout this work (i.e. log(\\Mstar) $>$ 10.777).\n\nThe model $w_p(r_p)$~ratio shown in Figure \\ref{fig:HOD} is quite similar to the measured ratios shown in Figures \\ref{fig:2ptVdSM}, \\ref{fig:2ptVddM} and \\ref{fig:2ptMdenSM}, showing a constant increase in the cross-correlation amplitude on intermediate and large scales and then an increasing amplitude at small scales. It is always tempting to interpret changes in small scale (1-halo) clustering as reflecting changes in the satellite population, but this is a clear example of how this is not necessarily the case. The satellite population has not changed between the mock low and high \\Vdisp~samples, and it is the change in the central galaxies which is causing a change in the numbers of central-satellite pairs that causes the increased clustering amplitude on small scales. More explicitly, at a given stellar mass the higher \\Vdisp~galaxies are in more massive halos, which typically contain more satellite galaxies. This results in an increase in the number of central-satellite pairs, thus increasing the small scale clustering amplitude. The converse is true for lower dispersion centrals i.e. they are in less massive halos, with fewer satellites and so fewer central-satellite pairs. The satellite-satellite pairs remain unchanged, but since for these massive galaxies the satellite fraction is low so there may only be at most one or two satellites in a given halo the central-satellite pairs make up a significant contribution to the small scale clustering amplitude. \n\nThis exercise clearly demonstrates that if there is a much tighter correlation between \\Vdisp~and halo mass than between \\Mstar~and halo mass then the clustering ratios we measure would be expected. There is, of course, some theoretical expectation that this would be the case. The stellar mass of a central galaxy depends on a large number of complicated astrophysical baryonic processes such as gas cooling and feedback as well as mergers from satellite galaxies, which may give rise to a large scatter between halo mass and stellar mass. \\Vdisp, on the other hand, is tracing the depth of the potential at the center of the halo and whilst this may be modified by the evolution of the baryons it is likely to be much more directly linked to the halo properties and in particular its mass. \n\n\\subsection{Satellite galaxy halo mass correlations}\n\\label{sec:satmass}\nWhilst our example halo model shows that our results can be produced just by central galaxies, it is of course possible to produce similar clustering trends by modifying the satellite distribution or both the central and satellite distributions. To increase the clustering amplitude of galaxies at fixed mass by modifying the satellites requires that higher \\Vdisp~satellites are preferentially found in more massive halos. \n\nOne possible process that could create such an effect is the stripping of the outer regions of a satellite galaxy by tidal interactions as it orbits it parent halo. This would likely have a minimal effect on \\Vdisp~whilst reducing the mass and size of the galaxy. The likelihood of a galaxy getting disrupted in this way increases as the halo mass increases but decreases as the mass of the galaxy increases. Since, the galaxies in this study are all more massive than $6\\times10^{10}M_\\odot$ it would only be effective for satellites passing very close to the center of the most massive halos an occurrence that would be quite rare. \n\nHowever, assuming that this process was an effective one we would expect to see its affects on the clustering both when the mass is held fixed and \\Vdisp~varied and visa-versa. Satellite galaxies at fixed mass would have higher \\Vdisp~in higher mass halos, since they would have been stripped more. This would mean that the satellite galaxies in the higher \\Vdisp~samples would be preferentially found in higher mass halos and hence would cluster more strongly, just as we have observed. The opposite would be true of satellite galaxies with fixed \\Vdisp; one would expect them to have lower masses in more massive halos since again they would have experienced more tidal stripping. \n\nOne could then imagine a scenario that could explain our observations. If we assume that the more massive a central galaxy or the higher its \\Vdisp~then the more massive its parent halo, then for centrals one would observe higher clustering amplitudes both for higher mass galaxies at fixed \\Vdisp~and higher \\Vdisp~galaxies at fixed mass. At fixed \\Vdisp~the higher mass satellites would typically be in lower mass halos (where they haven't been stripped) and visa-versa, resulting in an overall reduction in the clustering ratio between the high and low mass samples. The opposite would be true at fixed mass, where the higher \\Vdisp~satellite galaxies would preferentially be in higher mass halos (where the stripping is effective), increasing the clustering ratio between high and low \\Vdisp~samples. A similar effect could also occur when considering \\Mdyn~rather than \\Mstar~since the size of a galaxy would be reduced as it is stripped, thus reducing \\Mdyn. This does seem to be a viable explanation of our results, although we note that the fraction of satellite galaxies should be relatively low in our samples, $<$ 20\\%.\n\n\n\\subsection{Central galaxy halo age correlations}\n\\label{sec:age}\nFinally, as already mentioned, there is an alternative to the halo mass dependence outlined above, whereby instead of a tight correlation between central galaxy \\Vdisp~and halo mass there is a residual correlation between \\Vdisp~and a halo's formation history or age. N-body simulations show that at a given mass halos that formed earlier (and are more concentrated) are more clustered than younger halos \\citep{Gao05,Wechsler06,Wetzel07,Gao07,Jing07,Li08}. This is an appealing explanation since it would link together both the correlation between \\Vdisp~and clustering amplitude as well as that between \\Vdisp~and the star formation history of galaxies. At a given mass galaxies with higher \\Vdisp~exhibit older stellar populations which would be naturally explained by the earlier formation of their parent halos and thus their higher clustering amplitude. \n\n\\subsection{Consequences of these relationships}\n\\begin{figure}\n\\vspace{11cm}\n\\special{psfile=SMVdispatMdyn.ps hoffset=0 voffset=180 hscale=30 vscale=30 angle=-90}\n\\special{psfile=ReVdispatSM.ps hoffset=0 voffset=330 hscale=30 vscale=30 angle=-90}\n\\caption{\\small Top: The relationship between \\Re~and \\Vdisp~at fixed \\Mstar~($10.85 <$ log(\\Mstar) $< 10.9$). The blue points show the running median. At a given \\Mstar~galaxies with higher \\Vdisp~are smaller. Bottom: The relationship between \\Mstar~and \\Vdisp~at fixed \\Mdyn~($11.35 <$ log(\\Mdyn) $< 11.45$), showing very little trend.\n\\label{fig:ReatVdisp}}\n\\end{figure}\n\nAssuming that a tighter relationship exists between \\Vdisp~and halo properties than between galaxy mass and halo properties for central galaxies, as outlined in Sections \\ref{sec:cenmass} and \\ref{sec:age}, we can use our measurements to infer how \\Vdisp, \\Mstar, \\Mdyn~and \\Re~interrelate with one another, and with halo mass and concentration. We expect that \\Vdisp, \\Mstar~and \\Mdyn~all depend on halo mass and each other as follows: \n\\begin{equation}\n\\sigma^2 \\sim \\frac{GM_{dyn}}{R_e} \\propto \\frac{M_{star}}{R_e} + \\frac{M_{halo}}{R_{vir}\/c}\n\\end{equation}\nwhere \\Mhalo~is the halo mass, $R_{vir}$ is the halo virial radius and $c$ is the halo concentration \\citep{Hopkins09}. \n Therefore, when \\Mstar~is fixed and \\Vdisp~varied either \\Re, \\Mhalo~or $c$ must vary. Since the clustering amplitude does change then either \\Mhalo~or $c$ must also change, such that higher dispersion galaxies must live in more massive or more concentrated (older) halos. Whilst it is the case that galaxies with higher \\Vdisp~at a fixed \\Mstar~typically have a smaller \\Re~(see Figure \\ref{fig:ReatVdisp}) they must also occupy more massive or more concentrated halos. Conversely as \\Mstar~is increased at fixed \\Vdisp~\\Re~must be changing as well such that more massive galaxies have larger \\Re, since there is no change in the clustering amplitude so \\Mhalo~and $c$ aren't varying. This means that at fixed \\Vdisp~increasing \\Mstar~increases the ratio between \\Mstar~and \\Mhalo. \n\nWe see a similar effect with \\Mdyn. When \\Mdyn~is held fixed and \\Vdisp~varied \\Re~must also vary such that higher \\Vdisp~leads to lower \\Re. We also know that either \\Mhalo~or $c$ increases with \\Vdisp~at fixed \\Mdyn~since the clustering amplitude increases. \\Mstar~could potentially vary in either direction with the size of the change depending on the ratio of the contributions of dark and stellar mass to the potential and the magnitude of the change in \\Mhalo~or $c$ with \\Vdisp. In fact Figure \\ref{fig:ReatVdisp} shows that at fixed \\Mdyn~there is very little average change in \\Mstar~as \\Vdisp~increases, indicating that \\Mhalo~or $c$ have to increase with \\Vdisp~at fixed \\Mdyn~just as we observe.\nWhen \\Mdyn~is varied at fixed \\Vdisp~\\Re~must also change, increasing as \\Mdyn~increases. Since the clustering amplitude does not change, \\Mhalo~and $c$ do not change and so \\Mstar~must be increasing. This implies that at fixed \\Vdisp~galaxies with higher \\Mdyn~will have both larger \\Mstar~and \\Re. \n\n\n\n\\section{Summary and Conclusions}\n\nWe have investigated how the clustering of galaxies depends on stellar mass, dynamical mass, velocity dispersion, surface mass density, color and morphology using large samples of massive galaxies from the SDSS. We split the samples into narrow ranges in mass, \\Vdisp~or \\Mden~and measure any residual clustering dependence on the other parameters. We find the following:\n\n\\begin{enumerate}\n\\item When \\Mstar~or \\Mdyn~are fixed there is a significant dependence of the clustering amplitude on \\Vdisp~on all scales, such that galaxies with higher \\Vdisp~are more strongly clustered. Conversely when \\Vdisp~is fixed there is no dependence of the clustering amplitude on either \\Mstar~or \\Mdyn. These trends remain when we limit the samples to galaxies that are red or morphologically elliptical. We see a similar trend using \\Mden~instead of \\Vdisp.\n\n\\item When \\Mstar~or \\Mdyn~are fixed there is a weak dependence of the clustering amplitude on morphology. When \\Vdisp~ is fixed there is no significant morphological dependent clustering. \n\n\\item There always remains a strong dependence of the clustering amplitude on a galaxy's $g-r$ color at fixed mass or \\Vdisp. This is most likely caused by satellite galaxies in massive halos always being preferentially redder than central galaxies of the same mass or \\Vdisp.\n\n\\end{enumerate}\n\nWe suggest that for our main finding, point 1. above, that there are three possible explanations; the relationship between \\Vdisp~or \\Mden~and halo mass for central galaxies is tighter than that between \\Mstar~or \\Mdyn~and halo mass; the relationship between \\Vdisp~or \\Mden~and halo age (or concentration) for central galaxies is tighter than that between \\Mstar~or \\Mdyn~and halo age (or concentration); tidal stripping of satellite galaxies reduces the size and mass of the galaxy whilst having a minimal effect on \\Vdisp~and is more effective in more massive halos. Or it could be a combination of all three effects. If the host halo mass or age are indeed better correlated with \\Vdisp~and \\Mden~than with \\Mstar, these halo properties may also drive the star formation history of galaxies, explaining why \\citet{Kauffmann03b} found that star formation rate is better correlated with \\Mden~than \\Mstar, and why \\citet{Franx08} found that color and star formation rate are better correlated with \\Vdisp~than \\Mstar. \n \n\\acknowledgments{\n\nDAW would like to thank Nikhil Padmanabhan, Carlton Baugh, Violeta Gonzalez, Rachel Bezanson, Britt Lundgren and Ravi Sheth for helpful discussions on this work. We would like to thank the NYU and MPA-JHU groups for making their value added data products freely available. \n\nThis work makes use of data from the Sloan Digital Sky Survey (SDSS). Funding for the Sloan Digital Sky Survey (SDSS) has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Science Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Society. The SDSS Web site is http:\/\/www.sdss.org\/.\n\nThe SDSS is managed by the Astrophysical Research Consortium (ARC) for the Participating Institutions. The Participating Institutions are The University of Chicago, Fermilab, the Institute for Advanced Study, the Japan Participation Group, The Johns Hopkins University, Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, University of Pittsburgh, Princeton University, the United States Naval Observatory, and the University of Washington. \n}\n\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\\section{Introduction}\n\nUncertainty quantification is a fundamental desiderata of statistical inference and data science. In supervised learning settings it is common to quantify uncertainty with either conditional entropy or mutual information (MI). \nSuppose we are given a pair of random variables $(X, Y)$, where $X$ is $d$-dimensional vector-valued and $Y$ is a categorical variable of interest. Conditional entropy $H(Y|X)$ measures the uncertainty in $Y$ on average given the value of $X$. On the other hand, mutual information quantifies the shared information between $X$ and $Y$.\nAlthough both parameters are readily estimated when $X$ and $Y$ are low-dimensional and ``nicely'' distributed, an important problem arises in measuring these quantities from higher-dimensional data in a nonparametric fashion \\citep{mines}. \n\nWe address the problem of high-dimensionality by proposing a decision forest method for estimating conditional entropy under the framework that $X$ is any $d$-dimensional random vector and $Y$ is categorical. Because we restrict $Y$ to be categorical, we can easily compute the maximum-likelihood estimate of $H(Y)$ and hence use our estimator to compute mutual information.\n\nWe present Uncertainty Forests (UF) to estimate these information-theoretic quantities. UF combines quantile regression forests~\\citep{Meinshausen06} with honest sampling~\\citep{denil14}, and introduces a finite sample correction to improve performance while preserving consistency. \nWe prove that our estimation technique is consistent (meaning that it converges in probability to the true estimates of conditional entropy and mutual information, under mild conditions on the joint distribution of $X$ and $Y$), and demonstrate via simulations that UF performs well in both low- and high-dimensional settings when estimating conditional distributions, conditional entropy, and mutual information. \n\nFinally, we provide a real-world application of our estimator by measuring information about the various properties of \\textit{Drosophila} neurons that are contained in neuron cell types. Scientific prior knowledge poses various relationships between particular features and the cell type, which are supported by the mutual information and conditional mutual information estimates of UF.\n\n\\section{Background}\n\\subsection{Problem Formulation}\n\nSuppose we are given two random variables $X$ and $Y$ with support sets $\\mathcal{X}$ and $\\mathcal{Y}$, respectively. Let $x$, $y$ denote specific values that the random variables take on and $p(x), p(y)$ be the probabilities of $X=x$ and $Y=y$. The unconditioned Shannon entropy of $Y$ is $H(Y) = -\\sum_{y \\in \\mathcal{Y}} p(y) \\log p(y)$.\nAnalogously, conditional entropy is $ H(Y|X) = \\sum_{x \\in \\mathcal{X}}{p(x)H(Y|X = x)} = -\\sum_{x \\in \\mathcal{X}}{p(x)\\sum_{y \\in \\mathcal{Y}}{p(y|x)\\log{p(y|x)}}}$, where $p(y|x)$ is the conditional probability that $Y=y$ given $X=x$ and $H(Y|X = x)$ is the entropy of $Y$ conditioned on $X$ equaling particular value $x$. This quantity represents the average uncertainty in $Y$ having observed $X$. In the case of a continuous random variable, the sum over the corresponding support is replaced with an integral, and probability mass functions are replaced by densities. For the remainder of this work, we assume that $\\mathcal{X} = \\mathbb{R}^d$ and $\\mathcal{Y} = [K] = \\{1, ..., K\\}$ for positive integers $d$ and $K$.\nMutual information, $I(X; Y)$, can be computed from conditional entropy $ I(X; Y) = H(Y) - H(Y|X) = H(X) - H(X|Y)$.\nMutual information has many appealing properties, such as symmetry, and is widely used in data science applications \\citep{mixedksg}. \n\n\\subsection{Related Work}\n\nA common approach to estimating mutual information relies on the \\textit{3-H} principle \\citep{mixedksg},\n$I(X; Y) = H(Y) + H(X) - H(X, Y)$,\nwhere $H(X, Y) = -\\sum_{x \\in \\mathcal{X}} \\sum_{y \\in \\mathcal{Y}} p(x,y) \\log p(x,y)$ is the Shannon entropy of the pair $(X, Y)$. Procedures typically estimate unconditional entropy for the three separate random variables. Examples of these include kernel density estimates and ensembles of $k$-NN estimators \\citep{Beirlant1997, Leonenko08estimationof, berrett2019, sricharan}. One method in particular, the KSG estimator, popular because of its excellent empirical performance, improves $k$-NN estimates via heuristics \\citep{ksg}. Other approaches include binning~\\citep{binning} and von Mises estimators~\\citep{vonmises}. Unfortunately, many modern datasets contain mixtures of continuous inputs $X$ and discrete outputs $Y$. In these cases, few of the above methods work well, as while individual entropies $H(X)$ and $H(Y)$ are well-defined, $H(Y, X)$ is either ill-defined or not easily estimated, thus rendering the \\textit{3-H} approach intractable~\\citep{mixedksg}. \n\nA recent approach, referred to as Mixed KSG, modifies the KSG estimator to improve its performance in various settings, including mixed continuous and categorical inputs \\citep{mixedksg}. Computing both mutual information and conditional entropy becomes difficult in higher dimensional data. Numerical summations or integration become computationally intractable, and nonparametric methods (for example, $k$-nearest neighbor, kernel density estimates, binning, Edgeworth approximation, likelihood ratio estimators) typically do not scale well with increasing dimensions \\citep{mines, gaoetal}. While a recently proposed neural network approach, MINE \\citep{mines}, addresses high-dimensional data, the method does not return an estimate of the posterior $p(y \\mid x)$, which is useful in uncertainty quantification. Additionally, MINE requires that the network of choice be expressive enough so that the Donsker-Varadhan representation can be used to approximate the mutual information, an assumption that is difficult to inspect. Other theoretically analyzed deep approaches to mutual information estimation can make stringent assumptions on the distribution of learned weights \\citep{deep1}. Finally, in the interest of an ``out-of-the-box\" method, another common pitfall of deep approaches is sensitivity to hyperparameter choice.\n\nAnother popular ad-hoc approach relies on estimation of the posterior probability $p(y \\mid x)$. Calibration methods, such as isotonic regression, map classification scores, whether they are scalar projections of the data in linear discriminant analysis or empirical posteriors from random forests, to ``calibrated\" posteriors using a held-out set. A thorough theoretical analysis of such methods has yet to be addressed \\citep{isotonic}.\n\n\\subsection{CART Random Forest} \\label{sec:cart} Classification and Regression Tree (CART) Random Forest is a robust, powerful algorithm that leverages ensembles of decision trees for classification and regression tasks \\citep{Breiman2001}. In a study of over 100 classification problems, F\\'ernandez-Delgado et al. \\citep{Delgado14} showed that random forests have the overall best performance when compared 178 other classifiers. Furthermore, random forests are highly scalable; efficient implementations can build a forest of 100 trees from 110 Gigabyte data ($n$ = 10,000,000, $d$ = 1000) in little more than an hour \\citep{scalablerf}.\n\nRandom forest classifiers are instances of a bagging classifiers, in which the base classifiers are decision trees. A decision tree first learns a partition of feature space, then learns constant functions within each part to perform classification or regression. Precisely, $ \\mc{L} $ is called a partition of feature space $ \\mc{X} $, if for every $ L, L' \\in \\mc{L}$ with $L \\neq L'$, $L \\cap L' = \\varnothing$, and $ \\bigcup_{L \\in \\mc{L}} L = \\mc{X} $. This $ \\mc{L} $ is learned on data $\\{X_i, Y_i\\}_{i=1}^n$ by recursively splitting a randomly selected subsample along a single dimension of the input data based on an impurity measure \\citep{Breiman2001}, such as Gini impurity or entropy. The trees are grown until nodes reach a certain criterion (for example, a minimum number of samples). The bottom-most nodes are called ``leaf nodes''. Additionally, only a random subset of dimensions of $X$ are considered for the split at each node.\n\nGiven a partition $ \\mc{L} $, let $L(x)$ be the part of $ \\mc{L} $ to which $x \\in \\mc{X}$ belongs. Letting $\\ensuremath{\\mathbbm{1}}[\\cdot]$ be the indicator function, a possible predictor function $\\hat{g}$ for classification is $\\hat{g}(x) = \\operatornamewithlimits{argmax}_{y \\in \\mc{Y}} \\sum_{i=1}^n \\ensuremath{\\mathbbm{1}}[Y_i = y \\text{ and } X_i \\in L(x)]$. This is the plurality vote among the training data in the leaf node of $x$. For a regression task, a corresponding predictor $\\hat{\\mu}$ is $\\hat{\\mu}(x) = \\frac{\\sum_{i = 1}^n Y_i \\cdot \\ensuremath{\\mathbbm{1}}[X_i \\in L(x)]}{\\sum_{i = 1}^n \\ensuremath{\\mathbbm{1}}[X_i \\in L(x)]}$, which is the average $Y$ value for the training data in $L(x)$. For a forest, $B$ trees are learned on randomly subsampled points. These points used in tree construction are called the `in-bag' samples for that tree, while those that are left out are called `out-of-bag' samples.\n\n\\vspace{-6pt}\n\n\\subsection{Honest Sampling, Balanced Trees, and Random-Split Trees} In decision forest algorithms, because data are typically split to maximize purity within child nodes, the posteriors estimated in each cell tend to be biased toward certainty. Honesty, as shown in \\citet{cart}, helps in bounding the bias of tree-based estimates of posterior class probabilities \\citep{Athey2019-uw,Wager2018}. \\citet{Wager2018} describes honest trees as those for which any particular training example $(X_i, Y_i)$ is used to partition feature space, or to estimate the quantity of interest, but not both. This property can be achieved in (at least) two ways. The first method is splitting the observed sample into two sets, one set for learning the partitions of feature space $\\mathcal{X}$, and one set to make vote on the plurality or average within each leaf node. \nWe refer to them as the ``partition\" set $\\mathcal{D}^{\\text{P}}$ and ``voting\" set $\\mathcal{D}^{\\text{V}}$, respectively.\n(\\citet{denil14} called them ``structure'' and ``estimation'', which we avoid because both sets are used to estimate different quantities).\n\nFor example, say we wish to estimate the conditional mean function $\\mu(x) = \\mathbb{E}[Y \\mid X = x]$. Let $\\mathcal{L}$ be a partition of feature space $\\mc{X}$, as described in Section \\ref{sec:cart}. Letting $m < n$, such an $\\mc{L}$ can be learned via a decision tree with $\\mathcal{D}^{\\text{P}} = \\{(X_1,Y_1),...,(X_m,Y_m)\\}$. This leaves $\\mathcal{D}^{\\text{V}} = \\{(X_{m+1},Y_{m+1}),...,(X_n,Y_n)\\}$. Letting $L(x)$ be the part of $\\mathcal{L}$ to which $x$ belongs, the conditional mean estimate can be $\\hat{\\mu}(x) = \\frac{\\sum_{\\mathcal{D}^{\\text{V}}} Y_i \\cdot \\ensuremath{\\mathbbm{1}}[X_i \\in L(x)]}{\\sum_{\\mathcal{D}^{\\text{V}}} \\ensuremath{\\mathbbm{1}}[X_i \\in L(x)]} = \\frac{\\sum_{i = m + 1}^n Y_i \\cdot \\ensuremath{\\mathbbm{1}}[X_i \\in L(x)]}{\\sum_{i = m + 1}^n \\ensuremath{\\mathbbm{1}}[X_i \\in L(x)]}$.\nThis method can be applied at the forest level, that is, using the same partition points to learn every decision tree, or at the tree level, in which the data is randomly partitioned into partition and voting sets in each tree.\n\\citet{denil14} has shown that tree level splitting increases performance for low sample sizes, while for higher sample sizes the performance difference is indistinguishable. A second way of achieving honesty is by ignoring the responses of interest $Y_i$ and only using $X_i$ and any auxiliary variables $W_i$ to place splits in each decision tree. \\citet{Wager2018} proposes an example of this method in estimating heterogeneous treatment effects via random forest. \n\nBalanced trees are described by \\citet{Wager2018} and \\citet{Athey2019-uw} as $\\alpha$-regular, meaning that at each split in a tree, at least an $\\alpha$ fraction of the data are placed in each child node. Finally, trees for which each dimension has a nonzero chance of being used for the split are called random-split. This can be achieved, for example, by randomly choosing one candidate dimension uniformly at each split. For honest, balanced, random-split trees, consistency can be shown for a very general class of random forest estimates \\citep{Athey2019-uw}.\n\n\\section{Uncertainty Forests}\n\\label{sec:uf}\n\nGiven observations $\\mathcal{D}_n = \\{(X_1, Y_1), ..., (X_n, Y_n)\\}$, the goal is to estimate conditional entropy $H(Y|X) = \\mathbb{E}_{X'} [H(Y|X = X')$ and use that to estimate mutual information. UF provides a consistent estimate of the conditional entropy, and an empirically non-biased estimate with sufficient sample size. \n\nUF partitions the data into three sets: a partition set $\\mathcal{D}^{\\text{P}}$, a voting set $\\mathcal{D}^{\\text{V}}$, and evaluation set $\\mathcal{D}^{\\text{E}}$. $\\mathcal{D}^{\\text{P}}$ and $\\mathcal{D}^{\\text{V}}$ are used to learn low-bias posteriors $\\hat{p}(y \\mid x)$, and consequently the function $\\hat{H}(Y \\mid X = x)$. $\\mathcal{D}^{\\text{E}}$ will be used to estimate its expectation $\\ensuremath{\\mathbb{E}}_{X'}[H(Y \\mid X = X')]$. Note that for forest-level evaluation, $\\mathcal{D}^{\\text{E}}$ {\\bf need not be labeled}, as only the $x$ values are necessary to evaluate the function $H(Y \\mid X = x)$. If data is difficult to label, UF can effectively leverage unlabelled data in a semisupervised fashion.\n\n\\subsection{Algorithm}\n\n\\begin{enumerate}[start=0]\n \\item Set hyper-parameters including tree convergence criteria, number of trees $B$, and finite sample correction coefficient $\\kappa$, and (possibly unlabelled) evaluation set $\\mathcal{D}^{\\text{E}}$.\n \\item \\textbf{Build Trees}\\\\ For each tree $b$ from 1 to $B$ (the maximum number of trees):\n \\begin{enumerate}\n \\item Randomly subsample $s \\leq n$ data points from $\\mathcal{D}_n - \\mathcal{D}^{\\text{E}}$.\n \\item Randomly split the $s$ data points into a partition set $\\mathcal{D}^{\\text{P}}_b$ and voting set $\\mathcal{D}^{\\text{V}}_b$.\n \\item Using $\\mathcal{D}^{\\text{P}}_b$, learn a decision tree partition $ \\mc{L}_b $. $L_b(x)$ is the leaf node of $ \\mc{L}_b $ that $x$ ``falls\" into. \n \\item Using the $\\mathcal{D}^{\\text{V}}_b$, letting $N_b(x) = \\sum_{i \\in \\mathcal{D}^{\\text{V}}}\\ensuremath{\\mathbbm{1}}[X_i \\in L_b(x)]$ be the leaf size of $L_b(x)$, estimate the conditional probability of $Y = y$ given $X = x$ by $\\tilde{p}_b(y \\mid x) = \\frac{1}{N_b(x)} \\sum_{i \\in \\mathcal{D}^{\\text{V}}} \\ensuremath{\\mathbbm{1}}[Y_i = y \\text{ and } X_i \\in L_b(x)]$.\n This is the empirical frequency of $y$ (given by the voting data) in the leaf node of $x$.\n \\item When $Y$ is categorical, all samples in a leaf estimator may belong to one class even though the probabilities for other classes are nonzero, which biases finite-sample estimates of the conditional distribution. UF remedies this by adapting a robust finite sampling technique described in Vogelstein et al. \\citep{Vogelstein13}. We first replace all zero probabilities with $1\/ (\\kappa N_b(x))$ (where $\\kappa > 0$ is some suitably chosen constant), and renormalize the probabilities. The finite-sample corrected estimate is thus $\\hat{p}_b(y \\mid x) = \\frac{\\tilde{p}_b(y \\mid x)}{\\sum_{k=1}^K \\tilde{p}_b(k \\mid x)}$.\n If $N_b(x) \\overset{P}{\\rightarrow} \\infty$, then $\\hat{p}_b(y \\mid x)$ approaches $\\tilde{p}_b(y \\mid x)$.\n \\end{enumerate}\n \\item \\textbf{Estimate conditional entropy function} \n \\begin{enumerate}\n \\item Average all of the posterior estimates from each tree: $\\hat{p}(y \\mid x) = \\frac{1}{B} \\sum_{b = 1}^B \\hat{p}_b(y \\mid x)$.\n \\item Set the conditional entropy function estimator $\\hat{H}(Y\\mid X = x) = -\\sum_{y \\in [K]} \\hat{p}(y \\mid x) \\log \\hat{p}(y \\mid x)$.\n \\end{enumerate}\n \\item \\textbf{Compute conditional entropy and mutual information} \n \\begin{enumerate}\n \\item Evaluate at every point in $\\mathcal{D}^{\\text{E}}$ and average to yield $\\hat{H}(Y \\mid X) = \\frac{1}{|\\mathcal{D}^{\\text{E}}|} \\sum_{i \\in \\mathcal{D}^{\\text{E}}} \\hat{H}(Y\\mid X = \n X_i)$.\n \\item Letting $\\hat{p}(y) = \\frac{1}{n} \\sum_{i \\in \\mathcal{D}_n} \\ensuremath{\\mathbbm{1}}[Y_i = y]$, estimate $H(Y)$ with $\\hat{H}(Y) = -\\sum_{y \\in [K]} \\hat{p}(y) \\log \\hat{p}(y)$.\n \\item Let $\\hat{I}(X; Y) = \\hat{H}(Y) - \\hat{H}(Y \\mid X)$.\n \\end{enumerate}\n\\end{enumerate}\n\nUF uses basic CART for tree construction, which has a number of hyper-parameters. These include which objective function to optimize for each split, how many samples to use to generate each tree ($s$), \nminimum samples in a leaf before splitting ($k$), \nmaximum tree depth, and number of trees ($B$). Random forest has been shown in practice to be very robust to hyperparameters \\citep{probstTunability}. For UF, we do not impose a maximum depth and use general rules-of-thumb for the other choices (minimize Gini impurity, $k=1$, $B = 300$). We use $|\\mc{D}^{\\text{P}}| = 0.4 \\cdot n$, $|\\mc{D}^{\\text{V}}| = |\\mc{D}^{\\text{E}}| = 0.3 \\cdot n$ in experiments ($s = |\\mc{D}^{\\text{P}}| + |\\mc{D}^{\\text{V}}|$). The UF pseudocode is described in detail in the supplementary material. In experiments, we find that letting the evaluation set $\\mathcal{D}^{\\text{E}}$ be randomly sampled at the tree-level performs better than holding out a set that is common for all trees. We also use $\\kappa = 3$, after sweeping over choices in $[0.1, 10]$. Letting $T$ be the number of threads, the learning time complexity is $O(Bdn \\log^2(n))$, the memory complexity is $O(B(nd + K))$, and the storage complexity is $O(Bn)$. \\cite{sporf}\n\n\\subsection{Consistency of Uncertainty Forest Estimates}\n\nUnder mild conditions, UF provides a consistent estimate of conditional probability, conditional entropy, and mutual information. All proofs are collected in the supplementary material. Assume that $\\mathcal{Y}$ is discrete ($\\mathcal{Y} = [K]$), and that $\\mathcal{X} = \\mathbb{R}^d$. Let $\\hat{H}_n(Y \\mid X)$ be the conditional entropy estimate and $\\hat{I}_n(X; Y)$ be the mutual information estimate, now indexed by $n$ for explicitness. Suppose that $|\\mathcal{D}^{\\text{E}}| = \\gamma n$ for $0 < \\gamma < 1$. Suppose that the subsample size $s_n$ is chosen such that $s_n \\rightarrow \\infty$ and $\\frac{s_n}{n} \\rightarrow 0$. Consider the following assumptions.\n\\begin{assumption}\n \\label{thm:support}\n Suppose that $X$ is supported on $\\mathbb{R}^d$ and has a density which non-zero almost everywhere. This is equivalent to have $X$ be uniformly distributed in $[0, 1]^d$ due to the monotone invariance of trees \\citep{denil14}.\n\\end{assumption}\n\\begin{assumption}\n \\label{thm:lipshitz}\n Suppose that for each $y \\in [K]$ the conditional probability $p(y \\mid x)$ is Lipshitz continuous on $\\mathbb{R}^d$.\n\\end{assumption}\n\\begin{assumption}\n \\label{asm:bigcells}\n Each tree $b$ is constructed such that for all $x \\in \\mathcal{X}, \\epsilon > 0$, $P[N_b(x) < \\epsilon] \\rightarrow 0$ as $n \\rightarrow \\infty$.\n\\end{assumption}\n\\begin{theorem}[Consistency of the conditional entropy estimate] Given Assumptions \\ref{thm:support}-\\ref{asm:bigcells}, the conditional entropy estimate is consistent as $n \\rightarrow \\infty$, that is, $\\hat{H}_n(Y \\mid X) \\overset{P}{\\rightarrow} H(Y \\mid X)$.\n \\label{thm:cond_ent}\n\\end{theorem}\nThis result states that the estimate $\\hat{H}(Y \\mid X)$ is arbitrarily close in probability to the true $H(Y \\mid X)$ for sufficiently large $n$. As a simple consequence, due to the consistency of the maximum-likelihood estimate $\\hat{H}(Y)$ for $H(Y)$ based on empirical frequencies, we have Theorem \\ref{thm:mutual_info}. Both results rely heavily on recent theoretical work by \\citet{Athey2019-uw}.\n\\begin{theorem}[Consistency of the mutual information estimate] With the same assumptions as in Theorem~\\ref{thm:cond_ent}, $\\hat{I}_n(X;Y)\\overset{P}{\\rightarrow} I(X; Y)$ as $n \\rightarrow \\infty$.\n\\label{thm:mutual_info}\n\\end{theorem}\n\n\\section{Simulation Results}\n\n\\subsection{Posterior Probability Experiments}\n\n\nConsider the following setting: let each $Y_i$ be Bernoulli with 50\\% probability to be either $+1$ or $-1$; let each $X_i$ be normally distributed with mean $Y_i \\times \\mu$ and variance one, where $\\mu$ is a parameter controlling effect size. A CART random forest, a CART forest with isotonic regression calibration \\citep{isotonic}, and UF (using both honest sampling and finite sample correction) are trained on data drawn from the above distribution with $\\mu = 1$. We estimate posterior distributions and plot the posterior for class $1$ in Figure \\ref{posterior}. \n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{fig1.pdf}\n \\caption{Comparison of estimated posterior distributions using random forest algorithms. Left plots show posterior distribution of $Y=1$ given $x$ from CART, honest, and UF. Ten trials are plotted for each algorithm, the mean is highlighted. \n Right-most plot shows variance, over 100 trials, of posterior estimates vs $x$. $\\mu = 1, n = 6000$ for all plots. }\n \\label{posterior}\n\\end{figure}\nAs $x$ increases, the true probability that $Y$ is one givne $X = x$ increases. Thus, unsurprisingly, all random forest algorithms have $\\hat{P}(Y = 1 \\mid X = x)$ decrease to $0$ as $x$ becomes more negative, and increase to $1$ as $x$ becomes more positive.\nHowever, the posterior estimated from UF has significantly lower variance than both normal CART forests and isotonic regression forests (Figure \\ref{posterior}, right). \n\n\\subsection{Conditional Entropy Experiments}\n\nThese better posterior estimates from UF carry over to better estimates of conditional entropy. Figure \\ref{fig:convergence}A shows that UF estimates converge to the truth as sample size increases, whereas isotonic regression forest estimates and CART forest estimates are biased even for large sample sizes in the low-dimensional setting. All three algorithms behave as expected when the effect size ($\\mu$) increases: they all approach zero conditional entropy, as seen in Figure \\ref{fig:convergence}B. CART forests and isotonic regression forest however, exhibit a bias for small effect size.\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.75\\linewidth]{fig2.pdf}\n \\caption{Behavior of random forest estimates for conditional entropy. Top plots are for $d=1$; bottom plots are for $d=20$. The left plot shows estimates vs. increasing sample size ($\\mu = 1$). Twenty trials are plotted with high transparency to show variance. Right plot shows estimates vs. increasing $\\mu$ ($n = 3000$ for $d=1$ and $n = 6000$ for $d=20$). UF is consistent, and other approaches remain biased for very large sample sizes. UF is also closest to the truth for low effect sizes.\n }\n \\label{fig:convergence}\n\\end{figure}\nFor the high-dimensional experiment, $X_i$'s are multivariate Gaussians, where the mean of the first dimension is still $Y_i \\times \\mu$ but each additional dimension has mean $0$ and identity covariance: $X_i \\sim \\mathcal{N}(\\underbrace{(Y_i \\mu, 0, \\dots, 0)\\T}_{d}, \\underbrace{I_d}_{\\text{Identity matrix}})$,\nwhere $\\mathcal{N}(\\theta, \\Sigma)$ refers to the Gaussian distribution with mean $\\theta \\in \\mathbb{R}^d$ and covariance matrix $\\Sigma \\in \\mathbb{R}^{d \\times d}$. \nBecause each added dimension is noise, the conditional entropy does not change. This allows us to compare behavior of our forest estimates to truth \\citep{Ramdas15}.\nFigure \\ref{fig:convergence}C show that when $d=20$, the UF estimate still converges to truth as sample size increases. Interestingly, the bias of isotonic regression forests decreases in the high-dimensional setting, suggesting that in this setting the true posteriors might be (approximately) monotonic in the learned posteriors. When $d = 1$, the bias of IRF becomes worse as sample size increases.\nFigure \\ref{fig:convergence}D demonstrates that CART forests remain highly biased, whereas the other approaches lack bias in these high-dimensional settings. \n\n\n\\subsection{Mutual Information Experiments}\nWe compare UF to the KSG, mixed KSG, and isotonic regression estimators of mutual information~\\citep{ksg, mixedksg, isotonic}. Consider three simulation settings, all based on mixtures of Gaussians with various parameters. We compute normalized mutual information, $I(X; Y)\/\\min\\{H(X), H(Y)\\}$. For each setting, we consider both dimensionality $d = 1$ and increasing dimension up to $d=20$. Only the first dimension depends on the class label, each additional dimension is an independent, standard Gaussian. Because only the first dimension contains any signal, mutual information does not change with increasing dimensionality \\citep{Ramdas15}. In the case of two classes, $Y$ is drawn Bernoulli with probability $\\pi$. For the three class case, the classes are drawn multinomial from the vector $(\\pi, \\frac{1-\\pi}{2}, \\frac{1-\\pi}{2})$. We explore the robustness of the estimators both with $d = 2$ and changing class prior $\\pi$, as well as increasing dimensionality when $\\pi = \\frac{1}{2}$.\n\\begin{description}\n\\item[Spherical Gaussians] A mixture of two Gaussians, the same distribution as in Figure \\ref{fig:convergence}. The $\\mu$ parameter controls the effect size while $y \\in \\{-1, +1\\}$ controls the class label: $X \\mid Y=y \\sim \\mathcal{N}(y(\\mu , 0)\\T, I)$.\n\\item[Elliptical Gaussians] To quantify performance of MI estimators when the Bayes optimal decision boundary is not axis-aligned, consider elliptical Gaussians: $X \\mid Y=y \\sim \\mathcal{N}\\left(y (\\mu , 0)\\T, \\Sigma_y\\right)$,\nwhere $\\Sigma_{-1} =\n\\begin{bmatrix} \n3 & 0 \\\\\n0 & 1 \n\\end{bmatrix}$, and $\\Sigma_{+1} = I$.\n\\item[Three Classes] To quantify performance for greater than two classes, let $y \\in \\{0,1,2\\}$, and $X \\mid Y = y \\sim \\mathcal{N}(\\mb{\\mu}_y, I)$\nwhere $\\mb{\\mu}_0 = (0,\\mu)\\T, \\enskip \\mb{\\mu}_1= (\\mu, 0)\\T, I), \\enskip \\mb{\\mu}_2 = (-\\mu, 0)\\T$.\n\n\\end{description}\n\n\nFigure \\ref{mi} shows the performance of each estimator. When $d=2$, UF, KSG, mixed KSG, and IRF all do reasonably well, though IRF is heavily biased for various level of class balance. As dimension increases, the KSG and Mixed KSG estimators suffer a significant performance degradation. \nIn the three class case, the KSG suffers a worse bias in high-dimensional settings. On the other hand, the UF estimator maintains performance as dimensionality gets high, but incurs more bias amid severe imbalance. For these Gaussian settings, IRF actually improves its performance from low-to-high dimensional data. In comparison to other methods, UF shows strong performance in high-dimensional and moderately imbalanced settings.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.75\\linewidth]{fig3.pdf}\n \\caption{Mutual information estimates in three different (``near'') Gaussian settings. Left: an example sample for each setting. \\emph{Center}. Normalized mutual information for $d=2$ and $n = 6000$. \\emph{Right}. Normalized mutual information estimates at $n = 6000$ for various dimensions. The mean of twenty trials is plotted. Both KSG and Mixed KSG break down in the high-dimensional setting, past $d = 8$. Similar to \\ref{fig:convergence}, isotonic regression is biased in the low-dimensional setting, but improves for these distributions for high dimensions. For the two-class settings, UF successfully recovers the mutual information, while is suffers slight bias in the three-class setting. See main text for equations.\n }\n \\label{mi}\n\\end{figure}\n\n\n\n\\section{Mutual Information in Drosophila Neural Network (Connectome)}\n\nAn immediate application of our random forest estimate of conditional entropy is measuring information contained in neuron types for the larval \\textit{Drosophila} mushroom body (MB) connectome \\citep{mb}. This dataset, obtained via serial section transmission electron microscopy, provides a real and important opportunity for investigating synapse-level structural connectome modeling \\citep{Vogelstein2019-om}. This connectome consists of 213 different neurons ($n = 213$) in four distinct cell types: Kenyon Cells, Input Neurons, Output Neurons, and Projection Neurons. The connectome adjacency matrix is visualized in Figure \\ref{fig:app} (left).\n\\begin{figure}\n \\centering\n \\raisebox{-0.425\\totalheight}{\\includegraphics[width=0.35\\linewidth]{adj_matrix.pdf}}\n \\begin{tabular}{|c | c c c|}\n \\hline\n $X_{\\text{in}}$ & $\\hat{I}(Y, X_{\\text{in}})$ & $\\hat{I}(Y, X_{\\text{out}} \\mid X_{\\text{in}})$ & $\\hat{I}(Y,X)$ \\\\\n \\hline\n claw & {\\bf 0.294} & 0.622 & 0.917 \\\\\n dist & {\\bf 0.474} & 0.442 & 0.917 \\\\\n age & {\\bf 0.595} & 0.321 & 0.917 \\\\\n cluster & {\\bf 0.800} & 0.116 & 0.917 \\\\ \\hline \n cluster, claw & 0.802 & 0.114 & 0.917 \\\\\n cluster, dist & 0.845 & 0.071 & 0.917 \\\\\n cluster, age & 0.913 & 0.003 & 0.917 \\\\\n cluster, claw, dist & 0.847 & 0.069 & 0.917 \\\\\n cluster, claw, age & 0.917 & 0.000 & 0.917 \\\\\n cluster, dist, age & 0.587 & 0.329 & 0.917 \\\\\n \\hline\n \\end{tabular}\n \\caption{\\emph{Left} \\textit{Drosophila} larva right hemisphere connectome. Groups of neurons are labelled $K$ for Kenyon Cells, $I$ for Input Neurons, $O$ for Output Neurons, and $P$ for Projection Neurons (PN). Black cells represent the presence of an edge between the two corresponding nodes.\n \\emph{Right} Adjacency spectral embedding applied to the MB connectome shows clear cluster groups for each neuron type. This suggests a strong dependency between neuron type and neural features.}\n \\label{fig:app}\n\\end{figure}\nEach neuron comes with a mixture of categorical and continuous features: ``claw'' refers to the integer number of dendritic claws for Kenyon cells, ``dist'' refers to real distance from the neuron to the neuropil, ``age\" refers to neuron (normalized) age as a real number between -1 and 1, and ``cluster\" refers to the community detected by the latent structure model as in \\citet{lsm}. \nWe compute mutual information with $Y$ as the neuron type and $X$ as various subsets of the features. Because neuron type has been a subjective categorical assignment based on gross morphological features, we expect mutual information to be high for the entire feature vector. Running a permutation test with 1000 replicates, the test statistic $\\hat{I}_n(Y, X) = 0.913$ was found to be statistically greater than zero with a $p$-value of $0.001$. Regarding the individual features, scientific prior knowledge posits a few relationships that are confirmed by the mutual information estimates. Letting $X_{\\text{in}}$ be the subset of features under consideration, we compute $\\hat{I}(Y, X_{\\text{in}})$, the mutual information between $Y$ and the ``in\" features, and $\\hat{I}(Y, X_{\\text{out}} \\mid X_{\\text{in}})$, the additional information given by the ``out\" features (Figure \\ref{fig:app} (right)). We confirm that the estimates $\\hat{I}(Y, X_{\\text{in}}) + \\hat{I}(Y, X_{\\text{out}} \\mid X_{\\text{in}}) = \\hat{I}(X,Y) $\nadhering to the chain rule of mutual information. \nBecause only Kenyan cells have dendritic claws, this feature is unable to discriminate between the other three classes, explaining why it has the lowest mutual information estimate among the features. \nAge is typically computed using distance from the neuropil, which explains why both of these features yielded similar information regarding $Y$. \nFigure 4 of \\citet{lsm} presents compelling evidence of latent structure model clusters being closely related to cell type, which is corroborated by its highest mutual information estimate among neuron features. \nMoreover, adding one or two additional features increases our estimated MI with respect to neuron class. \n\n\\section{Conclusion}\nWe present Uncertainty Forest (UF), a nonparametric method of consistently estimating conditional entropy through randomized decision trees. Empirically, UF performs well in low- and high-dimensional settings. Furthermore, when extending our estimator to estimate mutual information, UF performs better than the mixed KSG and KSG estimators in a variety of settings. UF has strong theoretical justification in comparison to calibration methods such as isotonic regression forests. In machine learning and statistics, this tool could be adapted for feature selection, while in biology and neuroscience, scientists can find quantify uncertainty between various properties and labels.\n\nThe main limitation of this work is that it is only able to estimate uncertainty quantities for categorical $Y$; that said, the UF algorithm can be modified for continuous $Y$ as well.\nComputing the posterior distribution $\\hat P(Y|X = x)$ when $Y$ is continuous can be accomplished with a kernel density estimate instead of simply binning the probabilities. A theoretical and empirical analysis of this extension is of interest. When $Y$ is multivariate, a heuristic approach such as subsampling $Y$ dimensions or using multivariate random forests can be explored.\n\nOn the theoretical side, important next steps include rigorous proofs for convergence rates. Studying the behavior of UF estimates in more complicated nonlinear, high-dimensional settings should be explored as well. Practical applications such as dependence testing and $k$-sample testing for high-dimensional, nonlinear data will be natural applications for these information theoretic estimates.\n\n\\paragraph{Data and Code Availability Statement}\n\\label{sec:repo}\n\nThe implementation of UF, the simulated and real data experiments, and their visualization can be reproduced completely with the instructions in code available on \\href{https:\/\/github.com\/neurodata\/uncertainty-forest}{GitHub}. The experiments were run in parallel on a 1TB RAM machine with 96-cores.\n\n\\nocite{*}\n\n\\section*{Acknowledgements}\n\n\\input{acknowledgements}\n\n\\vspace{5mm}\n\n\n\n\\section{Pseudocode}\n\\label{sec:code}\nThe \\textproc{FitRandomForest}, \\textproc{GetInBagSamples}, and \\textproc{ApplyTree} operations, as well as the \\textproc{NumClasses} and \\textproc{NumLeaves} fields are all standard functions in the \\texttt{scikit-learn} decision tree and bagging classifier modules \\citep{scikit-learn}. \n\\begin{algorithm}\n \\caption{Uncertainty Forest (Forest-level)}\n \\label{alg:cerfestimate}\n\\begin{algorithmic} \n\\Require Training set $\\mathcal{D}_n$ and evaluation set $\\mathcal{D}^{\\text{E}}$, $\\theta= \\{$number of trees $B$, minimum leaf size $k$, subsample size $s$ $\\}$, finite correction constant $\\kappa$.\n\\Ensure Conditional entropy estimate $\\hat{H}(Y \\mid X)$.\n\\Function{UncertaintyForest}{$\\mathcal{D}_n$, $\\mathcal{D}^{\\text{E}}$, $\\theta$, $\\kappa$}\n \\State $\\mc{T} = \\mathcal{D}_n - \\mathcal{D}^{\\text{E}}$\n \\State model = \\textproc{FitRandomForest}$(\\mc{T}, \\theta)$.\n \\State posteriors = [].\n \\For{tree $b$ \\textbf{in} model}\n \\State posterior = \\textproc{EstimatePosterior}($\\mc{T}$, model, $\\kappa$, $b$)\n \\State posteriors.append(posterior)\n \\EndFor\n \\State $\\hat{H}(Y \\mid X)$ = \\textproc{EvaluatePosteriors}($\\mathcal{D}^{\\text{E}}$, model, posteriors).\n \\State \\Return $\\hat{H}(Y \\mid X)$.\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}\n \\caption{Uncertainty Forest (Tree-level)}\n \\label{alg:cerfestimate2}\n\\begin{algorithmic} \n\\Require Training set $\\mathcal{D}_n$ and evaluation set $\\mathcal{D}^{\\text{E}}$, $\\theta= \\{$number of trees $B$, minimum leaf size $k$, subsample size $s$ $\\}$, finite correction constant $\\kappa$.\n\\Ensure Conditional entropy estimate $\\hat{H}(Y \\mid X)$.\n\\Function{UncertaintyForest}{$\\mathcal{D}_n$, $\\mathcal{D}^{\\text{E}}$, $\\theta$, $\\kappa$}\n \\State $\\mc{T} = \\mathcal{D}_n - \\mathcal{D}^{\\text{E}}$\n \\State model = \\textproc{FitRandomForest}$(\\mc{T}, \\theta)$.\n \\State conditional\\_entropies = []\n \\For{tree $b$ \\textbf{in} model}\n \\State posterior = \\textproc{EstimatePosterior}($\\mc{T}$, model, $\\kappa$, $b$)\n \\State conditional\\_entropies.append(\\textproc{EvaluatePosteriors}($\\mathcal{D}^{\\text{E}}$, [model[$b$]], [posterior])).\n \\EndFor\n \\State $\\hat{H}(Y \\mid X)$ = conditional\\_entropies.\\textproc{Mean}().\n \\State \\Return $\\hat{H}(Y \\mid X)$.\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}\n \\caption{Posterior Estimation}\n \\label{alg:est_posterior}\n\\begin{algorithmic} \n\\Require training set $\\mc{T}$ (full dataset without evaluation), fitted random forest model, finite correction constant $\\kappa$, tree index $b$\n\\Ensure Posterior probability estimate tree $b$.\n\\Function{EstimatePosterior}{$\\mc{T}$, model, $\\kappa$, $b$}\n \\State $K$ = model.\\textproc{NumClasses}.\n \\State $\\mathcal{D}_\\text{P}$ = \\textproc{GetInBagSamples}(model, $b$)\n \\State $\\mathcal{D}_\\text{V} = \\mc{T} - \\mathcal{D}_\\text{P}$.\n \\State $L$ = model[$b$].\\textproc{NumLeaves}.\n \\State vote\\_counts = $[0]^{L \\times K}$.\n \\For{observation $(x, y)$ \\textbf{in} $\\mathcal{D}_\\text{V}$}\n \\State $l$ = model[$b$].\\textproc{ApplyTree}($x$)\n \\State vote\\_counts[$l$, $y$] = vote\\_counts[$l$, $y$] + 1.\n \\EndFor\n \\State leaf\\_sizes = \\textproc{RowSum}(vote\\_counts).\n \\State posterior = $[0]^{L \\times K}$.\n \\For{leaf index $l$ \\textbf{in} $[L]$}\n \\For{class $y$ \\textbf{in} $[K]$}\n \\State posterior[$l$, $y$] = vote\\_counts[$l$, $y$] \/ leaf\\_sizes[$l$].\n \\EndFor\n \\EndFor\n \\State posterior = \\textproc{FiniteSampleCorrect}(posterior, $\\kappa$, leaf\\_sizes)\n \\State \\Return posterior\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}\n \\caption{Posterior Evaluation}\n \\label{alg:eval_posterior}\n\\begin{algorithmic} \n\\Require evaluation set $\\mathcal{D}^{\\text{E}}$, fitted random forest model, posteriors for each tree\n\\Ensure Conditional entropy estimate $\\hat{H}(Y \\mid X)$.\n\\Function{EvaluatePosteriors}{$\\mathcal{D}^{\\text{E}}$, model, posteriors}\n \\State $B$ = model.\\textproc{NumTrees}.\n \\For{observation $x_i$ \\textbf{in} $\\mathcal{D}^{\\text{E}}$}\n \\For{tree $b$ \\textbf{in} model}\n \\State $l$ = model[$b$].\\textproc{ApplyTree}($x$).\n \\For{class $y$ \\textbf{in} $[K]$}\n \\State posterior = posteriors[$b$].\n \\State $p_b(y \\mid x_i)$ = posterior[$l$, $y$].\n \\EndFor\n \\EndFor\n \\State $p_i(y \\mid x) = \\frac{1}{B} \\sum_{b=1}^B p_b(y \\mid x_i)$.\n \\State $\\hat{H}_i(Y \\mid X) = -\\sum_{y \\in [K]}p_i(y \\mid x) \\log p_i(y \\mid x)$.\n \\EndFor\n \\State $\\hat{H}(Y \\mid X) = \\frac{1}{|\\mathcal{D}^{\\text{E}}|}\\sum_{i \\in \\mathcal{D}^{\\text{E}}} \\hat{H}_i(Y \\mid X)$.\n \\State \\Return $\\hat{H}(Y \\mid X)$.\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}\n \\caption{Finite Sample Correction}\n \\label{alg:finite}\n\\begin{algorithmic} \n\\Require posterior, finite correction constant $\\kappa$, leaf\\_sizes\n\\Ensure Corrected posterior.\n\\Function{FiniteSampleCorrect}{posterior, $\\kappa$, leaf\\_sizes}\n \\State $(L, K)$ = posterior.\\textproc{Shape}.\n \\For{node index $l$ \\textbf{in} $[L]$}\n \\For{class $y$ \\textbf{in} $[K]$}\n \\If{posterior[$l$, $y$] == 0} \n \\State posterior[$l$, $y$] = $\\frac{1}{\\kappa \\cdot \\text{leaf\\_sizes}[l]}$\n \\EndIf\n \\EndFor\n \\EndFor\n \\State normalizing\\_factor = \\textproc{RowSum}(posterior).\n \\For{node index $l$ \\textbf{in} $[L]$}\n \\For{class $y$ \\textbf{in} $[K]$}\n \\State posterior[$l$, $y$] = posterior[$l$, $y$] \/ normalizing\\_factor[$l$]\n \\EndFor\n \\EndFor\n \\State \\Return posterior.\n\\EndFunction\n\\end{algorithmic}\n\\end{algorithm}\n\nNote that $ \\mathcal{D}^{\\text{E}}$ can contain unlabelled data, as only the $x_i$ values are used.\n\\section{Proofs}\n\\label{sec:proof}\n\\setcounter{theorem}{0}\n\\setcounter{assumption}{0}\n\nIn this section, we present consistency results regarding estimation of conditional entropy and mutual information via Uncertainty Forest. The argument follows as a nearly direct consequence of \\citet{Athey2019-uw}, in which random forests that are grown according to some specifications, and solve locally weighted estimating equations are consistent and asymptotically Gaussian. We review the assumptions.\n\\begin{assumption}\n Suppose that $X$ is supported on $\\mathbb{R}^d$ and has a density which non-zero almost everywhere. This is equivalent to have $X$ be uniformly distributed in $[0, 1]^d$ due to the monotone invariance of trees \\citep{denil14}.\n\\end{assumption}\n\\begin{assumption}\n Suppose that for each $y \\in [K]$ the conditional probability $p(y \\mid x)$ is Lipshitz continuous on $\\mathbb{R}^d$.\n\\end{assumption}\n\\begin{assumption}\n Each tree $b$ is constructed such that for all $x \\in \\mathcal{X}, \\epsilon > 0$, $P[N_b(x) < \\epsilon] \\rightarrow 0$ as $n \\rightarrow \\infty$.\n \n\\end{assumption}\n\nNext, we summarize the notation and main result of \\citet{Athey2019-uw}. Let $\\theta(x)$ be a parameter that is implicitly defined as the solution to some equation $M_\\theta(x) = 0$. For example, in estimating the conditional mean $\\theta(x) = \\ensuremath{\\mathbb{E}}[Y \\mid X = x]$, we can use\n\\begin{align*}\n M_\\theta(x) = \\ensuremath{\\mathbb{E}}[Y \\mid X = x] - \\theta(x) = 0.\n\\end{align*}\nTo simplify notation, we suppress the dependency of $x$ in $\\theta$ and simply write $\\theta=\\theta(x)$ wherever this dependency can be deduced from the context. To develop a sample estimate for $\\theta$, we start with a sample score function $\\psi_\\theta$ such that\n\\begin{align*}\n \\ensuremath{\\mathbb{E}}[\\psi_\\theta(Y) \\mid X = x] = M_\\theta(x).\n\\end{align*}\nNow, let us be given a dataset $\\{(X_1, Y_1), ..., (X_n, Y_n)\\}$. The sample estimate $\\hat{\\theta}$ solves a locally weighted estimating equation\n\\begin{align*}\n \\sum_{i=1}^n \\alpha_i(x) \\psi_\\theta(Y_i) = 0.\n\\end{align*}\nIn the case of conditional mean estimation, we will have $\\psi_\\theta(Y) = Y - \\theta$. The $\\alpha_i(x)$'s weigh highly observations with $X_i$ close to $x$, as to approximate the conditional expectation of $\\psi_\\theta(Y)$, or $M_\\theta(x)$. In a random forest, these weights amount to being the empirical probability that the test point $x$ shares a leaf with each training point $X_i$. Under mild assumptions, \\citet{Athey2019-uw} have shown such a method to be consistent for the estimation of $\\theta(x)$. For such a result to be used for our purpose of estimating conditional entropy, we define $M_\\theta(x)$ as \n\\begin{align*}\n M_\\theta(x) &= p(y \\mid x) - \\theta\n\\end{align*}\nUsing this definition, we can derive consistency of $\\hat{H}(Y \\mid X = x)$. Given this function of $x$, we have the conditional entropy written as\n\\begin{align*}\n H(Y \\mid X) &= \\ensuremath{\\mathbb{E}}_{X'}[H(Y \\mid X = X')]\n\\end{align*}\n\nWe first confirm that the finite-sample corrected posterior $\\hat{p}_b(y \\mid x)$ approaches the uncorrected $\\tilde{p}_b(y \\mid x)$ for large $n$. \n\\begin{lemma} \n\\label{thm:finite}\nAs $n \\rightarrow \\infty$\n\\begin{align*}\n |\\hat{p}(y \\mid x) - \\tilde{p}(y \\mid x)| & \\overset{P}{\\rightarrow} 0\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\n We index the estimates with $n$ to make explicit the dependence on sample size. For tree $b$, in the cases where $0 < \\tilde{p}_{b,n}(y \\mid x) < 1$ for all $y$, we have |$\\hat{p}_{b,n}(y \\mid x) - \\tilde{p}_{b,n}(y \\mid x)| = 0$. Otherwise, for $\\mathcal{N} = \\{y : \\tilde{p}_{b,n}(y \\mid x) = 0\\}$ and $y \\in \\mathcal{N}$, then\n \\begin{align*}\n |\\hat{p}_{b,n}(y \\mid x) - \\tilde{p}_{b,n}(y \\mid x)| &= \\frac{1}{\\kappa N_{b,n}(x)}.\n \\end{align*}\n By assumption \\ref{asm:bigcells}, we have that $N_{b,n}(x) \\overset{n}{\\rightarrow} \\infty$ for all $x$. Thus,\n \\begin{align*}\n \\frac{1}{\\kappa N_{b,n}(x)} \\overset{P}{\\rightarrow} 0\n \\end{align*}\n Similarly, the constant $c = \\frac{|\\mathcal{N}|}{\\kappa N_{b,n}(x)} \\leq \\frac{K}{\\kappa N_{b,n}(x)} \\rightarrow 0$ in probability. Thus, for $y \\not\\in$ $\\mathcal{N}$,\n \\begin{align*}\n |\\hat{p}_{b,n}(y \\mid x) - \\tilde{p}_{b,n}(y \\mid x)|\n &= |(1-c)\\tilde{p}_{b,n}(y \\mid x) - \\tilde{p}_{b,n}(y \\mid x)|\\\\\n &= |c \\cdot \\tilde{p}_{b,n}(y \\mid x)| \\leq c \\rightarrow 0\n \\end{align*}\n as $n \\rightarrow \\infty$. The estimate\n $\\hat{p}_n(y \\mid x) = \\frac{1}{B} \\sum_{b=1}^B \\hat{p}_{b,n}(y \\mid x)$ is a finite average of the $\\hat{p}_{b,n}(y \\mid x)$. Given $\\epsilon > 0$,\n \\begin{align*}\n P[|\\hat{p}_n(y \\mid x) - \\tilde{p}_n(y \\mid x)| > \\epsilon] \n &= P[|\\tfrac{1}{B}\\sum_{b=1}^B \\hat{p}_{b,n}(y \\mid x) - \\tilde{p}_{b,n}(y \\mid x)| > \\epsilon]\\\\ \n &\\leq P[\\sum_{b=1}^B |\\hat{p}_{b,n}(y \\mid x) - \\tilde{p}_{b,n}(y \\mid x)| > B\\epsilon]\\\\ \n &\\leq P\\left[\\bigcup_{b=1}^B |\\hat{p}_{b,n}(y \\mid x) - \\tilde{p}_{b,n}(y \\mid x)| > \\epsilon\\right]\\\\\n &\\leq \\sum_{b=1}^B P[|\\hat{p}_{b,n}(y \\mid x) - \\tilde{p}_{b,n}(y \\mid x)| > \\epsilon]\\\\\n &= B \\cdot P[|\\hat{p}_{1,n}(y \\mid x) - \\tilde{p}_{1,n}(y \\mid x)| > \\epsilon]\\\\\n &= B \\cdot \\big( P[|\\hat{p}_{1,n}(y \\mid x) - \\tilde{p}_{1,n}(y \\mid x)| > \\epsilon \\cap \\tilde{p}_{1,n}(y \\mid x) = 0]\\\\\n &\\quad + P[|\\hat{p}_{1,n}(y \\mid x) - \\tilde{p}_{1,n}(y \\mid x)| > \\epsilon \\cap \\tilde{p}_{1,n}(y \\mid x) \\neq 0] \\big)\\\\\n &\\overset{n}{\\rightarrow} 0\n \\end{align*}\n\\end{proof}\n\nSecond, we require that honestly estimated posteriors $\\tilde{p}_n(y \\mid x)$ (without the finite sample correction) converge to the true posteriors. This can be done by application of Theorem 3 from \\citet{Athey2019-uw}.\n\n\\begin{lemma}\n \\label{thm:prob}\n For all $y \\in [K]$ and $x \\in \\mathbb{R}^d$, $\\tilde{p}_n(y \\mid x) \\overset{P}{\\rightarrow} p(y \\mid x)$ as $n \\rightarrow \\infty$.\n\\end{lemma}\n\\begin{proof}\n For a fixed $(x, y)$, we can define $\\theta$ as the solution to\n \\begin{align*}\n M_\\theta(x) = p(y \\mid x) - \\theta = 0.\n \\end{align*}\n The sample equivalent is thus\n \\begin{align*}\n \\psi_\\theta(Y) = \\ensuremath{\\mathbbm{1}}[Y = y] - \\theta,\n \\end{align*}\n and we have that\n \\begin{align*}\n \\ensuremath{\\mathbb{E}}[\\psi_\\theta(Y) \\mid X = x] = M_\\theta(x).\n \\end{align*}\n Because our forest is learned according to Specification 1 in \\citet{Athey2019-uw}, we must confirm Assumptions 1 - 6 from Section 3 of that paper to apply their result. Only Assumption 1 is a proper assumption on the distribution, whereas the remaining are confirmed true based on our choice of $M_\\theta$ and $\\psi_\\theta$.\n \\begin{enumerate}\n \\item For fixed $\\theta$, $M_\\theta(x) = p(y \\mid x) - \\theta$ is Lipschitz in $x$. We took this as a standard assumption on the joint distribution of $(X,Y)$.\n \\item For fixed $x$, $M_\\theta(x)$ is twice continuously differentiable in $\\theta$, $\\frac{\\partial^2 M_\\theta}{\\partial \\theta^2} = 0$ and $\\frac{\\partial}{\\partial \\theta}M_\\theta = -1$, which is invertible.\n \\item The function $\\psi_\\theta$ satisfies\n \\begin{align*}\n \\sup_{x \\in \\mathcal{X}}(\\text{Var}[\\psi_\\theta(Y) - \\psi_\\theta(Y)]) &= \\sup_{x \\in \\mathcal{X}}(\\text{Var}[\\ensuremath{\\mathbbm{1}}[Y = y] - \\theta - (\\ensuremath{\\mathbbm{1}}[Y = y] - \\theta')])\n &= \\sup_{x \\in \\mathcal{X}}(\\text{Var}[\\theta' - \\theta'])\n &= 0,\n \\end{align*}\n and hence $\\sup_{x \\in \\mathcal{X}}(\\text{Var}[\\psi_\\theta(Y) - \\psi_{\\theta'}(Y)]) \\leq L||\\theta - \\theta'||$ for all $\\theta, \\theta'$ and some $L \\geq 0$.\n \\item The function $\\psi_\\theta$ is itself Lipschitz in $\\theta$.\n \\item The solution of\n \\begin{align*}\n \\sum_{i=1} \\alpha_i(x) \\psi_{\\theta}(Y_i) = 0\n \\end{align*}\n in $\\theta$ exists, and is equal to\n \\begin{align*}\n \\hat{\\theta}(x) = \\sum_{i=1}^n \\alpha_i(x) \\ensuremath{\\mathbbm{1}}[Y_i = y]\n \\end{align*}\n \\item The function $\\psi_\\theta$ is the negative subgradient of a convex function, and $M_\\theta$ is the negative subgradient of a strongly convex function (with respect to $\\theta$). Choose\n \\begin{align*}\n \\bf{\\Psi}(\\theta) &= \\frac{1}{2}(\\ensuremath{\\mathbbm{1}}[Y = y] - \\theta)^2,\\\\\n \\bf{M}(\\theta) &= \\frac{1}{2}(p(y \\mid x) - \\theta)^2\n \\end{align*}\n as these functions.\n \\end{enumerate}\n Granted the above assumptions, by Theorem 3 of \\citet{Athey2019-uw} we have for every $x, y$,\n \\begin{align*}\n \\hat{p}_n(y \\mid x) \\overset{P}{\\rightarrow} p(y \\mid x).\n \\end{align*}\n\\end{proof}\n By Lemma \\ref{thm:prob} and continuity, we have the consistency of the honest forest estimate of conditional entropy.\n Let $\\tilde{H}(Y \\mid X = x) = -\\sum_{y \\in [K]} \\tilde{p}_n(y \\mid x) \\log \\tilde{p}_n(y \\mid x)$.\n\\begin{corollary} For each $x\\in \\mathbb{R}^d$,\n\\label{thm:no_finite}\n\\begin{align*}\n \\tilde{H}(Y \\mid X = x) \\overset{P}{\\rightarrow} H(Y \\mid X = x).\n\\end{align*}\n\\end{corollary}\n\\begin{proof}\n The function \n \\begin{align*}\n h(p) = \\begin{cases}\n 0 &\\text{ if } p = 0\\\\\n -p \\log p &\\text{ otherwise}\n \\end{cases}\n \\end{align*}\n is continuous on $[0,1]$. Similarly, the finite sum $\\sum_{k=1}^K h(p_k)$ is continuous on $\\{(p_1, ..., p_K) : 0 \\leq p_k \\leq 1, \\sum_{k=1}^K p_k = 1\\}$. Thus, by Lemma \\ref{thm:prob} and the continuous mapping theorem, we have the desired result.\n\\end{proof}\n\n\\begin{lemma}\n \\label{thm:function}\n The Uncertainty Forest function estimate $\\hat{H}(Y \\mid X = x)$ converges in probability to $H(Y \\mid X = x)$ for each $x \\in \\mathbb{R}^d$.\n\\end{lemma}\n\\begin{proof}\n By continuity of $\\sum_{k=1}^K h(p_k)$ on $\\{(p_1, ..., p_K) : 0 \\leq p_k \\leq 1, \\sum_{k=1}^K p_k = 1\\}$, and Lemma \\ref{thm:finite} we have that for any $x$,\n \\begin{align}\n \\left|\\hat{H}(Y \\mid X = x) - \\tilde{H}(Y \\mid X = x)\\right| \\overset{P}{\\rightarrow} 0\n \\label{eqn:entropy_correction}\n \\end{align}\n Then, given $\\epsilon > 0$,\n \\begin{align*}\n P[|\\hat{H}(Y \\mid X = x) - H(Y \\mid X = x)| > \\epsilon] &\\leq P[|\\hat{H}(Y \\mid X = x) - \\tilde{H}(Y \\mid X = x)|\\\\ \n &\\quad + |\\tilde{H}(Y \\mid X = x) - H(Y \\mid X = x)| > \\epsilon]\\\\\n &\\leq P[|\\hat{H}(Y \\mid X = x) - \\tilde{H}(Y \\mid X = x)| > \\frac{\\epsilon}{2}]\\\\\n &\\quad + P[|\\tilde{H}(Y \\mid X = x) - H(Y \\mid X = x)| > \\frac{\\epsilon}{2}]\\\\\n &\\overset{n}{\\rightarrow} 0,\n \\end{align*}\n by Lemma \\ref{thm:no_finite} and \\ref{eqn:entropy_correction}.\n\\end{proof}\n\\setcounter{theorem}{0}\n\nFor simplicity of notation, let $n$ refer only to the total sample size of the partition $\\mathcal{D}^{\\text{P}}$ and vote $\\mathcal{D}^{\\text{V}}$ sets. Let $\\nu$ be the size of the evaluation set $\\mathcal{D}^{\\text{E}}$. If a subset of the labelled data is being used, then this may be some fraction $0 < \\gamma < 1$ of the size of the full training data. In this case, we consider any growing evaluation set size $\\nu$. \n\n\\begin{theorem}[Consistency of the conditional entropy estimate] Suppose the conditions of the preceding lemmas. Then the conditional entropy estimate is consistent as $n, \\nu \\rightarrow \\infty$, that is,\n \\begin{equation*}\n \\hat{H}_{n, \\nu}(Y \\mid X) \\overset{P}{\\rightarrow} H(Y \\mid X).\n \\end{equation*}\n \n\\end{theorem}\n\\begin{proof}[Proof of Theorem \\ref{thm:cond_ent}]\n By Lemma \\ref{thm:function} we have the pointwise consistency of $\\hat{H}_n(Y \\mid X = x)$ (now indexed by $n$ explicitly) for $H(Y \\mid X = x)$. We compute the limits with respect to $n$ and $\\nu$ in either order to achieve the result. Let $\\hat{H}_{n, \\nu}(Y \\mid X) = \\frac{1}{\\nu} \\sum_{i \\in \\mathcal{D}^{\\text{E}}} \\hat{H}_n(Y \\mid X = X_i)$.\n \\begin{align*}\n \\lim_{\\nu \\rightarrow \\infty} \\lim_{n \\rightarrow \\infty} \\hat{H}_{n, \\nu}(Y \\mid X) &= \\lim_{\\nu \\rightarrow \\infty} \\frac{1}{\\nu} \\sum_{i \\in \\mathcal{D}^{\\text{E}}} \\lim_{n \\rightarrow \\infty} \\hat{H}_n(Y \\mid X = X_i)\\\\\n &= \\lim_{\\nu \\rightarrow \\infty} \\frac{1}{\\nu} \\sum_{i \\in \\mathcal{D}^{\\text{E}}} H(Y \\mid X = X_i)\\\\\n &= H(Y \\mid X)\n \\end{align*}\n by the Weak Law of Large Numbers, where the limits are taken in probability. On the other hand,\n \\begin{align*}\n \\lim_{n \\rightarrow \\infty} \\lim_{\\nu \\rightarrow \\infty} \\hat{H}_{n, \\nu}(Y \\mid X) &= \\lim_{n \\rightarrow \\infty} \\lim_{\\nu \\rightarrow \\infty} \\frac{1}{\\nu} \\sum_{i \\in \\mathcal{D}^{\\text{E}}} \\hat{H}_n(Y \\mid X = X_i)\\\\\n &= \\lim_{n \\rightarrow \\infty} \\ensuremath{\\mathbb{E}}_{X'}[\\hat{H}_n(Y \\mid X = X')]\\\\\n &= \\lim_{n \\rightarrow \\infty} \\int_{x \\in \\mathbb{R}^d} \\hat{H}_n(Y \\mid X = x) \\ dF_X,\n \\end{align*}\n also by the Weak Law. Because of the finiteness of $[K]$, the entropy-like term $|\\hat{H}_n(Y \\mid X = x)|$ is bounded by $\\log K$ for all $n$ and $x$. Therefore, by the Dominated Convergence Theorem,\n \\begin{align*}\n \\lim_{n \\rightarrow \\infty} \\int_{x \\in \\mathbb{R}^d} \\hat{H}_n(Y \\mid X = x) \\ dF_X &= \\int_{x \\in \\mathbb{R}^d} \\lim_{n \\rightarrow \\infty} \\hat{H}_n(Y \\mid X = x) \\ dF_X\\\\\n &= \\int_{x \\in \\mathbb{R}^d} H(Y \\mid X = x) \\ dF_X\\\\\n &= \\ensuremath{\\mathbb{E}}_{X'}[H(Y \\mid X = X')]\\\\\n &= H(Y \\mid X)\n \\end{align*}\n Thus, $\\hat{H}(Y \\mid X)$ is consistent for $H(Y \\mid X)$.\n\\end{proof}\nLetting the $n$ and $\\nu$ be a fixed fraction of the training set size is a special case of both growing to infinity.\n\nThe second result concerns the mutual information estimate. The entropy $H(Y)$ is estimated in the natural way, as in\n\\begin{align*}\n \\hat{H}(Y) &= -\\sum_{y \\in [K]} \\hat{p}(y) \\log \\hat{p}(y)\n\\end{align*}\nwhere $\\hat{p}(y) = \\frac{1}{n}\\sum_{i=1}^n \\ensuremath{\\mathbbm{1}}\\{Y_i = y\\}$ (and set to zero when appropriate). Consequently, the mutual information is estimated as\n\\begin{align*}\n \\hat{I}_{n, \\nu}(X,Y) &= \\hat{H}(Y) - \\hat{H}(Y \\mid X).\n\\end{align*}\nConsider the same assumptions as Theorem \\ref{thm:cond_ent}.\n\\begin{theorem}[Consistency of the mutual information estimate]\n $\\hat{I}_{n, \\nu}(X,Y)\\rightarrow_p I(X; Y)$ as $n, \\nu \\rightarrow \\infty$.\n\\end{theorem}\n\\begin{proof}\n For $i.i.d.$ observations of $Y_i$, it is clear that $\\hat{H}(Y)$ is a consistent estimate for $H(Y)$. By Theorem \\ref{thm:cond_ent} we have the consistency of $\\hat{H}(Y \\mid X)$ for $H(Y \\mid X)$. Thus, the consistency of $\\hat{I}(X,Y)$ follows immediately.\n\\end{proof}","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfnat b/data_all_eng_slimpj/shuffled/split2/finalzzfnat new file mode 100644 index 0000000000000000000000000000000000000000..de95598e8fc15c9df8d161102cd3440152431f6a --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzfnat @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nStochastic models have been extensively used in theoretical neuroscience since the pioneer work by Gerstein and Mandelbrot in 1964 \\cite{GernstenMandelbrot}. There they considered a Wiener process (also known as Brownian motion or Perfect-Integrate-and-Fire model) to model the voltage across the membrane. An action potential, also known as spike, is generated whenever the membrane potential reaches a certain constant threshold. After that, the membrane voltage is reset to its resting value and the evolution restarts. From a mathematical point of view, a spike is the first passage time (FPT) of a stochastic process to a constant threshold. The collection of spike epochs of a neuron, called spike train, defines a renewal process, with independent and identically distributed inter-spike intervals (ISIs). Despite the excellent fit with some experimental data, the Gerstein-Mandelbrot model was criticized because it disregards features involved in neuronal coding.\n\nA first extension, combining both mathematical tractability and biological realism, is represented by \\emph{Leaky-Integrate-and-Fire} (LIF) models \\cite{ReviewSac,Tuckwell88}. Despite some criticisms on the lack of fit of experimental data \\cite{Jolivetetal,Shinomoto}, these models are still largely used.\n\nAnother common generalization is represented by Wiener processes (or more generally LIF models) with \\emph{time-dependent threshold} \\cite{Tuckwell78,TuckwellWam}. These models can be chosen to reproduce biological features such as the afterhyperpolarization in neurons. For exponentially decaying thresholds, these processes can be used to model a neuron with an exponential time-dependent drift, as shown by Lindner and Longtin \\cite{LindnerLongtin}. They investigated the effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron \\cite{LindnerLongtin}. Using a perturbation method\n\\cite{Lindner2004b}, they derived analytical expressions of the firing statistics under the assumption that the amplitude $\\epsilon$ of the time-dependent change in the threshold is small. These statistics are useful to characterize the spontaneous neural activity and to investigate the neuronal signal transmission. In particular, they can suggest under which conditions a decaying threshold may facilitate or deteriorate signal processing by stochastic neurons. For a Wiener process, these quantities can also be obtained using the approach in \\cite{Urdapilleta}. Also this method assumes a small amplitude $\\epsilon$, but it has the advantage of providing an explicit approximation of the FPT density.\n\nHere we consider a Wiener process with exponentially decaying threshold. The first aim of the paper is to provide an alternative method to approximate the firing statistics and the FPT density for any possible amplitude $\\epsilon$, extending the results in \\cite{LindnerLongtin, Urdapilleta}. Different estimators are proposed, as mentioned in Section \\ref{Sec1a} and discussed in Section \\ref{Sec4b}. Means, variances, coefficients of variation (CVs) and distributions of the FPTs are compared on simulated data and the most suitable are recommended. A comparison with the results in \\cite{LindnerLongtin, Urdapilleta} under the assumption of a small amplitude $\\epsilon$ is also performed. The second aim of this work is the estimation of drift and diffusion coefficients of the Wiener process. Maximum likelihood and moment estimators are derived and evaluated on simulated data. Our results show a good approximation of both firing statistics and parameters of the underlying model.\n\nAlthough the considered model generates a renewal process, the proposed method can also be applied to non-renewal processes, e.g. adaptive threshold models \\cite{Chacron3,Kobayashi1}. Recently, an increasing interest arose towards these models, interest motivated by the excellent fit of the firing statistics of electrosensory neurons \\cite{Chacron2,Chacron1}. The novelty of these models is that the threshold has a jump immediately after a spike. Since the boundary depends on the previous firing epochs, the ISIs are not independent anymore. However, the distribution between two consecutive spikes, conditioned on the initial position of the threshold, is the same of that studied here. Hence, our results may represent a first step towards an understanding of the more complicated adapting-threshold models.\n\n\\subsection{Mathematical background} \\label{Sec1a}\nFPTs of diffusion processes to constant or time-dependent thresholds have been extensively studied in the literature. Explicit expressions for constant thresholds are available for the Wiener process \\cite{InverseGaussianBook,coxMiller}, for a special case of the Ornstein Uhlenbeck (OU) process \\cite{Ricciardi}, for the Cox-Ingersoll-Ross process \\cite{CapRic}, and for those processes which can be obtained from the previous through suitable measure or space-time transformations, see e.g. \\cite{Alili,CapRic,RicciardiW}. For most of the processes arising from applications and for time-varying thresholds, analytical expressions are not available.\nNumerical algorithms based on solving integral equations have been proposed in \\cite{BCCP,BNR,RicciardiNip,STZ3,Taillefumier,Telve},\nwhile approximations based on Monte-Carlo path-simulation methods in \\cite{GS, GSZ,Metzler}.\n\nA different approach to tackle the FPT problem consists in focusing directly on the two-sided boundary crossing probability (BCP), i.e. the probability that a process is constrained to be between two boundaries. If one of the boundary is set to $-\\infty$, the resulting one-sided BCP equals the survival probability of the FPT to the other boundary \\cite{Wang1997}. Explicit formulas for the BCP of a standard Wiener process for continuous and piecewise-linear thresholds are known (see \\cite{BorovkovNovikov,Novikovetal,Wang2001,Wang1997,Wang2007}). In general, the BCP of a diffusion process through an exponential decaying threshold is available only for those processes which can be expressed as a piecewise monotone functional of a standard Brownian motion. Examples are the OU process or the geometric Brownian motion with time dependent drift for specific parameter values \\cite{Wang2007}.\nThe simple but powerful idea is to approximate both one and two-sided curvilinear BCPs by similar probabilities for close boundaries of simpler form, namely $n$ piecewise-linear thresholds, whose computation of the BCP for Wiener is feasible. Under some mild assumptions, the approximated two-sided BCP converges to the original one when $n\\to\\infty$ \\cite{Wang2007}, with rate of convergence given in \\cite{BorovkovNovikov,Wang1997}.\n\nFor the exponential decaying threshold considered in this paper,\nthe convergence can be obtained by choosing piecewise linear thresholds approximating the curved boundary from above and below, with approximation accuracy given by their distance \\cite{Wang2007}. However, all the available formulas for the BCPs require either Monte-Carlo simulation methods or heavy numerical approximations.\n\nHere we consider a two-piecewise linear threshold as an approximation of the curvilinear boundary. Since $n=2$, the asymptotic convergence of the BCPs does not hold. However, we can derive analytical expression for the FPT density to the two-piecewise linear boundary, and use it as an approximation of the unknown FPT density. Four possible piecewise thresholds are proposed and optimized to minimize the distance to the original threshold.\n\n\\section{Model}\\label{Sec2}\nWe describe the membrane potential evolution of a single neuron by a Wiener process $X(t)$, starting at some initial value $x_0$. We assume $X(t)$ given as the solution to a stochastic differential equation\n\\begin{equation}\\label{model}\n\\left\\{\n\\begin{array}{l}\ndX(t)=\\mu dt + \\sigma dW(t),\\\\\nX(t_0)=x_0, \\qquad t>t_0,\n\\end{array}\n\\right.\n\\end{equation}\nwhere $W(t)$ is a standard (driftless) Wiener process. The drift $\\mu>0$ and the diffusion coefficient $\\sigma>0$ represent input and noise intensities, respectively. A spike occurs when the membrane potential $X(t)$ exceeds the exponentially decaying threshold\n\\begin{equation}\\label{b}\nb^*(t)=b_0+\\epsilon \\exp\\left[-\\lambda (t-\\delta_k)\\right]\n\\end{equation}\nfor the first time. Here, $\\delta_k$ denotes the time of the $k$th spike for $k>0$, and can be interpreted as a relative refractory period. We set $\\delta_0$ to be the starting time of the process, i.e. $\\delta_0=t_0$. The term $\\lambda$ represents the decay rate of the threshold, while $\\epsilon$ is interpreted as the amplitude of the time-dependent change in the boundary. After a spike, the membrane potential is reset to its resting position $x_0t_0: X(t)\\geq b(t)\\}.\n\\end{equation*}\nQuantities of interest are the probability density function (pdf) and the cumulative distribution function (cdf) of $T_b$, denoted by $f_{T_b}$ and $F_{T_b}$, respectively. Another relevant quantity is the two-sided BCP given by\n\\begin{equation*\n\\mathbb{P}_X(a,c,\\tau)=\\mathbb{P}\\left(a(t)t_0$ is fixed, boundaries $a(t)$ and $c(t)$ are real functions satisfying $a(t)\\tau)=1-F_{T_c}(\\tau),\n\\]\nwhich corresponds to the survival probability of $T_c$. For a standard Wiener process $W$, Wang and P\\\"{o}tzelberger \\cite{Wang2007} showed that, if the sequences of piecewise linear functions $a_n$ and $c_n$ converge uniformly to $a(t)$ and $c(t)$ on $[t_0,\\tau]$ respectively, then, for the continuity property of probability measure, it holds\n\\begin{equation*\n\\lim_{n\\to \\infty} P_W(a_n,c_n,\\tau)=P_W(a,c,\\tau).\n\\end{equation*}\nWhen $a(t)=-\\infty$ and $c(t)=b(t)$, the convergence of $\\mathbb{P}(T_{b_n}>\\tau)$ to $\\mathbb{P}(T_b>\\tau)$ can be obtained by choosing piecewise linear thresholds approximating $b(t)$ from above, denoted by $b_n^+(t)$, or from below, $b_n^-(t)$. That is, $b_n^+(t)\\geq b(t)$ and $b_n^-(t)\\leq b(t), \\ \\forall t\\in[t_0,\\tau]$, respectively. Since the considered curved boundary is convex, we have \\cite{Wang2001}\n\\begin{equation}\\label{updown}\n\\mathbb{P}_X(-\\infty,b_n^-,\\tau)\\leq \\mathbb{P}_X(-\\infty,b,\\tau)\\leq \\mathbb{P}(-\\infty,b_n^+,\\tau),\n\\end{equation}\ni.e.\n\\begin{equation*\n\\mathbb{P}(T_{b^+_n}\\leq \\tau)\\leq \\mathbb{P}(T_b\\leq \\tau)\\leq \\mathbb{P}(T_{b^-_n}\\leq \\tau).\n\\end{equation*}\nThe approximation accuracy is given by $\\mathbb{P}_X(-\\infty,b_n^+,\\tau)-\\mathbb{P}_X(-\\infty,b_n^-,\\tau)=F_{T_{b^-_n}}(\\tau)-F_{T_{b^+_n}}(\\tau)$, with bounds given in \\cite{BorovkovNovikov}. Obviously, the accuracy in the BCP increases when the distance between the two thresholds decreases.\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{FPTIF}\n\\caption{Schematic illustration of the single trial of a Wiener process in presence of exponentially decaying threshold $b(t)=b_0+\\epsilon\\exp(-\\lambda (t-\\delta_k))$, where $\\delta_k$ denotes the $k$th spike. The membrane potential starts in $x_0=0$ at time $\\delta_0:=t_0=0$, and it evolves until it hits the boundary for the first time. Then, a spike is generated, the voltage $X(t)$ is reset to its resting potential $x_0$, the threshold is reset to $b_0+\\epsilon$ and the evolution restarts. For the considered spiking generation rule, all ISIs are independent and identically distributed. Here the parameters are $\\mu=1, \\sigma^2=1, b_0=1, \\lambda=1$ and $\\epsilon=5$.}\n\\label{FigFPT}\n\\end{figure}\n\n\\section{FPT to continuous piecewise linear threshold}\\label{Method}\nThe transition density function of a standard Brownian motion in $x_1, x_2,\\ldots , x_n$ at time $t_1, t_2, \\ldots, t_n$, constrained to be below the absorbing threshold $c(t)$ defined by $n$ piecewise-linear threshold over $[t_0,t_n]$, is given in \\cite{Wang1997}. Extending that result to the case of a Brownian motion with drift $\\mu$ and diffusion coefficient $\\sigma$, starting in $x_0t_n)=\\int_{C_1}\\cdots\\int_{C_n}p_{c}(x_1,t_1;\\ldots;x_n,t_n|x_0,t_0)dx_1\\cdots dx_n,\n\\end{align}}}\nfor any Borel set $C_i\\subseteq (-\\infty,c_i), 1\\leq i \\leq n$. If $C_i=(-\\infty,c_i)$, then \\eqref{eq2.10} is equal to $\\mathbb{P}(T_c>t_n)$, and it holds\n\\begin{equation}\\label{fpt}\nf_{T_c}(t)=-\\frac{\\partial }{\\partial t_n} \\int_{-\\infty}^{c_1}\\cdots\\int_{-\\infty}^{c_n} p_{c}(x_1,t_1;\\cdots,x_n,t_n|x_0,t_0) dx_1\\cdots dx_n.\n\\end{equation}\n\nWhen $n=1$, the pdf $f_{T_c}$ is known. Since $X$ is a Wiener process with positive drift, the distribution of the FPT to $c(t)=\\alpha+\\beta (t-t_0)$ is inverse Gaussian,\n$T_{c}\\sim IG\\left[(\\alpha-x_0)\/(\\mu-\\beta), (\\alpha-x_0)^2\/\\sigma^2\\right]$, with pdf\n\\begin{equation}\\label{fpt3}\nf_{T_c}(t)=\\frac{\\alpha-x_0}{\\sqrt{2\\pi\\sigma^2(t-t_0)^3}}\\exp\\left(-\\frac{\\left[\\alpha-x_0-(\\mu-\\beta)(t-t_0)\\right]^2}{2\\sigma^2(t-t_0)}\\right),\n\\end{equation}\nmean $\\mathbb{E}[T_{c}]=(\\alpha-x_0)\/(\\mu-\\beta)$ and variance $\\textrm{Var}(T_{c})=(\\alpha-x_0)\\sigma^2\/(\\mu-\\beta)^3$ \\cite{InverseGaussianBook,coxMiller}. Note that the distribution of $T_c$ is the same of that of the FPT of a Wiener process with positive drift $\\mu-\\beta$ to a constant threshold $c(t)=\\alpha$. In general, the approximation of $F_{T_b}$ by $F_{T_c}$ when $n=1$ is too rough. However, when $\\lambda$ is very small, $\\exp(-\\lambda t)\\approx 1-\\lambda t$, yielding $b(t)\\approx b_0+\\epsilon-\\lambda \\epsilon t$. Hence, $T_b$ can be approximated by $T_c$ with $\\alpha=b_0+\\epsilon$ and $\\beta=-\\lambda \\epsilon$.\n\nSince we approximate $b(t)$ by means of a continuous two-piecewise linear threshold, we denote by $\\tilde b$ the linear threshold $c(t)$ when $n=2$. We have\n\\begin{equation}\\label{St}\n\\tilde{b}(t)=\\tilde b_1(t)\\mathbbm{1}_{\\{t\\leq t_1\\}}+\\tilde b_2(t) \\mathbbm{1}_{\\{t>t_1\\}}=\\left\\{\n\\begin{array}{ll}\n\\alpha_1+\\beta_1 (t-t_0) & \\textrm{ if } t_0\\leq t\\leq t_1\\\\\n\\alpha_2+\\beta_2(t-t_1) & \\textrm{ if } t> t_1\n\\end{array}\n\\right. ,\n\\end{equation}\nwhere $\\mathbbm{1}_A$ denotes the indicator function of the set $A$ and $\\alpha_1,\\alpha_2,\\beta_1,\\beta_2\\in \\mathbb{R}$. Throughout the paper, we set $\\alpha_2=\\alpha_1+\\beta_1 (t_1-t_0)$ to guarantee the continuity of $\\tilde b(t)$. This allows to provide analytical expressions of \\eqref{eq2.9} and \\eqref{fpt}, which we use as an approximation of $f_{T_b}$. In particular, we have\n\\begin{align}\\label{fpt2}\n\\nonumber\\mathbb{P}(T_{\\tilde{b}}t_1)\\\\\n\\nonumber=&\\ \\mathbb{P}(T_{\\tilde{b}_1} <\\min(t_1,t)) + \\int_{-\\infty}^{\\tilde{b}(t_1)} \\mathbb{P}(T_{\\tilde{b}_2} t_1\\}}\n\\frac{1}{\\sqrt{2\\pi\\sigma^2(t-t_0)^3}}\\exp\\left(-\\frac{(\\alpha_2-x_0-(\\mu-\\beta_2)(t-t_1)-\\mu(t_1-t_0))^2}{2\\sigma^2(t-t_0)}\\right)\\\\\n\\nonumber&\\times \\left\\{[\\alpha_2-x_0-\\beta_2(t_1-t_0)]\\Phi\\left(\\frac{(\\alpha_2-x_0-\\beta_2(t_1-t_0))\\sqrt{(t-t_1)}}{\\sqrt{\\sigma^2(t_1-t_0)(t-t_0)}}\\right)\\right.\\\\\\nonumber& \\left.-(\\alpha_2+x_0-\\beta_2(t_1-t_0)-2\\alpha_1\n\\exp\\left(-\\frac{2(t-t_1)(\\alpha_1-x_0)(\\alpha_2-\\alpha_1-\\beta_2(t_1-t_0))}{\\sigma^2(t_1-t_0)(t-t_0)}\\right)\\right.\\\\& \\times\\left.\\Phi\\left(\\frac{(\\alpha_2+x_0-\\beta_2(t_1-t_0)-2\\alpha_1)\\sqrt{(t-t_1)}}{\\sqrt{\\sigma^2(t_1-t_0)(t-t_0)}}\\right)\\right\\}\n\\end{align}}This result extends that for a driftless Brownian motion, see e.g. \\cite{Abundo,Scheike}.\nAs expected, setting $\\alpha_1=\\alpha_2=\\alpha$ and $\\beta_1=\\beta_2=\\beta$ yields the pdf of the FPT of a Wiener process to a linear threshold $c(t)=\\alpha+\\beta(t-t_0)$. By definition, the first two moments and variance of $T_{\\tilde b}$ are given by\n\\begin{equation}\\label{EVT}\n\\mathbb{E}[T_{\\tilde b}]=\\int_0^\\infty tf_{T_{\\tilde b}}(t)dt, \\qquad\n\\mathbb{E}[T^2_{\\tilde b}]=\\int_0^\\infty t^2 f_{T_{\\tilde b}}(t)dt, \\qquad \\textrm{Var}[T_{\\tilde b}]=\\mathbb{E}[T_{\\tilde b}^2]-\\mathbb{E}[T_{\\tilde b}]^2,\n\\end{equation}\nand can be numerically computed.\n\n\\section{Parameter estimation}\\label{Sec4}\n\\subsection{Parameter estimation of the piecewise-linear threshold}\\label{Sec4a}\nThe primary aim of this paper is the approximation of the FPT distribution (and relevant statistics) for a curved boundary $b(t)$, by means of the FPT distribution for a continuous two-piecewise linear threshold $\\tilde b(t)$. As discussed in Section \\ref{Sec2}, the quality of the approximation improves when the distance between $\\tilde b$ and $b$ decreases.\n\nDenote by $\\theta=(\\alpha_1,\\beta_1,\\beta_2,t_1)$ the parameters of $\\tilde b$ in \\eqref{St}, with $\\alpha_2=\\alpha_1+\\beta_1(t_1-t_0)$. We are interested in determining the estimator $\\hat\\theta$ which minimizes $|\\tilde b(t)-b(t)|$ on $[\\tau_0,\\tau_*] $, with $t_0<\\tau_0t)=\\mathbb{P}(X(s)b_0$, it follows that\n\\[\n\\mathbb{P}(X(t)\\geq b(t)) \\leq \\mathbb{P}(T_b\\leq t)\\leq \\mathbb{P}(T_{b_0} \\leq t),\n\\]\nwith $T_{b_0}\\sim IG((b_0-x_0)\/\\mu, (b_0-x_0)^2\/\\sigma^2)$. Since $X$ is a Wiener process, $X(t)\\sim N(\\mu t, \\sigma^2 t)$. Then, we choose $\\tau_0$ and $\\tau_*$ such that\n\\[\n\\mathbb{P}(T_{b_0}\\leq\\tau_0)=0.005, \\qquad\n\\mathbb{P}(X(\\tau_*)\\geq b(\\tau_*))=0.995,\n\\]\nyielding the desired probability \\eqref{mah}.\n\n\\begin{figure}\n\\includegraphics[width=1.0\\textwidth]{threshold}\n\\caption{Curved threshold $b(t)$ (continuous line) and four proposed approximating piecewise-linear thresholds: $b_+(t)$ from above (dashed lines) ; $b_-(t)$ from below (dashed-dotted line); $b_\\textrm{betw}(t)$ which is equidistant from $b_+$ and $b_-$ (gray dashed line); $b_\\textrm{free}(t)$ with no restrictions (gray continuous line). For each type of linear threshold, the best approximation is given by the line minimizing a function of the distance from $b$ on $[t_0=0,\\tau]$ (left figure) and on $[\\tau_0,\\tau_*]\\subseteq [t_0=0,\\tau]$ (right figure). As discussed in Section \\ref{Sec4a}, a shorter time interval provides a better approximation of $b$.}\n\\label{thresh}\n\\end{figure}\nThroughout the paper, we consider four possible continuous two-piecewise linear boundaries on $[\\tau_0,\\tau_*]$, as illustrated in Fig. \\ref{thresh}:\n\\begin{enumerate}\n\\item Threshold $b_+$ approximating $b$ from above on $[\\tau_0,\\tau_*]$, passing through $(\\tau_0,\\break b(\\tau_0))$, $(t_1,b(t_1))$ and\n $(\\tau_*,b(\\tau_*))$,\n\\begin{equation*}\\label{th2}\nb_+(t)=b(\\tau_0)+\\frac{b(t_1)-b(\\tau_0)}{t_1-\\tau_0}(t-\\tau_0)\\mathbbm{1}_{\\{t\\leq t_1\\}}+\n\\frac{b(\\tau_*)-b(t_1)}{\\tau_*-t_1}(t-t_1)\\mathbbm{1}_{\\{t>t_1\\}},\n \\end{equation*}\n i.e.\n \\[\n \\alpha_1=b_+(t_0), \\quad \\beta_1= \\frac{b(t_1)-b(\\tau_0)}{t_1-\\tau_0}, \\quad \\beta_2=\\frac{b(\\tau_*)-b(t_1)}{\\tau_*-t_1}(t-t_1).\n \\]\nDue to the assumptions, for given $\\tau_0$ and $\\tau_*$, $t_1$ is the only unknown quantity.\n\\item Threshold $b_-$ approximating $b$ from below on $[\\tau_0,\\tau_*]$. We assume that $b_-$ is tangent to $b(t)$ in both $\\tilde t_1$ and $\\tilde t_2>\\tilde t_1$, with $t_1$ intersection time point of the two tangent lines\n\\[\ny_i(t)=b(\\tilde t_i)-\\lambda \\epsilon \\exp(-\\lambda \\tilde t_i)(t-\\tilde t_i),\n\\]\nfor $i=1,2$. Setting $y_1(t_1)=y_2(t_1)$, we get\n\\[\nt_1=\\frac{\\exp(-\\lambda \\tilde t_1)[1+\\lambda \\tilde t_1]-\\exp(-\\lambda \\tilde t_2)[1+\\lambda \\tilde t_2]}{\\lambda[\\exp(-\\lambda \\tilde t_1)-\\exp(-\\lambda \\tilde t_2)]}.\n\\]\nThen, the desired threshold $b_-(t)$ is\n\\[\nb_-(t)=y_1(\\tilde t_1)+\\frac{y_1(t_1)-y_1(\\tilde t_1)}{t_1-\\tilde t_1}(t-\\tilde t_1)\\mathbbm{1}_{\\{t\\leq t_1\\}}+\\frac{y_2(\\tilde t_2)-y_2(t_1)}{\\tilde t_2-t_1}(t- t_1)\\mathbbm{1}_{\\{t>t_1\\}},\n\\]\nwith\n\\[\n\\alpha_1=b_-(t_0),\\quad \\beta_1=\\frac{y_1(t_1)-y_1(\\tilde t_1)}{t_1-\\tilde t_1}, \\quad\n\\beta_2=\\frac{y_2(\\tilde t_2)-y_2(t_1)}{\\tilde t_2-t_1}.\n\\]\nFor fixed $\\tau_0$ and $\\tau_*$, the unknown parameters are $\\tilde t_1$ and $\\tilde t_2$.\n\\item Threshold $b_\\textrm{betw}(t)$ constrained to be between $b_+(t)$ and $b_-(t)$ on $[\\tau_0,\\tau_*]$, i.e.\\break $b_-(t)\\leq b_\\textrm{betw}(t)\\leq b_+(t)$.\n\\item Threshold with no constraints, denoted by $b_\\textrm{free}(t)$.\n\\end{enumerate}\nDenote by $\\hat\\theta_+, \\hat\\theta_-,\\hat\\theta_\\textrm{betw}$ and $\\hat\\theta_\\textrm{free}$ the estimators of $\\theta$ from the boundaries $b_+, b_-,b_\\textrm{betw}$ and $b_\\textrm{free}$, respectively.\nFrom \\eqref{updown}, it follows that the best approximation of $\\mathbb{P}_X(-\\infty,b,\\break\\tau)$ is obtained when the distance between $b_+$ and $b_-$ is minimized. For this reason, we define $\\hat\\theta_+$ and $\\hat\\theta_-$ as the estimators minimizing the area of the squared distance between the two boundaries, i.e.\n\\begin{equation*\n(\\hat\\theta_+,\\hat\\theta_-)=\\arg\\min_{(\\theta_+,\\theta_-)}\\left[ \\int_{\\tau_0}^{\\tau{^*}} |b_+(t)-b_-(t)|^2dt\\right],\n\\end{equation*}\nwith $\\hat\\theta_+$ and $\\hat\\theta_-$ satisfying the conditions $b_+(t)>b(t)$ and $b_-(t)x_0$, we have\n\\begin{eqnarray}\n\\label{ET} \\widehat{\\mathbb{E}[T_b]}&=& \\frac{b_0}{\\mu}+\\frac{\\epsilon}{\\mu}\\exp\\left(\\frac{b_0\\left(\\mu-\\sqrt{\\mu^2+2\\lambda\\sigma^2}\\right)}{\\sigma^2}\\right),\\\\\n\\nonumber \\widehat{\\textrm{Var}(T_b)}&=&\\frac{b_0\\sigma^2}{\\mu^3}+\\frac{\\sigma^2\\epsilon}{\\mu^3}(b_0-1)\\\\\n&+&\\frac{2\\epsilon}{\\mu^2}\\left(\\frac{\\mu b_0}{\\sqrt{\\mu^2+2\\lambda\\sigma^2}}+\\frac{\\sigma^2}{2\\mu}-\\theta_0\\right)\\exp\\left(\\frac{b_0\\left(\\mu-\\sqrt{\\mu^2+2\\lambda\\sigma^2}\\right)}{\\sigma^2}\\right).\\quad\n\\label{VT}\n\\end{eqnarray}\nWe denote by $\\hat\\phi_{\\textrm{ME}}^\\epsilon$ the moment estimator of $\\phi$ obtained from \\eqref{ET} and \\eqref{VT} when $\\epsilon$ is small.\n\n\\section{Simulation study}\n\n\\subsection{Monte Carlo simulations} We simulate FPTs of the Wiener process $X$ to $b(t)$ as described in \\cite{LindnerLongtin,ReviewSac}. Applying the Euler-Maruyama scheme to the stochastic differential equation \\eqref{model}, we generate realizations of $X$, denoted by $x_i:=X(s_i)$, at discrete times $s_i=i \\Delta s, i\\geq1$. We set $X_0=x_0=0$ and $\\Delta s=0.001$ as time step. To avoid the risk of not detecting a crossing of the boundary due to the discretization of the sample path, at each iteration step we compute the probability that the bridge process $X^{[s_i,s_{i+1}]}=\\left\\{X_s^{[s_i,s_{i+1}]},s\\in [s_i,s_{i+1}]\\right\\}$, originated in $x_iu_i$. In this case, the mid-point $(s_i+s_{i+1})\/2$ is chosen as simulated FPT. Samples of size $100$ are simulated for different values of $\\sigma^2, \\lambda$ and $\\epsilon$ when $b_0=1$ and $\\mu=1$. In particular, we consider $\\sigma^2=0.2, 0.4, 1$; $\\epsilon=0.05, 0.1, 0.2, 1, 5, 10$ and $\\lambda=0.02, 0.04, 0.08, 0.15, 0.30, 0.60, 1.00, 3.00, 5.00, 10.00$. These parameter values are chosen to cover and extend the cases of small values of $\\epsilon$ considered in \\cite{LindnerLongtin,Urdapilleta}. Finally, for each value of $\\sigma^2,\\epsilon$ and $\\lambda$, we repeat simulation of data set 1000 times, obtaining 1000 statistically indistinguishable and independent trials.\n\n\\subsection{Set up} In the simulations we are mainly concerned to illustrate the performance of our method under the assumption that the threshold $b(t)$ is known, i.e. $b_0$, the rate $\\lambda$ and the amplitude $\\epsilon$ are given. Two scenarios are considered: both $\\mu$ and $\\sigma^2$ are known; no information about the parameter of the Wiener process is given.\nIn the first case it is of interest to evaluate the error in the estimation of mean, variance, CV and cdf of $T_b$ by comparing theoretical \\eqref{EVT} and empirical firing statistics. When $\\epsilon$ is small, a further comparison with \\eqref{ET} and \\eqref{VT} is carried out. To measure the error in the estimation of $F_{T_b}$, we consider the relative integrate absolute error ($R_\\textrm{IAE}$), defined as\n\\begin{equation}\\label{riae}\nR_{\\textrm{IAE}}(\\hat F_{T_b})=\\frac{\\int_0^{\\infty} | \\hat F_{T_b}(t)-F_{T_b}(t)| dt}{\\mathbb{E}[T_b]}.\n\\end{equation}\nWe replace the unknown quantities $F_{T_b}$ and $\\mathbb{E}[T_b]$ by their empirical estimators, defined by $F_n(t)=\\frac{1}{n}\\sum_{i=1}^n \\mathbbm{1}_{\\{T_{b_i}\\leq t\\}}$ and $\\bar t=\\sum_{i=1}^n T_{b_i}\/n$, respectively. Both empirical quantities are based on $n=1000 000$ simulated FPTs, ensuring the closeness to the theoretical counterparts by the law of large numbers. This first scenario is meant to understand the goodness of our approximation through simulations.\n\nAnother relevant question is the performance of the MLEs and MEs of $\\mu$ and $\\sigma^2$, as described in Section \\ref{Sec4b}. To compare different estimators, we use the relative mean error $R_\\textrm{ME}$ to evaluate the bias and the relative mean square error $R_{\\textrm{MSE}}$, which incorporates both the variance and the bias. They are defined as the average over the $1000$ repetitions of the quantities\n\\[\nE_\\textrm{rel}(\\hat\\mu)=\\frac{\\hat\\mu-\\mu}{\\mu}, \\qquad E_\\textrm{rel sq}(\\hat\\mu)=\\frac{(\\hat\\mu-\\mu)^2}{\\mu^2},\n\\]\nand likewise for $\\sigma^2$.\n\\begin{figure}\n\\includegraphics[width=1.0\\textwidth]{allb}\\\\%\\includegraphics[width=1.3\\textwidth]{all}\n\\includegraphics[width=1.0\\textwidth]{all3b}\\\\\n\\includegraphics[width=1.0\\textwidth]{all5}\n\\caption{Mean (left panels), variance (central panels) and CV (right panels) of the FPT $T_b$\n as a function of the decay rate $\\lambda$ of the threshold for small values of the amplitude $\\epsilon$ when $\\mu=1$. Top panels: $\\sigma^2=0.2$. Central panels: $\\sigma^2=0.4$. Bottom panels: $\\sigma^2=1$. Empirical quantities from simulations (symbols), theoretical quantities given by \\eqref{EVT} for the piecewise linear threshold $b_\\textrm{free}$ (solid lines), and theoretical quantities \\eqref{ET} and \\eqref{VT} when $\\epsilon$ is small (solid gray lines). Also shown are the firing statistics of $T_b$ when $\\epsilon=0$ (horizontal dashed lines).}\n\\label{Figall}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=1.0\\textwidth]{all2}\n\\caption{Mean (left panel), variance (central panel) and CV (right panel) of the FPT $T_b$\n as a function of the decay rate $\\lambda$ of the threshold for large values of the amplitude $\\epsilon$ when $\\mu=1$ and $\\sigma^2=0.2$. Empirical quantities from simulations (symbols), theoretical quantities given by \\eqref{EVT} for the piecewise linear threshold $b_\\textrm{free}$ (solid lines), and theoretical quantities \\eqref{ET} and \\eqref{VT} when $\\epsilon$ is small (solid gray lines). Also shown are the firing statistics of $T_b$ when $\\epsilon=0$ (horizontal dashed lines).}\n\\label{Figall2}\n\\end{figure}\n\n\n\\subsection{Theoretical results for cdf and firing statistics of $T_b$} In Fig. \\ref{Figall} are reported theoretical and empirical means, variances and CVs of $T_b$ as a function of the rate $\\lambda$, for small values of the amplitude $\\epsilon$ and for $\\sigma^2=0.2, 0.4$ and $1$. The given theoretical quantities are obtained from \\eqref{EVT} for the piecewise linear threshold $b_\\textrm{free}$. Note how the mean of $T_b$ does not depend on $\\sigma^2$, as it also happens for a linear threshold, while both variance and CV increase with growing $\\sigma^2$. We refer to \\cite{LindnerLongtin} for a detailed discussion on other qualitative features of the firing statistics, e.g. monotonic decrease on the mean with growing $\\lambda$, existence of a minimum value for the variance, etc. What is relevant to emphasize is the excellent fit of the firing statistics provided by our method for any $\\lambda$, and for both small (cf. Fig. \\ref{Figall}) and large (cf. Fig. \\ref{Figall2}) values of $\\epsilon$. When $\\epsilon$ is small, our theoretical firing statistics are at least as good as those in \\cite{LindnerLongtin, Urdapilleta}, outperforming them when $\\epsilon$ grows. The firing statistics of $T_{b_\\textrm{betw}}$ are almost identical to those of $T_{b_\\textrm{free}}$, while those of $T_{b_+}$ and $T_{b_-}$ are slightly different for increasing $\\epsilon$. This can be seen in Fig. \\ref{Figriae}, left panel, where the percentages of the $R_\\textrm{IAE}(\\hat F_T)$ for the four proposed estimators are given. As expected, the best approximation of $F_{T_b}$ is provided by $F_{T_{b_\\textrm{free}}}$, since $b_\\textrm{free}$ is the only threshold whose parameters are obtained from a non-constrained optimization problem. The performance of the estimators gets worse for large $\\sigma^2$ and $\\epsilon$. The highest error is observed for the value of $\\lambda$ that minimizes the variance of $T_b$. However, all errors are smaller than $2\\%$, confirming the good performance of the proposed estimators.\n\n\\subsection{Parameter estimation of $(\\mu,\\sigma^2)$}\nWe have seen that $T_{b_\\textrm{free}}$ yields the best approximation of $T_b$ in terms of both cdf and firing statistics. For this reason, we limit our study to the estimators $\\hat\\phi$ based on $b_\\textrm{free}$. In Fig. \\ref{Figphi1} the $R_\\textrm{ME}$ and the $R_\\textrm{MSE}$ of $\\hat\\mu$ and $\\hat\\sigma^2$ are reported. As expected, the MLE provides the best estimate of $\\phi$, while both MEs are acceptable only for small values of $\\epsilon$. The performance of $\\hat\\mu$ is highly satisfactory, with $R_\\textrm{ME}(\\hat\\mu)$ smaller than $0.5\\%$, and $R_\\textrm{MSE}(\\hat\\mu)<0.2\\%$. Larger but still good $R_\\textrm{ME}$ and $R_\\textrm{MSE}$ are observed for $\\hat\\sigma^2$. The performance of $\\hat\\phi_\\textrm{MLE}$ gets worse for growing $\\sigma^2$, as shown in Fig. \\ref{Figphi2}. However, except the $R_\\textrm{ME}(\\hat\\sigma^2)$ for large values of $\\epsilon$, all errors are between $0$ and $2-3\\%$. Two last remarks should be done: first, the $R_\\textrm{MSE}$ of $\\hat\\mu$ for small values of $\\epsilon$ approaches the corresponding values of $\\sigma^2$.\nSecond, $R_\\textrm{MSE}(\\hat\\sigma^2)$ seems not to depend on $\\lambda, \\epsilon$ and $\\sigma^2$, but to be equal to $2\\%$. This error decreases when increasing the sample size. For example, the $R_\\textrm{MSE}(\\hat\\sigma^2)\\approx 1\\%$ when $n=200$ (results not shown).\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{riaeb}\n\\caption{$R_{\\textrm{IAE}}(\\hat F_{T_b})$ (in percentage) given by \\eqref{riae} for different values of $\\lambda$ and $\\epsilon$ when $\\mu=1$. Left panel: $R_{\\textrm{IAE}}(\\hat F_{T_b})$ from threshold $b_\\textrm{free}$ (circles), $b_-$ (triangles), $b_+$ (rhombuses) and $b_\\textrm{betw}$ (gray circles) when $\\epsilon=1$ and $\\sigma^2=0.2$. Right panel: $R_\\textrm{IAE}(\\hat F_{T_{b_\\textrm{free}}})$ for $\\sigma^2=0.2$ (circles), $\\sigma^2=0.4$ (triangles) and $\\sigma^2=1$ (gray circles). The values of $\\epsilon$ between consecutive vertical dotted lines are fixed and equal to $0.05, 0.1, 0.2, 1, 5,10$, while $\\lambda$ varies between $0.02$ and $10$.}\n\\label{Figriae}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{est_phi}\n\\caption{Dependence of $R_\\textrm{ME}(\\hat\\mu), R_\\textrm{MSE}(\\hat\\mu), R_\\textrm{ME}(\\hat\\sigma^2)$ and $R_\\textrm{MSE}(\\hat\\sigma^2)$ (average over $1000$ simulations) on $\\lambda$ and $\\epsilon$ when $X$ is a Wiener process with $\\mu=1$ and $\\sigma^2=0.2$. Different estimators of $\\phi=(\\mu,\\sigma^2)$ are considered: maximum likelihood estimator $\\hat\\phi_\\textrm{MLE}$ (solid lines with triangles), moment estimator $\\hat\\phi_\\textrm{ME}$ (dashed lines with circles) and moment estimator from \\eqref{ET} and \\eqref{VT} when $\\epsilon$ is small, $\\hat\\phi_{ME}^\\epsilon$ (gray solid lines with gray circles). The values of $\\epsilon$ between consecutive vertical dotted lines are fixed and equal to $0.05, 0.1, 0.2, 1, 5,10$, while $\\lambda$ varies between $0.02$ and $10$.\n}\n\\label{Figphi1}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{est_phi2}\n\\caption{Dependence of $R_\\textrm{ME}(\\hat\\mu), R_\\textrm{MSE}(\\hat\\mu), R_\\textrm{ME}(\\hat\\sigma^2)$ and $R_\\textrm{MSE}(\\hat\\sigma^2)$ (average over $1000$ simulations) on $\\lambda, \\epsilon$ and $\\sigma^2$ when $X$ is a Wiener process with $\\mu=1$ and $\\sigma^2$ equal to $0.2$ (solid lines with circles), $0.4$ (dashed lines with triangles) and $1$ (gray solid lines with gray circles). Here only the maximum likelihood estimator $\\hat\\phi_\\textrm{MLE}$ of $\\phi=(\\mu,\\sigma^2)$ is considered. The values of $\\epsilon$ between consecutive vertical dotted lines are fixed and equal to $0.05, 0.1, 0.2, 1, 5,10$, while $\\lambda$ varies between $0.02$ and $10$.}\n\\label{Figphi2}\n\\end{figure}\n\n\\section{Discussion} As a consequence of the recent increasing interest towards adapting-threshold models for the description of the neuronal spiking activity, a need of suitable mathematical tools to deal with hitting times of diffusion processes to time-varying thresholds arises. The mathematical literature on the FPT problem is rich and extensive. Unfortunately, analytical solutions are not available even for a problem as simple (compared to others) as the one considered here, i.e. Wiener process to an exponentially decaying threshold. The closest result in this direction is represented by the work of Wang and P\\\"{o}tzelberger, who provide an explicit expression which should then be evaluated through Monte-Carlo simulations. The idea behind the works of Lindner and Longtin and of Urdapilleta, was to simplify some mathematical difficult equations arising from the study of the FPT by linearizing them in $\\epsilon$, the amplitude of the decaying threshold. As a consequence, the quality of the approximation rapidly decreases when $\\epsilon$ increases.\n\nThe method proposed here has no restriction on the parameter of the thresholds and it is based on the simple idea of replacing the boundary by a continuous two-piecewise linear threshold. This allows us to derive the analytical expression of the FPT density to the two-piecewise threshold, and to use it to approximate the desired distribution. To some extent, the presence of two linear thresholds can be considered as a second order approximation of the problem.\n\nNumerical simulations show a good performance of the proposed method both when computing the main firing statistics, such as means, variances and CVs, and when calculating the FPT distribution. Different approximating thresholds have been proposed. We suggest choosing the one minimizing the distance with the curvilinear threshold and to restrict the interval where to perform the optimization as described in the paper. Among the estimators of the drift and diffusion coefficients of the Wiener process, we suggest applying MLE which always estimates the parameters reasonably well.\n\nThe method proposed here may yield several interesting developments. First of all, it can be used to characterize the firing statistics of the Wiener process to the exponential decaying threshold, extending the previous considerations obtained for small values of $\\epsilon$. Then, it may be extended to the case of a Wiener process with an adapting decaying threshold, as suggested in the introduction. Finally, our results may also be applied to all those processes which can be expressed as a piecewise monotone functional of a standard Brownian motion \\cite{Wang2007}, as well as to Wiener processes with time-varying drift \\cite{LindnerLongtin,Molini2011}.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{Intro}\n\nIn their pioneering papers \\cite{wick,cutk}, Wick and Cutkosky (W-C) have\nfound the solutions of the Bethe-Salpeter (BS) equation \\cite{bs} for\ntwo scalar particles interacting by the exchange of a massless scalar\nparticle. In addition to the states which, in the non-relativistic\nlimit, reproduce the spectrum of the Schr\\\"odinger equation with the\nCoulomb potential, there was found another set of solutions which do\nnot have any non-relativistic counterparts. These solutions were\ncalled ``abnormal''.\nTheir discovery triggered the discussion as to whether they do\nindicate a mathematical inconsistency of the W-C model or\nof the BS equation, or whether they represent new physical systems,\nwhose existence does not contradict any physical principles, \nalthough they are not covered by the Schr\\\"odinger equation.\nIn the latter case, they might provide examples of relativistic\nsystems which could exist in nature, but which would not be\ndescribed by continuous extensions of non-relati\\-vis\\-tic quantum\nmechanics. A thorough discussion of this issue can be found in Ref.\n\\cite{nak69} (sections 6 and 8).\n\nOne would hope that a complementary lighting to the above questioning\nmight come from experimental data. Unfortunately the conditions of the\nemergence of abnormal states are not easy to realize.\nConsidering the W-C model as a simplified model of QED,\nabnormal states would appear as highly excited states for values of\nthe fine\nstructure constant $\\alpha$ above $0.5\\div 1$. Such values might be\nreached with the\naid of heavy ions and dedicated electron-ion scattering experiments\nmight be envisaged. However, to have a clear experimental distinction\nof abnormal bound states from the ionization threshold, one actually\nwould need to increase the values of $\\alpha$ up to $4\\div 5$, which\nthen further reduce the probability of an experimental success.\nAnother possibility is an analogy of the model with hadron dynamics,\nwhere hadrons mutually interact by means of the exchange of light\nparticles, like the pions, and where the coupling constants might\nlie in the range of values needed for the existence of abnormal\nstates. However, here, the exchanged particles being massive,\ndrastic changes occur with respect to the massless case: the\ninteraction forces become of short-range and one realizes that\nabnormal states are produced only with very small mass values of\nthe exchanged particle, much smaller than the pion mass. \nThe framework of Quantum Chromodynamics, where quarks\nmutually interact by means of exchange of massless gauge particles,\nthe gluons, with sufficiently strong forces, might provide another\ndomain to search for possible evidences of abnormal solutions.\n\n\nIn the absence of any direct experimental indication about the\nexistence or nonexistence of abnormal states, one is entitled to\nexplore all possible theoretical paths that might provide\ncomplementary information about their properties. From this point\nof view, we have found that an analysis of the Fock-space content\nof the abnormal, as well as normal, states would be of great help.\nThe BS amplitude allows one to extract the wave function related\nto the two-body sector of the Fock space \\cite{cdkm}. \nIts norm, which is positive and bounded by 1, is then\ninterpreted as the weight of that sector in the whole Fock space.\n\nComplementary information to the above analysis comes from the\nknowledge of the electromagnetic form factors, assuming that one\nof the massive constituents of the bound state is charged. Their\nasymptotic behavior qualitatively probes the compositeness of the\nstates: a rapid decrease would be the signature of a many-body\nstructure \\cite{matvmurtavk,brodsfarr,radyush}. \n\nOur calculations, as well as the results of ref. \\protect{\\cite{dshvk}},\nshow that, in the window of allowed coupling constants, the normal\nsolutions are essentially dominated by the two-body sector of the Fock\nspace. \nWe will show in the present work that, on the contrary, \nthe abnormal solutions have a two-body contribution that vanishes in\nthe limit of zero binding energies and remains small (less than\n10\\%) in all its domain of existence.\nThey are therefore dominated by the many-body sectors, composed of the\ntwo massive constituents and of several massless exchange\nparticles. This feature explains why the abnormal\nsolutions disappear from the spectrum in the non-relativistic\nlimit, the latter being formulated in the two-body sector alone,\nwhile the other sectors, containing massless particles, are by\nessence relativistic.\n\nThe asymptotic behaviors of the form factors also corroborate the\nabove conclusions. The form factors of the abnormal solutions\nasymptotically decrease, for spacelike momenta, faster, by factors\nof the order of $10^{3}$, than those of the normal solutions.\nAlso, the transition form factors between normal and abnormal\nsolutions display global suppressions, by factors of $5\\div 10$, with\nrespect to the normal-normal or abnormal-abnormal transition form\nfactors, signalling a different nature of the normal and abnormal\nsolutions.\n\nThese results suggest that the abnormal solutions might correspond\nto states called ``hybrids'' in the literature. In the present model,\nthey are dominated\nby Fock space sectors containing two massive constituents and several\nor many massless constituents, corresponding to the exchanged-particle\nfields.\nThey could be considered as the Abelian scalar analogs of the QCD\nhybrids, which, in the mesonic sector, are dominated by their\ncoupling to the set of fields made of a quark, an antiquark and one or\nseveral gluon fields.\n\nFinally, the question of the validity of the ladder approximation\nof the model, because of the necessity of having large values of the\ncoupling constant to create abnormal states, still remains an open issue.\n\nThe plan of the paper is the following. Sec. \\ref{def} is devoted to an\nintroductory definition of the BS amplitude and of the Fock-space sectors. \nIn Sec. \\ref{WCsol}, the properties of the solutions of the\nW-C model are displayed and some solutions are found numerically. \nIn Sec. \\ref{FFs} the elastic and transition electromagnetic form factors\nare expressed through the BS amplitudes and are calculated numerically,\nwith special emphasis put on their asymptotic behavior. Concluding remarks\nfollow in Sec. \\ref{concl}. \nThree appendices give technical details\nabout some of the formulas used in the main text. Preliminary results\nof the present study were presented in Ref. \\cite{LC2019}.\n\n\\par\n\\section{Fock space sectors} \\label{def}\n\nThe BS amplitude, satisfying the BS equation, is defined as\n\\begin{equation}\\label{bs1}\n\\Phi(x_1,x_2;p)\\equiv \\langle 0 |T[\\phi_1(x_1)\\phi_2(x_2)]|p\\rangle,\n\\end{equation}\nwhere $\\phi_a(x)$ ($a=1,2)$ are Heisenberg field operators,\n$T$ means time ordering, $|p\\rangle$ is the state vector of the\nbound system and $\\langle 0 |$ is the vacuum state vector.\nSince the amplitude $\\Phi(x_1,x_2;p)$ depends on two 4D variables,\n$x_1$ and $x_2$, it is usually called ``two-body'' BS amplitude,\nthough this terminology, to some extent, is misleading.\nAsking the questions ``what is the content of a system?'' or ``is it\ntwo-body or many-body?'' requires that we analyze the state\nvector $|p\\rangle$ of this system, entering in the matrix element\n(\\ref{bs1}), by decomposing it onto the states $| n\\rangle$ with\ndefinite numbers $n$ of particles (the Fock sector decomposition),\nschematically:\n\\begin{equation}\\label{p}\n|p\\rangle=\\sum_{n=2}^{\\infty}\\psi_n^{}|n\\rangle,\n\\end{equation}\nand studying the contributions of the two-body component $\\psi_2$,\nthe three-body component $\\psi_3$, etc., in the full normalization\nintegral. The answer to the above questions depends on which component\n(or sum of components) is dominant. It should be mentioned that the state vector $|p\\rangle$\n is usually defined on a $t$-constant plane in the 4D space. There are however some advantadges to choose the so called\nlight-front plane \n$t+z=0$ (or light-front plane of general orientation, see \\cite{cdkm}). In this case, the corresponding Fock components \n$\\psi_n$ are called the light-front wave functions. The two-body light-front wave function $\\psi_2$ is related to the BS \namplitude (\\ref{bs1}) by eq. (\\ref{lfwf}) from \\ref{appN2}.\n\\par\n\nIn the W-C model, in the ladder approximation, the\nFock decomposition can contain two constituent (massive) particles\nand any number of exchange (massless) particles. The state\n$|2\\rangle$ (two-body sector) contains two constituents only, the\nstate $|3\\rangle$ (three-body sector) contains two constituents and\none exchange particle, the state $|n\\rangle$ ($n$-body sector)\ncontains two constituents and $(n-2)$ exchange particles, etc. \nAssuming that the state vectors $|n\\rangle$ ($n=2,3,\\ldots$) are\nnormalized to unity, the state vector $|p\\rangle$ is then normalized\nas\n\\begin{equation}\\label{eq2}\n\\langle p|p\\rangle=\\sum_{n=2}^{\\infty}N_n=1,\n\\end{equation}\nwhere, schematically, $N_n=\\int |\\psi_n|^2\\ldots$ \nis the contribution of the $n$-body Fock sector\n(see eq. (\\ref{N2}) for the exact definition of $N_2$).\nIn practice, knowing the BS amplitude $\\Phi(x_1,x_2;p)$ we are able to find $N_2$ only. \nThe calculation of $N_2$ from the BS amplitude is presented in\n\\ref{appN2}. If it is dominant, this would mean that the\ncontribution of the other sectors, containing exchange\nparticles, is small. The limiting case, when it is enough to keep the\ntwo-body state only (the case corresponding to $N_2=1$), whereas the\nstates containing exchange particles can all be omitted, is realized\nin non-relativistic systems. On the contrary, when the two-body\ncontribution $N_2$ is small, the system is dominated by \ntwo constituents with an indefinite number of exchange massless \nparticles, whose contribution $\\sum_{n=3}^{\\infty}N_n$ is close to 1.\n\\par\n\nFor the normal solutions of the W-C model (in the\nequal-mass case), the above analysis\nhas been made in Ref. \\cite{dshvk}. It was found that for small binding\nenergies the two-body (constituent) sector dominates, as expected. When\nthe binding energy increases (i.e., the total mass $M$ decreases), the\ntwo-body contribution $N_2$ decreases in parallel. However, it still\ndominates and as $M\\to 0$, it tends, in this model, to 64\\%.\nThat is, the sectors $|n\\rangle$ with $n\\geq 3$ contribute in total\nto 36\\% of the total normalization of the normal state vector.\nIn the present paper, we will carry out the same analysis for the\nabnormal states.\n\n\n\\section{Wick-Cutkosky solutions}\\label{WCsol}\n\nThe BS equation \\cite{bs} for the amplitude (\\protect{\\ref{bs1}})\ncontaining two spinless fields, \nrestricted to the equal-mass case $m_1=m_2=m$, reads, in momentum space,\n\\begin{eqnarray}\\label{bs}\n\\Phi(k,p)&=&\\frac{i^2}{\\left[(\\frac{p}{2}+k)^2-m^2\n+i\\epsilon\\right]\\left[(\\frac{p}{2}-k)^2-m^2+i\\epsilon\\right]}\n\\nonumber\\\\\n&\\times&\\int \\frac{d^4k'}{(2\\pi)^4}iK(k,k',p)\\Phi(k',p),\n\\end{eqnarray}\nwhere $p$ and $k$ are the total and relative four-momenta,\nrespectively, and $K$ is the interaction kernel. The bound state\nmass squared is $M^2=p^2$. For nonconfining interactions, the mass $M$\nis smaller than $2m$, allowing the introduction of the binding energy\n$B$ (defined positive) through the relation $M^2=(2m-B)^2$.\nIn the ladder approximation of the kernel, represented by the exchange\nof a scalar particle with mass $\\mu$, the kernel has the form\n\\begin{equation}\\label{ladder}\niK(k,k',p)=\\frac{i(-ig)^2}{(k-k')^2-\\mu^2+i\\epsilon},\n\\end{equation} \nleading to an attractive interaction and the possible emergence of\nbound states.\n\n\n\\subsection{General properties of the solutions} \\label{genprop}\n\nThe W-C model corresponds to the case\n$\\mu=0$ in Eq. (\\ref{ladder}). In the non-relativistic limit, this\nmodel leads to the well-known Coulomb bound state spectrum.\nCutkosky showed that \nin the relativistic case,\nthe BS amplitude,\nhenceforth limited to $S$-wave states, characterized by a principal\nquantum number $n=1,2,\\ldots$, can be represented in terms of $n$\nfunctions $\\left\\{ g_n^{\\nu} \\right\\}_{{\\nu}=0,1\\ldots n-1}$, depending on\na single scalar argument $z\\in[-1,+1]$, as \n\\begin{eqnarray}\\label{Phi}\n&& \\Phi_n(k,p)=- {i \\over \\sqrt{N_{tot}}}\n\\sum_{{\\nu}=0}^{n-1}\\int_{-1}^1g_{n}^{\\nu}(z)dz \\nonumber \\\\\n&& \\times\\frac{m^{2(n-{\\nu})+1}}\n{[m^2-\\frac{1}{4}M^2 -k^2-p\\makebox[0.08cm]{$\\cdot$} k\\,z-\\imath\\epsilon]^{2+n-{\\nu}}},\n\\ n=1,2,\\ldots\\ . \\nonumber \\\\\n&& \n\\end{eqnarray}\n$N_{tot}$ is a dimensionless normalization factor, deter\\-min\\-ed in\n\\ref{exprssff},\nensuring the condition $F_{el}(0)=1$ for the elastic form factor . \nThe factor $m^{2(n-{\\nu})+1}$ in the numerator is introduced to deal\nwith dimensionless $g_{n}^{\\nu}(z)$ functions.\n\nBy inserting (\\ref{Phi}) in the BS equation (\\ref{bs}), Cutkosky\nobtained [Eq. (14) of Ref. \\cite{cutk}] a system of homogeneous coupled\nintegral equations for the functions $g_n^{\\nu}$.\nFor S-waves, it reads\\footnote{A typo seems to exist in Eq. (14) of\nRef. \\cite{cutk}: the integration with respect to $t$ goes from $-1$\nto $+1$, and not from $0$ to $+1$, as can be verified from Eq. (13)\nof that reference. Notice that we use a slightly different notation, \nreplacing $k \\to \\nu$, with respect to the original work \\cite{cutk}.}:\n\\begin{small}\n\\begin{eqnarray}\\label{ECut}\n&&g_n^{\\nu}(z)= {\\lambda\\over2} \\sum_{{\\nu}'=0}^{\\nu} \\frac{ (n-{\\nu}+1)(n-{\\nu})}\n { (n-{\\nu}'+1) (n-{\\nu}')} \\int_{-1}^1dt \\int_0^1 dx\\nonumber \\\\\n &\\times& x(1-x)^{n-{\\nu}-1} \\int_{-1}^{+1}dz'\n \\frac{\\delta[ z -xt-(1-x)z'] }{ [1-\\eta^2(1-z^2) ]^{{\\nu}-{\\nu}'+1}}\n g_n^{{\\nu}'}(z'), \\nonumber\\\\\n&& \n\\end{eqnarray}\n\\end{small}\nwhere $\\lambda$ is related to the coupling constant $g^2$ of the\ninteraction kernel (\\ref{ladder}) by\n$\\lambda= {g^2\\over 16 \\pi^2m^2}, $\nand the total mass square $M^2$, eigenvalue of the system (\\ref{ECut}),\nappears through the parameter\n\\[ \\eta^2={M^2\\over 4m^2}. \\]\nIntegrating Eq. (\\ref{ECut}), first with respect to $t$ through\nthe $\\delta$ function, taking into account the bounds to be\nsatisfied by $t$ and $x$, and distinguishing the two cases, $z>z'$\nand $z'>z$, one obtains the set of equations\n\\begin{eqnarray}\\label{ECut2}\n g_n^{\\nu}(z)&=& {\\lambda\\over2} \\sum_{{\\nu}'=0}^{{\\nu}} c_n^{{\\nu}{\\nu}'} \\;\n \\int_{-1}^{+1} dz' \\; {[R(z,z')]^{n-{\\nu}} \\over [Q(z')]^{{\\nu}-{\\nu}'+1}} \\;\n g_n^{{\\nu}'} (z'), \\nonumber \\\\\n && n=1,2,\\ldots , \\; \\; {\\nu}=0,1,...n-1, \n\\end{eqnarray}\nwhere we have introduced\n\\[ c_n^{{\\nu}{\\nu}'}=\\frac{ (n-{\\nu}+1)} { (n-{\\nu}'+1) (n-{\\nu}')}, \\]\n\\begin{equation}\\label{Q}\n Q(z)=1-\\eta^2(1-z^2), \n \\end{equation}\nand\n\\begin{equation}\\label{eq1bb}\nR(z,z')=\\left\\{\n\\begin{array}{ll}\n\\frac{1-z}{1-z'},& \\mbox{for $z'z$.}\n\\end{array}\n\\right.\n\\end{equation}\nBy expanding Eq. (\\ref{ECut2}), one is left with a $n\\times n$\ntriangular system of one-dimensional integral equations of the form\n\\begin{small}\n\\begin{eqnarray}\ng_n^0(z) & =& {\\lambda\\over2}\n \\left[ c_n^{00} \\int_{-1}^{+1} dz' \\; { [R(z,z')]^n \\over Q(z')}\n \\; g_n^0 (z') \\right], \\label{ECut2_0} \\\\\ng_n^1(z) &=& {\\lambda\\over2}\n \\left[ c_n^{10} \\int_{-1}^{+1} dz' \\; {[R(z,z')]^{n-1} \\over [Q(z')]^2}\n \\; g_n^0 (z') \\right.\n \\nonumber \\\\\n &+& \\left.c_n^{11} \\int_{-1}^{+1} dz' \\;\n { [R(z,z')]^{n-1} \\over Q(z') } \\; g_n^1 (z') \\right], \n \\nonumber\\\\\ng_n^2(z) & =& {\\lambda\\over2}\n \\left[ c_n^{20} \\int_{-1}^{+1} dz' \\; {[R(z,z')]^{n-2} \\over [Q(z')]^3}\n \\; g_n^0 (z') \\right.\n \\nonumber\\\\\n &+& \\left. c_n^{21} \\int_{-1}^{+1} dz' \\;\n {[R(z,z')]^{n-2} \\over [Q(z')]^2 } \\; g_n^1 (z')\\right.\n \\nonumber \\\\\n&+&\\left. c_n^{22} \\int_{-1}^{+1} dz' \\;\n { [R(z,z')]^{n-2} \\over Q(z') } \\; g_n^2 (z')\\right], \n \\nonumber\\\\ \n& & \\ldots \\ \\ \\ \\ \\ \\ \\ \\ldots \\ \\ \\ \\; \\ldots \\ \\ \\ \\ldots \\;\n \\ \\ \\ \\ldots\\ , \n \\nonumber\\\\\ng_n^{n-1}(z) &=& {\\lambda\\over2}\n \\left[ c_n^{n-1,0} \\int_{-1}^{+1} dz' \\; { R(z,z') \\over [Q(z')]^n}\n \\; g_n^0 (z') + \\ldots \\right. \n \\nonumber\\\\\n &+& \\left.c_n^{n-1,n-1}\n \\; \\int_{-1}^{+1} dz' \\; { R(z,z') \\over Q(z')} \\; g_n^{n-1}(z')\n \\right]. \\nonumber\n\\end{eqnarray}\n\\end{small}\n\nRemarkably, the function $g_n^0$, which allows the calculation of the\nenergy spectrum via the $M^2$-dependence of $Q$ [Eq. (\\ref{Q})], is\ntotally decoupled from the rest of the system.\nIt fulfills the single equation (\\ref{ECut2_0}), that we will hereafter\nwrite in terms of the fine structure coupling constant $\\alpha$, usual\nin the Coulomb problems:\n\\begin{eqnarray} \\label{gn}\n g_n^0(z)&=&\\frac{\\alpha}{2\\pi n} \\int_{-1}^1 {[R(z,z')]^n \\over Q(z')}\n \\; g_n^0 (z'), \n \\\\\n \\alpha&=&\\pi \\lambda = {g^2\\over 16\\pi m^2}.\n \\nonumber\n\\end{eqnarray}\n\nThe remaining equations allow the determination of $g_n^{k>0}$ -- and\nso of the BS amplitude (\\ref{Phi}) -- by solving an inhomogeneous\nproblem with an inhomogeneous term given by $g_n^0$.\nNotice that it is a quite unusual situation in Quantum Mechanics that\na part of the total system wave function, which, as we will see in what\nfollows is far from being dominant, \ndetermines the full spectrum of the system.\n\nAlthough the results presented here are limited to $S$-wave only, it is worth noticing that for $l\\neq 0$\nthe corresponding spherical function $Y_{lm}$ would appear as a prefactor\nin Eq. (\\ref{Phi}). \nThe angular momentum $l$ would enter in the system of equations (\\ref{ECut2}) and (\\ref{ECut2_0}), \nbut it turns out to be absent in the first equation (\\ref{ECut2_0}) determining the spectrum.\nAs a consequence, the BS amplitude would depend on $l$, while the spectrum would remain $l$- degenerate. \n\nIn view of its numerical solution, it is interesting to write Eq.\n(\\ref{gn}) in a differential form:\n\\begin{eqnarray}\\label{gndf}\n&&g_n^{0\\,\\prime\\prime}(z)+2(n-1)z(1-z^2)^{-1}g_n^{0\\,\\prime}(z)\n\\nonumber \\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ -n(n-1)(1-z^2)^{-1}g_n^0(z)\\nonumber \\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ +\\frac{\\alpha}{\\pi}\\frac{1}{(1-z^2)Q(z)}g_n^0(z)=0,\n\\end{eqnarray}\nwith the boundary conditions $g_n^0(\\pm 1)=0$.\n\\par\n\nFor a fixed $n$, Eq. (\\ref{gndf}) has an infinite number of\nsolutions, labeled by an additional quantum number\n$\\kappa=0,1,2,\\ldots$, which also labels the corresponding discrete\nspectrum of mass squared eigenvalues $M_{n\\kappa}^{2}$. \nWe will use the notation $g^{\\nu}_{n\\kappa}$ to identify a particular\nsolution.\nThe function {$g_{n\\kappa}^0$} has $\\kappa$ nodes within the interval\n\\mbox{$]-1,+1[$} and a well-defined parity given by $\\kappa$ \\cite{cutk}:\n\\[g^0_{n\\kappa} (-z)= (-1)^{\\kappa} g^0_{n\\kappa}(z) \\]\nThe parity is also preserved inside the ensemble $\\{ g_{n\\kappa}^{\\nu} \\}$\nwhen varying $\\nu=0,1,\\ldots$ and this entails, through Eq. (\\ref{Phi}),\nthat for even (odd) values of $\\kappa$,\nthe BS amplitude $\\Phi(k,p)$ is an even (odd) function of the relative\nenergy $k_0^{}$ in the c.m. frame. \\par\n\nThe mass squared $M^2$ of the ground state $g_{10}^0$ as function of the\ncoupling constant $\\alpha$ is shown in Fig. \\ref{Fig_M2_alpha_10}.\nIts value vanishes for $\\alpha=2\\pi$. In the range $\\alpha \\in [0,2\\pi]$,\n$M^2\\ge 0$, the total mass of the system $M$ is well defined as well as\nits binding energy $B=2m-M>0$.\nThis determines the domain where this model is physically consistent\nwith a well-defined ground state.\nIt is worth mentioning, however, that the solutions of the BS equation,\nas well as its spectral parameter $M^2$, can be analytically continued\nfor $\\alpha>2\\pi$ \nwithout encountering any kind of singularity. This is illustrated with\nthe dashed line in the lower right corner of the figure. \nAll excited states lie above the $M^2(\\alpha)$ curve and thus can have\na well-defined $M$ even in the unphysical region.\n\n \\vspace{.5cm}\n\\begin{figure}[h!]\n\\begin{center}\n\\mbox{\\epsfxsize=8.cm\\epsffile{M2_alphaext_1_0.eps}} \n\\hspace{.5cm}\n\\end{center}\n\\caption{(Color online) Dependence of the squared mass of the ground\n state ($n=1, \\kappa=0)$ on the coupling constant $\\alpha$. \n Beyond the critical value $\\alpha=2\\pi$, the solutions are smootly\n continued without any singularity but having negative values of\n $M^2$. \n The physical region, where the system has well-defined ground state\n mass and binding energy, is thus limited to $\\alpha\\in[0,2\\pi]$.}\n\\label{Fig_M2_alpha_10}\n\\end{figure}\n\nAmong the infinity of solutions existing for a given $n$, the one with\n$\\kappa=0$ coincides, in the limit \nof small binding energies $B\/m\\ll 1$, with the solution of the\nnon-relativistic Coulomb problem with main quantum number $n$. \nThis solution is called, following the original works of\nWick and Cutkosky \\cite{wick,cutk}, ``normal''.\nIndeed, these authors, analyzing in this limit the system of equations\nfor the functions $g_n^k(z)$ determining the BS amplitude (\\ref{Phi}), \nreproduced, for $\\kappa=0$, the Coulomb spectrum, i.e. the Balmer series \n\\begin{equation}\\label{Balmer}\nB_n=\\frac{m\\alpha^2}{4n^2}.\n \\end{equation}\nThis result corresponds to the Schr\\\"odinger equation with the potential\n$V(r)=-\\frac{\\alpha}{r}$. \nThe relativistic perturbative correction to\nthe binding energy (\\ref{Balmer}) was found in \\cite{FFT}; the binding\nenergy, incorporating it, reads\n\\begin{equation}\\label{B_Pert}\n B_n=\\frac{m\\alpha^2}{4n^2}\\left[ 1 - {4\\alpha\\over\\pi}\n \\ln\\left({1\\over\\alpha}\\right) + {\\mathcal O}(\\alpha) \\right]. \n \\end{equation}\n\nOn the contrary, the solutions corresponding to non-zero values of\n$\\kappa$ ($\\kappa=1,2,\\ldots$), have a spectrum totally decoupled\nfrom the non relativistic one.\nThey are genuinely of relativistic nature, without non-relativistic\ncounterparts, and were named \"abnormal\" by Wick.\n\nThese different behaviours are illustrated in Fig. \\ref{lambda_B} where\nwe have displayed the dependence of the coupling constant\n$\\lambda={\\alpha\\over\\pi}$ on the binding energy $B$\nfor the lowest solutions of the W-C model. \nUpper panel contains only the n=1 states with $\\kappa=0,1,2,3,4$. \nThe curve corresponding to $\\kappa=0$ (black solid line) is tangent\nto the non-relativistic (NR) one (black dashed line) from which\nit departures logarithmically, as it is visible, starting at\n$B\\approx 0.001$.\nThe perturbative results, provided by Eq. (\\ref{B_Pert}), are\nindistinguishable from the exact ones in the considered energy range.\nThose corresponding to $\\kappa>0$ (colored solid lines) do not have any\nnon-relativistic counterparts. \nLower panel represents the spectrum for $n\\ge 1$ states and different\nvalues of $\\kappa=0,1,2,3$.\nThe horizontal line $\\lambda=2$ ($\\alpha=2\\pi$) indicates the maximal\nvalue of the coupling constant ensuring a well-defined ground state. \n\n\\begin{figure}[h!]\n\\vspace{.5cm}\n\\begin{center}\n\\mbox{\\epsfxsize=8.2cm\\epsffile{lambda_B_mu_0.00_n_1_LogLog.eps}} \\\\\n\\vspace{1.cm}\n\\mbox{\\epsfxsize=8.cm\\epsffile{lambda_B_mu_0.00_n_LogLog_0.001_0.5.eps}} \n\\end{center}\n\\caption{(Color online) Spectrum of the W-C model as a function of the\n coupling constant $\\lambda(B)$ in the low energy limit. Upper panel\n corresponds to n=1 states with different values of $\\kappa=0,1,..$ .\n The case $\\kappa=0$ is compared\nto the non relativistic solution. Horizontal lines correspond respectively \nto the maximal values of the coupling constant for a well-defined\nground state ($\\lambda=2$) and to the minimal value for\nwhich the abnormal solutions exist ($\\lambda=1\/4$).\nLower panel contains the full spectrum for $n\\le6$ and $\\kappa\\le3$\nto make explicit the different crossings. Notice that the unphysical\nsolutions with odd $\\kappa$ (giving no contribution to the S-matrix)\nare naturally inserted in the spectrum.\n\\label{lambda_B}}\n\\end{figure}\n\n\nThe normal and abnormal solutions have also different domains of\nexistence with respect to the coupling constant $\\alpha$. \nAs a mathematical solution of the BS equation (\\ref{bs}), the normal\nsolutions exist for any (positive) values of $\\alpha$, although, as we\nhave already discussed,\nthey have a clear physical meaning only in the range $\\alpha\\in[0,2\\pi]$,\nwhere $M^2\\ge 0$.\nHowever, the very existence of the abnormal solutions (all of them)\nrequires a coupling constant greater than some critical value,\n$\\alpha\\ge{\\pi\\over4}$ ($\\lambda\\ge1\/4$). \nThis can be clearly seen from the results of Fig. \\ref{lambda_B}, where\nall the abnormal states (color line) were\nfound above the horizontal $\\lambda=1\/4$ line. \n\nWick and Cutkosky \\cite{wick,cutk} found the following approximate\nanalytic expression for the abnormal spectrum near the continuum\nthreshold ($B\\to 0$): \n\\begin{equation} \\label{abnspectr}\n M_{n\\kappa}^2\\simeq 4m^2\\left( 1 - e^{ -{(\\kappa-1)\\pi\/\n \\sqrt{{\\alpha\\over \\pi}-{1\\over 4}}}}\\right),\\;\\kappa=2,3,\\ldots\\ , \n\\end{equation}\nwhere the condition $\\alpha>\\pi\/4$ is explicitly obtained. \nAt $B\/m\\ll 1$ this spectrum vs. $\\kappa$ does not depend on $n$.\nIt also indicates that to be able to distinguish abnormal states from the\ncontinuum threshold on experimental grounds, the coupling constant\nshould be increased at least up to values of $\\alpha\\approx 4\\div 5$;\notherwise, for values of $\\alpha$ very close to $\\pi\/4$, the\nexponential in Eq. (\\ref{abnspectr}) is nearly zero and the discrete\nspectrum becomes hardly distinguishable from the continuum.\n\nIt is worth noting that the existence of a lower bound of the\ncoupling constant for the abnormal solutions is reminiscent of the\nmassive-exchange case, i.e. $\\mu\\ne0$ in the kernel (\\ref{ladder}),\nwhich was considered with some detail in \\cite{MC_PLB474_2000} both in\nthe BS and the Light-Front Dynamics frameworks.\nThe $B_{\\mu}(\\alpha)$ dependences (Figs. 5 and 7 of\n\\cite{MC_PLB474_2000}) are similar to those displayed in Fig.\n\\ref{lambda_B}), what suggest the possibility \nto associate a mass with the abnormal states.\nHowever, essential differences in the number of bound states for a\ngiven $\\mu$ remain: infinite in the W-C model and (at most) finite\nin the massive case.\n\nIn summary, the range of the coupling constants to be considered in\nthis model is $0<\\alpha<2\\pi$ ($0<\\lambda<2$) for the\nnormal states, i.e. $\\kappa=0$, and ${\\pi\\over 4}<\\alpha<2\\pi$\n($ 1\/4<\\lambda<2$) for the abnormal ones ($\\kappa>0$).\nOn another hand, according to Refs. \\cite{cia,nai}, the abnormal\nsolutions with odd\nvalues of $\\kappa$ do not contribute to the $S$-matrix and therefore only\nthose with even $\\kappa$ can have a physical meaning. \nIn the subsequent part of this work, we will concentrate on the latter case.\nFurthermore we will restrict ourselves, to the $l=0$ states with $n=1$\nand $n=2$.\n\nFor $n=1$ states, the sum (\\ref{Phi}) is reduced to a single term\ninvolving only the function $g_1^0$ satisfying the homogeneous integral\nequation\n\\begin{equation}\\label{g10_Int}\n g_1^0(z)=\\frac{\\alpha}{2\\pi}\\int_{-1}^1 \\frac{R(z,z')}{Q(z')} \\,\n g_1^{0}(z')dz',\n\\end{equation}\nor, equivalently, in its differential form\n\\begin{equation}\\label{g0}\n{g''}_1^0(z)+\\frac{\\alpha}{\\pi}\\frac{1}{(1-z^2)Q(z)}g_1^0(z)=0,\n\\end{equation}\nwith the boundary conditions $g_1^0(\\pm 1)=0$.\nThe corresponding BS amplitude is expressed in terms of $g_1^0$ as\n\\begin{equation}\\label{Phi10}\n \\Phi_1(k,p)=\\int_{-1}^1\\frac{-im^3 \\,g_{1}^0(z)dz}{[m^2-\\frac{1}\n {4}M^2 -k^2-p\\makebox[0.08cm]{$\\cdot$} k\\,z-\\imath\\epsilon]^{3}}.\n\\end{equation}\n\n\\bigskip\nFor $n=2$ states, the sum (\\ref{Phi}) involves two functions $g_2^0$\nand $g_2^1$. The function $g_2^0$ satisfies the homogeneous integral\nequation:\n\\begin{equation}\\label{g20_Int}\n g_2^0(z)=\\frac{\\alpha}{4\\pi}\\int_{-1}^1 \\frac{R^2(z,z')}{Q(z')} \\,\n g_2^{0}(z')dz',\n\\end{equation}\nand in differential form\n\\begin{eqnarray}\\label{eq1c}\n{g''_2}^0(z)&+&\\frac{2z}{(1-z^2)}{g'}_2^0(z)-\\frac{2}{(1-z^2)}g_2^0(z)\n\\nonumber\\\\\n&+&\\frac{\\alpha}{\\pi}\\frac{1}{(1-z^2)Q(z)}g_2^0(z)=0,\n\\end{eqnarray}\nwith the boundary conditions $g_2^0(\\pm 1)=0$, while $g_2^1$ is determined\nfrom $g_2^0$ through the integral equation\n\\begin{eqnarray}\\label{g21}\n g_2^1(z)&=& \\frac{\\alpha}{6\\pi}\\int_{-1}^1\\frac{R(z,z') }\n {[Q(z')]^2}\n \\,g_2^{0}(z')dz' \n \\nonumber\\\\\n &+& \\frac{\\alpha}{2\\pi}\\int_{-1}^1\\frac{R(z,z')}\n {Q(z')} \\, g_2^{1}(z')dz',\n\\end{eqnarray}\nwhich can also be rewritten in the form of an inhomogeneous\ndifferential equation:\n\\begin{eqnarray}\\label{eq21c}\n&& {g''}_2^1(z)+\\frac{\\alpha}{\\pi}\\frac{1}{(1-z^2)Q(z)}g_2^1(z)\n\\nonumber \\\\\n&&\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n= - \\frac{\\alpha}{3\\pi}\\frac{1}{(1-z^2)[Q(z)]^2}g_2^0(z).\n\\end{eqnarray}\n\\par\nThe BS amplitude (\\ref{Phi}) is now expressed in terms of two functions\n$g_2^1$ and $g_2^0$:\n\\begin{eqnarray}\\label{Phi2}\n \\Phi_2(k,p)&=&\\int_{-1}^1\\frac{-im^3 \\,g_{2}^1(z)dz}\n {[m^2-\\frac{1}{4}M^2 -k^2-p\\makebox[0.08cm]{$\\cdot$} k\\,z-\\imath\\epsilon]^{3}}\n \\nonumber\\\\\n &+&\\int_{-1}^1\\frac{-i m^5\\,g_{2}^0(z)dz}\n {[m^2-\\frac{1}{4}M^2 -k^2-p\\makebox[0.08cm]{$\\cdot$} k\\,z-\\imath\\epsilon]^{4}}.\n\\end{eqnarray}\n\n\n\\subsection{Numerical solutions for some selected states $g_{n\\kappa}^k$}\\label{g_num}\n\nWe present, in this subsection, the numerical results concerning the\nfirst states of the W-C spectrum. \nWe fix hereafter $m=1$ and the coupling constant to the value $\\alpha=5$.\nWe will consider along the work an ensemble of states with $n=1,2$ and\n$\\kappa=0,2,4$ that,\nfor the sake of simplicity in notation, will be numbered with No. 1-6\nin the Tables \\ref{tab1} and \\ref{tab2}.\n\n\\begin{table}[h!]\n\\begin{center}\n\\caption{Binding energy $B$ and two-body norm ($N_2$)\nof the low-lying normal ($\\kappa=0$) and abnormal\n($\\kappa=2$) states, for the coupling\nconstant value $\\alpha=5$.}\\label{tab1}\n\\begin{tabular}{cccll}\n\\hline\\noalign{\\smallskip} \nNo.& n & $\\kappa$ & $B$ & $N_2$ \\\\\n\\noalign{\\smallskip}\\hline \n\\noalign{\\smallskip}\n1&1&0 & 0.999259 &0.65 \\\\\n2&2&0 & 0.208410 & 0.61 \\\\\n3& 1 &2 &\n$3.51169 \\cdot 10^{-3}$\n& 0.094 \\\\\n4& 2 &2 &\n$1.12118 \\cdot 10^{-3}$\n& 0.077 \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nThe binding energies for the lowest $n=1,2$ normal ($\\kappa=0$) and\nabnormal ($\\kappa=2$) states $g_{n\\kappa}^{\\nu}$ are presented in Table\n\\ref{tab1}.\nAll $\\nu=0$ components are arbitrarily normalized to $g_{n\\kappa}^0(0)=1$. \nThe corresponding solutions for the $n=1$ states -- $g_{10}^0$ and\n$g_{12}^0$ -- are displayed in Figs. \\ref{fig1} and \\ref{fig1p}.\nThey have comparable sizes\nand their nodal structure is determined by $\\kappa$ only.\n\n\\begin{figure}[h!]\n\\vspace{.8cm}\n\\begin{minipage}[h!]{8.7cm}\n\\begin{center}\n\\epsfxsize=7.cm\\epsfysize=5.cm\\epsfbox{g100_Nb1.eps}\n\\caption{$g_{10}^0$ for the normal state No. 1 of Table \\ref{tab1}.} \\label{fig1}\n\\end{center}\n\\end{minipage}\n\n\\begin{minipage}[h!]{8.7cm}\n\\vspace{.9cm}\n\\begin{center}\n\\epsfxsize=7.cm\\epsfysize=5.cm\\epsfbox{g120_Nb3.eps}\n\\caption{$g_{12}^0$ for the abnormal state No. 3 of Table \\ref{tab1}.}\\label{fig1p}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\nThe two components $g_{2\\kappa}^0$ and $g_{2\\kappa}^1$ of the $n=2$\nstates are plotted in Fig. \\ref{g2} (state No. 2 with $\\kappa=0$)\nand Fig. \\ref{g2ab} (state No. 4 with $\\kappa=0$). \nThe component $\\nu=1$ is dominant in both cases, but for the state No. 4\nit is $\\sim 1000$ times larger (see the scaling factor in Fig. \\ref{g2ab}).\nThis enhancement is due to the $Q^2$ factor in the denominator of the\nright-hand-side of Eq. (\\ref{eq21c}), \nwhich, in the limit $B\\to 0$ and around $z=0$, behaves as\n$Q^2\\approx B^2 $.\nThus, for $n>1$ states with small binding energies, the component\n$g_{n\\kappa}^0$ that determines $M^2$ is\nnegligibly small with respect to the other components.\nWe will see, however, in the following section that they all\nplay equivalent roles in the construction of the BS amplitude itself \nand in the form factors.\n\n\\begin{figure}[h!]\n\\vspace{0.8cm}\n\\begin{center}\n\\epsfxsize=7cm\\epsfysize=5cm\\epsfbox{g200_g201_Nb2.eps}\n\\caption{Components $g_{20}^0$ and $g_{20}^1$ of the normal state No. 2 from Table \\ref{tab1}.}\\label{g2}\n\\end{center}\n\\end{figure}\n\\vspace{0.5cm}\n\n\\begin{figure}\n\\vspace{0.8cm}\n\\begin{center}\n\\epsfxsize=7cm\\epsfysize=5cm\\epsfbox{g220_g221_Nb4.eps}\n\\caption{$g_{22}^0$ and $g_{22}^1$ (scaled by a factor $10^3$) of the abnormal state No. 4 (Table \\ref{tab1}).}\n\\label{g2ab}\n\\end{center}\n\\end{figure}\n\n\\vspace{0.5 cm}\nIn the rightest column of Table \\ref{tab1} we have also included the\nnorm $N_2$ of the two-body contributions in the Fock space, as it is defined in \\ref{appN2}.\nFor the states with $n=1$, $N_2$ is given by Eq. \nof \\ref{appN2}\nand for $n=2$, by Eqs. ({\\ref{norm2}). We remark therein that the\ntwo-body norm $N_2$ for the abnormal ($\\kappa=2$) states is much smaller\nthan for the normal ($\\kappa=0$) ones.\nThis comparison concerns however states covering the two extreme cases\nin the spectrum: deeply bound states (Nos. 1 and 2) and\nnearthreshold ones (Nos. 3 and 4).\nTo better understand this difference, we have \nstudied the dependence of $N_2$ on the binding energy for the first\nnormal and abnormal states. Results are displayed in Fig. \\ref{N2_B}.\n\n\\begin{figure}[h!]\n\\vspace{0.9cm}\n\\begin{center}\n\\epsfxsize=7. cm\\epsfysize=5cm\\epsfbox{N2_B_N.eps}\\\\\\vspace{1.cm}\n\\epsfxsize=7. cm\\epsfysize=5cm\\epsfbox{N2_B_A.eps}\n\\caption{(Color online) $N_2$-dependence on the binding energy for\n the $n=1$ and $n=2$ states: normal (upper panel) and abnormal (lower\n panel).}\n\\label{N2_B}\n\\end{center}\n\\end{figure}\n\n\nThe upper panel concerns the normal states. \nThe behaviours of the the $n=1$ (black solid line) and $n=2$ (red\nsolid line) states are quite similar: \n$N_2$ decreases monotonically from $N_2\\approx 1$ when $B\\approx 0$\ndown to an asymptotic value when $B\\to 2m$. We found numerically\n$N_2(2m)\\approx 0.64$ for n=1 and $N_2(2m)\\approx 0.59$ for n=2.\nFor the ground state n=1, these limiting values were found analytically\nin \\cite{dshvk}, as well as the perturbative expansion\nin their vicinity. Thus, the limit $B\\to0$ is described by Eq.\n(\\ref{N2_1a}), i.e., \n\\[ N_2(B)= 1+{1\\over\\pi}\\sqrt{{4B\\over m}}\\ln\\left({4B\\over m}\\right). \\]\nAt $B\/m=10^{-5}$, this perturbative expansion gives $N_2=0.980$ in\nclose agreement with the black curve of Fig. \\ref{N2_B}.\nWe conclude from this study that the normal states are dominated by\ntwo-body norms. This is particularly true in the limit $B\\to 0$,\nwhere $N_2\\to 1$, but remains also true in all the energy domain,\nalthough decreasing with increasing $B$.\n\nA very different behaviour is observed with the abnormal states,\nrepresented in the lower panel of Fig. \\ref{N2_B}.\nAs one can see, the two body norm $N_2$ of these states not only remains\ncomparatively very small, but also vanishes in the non-relativistic\nlimit, making them, in this region, genuine many-body states.\nThe one order of magnitude observed in Table \\ref{tab1} for the\nbinding energies hides in fact\na deeper and striking difference between normal and abnormal BS states,\nindependent of their comparison with the non-relativistic spectrum.\nIt is provided by their two-body content: abnormal states do not have\nin the limit $B\\to 0$ any two-body contribution and have, thus,\ngenuine many-body structures.\nBeyond this limit the norm of the two-body sector remains extremely\nsmall. This is the reason why they are absent in the non-relativistic\nlimit reduced to the two-body Schr\\\"odinger equation.\n\n\nThe results presented in Table \\ref{tab1} are completed in\nTable \\ref{tab2} by studying the $\\kappa=4$ excitations of $n=1,2$ \nstates. The same conclusion holds, even in a more dramatic way.\nTheir two body norms are one order of magnitude smaller than for the\n$\\kappa=2$ states of Table \\ref{tab1}.\nThis can be expected due to their smaller binding energies and in view\nof the behaviour described in the lower panel of Fig. \\ref{N2_B}.\n\n\n\\begin{table}[h!]\n\\begin{center}\n \\caption{Same as in Table \\ref{tab1}, for the abnormal states with $\\kappa=4$.} \\label{tab2}\n\\begin{tabular}{cccll}\n\\hline\\noalign{\\smallskip} \nNo.& n & $\\kappa$ & $B$ & $N_2$ \\\\\n\\noalign{\\smallskip} \\hline\n\\noalign{\\smallskip}\n5&1 &4&\n$1.54091\\cdot 10^{-5}$ & $6.19\\cdot 10^{-3}$ \\\\\n6&2 &4&\n$4.95065 \\cdot 10^{-6}$ &\n$2.06\\cdot 10^{-5}$\\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe abnormal solution $g_{14}^0$} for $n=1, \\kappa=4$ state (No. 5\nin Table \\ref{tab2}), is shown in Fig. ~\\ref{fig10_14}, displaying its\nmore involved nodal structure (4 zeros in $]-1,+1[$). \nThe functions {$g_{24}^{\\nu}$ of the abnormal state $n=2, \\kappa=4$\n(No. 6 in Table \\ref{tab2}), are plotted in Fig. \\ref{g240_g241_Nb6}.\nThe extreme smallness of the binding energy of this state\ngenerates a huge enhancement factors in the inhomogeneous equation\n(\\ref{eq21c}) (through the factor Q) which results in \na huge dominance of the $\\nu=1$ component in the full BS amplitude.\nNotice that $g_{24}^1$ has been reduced by a factor $10^5$ to become\ncomparable with $g_{24}^0$.\n\n\\begin{figure}[h!]\n\\vspace{1.2cm}\n\\begin{minipage}[h!]{8.5cm}\n\\begin{center}\n\\epsfxsize=8.cm\\epsfysize=5cm\\epsfbox{g140_Nb5.eps}\n\\caption{$g_{14}^0$ from state No. 5 in Table \\ref{tab2}.}\n\\label{fig10_14}\n\\end{center}\n\\end{minipage}\n\\vspace{1cm}\\\\\n\\begin{minipage}[h!]{8.5cm}\n\\begin{center}\n\\epsfxsize=8.cm\\epsfysize=5cm\\epsfbox{g240_g241_Nb6.eps}\n\\caption{(Color online) $g_{24}^{\\nu}$ from state No. 6 in Table \\ref{tab2}.}\n\\label{g240_g241_Nb6}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\n\n\n\n\\section{Electromagnetic form factors}\\label{FFs}\n\nWe suppose that one of the two constituent particles is charged. \nThe electromagnetic form factor of the system can be expressed in\nterms of its BS amplitude. It is enough to consider inelastic\ntransitions from an initial $| i\\rangle$ to a final $| f\\rangle$ state.\nThe elastic form factors are obtained from them as a particular case,\nwith $f=i$.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics{triangle.eps}\n\\caption[*]{Feynman diagram for the electromagnetic form factor.\n\\label{triangle}}\n\\end{figure}\n\\par\nThe electromagnetic vertex $J_{\\mu}$, corresponding to a transition\n$| i\\rangle\\to | f\\rangle$ is shown graphically in Fig.~\\ref{triangle}. \nThe corresponding vertex amplitude reads (we use Itzykson and Zuber\n\\cite{IZ} conventions for the Feynman rules):\n\\begin{eqnarray}\\label{ffGam}\niJ_{\\mu}&=&\\int \\frac{d^4k}{(2\\pi)^4}\\,\n\\frac{i[-i(p+p'-2k)_{\\mu}]}{(k^2-m^2+i\\epsilon)}\n\\nonumber\\\\\n&\\times&\\frac{i\\left[-i\\overline{\\Gamma}\n\\left(\\frac{1}{2}p' -k,p'\\right)\\right]}\n{[(p'-k)^2-m^2+i\\epsilon]}\n\\frac{i\\left[i\\Gamma \\left(\\frac{1}{2}p-k,p\\right)\\right]}\n{[(p-k)^2-m^2+i\\epsilon]},\n\\end{eqnarray}\nwhere $\\Gamma(k,p)$ is the vertex function, related to\nthe BS amplitude by the equation\n\\begin{eqnarray}\\label{PhiG}\n& &\\Phi(k,p)=\\frac{\\Gamma(k,p)}\n{\\left[(\\frac{p}{2}+k)^2-m^2+i\\epsilon\\right]\n\\left[(\\frac{p}{2}-k)^2-m^2+i\\epsilon\\right]}.\\nonumber \\\\\n& &\n\\end{eqnarray}\n$\\overline{\\Gamma}$ is the conjugate of $\\Gamma$, obtained from the\nlatter by complex conjugation and use of the anti\\-chrono\\-logical\nproduct. A similar definition also holds for the BS amplitude\n$\\overline{\\Phi}$\n\\cite{nak69}.\n\\par\nThe electromagnetic vertex for the transition $i\\to f$ is expressed in\nterms of the BS amplitude as\n(see e.g. Eq. (7.1) in \\cite{cdkm}):\n\\begin{eqnarray}\\label{ffbs}\nJ_{\\mu}\n&=&i\\int \\frac{d^4k}{(2\\pi)^4}(p+p'-2k)_\\mu \\; (k^2-m^2)\n\\nonumber\\\\\n&\\times&\\overline{\\Phi}_f\\left(\\frac{1}{2}p'-k,p'\\right)\\Phi_i\n\\left(\\frac{1}{2}p-k,p\\right). \n\\end{eqnarray}\nIt has the following general decomposition in terms of two scalar\nfunctions (see\\footnote{We change the notations in comparison to Ref.\n\\cite{tff}, where the form factor $G$ was denoted $F'$.} \\cite{tff}):\n$F(Q^2)$ and $G(Q^2)$ \n\\begin{eqnarray}\\label{ffc}\nJ_{\\mu}&=&\\left[(p_{\\mu}+{p'}_{\\mu})+ ({p'}_{\\mu}-p_{\\mu})\\frac{Q_c^2}\n{Q^2}\\right]F(Q^2)\n\\nonumber\\\\\n&-& ({p'}_{\\mu}-p_{\\mu})\\frac{Q_c^2}{Q^2}G(Q^2).\n\\end{eqnarray}\nHere, $q=p'-p$, $Q^2=-q^2=-(p'-p)^2$ and\n\\begin{equation} \\label{Qc2}\nQ_c^2={M_f}^2-M_i^2,\n\\end{equation} \n$M_i$ and $M_f$ being the masses of the initial and final states,\nrespectively.\n\nSince $q\\makebox[0.08cm]{$\\cdot$} J=Q_c^2 G(Q^2)$, the above decomposition does not suppose,\nin general, current conservation $q\\makebox[0.08cm]{$\\cdot$} J=0$, which implies \n\\begin{equation}\\label{G}\nG(Q^2) \\equiv 0.\n\\end{equation} \nA direct proof of this result from the BS equation is presented in\n \\ref{proofG}.\nThe current conservation becomes a stringent self-consistency criterion\nof our results: the calculated form factor $G(Q^2)$ should\nbe identically zero, or very small within the numerical uncertainties.\n \nWe also note that since $J_{\\mu}$ [Eq. (\\ref{ffc})] is not singular\nat $Q^2=0$, the following relation should hold:\n\\begin{equation}\\label{FeqFp}\nF(0)=G(0). \n\\end{equation}\nEquation (\\ref{G}) then implies that $F(0)=0$ for the transition form\nfactors (for which $Q_c^2\\neq 0$).\n\\par\nFrom Eq. (\\ref{ffc}), the form factors are expressed throu\\-gh\n$J_{\\mu}$ as\n\\begin{eqnarray}\\label{FFp}\n& &F(Q^2) = \\frac{ (p+p')\\makebox[0.08cm]{$\\cdot$} J\\,Q^2 + q\\makebox[0.08cm]{$\\cdot$} J \\, Q_c^2}\n{[(M_f-M_i)^2 +Q^2][(M_f+M_i)^2+Q^2]},\\nonumber \\\\\n& &G(Q^2) = \\frac{q\\makebox[0.08cm]{$\\cdot$} J}{Q_c^2}.\n\\end{eqnarray}\nConsidering these formulas at $Q^2=0$ (which does not imply $q=0$),\none gets the relation\n$F(0)=\\left.\\frac{q\\makebox[0.08cm]{$\\cdot$} J}{Q_c^2}\\right|_{Q^2=0}=G(0)$, which\nreproduces Eq. (\\ref{FeqFp}). \n\\par\nThe expressions for the form factors are obtained by substituting\nin Eqs. (\\ref{FFp}) the current $J$ [Eq. (\\ref{ffbs})], then\nsubstituting the BS amplitudes (\\ref{Phi}), using the Feynman\nparametrization and integrating over $k$. In this way, we find the\nform factors in the form of integrals over products of functions\n$g_n^{\\nu}(z)$ and $g_{n'}^{\\nu'}(z')$. (Details of similar calculations can\nbe found in Ref. \\cite{ckm_ejpa}.) \nThe result for the transition form factor $F(Q^2)$ can be written\nin the form:\n\\begin{equation}\\label{ff}\nF(Q^2)=\\sum_{\\nu=0}^{n-1}\\sum_{\\nu'=0}^{n'-1}F_{nn'}^{\\nu\\nu'},\n\\end{equation}\nand similarly for $G(Q^2)$.\nThe expressions of the functions of the right-hand-side of this\nequation for the cases $n=n'=1$ and $n=n'=2$ are given in\n\\ref{exprssff}.\n\n\nIn the following subsections, we examine the numerical results for the\nelastic and transition form factors for some states from Tables\n\\ref{tab1} and \\ref{tab2}, corresponding to the coupling constant\n$\\alpha=5$. \n\n\n\\subsection{Elastic form factors $F_e(Q^2)$}\\label{el}\n\nThere are two regions of interest in studying the elastic form factors\n($F_e$):\nthe region near the origin, giving insight into the size of the system,\nand the asymptotic region $Q^2\\gg m^2$, related to the many-body structure\nof the wave function \\cite{matvmurtavk,brodsfarr,radyush}.\n\nWith the elastic form factor of a bound state, normalized to\n$F_e(Q^2=0)=1$, the squared radius is given by\n\\begin{equation}\\label{r2}\n= -6 \\left(dF_e\\over dQ^2\\right)_{Q^2=0} \n\\end{equation}\nand the root mean squared (r.m.s.) radius by $R=\\sqrt{}$. \nIn the non-relativistic theory, the size of a bound state scales, as\na function of its binding energy, as $R\\approx {1\\over\\sqrt{mB}}$. \n\nOn the other hand, as mentioned in the Introduction, the asymptotics\nof $F_e$ should qualitatively probe the compositeness of the state. \nAccording to \\cite{matvmurtavk,brodsfarr,radyush}, the elastic form\nfactors of a $n$-body system should decrease as $1\/(Q^2)^{n-1}$,\nwhere $n$ is the number of the constituents of the state.\nIt is, however, worth emphasizing here some essential differences of\nthe W-C model with the theoretical framework in which the above\nasymptotic behaviors have been obtained. The latter have been derived\nin theories characterized by dimensionless coupling constants, like\nQCD and the parton model. In the W-C model, bosonic fields interact\nby the exchange of a scalar particle; the coupling constant $g$ is\nthen dimensionful, having the dimension of mass.\nThis has an immediate consequence on the behavior of the BS\namplitude at large momenta. In the W-C model, as can be checked from\nEqs. (\\ref{bs}) and (\\ref{ladder}), the BS amplitude behaves at large\nmomenta as $1\/(k^2)^3$. In QCD, in the ladder approximation, where\nquarks interact by means of an exchange of a gluon field, the\nscalar part of the BS amplitude behaves at large momenta as\n$1\/(k^2)^2$, up to logarithms. Extending the comparison to bosonic\n$\\phi^4$-like theories, where the coupling constant is also dimensionless,\nthe interaction between two bosonic constituents is realized either\nby contact terms, or by the exchange of a two-particle loop; in both\ncases the large-momentum behavior of the BS amplitude is again \n$\\frac{1}{(k^2)^2}$, up to logarithms. The faster decrease of the\nBS amplitude in the W-C model affects the behaviors of the form factors: \none thus expects in this model behaviors of the type $1\/(Q^2)^n$, up\nto logarithms, instead of $1\/(Q^2)^{n-1}$. \n\nIn QCD, the number $n$ represents the number of valence quarks\n(including, eventually, the number of valence gluons, in the case\nof hybrids).\nThe Fock sectors of the state which contain the sea quarks, therefore\nwith higher $n$'s, are expected to decrease asymptotically faster and\nthus to display rapidly the asymptotic dominance of the valence quark\nsector, a phenomenon well observed on experimental grounds. In the\nW-C model, for the normal solutions, one indeed expects the dominance\nof the two-body sector, as was concluded in Sec. \\ref{WCsol}.\nHowever, for the abnormal solutions, the two-body sector is weakly\ncontributing to the composition of the corresponding states, which are\ndominated by higher sectors of the Fock space. One therefore expects\nhere a competition between the contributions of the higher sectors\nof the Fock space, which asymptotically decrease more rapidly, but\nhave large coefficients, and the contribution of the two-body sector,\nwhich dominates in the asymptotic region, but with a small coefficient.\n\nFor abnormal solutions, or hybrid states, of the \\mbox{W-C} model, a \nrefined analysis necessitates the distinction between three regimes in\nthe $Q^2$ evolution of the form factors, rather than two:\nthe very small $Q^2$ regime, determined by the binding energy (and \nequivalently by the rms radius), \nthe intermediate $Q^2$ region where the decrease is determined by\nthe many-body components (and therefore is fast)\nand the asymptotic region, where the many-body contribution is\nexhausted, and only the two-body contribution survives. \nSince in the W-C model the content of any state is a two-body component\nplus an indefinite number of exchange particles, \nthe asymptotic behavior of all the elastic form factors is finally\ndetermined by its two-body contribution. \nTherefore, the asymptotic $Q^2$-dependence should be the same, though\nwith different coefficients, for all elastic form factors. \nIn particular, we predict that the ratios of all elastic form factors\nshould tend to constants at $Q^2\\to\\infty$.\n\nWe first examine the $n=1$ states. They are defined by a single component\n$g_{1\\kappa}^0$, the same that determines the binding energy.\nWe have plotted, in Fig. \\ref{Fel_1_3}, in solid lines, the elastic\nform factors for the two $n=1$ states of Table \\ref{tab1}: on top,\nthe normal state No. 1 ($\\kappa=0$ , $B=0.999$), and at bottom, the\nabnormal state No. 3 ($\\kappa=2$, $B=0.00359$). \n\n\\vspace{0.5cm}\n\\begin{figure}[h!]\n\\begin{center}\n\\epsfxsize=7.5cm\\epsfysize=5cm\\epsfbox{FQ2_Nb1.eps}\\vspace{1cm}\\\\\n\\epsfxsize=7.5cm\\epsfysize=5cm\\epsfbox{FQ2_Nb3.eps}\n\\end{center}\n\\caption{Elastic form factors of $n=1$ states of Table \\ref{tab1}\n(solid lines): No. 1 (normal) on top and No. 3 (abnormal) at bottom;\nthe dashed line corresponds to the state with $\\kappa=0$ with the same\nbinding energy (and rms radius) as the state No. 3 with\n$\\kappa=2$.}\n\\label{Fel_1_3}\n\\end{figure}\n\nThe corresponding rms radii are $R_1=1.16$ fm and $R_3=15.7$ fm,\nrespectively, which roughly scale as $\\sim{1\\over \\sqrt{mB}}$.\nBoth states have only one component $g_{n\\kappa}^0$, which in turn\ndetermines the energy. It is then natural that they have a similar\nbehaviour, close to that of the non-relativistic one.\nAt this level, one cannot see any drastic difference between a normal\nand an abnormal state. \nThe behaviours of their form factors are quite similar, however with\na much faster decrease\nfor the state No. 3, as expected from its smaller binding energy\nand its many-body structure. At $Q^2=1$, the value of $F_e$ for the\nstate No. 3 is three orders of magnitude smaller than that of the\nstate No. 1.\nIn order to disentangle the contributions of the binding\nenergy and of the asymptotic behaviour,\nwe have adjusted, at a second step, $\\alpha$ of the state No. 1 to\nhave the same binding energy as the state No. 3. \nThe result is displayed in dashed line on the lower panel: both curves\nare tangent to each other at the origin, implying that the rms radii\nare the same, but one notices that the abnormal state form factor still\ndecreases much faster than that of the normal one, by a factor 10 at\n$Q^2=1$. \n\n\\newcommand\\egal{\\mathop{=}}\nOne can show that the elastic form factors behave, when\n$Q^2\/m^2\\to \\infty$ as\n\\begin{eqnarray}\\label{Fe_asymp} \n F_e(Q^2)&=&\\Big({m^2\\over Q^2}\\Big)^2\n \\left[c_2 \\ln \\left({Q^2\\over m^2}\\right) + c_0 \\right] \\nonumber\\\\\n &+& {\\mathcal O}\\left(\\Big({m^2\\over Q^2}\\Big)^3\\ln \\left({Q^2\\over m^2}\\right),\n \\Big({m^2\\over Q^2}\\Big)^3\\right).\n\\end{eqnarray}\nFor the states $n=1$, the coefficient $c_2$ has a simple expression\nin terms of the BS amplitude:\n\\begin{equation} \\label{c_2n=1}\nc_2(n=1)=\\frac{1}{4\\pi^2}\\frac{[g_{1\\kappa}^{0\\,\\prime}(-1)]^2}{N_{tot}},\n\\end{equation}\nwhere $g'(-1)$ is the derivative of $g$ with respect to $z$ at $z=-1$\nand $N_{tot}$ is the normalization factor that ensures the condition\n$F_e(0)=1$ when $g$ is arbitrarily normalized (its expression is given\nin Eq. (\\ref{Ntot_1})). The expression of $c_0$ is more complicated and\ndepends on the bound state mass $M$, as well as on the function\n$g$ over the whole region of $z$ in the interval $[-1,+1]$.\n\nFor the normal state $n=1,\\kappa=0$, $c_0$ is negative in general,\nbut changes sign and becomes positive for small binding energies.\nThe expression of $g$ takes a simple\nform in the two extreme cases of non-relativistic limit and maximal\nbinding energy ($M=0$). Normalizing $g$ so that $g(0)=1$,\none has in the first case $g(z)=(1-|z|)$, with\n$N_{tot}=1\/(32\\pi\\alpha^5)$, and in the second case $g(z)=(1-z^2)$,\nwith $N_{tot}=1\/(270\\pi^2)$ \\cite{wick,cutk}. One obtains, in these\ntwo extreme cases, the asymptotic behaviors \\cite{dshvk}\n\\begin{eqnarray}\n\\label{ff_NR} \nF_e(Q^2)&\\simeq& \\frac{8\\alpha^5}{\\pi}\\Big({m^2\\over Q^2}\\Big)^2\n\\left[\\ln \\left({Q^2\\over m^2}\\right) + \\frac{2\\pi}{\\alpha} \\right]\n\\nonumber\\\\\n& & (\\mathrm{non-relativistic\\ limit}\\; M\\to 2m), \\\\ \n\\label{ff_M0}\nF_e(Q^2)&\\simeq& 270 \\Big({m^2\\over Q^2}\\Big)^2\n\\left[\\ln \\left({Q^2\\over m^2}\\right)- 4\\right] \\nonumber \\\\ \n& & (\\mathrm{ultra-relativistic\\ case}\\; M=0).\n\\end{eqnarray}\nNotice that in Eq. (\\ref{ff_NR}), we have neglected\n$\\alpha$-indepen\\-dent\nconstant factors in front of the additive term $2\\pi\/\\alpha$.\nThe fact that $c_0$ is generally negative, except for small binding\nenergies, where it can, however, take a large value (proportional to\n$1\/\\alpha$), has as a main consequence the screening of the logarithmic\ntail, requiring, for a numerical analysis of the asymptotic\nbehaviors, very large values of $Q^2$ ($Q^2\\gg 100\\div 1000\\ m^2$). \n\nIn order to put in evidence the asymtptotic behavior (\\ref{Fe_asymp})\nand to determine its leading terms we have computed the\n``reduced form factor'' $\\bar{F}_e(Q^2)$, defined as \n\\begin{eqnarray}\\label{Coefs_asymp} \n\\bar{F}_e(Q^2) &\\equiv& {1 \\over \\ln\\left({Q^2\\over m^2}\\right) } \\;\n\\Big(\\frac{Q^2}{m^2}\\Big)^2 \\; F_e(Q^2) \\nonumber\\\\\n&_{\\stackrel{{\\displaystyle=}}\n{Q^2\\to\\infty}}&\nc_2 + {c_0 \\over \\ln\\left({Q^2\\over m^2}\\right)}, \n\\end{eqnarray}\nwhich should tend, when $Q^2\\to \\infty$, to a\nconstant (up to logarithmic corrections).\nThe asymptotic coefficients $c_i$ can be extracted from $\\bar{F}_e(Q^2)$\nand its derivative at a given $Q^2$ with the relations\n\\begin{equation}\\label{c_0}\n c_0= - \\; { d\\bar{F}(Q^2)\\over dQ^2} \\; Q^2\n \\ln^2\\left({Q^2\\over m^2}\\right)\n\\end{equation}\nand \n\\begin{equation}\\label{c_2}\nc_2= \u00a0\\bar{F}(Q^2) - {c_0 \\over \\ln\\left({Q^2\\over m^2}\\right)} \n\\end{equation}\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\vspace{0.7cm}\n\\epsfxsize=7.5cm\\epsfysize=5.5cm\\epsfbox{QnuFQ2_Nb1_1000.eps}\n\\vspace{0.8cm}\\\\\n\\epsfxsize=7.5cm\\epsfysize=5.5cm\\epsfbox{c0_c2_Nb1_1000_NON.eps}\n\\end{center}\n\\caption{Asymptotic behaviours of the elastic form factor of the $n=1$\nstate No. 1 of Table \\ref{tab1}. Upper panel: the elastic form factor\n$F_e$ multiplied by $Q^{\\sigma}$ with $\\sigma=2$ (red), $\\sigma=4$ (blue), and $\\sigma=4$\ndivided by $\\ln(Q^2)$. \nLower panel: the asymptotic coefficients $c_0$ and $c_2$\ndefined in Eq. (\\ref{Coefs_asymp}).}\\label{QnuFQ2_1}\n\\end{figure}\n\n\nThe results for the $n=1$ state No. 1 from Table \\ref{tab1} are\ndisplayed in Fig. \\ref{QnuFQ2_1}.\nThe upper panel represents the elastic form factor multiplied by\n$Q^{\\sigma}$ with $\\sigma=2$ (red line), $\\sigma=4$ (blue line)\nand $\\sigma=4$ divided by the logarithmic term, as in Eq.\n(\\ref{Coefs_asymp}), to exhibit the asymptotic behaviour derived in\nEq. (\\ref{Fe_asymp}) \nand the important contribution that the logarithmic term can have\nin the domain ${Q^2}\\in[0,1000]$, even at $Q^2 \\sim 1000$. The\nlatter is seen in the difference between the blue and black lines,\nwhich exactly correspond to $\\bar{F}_e$.\nIn the lower panel we have plotted the coefficients $c_0$ and $c_2$\nin the ``asymptotic'' domain ${Q^2}\\in[100,1000]$, together\nwith the full reduced form factor $\\bar{F}_e$. They already show a nice\nconvergence at ${Q^2}=1000$, but the difference between $\\bar{F}_e$\nand its asymptotic value $c_2$ remains sizeable, due to the large\ncontribution of $c_0$, which decreases very slowly. We have also\nchecked the stability of our results with respect to the number\nof grid points ($n_{grid}=400,800,1600$) used in computing the form factors:\nthe sensitivity is not visible by eyes and is not significant in our analysis.\n\n\n\\begin{figure}[h!]\n\\begin{center}\n\\vspace{0.5cm}\n\\epsfxsize=7.cm\\epsfysize=5.cm\\epsfbox{QnuFQ2_Nb3.eps}\\vspace{0.8cm}\\\\\n\\epsfxsize=7.cm\\epsfysize=5.cm\\epsfbox{c0_c2_Nb3_100.eps}\n\\end{center}\n\\caption{Same as in Fig. \\ref{QnuFQ2_1}, but for the $n=1$ abnormal\nstate No. 3 from Table \\ref{tab1} .}\\label{QnuFQ2_3}\n\\end{figure}\n\nFig. \\ref{QnuFQ2_3} contains the same results for the $n=1$ abnormal\nstate No. 3.\nDue to the faster decrease of the corresponding elastic form factor,\n(see the lower panel of Fig. \\ref{Fel_1_3}) the asymptotic regime is\nreached at $Q^2\\approx 50$, with the asymptotic constant\n$c_2\\approx 0.0004$, which is\nseven orders of magnitude smaller than for the normal state No. 1.\n\n\n\nFigures \\ref{Fel_2} and \\ref{Fel_4} contain the elastic form factors\nof the two $n=2$ states.\nThe result for the normal state No. 2 ($\\kappa=0, B=0.2084$) is\nrepresented in the upper panel of Fig. \\ref{Fel_2}.\nOne first observes a much faster decrease than for the $n=1$ state No. 1\n(upper panel of Fig. \\ref{Fel_1_3}), having comparable binding energies:\none order of magnitude at $Q^2 =1$.\nOne also remarks the appearance of two zeroes in the form factor, which\nbecomes negatives in the range $Q^2\\in[1.5,3.0]$. This is\na consequence of the complex structure of the state.\nIndeed, the different contributions $F^{\\nu\\nu'}$ depending on\n$g_{20}^\\nu g_{20}^{\\nu'}$ are indicated in the lower panel.\nAs one can see, the physical $F_e$ results from strong cancellations\nof terms which have opposite signs and are one order of magnitude larger\nthan the physical value they build. \nAlthough these components are of the same order, the contribution due\nto $g^0_{20}$ -- which determines the binding energy of the state --\nis far from being dominant.\n\n\\begin{figure}[h!]\n\\vspace{0.6cm}\n\\begin{center}\n\\epsfxsize=7.cm\\epsfysize=5cm\\epsfbox{FQ2_Nb2_20.eps}\\vspace{0.5cm}\\\\\n\\epsfxsize=7.cm\\epsfysize=5cm\\epsfbox{FQ2_Nb2_Dec.eps}\n\\caption{Upper panel: elastic form factor of the normal state No. 2 of\nTable \\ref{tab1} ($n=2$, $\\kappa=0$). The different contributions\n$F^{\\nu\\nu'}$ depending on $g_{20}^\\nu g_{20}^{\\nu'}$ are indicated in the lower\npanel.}\n\\label{Fel_2}\n\\end{center}\n\\vspace{0.5cm}\n\\end{figure}\n\nThe elastic form factor of the abnormal state No. 4\n($\\kappa=2, B=0.00112$) is displayed in Fig. \\ref{Fel_4}.\nThe same remark concerning the faster decrease than the n=1 state\nNo. 3 with comparable binding energy holds.\nNotice also the non trivial structure -- similar to a diffraction\npattern -- seen at $Q^2\\approx 0.01$ and detailed in the lower panel.\nSuch a structure, as well as the zeroes in upper panel of Fig \\ref{Fel_2},\nis totally unusual in a two-scalar system interacting by the simple\nkernel (\\ref{ladder}) and indicates the complexity of the wave function\nfor any state solution with $n>1$, be it normal or abnormal.\nThe decomposition of $F_e$ in terms of the different components\n$F^{\\nu\\nu'}$ is similar than for the state No. 2, i.e., strong\ncancellations occur among opposite sign larger terms. \n \n\\begin{figure}[h!]\n\\vspace{0.6cm}\n\\begin{center}\n\\epsfxsize=7cm\\epsfysize=5.5cm\\epsfbox{FQ2_Nb4.eps}\\vspace{1cm}\\\\\n\\epsfxsize=7cm\\epsfysize=5.5cm\\epsfbox{FQ2_Nb4_Zoom2.eps}\n\\caption{Elastic form factor of the abnormal state No. 4 of Table\n\\ref{tab1} ($n=2$, $\\kappa=2$). The non-trivial structure at\n$Q^2\\approx 0.01$ is detailed in the lower panel.}\\label{Fel_4}\n\\end{center}\n\\end{figure}\n\nThe corresponding rms radii, extracted using Eq. (\\ref{r2}), are\n$R_2=3.8$ fm (state No. 2) and $R_4=49.0$ fm (state No.4).\nJust on the basis of their binding energies one should expect\ntwice smaller values.\nThe reason is again the complex structure of the BS amplitude of\n$n>1$ states, with a $\\nu>0$ dominating component (by a factor of\n$10^3$ in state No. 4) that plays no role in determining the\nbinding energy -- and so for the spatial extension -- of the system.\n\n\\begin{figure}[h!]\n\\vspace{0.6cm}\n\\begin{center}\n\\epsfxsize=7.cm\\epsfysize=5.5cm\\epsfbox{QnuFQ2_Nb2_1000.eps}\\vspace{1cm}\\\\\n\\epsfxsize=7.cm\\epsfysize=5.5cm\\epsfbox{QnuFQ2_Nb4_1000.eps}\n\\end{center}\n\\caption{Asymptotic behaviour of the elastic form factors of $n=2$\nstates of Table \\ref{tab1} (solid lines): No. 2 (normal) on top and No. 4 (abnormal) at bottom.}\\label{QnuFQ2_24}\n\\end{figure}\n\nAs it was the case for the $n=1$ states, the comparison of the elastic\nform factors of the $n=2$ normal and abnormal\nstates (upper panels of Figs. \\ref{Fel_2} and \\ref{Fel_4}) shows that\nthe abnormal state form factors decrease faster than the normal ones\nas functions of $Q^2$.\nAt $Q^2=1$ the ratio normal\/abnormal is three orders of magnitude.\nEven after adjusting the coupling constant of state No. 2, to have\nthe same binding energy than the state No. 4, the conclusion remains\nunchanged.\n\nIt is also interesting to examine the asymptotic behaviour, which is\nsupposed to have the same form (\\ref{Coefs_asymp}) than for $n=1$ states.\nThis is done in the two panels of Figure \\ref{QnuFQ2_24}. \nThey show again the importance of logarithmic corrections and the\ndifferent orders of magnitudes of the asymptotic constant $c_2$ between\nnormal and abnormal sates. Notice the scaling factor introduced in some\nof the plots to include the comparison in the same frame.\n\nThe ratios of form factors for the normal states Nos. 1 and 2 and for\nthe abnormal ones 3 and 4 are shown in Fig. \\ref{rat1234} (upper panel).\nThey indeed tend to constants. The value of this constant is $\\sim 7$.\nThe ratio of form factors for the abnormal states No. 3 and the normal\none No. 1 is shown in the lower panel of the same figure. It also tends\nto a constant. The value of this constant is $\\sim 10^{-5}$. \nNote that the ratio of form factors of different nature (abnormal\/normal)\nis much smaller than normal\/normal and abnormal\/abnormal, as expected.\nSurprisingly, the ratios normal\/normal and abnormal\/abnormal are the same.\nThese asymptotic behaviors of the elastic form factors bring additional\narguments in favor of the interpretation of the abnormal states of the\nW-C model as hybrids. \n\n\\begin{figure}[h!]\n\\begin{center}\n\\epsfxsize=7.cm\\epsfysize=5cm \\epsfbox{rat_12_34.eps} \\vspace{0.5cm}\\\\\n\\epsfxsize=7.cm\\epsfysize=5cm\\epsfbox{rat_31.eps} \n\\caption{ (Color online) \nUpper panel: Dotted curve is the ratio of the form factors $F(Q^2)$ for\nthe normal ground state No. 1 and for the normal excited state No. 2. \nSolid curve is the same for the abnormal states No. 3 and No. 4.\nLower panel: The ratio of the form factors $F(Q^2)$ for the abnormal\nstate No. 3 from Table 1 and the normal one No. 1.}\\label{rat1234}\n\\end{center}\n\\end{figure}\n \n\\subsection{Transition form factors $F_{if}(Q^2)$}\\label{trF}\n\nFor the sake of completness in the study of abnormal solutions of the\nW-C model we present here the results for the transition form factors.\nThere are four states in Table \\ref{tab1} and, hence, six possible\ntransitions between them. \nThe corresponding transition form factors $F$ are shown in Figs.\n\\ref{Ftr1_234} and \\ref{F34_23_24}.\nThe comparison reveals a hierarchy of the transition form factors.\nIn Fig. \\ref{Ftr1_234} we can see that the form factor for the\ntransition between two normal states, No. 1\n($n=1,\\kappa=0$) $\\to$ No. 2 ($n=2,\\kappa=0)$ (upper panel), dominates, \nby a factor $\\sim 100$, over the maximal values of the normal $\\to$\nabnormal transitions (central and lower panels).\n\nThe transition form factor $F(Q^2)$ between two abnormal states, No. 3\n($n=1,\\kappa=2$) $\\to$ No. 4 ($n=2,\\kappa=2$), is displayed in Fig.\n\\ref{F34_23_24} (upper panel). Its maximal value has the same\norder of magnitude as the normal-normal one, \nthough it decreases much faster. At last, the form factors for the\ntransitions between the normal and abnormal states \n(Figs. \\ref{Ftr1_234} and \\ref{F34_23_24}, both central and bottom\npanels) are approximately 100 times smaller than the\nabnormal$\\leftrightarrow$abnormal form factor. This hierarchy is\napparently related to the need of rebuilding the state structure for\nthe normal$\\leftrightarrow$abnormal transitions.\n\nAt first glance, this dominance can be simply due to the very different\nbinding energies: two such states will have a small overlap,\nwithout invocating any abnormal character. Indeed, one can hardly\nseparate unambiguously the effect of different structures from the\ndifferent binding energies (the latter result in different wave functions).\nThat is why we have carried out the complex analysis based on the behavior\nof the form factor (elastic and inelastic) and on the content of the\nFock sectors.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\epsfxsize=5.cm\\epsfysize=4.cm\\epsfbox{Ftr12.eps}\\vspace{0.5cm}\\\\\n\\epsfxsize=5.cm\\epsfysize=4.cm\\epsfbox{Ftr13.eps}\\vspace{0.5cm}\\\\\n\\epsfxsize=5.cm\\epsfysize=4.cm\\epsfbox{F14.eps}\n\\caption{Transition form factors from normal state No. 1\n($n=1,\\kappa=0$) to other states listed in Table \\ref{tab1}.\n$1\\to2$: No.2 ($n=2, \\kappa=0$, normal) (upper panel);\n$1\\to3$: No. 3 ($n=1, \\kappa=2$, abnormal) (central panel) and\n $1\\to4$: No.4 ($n=2,\\kappa=2$, abnormal) (lower panel).}\n\\label{Ftr1_234}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\epsfxsize=5.cm\\epsfysize=4cm\\epsfbox{F34.eps} \\vspace{0.5cm}\\\\\n\\epsfxsize=5.cm\\epsfysize=4cm\\epsfbox{Ftr20_12.eps}\\vspace{0.5cm}\\\\\n\\epsfxsize=5.cm\\epsfysize=4cm\\epsfbox{Ftr24a.eps}\n\\caption{Same as in Fig. \\ref{Ftr1_234} between \n$3\\to 4$: No. 3($n=1,\\kappa=2$, abnormal) and No. 4 ($n=2,\\kappa=2$,\nabnormal)\n(upper panel); $2\\to 3$: the state No. 2 ($n=2,\\kappa=0$, normal) and\nthe No. 3 ($n=1,\\kappa=2$, abnormal) (central panel) and\n$2\\to 4$: the state No. 2 ($n=2, \\kappa=0$, normal) and \nNo. 4 ($n=2,\\kappa=2$, abnormal) (lower panel).}\\label{F34_23_24}\n\\end{center}\n\\end{figure}\n\nFor all the transitions that were considered in this section we have\ncalculated simultaneously the transition form factor $G(Q^2)$.\nThe contraction of the electromagnetic current $J$ with the momentum\ntransfer $q$ results in $q\\makebox[0.08cm]{$\\cdot$} J=Q_c^2G(Q^2)$ [Eq. (\\ref{FFp})].\nTherefore, the current conservation implies $G(Q^2)\\equiv 0$ for any $Q^2$\n[Eq. (\\ref{G})]. A formal proof of this equality, using the BS\nequation, is given in \\ref{proofG}.\nComputing a quantity that we know from the first principles (current\nconservation) that should be identically zero could be in principle\nconsidered, at most, as being superfluous.\nHowever, as is seen from the derivation given in \\ref{proofG}, \nthis property is directly related to the fact that $g$ is\nindeed a solution of the BS equation\nand the form factors, mainly the 3D integrals (\\ref{Fe_kkp}) and\n(\\ref{Fif_kkp}), have been accurately computed.\nIt thus constitutes a test for our numerical solutions.\nSimilar tests were successfully carried out in Ref. \\cite{tff} for the\nform factors corresponding to the electro-desintegration of a bound\nsystem (the transition discrete $\\to$ continuous spectrum).\n\nThe kind of results obtained in computing $G(Q^2)$ is illustrated in Figs.\n\\ref{G_23}, in a single example corresponding to the transition between \nNo. 2 $(n=2,\\kappa=0)$ $\\to$ No. 3 $(n=1,\\kappa=2)$ states.\nAs in the elastic case, the form factor $G(Q^2)$ results from the sum\nof two terms: $G^{10}(Q^2)$ (proportional to $g_2^1g_1^0$ ) and\n$G^{00}(Q^2)$ (proportional to $g_2^0g_1^0$). They are indicated\nrespectively at upper panel in dashed ($G^{10}(Q^2)$) and \nin dotted ($G^{00}(Q^2)$) lines. The sum of them, i.e., the full form\nfactor $G(Q^2)$, is indicated by a thick solid line and is\nindistinguishable from zero at the scale of the figure.\nIt is in fact $\\approx 10^{-6}$.\n\nNote however that the sensitivity of $G(Q^2)$ to the accuracy in solving\nthe BS equation, that is, in computing the functions $g(z)$ and in\ncalculating the 3D integrals (\\ref{Fif_kkp}), is very high. \nThe result $G(Q^2)\\equiv 0$ is due to delicate cancellations\nbetween terms which are several orders of magnitude greater.\nA small error in these calculations results in non-zero $G(Q^2)$. This\nis demonstrated in the lower panel of Fig. \\ref{G_23}.\nThus, an error in computing the binding energy, e.g., \nsetting $B_i= 3.512 \\cdot 10^{-3}$ instead of $B_i= 3.51169 \\cdot 10^{-3}$\nand $B_f=0.2084$ instead of $B_f = 0.2084099$, provides\n$G(Q^2) \\approx -0.005$.\nThis value is comparable with the maximum value of the corresponding\nform factor $F(Q^2)$ (in central panel of Fig. \\ref{F34_23_24}).\nAn apparent violation of current conservation would hide in fact a lack\nof accuracy in the computational procedure.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\epsfxsize=6.5cm\\epsfysize=5cm\\epsfbox{Fp23b.eps}\\vspace{0.5cm}\\\\\n\\epsfxsize=6.5cm\\epsfysize=5cm\\epsfbox{Fp23aa.eps}\n\\caption{(Color online) Contributions to the $2\\to 3$ transition\nform factor $G(Q^2)$: \nthe dashed line is $G^{10}(Q^2)$ and the dotted line $G^{00}(Q^2)$.\nThe sum of them (solid line) is the full form factor $G(Q^2)$\n(upper panel):\nit is of the order of $10^{-6}$ and results from contributions of\nseveral orders of magnitude greater.\nA small error in computing the binding energy, e.g., \n$B_i= 3.512 \\cdot 10^{-3}$ instead of $B_i= 3.51169 \\cdot 10^{-3}$\nand $B_f=0.2084$ instead of $B_f = 0.2084099$, would provide\n$G(Q^2) \\approx -0.005$ (lower panel), a size comparable with $F(Q^2)$ in\nFig. \\ref{F34_23_24}.}\\label{G_23}\n\\end{center}\n\\end{figure}\n\nTo summarize this section, the comparison of the elastic form factors\npresented in Figs. \\ref{Fel_1_3}, \\ref{Fel_2}, \\ref{Fel_4},\nof the normal states with the abnormal ones,\nshows that the elastic form factors of the abnormal states vs.\n$Q^2$ decrease much faster than for the normal ones.\nAt $Q^2\\sim 1$, the abnormal form factors are about $10^3$ times\nsmaller than the normal ones, whereas the behaviors of the elastic\nform factors of the normal states with $n=1$ and $n=2$ remain very\nclose to each other. These observations confirm that the abnormal\nstates are dominated by the many-body Fock states\n\\cite{matvmurtavk,brodsfarr,radyush}. \n\\par\nThe transitions between the normal and abnormal states, in comparison\nto the normal-normal and abnor\\-mal-abnormal transitions, are also\nsuppressed. This suppression indicates that the normal and abnormal\nstates have different structures and the transitions between them\nrequire the rebuilding of the states.\n\\par\nThe quality of our numerical calculation is quite sufficient to\njustify the above conclusions. This is demonstrated in \nFig. \\ref{G_23} for the transition form\nfactor $G(Q^2)$. As is explained in Sec. \\ref{FFs} and proved in\n \\ref{proofG}, the electromagnetic current conservation\nrequires $G(Q^2)=0$. This is indeed observed in Fig. \\ref{G_23}, top,\n(solid line) as a result of rather delicate\ncancellations of several contributions. \nNumerical changes of $B$ and $g(z)$, which seem insignificant,\nmay noticeably change the value of $G(Q^2)$ (Fig. \\ref{G_23}, bottom). \n\nThe calculations of the form factors for the transitions to the states\ngiven in the Table \\ref{tab2}, with the precision used so far, are\nunstable and require much higher precision. We do not present them here.\n\n\n\\section{Concluding remarks}\\label{concl}\n\nOur present analysis shows that the abnormal solutions\nof the W-C model have a different internal structure than\nthe normal ones, which can be traced back to their decomposition\nproperties into Fock space sectors on light-front planes. \nThis constitutes a genuine property of these states and we propose it as\nan alternative characteristic to the traditional explanation in terms of\ntemporal degrees of freedom excitations.\nWhereas the normal solutions are dominated by the two-body Fock sector\nmade of the two massive\nconstituents, the abnormal ones are dominated by the Fock sectors made of\nthe two massive constituents and several or many massless exchange\nparticles. This feature is also manifested through the fast decrease\nof the electromagnetic form factors of the abnormal states, signalling\ntheir many-body compositeness. Therefore, the abnormal states do\nnot appear as pathological solutions of the BS equation, but rather as\nsolutions having specifically a relativistic origin, through the\ndominance, in their internal structure, of the massless exchange\nparticles. \n\nAnother particular feature of the abnormal solutions is the\nrelatively large value of the coupling constant needed for their\nexistence ($\\alpha>\\pi\/4$). While the stability condition of the\nW-C model also requires that $\\alpha$ be bounded by the\nupper value $2\\pi$, the corresponding window of permissible values\ndoes not belong to the domain of perturbation theory and\nthe question of the validity of the ladder approximation can be raised.\nThis question has been examined in Ref. \\cite{alkofer} in the light\nof the incorporation into the model of the renormalization effects.\nIt turns out that the above\ndomain of values of the coupling constant is incompatible with\na consistent treatment of such effects. The renormalization constants \nviolate the inequalities which follow from the positivity conditions\ncoming from the spectral functions of the renormalized fields.\n\nThe latter result brings us to questioning the effect of the\nhigher-order multiparticle exchange diagrams. This problem has been\ndealt with in Ref. \\cite{jalsazdj} in a model of QED, where\nthe two massive constituents are static and tied at fixed positions in\nthree-dimensional space.\nThe abnormal solutions corresponding essentially to excitations of\ndegrees of freedom described by\nthe relative time variable (or equivalently, of the relative energy\nvariable), this model should rather preserve their possible existence.\nIt turns out that in this configuration, the two-particle Green's\nfunction is exactly calculable: it does not display any abnormal\ntype of bound state in its structure; only the normal ground-state\nis present in the spectrum. On the other hand, the BS equation, in\nthe ladder approximation, still continues exhibiting abnormal\nsolutions. A similar conclusion is also obtained\nfrom a different approach, based on a three-dimensional reduction\nof the BS equation with the inclusion of multiparticle exchange\ndiagrams \\cite{bijteb}.\n\nThe above considerations \nsupport, as mentioned, another\ninterpretation of the abnormal solutions of the W-C model.\nIt is possible that excitations of the degrees of freedom described \nby the relative time variable correspond, from the point of view of\nthe Fock decomposition, to filling of the higher Fock sectors.\nThese two interpretations may not contradict each other, but rather\nbe compatible.\n\n\nIn the W-C model, the even-relative-ener\\-gy abnormal\nsolutions appear as theo\\-retically acceptable states, {and are a\nconstitutive part of the corresponding S-matrix. The fact that\nthey are dominated in Fock space by the many-body massless exchange\nparticles may suggest that they are a kind of ``hybrid'' states.\nThey might represent the Abelian scalar analogs of the hybrids that\nare searched for in QCD, which are coupled essentially to a pair of\nquark-antiquark and one or several gluon fields. Here, however,\nthe non-Abelian property of the gauge group, as well as the existence\nof gluon self-interactions, make the latter states better adapted\nfor experimental, as well as theoretical, investigations. On the\nother hand, the possible relevance of the multiparticle exchange\ndiagrams in the kernel of the BS equation remains a key ingredient\nfor the ultimate conclusion as to whether the W-C model\nmay have any experimental impact. Note, however, that it is natural\nto expect that since the multiparticle exchanges add extra exchange\nparticles in the intermediate states, \nthey do not reduce but rather increase the higher Fock components. \n\nIn any event, in spite of the fact that the W-C model\nis an oversimplified model, it nevertheless contains the phenomenon\nof the particle creation and gives an interesting example of natural\ngeneration of hybrid states.\nTherefore, the hybrid systems can naturally exist in more sophisticated\nfield theories and be detected in appropriate experiments.\n\nThe question of a possible existence of abnormal states in the case\nof massive-particle exchanges is currently under investigation by\nthe present authors.\n\n\\par\n\\bigskip\n\n\\begin{acknowledgements}\nV.A.K. is grateful to the theory group of the Institut de Physique\nNucl\\'eaire d'Orsay (IPNO) for the kind hospitality and financial\nsupport during his visits.\nH.S. acknowledges support from the EU research and innovation program\nHorizon 2020, under Grant Agreement No. 824093.\n\\end{acknowledgements}\n\\par\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n\nHom-Lie algebras are a generalization of Lie algebras, formalizing an algebraic structure which appeared first in quantum deformations of Witt and Virasoro algebras, see for example \\cite{AizawaSaito,ChaiIsKuLuk,CurtrZachos1,Kassel1,Hu}. A quantum deformation or a $q$-deformation of an algebra of vector fields is obtained when replacing a usual derivation by a $\\sigma$-derivation $d_\\sigma$ that satisfies a twisted Leibniz rule $d_\\sigma(f g)=d_\\sigma(f)g+\\sigma(f) d_\\sigma(g) $, where $\\sigma$ is an algebra endomorphism of a commutative associative algebra. An example of a $\\sigma$-derivation is the Jackson derivative on polynomials in one variable. A general construction of quasi-Lie algebras and the introduction of Hom-Lie algebras were given in \\cite{HLS}. The corresponding associative algebras, called Hom-associative algebras, were introduced in \\cite{MS}, where it is shown that they are Hom-Lie admissible, while the enveloping algebra of a Hom-Lie algebra was constructed in \\cite{Yau08}. Moreover, Hom-bialgebras and Hom-Hopf algebras were studied in \\cite{MS2009,MS2010a,Yau4}. Further results could be found in \\cite{AM,AEM,BEM,E,LS,MS2,MS2010,Sh,Yau09,Yau10}.\n\n\nThe purpose of this article is two-fold. First, we construct explicitly the universal enveloping algebra of a multiplicative Hom-Lie algebra and show that it is a Hom-Hopf algebra. Then, we intend to mimic the construction of a Lie group integrating a Lie algebra $\\gg$ obtained by choosing, as a candidate for integrating the Lie algebra $\\gg$, group-like elements of the universal enveloping algebra $ {\\mathcal U}\\gg$.\n\nThis task, in particular the second step, is not trivial, and forced us to reconsider the way antipodes are generally defined on bialgebras, as well as the definition of invertible elements on Hom-groups. Still, we are able to present a Hom-group that we claim to be the integration of a Hom-Lie algebra, but it is more involved than simply group-like elements in the universal enveloping algebra of a Hom-Lie algebra.\nBefore describing our construction step by step, let us discuss what inverse means in the context of Hom-algebras.\n\n\n\\subsection*{Invertibility and inverse on Hom-algebras or Hom-groups}\n\nThere is one natural manner, which was already considered by \\cite{Yau08}, to construct the universal enveloping algebra $ {\\mathcal U}\\gg$ of a multiplicative Hom-Lie algebra $\\gg$. This object is, as expected, a Hom-bialgebra. But, as we shall see while proving Theorem \\ref{thm:HopfAlgOnTrees}, it turns out \\emph{not} to be a Hom-Hopf algebra in the sense of \\cite{MS2010a}, because the antipode is not an inverse of the identity map for the convolution product.\n\nThis failure is, however, productive, in the sense that it paves the way for what seems to be a definition of a Hom-Hopf algebra suitable in this context. Let us explain the situation. In the Hom-bialgebra $ ({\\mathcal U}\\gg,\\vee, \\alpha,\\Delta, \\eta, \\epsilon)$ the usual antipode $S$ (for Hopf algebra structure) does not satisfy the axiom:\n$$ S \\star id = id \\star S = \\eta \\circ \\epsilon $$\nwith $\\star $ being the convolution product, defined by $S \\star T = \\vee \\circ (S \\otimes T) \\circ \\Delta $\nfor two arbitrary linear endomorphisms $S,T$ of ${\\mathcal U}\\gg$, as in the classical case. The antipode $S$ only satisfies a weakened condition: for any $ x \\in {\\mathcal U}\\gg$, there exists an integer $ k \\in {\\mathbb N}$ such that:\n \\begin{equation}\\label{eq:weaker} \\alpha^k \\circ \\vee \\circ (S \\otimes id) \\circ \\Delta \\, (x) = \\alpha^k \\circ \\vee \\circ (id \\otimes S) \\circ \\Delta \\, (x) = \\eta \\circ \\epsilon\\, (x) .\n\t\t\t\\end{equation}\nThe smallest such integer $k$ is called the invertibility index of $x$.\nWe define Hom-Hopf algebras as bialgebras satisfying this weakened condition.\n\nThis definition is indeed not surprising. Given a Hom-associative algebra $ ( {\\A} , \\vee , \\alpha )$ that admits a unit ${\\mathds{1}} $, it is tempting to define invertible elements as being elements $x$ in $ {\\A}$ such that there exists $ y \\in {\\A}$ with $ x \\vee y=y \\vee x= {\\mathds{1}}$.\nAlternatively, it may be tempting to define invertible elements to be those elements $x$ in $ {\\A}$ such that there exists $ y \\in {\\A}$ with $ \\alpha(x \\vee y)=\\alpha(y \\vee x)= {\\mathds{1}}$\nas in \\cite{F}. However, there is an issue with both definitions: invertible elements in any of these two senses are in general not stable under $\\vee $.\n\nIn order to get a notion of invertible elements that would allow those to be invariant under $\\vee $, we say that an element $x \\in {\\A}$ is \\textbf{hom-invertible} if and only if there exists $y \\in {\\A}$ (not necessarily unique) called a \\textbf{hom-inverse} and an integer $k \\in {\\mathbb N}$ such that\n$$ \\alpha^k ( x \\vee y ) = \\alpha^k ( y \\vee x ) = {\\mathds{1}} .$$\nThis definition is consistent with Equation (\\ref{eq:weaker}). The antipode $S$ that we have constructed on the Hom-bialgebra $ ({\\mathcal U}\\gg,\\vee, \\Delta, \\eta, \\epsilon)$ becomes now a kind of hom-inverse of the identity for the convolution product, making $ ({\\mathcal U}\\gg,\\vee, \\Delta, \\eta, \\epsilon, S)$ a Hom-Hopf algebra.\n\nThis issue being solved, we intend to find a Hom-group integrating a Hom-Lie algebra. Having modified the definition of a hom-inverse, we have to modify the definition of Hom-Lie group accordingly.\n\n\\begin{defn}\\label{def:ourHomGroup}\nA \\textbf{Hom-group} is a set $(G,\\vee,\\alpha,{\\mathds{1}})$ equipped with a Hom-associative product with unit ${\\mathds{1}} $ and an anti-morphism $g \\to g^{-1}$ such that, for any $g \\in G$, there exists an integer $k \\in {\\mathbb N}$ satisfying\n$$ \\alpha^k ( g\\vee g^{-1}) = \\alpha^k ( g^{-1} \\vee g ) = {\\mathds{1}} .$$\nThe smallest such integer $k$ is called the invertibility index of $g$.\n\\end{defn}\n\n\n\\subsection*{From Hom-Lie algebras to Hom-groups}\n\nGroup-like elements in a Hom-Hopf algebra $ {\\A}$ (equipped with an antipode satisfying the weakened assumption (\\ref{eq:weaker})) form a Hom-group (group-like elements being defined as formal series $ g(\\nu )\\in {\\A}[[\\nu]]$\nwith some assumptions). It is therefore tempting to define the object integrating the Hom-Lie algebra $ \\gg$ as being the set of group-like elements in the Hom-Hopf algebra $ {\\mathcal U}\\gg[[\\nu]]$. However, this definition is irrelevant: there is in general very little group-like elements in $ {\\mathcal U}\\gg[[\\nu]]$, except for the unit $ {\\mathds{1}}$ itself.\n\nIt is possible, fortunately, to go around this difficulty by defining, for all $p \\in {\\mathbb N}$, {$p$-order formal group-like elements} as being elements in $ {\\mathcal U}\\gg[[\\nu]]$ satisfying\n$ g(0)={\\mathds{1}} $ and:\n $$ \\Delta g(\\nu) = g(\\nu) \\otimes g(\\nu) \\hspace{1cm} \\left[\\nu^{p+1}\\right]$$\n(where $\\left[\\nu^{p+1}\\right]$ means \"modulo $ \\nu^{p+1} $\").\nIt is routine to check that $p$-order formal group-like element do form a Hom-group, with inverse given by the antipode.\nThen we consider sequences $ (g_p(\\nu))_{p \\geq 0}$, with $g_p(\\nu)$ a $p$-order formal group-like element,\nsuch that the quotient of $ g_{p+1} (\\nu)$ modulo $ \\nu^{p+1}$ is $\\alpha ( g_p(\\nu) ) $ for all $p \\in {\\mathbb N}$.\n We call these sequences {formal group-like sequences} when their invertibility index is bounded. We show that formal group-like sequences do form a Hom-group, with inverse again induced by the antipode. Moreover an exponential map valued in formal group-like sequence can be constructed making this Hom-group a reasonable candidate for being considered as a Hom-group integrating the Hom-Lie algebra $ \\gg$, as several theorems will show at the end of this article.\n\n\n\\section{Hom-Lie algebras, Hom-associative algebras and Hom-bialgebras}\n\nGiven $\\gg$ a vector space and a bilinear map $ \\brr{\\, , \\, }: \\gg \\otimes \\gg \\to \\gg$, by \\textbf{endomorphism of\n$(\\gg,\\brr{\\, , \\, } )$}, we mean a linear map $\\alpha: \\gg \\to \\gg$ such that\n$$\n\\alpha (\\brr{x,y}) = \\brr{\\alpha (x), \\alpha (y)}\n$$\nfor all $x,y \\in \\gg$. We now define Hom-Lie algebras, sometimes called multiplicative Hom-Lie algebras.\n\n\\begin{defn}\\label{def:hom:Lie:algebra} \\cite{HLS}\nA \\textbf{Hom-Lie algebra} is a triple $(\\gg, \\brr{\\, , \\, }, \\alpha)$ with $\\gg$ a vector space equipped with a\nskew-symmetric bilinear map $ \\brr{\\, , \\, }:\\gg \\otimes \\gg \\to \\gg$ and an endomorphism\n$\\alpha$ of $(\\gg,\\brr{\\, , \\, })$ such that:\n\\begin{equation}\n\\brr{\\alpha (x),\\brr{y,z}}+\\brr{\\alpha (y),\\brr{z,x}}+\\brr{\\alpha\n(z),\\brr{x,y}}=0, \\quad \\forall x,y,z\\in\\gg \\quad \\hbox{(Hom-Jacobi identity)}. \\label{eq:hom:Jacobi:algebra}\n\\end{equation}\nA \\textbf{morphism} between Hom-Lie algebras $(\\gg,\\brr{\\, , \\, }_\\gg,\\alpha)$ and $(\\hh,\\brr{\\, , \\, }_\\hh,\\beta)$ is a linear map $\\psi:\\gg \\to \\hh$\nsuch that $\\displaystyle \\psi(\\brr{x,y}_\\gg)=\\brr{\\psi( x),\\psi (y)}_\\hh$ and $\\psi (\\alpha(x))=\\beta(\\psi(x))$\nfor all $x,y \\in \\gg$.\nWhen $\\hh$ is a vector subspace of $\\gg$ and $\\psi$ is the inclusion map, one speaks of \\textbf{Hom-Lie subalgebra}.\n\\end{defn}\n\n\\begin{rem}\\label{Rmk:IdealAndInverse}\nLet $(\\gg, \\brr{\\, , \\, }, \\alpha)$ be a Hom-Lie algebra. The subspace ${\\mathfrak k} \\subset \\gg$ of elements $x \\in \\gg$ such that there exists an integer $k$ with $\\alpha^k (x)=0$ is a Hom-Lie subalgebra. It is even a Hom-Lie ideal, i.e. the quotient $\\gg\/{\\mathfrak k} $ inherits a structure of Hom-Lie algebra. By construction, its induced morphism $\\underline{\\alpha}: \\gg\/{\\mathfrak k} \\to \\gg\/{\\mathfrak k}$ is invertible. The induced bracket $\\underline{\\alpha}^{-1} \\circ \\underline{\\brr{\\, , \\, }}$ is therefore a Lie algebra bracket, see \\cite{G}. Also, the natural projection $\\gg\/{\\mathfrak k} $ is a morphism of Hom-Lie algebra.\n\\end{rem}\n\n\n\n\\begin{defn} \\label{hom-associative}\\cite{MS}\nA \\textbf{Hom-associative algebra} is a triple $(\\A, \\mu, \\alpha)$ consisting of a vector space $\\A$, a bilinear map $\\mu: \\A \\otimes \\A \\to \\A$ and an endomorphism $\\alpha$ of $( \\A, \\mu) $ satisfying\n$$\n\\mu(\\alpha(x), \\mu(y,z))= \\mu (\\mu(x,y), \\alpha (z)), \\quad \\forall x,y,z \\in \\A \\hbox{ (Hom-associativity)}.\n$$\nA Hom-associative algebra $(\\A, \\mu, \\alpha)$ is called \\textbf{unital} if there exists a linear map $\\eta: \\mathbb K \\to \\A$ such that\n$$\n\\mu \\circ (id_{\\A} \\otimes \\eta) = \\mu \\circ (\\eta \\otimes id_{\\A} )= \\alpha \\hbox{ and } \\alpha \\circ \\eta = \\eta.\n$$\nWe denote a unital Hom-associative algebra by a quadruple $(\\A, \\mu, \\alpha, \\eta)$. The unit element (or unit, for simplicity) is $\\mathds{1}= \\eta(1_{\\mathbb K})$.\n\\end{defn}\n\nNotice that a Hom-associative algebra $(\\A, \\mu, \\alpha)$ is unital, with unit $\\mathds{1} \\in \\A$,\nwhen $ \\mu(x, \\mathds{1})= \\mu(\\mathds{1},x) =\\alpha(x) $ and $ \\alpha(\\mathds{1})=\\mathds{1}$.\n\nMorphisms between Hom-associative algebras are defined in the similar way as Hom-Lie algebras. For unital Hom-associative algebras, the image of the unit is a unit.\n\\begin{ex}\\cite{Yau09,MS}\n\\label{ex:composition}\nGiven a vector space $\\gg$ equipped with a bilinear map $ \\brr{\\, , \\, }:\\gg \\otimes \\gg\n\\to \\gg$ and an endomorphism $\\alpha:\\gg\\to \\gg$ of $(\\gg, \\brr{\\, , \\, })$. Define\n$ \\brr{\\, , \\, }_{\\alpha}:\\gg \\otimes \\gg\n\\to \\gg$ by\n$$ \\brr{x,y}_\\alpha=\\alpha (\\brr{x,y}), \\quad \\hbox{ $\\forall x,y \\in \\gg$.} $$\n Then $(\\gg, \\brr{\\, , \\, }_{\\alpha}, \\alpha)$ is a Hom-Lie algebra (resp. a Hom-associative algebra, resp. unital Hom-associative algebra) if and only if the restriction of $\\brr{\\, , \\, }$\nto the image of $\\alpha^2 $ is a Lie bracket (resp. an associative product, resp. a unital associative product).\nIn particular, Hom-Lie structures are naturally associated to Lie algebras equipped with a Lie algebra endomorphism \\cite{Yau09}.\nSuch Hom-Lie structures are said to be \\textbf{obtained by composition} or \\textbf{Twisting principle}.\n\\end{ex}\n\n\n\\begin{ex} \\label{ex:composition2}\nAs one can expect, the commutator of a Hom-associative algebra is a Hom-Lie algebra\n\\cite{MS}. More precisely, for every Hom-associative algebra $(\\A, \\mu, \\alpha)$ (see Definition \\ref{hom-associative} above), the triple $(\\A, \\brr{\\, , \\, }, \\alpha)$ is a Hom-Lie algebra, where\n$$\n\\brr{x,y}:= \\mu(x,y)- \\mu (y,x)\n$$\nfor all $x,y \\in \\A$.\n\\end{ex}\n\n\\begin{defn}\\label{def:our_inverse}\nAn element $x$ in a unital Hom-associative algebra $(\\A, \\mu, \\alpha,\\mathds{1}) $ is said to be \\textbf{hom-invertible} if there exists an element $x^{-1}$ and a non-negative integer $k \\in {\\mathbb N}$, such that\n\\begin{equation}\\label{eq:def_inverse}\n\\alpha^k \\circ \\mu(x, x^{-1})= \\alpha^k \\circ \\mu(x^{-1}, x) = \\mathds{1}.\n\\end{equation}\n The element $x^{-1}$ is called a \\textbf{hom-inverse} and the smallest $k$ is the \\textbf{invertibility index} of $x$.\n\\end{defn}\nWhen it exists, the hom-inverse may not be unique, which prevents hom-invertible elements to be a Hom-group in the sense of Definition \\ref{def:ourHomGroup}. However, the following can be shown.\n\n\\begin{prop}\\label{prop:stableByProduct}\nFor every unital Hom-associative algebra $(\\A, \\mu, \\alpha,\\mathds{1}) $, the unit $\\mathds{1}$ is hom-invertible,\nthe product of any two hom-invertible elements is hom-invertible and every inverse of a hom-invertible element is hom-invertible.\n \\end{prop}\n\\begin{proof}\n The only non-trivial point is that $\\mu(x , x')$ is a hom-invertible element if both $x$ and $x'$ are, i.e.\n if there exists $y,y' \\in \\A$, $k,k' \\in {\\mathbb N}$ such that\n $$\\alpha^k \\circ \\mu(x, y)= \\alpha^k \\circ \\mu(y,x) = \\alpha^{k'} \\circ \\mu(x', y')= \\alpha^{k'} \\circ \\mu(y', x') = \\mathds{1}.$$\n Hom-associativity implies\n $$ \\alpha \\circ \\mu(\\mu(x,x') , \\mu(y',y)) = \\mu( \\alpha^2(x) , \\mu(\\mu(x',y'), \\alpha(y) ) .\n $$\n\n So that\n \\begin{eqnarray*} \\alpha^{k+k'+1} \\circ \\mu(\\mu(x,x') , \\mu(y',y)) & = & \\alpha^k ( \\mu( \\alpha^{k'+2}(x) , \\mu(\\alpha^{k'} \\mu(x',y'), \\alpha^{k'+1}(y) )\n \\\\ & =& \\alpha^k ( \\mu( \\alpha^{k'+2}(x) , \\mu(\\mathds{1}, \\alpha^{k'+1}(y) ) \\\\\n & =& \\alpha^k ( \\mu( \\alpha^{k'+2}(x) , \\alpha^{k'+2}(y) ) \\\\ &=& \\alpha^{k+k'+2} ( \\mu( x , y ) ) \\\\\n &=& \\alpha^{k'+2}(\\mathds{1}) = \\mathds{1}.\n \\end{eqnarray*}\n This completes the proof.\n\\end{proof}\n\n\\begin{rem}\nFor every unital Hom-associative algebra $(\\A, \\mu, \\alpha,\\mathds{1}) $, the subspace ${\\mathfrak k}$ of all elements $x \\in \\A$ such that $ \\alpha^k (x)=0$ for some integer $k \\in {\\mathbb N}$ is an ideal, i.e. the quotient map\n$\\A\/{\\mathfrak k}$ is a unital Hom-associative algebra for which the induced map $\\underline{\\alpha}$ is invertible\n for the induced product $ \\underline{\\mu}$.\nIn particular $ \\A\/{\\mathfrak k}$ equipped with the product $\\underline{\\alpha}^{-1} \\circ \\underline{\\mu} $ is an algebra.\n\nAn element $x\\in \\A$ is invertible in $(\\A, \\mu, \\alpha,\\mathds{1}) $ if and only if its image in $\\A\/{\\mathfrak k}$ is invertible in the usual sense, which gives an alternative proof of Proposition \\ref{prop:stableByProduct}.\n\\end{rem}\n\nWe now recall the notion of Hom-coalgebras.\n\n\n\n\n\n\\begin{defn} \\cite{MS}\nA \\textbf{Hom-coalgebra} is a triple $(A, \\Delta, \\beta)$ where $A$ is a vector space and $\\Delta: A \\to A\\otimes A$, $\\beta: A \\to A$ are linear maps.\n\nA \\textbf{Hom-coassociative coalgebra} is a Hom-coalgebra $(A, \\Delta, \\beta)$ satisfying\n$$\n(\\beta \\otimes \\Delta) \\circ \\Delta=(\\Delta \\otimes \\beta) \\circ \\Delta.\n$$\nA Hom-coassociative coalgebra is said to be \\textbf{co-unital} if there exists a linear map $\\epsilon: A \\to \\mathbb K$ satisfying\n$$\n(id \\otimes \\epsilon) \\circ \\Delta= \\beta \\hbox{, } (\\epsilon \\otimes id) \\circ \\Delta = \\beta\n \\hbox{ and } \\epsilon \\circ \\beta = \\epsilon.\n$$\nWe refer to a counital Hom-coassociative coalgebra with a quadruple $(A,\\Delta, \\beta, \\epsilon)$.\n\\end{defn}\n\nLet $(A, \\Delta, \\beta)$ and $(A', \\Delta', \\beta')$ be two Hom-coalgebras (resp. Hom-coassociative algebras). A linear map $f: A \\to A'$ is a morphism of Hom-coalgebras (resp. Hom-coassociative coalgebras) if\n$$\n(f \\otimes f) \\circ \\Delta = \\Delta' \\circ f \\qquad f \\circ \\beta = \\beta' \\circ f.\n$$\nIt is said to be a weak Hom-coalgebras morphism if it holds only the first condition. If furthermore the Hom-coassociative coalgebras admit counits $\\epsilon$ and $\\epsilon'$, we have moreover $\\epsilon = \\epsilon' \\circ f$.\n\nThe category of coassociative Hom-coalgebras is closed under weak Hom-coalgebra morphisms.\n\n\\begin{ex}\n\\cite{MS2009}\nThe dual of a Hom-algebra $(A,\\mu,\\alpha)$ is not always a Hom-coalgebra, because the coproduct does not land in the good space $\\mu^*: A^* \\to (A \\otimes A)^* \\supsetneq A^* \\otimes A^*$.\nNevertheless, it is the case if the Hom-algebra is finite-dimensional, since $(A \\otimes A)^*= A^* \\otimes A^*$.\nThe converse always holds true. Let $(A, \\Delta, \\beta)$ be a Hom-coassociative coalgebra. Then its dual vector space is provided with a structure of Hom-associative algebra $(A^*, \\Delta^*, \\beta^*)$ where $\\Delta^*$, $\\beta^*$ are the transpose maps.\nMoreover, the Hom-associative algebra is unital whenever $A$ is counital.\n\n\n\\cite{MS2010a} Let $(A,\\Delta, \\beta, \\epsilon)$ be a counital Hom-coassociative coalgebra and $\\alpha: A \\to A$ be a weak Hom-coalgebra morphism. Then $(A,\\Delta_{\\alpha}= \\Delta \\circ \\alpha, \\beta \\circ \\alpha, \\epsilon)$ is a counital Hom-coassociative coalgebra.\nIn particular, let $(A,\\Delta,\\epsilon)$ be a coalgebra and $\\beta: A \\to A$ be a coalgebra morphism. Then $(A,\\Delta_{\\beta}, \\beta, \\epsilon)$ is a counital Hom-coassociative coalgebra\n\\end{ex}\n\n\n\n\\begin{defn} \\cite{MS2009}\nAn $(\\alpha,\\beta)$-\\textbf{Hom-bialgebra} (simply called Hom-bialgebra when there is no ambiguity) is a heptuple $(A,\\mu,\\alpha, \\eta,\\Delta, \\beta, \\epsilon)$ where\n\\begin{enumerate}\n \\item[(i)] $(A,\\mu,\\alpha, \\eta)$ is a Hom-associative algebra with unit $\\eta$ and unit element $\\mathds{1}$,\n \\item[(ii)] $(A,\\Delta, \\beta, \\epsilon)$ is a Hom-coassociative coalgebra with a counit $\\epsilon$,\n \\item[(iii)] the linear maps $\\Delta$ and $\\epsilon$ are compatible with the multiplication $\\mu$ and the unit $\\eta$, that is for $x,y \\in A$\n\\begin{enumerate}\n \\item $\\displaystyle{\\Delta(\\mu(x \\otimes y))= \\Delta(x) \\cdot \\Delta(y)= \\sum_{(x)(y)} \\mu(x_1 \\otimes y_1) \\otimes \\mu(x_2 \\otimes y_2)}$, where $\\cdot$ denotes the multiplication on the tensor algebra $A \\otimes A$,\n \\item $\\Delta(\\mathds{1})= \\mathds{1} \\otimes \\mathds{1}$,\n \\item $\\epsilon(\\mathds{1})= 1$,\n \\item $\\epsilon(\\mu(x \\otimes y))= \\epsilon(x) \\epsilon(y)$,\n \\item $\\epsilon \\circ \\alpha(x)= \\epsilon(x)$.\n\\end{enumerate}\n\\end{enumerate}\nIf $\\alpha=\\beta$ the $(\\alpha,\\alpha)$-Hom-bialgebra is denoted by the hexuple $(A, \\mu, \\eta, \\Delta, \\epsilon, \\alpha)$.\n\\end{defn}\n\nA Hom-bialgebra morphism is a linear endomorphism which is simultaneously a Hom-algebra and Hom-coalgebra morphism.\n\n\\begin{ex}\\label{TwistingPrincipleBialgebra}\nLet $(A,\\mu, \\eta,\\Delta,\\epsilon, \\alpha)$ be a Hom-bialgebra and $\\beta:A \\to A$ be a Hom-bialgebra morphism. Then $(A, \\mu_{\\beta} = \\beta \\circ \\mu, \\eta,\\Delta_{\\beta} = \\Delta \\circ \\beta, \\epsilon, \\beta \\circ \\alpha)$ is a Hom-bialgebra.\nIn particular, if $(A,\\mu, \\eta,\\Delta, \\epsilon)$ is a bialgebra and $\\beta: A \\to A$ is a bialgebra morphism then $(A, \\mu_{\\beta}, \\eta,\\Delta_{\\beta}, \\epsilon, \\beta )$ is a Hom-bialgebra.\nThis construction method of Hom-bialgebra, starting with a given Hom-bialgebra or a bialgebra and a morphism, is called composition method or twisting principle \\cite{MS2010a}. We can also define an $(\\alpha, \\beta)$ twist. If $(A,\\mu, \\eta,\\Delta, \\epsilon)$ is a bialgebra and $\\alpha, \\beta: A \\to A$ are bialgebra morphisms which commute, that is $\\alpha \\circ \\beta = \\beta \\circ \\alpha$,\nthen $(A, \\mu_{\\alpha}=\\alpha \\circ \\mu, \\alpha,\\eta,\\Delta_{\\beta}=\\Delta \\circ \\beta, \\beta,\\epsilon )$ is a Hom-bialgebra. In particular we can consider one of the morphisms equal to identity.\n\\end{ex}\n\n\n\\section{Hom-Hopf algebras}\n\nThe following theorem holds and the proof goes through a direct verification of the axioms.\n\n\\begin{thm} \\label{conv-product}\n\\cite{MS2009,MS2010a}\nLet $(A,\\mu,\\alpha, \\eta,\\Delta, \\beta, \\epsilon)$ be an $(\\alpha, \\beta)$-Hom-bialgebra. Then $({\\rm Hom}(A,A),\\star,\\gamma)$ is a unital Hom-associative algebra, with $\\star$ being the multiplication given by the convolution product defined by\n$$\nf\\star g = \\mu \\circ (f \\otimes g) \\circ \\Delta\n$$\nand $ \\gamma$ being the homomorphism of ${\\rm Hom}(A,A)$ defined by $\\gamma(f)= \\alpha \\circ f \\circ \\beta$. The unit is $\\eta \\circ \\epsilon$.\n\\end{thm}\n\n\n\n\nWe now define Hom-Hopf algebra in a manner that differs from \\cite{MS2009,MS2010a}. In those works, an antimorphism $S$ of $A$ is said to be an antipode if it is an inverse (in the usual sense) of the identity\nover $A$ for the Hom-associative algebra ${\\rm Hom} (A,A)$ with the multiplication given by the convolution product, i.e.\n $ S \\star id =id \\star S = \\eta \\circ \\epsilon $.\nThis definition matches examples given by twisting principle out of a Hopf algebra but does not match our examples.\nFor this reason, and also in view of the Definition \\ref{def:our_inverse} of an invertible element in the context of Hom-algebras, we prefer to define an antipode as being an anti-morphism $S$ of $A$ which is a relative hom-inverse of the identity, defined as follows. We say that $S \\in {\\rm Hom} (A,A)$ is a \\textbf{relative hom-inverse} of $T \\in {\\rm Hom} (A,A)$ if and only if for every $x \\in A$ there exists an integer $k$ (depending on $x$) such that:\n$$ \\alpha^{k} (S \\star T ) \\, (x) = \\alpha^k (T \\star S ) \\, (x) = \\eta \\circ \\epsilon \\, (x).$$\n\n\\begin{rem}\nNotice that $S$ is not a hom-inverse of $T$ in the sense of Definition \\ref{def:our_inverse}.\nHowever, if there exists an integer\n$\\bar{k}$ such that $ \\alpha^{\\bar{k}} (S \\star T ) \\, (x) = \\alpha^{\\bar{k}} (T \\star S ) \\, (x) = \\eta \\circ \\epsilon \\, (x)$ for all $x\\in A$, then $S$ is a hom-inverse of $T$ and the smallest $\\overline{k}$ is the invertibility index of $S$.\\end{rem}\n\n\nThis amounts to the following definition:\n\n\\begin{defn} \\label{def:hom-Hopf-algebra_in_our_sense}\nLet $(A,\\mu,\\alpha, \\eta,\\Delta, \\beta, \\epsilon)$ be an $(\\alpha, \\beta)$-Hom-bialgebra.\nAn anti-homomorphism $S$ of $A$ is said to be an \\textbf{antipode} if\n\\begin{enumerate}\n\\item[a)] $ S \\circ \\alpha = \\alpha \\circ S$,\n\\item[b)] $ S \\circ \\eta= \\eta $ and $ \\epsilon \\circ S = \\epsilon$,\n\\item[c)] $S$ is a relative Hom-inverse of the identity map $id :A \\to A$ for the convolution product given as in Theorem \\ref{conv-product}.\n\\end{enumerate}\n An $(\\alpha, \\beta)$-\\textbf{Hom-Hopf algebra} is an $(\\alpha, \\beta)$-Hom-bialgebra admitting an antipode.\n\\end{defn}\n\nRecall that condition (c) means equivalently that, for every $x \\in A $, there exists $ k \\in {\\mathbb N}$ such that:\n \\begin{equation}\\label{eq:antipode_condition}\\alpha^k \\circ( S \\star id) (x)= \\alpha^k \\circ (id \\star S) (x) = \\eta \\circ \\epsilon (x).\n \\end{equation}\n\n\nNotice that we do not need to assume that $ S $ and $ \\beta $ commute (in most examples,\n$\\beta$ is either the identity or coincides with $\\alpha$).\n\n\\begin{ex} \\label{Hom-Hopf-Twist}\nLet $(A,\\mu, \\eta,\\Delta, \\epsilon, S)$ be a Hopf algebra, and $\\alpha, \\beta: A \\to A$ be commuting bialgebra morphisms satisfying $S\\circ \\alpha=\\alpha \\circ S$. Then\n$$\n(A, \\mu_{\\alpha}=\\alpha\\circ \\mu,\\alpha, \\eta,\\Delta_{\\beta}=\\Delta\\circ\\beta, \\beta,\\epsilon ,S)\n$$\nis an $(\\alpha, \\beta)$-Hom-Hopf algebra, called the \\textbf{$(\\alpha,\\beta)$-twist} of the Hopf algebra $A$.\nMore generally, the same idea turns a $(\\alpha', \\beta')$-Hom-Hopf algebra in a $(\\alpha \\circ \\alpha', \\beta \\circ \\beta')$-Hom-Hopf algebra.\n\nIndeed, according to Example \\ref{TwistingPrincipleBialgebra}, $\n(A, \\mu_{\\alpha}=\\alpha\\circ \\mu,\\alpha, \\eta,\\Delta_{\\beta}=\\Delta\\circ\\beta, \\beta,\\epsilon )\n$ is a Hom-bialgebra. It remains to show that $S$ is still an antipode for the Hom-bialgebra. We have\n$$S(\\mu_\\alpha(x,y))=S(\\alpha(\\mu(x,y)))=\\alpha(S(\\mu(x,y)))=\\alpha(\\mu(S(y),S(x)))=\\mu_\\alpha(S(y),S(x)),\n$$\nand\n$$\\mu_\\alpha\\circ(S\\otimes \\id)\\circ\\Delta_\\beta=\\alpha\\circ\\mu\\circ(S\\otimes \\id)\\circ\\Delta\\circ\\beta=\n\\mu\\circ(S\\otimes \\id)\\circ\\Delta\\circ\\alpha\\circ\\beta=\\eta\\circ\\epsilon\\circ\\alpha\\circ\\beta=\\eta\\circ\\epsilon,\n$$\nwhich complete the proof. The proof for the general case is similar.\n\\end{ex}\n\n\n\n\n\\begin{rem}\n\\label{rmk:SeveralPoints}\nFor every $(\\alpha,\\beta)$-Hom-Hopf algebra, the following properties hold:\n\\begin{enumerate}\n \\item Using counitality, we have (in Sweedler's notation):\n $ \\beta(x) = \\sum x_1 \\epsilon (x_2) = \\sum \\epsilon (x_1) \\, x_2$.\n \\item Let $x$ be a primitive element (which means that $\\Delta(x)= \\mathds{1}\\otimes x + x \\otimes \\mathds{1}$), then $\\epsilon(x)= 0$.\n \\item If $x$ and $y$ are two primitive elements in $A$, then we have $\\epsilon(x)=0$ and the commutator $[x,y]= \\mu(x \\otimes y) - \\mu (y \\otimes x)$ is also a primitive element.\n \\item The set of all primitive elements of $A$, denoted by ${\\rm Prim}(A)$, admits a natural structure of Hom-Lie algebra, with bracket given by the commutator $ [x,y]:= \\mu(x \\otimes y) - \\mu (y \\otimes x)$, see \\cite{MS2009,MS2010a}.\n \\item If $x,y,z$ are primitive elements in $A$, then the Hom-associator $\\mu(\\alpha(x),\\mu(y,z))-\\mu(\\mu(x,y),\\alpha(z))$ is a primitive element.\n\t \\item Using counitality and unitality, we have $ S \\star (\\eta \\circ \\epsilon) = \\alpha \\circ S \\circ \\beta =\\gamma(S) $,\n\tand more generally $ (\\alpha^p \\circ S \\circ \\beta^q )\\star (\\eta \\circ \\epsilon) = \\alpha^{p+1} \\circ S \\circ \\beta^{q+1} $.\n\t\\item For all linear endomorphism $ S,T $ of $ A$, we have $\\alpha ( S \\star T) = (\\alpha \\circ S) \\star (\\alpha \\circ T) $.\n\t\\item The antipode condition (\\ref{eq:antipode_condition}) can be stated as $ (\\alpha^k \\circ S) \\star \\alpha^k = \\alpha^k \\star (\\alpha^k \\circ S) = \\eta \\circ \\epsilon$.\n\\end{enumerate}\n\\end{rem}\n\n\n\n\n\\begin{prop}\nLet $(A,\\mu,\\alpha, \\eta,\\Delta, \\beta, \\epsilon) $ be an $(\\alpha,\\beta)$-Hom-bialgebra. Assume that $S $ and $S'$ are two antipodes. Let $x\\in A$ and $k,k' \\in {\\mathbb N}$\n such that:\n $$ \\alpha^{k} ( S \\star \\id )(x) = \\alpha^{k} ( \\id \\star S ) (x)= \\eta \\circ \\epsilon(x) \\hbox{ and }\n\t\\alpha^{k'} ( S' \\star \\id )(x) = \\alpha^{k'} ( \\id \\star S' )(x) = \\eta \\circ \\epsilon(x).$$\n\tThen, the following relation holds\n $$ \\alpha^{K+2} \\circ S \\circ \\beta^2(x) = \\alpha^{K+2} \\circ S' \\circ \\beta^2 (x) $$\nwith $K ={\\rm max}(k,k')$.\n\\end{prop}\n\\begin{proof}We assume that $k'\\leq k$.\nRecall that for any $f$, $f\\star \\eta\\circ \\epsilon =\\alpha \\circ f\\circ \\beta$. For simplicity, we omit the composition circle.\n\\begin{align*}\n\\alpha^{k+2}S'\\beta^2&=\\alpha(\\alpha^{k+1}S'\\beta)\\beta=(\\alpha^{k+1}S'\\beta)\\star (\\eta \\epsilon)=(\\alpha^{k+1}S'\\beta)\\star(\\alpha^k(\\id \\star S))=(\\alpha^{k+1}S'\\beta)\\star(\\alpha^k \\star \\alpha^k S).\n\\end{align*}\nBy Hom-associativity we have\n\\begin{align*}\n\\alpha^{k+2}S'\\beta^2&=(\\alpha^{k}S'\\star\\alpha^k)\\star(\\alpha^{k+1} S\\beta)\n=(\\alpha^{k-k'}(\\alpha^{k'}(S'\\star\\id))\\star(\\alpha^{k+1} S\\beta)\n=(\\alpha^{k-k'}\\eta\\epsilon))\\star(\\alpha^{k+1} S\\beta).\n\\end{align*}\nSince $\\alpha\\eta=\\eta$ and $\\epsilon\\beta=\\epsilon$, we have\n\\begin{align*}\n\\alpha^{k+2}S'\\beta^2&\n=(\\eta\\epsilon)\\star(\\alpha^{k+1} S\\beta)=(\\alpha^{k+1}\\eta\\epsilon\\beta)\\star(\\alpha^{k+1} S\\beta)=\\alpha^{k+1}(\\eta\\epsilon\\star S)\\beta=\\alpha^{k+2}S\\beta^2.\n\\end{align*}\n\\end{proof}\n\\begin{rem}\nThis proposition means that the antipode is in some sense unique.\nIndeed, when $\\alpha$ and $\\beta$ are invertible, the antipode is unique when it exists.\n\\end{rem}\n\n\n\\section{Elements of group-like type in an $(\\alpha,\\beta)$-Hom-Hopf algebra}\n\\label{sec:grouplike}\n\nFor $( {A},\\mu,\\alpha,\\mathds{1},\\Delta,\\beta, \\epsilon, S) $ an $(\\alpha,\\beta)$-Hopf-algebra, all the structural maps\nextend by ${\\mathbb K}[[\\nu]] $-linearity to yield an $(\\alpha,\\beta)$-Hom-bialgebra structure on the space ${A}[[\\nu]]$ of formal series with coefficients in $A$. However, it may not be an $(\\alpha,\\beta)$-Hom-Hopf algebra because\n$S$ may not be a relative-inverse of the identity map.\nHowever, for formal series $g(\\nu)=\\sum_{i \\geq 0} g_i \\nu^i$ such that the invertibility indexes of the elements $(g_i)_{i \\in {\\mathbb N}}$ are bounded, there exits $k\\in {\\mathbb N}$ such that\n$$ \\alpha^k\\circ(S\\star \\id )(g_i)= \\alpha^k\\circ(\\id \\star S )(g_i)=\\eta \\circ \\epsilon (g_i).$$\nSumming up these relations, we obtain:\n\n\n\\begin{prop}\nThe space $A_{b}[[\\nu]]$ of formal series $g(\\nu)=\\sum_{i \\geq 0} g_i \\nu^i$, such that the invertibility indexes of the elements $(g_i)_{i \\in {\\mathbb N}}$ are bounded,\nis an $(\\alpha,\\beta)$-Hom-Hopf algebra.\n\\end{prop}\nIn particular, polynomial elements are in $ A_{b}[[\\nu]] $.\n\n\n\\begin{defn}\n\\label{def:groupLikeAndTheLikes}\nLet $( {\\A},\\mu,\\alpha,\\mathds{1},\\Delta,\\beta , \\epsilon, S) $ be an $(\\alpha,\\beta )$-Hopf-algebra.\nAn element $g\\in \\A$ is a {\\textbf{ group-like element}} if\n$ \\Delta (g) = g \\otimes g \\hbox{ and } \\epsilon(g)=1$.\nLet $\\nu $ be a formal parameter.\n\\begin{enumerate}\n \\item[(i)] We call a {\\textbf{formal group-like element}} a formal series $g(\\nu) \\in {\\A}_b[[\\nu]] $ such that:\n\\begin{equation}\\label{eq:group-like} \\Delta (g(\\nu)) = g(\\nu) \\otimes g(\\nu) \\hbox{ and } \\epsilon(g(\\nu))=1 .\\end{equation}\n \\item[(ii)]\nElements in ${A}[\\nu] $ (i.e. polynomials in $\\nu$) are called {\\textbf{$p$-order group-like elements}}\nwhen:\n\\begin{equation}\\label{eq:group-like-order-n} \\Delta (g(\\nu)) = g(\\nu) \\otimes g(\\nu) \\hbox{ modulo } \\nu^{p+1} \\hbox{ and } \\epsilon(g(\\nu))=1.\\end{equation}\n \\item [(iii)]\nA {\\textbf{formal group-like sequence}} is a sequence $ (g_p (\\nu))_{p \\in {\\mathbb N}} $ of elements in ${\\A}[\\nu] $ such that\n \\begin{enumerate}\n \\item[a)] for all $p \\geq 1$, $g_p(\\nu) $ is a $p$-order formal group-like element,\n \\item[b)] $g_{p+1} (\\nu)= \\alpha (g_p (\\nu))$ modulo $\\nu^{p+1} $,\n \\item[c)] there exists an integer $k$ such that the invertibility index of $g_p(\\nu) $\nis less or equal to $k$ for all $p \\in {\\mathbb N}$.\n \\end{enumerate}\n We denote by $G_{seq}(A)$ the set of all formal group-like sequences of ${\\A} $.\n\\end{enumerate}\n\\end{defn}\n\nIt is classical that group-like elements form a group.\nMore generally:\n\n\\begin{prop}\n\\label{prop:grouplike}\nLet $( {A},\\mu,\\alpha,\\mathds{1},\\Delta,\\beta , \\epsilon, S) $ be an $(\\alpha,\\beta)$-Hom-Hopf algebra.\nGroup-like elements, formal group-like elements, $p$-order group-like elements for an arbitrary $p \\in {\\mathbb N}$, and formal group-like sequences form a Hom-group. Its product is $\\mu$, the inverse of $g$ is $S(g)$, and the unit is ${\\mathds{1}}$.\n\\end{prop}\n\\begin{proof}\n Stability under $\\mu$ of group-like elements, $k$-order group-like elements and formal group-like sequences is an immediate consequence of the compatibility relation between the multiplication and the comultiplication.\n The fact that group-like elements are hom-invertible and that $S(g)$ is a hom-inverse of a group-like element $g$ follows from \\eqref{eq:antipode_condition}, which implies that there exists $k \\in { \\mathbb N}$ such that\n $$ \\alpha^k \\circ \\mu \\,( S \\otimes \\id)\\circ \\Delta (g) = \\alpha^k \\circ \\mu \\, ( \\id \\otimes S)\\circ \\Delta (g) = \\eta \\circ \\epsilon \\, (g) ,$$\n\twhich amounts the relation\n\t$$ \\alpha^k \\circ \\mu \\, (S(g), g ) = \\alpha^k \\circ \\mu \\, (g , S(g) ) = {\\mathds{1}} .$$\n The same applies to $p$-order group-like elements, by taking the previous relation modulo $\\nu^{p+1}$.\nIt then applies for formal group-like sequences, upon noticing that the assumption\n(iii), c) in Definition \\ref{def:groupLikeAndTheLikes}\nguaranties that $ S(g_p (\\nu)) $, which is a Hom-inverse of $g_p(\\nu)$, has\nan invertibility index bounded independently from $p$, hence that $ (S(g_p (\\nu)))_{p \\in {\\mathbb N}} $ is a Hom-inverse of $ (g_p (\\nu))_{p \\in {\\mathbb N}} $.\n For formal group-like elements, the proof follows the same lines, upon noticing that for every formal group-like element $g(\\nu)=\\sum_{i= 0}^\\infty g_i \\nu^i$, there is by assumption an integer $k$ such that for all $i \\in {\\mathbb N}$:\n$$ \\alpha^k \\circ \\mu ( S \\otimes \\id)\\circ \\Delta (g_i) = \\alpha^k \\circ \\mu ( \\id \\otimes S)\\circ \\Delta (g_i) = \\eta \\circ \\epsilon \\, (g_i) = \\mathds{1} ,$$\nso that $S(g(\\nu))=\\sum_{i= 0}^\\infty S(g_i) \\nu^i$ is a Hom-inverse of $g(\\nu)$.\n\\end{proof}\n\n\\begin{rem}\\label{rem:functorgroup}\nAlso, any morphism $\\Phi$ of $(\\alpha,\\beta)$-Hom-Hopf algebra induces in an obvious manner a morphism $\\underline{\\Phi}$ of Hom-groups between their respective group-like elements, formal group-like elements, $p$-order group-like elements for an arbitrary $p \\in {\\mathbb N}$, and formal group-like sequences.\nAssigning to an $(\\alpha,\\beta)$-Hom-Hopf algebra any of the previous types of Hom-groups, one obtains therefore a functor.\nWe call $G_{seq}$ the functor which associates to an $(\\alpha,id)$-Hom-Hopf algebra its Hom-group of formal group-like sequences.\n\\end{rem}\n\\section{Weighted trees and universal enveloping algebras as Hom-Hopf algebras}\n\\label{univ_envel_algebra}\n\nDonald Yau \\cite{Yau08,Yau4} associated to any Hom-Lie algebra $(\\gg, \\brr{\\, , \\, }, \\alpha)$ a Hom-associative algebra $ (U\\gg,\\mu,\\alpha_F)$, that he called the universal enveloping algebra of $\\gg$ and proved to be a Hom-bialgebra. His construction went through two steps: he first associated to any vector space $E$, equipped with a linear map $\\alpha: E \\to E$, the \\textbf{free Hom-nonassociative algebra} $(F_{HNAs}(E), \\mu_F, \\alpha_F)$,\nthen, he considered the quotient of this algebra through the ideal $I^{\\infty}=\\bigcup_{n \\geq 1} I^n$ where $I^1$ is the two-sided ideal\n$$\nI^1= \\langle {\\rm Im}(\\mu_F \\circ (\\mu_F \\otimes \\alpha_F - \\alpha_F \\otimes \\mu_F)); [x,y] -(xy-yx) \\mbox{ for } x,y \\in \\gg \\rangle, $$\nand $(I^n)_{n \\in {\\mathbb N}} $ is given by the recursion relation $ I^{n+1}= \\langle I^n \\cup \\alpha(I^n)\\rangle $. He did not show that the henceforth obtained Hom-bialgebra comes with an antipode.\n\nIn the multiplicative case, a more direct construction exists, that we now present. We need first to introduce some generalities about weighted trees, and to define the free Hom-associative multiplicative algebra.\nMoreover, we will see that it carries a structure of Hom-Hopf algebra.\n\n\\subsection{Weighted trees as the free Hom-associative algebra with $1$-generator}\n\nA planar tree is an oriented graph drawn on a plane with only one root. It is called binary when any vertex is trivalent, i.e., one root and two leaves. Usually we draw the root at the bottom of the tree and the leaves are drawn at the top of it:\n\\begin{center}\n\\begin{tikzpicture}[xscale=0.2, yscale=0.2]\n\n\\draw[line width=1pt] (0,-1) -- (0,2) -- (4,6);\n\\draw[line width=1pt] (3,5) -- (2,6);\n\\draw[line width=1pt] (0,2) -- (-4,6);\n\\draw[line width=1pt] (-2,4) -- (0,6);\n\\draw[line width=1pt] (-1,5) -- (-2,6);\n\n\n\\draw (-2,0) node {root};\n\\draw (-7,6) node {leaves};\n\n\\end{tikzpicture}\n\\end{center}\n\n\nFor any natural number $n\\geq 1$, let $T_n$ denote the set of planar binary trees with $n$ leaves and one root. For $n=1$, $T_1$ admits only one element, namely the unique tree with one leaf and the root. Below are the sets $T_n$ for $n=1,2,3,4$:\n$$\nT_1=\\left\\{ \\mbox{\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\end{tikzpicture}} \\: \\right\\}, \\quad T_2= \\left\\{ \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture}\n} \\right\\} , \\quad T_3=\\left\\{ \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture}\n}, \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (-0.5,1.5) -- (0,2);\n\\end{tikzpicture}\n} \\right\\} $$\n$$ T_4=\\left\\{ \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.3,1.3) -- (-0.2,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.35,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture}\n}, \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.3,1.3) -- (-0.2,2);\n\\draw[line width=1pt] (0.07,1.6) -- (0.45,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture}\n},\\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.2,2);\n\\end{tikzpicture}\n}, \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (-0.3,1.3) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.35,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture}\n}, \\mbox{\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (-0.3,1.3) -- (0.2,2);\n\\draw[line width=1pt] (-0.07,1.6) -- (-0.45,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture}\n}\\right\\}.\n$$\nAn element $\\varphi \\in T_n$ shall be called an $n$-tree for short. When necessary we label the leaves of an $n$-tree by $1,2,3, \\dots, n$ from left to right.\n\nFor $\\varphi \\in T_n$ and $\\psi \\in T_m$ be a pair of trees, the $(n+m)$-tree $\\varphi \\vee \\psi$, called the \\emph{grafting of $\\varphi$ and $\\psi$}, is obtained by joining the roots of $\\varphi $ and $\\psi$ to create a new root. For instance,\n\\begin{center}\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (-0.5,1.5) -- (0,2);\n\\draw (2,1) node {$\\vee$};\n\\end{tikzpicture}\n \\begin{tikzpicture}[xscale=0.4, yscale=0.4]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (2,1) node {$=$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[xscale=0.3, yscale=0.3]\n\\draw[line width=1pt] (1,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (-0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (1,0) -- (1,-1);\n\\draw[line width=1pt] (1,0) -- (3,2);\n\\draw[line width=1pt] (2.5,1.5) -- (2,2);\n\\end{tikzpicture}\n\\end{center}\nNote that grafting is neither an associative nor a commutative operation. For any tree $\\varphi \\in T_n$, there are unique integers $p$ and $q$ with $p+q=n$ and trees $\\varphi_1 \\in T_p$ and $\\varphi_2 \\in T_q$ such that $\\varphi= \\varphi_1 \\vee \\varphi_2$. It is clear that any tree in $T_n$ can be obtained from \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1);\n\\end{tikzpicture} , the 1-tree, by sucessive graftings.\n\n Yau's construction of $U\\gg $ used weighted trees. In the sequel, since we are dealing with multiplicative case, it suffices to work with leaf weighted trees:\n\n\\begin{defn}\nA \\textbf{leaf weighted $n$-tree} is a pair $(\\varphi, a)$ where:\n\\begin{itemize}\n\\item $\\varphi \\in T_n$ is a $n$-tree,\n\\item $a$ is an $n$-tuple $(a_1, a_2, \\dots,a_n) \\in {\\mathbb N}^n$ of non-negative integers.\n\\end{itemize}\nWe call the tree $\\varphi$ the underlying tree of the leaf weighted $n$-tree $(\\varphi,a)$ while,\nfor all $ i=1 \\dots,n$, the integer $a_i$ shall be referred to as the \\textbf{weight} of the leaf $i$.\n\\end{defn}\n\n\nWe will indeed barely use the notation $ (\\varphi,a) $ at all, and find more convenient to picture a leaf weighted $n$-tree $(\\varphi, a_1,a_2, \\dots, a_n)$ by drawing the tree $\\varphi$ and putting the weight $a_i$ next to each leaf. For example, here are two leaf weighted $3$-trees:\n\\begin{center}\n\\begin{tikzpicture}[xscale=0.25, yscale=0.25]\n\\draw[line width=1pt] (0,0) -- (0,2) -- (2,4);\n\\draw[line width=1pt] (0,2) -- (-2,4);\n\\draw[line width=1pt] (-1,3) -- (0,4);\n\n\n\\draw (-2,4.7) node {$0$};\n\\draw (0,4.7) node {$2$};\n\\draw (2,4.7) node {$1$};\n\\end{tikzpicture}\\hspace*{2cm}\n\\begin{tikzpicture}[xscale=0.25, yscale=0.25]\n\n\\draw[line width=1pt] (0,0) -- (0,2) -- (-2,4);\n\\draw[line width=1pt] (0,2) -- (2,4);\n\\draw[line width=1pt] (1,3) -- (0,4);\n\\draw (-2,4.7) node {$0$};\n\\draw (0,4.7) node {$1$};\n\\draw (2,4.7) node {$0$};\n\\end{tikzpicture}\n\\end{center}\n\nThe grafting operation extends to leaf weighted $n$-trees. For example:\n\\begin{center}\n\\begin{tikzpicture}[xscale=0.25, yscale=0.25]\n\\draw[line width=1pt] (0,0) -- (0,2) -- (2,4);\n\\draw[line width=1pt] (0,2) -- (-2,4);\n\\draw[line width=1pt] (-1,3) -- (0,4);\n\n\\draw (-2,4.7) node {$0$};\n\\draw (0,4.7) node {$2$};\n\\draw (2,4.7) node {$1$};\n\\draw (3,2) node {$\\vee$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[xscale=0.25, yscale=0.25]\n\\draw[line width=1pt] (0,0) -- (0,2) -- (-2,4);\n\\draw[line width=1pt] (0,2) -- (2,4);\n\\draw[line width=1pt] (1,3) -- (0,4);\n\n\\draw (-2,4.7) node {$0$};\n\\draw (0,4.7) node {$1$};\n\\draw (2,4.7) node {$0$};\n\\draw (3,2) node {$=$};\n\\end{tikzpicture}\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.5,3) -- (2.5,4);\n\\draw[line width=1pt] (1.5,3) -- (0.5,4);\n\\draw[line width=1pt] (2,3.5) -- (1.5,4);\n\n\n\\draw (-2.5,4.4) node {$0$};\n\\draw (-1.5,4.4) node {$2$};\n\\draw (-0.5,4.4) node {$1$};\n\\draw (0.5,4.4) node {$0$};\n\\draw (1.5,4.4) node {$1$};\n\\draw (2.5,4.4) node {$0$};\n\\end{tikzpicture}\n\\end{center}\n\nFor all $n \\geq 1$, we let $B_n$ denote the set of leaf weighted $n$-trees. Let $B$ denote the union over $n \\in {\\mathbb N}$ of the sets $B_n$ together with an element that we call the unit and denote by $\\mathds{1}$. Note that the element $\\mathds{1}$ is different from the leaf weighted 1-tree $\\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,0.4);\n\\draw (0,0.6) node {$0$};\n\\end{tikzpicture}$. We then consider the free vector space ${\\mathds{T}}$ generated by the set~$B$.\n\nWe define on ${\\mathds{T}}$ two natural linear maps:\n\\begin{enumerate}\n \\item[(i)] $\\alpha:{\\mathds{T}} \\to {\\mathds{T}} $ sending $\\mathds{1}$ to $\\mathds{1}$, i.e. $\\alpha(\\mathds{1})=\\mathds{1}$ and sending a leaf weighted $n$-tree to the leaf weighted $n$-tree obtained adding $+1$ to all the weights of the leaves, i.e. $\\alpha((\\varphi, a_1,a_2, \\dots, a_n))=((\\varphi, a_1+1,a_2+1, \\dots, a_n+1))$;\n \\item[(ii)] a product $ \\vee $ that, for any pair of leaf weighted trees, is just the grafting of these trees and such that for any weighted tree $\\varphi $\n $$\n \\varphi \\vee \\mathds{1} = \\mathds{1} \\vee \\varphi = \\alpha (\\varphi)\n $$\nand $\\mathds{1}\\vee \\mathds{1} = \\mathds{1}$.\n\\end{enumerate}\n\n\nFor example\n$$\\alpha\\left( \\mbox{\\begin{tikzpicture}[xscale=0.2, yscale=0.2,baseline={([yshift=-.8ex]current bounding box.center)}]\n\n\\draw[line width=1pt] (0,0) -- (0,2) -- (2,4);\n\\draw[line width=1pt] (0,2) -- (-2,4);\n\\draw[line width=1pt] (-1,3) -- (0,4);\n\n\n\\draw (-2,4.8) node {$0$};\n\\draw (0,4.8) node {$2$};\n\\draw (2,4.8) node {$1$};\n\\end{tikzpicture}} \\right)= \\mbox{ \\begin{tikzpicture}[xscale=0.2, yscale=0.2,baseline={([yshift=-.8ex]current bounding box.center)}]\n\n\\draw[line width=1pt] (0,0) -- (0,2) -- (2,4);\n\\draw[line width=1pt] (0,2) -- (-2,4);\n\\draw[line width=1pt] (-1,3) -- (0,4);\n\n\\draw (-2,4.8) node {$1$};\n\\draw (0,4.8) node {$3$};\n\\draw (2,4.8) node {$2$};\n\\end{tikzpicture}}\n$$\n\nThe proof of the following lemma is trivial:\n\n\\begin{lem} \\label{alphavee}\nThe map $\\alpha$ defined in item (i) above is a morphism for the grafting of trees, that is\n$$\n\\alpha(\\varphi \\vee \\psi)= \\alpha(\\varphi)\\vee \\alpha(\\psi)\n$$\nfor any $\\varphi, \\psi \\in {\\mathds{T}}$.\n\\end{lem}\n\n\n\nWe now define an important operation which consists in eliminating leaves while changing weights of the remaining ones:\n\n\\begin{defn}\nLet $\\varphi \\in B_n$ and $I \\subset \\{ 1,2, \\dots, n \\}$, we will denote by $\\varphi_I$ the tree obtained by replacing\nall the leaves in $\\{ 1,2, \\dots, n \\} \\backslash I$ by $\\mathds{1}$. In particular, $\\varphi_{\\emptyset}=\\mathds{1}$ and $\\varphi_{\\{ 1,2, \\dots, n \\}}= \\varphi$.\n\\end{defn}\n\nAs an example, if $\\varphi$ is the tree \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.5,3) -- (2.5,4);\n\\draw[line width=1pt] (1.5,3) -- (0.5,4);\n\\draw[line width=1pt] (2,3.5) -- (1.5,4);\n\n\n\\draw (-2.5,4.5) node {$2$};\n\\draw (-1.5,4.5) node {$4$};\n\\draw (-0.5,4.5) node {$0$};\n\\draw (0.5,4.5) node {$3$};\n\\draw (1.5,4.5) node {$1$};\n\\draw (2.5,4.5) node {$2$};\n\\end{tikzpicture} and $I=\\{3,5,6\\}$, then\n\n$$\\varphi_I = \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.5,3) -- (2.5,4);\n\\draw[line width=1pt] (1.5,3) -- (0.5,4);\n\\draw[line width=1pt] (2,3.5) -- (1.5,4);\n\n\n\\draw (-2.5,4.5) node {$\\mathds{1}$};\n\\draw (-1.5,4.5) node {$\\mathds{1}$};\n\\draw (-0.5,4.5) node {$0$};\n\\draw (0.5,4.5) node {$\\mathds{1}$};\n\\draw (1.5,4.5) node {$1$};\n\\draw (2.5,4.5) node {$2$};\n\\end{tikzpicture} =\n\\begin{tikzpicture}[xscale=0.8, yscale=0.8,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0.7) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.2,2);\n\\draw (-1,2.2) node {$\\mathds{1}$};\n\\draw (-0.2,2.25) node {$0$};\n\\draw (0.2,2.25) node {$2$};\n\\draw (1,2.25) node {$3$};\n\\end{tikzpicture} =\n\\begin{tikzpicture}[xscale=0.8, yscale=0.8,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0.7) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw (-1,2.2) node {$1$};\n\\draw (0.2,2.25) node {$2$};\n\\draw (1,2.25) node {$3$};\n\\end{tikzpicture}\n$$\n\n\n\\\n\n\nWe then define a coproduct $\\Delta: {\\mathds{T}} \\longrightarrow {\\mathds{T}} \\otimes {\\mathds{T}} $ by\n\\begin{equation}\\label{def:coproduct}\n\\Delta \\varphi= \\sum_{\\substack{I\\cup J= \\{1 , \\dots, n\\} \\\\ I \\cap J= \\emptyset}} \\varphi_I \\otimes \\varphi_J\n\\end{equation}\nfor a leaf weighted $n$-tree $\\varphi$, extended by linearity, and $\\Delta (\\mathds{1}) = \\mathds{1} \\otimes \\mathds{1} $.\n\n\\begin{lem}\\label{lem:coproduct-properties}\nThis coproduct satisfies the following:\n\\begin{enumerate}\n \\item[(i)] $(\\Delta \\otimes id) \\circ \\Delta = ( id \\otimes \\Delta) \\circ \\Delta$, i.e., $\\Delta$ is coassociative.\n \\item[(ii)] $\\sigma_{12} \\circ \\Delta = \\Delta$, i.e., $\\Delta$ is cocommutative (with $\\sigma_{12}: A \\otimes B \\longrightarrow B \\otimes A$ being the twist map defined by $\\sigma_{12}(a \\otimes b)= b \\otimes a$.)\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\n(i) Using Definition \\ref{def:coproduct} of $\\Delta$, we have\n$$\n(\\Delta \\otimes id) \\circ \\Delta \\varphi= \\sum_{\\substack{I \\cup J \\cup K= \\{1,\\dots, n\\} \\\\ I \\cap J= \\emptyset \\\\ I \\cap K = \\emptyset \\\\ J \\cap K = \\emptyset}} \\varphi_I \\otimes \\varphi_J \\otimes \\varphi_K =( id \\otimes \\Delta) \\circ \\Delta \\varphi\n$$\nfor any $\\varphi \\in B_n$.\n\n(ii) Trivial in view of (\\ref{def:coproduct}).\n\\end{proof}\n\nWe also define a counit $\\epsilon: {\\mathds{T}} \\longrightarrow \\mathbb K$ by $\\epsilon(\\mathds{1})= 1$ and $\\epsilon( \\varphi)=0$, for any $\\varphi \\in B_n$, using then linearity. Using the above Lemma \\ref{lem:coproduct-properties}, it is easy to prove that:\n\n\\begin{prop}\\label{prop:coalgebra}\nThe triple $({\\mathds{T}}, \\Delta,\\epsilon)$ is a co-unital coassociative cocommutative coalgebra.\n\\end{prop}\n\nThis co-unital coassociative cocommutative coalgebra is compatible with the product $\\vee$ in the following sense:\n\n\\begin{lem} \\label{Deltavee}\nFor all $\\varphi, \\psi \\in {\\mathds{T}} $ the compatibility relation\n\\begin{equation}\\label{compatibility}\n\\Delta( \\varphi \\vee \\psi)= \\Delta(\\varphi) \\vee \\Delta(\\psi),\n\\end{equation}\nholds with the understanding that the right hand side of (\\ref{compatibility}) is equipped with the natural algebra structure on the tensor algebra.\n\\end{lem}\n\n\\begin{proof}\nFor $I$ a subset of $\\{1, \\dots, n\\}$, let us denote by $I^c$ the complement set.\nFor all $\\varphi \\in B_n$ and $\\psi \\in B_m$, equation (\\ref{def:coproduct}) gives\n$$\n\\Delta( \\varphi \\vee \\psi) = \\sum_{I \\subseteq \\{1, \\dots, n+m\\}} (\\varphi \\vee \\psi)_I \\otimes (\\varphi \\vee \\psi)_{I^c}.\n$$\nLet $I_n= I \\cap \\{1,\\dots,n\\}$ and $I_m = I \\cap \\{n+1, \\dots, n+m\\}$, and define $I_n^c$ and $I_m^c$ to be the complements of $I_n $ and $ I_m$ in $\\{1,\\dots,n\\}$\nand $ \\{n+1, \\dots, n+m\\}$, respectively. A direct computation gives:\n\\begin{align}\n \\Delta( \\varphi \\vee \\psi)& = \\sum_{\\substack{I_n \\subseteq \\{1, \\dots, n\\} \\\\ I_m \\subseteq \\{n+1, \\dots, n+m\\}}} (\\varphi_{I_n} \\vee \\psi_{I_m}) \\otimes (\\varphi_{I_n^c} \\vee \\psi_{I_m^c}) \\nonumber \\\\\n& = \\sum_{\\substack{I_n \\subseteq \\{1, \\dots, n\\} \\\\ I_m \\subseteq \\{1, \\dots, m\\}}} (\\varphi_{I_n} \\otimes \\varphi_{I_n^c}) \\vee (\\psi_{I_m} \\otimes \\psi_{I_m^c}) \\nonumber \\\\\n& = \\left(\\sum_{I_n \\subseteq \\{1, \\dots, n\\}} \\varphi_{I_n} \\otimes \\varphi_{I_n^c}\\right) \\vee \\left(\\sum_{ I_m \\subseteq \\{1, \\dots, m\\}}\\psi_{I_m} \\otimes \\psi_{I_m^c} \\right) \\nonumber \\\\\n& = \\Delta(\\varphi) \\vee \\Delta(\\psi). \\nonumber\n\\end{align}\nIf $ \\varphi$ or $\\psi $ are equal to $\\mathds{1}$, (\\ref{compatibility}) is equivalent to:\n\\begin{equation}\n\\label{Deltaal}\n\\Delta \\circ \\alpha= (\\alpha \\otimes \\alpha) \\circ \\Delta,\n\\end{equation}\nwhich follows directly from (\\ref{def:coproduct}). This completes the proof.\n\\end{proof}\n\nBecause the map $\\vee$ is not associative the set $({\\mathds{T}}, \\vee, \\mathds{1}, \\Delta, \\epsilon)$ is not a bialgebra. Let us consider now the bilateral ideal ${\\mathcal I}$, with respect to $\\vee$, generated by the following elements:\n $$\n (\\phi \\vee \\psi) \\vee \\alpha (\\chi) -\\alpha (\\phi) \\vee (\\psi \\vee \\chi)\n $$\nwhere $\\phi,\\psi,\\chi $ are either arbitrary leaf weighted trees or the unit $\\mathds{1}$.\nNotice that in case $\\phi, \\psi$ or $\\chi$ are equal to $\\mathds{1}$, then $c(\\phi,\\psi,\\chi)=0$, using Lemma \\ref{alphavee}.\nWe call the quotient $\\mathds{T}\/{\\mathcal I}$, equipped with its structural maps $\\vee, \\alpha$ and the unit $\\mathds{1}$,\nthe \\textbf{free Hom-associative algebra with $1$-generator}.\n\n\n\\begin{rem}\\label{rem:universal}\nThe free Hom-associative algebra with $1$-generator should be considered as an algebra that encodes all the possible operations\nthat can be described purely in terms of the Hom-associative product and $\\alpha$. These operations can then\nbe applied to an arbitrary Hom-associative algebra $\\A$.\n\\end{rem}\n\n\n\\begin{prop}\nThe coproduct $\\Delta$ and the counit $\\epsilon$ of $\\mathds{T}$ go to the quotient, and endow\nthe free Hom-associative algebra with $1$-generator $\\mathds{T}\/{\\mathcal I}$ with a Hom-bialgebra\nstructure.\n\\end{prop}\n\nWe first prove an obvious lemma:\n\n\\begin{lem}\\label{lem:coideal1}\nLet $e,f,g,h$ be elements in $ {\\mathds{T}}$.\nIf $e - f \\in {\\mathcal I}$ and $g-h \\in {\\mathcal I}$, then $e \\otimes g - f \\otimes h \\in {\\mathcal I} \\otimes {\\mathds{T}} + {\\mathds{T}} \\otimes {\\mathcal I}$.\n\\end{lem}\n\\begin{proof}\nWe just have to use the algebraic identity $e \\otimes g - f \\otimes h= (e-f) \\otimes g + f \\otimes (g-h)$.\n\\end{proof}\n\n\\begin{lem}\\label{lem:coideal2}\nThe ideal ${\\mathcal I}$ is a coideal of $({\\mathds{T}}, \\Delta,\\epsilon)$.\n\\end{lem}\n\\begin{proof}\nWe have to check that:\n\\begin{itemize}\n \\item $\\epsilon({\\mathcal I})=0$.\n \\item $\\Delta$ maps ${\\mathcal I} $ to ${\\mathcal I} \\otimes {\\mathds{T}} + {\\mathds{T}} \\otimes {\\mathcal I}$.\n \\end{itemize}\n The first point is trivial by definition of $\\epsilon$.\n For the second point,\nsince $\\Delta$ and $\\vee$ are compatible (Lemma \\ref{Deltavee}) it suffices indeed to show that generating elements of ${\\mathcal I} $, i.e.\nelements of the form $c(\\phi,\\psi,\\chi)=( (\\phi \\vee \\psi) \\vee \\alpha (\\chi) -\\alpha (\\phi) \\vee (\\psi \\vee \\chi) $,\n are mapped to ${\\mathcal I} \\otimes {\\mathds{T}} + {\\mathds{T}} \\otimes {\\mathcal I} $.\n\n\n\nFor arbitrary $\\phi \\in B_n$, $\\psi \\in B_m$ and $\\chi \\in B_k$, we have on the one hand:\n\\begin{align}\n \\Delta\\left( (\\phi \\vee \\psi) \\vee \\alpha (\\chi) \\right)\n & = \\left(\\Delta(\\phi) \\vee \\Delta(\\psi) \\right) \\vee \\Delta\\left( \\alpha (\\chi) \\right), \\quad \\mbox{by Lemma \\ref{Deltavee}} \\nonumber \\\\\n &= \\left(\\Delta(\\phi) \\vee \\Delta(\\psi) \\right) \\vee \\alpha ^{\\otimes^2}\\left( \\Delta (\\chi) \\right), \\quad \\mbox{by equation (\\ref{Deltaal})} \\nonumber \\\\\n& = \\sum_{\\substack{I \\subseteq \\{1,\\dots, n\\} \\\\ J \\subseteq \\{1,\\dots, m\\} \\\\K \\subseteq \\{1,\\dots, k\\}}} \\left( \\left( \\phi_I \\otimes \\phi_{I^c} \\right) \\vee \\left(\\psi_J \\otimes \\psi_{J^c} \\right) \\right) \\vee \\left( \\alpha (\\chi_K) \\otimes \\alpha (\\chi_{K^c})\\right) \\nonumber \\\\\n& = \\sum_{\\substack{I \\subseteq \\{1,\\dots, n\\} \\\\ J \\subseteq \\{1,\\dots, m\\} \\\\K \\subseteq \\{1,\\dots, k\\}}} \\left( \\left( \\phi_I \\vee \\psi_J \\right) \\vee \\alpha (\\chi_K) \\right) \\otimes \\left( \\left( \\phi_{I^c} \\vee \\psi_{J^c} \\right) \\vee \\alpha (\\chi_{K^c})\\right) \\nonumber,\n\\end{align}\nwhile on the other hand, we have\n\\begin{align}\n \\Delta\\left( \\alpha(\\phi) \\vee (\\psi \\vee \\chi) \\right) = \\sum_{\\substack{I \\subseteq \\{1,\\dots, n\\} \\\\ J \\subseteq \\{1,\\dots, m\\} \\\\K \\subseteq \\{1,\\dots, k\\}}} \\left( \\alpha( \\phi_I) \\vee \\left( \\psi_J \\vee \\chi_K \\right)\\right) \\otimes \\left( \\alpha(\\phi_{I^c}) \\vee \\left( \\psi_{J^c} \\vee \\chi_{K^c} \\right)\\right). \\nonumber\n\\end{align}\nApplying Lemma \\ref{lem:coideal1} to $e= \\left( \\phi_I \\vee \\psi_J \\right)\\vee \\alpha (\\chi_K) , f=\\left( \\phi_{I^c} \\vee \\psi_{J^c} \\right) \\vee \\alpha (\\chi_{K^c}),\ng= \\alpha( \\phi_I) \\vee \\left( \\psi_J \\vee \\chi_K \\right), h= \\alpha(\\phi_{I^c}) \\vee \\left( \\psi_{J^c} \\vee \\chi_{K^c}\\right)$,\nwe see that the difference between $ \\Delta\\left( (\\phi \\vee \\psi) \\vee \\alpha (\\chi) \\right)$ and $ \\Delta\\left( \\alpha(\\phi) \\vee (\\psi \\vee \\chi) \\right) $\nis an element in ${\\mathcal I} \\otimes {\\mathds{T}} + {\\mathds{T}} \\otimes {\\mathcal I} $, which completes the proof.\n\\end{proof}\n\n\n\\begin{proof}[Proof of the proposition]\nBy definition of the ideal ${\\mathcal I}$, the quotient space $\\mathds{T}\/{\\mathcal I}$, is a Hom-associative algebra\nwhen equipped with $\\vee$, and ${\\mathds{1}} $ is a unit. From Proposition \\ref{prop:coalgebra}, it also follows that $({\\mathds{T}}\/{\\mathcal I}, \\Delta,\\epsilon)$\nis a coassociative algebra with counit $\\epsilon$. According to Lemma \\ref{Deltavee}, these induced structures are compatible.\n\\end{proof}\n\nWe now intend to define an antipode on the free Hom-associative algebra with $1$-generator. We first define $S: {\\mathds{T}} \\longrightarrow {\\mathds{T}}$ by:\n\\begin{itemize}\n \\item $S(\\mathds{1})=\\mathds{1} $;\n \\item $S(|, a_1)= - (|, a_1)$, or $S\\left( \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\draw (0,1.8) node {$a_1$};\n\\end{tikzpicture}\\right) = - \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5);\n\\draw (0,1.8) node {$a_1$};\n\\end{tikzpicture}$, where $a_1$ is a non-negative integer;\n \\item $S$ is an antimorphism of $({\\mathds{T}},\\vee)$, i.e., $S(\\varphi \\vee \\psi)= S(\\psi) \\vee S(\\varphi)$, for any $\\varphi, \\psi \\in B$.\n\\end{itemize}\n\n\nExplicitly, $S$ maps a tree with $n$-leaves to $(-1)^n$ times the image of that tree through a vertical symmetry applied at each node, for example:\n$$\nS\\left(\n\\begin{tikzpicture}[xscale=0.35, yscale=0.35,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.75,4) ;\n\\draw[line width=1pt] (1.25,3.25) -- (0.5,4);\n\n\n\\draw (-2.5,4.5) node {$2$};\n\\draw (-1.5,4.5) node {$4$};\n\\draw (-0.5,4.5) node {$0$};\n\\draw (0.5,4.5) node {$3$};\n\\draw (1.75,4.5) node {$2$};\n\\end{tikzpicture}\n \\right)= (-1)^5\n \\begin{tikzpicture}[xscale=0.35, yscale=0.35,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (1.5,3) -- (2.5,4);\n\\draw[line width=1pt] (2,3.5) -- (1.5,4);\n\\draw[line width=1pt] (1.5,3) -- (0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (-1.75,4) ;\n\\draw[line width=1pt] (-1.25,3.25) -- (-0.5,4);\n\n\n\\draw (2.5,4.5) node {$2$};\n\\draw (1.5,4.5) node {$4$};\n\\draw (0.5,4.5) node {$0$};\n\\draw (-0.5,4.5) node {$3$};\n\\draw (-1.75,4.5) node {$2$};\n\\end{tikzpicture}\n$$\n\n\nThe map $S$ clearly goes to the quotient and induces an endomorphism of the free Hom-associative algebra with $1$-generator. The main result of this section is:\n\n\\begin{thm}\\label{thm:HopfAlgOnTrees}\n The free Hom-associative algebra with $1$-generator is an $(\\alpha,id )$-Hom-Hopf algebra\n (in the sense of Definition \\ref{def:hom-Hopf-algebra_in_our_sense}) when equipped with the antipode $S$.\n\\end{thm}\n\\begin{proof}\nIt is obvious that $S$ commutes with $\\alpha$ and $S({\\mathcal I}) \\subset {\\mathcal I}$, so $S$ goes to the quotient.\nThe only non-immediate result is that $S$ satisfies (\\ref{eq:antipode_condition}).\nThe result is true for $\\mathds{1}$ and for any element in $B_1$, i.e., of the form \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\draw (0,1.8) node {$a_1$};\n\\end{tikzpicture} where $a_1$ is a non-negative integer.\nLet us prove that if the result holds true for leaf weighted trees $\\varphi,\\psi $ it also holds true for $\\varphi \\vee \\psi $. Let $k$ be be the non-negative number such that\n$\\alpha^k \\circ \\vee \\circ (S \\otimes id) \\circ \\Delta (\\varphi)=0$, then\n\\begin{align}\n & \\alpha^{k+1} \\circ \\vee \\circ (S \\otimes id) \\circ \\Delta (\\varphi \\vee \\psi) = \\nonumber \\\\\n & \\qquad = \\sum_{J,I} \\alpha^{k+1} \\circ \\vee \\circ (S \\otimes id) (\\varphi_J \\vee \\psi_I) \\otimes (\\varphi_{J^c} \\vee \\psi_{I c}) \\nonumber \\\\\n & \\qquad = \\sum_{J,I} \\alpha^{k+1}\\left( (S(\\psi_I) \\vee S(\\varphi_J)) \\vee (\\varphi_{J^c} \\vee \\psi_{I c})\\right) \\nonumber \\\\\n& \\qquad = \\sum_{J,I} \\alpha^{k}\\left( \\alpha^2(S(\\psi_I)) \\vee \\left( \\alpha (S(\\varphi_J)) \\vee (\\varphi_{J^c} \\vee \\psi_{I c}) \\right) \\right), \\mbox{ using Hom-associativity} \\nonumber \\\\\n& \\qquad = \\sum_{J,I} \\alpha^{k} (\\alpha^2(S(\\psi_I)) \\vee \\left( ( S(\\varphi_J) \\vee \\varphi_{J^c}) \\vee \\alpha(\\psi_{I c}) \\right), \\mbox{ using again Hom-associativity} \\nonumber \\\\\n& \\qquad = 0. \\nonumber\n\\end{align}\nAn induction step completes the proof.\n\\end{proof}\n\nThe antipode $S$ is an inverse in the usual sense for quite a few trees.\nBy inverse in the usual sense we mean that we can take $k=0$ in (\\ref{eq:antipode_condition}).\nLet us call \\textbf{ferns} weighted trees whose underlying tree is such that each node is\nrelated to a leaf. For instance, the following trees are ferns:\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.3,1.3) -- (-0.2,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.35,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture} and\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.3,1.3) -- (-0.2,2);\n\\draw[line width=1pt] (0.07,1.6) -- (0.45,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\end{tikzpicture} while\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.2,2);\n\\end{tikzpicture}, is not a fern.\n\n\n\nWe still call \\textbf{space of ferns} the subspace of the free vector space $\\mathds{T} $ generated by ferns and the subspace of the free Hom-associative algebra with $1$-generator which is the image of ferns in $\\mathds{T} $ through the canonical projection from $ \\mathds{T} $\nto the free Hom-associative algebra with $1$-generator.\n\nLet us investigate the subspace of the free Hom-associative algebra with $1$-generator $\\mathds{T}\/{\\mathcal I}$ made of all elements with invertibility index $0$, i.e. the subspace of all elements $g \\in \\mathds{T}\/{\\mathcal I}$ such that the following relation holds\n\\begin{equation} \\label{eq:antipodeK0}\n \\vee \\circ (S \\otimes \\id) \\circ \\Delta (g)= \\vee \\circ( \\id \\otimes S) \\circ \\Delta (g)= \\eta \\circ \\epsilon \\, (g).\n\\end{equation}\n\n\\begin{prop}\\label{prop:invert-index-ferns}\nThe subspace of the free Hom-associative algebra with $1$-generator $\\mathds{T}\/{\\mathcal I}$ made of all elements with invertibility index $0$ is stable under left or right multiplication under elements in $B_1$.\nIt contains the space of ferns, as well as the spaces $B_i, i=1,2,3,4$.\n\\end{prop}\n\\begin{proof}\nLet $\\varphi$ be an element in $\\mathds{T}$ where (\\ref{eq:antipodeK0}) holds, then it also holds for any tree of the form \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.3) ;\n\\draw (0,1.5) node {$a_1$};\n\\end{tikzpicture} $\\vee \\varphi$ (in this case we need to use the Hom-associativity of $\\vee$):\n\n\\begin{align}\n \\vee \\circ (S \\otimes id ) \\circ \\Delta (\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.3) ;\n\\draw (0,1.5) node {$a_1$};\n\\end{tikzpicture} \\vee \\varphi) & = \\vee \\circ (S \\otimes id ) \\circ \\left( \\Delta (\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.3) ;\n\\draw (0,1.5) node {$a_1$};\n\\end{tikzpicture} ) \\vee \\Delta(\\varphi)\\right) \\nonumber \\\\\n& = \\vee \\circ (S \\otimes id ) \\circ \\left( (\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.3) ;\n\\draw (0,1.5) node {$a_1$};\n\\end{tikzpicture} \\otimes \\mathds{1} + \\mathds{1} \\otimes \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.3) ;\n\\draw (0,1.5) node {$a_1$};\n\\end{tikzpicture} ) \\vee (\\varphi_J \\otimes \\varphi_{J^c})\\right) \\nonumber \\\\\n& = \\vee \\circ (S \\otimes id ) \\circ \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {$\\varphi_J$};\n\\draw (-1,2.3) node {$\\mathds{1}$};\n\\end{tikzpicture} \\otimes \\alpha(\\varphi_{J^c}) + \\alpha(\\varphi_{J}) \\otimes \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {$\\varphi_{J^c}$};\n\\draw (-1,2.3) node {$\\mathds{1}$};\n\\end{tikzpicture} \\right) \\nonumber\n\\end{align}\nwhich proves the claim.\nIt turns out that this also proves that relations\n $ \\vee \\circ (S \\otimes id) \\circ \\Delta (\\varphi)= \\mu \\circ( id \\otimes S) \\circ \\Delta (\\varphi)=0$\nhold for all ferns, since any fern in $B_{n+1}$ is obtained out of\na fern in $B_n$ by either left or right grafting with an element in $B_1$.\nIn particular, this relation also holds for all elements in $B_2,B_3$, since those spaces are included\nin the space of ferns.\nTo check it for all elements in $B_4$, it suffices to check it in the unique element which is not a fern, i.e.\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.2,2);\n\\end{tikzpicture},\n which is a direct computation.\n\\end{proof}\n\nWe give an example of an element in the free Hom-associative algebra with $1$-generator for which the antipode does not satisfy Relation (\\ref{eq:antipodeK0}):\n\\begin{center}\n \\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] ((0,1.5) -- (1.5,3) -- (2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.75,3.7) -- (-2,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (-0.75,3.7) -- (-1,4);\n\\draw[line width=1pt] (1.35,2.8) -- (0.25,4);\n\\draw[line width=1pt] (0.7,3.5) -- (1.2,4);\n\\draw[line width=1pt] (0.95,3.7) -- (0.7,4);\n\\draw[line width=1pt] (2.25,3.7) -- (2,4);\n \\end{tikzpicture}\n \\end{center}\n\nProof: Left to the reader.\n\n\\subsection{Universal enveloping Lie algebra of a Hom-Lie algebra}\n\nLet $(\\gg, \\brr{\\, , \\, }, \\alpha)$ be a multiplicative Hom-Lie algebra. Let us apply the so-called Schur functor \\cite{Loday} to ${\\mathds{T}} $ and $ \\gg$. We define ${\\mathds{T}}^\\gg $ to be the vector space:\n$$\n {\\mathds{1}} \\oplus \\bigoplus_{n \\geq 1} B_n \\otimes \\gg^{\\otimes n} .\n $$\nElements of $B_n \\otimes \\gg^{\\otimes n}$ shall be pictured for every $ \\phi \\in B_n$, $x_1, \\dots,x_n \\in \\gg$,\nby inserting, for all $i=1, \\dots,n$, the element $ x_i$ at the top of the leaf with label $x_i$:\n $$ \\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.75,4) ;\n\\draw[line width=1pt] (1.25,3.25) -- (0.5,4);\n\n\n\\draw (-2.5,4.5) node {$2$};\n\\draw (-1.5,4.5) node {$4$};\n\\draw (-0.5,4.5) node {$0$};\n\\draw (0.5,4.5) node {$3$};\n\\draw (1.75,4.5) node {$1$};\n\\draw (-2.5,5.2) node {$x_1$};\n\\draw (-1.5,5.2) node {$x_2$};\n\\draw (-0.5,5.2) node {$x_3$};\n\\draw (0.5,5.2) node {$x_4$};\n\\draw (1.75,5.2) node {$x_5$};\n\\end{tikzpicture}$$\nFrom now, we only write down the weight of a given leaf of an element in $B_n \\otimes \\gg^{\\otimes n}$ when it is not equal to zero.\nSo \\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (0,2.5) node {$ 2$};\n\\end{tikzpicture}\nis a short hand for\\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.5) node {$0$};\n\\draw (0,2.5) node {$ 2$};\n\\draw (-1,2.5) node {$0$};\n\\end{tikzpicture}.\n\nThe operations $ \\vee,S,\\Delta,\\epsilon, \\alpha$ defined on $ {\\mathds{T}} $ have natural extensions\nto ${\\mathds{T}}^\\gg $, that we denote by the same symbols. The extension $\\Delta$ is still a coproduct with counit $\\epsilon$, which is still compatible with $\\vee $.\nMoreover, the subspace $ {\\mathcal I}^\\gg = \\oplus_{n \\geq 3} {\\mathcal I}_n \\otimes \\gg^{\\otimes n} $,\n(with $ {\\mathcal I}_n $ being the subspace of ${\\mathcal I} \\cap B_n $) is an ideal for $\\vee$\n and a coideal for $\\Delta$. It follows directly from Theorem \\ref{thm:HopfAlgOnTrees} that the sextuple\n$({\\mathds{T}}^\\gg\/{\\mathcal I}^\\gg,\\vee,\\Delta \\circ \\alpha,S,\\mathds{1},\\epsilon) $ is an $(\\alpha,id)$-Hom-Hopf algebra.\nWe call this Hom-Hopf algebra the \\textbf{free Hom-associative algebra with generators in~$\\gg$}.\n\nWe now consider the quotient of the latter Hom-Hopf algebra by the ideal $ {\\mathcal J}^\\gg$ of $({\\mathds{T}}^\\gg\/{\\mathcal I}^\\gg,\\vee,\\mathds{1})$ generated by:\n\\begin{enumerate}\n \\item[(i)] elements of the form \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture},\n(recall that for all $ y \\in {\\mathfrak g} $, \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$y$};\n\\end{tikzpicture} is a short hand for \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.95) node {$0$};\n\\draw (0,2.75) node {$y$};\n\\end{tikzpicture} )\n \\item[(ii)] elements of the form \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture}.\n\\end{enumerate}\n\nWe first establish the following result.\n\n\\begin{prop}\nThe ideal $ {\\mathcal J}^\\gg$ is a Hom-Hopf ideal of\nthe free Hom-associative algebra with generators in $\\gg$.\n\\end{prop}\n\\begin{proof}\nAgain, it suffices to show that the structural maps go to the quotient with respect to ${\\mathcal J}^\\gg $.\nSince $\\Delta $ and $\\vee $ are compatible in the sense of equation (\\ref{compatibility}), to show that $\\Delta $ goes to the quotient suffices to show that\nfor all $ x,y \\in \\gg$, both\n$ \\Delta \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture} \\right) $ and $\\Delta \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture} \\right) $\nare elements in\n${\\mathcal J}^\\gg \\otimes {\\mathds{T}}^\\gg\/ {\\mathcal{I}}^\\gg + {\\mathds{T}}^\\gg\/ {\\mathcal{I}}^\\gg\\otimes {\\mathcal J}^\\gg$.\nThis follows from the following two computations, obtained by using directly the definition (\\ref{def:coproduct}) of the coproduct :\n$$\\Delta\\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture} \\right)= \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture} \\right) \\otimes \\mathds{1} + \\mathds{1} \\otimes \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture} \\right) $$\nand\n$$\\Delta \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture} \\right)= \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture} \\right) \\otimes \\mathds{1} + \\mathds{1} \\otimes \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture} \\right).$$\nThe antipode $S$ being an antimorphism of $\\vee $, it suffices to check that it preserves the set of generators of ${\\mathcal J}^\\gg $ to state that preserves ${\\mathcal J}^\\gg$.\nThis simply follows from the definition of $S$, since:\n $$ S\\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture} \\right) = -\\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)} ]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,1.8) node {$n$};\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$\\alpha^n(x)$};\n\\end{tikzpicture} \\right) $$\nand\\footnote{Notice that this is the first time that we are using the skew-symmetry of the bracket $[.,.]$ on ${\\mathfrak g}$.} $S\\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture} \\right) = - \\left( \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture} \\right) .$\n\nThis completes the proof.\n\\end{proof}\n\n\\begin{defn}\\label{def:Hom-Universal}\nThe {\\bf universal enveloping algebra of a multiplicative Hom-Lie algebra $(\\gg, \\brr{\\, , \\, }, \\alpha)$}\nis by definition the quotient of the free Hom-associative algebra with generators in $\\gg$ (which is an $(\\alpha,\\id)$-Hom-Hopf algebra) by the Hom-Hopf ideal ${\\mathcal J}^\\gg $. This quotient is itself an $(\\alpha,\\id)$-Hom-Hopf algebra that we denote by ${\\mathcal U}\\gg $, while we keep the usual symbols for its structural maps.\n\\end{defn}\n\n\\begin{ex}\n For $\\alpha=id$, Hom-Lie algebras are just Lie algebras. It is routine to check that the universal enveloping algebra of $(\\gg, \\brr{\\, , \\, }, id) $ coincides with the usual universal enveloping algebra.\n\n\n\nFor $\\alpha=0$ and $[.,.]$ an arbitrary skew-symmetric map, all weighted trees are in the ideal ${\\mathcal I}^{\\mathfrak g}$ unless the weight of each leaf is $0$ and the universal enveloping algebra is obtained by applying the Schur functor to the algebra of all binary trees, then dividing the outcome by the ideal (for grafting) generated by\n$$\n \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$x$};\n\\draw (1,2.5) node {$y$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.5) node {$y$};\n\\draw (1,2.5) node {$x$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$[x,y]$};\n\\end{tikzpicture}\n$$\nfor all $x,y \\in {\\mathfrak g}$. This is of course not associative (it is by construction Hom-associative with respect to a map that satisfies $\\alpha=0$ on (the image of) $\\bigoplus_{n \\geq 1} B_n \\otimes \\gg^{\\otimes n}$, which is not a strong constraint). Also, the coproduct is simply:\n$ \\Delta (\\phi) = \\phi \\otimes {\\mathds{1}}+ {\\mathds{1}} \\otimes \\phi$ for all $ \\phi \\in \\bigoplus_{n \\geq 1} B_n \\otimes \\gg^{\\otimes n}$.\n\\end{ex}\n\n\\begin{rem}\\label{rem:functor}\nAny Hom-Lie algebra morphism $\\psi:\\gg \\to \\gg'$ induces a natural $(\\alpha,\\id)$-Hom-Hopf algebra morphism\n${\\mathcal U}\\phi : {\\mathcal U}\\gg \\to {\\mathcal U}\\gg'$ by:\n $$ {\\mathcal U}\\phi : \\phi \\otimes (x_1 \\otimes \\dots \\otimes x_n ) \\mapsto \\phi \\otimes (\\psi(x_1) \\otimes \\dots \\otimes \\psi(x_n) ) $$\nfor all weighted $n$-tree $\\phi$ and $x_1, \\dots,x_n \\in \\gg$.\nAssociating to a Hom-Lie algebra $\\gg$ a universal envelopping algebra ${\\mathcal U}\\gg$,\none therefore obtains a functor ${\\mathcal U} $ from the category of Hom-Lie algebras to the category of\n$(\\alpha,\\id)$-Hom-Hopf algebras.\n\\end{rem}\n\n Notice that for Hom-Lie algebras constructed by composition out of a Lie algebra $\\gg $, equipped with a bracket $[.,.]_{Lie}$, through an endomorphism $\\alpha$, there are two natural Hom-Hopf algebra structures:\n\\emph{(i)} the universal enveloping algebra of the Hom-Lie algebra $ (\\gg,\\alpha \\circ [.,.],\\alpha)$\nas in Definiton \\ref{def:Hom-Universal} and \\emph{(ii)} the $(\\tilde{\\alpha},\\id)$-twist of the universal enveloping algebra (in the usual sense) $U^{Lie}(\\gg) $\n of the Lie algebra $ (\\gg, [.,.]_{Lie})$\nthrough the Hopf algebra morphism $\\tilde{\\alpha} $ associated to $\\alpha $ (this is an $(\\alpha,\\id) $-Hopf-algebra by Example \\ref{Hom-Hopf-Twist}).\n\nThese two Hom-Hopf algebra do not agree. This is easily seen when $\\alpha=0$, for\nthe product in simply $0$ for the Hom-Hopf algebra of item \\emph{(ii)} (since its product is $\\mu_{\\alpha}=\\alpha \\circ \\mu$\nwith $\\mu$ the product of $U^{Lie}(\\gg)$), while it is not zero for\nthe universal enveloping algebra of the Hom-Lie algebra $ (\\gg,\\alpha \\circ [.,.],\\alpha)$.\nFor instance, the product\n\\begin{center}\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$x$};\n\\end{tikzpicture}\n$\\vee$\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$x$};\n\\end{tikzpicture}\n\\end{center}\ndoes not vanish if $ x \\in {\\mathfrak g}$ a non-zero element. Notice that the spaces of which they are defined also do not coincide.\n\n\n\n\\subsection{Primitive elements on the free Hom-associative algebra with $1$-generator}\n\nBesides the properties of primitive elements of an $(\\alpha,\\beta)$-Hom-Hopf algebra described in Remark \\ref{rmk:SeveralPoints}, we aim to discuss the case of $\\mathds{T}\/{\\mathcal I}$.\nBy construction, $\\alpha :\\mathds{T} \\to \\mathds{T}$ is injective, and it is natural to ask\nif its induced map $\\alpha$ on the free Hom-associative algebra with $1$-generator $\\mathds{T}\/{\\mathcal I}$ is injective as well. The answer is negative, in view of the next proposition, which introduces a useful element in building counterexamples. It shows in particular that although all elements in $B_1$ (i.e. leaf weighted $1$-trees) are primitive for the coassociative coproduct (defined as in second item of Remark \\ref{rmk:SeveralPoints}), the transpose is not true: there are primitive elements which are not in $B_1$.\n\n\\begin{prop}\\label{prop:remarquable}\nConsider the following element in $\\mathds{T}$:\n$$\n\\begin{tikzpicture}[xscale=0.6, yscale=0.6, baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0.5) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.2,2);\n\\draw (-1,2.3) node {$0$};\n\\draw (-0.2,2.3) node {$1$};\n\\draw (0.2,2.3) node {$1$};\n\\draw (1,2.3) node {$0$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.6, yscale=0.6,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0.5) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.3,1.3) -- (-0.2,2);\n\\draw[line width=1pt] (0.07,1.6) -- (0.45,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.3) node {$1$};\n\\draw (-0.2,2.3) node {$0$};\n\\draw (0.45,2.3) node {$0$};\n\\draw (1,2.3) node {$0$};\n\\end{tikzpicture}.\n$$\nDenote by $u$ its class in the free Hom-associative algebra with $1$-generator $\\mathds{T}\/{\\mathcal I}$.\nThen $u \\neq 0$ but $\\alpha (u) =0$. Moreover, $u$ is a primitive element, and any element in the algebra (for $\\vee$) generated by $u $ is primitive.\n\\end{prop}\n\\begin{proof}\nIt is clear that the tree that defines $u$ is not contained in $ {\\mathcal I}$, which is therefore not equal to zero.\nAlso, $\\alpha(u)= \\begin{tikzpicture}[xscale=0.6, yscale=0.6, baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0.5) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw[line width=1pt] (0.6,1.6) -- (0.2,2);\n\\draw[line width=1pt] (-0.6,1.6) -- (-0.2,2);\n\\draw (-1,2.3) node {$1$};\n\\draw (-0.2,2.3) node {$2$};\n\\draw (0.2,2.3) node {$2$};\n\\draw (1,2.3) node {$1$};\n\\end{tikzpicture} - \\begin{tikzpicture}[xscale=0.6, yscale=0.6,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0.5) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.3,1.3) -- (-0.2,2);\n\\draw[line width=1pt] (0.07,1.6) -- (0.45,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (-1,2.3) node {$2$};\n\\draw (-0.2,2.3) node {$1$};\n\\draw (0.45,2.3) node {$1$};\n\\draw (1,2.3) node {$1$};\n\\end{tikzpicture}=0$, using Hom-associativity. By a direct computation, $\\Delta(u)= u \\otimes \\mathds{1} + \\mathds{1} \\otimes u$, i.e. $u$ is a primitive element.\n\nAs a consequence, $ u$ is a primitive element contained in the kernel of $ \\alpha$.\nNow, for every pair $ u_1,u_2$ of primitive elements contained in the kernel of $\\alpha$,\n$u_1 \\vee u_2$ is a primitive element, as follows from equation (\\ref{compatibility}):\n \\begin{eqnarray*} \\Delta (u_1 \\vee u_2 ) &=& \\Delta (u_1 ) \\vee \\Delta (u_2) \\\\\n &=& (u_1 \\otimes \\mathds{1} + \\mathds{1} \\otimes u_1) \\otimes ( u_2 \\otimes \\mathds{1} + \\mathds{1} \\otimes u_2) \\\\\n& = & (u_1 \\vee u_2) \\otimes \\mathds{1} + \\mathds{1} \\otimes (u_1 \\vee u_2) \\\\\n& & + ( u_1 \\vee \\mathds{1} ) \\otimes ( \\mathds{1} \\vee u_2 ) +( \\mathds{1} \\vee u_2) \\otimes (u_1 \\vee \\mathds{1} ) \\\\\n& = & (u_1 \\vee u_2) \\otimes \\mathds{1} + \\mathds{1} \\otimes (u_1 \\vee u_2) \\\\\n& & + \\alpha( u_1) \\otimes \\alpha( u_2 ) + \\alpha (u_2) \\otimes \\alpha (u_1) \\\\\n & =& (u_1 \\vee u_2) \\otimes \\mathds{1} + \\mathds{1} \\otimes (u_1 \\vee u_2) \\end{eqnarray*}\nMoreover, Equation (\\ref{alphavee}) implies that any element in the algebra generated by $u $ is in the kernel of $\\alpha$. Altogether, these properties imply that the space of elements with are primitive and in the kernel of $\\alpha$\nis stable under $\\vee$. In particular, every element in the algebra generated by $u$ is primitive.\n\\end{proof}\n\n\\begin{rem}\n In view of Remark \\ref{rem:universal}, Proposition \\ref{prop:remarquable} implies that for any Hom-associative algebra $(\\A,\\vee, \\alpha)$, and any\n $x,y,z,t \\in \\A$, the element\n $$ \\alpha(t) \\vee ((x \\vee y) \\vee t) - (t\\vee \\alpha(x)) \\vee (\\alpha(y) \\vee z)$$\nis in the kernel of $\\alpha$.\n\\end{rem}\n\n\n\n\\subsection{Canonical $n$-ary operations on Hom-associative algebras}\nLet $A$ be a Hom-associative algebra.\nOn $ \\A$, making the product of $n$ elements, $n \\geq 3$ depends on the order in which the products are taken. There is however a natural manner manner to define $n$-ary operations $\\A^{\\otimes n} \\to \\A $, as we will see in the sequel. Recall that by Remark \\ref{rem:universal}, operations of $n$ elements of $A$ are encoded by leaf weighted $n$-trees.\n\nGiven a $n$-tree $ \\varphi \\in T_n $ (without weights) and an integer $n \\leq k$,\nwe define a leaf weighted $n$-tree $\\varphi[k] $ by assigning to the leaf with label $i$ the integer\n$ k - \\ell (i)$ with $ \\ell$ being the length of the branch from the root to the leaf. The length of the leaf $i$ is then the length of the path from the root to the leaf,\nwhen the tree is seen as a graph.\nFor instance, let\n$\\varphi=\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.75,4) ;\n\\draw[line width=1pt] (1.25,3.25) -- (0.5,4);\n\\end{tikzpicture}\n$ then\n$\\varphi[7]=\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.75,4) ;\n\\draw[line width=1pt] (1.25,3.25) -- (0.5,4);\n\\draw (-2.5,4.5) node {$3$};\n\\draw (-1.5,4.5) node {$3$};\n\\draw (-0.5,4.5) node {$4$};\n\\draw (0.5,4.5) node {$4$};\n\\draw (1.75,4.5) node {$4$};\n\\end{tikzpicture}\n$ and $\\varphi[5]=\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (1.75,4) ;\n\\draw[line width=1pt] (1.25,3.25) -- (0.5,4);\n\\draw (-2.5,4.5) node {$1$};\n\\draw (-1.5,4.5) node {$1$};\n\\draw (-0.5,4.5) node {$2$};\n\\draw (0.5,4.5) node {$2$};\n\\draw (1.75,4.5) node {$2$};\n\\end{tikzpicture}\n$.\n\nLet us call \\textbf{right $n$-fern} the $n$-tree (without weights) obtained by successive graftings on the right of the $1$-tree and denote it by $F_n^r$, i.e. $$F_n^r= \\left(\\left(\\dots \\left( \\left(\\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture}\\right) \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\right) \\vee \\dots \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\right) \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture}\\right)= \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (-1.5,3) -- (-2.5,4);\n\\draw[line width=1pt] (-2,3.5) -- (-1.5,4);\n\\draw[line width=1pt] (-1.5,3) -- (-0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (2.2,4) ;\n\\draw[line width=1pt] (-0.3,1.75) -- (1.65,4) ;\n\\draw (0.4,4) node {$\\dots$};\n\\end{tikzpicture}$$\nOf course, the right $n$-fern is a fern. Similarly we call \\textbf{left $n$-fern} the $n$-tree (without weights) obtained by successive graftings on the left of the $1$-tree and denote it by $F_n^l$, i.e. $$F_n^l= \\left( \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\vee \\left( \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\vee \\dots \\vee \\left( \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\vee \\left( \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\end{tikzpicture} \\right) \\right) \\dots \\right) \\right)\n= \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,1) -- (0,1.5) -- (1.5,3) -- (2.5,4);\n\\draw[line width=1pt] (2,3.5) -- (1.5,4);\n\\draw[line width=1pt] (1.5,3) -- (0.5,4);\n\\draw[line width=1pt] (0,1.5) -- (-2.2,4) ;\n\\draw[line width=1pt] (0.3,1.75) -- (-1.65,4) ;\n\\draw (-0.4,4) node {$\\dots$};\n\\end{tikzpicture}$$\n\nWe first prove a lemma:\n\n\\begin{lem}\\label{KFougeres}\n Let $k,n $ be non-negative integers with $ n \\leq k$. The identity\n $$\n F_n^l[k]= F_n^r[k]\n $$\nholds in the free Hom-associative algebra with 1-generator ${\\mathds{T}}\/{\\mathcal I} $.\n\\end{lem}\n\n\\begin{proof}\nThis follows by a finite induction using definitions of these trees and Hom-associativity.\n\\end{proof}\nThe following Lemma is straightforward.\n\\begin{lem}\\label{lemmaKtoK-1}\nLet $\\varphi_1 \\in T_p, \\varphi_2 \\in T_q $ be two trees and $k$ an integer such that $k\\geq p+q$. Then\n$$\n(\\varphi_1\\vee \\varphi_2)[k]= \\varphi_1[k-1] \\vee \\varphi_2[k-1].\n$$\n\\end{lem}\nNow, we state the main result of this section showing that the operations on Hom-associative algebras encoded by the weighted $n$-trees $\\varphi[k]$ depends only on $n$ and $k$.\n\\begin{prop}\n\\label{prop:indifference}\n Let $k,n $ be two non-negative integers with $ n \\leq k$. For any $n$-trees $\\varphi,\\psi \\in T_n$, the identity $\\varphi[k] = \\psi[k]$ holds in the free Hom-associative algebra with 1-generator ${\\mathds{T}}\/{\\mathcal I} $.\n\\end{prop}\n\\begin{proof}\nIt suffices to show that, for any tree $\\varphi \\in T_n$, we have $\\varphi[k]= F_n^r[k]$. For $n=1,2,3$, it is routine to check that this identity is true (using Hom-associativity in case $n=3$). Let us suppose now that this equality holds for any tree in $T_p$ with $p< n$. Let $\\varphi$ be a tree in $T_n$ with $n>3$, then there exist $\\varphi_1 \\in T_p, \\varphi_2 \\in T_q $ such that $\\varphi= \\varphi_1 \\vee \\varphi_2$ and $p+q=n$. By Lemma \\ref{lemmaKtoK-1} we have\n$$\n\\varphi[k]= \\varphi_1[k-1] \\vee \\varphi_2[k-1]\n$$\nand applying the hypothesis $\\varphi_1[k-1]= F^r_p[k-1]$ and $\\varphi_2[k-1]= F^r_q[k-1]=F^l_q[k-1]$, using also the Lemma \\ref{KFougeres}. If $q=1$, then $\\varphi[k]=F^r_p[k-1] \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\draw (0,1.7) node {\\tiny $k-2$};\n\\end{tikzpicture} = F^r_n[k] $, if not, using in each step Hom-associativity, we have $\\varphi[k]=F^r_p[k-1] \\vee F^l_q[k-1] = F^r_{p+1}[k-1] \\vee F^l_{q-1}[k-1]= F^r_{p+2}[k-1] \\vee F^l_{q-2}[k-1]= \\dots=F^r_{p+q-1}[k-1] \\vee \\begin{tikzpicture}[xscale=0.3, yscale=0.3,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1.5) ;\n\\draw (0,1.7) node {\\tiny $k-2$};\n\\end{tikzpicture}= F^r_{p+q}[k] = F^r_{n}[k]$.\n\\end{proof}\n\nAs a consequence, for every non-negative integers $k,n $ with $ n \\leq k$, we denote by $\\lfloor e^n \\rfloor_k $ the element in ${\\mathds{T}}\/{\\mathcal I} $, defined by $\\varphi[k] \\in {\\mathds{T}}\/{\\mathcal I} $ for an arbitrary $n$-tree $ \\varphi \\in T_n$. It is called the \\textbf{$k$-weighted $n$-ary product}.\n\n\\begin{ex} $ \\lfloor e^1 \\rfloor_k= $\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$k-1$};\n\\end{tikzpicture} , $\\lfloor e^2 \\rfloor_k= $\\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {\\scriptsize$k-2$};\n\\draw (-1,2.2) node {\\scriptsize$k-2$};\n\\end{tikzpicture} , $\\lfloor e^3 \\rfloor_k=$ \\begin{tikzpicture}[xscale=1, yscale=1,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.1) node {\\scriptsize$k-3$};\n\\draw (0,2.1) node {\\scriptsize$ k-3$};\n\\draw (-1,2.1) node {\\scriptsize$ k-2$};\n\\end{tikzpicture}.\n\\end{ex}\n\\begin{lem}\\label{lem:coproduct-explicit}\n For all non-negative integers $k,n,m $ with $ n < k$ and $ m < k$, we have\n\\begin{enumerate}\n\\item\n $ \\Delta ( \\lfloor e^n \\rfloor_k) = \\sum_{i=0}^n\n\\left(\n\\begin{matrix}\n n \\\\\ni\n\\end{matrix}\n\\right)\n \\lfloor e^i\\rfloor_k \\otimes \\lfloor e^{n-i}\\rfloor_{k} ,$\n\\item\n $ \\lfloor e^n\\rfloor_k \\vee \\lfloor e^m\\rfloor _k = \\lfloor e^{n+m}\\rfloor_{k+1} =\\alpha (\\lfloor e^{n+m}\\rfloor_k), $\n \\item $S(\\lfloor e^{n}\\rfloor_{k} )=(-1)^n\\lfloor e^{n}\\rfloor_{k}.$\n \\end{enumerate}\n\\end{lem}\n\n\\\n\nGiven a formal power series in one variable with real coefficients $ f(\\nu)= \\sum_{i \\geq 0}^\\infty a_i \\nu^i$, we denote by $ \\widehat{f}_p (\\nu)$ and call it \\textbf{ $k$-weighted realization of $f$ } the element in ${\\mathds{T}}\/{\\mathcal I}[[\\nu]]$ modulo $\\nu^{p+1}$ given by:\n\\begin{equation}\\label{ChapFnu}\n\\widehat{f}_p(\\nu) =a_0 \\mathds{1}+ \\sum_{i \\geq 1}^p a_i \\nu^i \\lfloor e^i \\rfloor_p.\n\\end{equation}\nWe call the sequence $(\\widehat{f}_p(\\nu))_{p\\in \\mathbb{N}}$ the \\textbf{ realization of $f$ } and denote it by $\\widehat{f}(\\nu)$.\\\\\n\nWe provide the following properties:\n\n\\begin{prop}\\label{Prop:proprietiesWidehat}\nGiven two formal power series in one variable with real coefficients $ f(\\nu)= \\sum_{i \\geq 1}^\\infty a_i \\nu^i$,\n$ g(\\nu)= \\sum_{i \\geq 1}^\\infty b_i \\nu^i$. We have:\n\\begin{enumerate}\n \\item $\\widehat{f}(\\nu)\\vee \\widehat{g}(\\nu)=\\alpha( \\widehat{fg}(\\nu)),$\n \\item for all $k\\in \\mathbb{N}$, $\\widehat{f}_{p+1}(\\nu)=\\alpha(\\widehat{f}_p(\\nu))$ modulo $\\nu^{p+1}$,\n \\item $S(\\widehat{f}(\\nu))=\\widehat{f}(-\\nu)$,\n\t\\item the invertibility index of $\\widehat{f}_p(\\nu)$ is equal to $0$ for all $p \\in {\\mathbb N}$,\n\\end{enumerate}\nwhere $\\alpha$, $\\vee$ and $S$ are the structure operations of ${\\mathds{T}}\/{\\mathcal I} $ extended by $\\mathbb{R}[[\\nu]]$-linearity to ${\\mathds{T}}\/{\\mathcal I} [[\\nu]]$.\n\\end{prop}\n\\begin{proof}\nThe first assertion follows from the second item of Lemma \\ref{lem:coproduct-explicit}. The second one is obtained by considering the definition \\eqref{ChapFnu} and the relation $ \\lfloor e^{i}\\rfloor_{p+1} =\\alpha (\\lfloor e^{i}\\rfloor_p)$. The third one is a consequence of the third item of Lemma \\ref{lem:coproduct-explicit}.\nLet us prove the last assertion. Proposition \\ref{prop:indifference} implies that\n$$\\widehat{f}_p(\\nu) =a_0 \\mathds{1}+ \\sum_{i \\geq 1}^p a_i \\nu^i \\lfloor e^i \\rfloor_p$$\ncan be represented by ferns. The conclusion then follows from Proposition \\ref{prop:invert-index-ferns}.\n\\end{proof}\n\n\n\n\n\n\\section{A Hom-group integrating a Hom-Lie algebra}\n\nIn this section, we aim to associate to any Hom-Lie algebra a Hom-group.\nThis construction uses the study of the universal enveloping algebra\nand elements of group-like type.\n\n\n\\subsection{Group-like elements in the free Hom-associative algebra with $1$-generator}\n\nFor $g(\\nu)= {\\mathds 1}+\\sum_{i=1}^\\infty {g_i\\nu^i}$ a formal group-like element, $g_1$ is primitive. Unlike for the free associative algebra with $1$-generator, not any primitive element of ${\\mathds{T}}\/{\\mathcal I}$ could be the first order element of a formal group-like element with $g_0={\\mathds 1}$.\n\n\\begin{prop}\\label{prop:negativeGroupLike}\nThe $(\\alpha,\\id)$-Hom-Hopf algebra ${\\mathds{T}}\/{\\mathcal I}[[\\nu]]$ does not admit formal group-like elements\nof the form $g(\\nu)= {\\mathds 1}+\\sum_{i=1}^\\infty {g_i\\nu^i}$ where $g_1$ is the leaf weighted 1-tree $\\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,0.4);\n\\draw (0,0.6) node {$0$};\n\\end{tikzpicture}$.\n\\end{prop}\n\\begin{proof}\nAssume that $g(\\nu)= {\\mathds 1}+\\sum_{i=1}^\\infty {g_i\\nu^i}$ is a formal group element. For example, the coefficient of $\\nu^2$ in $\\Delta (g(\\nu))=g(\\nu)\\otimes g(\\nu) $ yields\n$$\\Delta (g_2)= g_1\\otimes g_1+ g_2\\otimes {\\mathds 1}+{\\mathds 1}\\otimes g_2.$$\nThis imposes that $g_2$ is in $B_2$ which is impossible, because the projection of $\\Delta (g_2)$ on $B_1\\otimes B_1$ is a linear combination of elements of the form\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$k$};\n\\end{tikzpicture}\n$\\otimes$\n\\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,-0.5) -- (0,1.5) ;\n\\draw (0,2) node {$l$};\n\\end{tikzpicture}\nwith $k$ or $l$ strictly positive.\n\\end{proof}\n\\begin{rem}\nThe proof of Proposition \\ref{prop:negativeGroupLike} gives indeed that\n there is no $2$-order formal group-like elements of the form\n$g(\\nu)= {\\mathds 1}+\\sum_{i=1}^p {g_i\\nu^i}$ where $g_1$ is the leaf weighted 1-tree $\\begin{tikzpicture}[baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,0.4);\n\\draw (0,0.6) node {$0$};\n\\end{tikzpicture}$.\n\\end{rem}\n\nLet us show that formal group-like sequence is a relevant object by showing that the free Hom-associative algebra with $1$-generator admits a $1$-parameter family of formal group-like sequences, although it admits very few\ngroup-like elements.\n\nFor all $s\\in {\\mathbb R}$, consider the realization $\\widehat{exp}(s)$ of the formal series $exp(s)=\\sum_{i=0}^\\infty{\\frac{s^i}{i!}\\nu^i}$. We call the assignment $s \\rightarrow \\widehat{exp}(s)$ the \\textbf{exponential sequence}.\n\n\nFor a better understanding of $\\widehat{exp}(s)$, we give its first terms:\n$$\n\\begin{array}{rcl}\n\\widehat{exp}_0(s)&=& {\\mathds{ 1}} \\\\ \\widehat{exp}_1(s) &=&{\\mathds{ 1}} + s \\nu \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$0$};\n\\end{tikzpicture} \\\\\n \\widehat{exp}_2(s) &=&{\\mathds{ 1}} + s \\nu \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$1$};\n\\end{tikzpicture}\n+ \\frac{s^2 \\nu^2}{2!}\n\\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {\\scriptsize$0$};\n\\draw (-1,2.2) node {\\scriptsize$0$};\n\\end{tikzpicture} \\\\\n\\widehat{exp}_3(s) &=&{\\mathds{ 1}} + s \\nu \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$2$};\n\\end{tikzpicture}\n+ \\frac{s^2 \\nu^2}{2!}\n\\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {\\scriptsize$1$};\n\\draw (-1,2.2) node {\\scriptsize$1$};\n\\end{tikzpicture}\n+\\frac{s^3 \\nu^3}{3!}\n \\begin{tikzpicture}[xscale=1, yscale=1,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.1) node {\\scriptsize$0$};\n\\draw (0,2.1) node {\\scriptsize$ 0$};\n\\draw (-1,2.1) node {\\scriptsize$ 1$};\n\\end{tikzpicture}.\n\\end{array}\n$$\nGenerally, we have\n$$\\widehat{exp}_p(s) =\\mathds{1}+ \\sum_{i = 1}^p \\frac{ s^i}{i!} \\nu^i \\lfloor e^i \\rfloor_p .\n$$\n\n\\begin{thm}\n\\label{theo:group-like_order_kappa}\n The exponential sequence $s \\rightarrow \\widehat{exp}(s)$ is valued in\n the Hom-group $G_{seq}({\\mathds T}\/{\\mathcal I}) $ of formal group-like sequence.\\\\\n Moreover,\n \\begin{enumerate}\n \\item $ \\widehat{exp}(0) $ is the unit element of the Hom-group of formal group-like sequences.\n \\item For all $s,t \\in {\\mathbb R}, k \\in {\\mathbb N}$,\n $ \\widehat{exp}(s) \\vee \\widehat{exp}(t) = \\alpha(\\widehat{exp}(s+t)) $.\n \\item $ S(\\widehat{exp}(s)) = \\widehat{exp}(-s)$ is a strict inverse, i.e.\n $$ \\widehat{exp}(s) \\vee \\widehat{exp}(-s) = \\widehat{exp}(-s) \\vee \\widehat{exp}(s) = \\mathds{1} . $$\n \\end{enumerate}\n\\end{thm}\n\\begin{proof}\nThe first item just follows from the definition of the realization of the formal series $f(\\nu)=e^{0\\nu}=1$.\nThe second item follows from the first item in Proposition \\ref{Prop:proprietiesWidehat} applied to the classical relation $e^{s \\nu} e^{t \\nu} = e^{(s+t)\\nu}$. The third item follows from the third item in Proposition \\ref{Prop:proprietiesWidehat}.\n\nIt remains to show that the exponential sequence is valued in formal group-like sequence, defined in item (iii)\nof Definition \\ref{def:groupLikeAndTheLikes}. The fourth item in Proposition \\ref{Prop:proprietiesWidehat} implies that $\\widehat{exp}_p(s)$ (i.e. the $p$-th term in the sequence $\\widehat{exp}(s)$) has invertibility index equal to $0$, so that condition c) holds.\nThe second item in Proposition \\ref{Prop:proprietiesWidehat} implies item b) in\nDefinition \\ref{def:groupLikeAndTheLikes}. We are left with the task of showing that\n$\\widehat{exp}_p(s)$ is a $p$-order group-like element. This follows from the following computation,\nwhich is done modulo $\\nu^{p+1}$:\n\\begin{eqnarray*} \\Delta (\\widehat{exp}_p(s)) &=& \\Delta(\\mathds{1})+ \\sum_{i = 1}^p \\frac{ s^i}{i!} \\nu^i \\Delta \\lfloor e^i \\rfloor_p \\\\\n&=& \\mathds{1} \\otimes \\mathds{1}+ \\sum_{i = 1}^p \\sum_{j = 1}^i \\frac{ s^i}{i!} \\nu^i \\left( \\begin{matrix}\n i \\\\\nj\n\\end{matrix}\\right) \\lfloor e^{j} \\rfloor_p \\otimes\n \\lfloor e^{i-j} \\rfloor_p \\\\\n&=& \\widehat{exp}_p(s) \\otimes \\widehat{exp}_p(s),\n\\end{eqnarray*}\nwhere the first item of Lemma \\ref{lem:coproduct-explicit} was used to go from the first to the second line.\n\\end{proof}\n\n\n\n\\subsection{Formal group-like sequences of the universal enveloping algebra}\n\nNow, we define the exponential map for a Hom-Lie algebra $(\\gg,[\\cdot,\\cdot],\\alpha)$, in order to achieve a construction of a functor from the category of Hom-Lie algebras to the category of Hom-groups.\n For all $x_1, \\dots,x_i \\in \\gg$, and $p,i \\in \\mathbb{N}$ with $ p \\geq i $,\ndefine an element in ${\\mathcal U}\\gg$ by:\n$$ \\lfloor x_1, \\dots,x_i \\rfloor_p = e^i_p \\otimes (x_1 \\otimes \\dots \\otimes x_i). $$\nIf $x_1= \\dots=x_i=x$, then we denote this product as $\\lfloor x^i\\rfloor_p $.\nFor $f(\\nu) = \\sum a_i \\nu^i$ a formal series, we call \\textbf{realization of $f(\\nu)$ evaluated at $x$} the sequence\n $$\\left(\\sum_{i=0}^p \\frac{s^i}{i!} \\lfloor x^i \\rfloor_p \\right)_{p \\in {\\mathbb N}}.$$\nFor $f=exp(\\nu)$ in particular, we define\nthe exponential map $\\widehat{exp}(sx)$ to be the sequence in ${\\mathcal U}\\gg[[\\nu]]$\nobtained by taking the realization evaluated at $x$ of the formal series $e^{s\\nu}$.\nBy construction, $\\widehat{exp}(sx)$ is obtained by applying the Schur construction to\n$\\widehat{exp}(s)$ and to the element $x$, and the following theorem\ncan be derived easily from Theorem \\ref{theo:group-like_order_kappa}.\n\n\\begin{thm}\n\\label{theo:group-like_x}\n For all $x \\in {\\gg} $ the exponential sequence $s \\rightarrow \\widehat{exp}(sx)$ is valued in\n the Hom-group $G_{seq}(\\gg) $. Moreover,\n \\begin{enumerate}\n \\item $ \\widehat{exp}(0 x) $ is the unit element $\\mathds{1} \\in G_{seq}(\\gg) $.\n \\item For all $s,t \\in {\\mathbb R}, k \\in {\\mathbb N}$,\n $ \\widehat{exp}(sx) \\vee \\widehat{exp}(tx) = \\alpha(\\widehat{exp}((s+t)x)) = \\widehat{exp}((s+t)\\alpha(x))$.\n \\item $ S(\\widehat{exp}(sx)) = \\widehat{exp}(-sx)$ is a strict inverse, i.e.\n $$ \\widehat{exp}(sx) \\vee \\widehat{exp}(-sx) = \\widehat{exp}(-sx) \\vee \\widehat{exp}(sx) = \\mathds{1} . $$\n \\end{enumerate}\n\\end{thm}\n\nFor a better understanding of $\\widehat{exp}(sx)$, we give its first terms:\n$$\n\\begin{array}{rcl}\n\\widehat{exp}_0(sx)&=& {\\mathds{ 1}} \\\\ \\widehat{exp}_1(sx) &=&{\\mathds{ 1}} + s \\nu \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$x$};\n\\end{tikzpicture} \\\\\n \\widehat{exp}_2(sx) &=&{\\mathds{ 1}} + s \\nu \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$\\alpha(x)$};\n\\end{tikzpicture}\n+ \\frac{s^2 \\nu^2}{2!}\n\\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {\\scriptsize$x$};\n\\draw (-1,2.2) node {\\scriptsize$x$};\n\\end{tikzpicture} \\\\\n\\widehat{exp}_3(sx) &=&{\\mathds{ 1}} + s \\nu \\begin{tikzpicture}[xscale=0.4, yscale=0.4,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,2);\n\\draw (0,2.5) node {$\\alpha^2(x)$};\n\\end{tikzpicture}\n+ \\frac{s^2 \\nu^2}{2!}\n\\begin{tikzpicture}[xscale=0.5, yscale=0.5,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.2) node {\\scriptsize$\\alpha(x)$};\n\\draw (-1,2.2) node {\\scriptsize$\\alpha(x)$};\n\\end{tikzpicture}\n+\\frac{s^3 \\nu^3}{3!}\n \\begin{tikzpicture}[xscale=1, yscale=1,baseline={([yshift=-.8ex]current bounding box.center)}]\n\\draw[line width=1pt] (0,0) -- (0,1) -- (1,2);\n\\draw[line width=1pt] (0.5,1.5) -- (0,2);\n\\draw[line width=1pt] (0,1) -- (-1,2);\n\\draw (1,2.1) node {\\scriptsize$x$};\n\\draw (0,2.1) node {\\scriptsize$ x$};\n\\draw (-1,2.1) node {\\scriptsize$ \\alpha(x)$};\n\\end{tikzpicture}.\n\\end{array}\n$$\n\nAn immediate consequence of the expression of $ \\widehat{exp}_1(sx) $ is the next proposition, that we invite the reader to see as saying that $G_{seq}(\\gg)$ is large enough to be meaningful.\n\\begin{prop}\\label{prop:expinjective}\nFor every $s \\neq 0$, the assignment\n$x \\mapsto \\widehat{exp}(sx)$\nis an injection from $\\gg$ to the Hom-group of formal group-like sequences $G_{seq}(\\gg)$.\n\\end{prop}\n\nIn order to integrate a Hom-Lie algebra into a Hom-group, we compose the following two functors:\n\\begin{enumerate}\n\\item The functor $ {\\mathcal U} $ from the category of Hom-Lie algebras to the category of\nHom-Hopf algebras which consists in assigning to a Hom-Lie algebra $\\gg$\nits universal algebra ${\\mathcal U}\\gg$, as in Remark \\ref{rem:functor}.\n\\item The functor $G_{seq}$ from the category of Hom-Hopf algebras to the category of Hom-groups,\nwhich consists in assigning to a Hom-Hopf algebra $A$ its formal group-like sequences,\nas in Remark \\ref{rem:functorgroup}.\n\\end{enumerate}\nTherefore the composition of these functors is a functor ${\\mathfrak G}$ from the category of Hom-Lie algebras to the category of Hom-groups. Proposition \\ref{prop:expinjective} implies that this functor is not trivial. Notice that it is compatible with the exponential map in the sense that for every morphism of Hom-Lie algebra $\\varphi: \\gg \\to \\gg'$,\nthe following diagram\n $$ \\xymatrix{ \\gg \\ar[r]^{\\varphi} \\ar[d]^{ \\widehat{exp}(s \\cdot) }& \\gg' \\ar[d]^{ \\widehat{exp}(s \\cdot) }\\\\ {\\mathfrak G} (\\gg) \\ar[r]^{{\\mathfrak G}(\\varphi)}& {\\mathfrak G}(\\gg').} $$\nis commutative.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{INTRO}}\n\n\\noindent{}A normal complex surface $X$ with at worst log terminal\nsingularities, i.e., quotient singularities, is called \\textit{log Del Pezzo}\n\\textit{surface} if its anticanonical divisor $-K_{X}$ is a $\\mathbb{Q}%\n$-Cartier ample divisor. The \\textit{index} of such a surface is defined to\nbe the smallest positive integer $\\ell$ for which $\\ell K_{X}$ is a Cartier\ndivisor. Every log Del \\ Pezzo surface is isomorphic to the\n\\textit{anticanonical model} (in the sense of Sakai \\cite{Sakai}) of the\nrational surface obtained by its minimal desingularization. The following\nTheorem is due to Nikulin \\cite{Nikulin2} (for related results cf. \\cite{Alex, Nikulin3}):\n\n\\begin{theorem}\n\\label{NIKTHM}Let $X$ be a log Del Pezzo surface of index $\\ell$ and\n$\\widetilde{X}\\longrightarrow X$ be its minimal desingularization. Then the\nPicard number $\\rho(\\widetilde{X})$ of $\\widetilde{X}$ \\emph{(}i.e., the rank\nof its Picard group\\emph{)} is bounded by%\n\\begin{equation}\n\\rho(\\widetilde{X})1\\right\\} .$ Its subset $\\{ \\left. \\text{orb}(\\sigma_{F})\\right\\vert\n\\,F\\in\\breve{I}_{Q}\\},$ with $\\breve{I}_{Q}$ defined to be $\\breve{I}%\n_{Q}:=\\left\\{ \\left. F\\in I_{Q}\\, \\right\\vert \\,p_{F}=1\\right\\} ,$ is the\nset of the \\textit{Gorenstein singularities} of $X_{Q}.$\n\nThe minimal desingularization of the surface $X_{Q}$ can be described as\nfollows: Equip the minimal generators of $\\Delta_{Q}$ with an order (e.g.,\nanticlockwise), and assume that for every $F\\in\\mathcal{F}(Q)$ the cone\n$\\sigma_{F}$ has $\\mathbf{n}^{(F)},\\mathbf{n}^{\\prime(F)}\\in\\mathbb{Z}^{2}$ as\nminimal generators ($\\sigma_{F}=\\mathbb{R}_{\\geq0}\\, \\mathbf{n}^{(F)}%\n+\\mathbb{R}_{\\geq0}\\, \\mathbf{n}^{\\prime(F)}$), with $\\mathbf{n}^{(F)}$ coming\nfirst w.r.t. this order. Next, for all $F\\in I_{Q},$ consider the\nnegative-regular continued fraction expansion of\n\\begin{equation}\n\\frac{q_{F}}{q_{F}-p_{F}}=\\left[ \\! \\! \\left[ b_{1}^{(F)},b_{2}^{(F)}%\n,\\ldots,b_{s_{F}}^{(F)}\\right] \\! \\! \\right] :=b_{1}^{(F)}%\n-\\cfrac{1}{b_{2}^{(F)}-\\cfrac{1}{\\begin{array} [c]{cc}\\ddots & \\\\ & -\\cfrac{1}{b_{s_{F}}^{(F)}}\\end{array} }}\\ \\ ,\n\\label{EXPPQCF}%\n\\end{equation}\nand define $\\mathbf{u}_{0}^{(F)}:=\\mathbf{n}^{(F)},$ $\\mathbf{u}_{1}%\n^{(F)}:=\\frac{1}{q_{F}}((q_{F}-p_{F})\\mathbf{n}^{(F)}+\\mathbf{n}^{\\prime\n(F)}),$ and lattice points $\\{ \\mathbf{u}_{j}^{(F)}\\left\\vert \\,2\\leq j\\leq\ns_{F}+1\\right. \\}$ by the formulae\n\\[\n\\mathbf{u}_{j+1}^{(F)}:=b_{j}^{(F)}\\mathbf{u}_{j}^{(F)}-\\mathbf{u}_{j-1}%\n^{(F)},\\ \\ \\forall j\\in\\{1,\\ldots,s_{F}\\}.\\\n\\]\nIt is easy to see that $\\mathbf{u}_{s_{F}+1}^{(F)}=\\mathbf{n}^{\\prime(F)},$\nand that the integers $b_{j}^{(F)}$ are $\\geq2,$ for all $j\\in\\{1,\\ldots\n,s_{F}\\}.$ The singularity orb$(\\sigma_{F})\\in U_{F}$ is resolved minimally by\nthe proper birational map induced by the refinement $\\{ \\mathbb{R}_{\\geq0}\\,\n\\mathbf{u}_{j}^{(F)}+\\mathbb{R}_{\\geq0}\\, \\mathbf{u}_{j+1}^{(F)}\\ \\left\\vert\n\\ 0\\leq j\\leq s_{F}\\right. \\}$ of the fan which is composed of the cone\n$\\sigma_{F}$ and its faces. The exceptional divisor is $E^{(F)}:=%\n{\\textstyle\\sum\\nolimits_{j=1}^{s_{F}}}\nE_{j}^{(F)},$ having%\n\\[\nE_{j}^{(F)}:=\\text{ }\\overline{\\text{orb}(\\mathbb{R}_{\\geq0}\\, \\mathbf{u}%\n_{j}^{(F)})}\\ (\\cong\\mathbb{P}_{\\mathbb{C}}^{1}),\\ \\forall j\\in\\{1,\\ldots\n,s_{F}\\},\n\\]\n(i.e., the closures of the $\\mathbb{T}$-orbits of the \\textquotedblleft\nnew\\textquotedblright\\ rays) as its components, with self-intersection number\n$(E_{j}^{(F)})^{2}=-b_{j}^{(F)}$ (see \\cite[Cor. 1.18 and Prop. 1.19, pp.\n23-25]{Oda}).\n\n\\begin{note}\n(i) If $F\\in\\mathcal{F}(Q),$ and ${\\boldsymbol{\\eta}}_{F}\\in(\\mathbb{Z}%\n^{2})^{\\vee}$ is its unique primitive outer normal vector, we define its\n\\textit{local index} to be the positive integer $l_{F}:=\\left\\langle\n{\\boldsymbol{\\eta}}_{F},F\\right\\rangle ,$ where\n\\[\n\\left\\langle \\cdot,\\cdot\\right\\rangle :\\text{Hom}_{\\mathbb{R}}(\\mathbb{R}%\n^{2},\\mathbb{R})\\times\\mathbb{R}^{2}\\longrightarrow\\mathbb{R}%\n\\]\nis the usual inner product. For $F\\in\\mathcal{F}(Q)\\mathbb{r}I_{Q}$ we have\nobviously $l_{F}=1.$ For $F\\in I_{Q},$ let $K(E^{(F)})$ be the \\textit{local\ncanonical divisor }of the minimal resolution of orb$(\\sigma_{F})\\in U_{F}$ (in\nthe sense of \\cite[p. 75]{Dais}). $K(E^{(F)})$ is a $\\mathbb{Q}$-Cartier\ndivisor (a rational linear combination of $E_{j}^{(F)}$'s), and\n\\begin{equation}\nl_{F}=\\text{ min}\\left\\{ \\left. \\xi\\in\\mathbb{N}\\ \\right\\vert \\ \\xi\nK(E^{(F)})\\text{ is a Cartier divisor}\\right\\} =\\tfrac{q_{F}}{\\text{gcd}%\n(q_{F},p_{F}-1)}. \\label{localind}%\n\\end{equation}\n(ii) If $F\\in I_{Q},$ denoting by $\\mathfrak{m}_{X_{Q},\\text{orb}(\\sigma_{F}%\n)}$ the maximal ideal of the local ring $\\mathcal{O}_{X_{Q},\\text{orb}%\n(\\sigma_{F})}$ of the singularity orb$(\\sigma_{F}),$ and by\n\\[\nm_{F}:=\\text{dim}_{\\mathbb{C}}((\\mathfrak{m}_{X_{Q},\\text{orb}(\\sigma_{F}%\n)})\/(\\mathfrak{m}_{X_{Q},\\text{orb}(\\sigma_{F})}^{2}))-1\n\\]\nits \\textit{multiplicity}, it is known (cf. \\cite[Satz 2.11, p. 347]%\n{Brieskorn}) that\n\\begin{equation}\nm_{F}=2+\\sum_{j=1}^{s_{F}}(b_{j}^{(F)}-2). \\label{multformula}%\n\\end{equation}\n\n\\end{note}\n\n\\begin{lemma}\n\\label{MULTIND}For all $F\\in I_{Q}$ we have\n\\[\nm_{F}\\leq2l_{F}.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nSee \\cite[Lemma 1.1 (iii), p. 235]{Nikulin1}.\n\\end{proof}\n\n\\begin{lemma}\n\\label{KESQUARE}For all $F\\in I_{Q}$ the self-intersection number of\n$K(E^{(F)})$ equals\n\\[\nK(E^{(F)})^{2}=-\\left( \\frac{2-\\left( p_{F}+\\widehat{p}_{F}\\right) }{q_{F}%\n}+(m_{F}-2)\\right) .\n\\]\n\n\\end{lemma}\n\n\\begin{proof}\nFollows from \\cite[Corollary 4.6, p. 96]{Dais} and formula\n(\\ref{multformula}).\n\\end{proof}\n\n\\noindent{}The minimal desingularization $\\varphi:\\widetilde{X}_{Q}%\n\\longrightarrow X_{Q}$ of $X_{Q}$ is constructed by means of the smooth\ncompact toric surface $\\widetilde{X}_{Q}$ which is defined by the fan\n\\[\n\\widetilde{\\Delta}_{Q}:=\\left\\{\n\\begin{array}\n[c]{c}%\n\\text{ the cones }\\left\\{ \\left. \\sigma_{F}\\ \\right\\vert \\ F\\in\n\\mathcal{F}(Q)\\mathbb{r}I_{Q}\\right\\} \\text{ and }\\\\\n\\left\\{ \\left. \\mathbb{R}_{\\geq0}\\, \\mathbf{u}_{j}^{(F)}+\\mathbb{R}_{\\geq\n0}\\, \\mathbf{u}_{j+1}^{(F)}\\ \\right\\vert \\ F\\in I_{Q},\\ j\\in\\{0,1,\\ldots\n,s_{F}\\} \\right\\} ,\\\\\n\\text{together with their faces}%\n\\end{array}\n\\right\\}\n\\]\n(refining each of the cones $\\left\\{ \\left. \\sigma_{F}\\ \\right\\vert \\ F\\in\nI_{Q}\\right\\} $ of $\\Delta_{Q}$ as mentioned above). Furthermore, the\ncorresponding \\textit{discrepancy divisor} equals%\n\\begin{equation}\nK_{\\widetilde{X}_{Q}}-\\varphi^{\\star}K_{X_{Q}}=\\sum_{F\\in I_{Q}}K(E^{(F)}).\n\\label{DISCREPANCY}%\n\\end{equation}\n(By $K_{X_{Q}},K_{\\widetilde{X}_{Q}}$ we denote the canonical divisors of\n$X_{Q}$ and $\\widetilde{X}_{Q},$ respectively.)\n\n\\begin{note}\nBy virtue of (\\ref{localind}) and (\\ref{DISCREPANCY}) the index $\\ell$ of\n$X_{Q}$ (as defined in \\S \\ref{INTRO}) equals%\n\\begin{equation}\n\\ell=\\text{ lcm}\\left\\{ \\left. l_{F}\\ \\right\\vert \\ F\\in\\mathcal{F}%\n(Q)\\right\\} . \\label{LCM}%\n\\end{equation}\n(For simplicity, sometimes $\\ell $ is referred as \\textit{index} of $Q.$) In fact, \\ if we denote by\n\\[\nQ^{\\ast}:=\\left\\{ \\left. \\mathbf{y}\\in\\text{Hom}_{\\mathbb{R}}(\\mathbb{R}%\n^{2},\\mathbb{R})\\ \\right\\vert \\ \\left\\langle \\mathbf{y},\\mathbf{x}%\n\\right\\rangle \\,\\leq\\, 1,\\ \\forall\\, \\mathbf{x}\\in Q\\right\\}\n\\]\nthe \\textit{polar} of the polygon $Q,$ the index $\\ell$ is nothing but\nmin$\\left\\{ \\left. k\\in\\mathbb{N}\\; \\right\\vert \\ \\mathcal{V}(kQ^{\\ast\n})\\subset\\mathbb{Z}^{2}\\right\\} ,$ where $kQ^{\\ast}:=$ $\\left\\{ \\left.\nk\\mathbf{y}\\right\\vert \\mathbf{y}\\in Q^{\\ast}\\right\\} .$ In other words,\n$\\ell$ equals the least common multiple of the (smallest) denominators of the\n(rational) coordinates of the vertices of $Q^{\\ast}.$\n\\end{note}\n\n\\section{Proof of main theorem}\n\n\\noindent{}The proof follows from suitable combination of the two upper bounds\ngiven in Lemmas \\ref{LemmaVQ} and \\ref{LemmaMINDES}. (Henceforth we use freely\nthe notation introduced in \\S \\ref{PRELIM}.)\n\n\\begin{lemma}\n\\label{LemmaVQ}Let $X_{Q}$ be a toric log Del Pezzo surface of index $\\ell\n\\geq1.$ Then%\n\\begin{equation}\n\\sharp(\\mathcal{V}(Q))\\leq4\\, \\text{\\emph{max}}\\left\\{ \\left. l_{H}%\n\\right\\vert \\,H\\in\\mathcal{F}(Q)\\right\\} +2\\leq4\\ell+2. \\label{firstineq}%\n\\end{equation}\nMoreover, $\\sharp(\\mathcal{V}(Q))= \\,4\\, \\text{\\emph{max}}\\left\\{ \\left.\nl_{H}\\right\\vert \\,H\\in\\mathcal{F}(Q)\\right\\} +2$, if and only if $\\ell=1$,\nand $Q$ is the unique hexagon \\emph{(}up to lattice-equivalence\\emph{)} with\none interior lattice point. This means, in particular, that for indices\n$\\ell\\geq2$ we have%\n\\begin{equation}\n\\sharp(\\mathcal{V}(Q))\\leq4\\ell+1. \\label{nicebound}%\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{proof}\nObviously, there exists a facet $F\\in\\mathcal{F}(Q)$ such that $\\sum\n_{\\mathbf{v}\\in\\mathcal{V}(Q)}\\mathbf{v}\\in\\sigma_{F}$ (this is a\n\\emph{special facet}, in the sense of \\cite[Sect.~3]{Oebro}). In addition, since\n$Q$ is two-dimensional, we have for all integers $j$:%\n\\[\n\\sharp\\left\\{ \\left. \\mathbf{v}\\in\\mathcal{V}(Q)\\right\\vert \\,\\left\\langle\n{\\boldsymbol{\\eta}}_{F},\\mathbf{v}\\right\\rangle =j\\right\\} \\leq2.\n\\]\nWriting $\\mathcal{V}(Q)$ as disjoint union $\\mathcal{V}(Q)=\\mathcal{V}_{\\geq\n0}^{\\left( F\\right) }(Q)\\,%\n{\\textstyle\\bigsqcup}\n\\,\\mathcal{V}_{<0}^{\\left( F\\right) }(Q),$ where%\n\\[\n\\mathcal{V}_{\\geq0}^{\\left( F\\right) }(Q):=\\left\\{ \\left. \\mathbf{v}%\n\\in\\mathcal{V}(Q)\\right\\vert \\,\\left\\langle {\\boldsymbol{\\eta}}_{F}%\n,\\mathbf{v}\\right\\rangle \\geq0\\right\\} \\text{ \\ and \\ \\ }\\mathcal{V}%\n_{<0}^{\\left( F\\right) }(Q):=\\left\\{ \\left. \\mathbf{v}\\in\\mathcal{V}%\n(Q)\\right\\vert \\,\\left\\langle {\\boldsymbol{\\eta}}_{F},\\mathbf{v}\\right\\rangle\n<0\\right\\} ,\n\\]\nwe observe that\n\\[\n\\sharp(\\mathcal{V}_{\\geq0}^{\\left( F\\right) }(Q))\\leq2\\left( l_{F}%\n+1\\right) ,\n\\]\nbecause $\\left\\langle {\\boldsymbol{\\eta}}_{F},\\mathbf{v}\\right\\rangle\n\\in\\{0,1,\\ldots,l_{F}\\}$ for all $\\mathbf{v}\\in\\mathcal{V}_{\\geq0}^{\\left(\nF\\right) }(Q)$. On the other hand,%\n\\begin{align*}\n0 & \\leq\\left\\langle {\\boldsymbol{\\eta}}_{F},\\sum\\nolimits_{\\mathbf{v}%\n\\in\\mathcal{V}(Q)}\\mathbf{v}\\right\\rangle =\\sum\\nolimits_{\\mathbf{v}%\n\\in\\mathcal{V}_{\\geq0}^{\\left( F\\right) }(Q)}\\left\\langle {\\boldsymbol{\\eta\n}}_{F},\\mathbf{v}\\right\\rangle +\\sum\\nolimits_{\\mathbf{v}\\in\\mathcal{V}%\n_{<0}^{\\left( F\\right) }(Q)}\\left\\langle {\\boldsymbol{\\eta}}_{F}%\n,\\mathbf{v}\\right\\rangle \\\\\n& =\\sum_{j=0}^{l_{F}}\\ \\sum\\limits_{\\left. \\{\\mathbf{v}\\in\\mathcal{V}%\n_{\\geq0}^{\\left( F\\right) }(Q)\\right\\vert \\,\\left\\langle {\\boldsymbol{\\eta}%\n}_{F},\\mathbf{v}\\right\\rangle =j\\}}\\left\\langle {\\boldsymbol{\\eta}}%\n_{F},\\mathbf{v}\\right\\rangle +\\sum\\limits_{\\mathbf{v}\\in\\mathcal{V}%\n_{<0}^{\\left( F\\right) }(Q)}\\left\\langle {\\boldsymbol{\\eta}}_{F}%\n,\\mathbf{v}\\right\\rangle \\\\\n& \\leq\\sum_{j=0}^{l_{F}}2j+\\sum\\limits_{\\mathbf{v}\\in\\mathcal{V}%\n_{<0}^{\\left( F\\right) }(Q)}\\left\\langle {\\boldsymbol{\\eta}}_{F}%\n,\\mathbf{v}\\right\\rangle .\n\\end{align*}\nThis implies%\n\\[\na:=-\\sum\\nolimits_{\\mathbf{v}\\in\\mathcal{V}_{<0}^{\\left( F\\right) }%\n(Q)}\\left\\langle {\\boldsymbol{\\eta}}_{F},\\mathbf{v}\\right\\rangle \\leq\n2\\binom{l_{F}+1}{2}.\n\\]\nSetting $\\mu:=\\sharp(\\mathcal{V}_{<0}^{\\left( F\\right) }(Q))$ we examine two\ncases: (i) If $\\mu=2\\lambda,$ for a $\\lambda\\in\\mathbb{N},$ then\n\\[\n\\sum_{j=0}^{\\lambda}2j\\leq a\\Longrightarrow2\\binom{\\lambda+1}{2}\\leq\n2\\binom{l_{F}+1}{2}\\Longrightarrow\\lambda\\leq l_{F}\\text{ and }\\mu\\leq2l_{F}.\n\\]\n(ii) If $\\mu=2\\lambda+1,$ for a $\\lambda\\in\\mathbb{Z}_{\\geq0},$ then\n$\\sum_{j=0}^{\\lambda}2j+\\left( \\lambda+1\\right) \\leq a,$ i.e.,%\n\\[\n2\\binom{\\lambda+1}{2}+\\left( \\lambda+1\\right) \\leq2\\binom{l_{F}+1}%\n{2}\\Longrightarrow\\lambda\\leq l_{F}-1\\text{ and }\\mu\\leq2l_{F}-1.\n\\]\nHence,%\n\\[\n\\sharp(\\mathcal{V}(Q))=\\sharp(\\mathcal{V}_{\\geq0}^{\\left( F\\right)\n}(Q))+\\sharp(\\mathcal{V}_{<0}^{\\left( F\\right) }(Q))\\leq2\\left(\nl_{F}+1\\right) +\\mu\n\\]%\n\\begin{equation}\n\\leq2\\left( l_{F}+1\\right) +2l_{F}=4l_{F}+2\\leq4\\,\\text{max}\\left\\{ \\left.\nl_{H}\\right\\vert \\,H\\in\\mathcal{F}(Q)\\right\\} +2, \\label{Approx}%\n\\end{equation}\nwith the latter upper bound $\\leq4\\ell+2$ (by (\\ref{LCM})), giving the\ninequality (\\ref{firstineq}). Finally, we deal with the case of equality:\nSuppose that $\\sharp(\\mathcal{V}(Q))=4\\ell^{\\prime}+2$, where\n\\[\n\\ell^{\\prime}:=\\max\\left\\{ \\left. l_{H}\\right\\vert \\,H\\in\\mathcal{F}%\n(Q)\\right\\} .\n\\]\nFrom (\\ref{Approx}) we see that $\\mu=2l_{F}$, and $\\lambda=l_{F}=\\ell^{\\prime\n}$. Therefore, by the equalities in (i) we have for the integers\n$j=-\\ell^{\\prime},\\ldots,0,\\ldots,\\ell^{\\prime}$:\n\\begin{equation}\n\\sharp\\left\\{ \\left. \\mathbf{v}\\in\\mathcal{V}(Q)\\right\\vert \\,\\left\\langle\n{\\boldsymbol{\\eta}}_{F},\\mathbf{v}\\right\\rangle =j\\right\\} =2.\n\\label{gleichheit}%\n\\end{equation}\nIn particular, $0=\\left\\langle {\\boldsymbol{\\eta}}_{F},\\sum\n\\nolimits_{\\mathbf{v}\\in\\mathcal{V}(Q)}\\mathbf{v}\\right\\rangle $, i.e.,\n$\\sum_{v\\in\\mathcal{V}(Q)}\\mathbf{v}=\\mathbf{0}$. Hence, the previous argument\nholds for \\textit{any} facet. Now let $F^{\\prime}$ be another facet of $Q$\nhaving a common vertex, say $\\mathbf{v,}$ with $F.$ If $\\mathcal{V}%\n(F)=\\{\\mathbf{u},\\mathbf{v}\\}$ and $\\mathcal{V}(F^{\\prime})=\\{\\mathbf{v}%\n,\\mathbf{w}\\},$ then applying (\\ref{gleichheit}) for \\textit{both} $F$ and\n$F^{\\prime}$ we get $\\left\\langle {\\boldsymbol{\\eta}}_{F},\\mathbf{w}%\n\\right\\rangle =\\ell^{\\prime}-1$ and $\\left\\langle {\\boldsymbol{\\eta}%\n}_{F^{\\prime}},\\mathbf{u}\\right\\rangle =\\ell^{\\prime}-1$. This implies\n$\\ell^{\\prime}=1=\\ell$, since otherwise the primitive vertex $\\mathbf{v}$\nequals $(\\ell^{\\prime}\/(\\ell^{\\prime}-1))(\\mathbf{w}+\\mathbf{u}-\\mathbf{v})$,\na contradiction. Consequently, $Q$ has to be the unique hexagon (up to\nlattice-equivalence) with just one interior lattice point (see \\cite[Proposition 2.1]{Nill}).\n\\end{proof}\n\n\\begin{lemma}\n\\label{LemmaMINDES}If $X_{Q}$ is a toric log Del Pezzo surface of index\n$\\ell\\geq2$ and $\\widetilde{X}_{Q}\\overset{\\varphi}{\\longrightarrow}X_{Q}$ its\nminimal desingularization, then%\n\\begin{equation}\n\\rho(\\widetilde{X}_{Q})<2\\, \\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})(\\ell-1)\n-\\frac{1}{\\ell} \\; \\sharp(\\mathcal{V}(Q)) +10. \\label{secineq}%\n\\end{equation}\n\n\\end{lemma}\n\n\\begin{proof}\nBy Noether's formula and (\\ref{DISCREPANCY}) we deduce\n\\[\n\\rho(\\widetilde{X}_{Q})=10-K_{\\widetilde{X}_{Q}}^{2}=10-K_{X_{Q}}^{2}%\n-\\sum_{F\\in I_{Q}}K(E^{(F)})^{2}.\n\\]\nSince $-\\ell K_{X_{Q}}$ is an ample Cartier divisor on $X_{Q},$ we can compute\nby \\cite[Proposition 2.10, p. 79]{Oda} its self-intersection number:\n\\[\n(-\\ell K_{X_{Q}})^{2}=2\\text{\\thinspace area}(\\ell Q^{\\ast})\\Longrightarrow\nK_{X_{Q}}^{2}=\\frac{2}{\\ell^{2}}\\,\\text{area}(\\ell Q^{\\ast})=2\\text{\\thinspace\narea}(Q^{\\ast}).\n\\]\nFor any facet $H$ of $\\ell Q^{\\ast}$ the primitive outer normal vector is given by\nsome vertex of $Q$, i.e., the lattice distance of $H$ from $\\mathbf{0}$ equals\n$\\ell$. This implies\n\\[\n\\text{area}(\\ell Q^{\\ast}) \\geq\\frac{1}{2} \\ell\\;\\sharp(\\mathcal{F}(\\ell\nQ^{\\ast})) = \\frac{1}{2} \\ell\\;\\sharp(\\mathcal{V}(Q)).\n\\]\nHence,\n\\[\n-K_{X_{Q}}^{2} = -\\frac{2}{\\ell^{2}} \\;\\text{area}(\\ell Q^{\\ast}) \\leq\n-\\frac{1}{\\ell} \\;\\sharp(\\mathcal{V}(Q)).\n\\]\nOn the other hand, by Lemma \\ref{KESQUARE} we infer that%\n\\[\n-\\sum_{F\\in I_{Q}}K(E^{(F)})^{2}=\\sum_{F\\in I_{Q}}\\left( \\frac{2-\\left(\np_{F}+\\widehat{p}_{F}\\right) }{q_{F}}+(m_{F}-2)\\right) .\n\\]\nTaking into account that $m_{F}=2$ for all $F\\in\\breve{I}_{Q},$ and that\n$p_{F}+\\widehat{p}_{F}\\geq2$ for all $F\\in I_{Q},$ which is valid as equality\nonly for $p_{F}=\\widehat{p}_{F}=1,$ i.e., whenever $F\\in\\breve{I}_{Q},$ we\nobtain%\n\\[\n-\\sum_{F\\in I_{Q}}K(E^{(F)})^{2}=-\\sum_{F\\in I_{Q}\\mathbb{r}\\breve{I}_{Q}%\n}K(E^{(F)})^{2}<\\sum_{F\\in I_{Q}\\mathbb{r}\\breve{I}_{Q}}(m_{F}-2)\n\\]%\n\\begin{align*}\n& \\leq\\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})\\,\\,\\text{max}\\left\\{ \\left.\nm_{F}-2 \\;\\right\\vert \\; F\\in I_{Q}\\mathbb{r}\\breve{I}_{Q}\\right\\} \\\\\n& \\leq\\,\\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})\\,\\,\\text{max}\\left\\{ \\left.\n2(l_{F}-1) \\;\\right\\vert \\; F\\in I_{Q}\\mathbb{r}\\breve{I}_{Q}\\right\\}\n\\leq2\\,\\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})(\\ell-1),\n\\end{align*}\nwhere the last but one inequality follows from Lemma \\ref{MULTIND}. Thus,\n$\\rho(\\widetilde{X}_{Q})$ is strictly smaller than the sum $10 -\\sharp\n(\\mathcal{V}(Q))\/\\ell+ 2\\,\\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})(\\ell-1).$\n\\end{proof}\n\n\\noindent{}\\textit{Proof of Theorem \\ref{main}}. If $\\ell=1,$ then\n$\\rho(\\widetilde{X}_{Q})\\leq7$ by the known classification of the reflexive\npolygons (see \\cite{KS} or \\cite[Proposition 2.1]{Nill}). If $\\ell\\geq2,$\napplying (\\ref{nicebound}) and (\\ref{secineq}), and the inequality\n$\\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})\\leq\\sharp(\\mathcal{V}(Q)),$ we get\n\\[\n\\rho(\\widetilde{X}_{Q})<2\\,\\sharp(I_{Q}\\mathbb{r}\\breve{I}_{Q})(\\ell\n-1)-\\frac{1}{\\ell}\\;\\sharp(\\mathcal{V}(Q))+10\n\\]%\n\\[\n\\leq\\sharp(\\mathcal{V}(Q))\\left( 2(\\ell-1)-\\frac{1}{\\ell}\\right)\n+10\\leq\\left( 4\\ell+1\\right) \\left( 2(\\ell-1)-\\frac{1}{\\ell}\\right) +10,\n\\]\ni.e., $\\rho(\\widetilde{X}_{Q})<8\\ell^{2}-6\\ell+4-\\frac{1}{\\ell}$, which yields the bound for $\\ell \\geq 2$.\n\\hfill{}$\\square$\n\n\\section{Discussion, improvements and examples}\n\n\\noindent{} First, let us note that from the proof of Theorem \\ref{main} we derive a \\emph{linear} upper bound on\n$\\rho(\\widetilde{X}_{Q})$, if the number of vertices of $Q$ is \\emph{fixed}. It is\ntherefore natural to ask for an example of an infinite family $\\{Q_{i}\\}$ of\nLDP-polygons with increasing number of vertices, for which $\\rho(\\widetilde\n{X}_{Q_{i}})$ exhibits a non-linear growth with respect to the indices of its members.\nTo the best knowledge of the authors, this seems to be an open question.\n\nNow, in some specific cases we can further improve the bound (\\ref{mainbound}).\nIf $Q$ is an LDP-polygon and $F\\in I_{Q},$ then, according to (\\ref{localind}%\n), there is a positive integer $\\beta_{F}$ such that%\n\\[\np_{F}-1=\\beta_{F}\\cdot\\frac{q_{F}}{l_{F}}\\Longrightarrow l_{F}\\,(p_{F}%\n-1)=\\beta_{F}\\,q_{F}.\n\\]\nSince $l_{F}(p_{F}-1) \\nu}\\sum_{\\sigma,\\sigma'}U'_{\\mu\\nu}n_{i\\mu\\sigma} n_{i\\nu\\sigma'}\n\\right.\\nonumber\\\\\n&&\\left.-\\sum_{\\mu\\neq\\nu}J_{\\mu\\nu}\\Vec{S}_{i\\mu}\\cdot\\Vec{S}_{i\\nu}\n+\\sum_{\\mu\\neq\\nu}J'_{\\mu\\nu}\nc_{i\\mu\\uparrow}^\\dagger c_{i\\mu\\downarrow}^\\dagger\nc_{i\\nu\\downarrow}c_{i\\nu\\uparrow}\n\\right).\n\\end{eqnarray}\nWe apply the fluctuation exchange (FLEX) approximation\\cite{Bickers1989,Dahm} \nusing multiorbital \nHubbard Hamiltonian. In FLEX, bubble and ladder type diagrams consisting of \nrenormalized Green's functions are summed up to obtain the susceptibilities, \nwhich are used to calculate the self energy. The renormalized Green's \nfunctions are then determined self-consistently from the Dyson's equation.\nThe obtained Green's function is plugged into the linearized Eliashberg \nequation, whose eigenvalue $\\lambda$ reaches unity at the superconducting \ntransition temperature $T=T_c$. Also, in order to investigate the correlation \nbetween superconductivity and magnetism, we obtain the Stoner factor $a_S$\nof the antiferromagnetism at the wave vector $(\\pi,0)$ in the unfolded \nBrillouin zone, which is defined as the largest eigenvalue of the matrix \n$U\\chi_0({\\bf{k}}=(\\pi,0),i\\omega_n=0)$, where $U$ is the interaction and \n$\\chi_0$ is the irreducible susceptibility matrices, respectively. \nThis value monitors the tendency towards stripe type antiferromagnetism and \nthe strength of the spin fluctuations at zero energy. Since the three \ndimensionality is not strong in Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$, \nwe take a two dimensional model where \nwe neglect the out-of-plane hopping integrals, and take $32 \\times 32$ \n$k$-point meshes and 4096 Matsubara frequencies.\n\nAs for the electron-electron interaction values, \nwe adopt the orbital-dependent interactions\nas obtained from first principles calculation \nin ref.\\onlinecite{Miyake_private} \nfor Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$, but multiply all of them \nby a constant reducing \nfactor $f$. The reason for introducing this factor is as follows.\nAs has been studied in refs.\\onlinecite{Ikeda_prb,Arita,Ikeda_jpsj} \nthe FLEX calculation for models obtained from LDA calculations tends to \noverestimate the effect of the \nself-energy because LDA already partially takes into account the \neffect of the self-energy in the exchange-correlation functional. \nWhen the electron-electron interactions as large as those evaluated \nfrom first principles are adopted in the FLEX calculation, \nthis double counting of the self-energy becomes so large \nthat the band structure largely differs from its original one. \nIn such a case, the spin fluctuations will develop around the wave vector\n$(\\pi,\\pi)$ rather than $(\\pi,0)$, which is in disagreement \nwith the experimental observations. \nIn the present study, we therefore \nintroduce the factor $f$ so as to reduce the electron-electron interactions,\nwhile maintaining the relative magnitude between interactions of different \norbitals.\n\n\\subsection{Bond angle}\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig6}\n\\caption{(a) The Eliashberg equation eigenvalue for \nsuperconductivity ($s\\pm$-wave pairing) (solid) and the Stoner factor \nat $(\\pi,0)$ (dashed) against the bond angle for \ntemperature $T = 0.005$. The interaction \nreduction factor is $f = 0.45$.\\label{fig:6}}\n\\end{figure}\n\nWe show the eigenvalue of the Eliashberg equation $\\lambda$ for \nthe s$\\pm$-wave superconductivity and the Stoner factor at $(\\pi,0)$ for \nthe hypothetical lattice structure of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$ varying \nthe bond angle while fixing the bond length(Fig.\\ref{fig:6}).\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig7}\n\\caption{The arrows indicate the wave vector of the \ndominant pairing interactions for the (a)$X^2-Y^2$ and \n(b) $XZ\/YZ$ portions of the Fermi surface in the case where \nthe inner hole Fermi surface ($\\alpha_1$) is barely present. \nIn this case, $\\alpha_1$ is a mixture of $X^2-Y^2$ and $XZ\/YZ$.\\label{fig:7}}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig8}\n\\caption{The gap function obtained by FLEX for the hypothetical \nlattice structures of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$. The bond angle \n$\\alpha$ is set to 110$^\\circ$ or 111$^\\circ$, while the bond length \nis fixed at the original value.\\label{fig:8}}\n\\end{figure}\n\nAs we decrease the bond angle from 115 to 110$^\\circ$, \neigenvalue of the Eliashberg equation $\\lambda$ increases, reflecting \nthe appearance of the $\\gamma$ Fermi surface around $(\\pi,\\pi)$.\nSuperconductivity is locally optimized around 110$^\\circ$, \nbut $\\lambda$ immediately goes down for larger bond angle. \nThis is in contrast to the case of LaFeAsO, \nwhere $\\lambda$ is broadly maximized around the regular tetrahedron \nbond angle.\nThis difference can be understood from the comparison between \nFig.\\ref{fig:2}(c) and Fig.\\ref{fig:4}(b).\nNamely, in the case of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$ with hypothetical \nbond angle, the Fermi surface configuration (b) \nwith the optimal Fermi surface configuration \nis missing, i.e., in the three Fermi surface regime, \n$\\alpha_1$ Fermi surface around $(0,0)$ is \nconstructed from a mixture of $X^2-Y^2$ and $XZ\/YZ$ orbital \ncharacters. In this configuration, The pair scattering takes place not only \nat $\\sim (\\pi,0)$ but also at $\\sim (\\pi,\\pi)$ \ndue to the same orbital character between \n$\\alpha_2$ and $\\gamma$ Fermi surfaces.\nSince these Fermi surfaces interact with repulsive pairing interactions, \na frustration arises in the sign of the superconducting gap \nas shown schematically in Fig.\\ref{fig:7}. \nIn addition to this, there can also be some $XZ\/YZ$ component remaining in the \n$\\alpha_1$ Fermi surface, and this portion tends to change the sign from\nthe $\\beta$ Fermi surfaces, making it another possible factor\nfor the frustration.\nThe effect of the frustration appears in the form of the superconducting gap.\nIn Fig.\\ref{fig:8}, we show the gap function for \nthe hypothetical lattice structure of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$ \nat the bond angles 110$^\\circ$ and 111$^\\circ$. \nThe sign of the gap function on $\\alpha_1$ is positive at 111$^\\circ$, \nbut is very small (barely positive) at 110$^\\circ$\\cite{commentSUST},\nreflecting the effect of the frustration.\nThe bond angle of 110$^\\circ$ is actually very close to that of \nCa$_4$Al$_2$O$_6$Fe$_2$P$_2$, so the appearance of a very small gap at \nthis bond angle may be related to the nodal gap structure \nsuggested experimentally for Ca$_4$Al$_2$O$_6$Fe$_2$P$_2$\\cite{Kinouchi2011}.\nAs the bond angle is further reduced, the $\\alpha_1$ Fermi surface \ndisappears but the effect of the frustration remains strong as far as the \ntop of the $\\alpha_1$ hole band does not sink far below the \nFermi level. In fact, the frustration effect can be very strong \nright after the Fermi surface disappears because the top of \nthis $\\alpha_1$ band (the closest point to the Fermi level) \nhas pure $X^2-Y^2$ orbital character.\nTherefore, $\\lambda$ is suppressed around the bond angle of \n105$^\\circ\\sim$ 108$^\\circ$. Meanwhile, the Fermi surface nesting itself\nbecomes very good in this regime because there are now two hole and two \nelectron Fermi surfaces with no doped carriers, so that the \naverage area of the hole and the electron Fermi surfaces becomes the same.\nIn particular, around the bond angle of 105$^\\circ$, the nesting \nbecomes nearly perfect, as shown in Fig.\\ref{fig:9}. Therefore, \nthe Stoner factor at $(\\pi,0)$ takes a local maximum around this bond angle.\nAs the bond angle is reduced even further, the $X^2-Y^2$ band \nsinks far below the Fermi level and the frustration effect \nbecomes small, so that $\\lambda$ increases once again to a value \ncomparable to that around the local maximum around the regular \ntetrahedron bond angle.\nAt the same time, the Fermi surface nesting becomes somewhat degraded, \nand the Stoner factor is reduced. \nFor smaller bond angle$<96^\\circ$ (which may not be realistic), \nthe Fermi surface becomes too large, and the \nsuperconductivity is degraded. The bottom line here is that \nsuperconductivity is favored at around two bond angles 102$^\\circ$ and \n110$^\\circ$, and antiferromagnetism is favored in the regime in between \nthese angles.\nThis is at least qualitatively consistent with the experimental \nobservations for Ca$_4$Al$_2$O$_6$Fe$_2$As$_{1-x}$P$_x$.\n\nThe important point here is that \nsuperconductivity is suppressed in the intermediate bond angle regime \ndue to the frustration effect. Apart from this, \nantiferromagnetism is favored around this bond angle \nregime due to a nearly perfect nesting of the Fermi surface.\n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig9}\n\\caption{The Fermi surface of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$ \nfor the hypothetical lattice structures with $\\alpha=105^\\circ$ \nand $102^\\circ$ (solid), superposed with the Fermi surface \nshifted by $(\\pi,0)$ (dashed).\\label{fig:9}}\n\\end{figure}\n\n\n\\subsection{Pnictogen height}\n\n\\begin{figure}\n\\includegraphics[width=6.5cm]{fig10}\n\\caption{The pnictogen height dependence of (a) the Eliashberg equation \neigenvalue and (b) the Stoner factor at $(\\pi,0)$ for the hypothetical \nlattice structure of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$. \nSeveral values of the reducing factor are taken for comparison.\\label{fig:10}\n(c) A schematic figure of the $x$ dependence of $\\lambda$ for \nsuperconductivity and $a_S$ for antiferromagnetism.}\n\\end{figure}\n\nWe have studied in the previous section the bond angle dependence of \nsuperconductivity and the spin fluctuations, and mentioned the \npossible relation between the calculation results and \nthe experimental observations for Ca$_4$Al$_2$O$_6$Fe$_2$As$_{1-x}$P$_x$.\nAs mentioned previously, \nthe actual lattice structure variation upon replacing As by P is \nmore close to the variance of the \npnictogen height $h_{\\rm Pn}$ rather than just the bond angle. \nThe increase of the bond length results in an increase in the density of \nstates, generally resulting in an enhancement of \nboth superconductivity and spin fluctuations\\cite{Usui2011}.\nIn Fig.\\ref{fig:10}, \nwe show the eigenvalue of the Eliashberg equation and \nthe Stoner factor at $(\\pi,0)$ for the hypothetical lattice \nstructure of Ca$_4$Al$_2$O$_6$Fe$_2$As$_2$ \nvarying solely the pnictogen height $h_{\\rm Pn}$. \nAround $h_{\\rm Pn}=1.3\\sim 1.35{\\rm \\AA}$, corresponding to the \nP content close to unity, \nthe height dependence of $\\lambda$ is weak (or $\\lambda$ is even suppressed \nwith the increase of $h_{\\rm Pn}$ for large $f$),\nwhile the Stoner factor rapidly increases with $h_{\\rm Pn}$. \nThis height regime corresponds to the bond angle regime of \n$110^\\circ\\sim 108^\\circ$, where \nsuperconductivity is suppressed due to the momentum space \nfrustration, and at the same time antiferromagnetism \nis favored due to the nearly perfect nesting (Fig.\\ref{fig:7}). \nHere in Fig.\\ref{fig:10}(a), \nthe enhancement of superconductivity \nby the increase of the density of states is canceled out due to the \nfrustration effect, so that the $h_{\\rm Pn}$ dependence of $\\lambda$ is weak.\nOn the other hand, the Stoner factor quickly grows due to the \ncooperation of the \ngood nesting and the increased density of states.\nAs the pnictogen height increases further beyond $1.35{\\rm \\AA}$, \n$\\lambda$ starts to \nincrease rapidly due to the reduction of the frustration \nand the increase of the \ndensity of states, while the Stoner factor tends to saturate because \nthe nearly perfect nesting is degraded.\nThis overall tendency is summarized in a schematic figure in \nFig.\\ref{fig:10}(c)\n\n\n\\section{Pressure experiment}\nOur theoretical study so far has \nshown that in the region where antiferromagnetism \nappears in the phase diagram, not only antiferromagnetism is \nenhanced due to the good Fermi surface nesting, but also superconductivity is \nsuppressed due to the momentum space frustration, and these two are \nindependent matters. Since superconductivity is suppressed \nregardless of whether antiferromagnetism is present or not,\nsuperconductivity may not take place \neven when antiferromagnetism is suppressed by applying pressure, as \nis often done in other iron based superconductors. \n\nTo actually see this experimentally, \nwe have applied hydrostatic pressure to \nCa$_4$Al$_2$O$_6$Fe$_2$(As$_{1-x}$P$_x$)$_2$. \nThe results are shown in Fig.\\ref{fig:11}.\nFor the end compounds $x=0$ and $x=1$, $T_c$ \nmonotonically decreases with increasing pressure.\nThis is most likely due to the decrease in the density of states.\nFor $x=0.75$, where antiferromagnetism takes place at ambient pressure, \nsuperconductivity is not found up to 12GPa, although the \nantiferromagnetic transition is smeared out at high pressures. \nThis is in contrast with cases where antiferromagnetism takes place at \nambient pressure, but gives way to superconductivity under \npressure.\nThe present experimental result supports the scenario that \nsuperconductivity in the intermediate $x$ regime is suppressed \nby momentum space frustration, apart from the presence of the \nantiferromagnetism itself.\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig11}\n\\caption{(a) The pressure dependence of the superconducting \ntransition temperature for various materials. The resistivity \nagainst pressure for Ca$_4$Al$_2$O$_6$Fe$_2$(As$_{1-x}$P$_x$)$_2$ for \n(b) $x=0$, (c)$x=0.75$ and (d)$x=1$. \\label{fig:11}}\n\\end{figure}\n\n\n\\section{Conclusion}\nIn the present paper, we studied the origin of the peculiar phase \ndiagram obtained for Ca$_4$Al$_2$O$_6$Fe$_2$As$_{1-x}$P$_x$ \nusing a five orbital model constructed from first principles \nband calculation.\nWhile the inner hole Fermi surface is absent at $x=0$\\cite{Miyake2010}, \nit is present at $x=1$, but the orbital character has strong\n $X^2-Y^2$ character rather than $XZ\/YZ$ as in LaFeAsO. \nThis gives rise to momentum space frustration of the \npairing interaction mediated by spin fluctuations, and degrades \nsuperconductivity. We propose this to be one of the reasons why \n$T_c$ is not so high in Ca$_4$Al$_2$O$_6$Fe$_2$P despite of the \nmaximized multiplicity of the hole Fermi surface.\nThe frustration effect remains strong \neven after the inner Fermi surface has disappeared for $x<1$ \nbecause the top of the band with $X^2-Y^2$ orbital character \nremains near the Fermi level. At the same time, the \ndisappearance of the most inner hole Fermi surface gives \nvery good nesting of the electron and hole Fermi surfaces due to the \nequal number of sheets, favoring antiferromagnetism in the \nintermediate regime of $x$. Finally for $x\\sim 1$, the top of the \nband sinks far below the Fermi level, and the frustration effect \nis reduced, so that superconductivity is favored once again.\nAlthough we cannot directly determine which one of the \nsuperconductivity and antiferromagnetism wins, the tendency \nobserved in the calculation is at least consistent with the \nexperimental observation, where nodeless and nodal superconducting \nphases are separated by an antiferromagnetic phase.\nFinally, we have performed hydrostatic pressure experiment, \nwhich further supports our scenario that superconductivity \nis suppressed by momentum space frustration in the intermediate \n$x$ regime.\n\n\\section{ACKNOWLEDGMENTS}\n\nWe are grateful to H. Mukuda, H. Kinouchi, and Y.Kitaoka \nfor fruitful discussions.\nThe numerical calculations were performed at the Supercomputer Center, \nISSP, University of Tokyo. This study has been supported by \nGrants-in-Aid for Scientific Research from JSPS. \nK.S acknowledges support from JSPS.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfojl b/data_all_eng_slimpj/shuffled/split2/finalzzfojl new file mode 100644 index 0000000000000000000000000000000000000000..3df61ef44aa2806be3404f2499ace8eae3a5f988 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzfojl @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThe standard paradigm in modeling is to postulate that measured quantities contain a contribution of ``accidental deviation'' \\cite{Spearman} from the otherwise ``uniformities'' that characterize an underlying law.\nTherefore, a key issue when identifying dependencies between variables is how to account for the contribution of noise in the data. Various assumptions on the structure of noise and of the possible dependencies lead to a number of corresponding methodologies.\n\nThe purpose of the present paper is to consider from a modern\ncomputational point of view, the important situation where the noise\ncomponents are assumed independent, and the consequences of this\nassumption --the data is typically abstracted into a corresponding\n(estimated) covariance statistic. This independence assumption\nunderlies the errors-in-variables model \\cite{Durbin,KlepperLeamer}\nand factor analysis\n\\cite{AndersonRubin,Ledermann,Harman1966,Joreskog1969,Shapiro}, and\nhas a century-old history \\cite{Frisch2,Reiersol,Koopmans}; see also\n\\cite{Kalman1982,Kalman1985,Los,Woodgate1,Guidorzi95,Soderstrom2007errors,Anderson2008,Forni2000}.\nAccordingly, given the large classical literature on this problem,\nthis paper will also have a tutorial flavor.\n\n\nThe precise formulation has its roots in the work of Ragnar Frisch\nin the 1930's. The central assumption is that the noise components\nare independent of the underlying variables and are also mutually\nindependent \\cite{Kalman1982,Kalman1985}. In addition, since several\nalternative linear relations are typically consistent with the data,\na maximal set of simultaneous dependencies is sought as a means to\nlimit uncertainty and to provide canonical models\n\\cite{Kalman1982,Kalman1985}. This particular dictum gives rise to a\n(non-convex) rank-minimization problem. Thus, it is somewhat\nsurprising that the special case where the maximal number of\npossible simultaneous linear relations is equal to $1$ can be\nexplicitly characterized --this was accomplished over half a\ncentury ago by Reiers{\\o}l \\cite{Reiersol}; see also\n\\cite{Kalman1982,KlepperLeamer}. To date no other case is known that\nadmits a precise closed-form solution.\n\nIn recent years, emphasis has been shifting from hard, non-convex\noptimization to convex regularizations, which in addition scale\nnicely with the size of the problem. Following this trend we revisit\nthe Frisch problem from several alternative angles. We first present\nan overview of the literature, and present several new insights and\nproofs. In the process, we also give an extension of Reiers{\\o}l's\nresult to complex matrices. Our main interest is in exploring\nrecently studied convex optimization problems that approximate rank\nminimization by use of suitable surrogates. In particular, we study\niterative schemes for treating the general Frisch problem and focus\non certificates that guarantee optimality. In parallel, we consider\na viewpoint that serves as an alternative to the Frisch problem\nwhere now, instead of a maximal number of simultaneous linear\nrelations, we seek a uniformly optimal estimator for the unobserved\ndata under the independence assumption of the Frisch scheme. The\noptimal estimator is obtained as a solution to a min-max\noptimization problem. Rank-regularized and min-max alternatives are\ndiscussed and an example is given to highlight the potential and limitations of the techniques.\n\nThe remainder of this paper is organized as follows. We first\nintroduce the errors-in-variables problem in\nSection~\\ref{sec:datastrcuture}. In Section~\\ref{sec:Frisch}, we\nrevisit the Frisch problem, and a related problem due to Shapiro,\nand provide a geometric interpretation of Reiers{\\o}l's result along\nwith a generalization to complex-valued covariances. In\nSection~\\ref{sec:MinTrace}, we present an iterative\ntrace-minimization scheme for solving the Frisch problem and provide\ncomputable lower-bounds for the minimum-rank. In\nSection~\\ref{correspondence}, we bring up the question of estimation\nin the context of the Frisch scheme and motivate a suitable a\nrank-regularized min-max optimization problem in\nSection~\\ref{sec:regularized}. Some concluding remarks are provided\nin Section~\\ref{sec:conclusion}.\n\n\n\\section{Notation}\n\n\n$\\;$\\\\[.1in]\n\\noindent\n\\begin{tabular}{ll}\n ${\\mathcal R}(\\cdot)$, ${\\mathcal N}(\\cdot)$ & range space, null space\\\\\n $\\Pi_{\\mathcal X}$ & orthogonal projection onto ${\\mathcal X}$\\\\\n $>0\\;\\; (\\geq 0)$ & positive definite (resp., positive semi-definite) \\\\\n ${\\mathbf S}_n$& $=\\;\\;\\left\\{M \\mid M\\in {\\mathbb R}^{n\\times n},\\; M=M' \\right\\}$\\\\\n ${\\mathbf S}_{n,+}$& $=\\;\\;\\left\\{M \\mid M\\in {\\mathbf S}_n,\\; M\\geq0 \\right\\}$\\\\\n ${\\mathbf H}_n$& $=\\;\\;\\left\\{M \\mid M\\in {\\mathbb C}^{n\\times n},\\; M=M^* \\right\\}$\\\\\n ${\\mathbf H}_{n,+}$& $=\\;\\;\\left\\{M \\mid M\\in {\\mathbf H}_n,\\; M\\geq0 \\right\\}$\\\\\n $[\\cdot ]_{k\\ell},\\;\\; ([\\cdot ]_{k})$ & $(k, \\ell)$-th entry (resp., $k$-th entry)\\\\\n $|M|$& determinant of $M\\in {\\mathbb R}^{n\\times n}$\\\\\n $n_+(\\cdot)$& number of positive eigenvalues\\\\\n ${\\operatorname{diag}}: {\\mathbb R}^{n\\times n} \\to {\\mathbb R}^n: M\\mapsto d$ & where $[d]_i=[M]_{ii}$ for $i=1, \\ldots n$\\\\\n ${\\operatorname{diag}}^*: {\\mathbb R}^{n} \\rightarrow {\\mathbb R}^{n\\times n}: d\\mapsto D$& where $D$ is diagonal and $[D]_{ii}=[d]_{i}$ for $i=1,\\ldots n$\\\\\n $M\\succ_{\\hspace*{-1pt}_e} 0\\;(\\succeq_{\\hspace*{-1pt}_e} 0,\\; \\prec_{\\hspace*{-1pt}_e} 0,\\;\\preceq_{\\hspace*{-1pt}_e} 0)$& the off-diagonal entries are $>0$ (resp.\\ $\\geq 0$, $<0$, $\\leq 0$),\\\\&or \n can be made so by changing the signs of selected\\\\&rows and corresponding columns\\\\\n\\end{tabular}\n\n\\section{Data and basic assumptions}\\label{sec:datastrcuture}\n\nConsider a Gaussian vector ${\\mathbf x}$ taking values in ${\\mathbb R}^{n\\times 1}$ having zero mean and covariance $\\Sigma$. We\nassume that it represents an additive mixture of a Gaussian ``noise-free'' vector ${\\hat{\\mathbf x} }$\nand a ``noise component'' ${\\tilde{\\mathbf x}}$, thus\n\\begin{equation}\\label{eq:xa}\n{\\mathbf x}={\\hat{\\mathbf x} }+{\\tilde{\\mathbf x}}.\n\\end{equation}\nThe entries of ${\\tilde{\\mathbf x}}$ are assumed independent of one another\nand independent of the entries of ${\\hat{\\mathbf x} }$ with both vectors having zero mean\nand covariances $\\hat\\Sigma$ and $\\tilde\\Sigma$, respectively.\nThus,\n\\begin{subequations}\\label{eq:firstsetofconstraints}\n\\begin{eqnarray}\n&&{\\mathcal E}({\\tilde{\\mathbf x}} {\\tilde{\\mathbf x}}') =: \\tilde\\Sigma \\mbox{ is diagonal} \\label{eq:xc}\\\\\n&&{\\mathcal E}({\\hat{\\mathbf x} } {\\tilde{\\mathbf x}}')=0. \\label{eq:xb}\n\\end{eqnarray}\nThroughout ${\\mathcal E}(\\cdot)$ denotes the expectation operation and $0$\ndenotes the zero vector\/matrix of appropriate size. The noise-free\nentries of ${\\hat{\\mathbf x} }$ are assumed to satisfy a set of $q$ simultaneous\nlinear relations. Hence, $M'{\\hat{\\mathbf x} }=0$, with $M\\in {\\mathbb R}^{n\\times q}$ and\n$n>{\\operatorname{rank}}(M)=q>0$. The problem is mainly to infer these relations.\nEquivalently, ${\\mathcal E}({\\hat{\\mathbf x} } {\\hat{\\mathbf x} }') =: \\hat\\Sigma$ has\n\\begin{eqnarray}\n&&{\\operatorname{rank}}(\\hat\\Sigma)= n-q \\label{eq:xd}\n\\end{eqnarray}\n\\end{subequations}\nand $\\hat\\Sigma M=0$. Statistics are typically estimated from\nobservation records. To this end, consider a sequence\n\\[\nx_t\\in{\\mathbb R}^{n\\times 1},\\; t=1,\\ldots,T\n\\]\nof independent measurements (realizations) of ${\\mathbf x}$\nand, likewise, let $\\hat x_t$ and $\\tilde x_t$ represent the corresponding values of the noise-free\nvariable and noise components. Denote by\n\\[\nX=\\left[\\begin{matrix} x_1\\;x_2\\; \\ldots\\; x_T\\end{matrix}\\right]\\in {\\mathbb R}^{n\\times T}\n\\]\nthe matrix of observations of ${\\mathbf x}$ and similarly denote by $\\hat X$\nand $\\tilde X$ the corresponding matrices of the noise-free and\nnoise entries, respectively. Data for identifying relations among\nthe noise-free variables are typically limited to the observation\nmatrix $X$ and, neglecting a scaling factor of $1\/T$, the data is\ntypically abstracted in the form of a sample covariance $XX^\\prime$.\nFor the most part we will assume that sample covariances are\naccurate approximations of true covariances, and hence the modeling\nassumptions amount to\n\\begin{subequations}\n\\begin{eqnarray}\n&& \\tilde X \\tilde X ^\\prime \\simeq \\mbox{ diagonal}\\label{eq:diagonal}\\\\\n&& \\hat X \\tilde X ^\\prime\\simeq 0 \\label{eq:orthogonality}\\\\\n&&{\\operatorname{rank}}(\\hat X) =n-q \\label{eq:rank}\n\\end{eqnarray}\n\\end{subequations}\nsince $M^\\prime \\hat X=0$.\n\nThe number of possible linear relations among the noise free\nvariables and the corresponding coefficient matrix need to be\ndetermined from either $X$ or $\\Sigma$. This motivates the Frisch\nand Shapiro problems discussed in Section~\\ref{sec:Frisch}. An\nalternative set of problems can be motivated by the need to\ndetermine $\\hat X$ from $X$ via suitable decomposition\n\\begin{equation}\\label{eq:decompose}\nX=\\hat X+\\tilde X\n\\end{equation}\nin a way that is consistent with the existence of a set of $q$\nlinear relations. We will return to this in\nSection~\\ref{sec:min-max}.\n\n\n\\section{The problems of Frisch and Shapiro}\\label{sec:Frisch}\n\nWe begin with the Frisch problem concerning the decomposition of a\ncovariance matrix $\\Sigma$ that is consistent with the assumptions\nin Section~\\ref{sec:datastrcuture}. The fact that, in practice,\n$\\Sigma$ is an empirical sample covariance motivates relaxing\n(\\ref{eq:xc}-\\ref{eq:xd}) in various ways. In particular, relaxation\nof the constraint $\\tilde \\Sigma\\geq 0$ leads to the Shapiro\nproblem.\n\n\n\\begin{problem}[\\em The Frisch problem]\\label{problem1} Given $\\Sigma\\in{\\mathbf S}_{n,+}$, determine\n\\begin{eqnarray}\\nonumber\n{\\operatorname{mr}}_+(\\Sigma)&:=&\\min\\{{\\operatorname{rank}}(\\hat\\Sigma) \\mid \\Sigma=\\tilde \\Sigma+\\hat \\Sigma,\\\\&&\\tilde\\Sigma, \\hat\\Sigma\\geq 0,\\;\\tilde\\Sigma \\mbox{ is diagonal}\\}.\\label{eq:mc}\n\\end{eqnarray}\n\\end{problem}\n\n\\begin{problem}[\\em The Shapiro problem]\\label{problemShapiro} Given $\\Sigma\\in{\\mathbf S}_{n,+}$, determine\n\\begin{eqnarray}\\nonumber\n{\\operatorname{mr}}(\\Sigma)&:=&\\min\\{{\\operatorname{rank}}(\\hat\\Sigma) \\mid \\Sigma=\\tilde \\Sigma+\\hat \\Sigma,\\\\&& \\hat\\Sigma\\geq 0,\\;\\tilde\\Sigma \\mbox{ is diagonal}\\}.\\label{eq:mc2}\n\\end{eqnarray}\n\\end{problem}\n\nThe Frisch problem was studied by several researchers, see e.g.,\n\\cite{Kalman1985,Los,Woodgate1,woodgate2} and the references therein. On the other hand, Shapiro \\cite{Shapiro} introduced the above relaxed\nversion, removing the requirement that $\\tilde \\Sigma\\geq 0$, in an\nattempt to gain understanding of the algebraic constraints imposed\nby the off-diagonal elements of $\\Sigma$ on the decomposition. We\nrefer to ${\\operatorname{mr}}_+(\\cdot)$ as the {\\em Frisch minimum rank} and\n${\\operatorname{mr}}(\\cdot)$ as the {\\em Shapiro minimum rank}. The former is lower\nsemicontinuous whereas the latter is not, as stated next. This\ndifference is crucial if one wants to apply this type of methodology\nto real data, namely some sort of continuity is necessary.\n\n\\begin{prop}\\label{lemma:lowersc}\n${\\operatorname{mr}}_+(\\cdot)$ is lower semicontinuous whereas ${\\operatorname{mr}}(\\cdot)$ is not.\n\\end{prop}\n\n\\begin{proof}\nAssume that for a given $\\Sigma>0$ there exists a sequence $\\Sigma_1,\\,\\Sigma_2,\\,\\ldots$ of positive definite matrices such that\n$\\Sigma_i\\rightarrow \\Sigma$\nwhile\n\\[\n{\\operatorname{mr}}_+(\\Sigma_i)<{\\operatorname{mr}}_+(\\Sigma)=r,\\; \\mbox{ for all }i=1,\\,2,\\,\\dots.\n\\]\nDecompose $\\Sigma_i=\\hat\\Sigma_i+D_i$ with ${\\operatorname{rank}}(\\hat\\Sigma_i)0$.\nClearly ${\\operatorname{mr}}(\\Sigma)=2$. Also $\\lim_{\\epsilon\\to 0}\\Sigma_\\epsilon=\\Sigma$. Yet $\\Sigma_\\epsilon=\n\\hat\\Sigma_\\epsilon+D_\\epsilon$ while $\\Sigma_\\epsilon$\nhas rank $1$ and $D_\\epsilon$ is diagonal ($\\not\\geq 0$). Hence ${\\operatorname{mr}}(\\Sigma_\\epsilon)=1$.\n\\end{proof}\n\nAssuming that the off-diagonal entries of $\\Sigma>0$ of size\n$n\\times n$ are known with absolute certainty, any ``minimum rank''\n(${\\operatorname{mr}}_+(\\cdot)$ and ${\\operatorname{mr}}(\\cdot)$) is bounded below by the so-called\nLederman bound, i.e.,\n\\begin{align}\\label{ledermann}\n\\frac{2n+1-\\sqrt{8n+1}}{2}\\leq {\\operatorname{mr}}(\\Sigma)\\leq {\\operatorname{mr}}_+(\\Sigma),\n\\end{align}\nwhich holds on a generic set of positive definite matrices $\\Sigma$,\nthat is, on a (Zariski open) subset of positive definite matrices.\nEquivalently, the set of matrices $\\Sigma$ for which ${\\operatorname{mr}}(\\Sigma)$\nis lower than the Lederman bound is non-generic --their entries\nsatisfy algebraic equations which fail under small perturbation. To\nsee this, consider any factorization\n\\[\\Sigma =FF^\\prime,\n\\]\nwith $F\\in{\\mathbb R}^{n\\times r}$. There are $(n-r)r + \\frac{r(r+1)}{2}$ independent entries in $F$ (when accounting for the action of a unitary transformation of $F$ on the right), whereas the value of the off-diagonal entries of $\\Sigma$ impose $\\frac{n(n-1)}{2}$ constraints. Thus, the number of independent entries in $F$ exceeds the number of constraints when $(n-r)^2\\geq n+r$ which then leads to the inequality $\\frac{2n+1-\\sqrt{8n+1}}{2}\\leq r$. The bound was first noted in \\cite{Ledermann} while the independence of the constraints has been detailed in \\cite{Bekker1997}.\nIn general, the computation of the exact value for ${\\operatorname{mr}}_+(\\Sigma)$ and ${\\operatorname{mr}}(\\Sigma)$ is a non-trivial matter.\nThus, it is rather surprising that an exact analytic result is available for both, in the special case when $r=n-1$.\nWe review this next in the form of two theorems.\n\n\\begin{thm}[\\em Reiers\\o l's theorem \\cite{Reiersol}] \\label{thm:Reiersol}\nLet $\\Sigma\\in {\\mathbf S}_{n,+}$ and $\\Sigma>0$, then\n\\[\n{\\operatorname{mr}}_+(\\Sigma)=n-1 \\Leftrightarrow \\Sigma^{-1} \\succ_{\\hspace*{-1pt}_e} 0.\n\\]\n\\end{thm}\n\n\n\\begin{thm}[\\em Shapiro's theorem \\cite{Shapiro1982b}]\\label{thm:Shapiro}\nLet $\\Sigma\\in{\\mathbf S}_{n,+}$ and irreducible,\n\\[\n{\\operatorname{mr}}(\\Sigma)=n-1\\Leftrightarrow \\Sigma\\preceq_{\\hspace*{-1pt}_e} 0.\n\\]\n\\end{thm}\n\nThe characterization of covariance matrices $\\Sigma$ for which\n${\\operatorname{mr}}_+(\\Sigma)=n-1$ was first recognized by T.~C.~Koopmans in 1937\n\\cite{Koopmans} and proven by Reiers\\o l \\cite{Reiersol} who used\nthe Perron-Frobenius theory to improve on Koopmans' analysis. Later\non, R.~E.~Kalman streamlined and completed the steps in\n\\cite{Kalman1982} relying again on the Perron-Frobenius theorem (see\nalso Klepper and Leamer \\cite{KlepperLeamer} for a detailed\nanalysis). Our treatment below takes a slightly different angle and\nprovides some geometric insight by pointing as a key reason that the\nmaximal number of vectors at an obtuse angle from one another can\nexceed the dimension of the ambient space by at most one\n(Corollary~\\ref{cor:numberofobtuseangles}). We provide new proofs\nwhere we also utilize a dual formulation with an analogous\ndecomposition of the inverse covariance.\n\n\\subsection{A geometric insight}\n\nWe begin with two basic lemmas for irreducible matrices in $M\\in{\\mathbf S}_{n,+}$. Recall that a matrix is reducible if by permutation of rows and columns can be brought into a block diagonal form, otherwise it is irreducible.\n\\begin{lemma}\\label{lemma:previous} Let $M>0$ and irreducible. Then,\n\\begin{eqnarray}\\label{eq:first}\nM\\preceq_{\\hspace*{-1pt}_e} 0 &\\Rightarrow & M^{-1}\\succ_{\\hspace*{-1pt}_e} 0.\n\\end{eqnarray}\n\\end{lemma}\n\\begin{lemma}\\label{lemma:next} Let $M\\geq 0$ and irreducible. Then,\n\\begin{eqnarray}\\label{eq:nullitybound}\nM\\preceq_{\\hspace*{-1pt}_e} 0\n&\\Rightarrow & {\\rm nullity}(M)\\leq 1.\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nIt is easy to verify that for matrices of size $2\\times 2$,\n(\\ref{eq:first}) holds true. Assume that the statement also holds true for matrices of size up to $k\\times k$, for a certain value of $k$,\nand consider a matrix $M$ of size $(k+1)\\times(k+1)$ with $M>0$ and $M\\preceq_{\\hspace*{-1pt}_e} 0$. Partition\n\\[M=\\left[\\begin{matrix}A &b\\\\b' &c\\end{matrix}\\right]\n\\]\nso that $c$ is a scalar and, hence, $A$ is of size $k\\times k$.\nPartitioning conformably,\n\\[M^{-1}=\\left[\\begin{matrix}F &g\\\\g' &h\\end{matrix}\\right]\n\\]\nwhere\n\\[F=(A-bc^{-1}b')^{-1}, ~g=-A^{-1}bh, \\mbox{ and }h=(c-b'A^{-1}b)^{-1}>0.\n\\]\n\nFor the case where $A$ is irreducible, because $A$ has size $k\\times\nk$ and $A\\preceq_{\\hspace*{-1pt}_e} 0$, invoking our hypothesis we conclude that\n$A^{-1}\\succ_{\\hspace*{-1pt}_e} 0$. Now, since $b$ has only non-positive entries and\n$b\\neq0$, $g=-A^{-1}bh$ has positive entries. Since\n$-bc^{-1}b'\\preceq_{\\hspace*{-1pt}_e} 0$ and $A\\preceq_{\\hspace*{-1pt}_e} 0$, then $A-bc^{-1}b'\\preceq_{\\hspace*{-1pt}_e}\n0$ is also irreducible. Thus $F=(A-bc^{-1}b')^{-1}$ has positive\nentries by hypothesis.\n\nFor the case where $A$ is reducible, permutation of columns and rows\nbrings $A$ into a block-diagonal form with irreducible blocks. Thus,\n$A^{-1}$ is also block diagonal matrix with each block entry-wise\npositive. Because $M$ is irreducible, $b$ must have at least one\nnon-zero entry corresponding to the rows of each diagonal blocks of\n$A$. Then $A-bc^{-1}b'$ is irreducible and $\\preceq_{\\hspace*{-1pt}_e} 0$. Also\n$A^{-1}b$ has all of its entries negative. Therefore\n$F=(A-bc^{-1}b')^{-1}$ and $g=-A^{-1}bh$ have positive entries.\nTherefore $M^{-1}\\succ_{\\hspace*{-1pt}_e} 0$.\n\\end{proof}\n\n\\begin{proof}\nRearrange rows and columns and partition\n\\[M=\\left[\\begin{matrix}A &B\\\\B' &C\\end{matrix}\\right]\n\\]\nso that $A$ is nonsingular and of maximal size, equal to the rank of $M$.\nThen\n\\begin{equation}\\label{eq:equality}\nC=B'A^{-1}B.\n\\end{equation}\n\nWe first show that $B'A^{-1}B\\succeq_{\\hspace*{-1pt}_e} 0$. Assume that $A$ is irreducible.\nThen $A^{-1}\\succ_{\\hspace*{-1pt}_e} 0$. At the same time $B$ has negative entries and not all zero (since $M$ is irreducible). In this case, $B'A^{-1}B\\succ_{\\hspace*{-1pt}_e} 0$.\nIf on the other hand $A$ is reducible, Lemma \\ref{lemma:previous} applied to the (irreducible) blocks of $A$ implies that $A^{-1}\\succeq_{\\hspace*{-1pt}_e} 0$.\nTherefore, in this case, $B'A^{-1}B\\succeq_{\\hspace*{-1pt}_e} 0$.\n\nReturning to \\eqref{eq:equality} and in view of the fact that $C\\preceq_{\\hspace*{-1pt}_e} 0$ while $B'A^{-1}B\\succeq_{\\hspace*{-1pt}_e} 0$ we conclude that, either $C$ is a scalar (and hence there are no off-diagonal negative entries), or both $C$ and $B'A^{-1}B$ are diagonal. The latter contradicts the assumption that $M$ is irreducible. Hence, the nullity of $M$ can be at most $1$.\n\\end{proof}\n\nLemma \\ref{lemma:next} provides the following geometric insight, stated as a corollary.\n\\begin{cor}\\label{cor:numberofobtuseangles} In any Euclidean space of dimension $n$,\nthere can be at most $n+1$ vectors forming an obtuse angle with one another.\n\\end{cor}\n\n\\begin{proof} The Grammian $M=[v_k'v_\\ell]_{k,\\ell=1}^{n+q}$ of a selection $\\{v_k\\mid k=1,\\ldots, n+q\\}$ of such vectors\nhas off-diagonal entries which are negative. Hence, by Lemma \\ref{lemma:next}, the nullity of $M$ cannot exceed $1$.\n\\end{proof}\n\n\nThe necessity part of Theorem \\ref{thm:Shapiro} is also a direct corollary of Lemma \\ref{lemma:next}.\n\\begin{cor}\\label{prop:weaker} Let $\\Sigma\\in{\\mathbf S}_{n,+}$ and irreducible. Then\n\\[\n\\Sigma \\preceq_{\\hspace*{-1pt}_e} 0 \\Rightarrow{\\operatorname{mr}}(\\Sigma)=n-1.\n\\]\n\\end{cor}\n\n\\begin{proof}\nLet\n$\\Sigma =\\hat \\Sigma+\\tilde\\Sigma$, with $\\tilde\\Sigma$ diagonal and $\\hat\\Sigma\\geq 0$. $\\hat\\Sigma$ is irreducible since $\\Sigma$ is irreducible. From Lemma \\ref{lemma:next}, the nullity of $\\hat\\Sigma$ is at most $1$. Thus ${\\operatorname{mr}}(\\Sigma)=n-1$.\n\\end{proof}\n\n\\subsection{A dual decomposition}\n\nThe matrix inversion lemma provides a correspondence between an\nadditive decomposition of a positive-definite matrix and a\ndecomposition of its inverse, albeit with a different sign in one of\nthe summands. This is stated next.\n\n\\begin{lemma}\\label{lemma:decompositions} Let\n\\begin{equation}\\label{eq:first_decomposition}\n\\Sigma=D+FF'\n\\end{equation}\nwith $\\Sigma,D\\in{\\mathbf S}_{n,+}$,\nwith $\\Sigma,D>0$ and $F\\in{\\mathbb R}^{n\\times r}$. Then\n\\begin{equation}\\label{eq:second_decomposition}\nS:=\\Sigma^{-1} = E - GG'\n\\end{equation}\nfor $E=D^{-1}$ and $G=D^{-1}F(I+F'D^{-1}F)^{-1\/2}$. Conversely, if (\\ref{eq:second_decomposition}) holds with $G\\in{\\mathbb R}^{n\\times r}$, then so does (\\ref{eq:first_decomposition}) for $D=E^{-1}$ and $F=E^{-1}G(I-G'E^{-1}G)^{-1\/2}$.\n\\end{lemma}\n\n\\begin{proof} This follows from the identity\n$(I\\pm MM')^{-1}=I\\mp M(I\\mp M'M)^{-1}M'$.\n\\end{proof}\n\n\nApplication of the lemma suggests the following variation to Frisch's problem.\n\\begin{problem}[\\em The dual Frisch problem]\\label{problem2} Given a positive-definite $n\\times n$ symmetric matrix $S$ determine\nthe {\\em dual minimum rank}:\n\\begin{eqnarray}\\nonumber\n{\\operatorname{mr_{dual}}}(S)&:=&\\min\\{{\\operatorname{rank}}(\\hat S \\mid S=E -\\hat S,\\\\&& \\hat S,E\\geq 0,\\;E \\mbox{ is diagonal}\\nonumber\\}.\\label{eq:mcdual}\n\\end{eqnarray}\n\\end{problem}\n\nClearly, if $S=\\Sigma^{-1}=E-GG^\\prime$ (as in (\\ref{eq:second_decomposition})), then $E>0$. Furthermore, a\ndecomposition of $S$ always gives rise to\na decomposition $\\Sigma=D+FF^\\prime$ (as in (\\ref{eq:first_decomposition})) with the terms $FF'$ and\n$GG'$ having the same rank. Thus, it is clear that\n\\begin{equation}\\label{eq:inequality}\n{\\operatorname{mr}}_+(\\Sigma)\\leq {\\operatorname{mr_{dual}}}(\\Sigma^{-1}),\n\\end{equation}\nand that the above holds with equality when an optimal choice of $D\\equiv\\tilde \\Sigma$ in (\\ref{eq:mc}) is invertible.\nHowever, if $D$ is allowed to be singular,\nthe rank of the summands $FF'$ and $GG'$ may not agree. This is can be seen using the following example. Take\n\\[\\Sigma=\\left[\\begin{matrix}\n2&1&1\\\\\n1&2&1\\\\\n1&1&1\\end{matrix}\\right].\n\\]\nIt is clear that $\\Sigma$ admits a decomposition\n$\\Sigma=\\tilde\\Sigma+\\hat\\Sigma$, in correspondence with\n(\\ref{eq:first_decomposition}), where\n$\\tilde\\Sigma=D={\\operatorname{diag}}\\{1,1,0\\}$\nwhile $\\hat\\Sigma=FF'$ as well as $F'=[1,\\,1,\\,1]$ are of rank\none. On the other hand,\n\\[\nS=\\Sigma^{-1}=\\left[\\begin{matrix}\n\\;\\;1&\\;\\;0&-1\\\\\n\\;\\;0&\\;\\;1&-1\\\\\n-1&-1&\\;\\;3\\end{matrix}\\right].\n\\]\nTaking $E={\\operatorname{diag}}\\{e_1,\\;e_2,\\;e_3\\}$ in (\\ref{eq:second_decomposition}), it is evident that the rank of\n\\[\nGG'=E-S=\\left[\\begin{matrix}\ne_1-1&0&1\\\\\n0&e_2-1&1\\\\\n1&1&e_3-3\\end{matrix}\\right]\n\\]\ncannot be less than $2$ without violating the non-negativity\nassumption for the summand $GG'$. The minimal rank for the factor\n$G$ is $2$ and is attained by taking $e_1=e_2=2$ and $e_3=5$.\n\nOn the other hand, in general, if we perturb $\\Sigma$ to $\\Sigma+\\epsilon I$ and, accordingly, $D$ to $D+\\epsilon I$, then\n\\begin{equation}\\label{eq:inequality2}\n{\\operatorname{mr_{dual}}}((\\Sigma+\\epsilon I)^{-1})\\leq {\\operatorname{mr}}_+(\\Sigma),~ \\forall \\epsilon>0.\n\\end{equation}\nEquality in \\eqref{eq:inequality2} holds for sufficiently small\nvalue of $\\epsilon$. Thus, ${\\operatorname{mr}}_+$ and ${\\operatorname{mr_{dual}}}$ are closely\nrelated. However, it should be noted that ${\\operatorname{mr_{dual}}}(\\cdot)$ fails to\nbe lower semi-continuous since a small perturbation of the\noff-diagonal entries can reduce ${\\operatorname{mr_{dual}}}(\\cdot)$. Yet,\ninterestingly, an exact characterization of the ${\\operatorname{mr_{dual}}}(S)=n-1$ can\nbe obtained which is analogous to those for ${\\operatorname{mr}}_+$ and ${\\operatorname{mr}}$ being\nequal to $n-1$; the condition for ${\\operatorname{mr_{dual}}}$ will be used to prove\nthe Reiers\\o l and Shapiro theorems.\n\n\\begin{thm}\\label{thm:dualreiersol} For $S\\in{\\mathbf S}_{n,+}$, with $S>0$ and irreducible,\n\\begin{equation}\\label{dualreiersol}\n{\\operatorname{mr_{dual}}}(S)=n-1 \\Leftrightarrow S \\succeq_{\\hspace*{-1pt}_e} 0.\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nIf $S\\succeq_{\\hspace*{-1pt}_e} 0$ and $E$ is diagonal satisfying $E\\geq S>0$, then $E-S=GG'\\preceq_{\\hspace*{-1pt}_e} 0$.\nBy invoking Lemma~\\ref{lemma:next} we deduce that if $E-S$ is singular, ${\\operatorname{rank}}(G)=n-1$. Hence, ${\\operatorname{mr_{dual}}}(S)=n-1$.\n\nTo establish that ${\\operatorname{mr_{dual}}}(S)=n-1\\Rightarrow S \\succeq_{\\hspace*{-1pt}_e} 0$, we\nassume that the condition $S\\succeq_{\\hspace*{-1pt}_e} 0$ fails and show that\n${\\operatorname{mr_{dual}}}(S)c$ and\n\\[\nM:=E-(A+b(e-c)^{-1}b')\\geq0.\n\\]\nThe nullity of $\\tilde S-S$ coincides with that of $M$. To prove our claim, it suffices to show\nthat $A_e:=A+b(e-c)^{-1}b'\\not\\succeq_{\\hspace*{-1pt}_e} 0$, or that $A_e$ is reducible for some $e>c$. (Since, in either case, by our hypothesis, the nullity of $M$ for a suitable $E$ exceeds $1$.)\n\nWe now consider two possible cases where $S\\succeq_{\\hspace*{-1pt}_e} 0$ fails. First, we consider the case where already $A\\not \\succeq_{\\hspace*{-1pt}_e} 0$.\nThen obviously $A_e\\not\\succeq_{\\hspace*{-1pt}_e} 0$ for $e-c$ sufficiently large.\nThe second possibility is $S\\not \\succeq_{\\hspace*{-1pt}_e} 0$ while $A\\succeq_{\\hspace*{-1pt}_e} 0$. But if $A$ is (transformed into) element-wise nonnegative, then $bb'$ must have at least one pair of negative off-diagonal entries. Then, consider $A_e=A+\\lambda bb'$ for $\\lambda=(e-c)^{-1}\\in(0,\\infty)$. Evidently, for certain values of $\\lambda$ entries of $A_e$ change sign. If a whole row becomes zero for a particular value of $\\lambda$, then $A_e$ is reducible. In all other cases, there are values of $\\lambda$ for which $A_e\\not\\succeq_{\\hspace*{-1pt}_e} 0$. This completes the proof.\n\\end{proof}\n\n\\subsection{Proof of Reiers{\\o}l's theorem (Theorem \\ref{thm:Reiersol})}\\label{sec:proofReiersol}\n\nWe first show that $\\Sigma^{-1}\\succ_{\\hspace*{-1pt}_e} 0$ implies\n${\\operatorname{mr}}_+(\\Sigma)=n-1$. From the continuity of the inverse,\n$(\\Sigma+\\epsilon I)^{-1}\\succ_{\\hspace*{-1pt}_e} 0$ for sufficiently small\n$\\epsilon>0$. Applying Theorem~\\ref{thm:dualreiersol}, we conclude\nthat\n\\[\n{\\operatorname{mr_{dual}}}((\\Sigma+\\epsilon I)^{-1})=n-1.\n\\]\nSince ${\\operatorname{mr}}_+(\\Sigma)\\geq {\\operatorname{mr_{dual}}}((\\Sigma+\\epsilon I)^{-1})$ as in\n\\eqref{eq:inequality2}, we conclude that ${\\operatorname{mr}}_+(\\Sigma)=n-1$.\n\nTo prove that ${\\operatorname{mr}}_+(\\Sigma) =n-1\\Rightarrow \\Sigma^{-1}\\succ_{\\hspace*{-1pt}_e} 0$,\nwe show that assuming $\\Sigma^{-1}\\not\\succ_{\\hspace*{-1pt}_e} 0$ and ${\\operatorname{mr}}_+(\\Sigma)\n=n-1$ together leads to a contradiction. From the continuity of the\ninverse and the lower semicontinuity of ${\\operatorname{mr}}_+(\\cdot)$ (Proposition\n\\ref{lemma:lowersc}), there exists a symmetric matrix $\\Delta$ and\nan $\\epsilon>0$ such that\n\\[\n(\\Sigma+\\epsilon \\Delta)^{-1} \\not \\succeq_{\\hspace*{-1pt}_e} 0, \\text{~and~} {\\operatorname{mr}}_+(\\Sigma+\\epsilon \\Delta)=n-1.\n\\]\nThen, from Theorem \\ref{thm:dualreiersol},\n$\n{\\operatorname{mr_{dual}}}((\\Sigma+\\epsilon \\Delta)^{-1})< n-1\n$\nwhile from \\eqref{eq:inequality}\n\\[\n{\\operatorname{mr}}_+(\\Sigma+\\epsilon \\Delta) \\leq {\\operatorname{mr_{dual}}}((\\Sigma+\\epsilon \\Delta)^{-1}).\n\\]\nThus, we have a contradiction and therefore $\\Sigma^{-1}\\succ_{\\hspace*{-1pt}_e} 0$. $\\Box$\n\n\\subsection{Proof of Shapiro's theorem (Theorem \\ref{thm:Shapiro})}\\label{sec:proofShapiro}\nGiven $\\Sigma\\geq 0$ consider $\\lambda>0$ such that $\\lambda I-\\Sigma\\geq0$, a diagonal $D$, and let $E:=\\lambda I-D$.\nSince\n$\\Sigma-D=E-(\\lambda I -\\Sigma)$,\n\\begin{align}\\label{eq:mrmrdual}\n{\\operatorname{mr}}(\\Sigma)={\\operatorname{mr_{dual}}}(\\lambda I-\\Sigma).\n\\end{align}\nIf $\\Sigma$ is irreducible and $\\Sigma\\preceq_{\\hspace*{-1pt}_e} 0$, then $\\lambda I-\\Sigma$ is irreducible and $\\lambda I-\\Sigma\\succeq_{\\hspace*{-1pt}_e} 0$. It follows (Theorem \\ref{thm:dualreiersol}) that\n${\\operatorname{mr_{dual}}}(\\lambda I-\\Sigma)=n-1$, and therefore ${\\operatorname{mr}}(\\Sigma)=n-1$ as well.\n\nFor the the reverse direction, if ${\\operatorname{mr}}(\\Sigma)=n-1$ then ${\\operatorname{mr_{dual}}}(\\lambda I-\\Sigma)=n-1$, which implies that\n$\\lambda I-\\Sigma\\succeq_{\\hspace*{-1pt}_e} 0$ and therefore that $\\Sigma\\preceq_{\\hspace*{-1pt}_e} 0$. $\\Box$\n\nThe original proof in \\cite{Shapiro1982b} claims that for any $\\Sigma\\geq 0$ of size $n\\times n$ with $n>3$ and $\\Sigma\\not \\preceq_{\\hspace*{-1pt}_e} 0$, there exists a $(n-1)\\times (n-1)$ principle minor that is $\\not\\preceq_{\\hspace*{-1pt}_e} 0$. This statement fails for the following sign pattern\n\\[\\footnotesize{\n\\left[\\begin{matrix}+&0&-&-\\\\0&+&-&+\\\\-&-&+&0\\\\-&+&0&+ \\end{matrix} \\right].}\n\\]\nThis matrix can not transformed to have all nonpositive off-diagonal entries, yet all its $3\\times 3$ principle minors $\\preceq_{\\hspace*{-1pt}_e} 0$.\n\n\n\\subsection{Parametrization of solutions under Reiers{\\o}l's and Shapiro's conditions}\\label{section:parametrization}\n\nFor either the Frisch or the Shapiro problem, a solution is not\nunique in general. The parametrization of solutions to the Frisch\nproblem when ${\\operatorname{mr}}_+(\\Sigma)=n-1$ has been known and is briefly\nexplained below (without proof). Interestingly, an analogous\nparametrization is possible for Shapiro's problem and this is given\nin Proposition~\\ref{shapiro_parametrization} that follows, and both\nare presented here for completeness of the exposition.\n\n\\begin{prop}\\label{lemma:parameter1}\nLet $\\Sigma\\in{\\mathbf S}_{n,+}$ with $\\Sigma>0$ and $\\Sigma^{-1}\\succe0$. The following hold:\n\\begin{itemize}\n\\item[i)] For $D\\geq0$ diagonal with $\\Sigma-D\\geq0$ and singular, there is a probability vector $\\rho$ ($\\rho$ has entries $\\geq 0$ that sum up to $1$) such that $(\\Sigma-D)\\Sigma^{-1}\\rho=0$.\n\\item[ii)] For any probability vector $\\rho$,\n\\[D={\\operatorname{diag}}^*\\left(\\left[\\frac{[\\rho]_i}{[\\Sigma^{-1}\\rho]_i}, i=1,\\ldots, n\\right] \\right)\n\\]\nsatisfies $\\Sigma-D\\geq0$ and $\\Sigma-D$ is singular.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof} See \\cite{Kalman1982,KlepperLeamer}.\n\\end{proof}\n\nThus, solutions of Frisch's problem under Reiers{\\o}l's conditions\nare in bijective correspondence with probability vectors. A very\nsimilar result holds true for Shapiro's problem.\n\n\\begin{prop}\\label{shapiro_parametrization}\nLet $\\Sigma\\in{\\mathbf S}_{n,+}$ be irreducible and have $\\leq 0$ off-diagonal entries. The following hold:\n\\begin{itemize}\n\\item[i)]\nFor $D$ diagonal with $\\Sigma-D\\geq0$ and singular, there is a strictly positive vector $v$ such that $(\\Sigma-D)v=0$.\n\\item[ii)] For any strictly positive vector $v\\in {\\mathbb R}^{n\\times 1}$,\n\\begin{align}\\label{eq:Dshapiro}\nD={\\operatorname{diag}}^*\\left(\\left[\\frac{[\\Sigma v]_i}{[v]_i}, i=1,\\ldots, n\\right] \\right)\n\\end{align}\nsatisfies that $\\Sigma-D\\geq0$ and $\\Sigma-D$ is singular.\n\\end{itemize}\n\\end{prop}\n\n\\begin{proof}\nTo prove $(i)$, we note that if $(\\Sigma-D)v=0$, then $v\\succ_{\\hspace*{-1pt}_e} 0$.\nTo see this consider $(\\Sigma-D+\\epsilon I)^{-1}$ for $\\epsilon>0$.\nFrom Lemma~\\ref{lemma:previous},\n\\[\n(\\Sigma-D+\\epsilon I)^{-1}\\succ_{\\hspace*{-1pt}_e} 0\n\\]\nand since $v$ is an eigenvector corresponding to its largest eigenvalue, a power iteration argument concludes that $v\\succ_{\\hspace*{-1pt}_e} 0$.\n\nTo prove $ii)$, it is easy to verify that the diagonal matrix $D$ in \\eqref{eq:Dshapiro} for $v\\succ_{\\hspace*{-1pt}_e} 0$\nsatisfies $(\\Sigma-D)v=0$. We only need to prove that $\\Sigma-D\\geq0$. Without loss of generality we assume that all the entries of $v$ are equal.\n(This can always be done by scaling the entries of $v$ and scaling accordingly rows and columns of $\\Sigma$.)\nSince $v$ is a null vector of $\\Sigma-D$ and since $M:=\\Sigma-D$ has $\\leq 0$ off-diagonal entries\n\\[\n[M]_{ii}=\\sum_{j\\neq i}|[M]_{ij}|.\n\\]\nGersgorin Circle Theorem (e.g., see \\cite{Varga2004})\nnow states that every eigenvalue of $M$ lies within at least one of the closed discs $\\left\\{{\\rm Disk}\\left([M]_{ii}, \\sum_{j\\neq i}|[M]_{ij}| \\right), i=1, \\ldots, n\\right\\}$. No disc intersects the negative real line. Therefore $\\Sigma-D\\geq0$.\n\\end{proof}\n\n\\subsection{Decomposition of complex-valued matrices}\n\nComplex-valued covariance matrices are commonly used in radar and\nantenna arrays \\cite{vantrees}. The rank of $\\Sigma-D$, for\nnoise covariance $D$ as in the Frisch problem, is an indication of\nthe number of (dominant) scatterers in the scattering field. If this\nis of the same order as the number of array elements (e.g., $n-1$),\nany conclusion about their location may be suspect. Thus, it is\nnatural to seek conditions for ${\\operatorname{mr}}_+(\\Sigma)=n-1$ analogous to\nthose given by Reiers{\\o}l, for the case of complex covariances, as\na possible warning. This we do next.\n\nConsider complex-valued observation vectors\n$\nx_t=y_t+ {\\rm i} z_t,~ t=1,\\ldots T,\n$\nwhere ${\\rm i}=\\sqrt{-1}$ and $y_t, z_t \\in {\\mathbb R}^{n\\times 1}$, and\nset\n\\[\nX=[x_1,\\; \\ldots x_T]=Y+ {\\rm i} Z\n\\]\nwith\n$Y=[y_1,\\; \\ldots y_T]$,\n$Z=[z_1,\\; \\ldots z_T]$.\nThe (scaled) sample covariance is\n\\begin{align*}\n\\Sigma=XX^*\n&=\\Sigma_{\\rm r}+{\\rm i} \\Sigma_{\\rm i}\\in{\\mathbf H}_{n,+},\n\\end{align*}\nwhere the real part\n$\\Sigma_{\\rm r}:=YY'+ZZ'$ is symmetric,\nthe imaginary part $\\Sigma_{\\rm i}:=ZY'-YZ'$ is anti-symmetric,\nand ``$*$'' denotes complex-conjugate transpose.\nAs before, we consider a decomposition\n\\[\n\\Sigma=\\hat\\Sigma+D\n\\]\nwith $\\hat\\Sigma\\geq 0$ singular and $D\\geq 0$ diagonal.\nWe refer to \\cite{Anderson1988,Deistler1989} for the special case where\n${\\operatorname{mr}}_+(\\Sigma)=1$. In this section we present a sufficient condition for a Reiers{\\o}l-case\nwhere ${\\operatorname{mr}}_+(\\Sigma)=n-1$.\n\nBefore we proceed we note that re-casting the problem in terms\nof the real-valued\n\\[\nR:=\\left[\n \\begin{array}{cc}\n \\Sigma_{\\rm r} & \\Sigma_{\\rm i} \\\\\n \\Sigma_{\\rm i}^\\prime & \\Sigma_{\\rm r} \\\\\n \\end{array}\n \\right]\\in{\\mathbf S}_{2n,+}\n\\]\ndoes not allow taking advantage of earlier results. The structure of $R$ with antisymmetric off-diagonal blocks implies that if $[a',\\;b']'$ is a null vector then so is\n$[-b',\\;a']'$ (since, accordingly, $a+ {\\rm i} b$ and ${\\rm i} a - b$ are both null vectors of $\\Sigma$). Thus, in general, the nullity of $R$ is not $1$ and the theorem of Reiers{\\o}l is not applicable. Further, the corresponding noise covariance is diagonal with repeated blocks.\n\nThe following lemmas for the complex case echo Lemma \\ref{lemma:previous} and Lemma \\ref{lemma:next}.\n\\begin{lemma}\\label{lemma:complexprevious}\nLet $M\\in{\\mathbf H}_{n,+}$ be irreducible. If the argument of each non-zero off-diagonal entry of $-M$ is in $\\left(-\\frac{\\pi}{2^n},~ \\frac{\\pi}{2^n} \\right)$, then\neach entry of $M^{-1}$ has argument in $\\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^n}, ~ \\frac{\\pi}{2}-\\frac{\\pi}{2^n}\\right)$.\n\\end{lemma}\n\n\\begin{proof}\nIt is easy to verify the lemma for $2\\times 2$ matrices. Assume that\nthe statement holds for sizes up to $n\\times n$ and consider an\n$(n+1)\\times (n+1)$ matrix $M$ that satisfies the conditions of the\nlemma. Partition\n\\[\nM=\\left[\n \\begin{array}{cc}\n A & b \\\\\n b^* & c \\\\\n \\end{array}\n \\right]\n\\]\nwith $A$ is of size $n\\times n$, and conformably,\n\\[\nM^{-1}=\\left[\n \\begin{array}{cc}\n F & g \\\\\n g^* & h \\\\\n \\end{array}\n \\right].\n\\]\nBy assumption non-zero entries of $-A$ and $-b$ have their argument in $\\left(-\\frac{\\pi}{2^{n+1}}, ~\\frac{\\pi}{2^{n+1}}\\right)$.\nThen, by bounding the possible contribution of the respective terms, it follows that for the argument of each of the entries of $-A+bc^{-1}b^*$ is in\n$\\left(-\\frac{\\pi}{2^n}, ~\\frac{\\pi}{2^n}\\right)$. Then, the argument of each entry of $F=(A-bc^{-1}b^*)^{-1}$ is in\n$\\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^n}, ~ \\frac{\\pi}{2}-\\frac{\\pi}{2^n}\\right)$; this follows by assumption since $F$ is $n\\times n$.\nClearly, $\\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^n}, ~ \\frac{\\pi}{2}-\\frac{\\pi}{2^n}\\right) \\subset \\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^{n+1}}, ~\\frac{\\pi}{2}-\\frac{\\pi}{2^{n+1}}\\right)$. Regarding $g$, by bounding the possible contribution of respective terms, we similarly conclude that\nthe argument of each of its non-zero entries is in $\\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^{n+1}}, ~ \\frac{\\pi}{2}-\\frac{\\pi}{2^{n+1}}\\right)$.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma:complexnext}\nLet $M\\in{\\mathbf H}_{n,+}$ be irreducible. If the argument of each non-zero off-diagonal entry of $-M$ is in $\\left(-\\frac{\\pi}{2^n},~\\frac{\\pi}{2^n}\\right)$,\nthen ${\\operatorname{rank}}(M)\\geq n-1$.\n\\end{lemma}\n\n\\begin{proof}\nFirst rearrange rows and columns of $M$, and partition as\n\\[\nM=\\left[\n \\begin{array}{cc}\n A & B \\\\\n B^* & C \\\\\n \\end{array}\n\\right]\n\\]\nso that $A$ is nonsingular and of size equal to the rank of $M$, which we denote by $r$. Then\n\\begin{equation}\\label{eq:CBAB}\nC=B^*A^{-1}B\n\\end{equation}\nand has size equal to the nullity of $M$. We now compare the\nargument of the off-diagonal entries of $C$ and $B^*A^{-1}B$, and\nshow they cannot be equal unless $C$ is a scalar. Since the\noff-diagonal entries of $-A$ have their argument in\n$\\left(-\\frac{\\pi}{2^n}, ~\\frac{\\pi}{2^n}\\right)\\subset\n\\left(-\\frac{\\pi}{2^r}, ~\\frac{\\pi}{2^r}\\right)$, the off-diagonal\nentries of $A^{-1}$ have their argument in\n$\\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^r}, ~\n\\frac{\\pi}{2}-\\frac{\\pi}{2^r}\\right)$ from Lemma\n\\ref{lemma:complexprevious}. Now, the $(k,\\ell)$ entry of\n$B^*A^{-1}B$ is\n\\begin{align*}\n[B^*A^{-1}B]_{k\\ell}=\\sum_{i,j}[B^*]_{ki}[A^{-1}]_{ij}[B]_{j \\ell}\n\\end{align*}\nand the phase of each summand is\n\\[\n\\arg([B^*]_{ki}[A^{-1}]_{ij} [B]_{j \\ell}) \\in\\left(-\\frac{\\pi}{2}+\\frac{\\pi}{2^r}-\\frac{\\pi}{2^{n-1}},~ \\frac{\\pi}{2}-\\frac{\\pi}{2^r}+\\frac{\\pi}{2^{n-1}}\\right).\n\\]\nThus, the non-zero off-diagonal entries of $B^*A^{-1}B$ have positive real part while\n \\[\n \\arg(-[C]_{k\\ell})\\in \\left(-\\frac{\\pi}{2^n},~\\frac{\\pi}{2^n}\\right) .\n \\]\nHence, either the off-diagonal entries of $B^*A^{-1}B$ and $C$ are\nzero, in which case these are diagonal matrices and $M$ must be\nreducible, or $B^*A^{-1}B$ and $C$ are both scalars. This concludes\nthe proof.\n\\end{proof}\n\n\\begin{thm}\\label{prop:complexReiersol}\nLet $\\Sigma\\in{\\mathbf H}_{n,+}$ be irreducible.\nIf the argument of each non-zero off-diagonal entry of $-\\Sigma$ is in\n$\\left(-\\frac{\\pi}{2^n},~ \\frac{\\pi}{2^n}\\right)$, then\n${\\operatorname{mr}}(\\Sigma)=n-1$.\n\\end{thm}\n\n\\begin{proof}\nThe matrix $\\Sigma-D$ is irreducible since $D$ is diagonal.\nIf $\\Sigma-D\\geq0$ and singular, and since the argument of each non-zero off-diagonal entry of $-(\\Sigma-D)$ is in\n$\\left(-\\frac{\\pi}{2^n},~ \\frac{\\pi}{2^n}\\right)$, Lemma \\ref{lemma:complexnext} applies and gives that ${\\operatorname{rank}}(\\Sigma-D)=n-1$.\n\\end{proof}\n\nClearly, since ${\\operatorname{mr}}_+(\\Sigma)\\geq{\\operatorname{mr}}(\\Sigma)$, under the condition of Theorem \\ref{prop:complexReiersol}, ${\\operatorname{mr}}_+(\\Sigma)=n-1$.\nIt is also clear that for $S\\in{\\mathbf H}_{n,+}$ irreducible with all non-zero off-diagonal entries having argument in $\\left(-\\frac{\\pi}{2^n},~ \\frac{\\pi}{2^n}\\right)$, we also conclude that\n${\\operatorname{mr_{dual}}}(S)=n-1$.\n\n\\section{Trace minimization heuristics}\\label{sec:MinTrace}\n\nThe rank of a matrix is a non-convex function of its elements and the problem to find the matrix of minimal rank within a given set is a difficult one, in general.\nTherefore, certain heuristics have been developed over the years to obtain approximate solutions.\nIn particular, in the context of factor analysis, trace minimization has been pursued as a suitable heuristic \\cite{Ledermann1940,Shapiro,Shapiro1982b} thereby relaxing the Frisch problem into\n\\begin{align}\\nonumber\n&\\min_{D: \\Sigma\\geq D\\geq0} {\\operatorname{trace}}(\\Sigma-D),\n\\end{align}\nfor a diagonal matrix $D$; with a relaxation of $D\\geq 0$ corresponding to Shapiro's problem. The theoretical basis for using the trace and, more generally, the nuclear norm for non-symmetric matrices, as a surrogate for the rank was provided by\nFazel {\\em etal.} \\cite{Fazel2001} who proved that these constitute convex envelops of the rank function on bounded sets of matrices.\n\nThe relation between minimum trace factor analysis and minimum rank factor analysis goes back to Ledermann in \\cite{Ledermann1939} (see \\cite{Della1982} and \\cite{Saunderson2012}). Herein we only refer to two propositions which characterize minimizers for the two problems, Frisch's and Shapiro's, respectively.\n\n\\begin{subequations}\n\\begin{prop}[\\cite{Della1982}]\\label{prop:mintrace}\nLet $\\Sigma=\\hat\\Sigma_1+D_1>0$ for a diagonal $D_1\\geq0$. Then,\n\\begin{align}\\label{trmc}\n &(\\hat\\Sigma_1,D_1)=\\arg\\min\\{ {\\operatorname{trace}}(\\hat\\Sigma) \\mid \\Sigma=\\hat\\Sigma+D>0,\\;\\hat\\Sigma\\geq 0,\\;\\mbox{diagonal }D\\geq 0\\}\\\\\n& \\Leftrightarrow~ \\exists~ \\Lambda_1 \\geq0 ~:~ \\hat\\Sigma_1 \\Lambda_1=0 \\text{~and~} \\left\\{\n \\begin{array}{ll}\n [\\Lambda_1]_{ii}=1, & \\text{~if~} [D_1]_{ii}>0, \\\\\n \\left[ \\Lambda_1\\right]_{ii}\\geq1, &\\text{~if~}[D_1]_{ii}=0.\\nonumber\n \\end{array}\n \\right.\n\\end{align}\n\\end{prop}\n\n\\begin{prop}[\\cite{Saunderson2012}]\\label{prop:MTFAShapiro}\nLet $\\Sigma=\\hat\\Sigma_2+D_2>0$ for a diagonal $D_2$.\nThen,\\begin{align}\\label{trmc2}\n &(\\hat\\Sigma_2,D_2)=\\arg\\min\\{ {\\operatorname{trace}}(\\hat\\Sigma) \\mid \\Sigma=\\hat\\Sigma+D>0,\\;\\hat\\Sigma\\geq 0,\\;\\mbox{diagonal }D\\}\\\\\n& \\Leftrightarrow~\\exists~ \\Lambda_2 \\geq0 ~:~ \\hat\\Sigma_2 \\Lambda_2=0 \\text{~and~} [\\Lambda_2]_{ii}=1~ \\forall i.\\nonumber\n\\end{align}\n\\end{prop}\n\\end{subequations}\n\n\\noindent Evidently, when the solutions to these two problems differ\nand $D_1\\neq D_2$, then there exists $k\\in\\left\\{1, \\ldots, n\\right\\}$ such that\n\\[\n[D_2]_{kk}<0 \\text{~and~} [D_1]_{kk}=0.\n\\]\nFurther, the essence of Proposition \\ref{prop:MTFAShapiro} is that\na singular $\\hat\\Sigma$ originates from such a minimization problem if and only if there is a correlation matrix in its null space. The matrices $\\Lambda_1$ and $\\Lambda_2$ appear as Lagrange multipliers in the respective problems.\n\n\\newcommand{{\\mathcal A}}{{\\mathcal A}}\n{Factor analysis is closely related to {\\em low-rank matrix completion} as well as to {\\em sparse and low-rank decomposition} problems. Typically, low-rank matrix completion asks for a matrix $X$ which satisfies a linear constraint ${\\mathcal A}(X)=b$ and has low\/minimal rank (${\\mathcal A}(\\cdot)$ denotes a linear map ${\\mathcal A}\\,:\\,{{\\mathbb R}}^{n\\times n}\\rightarrow {{\\mathbb R}}^p$). Thus, factor analysis corresponds to the special case where ${\\mathcal A}(\\cdot)$ maps $X$ onto its off-diagonal entries. In a recent work by Recht {\\em etal.}~\\cite{Recht2010guaranteed}, the nuclear norm of $X$ was considered as a convex relaxation of ${\\operatorname{rank}}(X)$ for such problems and a sufficient condition for exact recovery was provided. However, this sufficient condition amounts to the requirement that the null space of ${\\mathcal A}(\\cdot)$ contains no matrix of low-rank. Therefore, since in factor analysis diagonal matrices are in fact contained in the null space of ${\\mathcal A}(\\cdot)$ and include matrices of low-rank, the condition in \\cite{Recht2010guaranteed} does not apply directly. Other works on low-rank matrix completion (see, e.g., \\cite{Recht2010guaranteed,Candes2009exact}) mainly focus on assessing the probability of exact recovery and on constructing efficient computational algorithms for {\\em large-scale} low-rank completion problems \\cite{Keshavan2010matrix,Keshavan2010noisy}.\nOn the other hand, since diagonal matrices are sparse (most of their entries are zero), the work on matrix decomposition into sparse and low-rank components by Chandrasekaran {\\em etal.} \\cite{Chandrasekaran2011rank} is very pertinent. In this, the $\\ell_1$ and nuclear norms were used as surrogates for sparsity and rank, respectively, and a sufficient condition for exact recovery was provided which captures a certain ``rank-sparsity incoherence''; an analogous but stronger sufficient ``incoherence'' condition which applies to problem\n\\eqref{trmc2} is given in \\cite{Saunderson2012}.}\n\n\n\\subsection{Weighted minimum trace factor analysis}\n\nBoth ${\\operatorname{mr}}(\\Sigma)$ and ${\\operatorname{mr}}_+(\\Sigma)$ in \\eqref{eq:mc} and\n\\eqref{eq:mc2}, respectively, remain invariant under scaling of rows\nand the corresponding columns of $\\Sigma$ by the same coefficients.\nOn the other hand, the minimizers in \\eqref{trmc} and \\eqref{trmc2}\nand their respective ranks are not invariant under scaling. This\nfact motivates weighted-trace minimization,\n\\begin{align}\\label{eq:Dw}\n\\min\\left\\{ {\\operatorname{trace}}(W\\hat\\Sigma) \\mid \\Sigma=\\hat\\Sigma+D,~\\hat\\Sigma\\geq 0,~\\mbox{diagonal }D\\geq 0 \\right\\},\n\\end{align}\ngiven $\\Sigma>0$ and a diagonal weight $W>0$.\nAs before the characterization of minimizers relates to a suitable condition for the corresponding Lagrange multipliers:\n\n\\begin{prop}[{\\rm \\cite{Shapiro1982b}}]\\label{prop:WMTFAShapiro}\nLet $\\Sigma=\\hat\\Sigma_0+D_0>0$ for a diagonal matrix $D_0\\geq0$ and consider a diagonal $W>0$. Then,\n\\begin{align}\\label{trmc3}\n &(\\hat\\Sigma_0,D_0)=\\arg\\min\\{ {\\operatorname{trace}}(W\\hat\\Sigma) \\mid \\Sigma=\\hat\\Sigma+D>0,\\;\\hat\\Sigma\\geq\n 0,\\;\\mbox{diagonal }D\\geq 0\\}\\\\\n& \\Leftrightarrow~ \\exists~ \\Lambda_0 \\geq0 ~:~\n\\hat\\Sigma \\Lambda_0=0 \\text{~and~} \\left\\{\n \\begin{array}{ll}\n [\\Lambda_0]_{ii}=[W]_{ii}, & \\text{~if~} [D_0]_{ii}>0, \\\\\n \\left[ \\Lambda_0\\right]_{ii}\\geq [W]_{ii}, &\\text{~if~}[D_0]_{ii}=0.\\nonumber\n \\end{array}\n \\right.\\nonumber\n\\end{align}\n\\end{prop}\n\nA corresponding sufficient and necessary condition for $(\\hat\\Sigma, D)$ to be a minimizer in Shapiro's problem is that there exists a Grammian in the null space of $\\hat\\Sigma$ whose diagonal entries are equal to the diagonal entries of $W$.\n\nMinimum-rank solutions may be recovered as solutions to \\eqref{trmc3} using suitable choices of weight.\nHowever, these choices depend on $\\Sigma$ and are not known in advance --this motivates a selection of certain canonical $\\Sigma$-dependent\nweight as well as iteratively improving the choice of weight. One should note that since $D$ is diagonal, letting $W$ be a not-necessarily\ndiagonal matrix does not change the problem --only the diagonal entries of $W$ determine the minimizer.\n\nWe first consider taking $W=\\Sigma^{-1}$. A rationale for this\nchoice is that the minimal value in \\eqref{eq:Dw} bounds\n${\\operatorname{mr}}_+(\\Sigma)$ from below, since for any decomposition\n$\\Sigma=\\hat\\Sigma+D$,\n\\begin{align}\\nonumber\n{\\operatorname{rank}}(\\hat \\Sigma) =&~ {\\operatorname{trace}} (\\hat\\Sigma^\\sharp \\hat\\Sigma)\\\\\\nonumber\n\\geq&~ {\\operatorname{trace}}((\\hat\\Sigma+D)^{-1} \\hat\\Sigma)\\\\\n=&~ {\\operatorname{trace}}(\\Sigma^{-1} \\hat\\Sigma)\\label{eq:rankjustify}\n\\end{align}\nwhere $^\\sharp$ denotes the Moore-Penrose pseudo inverse. Continuing\nwith this line of analysis\n\\begin{align}\n{\\operatorname{rank}}(\\hat \\Sigma) =&~ {\\operatorname{trace}} (\\hat\\Sigma^\\sharp \\hat\\Sigma)\\nonumber\\\\\n\\geq&~ {\\operatorname{trace}}((\\hat\\Sigma+\\epsilon I)^{-1} \\hat\\Sigma)\\label{eq:ranktrace}\n\\end{align}\nfor any $\\epsilon>0$, suggests the iterative re-weighting process\n\\begin{align}\\label{minimizer_a}\nD_{(k+1)}:=&~\\arg\\min_{D}{\\operatorname{trace}}\\left((\\Sigma-D_{(k)} +\\epsilon I)^{-1}(\\Sigma-D)\\right)\n\\end{align}\nfor $k=1,\\,2,\\,\\ldots$ and $D_{(0)}:=0$.\nIn fact, as pointed out in \\cite{Fazel2003}, \\eqref{minimizer_a} corresponds to minimizing\n$\\log\\det(\\Sigma-D+\\epsilon I)$\nby local linearization.\n\nNext we provide a sufficient condition for $\\hat\\Sigma$ to be such a\nstationary point \\eqref{minimizer_a}, i.e., \nfor $\\hat\\Sigma$ to satisfy\n\\begin{align}\\label{stationary_a}\n\\arg\\min_{D}{\\operatorname{trace}}\\left((\\hat\\Sigma+\\epsilon I)^{-1}(\\hat\\Sigma-D)\\right)=0.\n\\end{align}\nThe notation $\\circ$ used below denotes the\nelement-wise product between vectors or matrices which is also known\nas \\emph{Schur product} \\cite{Horn1990matrix} and, likewise, for\nvectors $a, b \\in {\\mathbb R}^{n\\times 1}$, $a\\circ b\\in {\\mathbb R}^{n\\times 1}$\nwith $[a\\circ b]_i=[a]_i[b]_i$.\n\n\n\\begin{prop}\\label{prop:stationary_a}\nLet $\\hat\\Sigma\\in{\\mathbf S}_{n,+}$ and let the columns of $U$ form a basis of ${\\mathcal R}(\\hat\\Sigma)$. If\n\\begin{align}\\label{eq:stationary_a}\n{\\mathcal R}(U\\circ U) \\subset {\\mathcal R}(\\Pi_{{\\mathcal N}(\\hat\\Sigma)}\\circ\\Pi_{{\\mathcal N}(\\hat\\Sigma)} ),\n\\end{align}\nthen $\\hat\\Sigma$ satisfies \\eqref{stationary_a} for all $\\epsilon\\in(0,\\; \\epsilon_1)$ and some $\\epsilon_1>0$.\n\\end{prop}\n\nWe first need the following result which generalizes \\cite[Theorem 3.1]{Shapiro1985}.\n\\begin{lemma}\\label{lemma:trace}\nFor $A\\in {\\mathbb R}^{n\\times p}$ and $B\\in {\\mathbb R}^{n\\times q}$ having columns $a_1, \\ldots, a_p$ and $b_1, \\ldots, b_q$, respectively, we let\n\\begin{align*}\nC&=[a_1\\circ b_1, a_1\\circ b_2, \\ldots,a_2\\circ b_1\\dots a_p \\circ b_q]\\in {\\mathbb R}^{n\\times pq},\\\\\n\\phi &: ~{\\mathbb R}^n \\hspace*{.4cm}\\rightarrow {\\mathbb R}^n \\hspace*{.57cm} d \\mapsto {\\operatorname{diag}}(AA'{\\operatorname{diag}}^*(d)BB'), \\mbox{ and}\\\\\n\\psi &: ~{\\mathbb R}^{p\\times q} \\rightarrow {\\mathbb R}^n \\hspace*{.5cm} \\Delta \\mapsto {\\operatorname{diag}}(A\\Delta B').\n\\end{align*}\nThen ${\\mathcal R}(\\phi)={\\mathcal R}(\\psi)={\\mathcal R}((AA')\\circ (BB'))={\\mathcal R}(C)$.\n\\end{lemma}\n\n\\begin{proof}\nSince ${\\operatorname{diag}}(AA'{\\operatorname{diag}}^*(d)BB')=((AA')\\circ (BB'))d$, it follows that\n\\[{\\mathcal R}(\\phi)={\\mathcal R}((AA')\\circ (BB').\\]\nMoreover,\n${\\operatorname{diag}}(A\\Delta B')= \\sum_{i=1}^p\\sum_{j=1}^q a_i\\circ b_j [\\Delta]_{ij}$, and then\n${\\mathcal R}(\\psi)={\\mathcal R}(C)$.\nWe only need to show that ${\\mathcal R}(C)={\\mathcal R}((AA')\\circ (BB'))$. This follows from\n\\begin{align*}\n(AA')\\circ (BB') =&~\\sum_{i=1}^p\\sum_{j=1}^q (a_ia_i')\\circ (b_jb_j')\\\\\n =&~\\sum_{i=1}^p\\sum_{j=1}^q (a_i\\circ b_j) (a_i\\circ b_j)'\n =CC'.\n\\end{align*}\nThus ${\\mathcal R}(C)={\\mathcal R}((AA')\\circ (BB'))$.\n\\end{proof}\n\n\n\\begin{proof} {\\em [Proof of Proposition \\ref{prop:stationary_a}:]}\nAssume that $\\hat\\Sigma$ satisfies \\eqref{stationary_a}.\nIf ${\\operatorname{rank}}(\\hat\\Sigma)=r$, let $\\hat\\Sigma=USU'$ be the eigendecomposition of $\\hat\\Sigma$ with $S={\\operatorname{diag}}^*(s)$ with $s\\in {\\mathbb R}^r$. Let the columns of $V$ be an orthogonal basis of the null space of $\\hat\\Sigma$, i.e., $\\Pi_{{\\mathcal N}(\\hat\\Sigma)}=VV'$.\nThen\n\\begin{align*}\n(\\hat\\Sigma+\\epsilon I)^{-1}=(\\hat\\Sigma+\\epsilon \\Pi_{{\\mathcal R}(\\hat\\Sigma)}+\\epsilon \\Pi_{{\\mathcal N}(\\hat\\Sigma)})^{-1} =(\\hat\\Sigma+\\epsilon \\Pi_{{\\mathcal R}(\\hat\\Sigma)})^\\sharp+\\frac{1}{\\epsilon} \\Pi_{{\\mathcal N}(\\hat\\Sigma)},\n\\end{align*}\nand\n\\begin{align*}\n\\arg\\min_{D:\\hat\\Sigma\\geq D} {\\operatorname{trace}}\\left((\\hat\\Sigma+\\epsilon I)^{-1}(\\hat\\Sigma-D)\\right)& =\\\\\n&\\hspace*{-1.5cm}\\arg\\min_{D:\\hat\\Sigma\\geq D} {\\operatorname{trace}}\\left(\\left(\\epsilon( \\hat\\Sigma+\\epsilon \\Pi_{{\\mathcal R}(\\hat\\Sigma)})^\\sharp+\\Pi_{{\\mathcal N}(\\hat\\Sigma)}\\right)(\\hat\\Sigma-D)\\right).\n\\end{align*}\nFrom Proposition \\ref{prop:WMTFAShapiro}, \\eqref{stationary_a} holds if there is $M\\in {\\mathbf S}_{r,+}$ such that\n\\begin{align}\\label{stationaryaDiag}\n{\\operatorname{diag}}(VMV')={\\operatorname{diag}}\\left( \\epsilon( \\hat\\Sigma+\\epsilon \\Pi_{{\\mathcal R}(\\hat\\Sigma)})^\\sharp+\\Pi_{{\\mathcal N}(\\hat\\Sigma)}\\right).\n\\end{align}\nObviously, if $\\epsilon=0$\n $M=I$ satisfies the above equation. We consider the matrix $M$ of the form $M=I+\\Delta$. For \\eqref{stationaryaDiag} holds, we need ${\\operatorname{diag}}((\\hat\\Sigma+\\epsilon \\Pi_{{\\mathcal R}})^\\sharp)$ to be in the range of $\\psi$ for\n\\[\n\\psi: {\\mathbf S}_n \\rightarrow {\\mathbb R}^n \\hspace*{.57cm} \\Delta \\mapsto {\\operatorname{diag}}(V\\Delta V').\n\\]\nFrom Lemma \\ref{lemma:trace} that ${\\mathcal R}(\\psi)={\\mathcal R}(\\Pi_{{\\mathcal N}(\\hat\\Sigma)}\\circ\\Pi_{{\\mathcal N}(\\hat\\Sigma)})$. On the other hand, since\n\\[\n\\epsilon(\\hat\\Sigma+\\epsilon \\Pi_{{\\mathcal R}(\\hat\\Sigma)})^\\sharp=U{\\operatorname{diag}}\\left(\\left[\\frac{\\epsilon}{[s]_1+\\epsilon}, \\ldots, \\frac{\\epsilon}{[s]_r+\\epsilon} \\right]\\right)U',\n\\]\nthen ${\\operatorname{diag}}(\\epsilon(\\hat\\Sigma+\\epsilon\n\\Pi_{{\\mathcal R}(\\hat\\Sigma)})^\\sharp)\\in {\\mathcal R}(U\\circ U)$. So if\n\\eqref{eq:stationary_a} holds, there is always a $\\Delta$ such that\n$M=I+\\Delta$ satisfies \\eqref{stationaryaDiag}. Morover, it is also\nrequired that $I+\\Delta\\geq0$. Since the map from $\\epsilon$ to\n$\\Delta$ is continuous, for small enough $\\epsilon$, i.e. in a\ninterval $(0, \\epsilon_1)$ the condition $I+\\Delta$ can always be\nsatisfied.\n\\end{proof}\n\nWe note that \\eqref{eq:stationary_a} is a sufficient condition for $\\hat\\Sigma$ to be a stationary point of \\eqref{stationary_a} in both Frisch's and Shapiro's settings.\n\n\n\\section{Certificates of minimum rank}\\label{sec:CertifMinRank}\n\nWe are interested in obtaining bounds on the minimal rank for the\nFrisch problem so as to ensure optimality when candidate solutions\nare obtained by the earlier optimization approach in \\eqref{minimizer_a}.\n\nThe following two bounds were proposed in \\cite{Woodgate1}, and\nfollow from Theorem~\\ref{thm:Reiersol}. However, both of these\nbounds require exhaustive search which may be prohibitively\nexpensive when $n$ is large.\n\\begin{subequations}\n\\begin{cor}\\label{cor1}\nLet $\\Sigma\\in{\\mathbf S}_{n,+}$ and $\\Sigma>0.$ If there is an $s_1\\times s_1$\nprinciple minor of $\\Sigma$ whose inverse is positive, then\n \\begin{align}\n{\\operatorname{mr}}_+(\\Sigma)&\\geq s_1-1.\n \\end{align}\n If there is an $s_2\\times s_2$ principle\nminor of $\\Sigma^{-1}$ which is element-wise positive, then\n \\begin{align}\n{\\operatorname{mr}}_+(\\Sigma)&\\geq s_2-1.\n \\end{align}\n\\end{cor}\n\nNext we discuss three other bounds that are computationally\nmore tractable --the first two were proposed by Guttman\n\\cite{Guttman1954}.\nGuttman's bounds are based on a conservative assessment for the admissible range of each of the diagonal entries of $D=\\Sigma-\\hat\\Sigma$.\n\n\\begin{prop}\\label{prop:Guttman}\n Let $\\Sigma\\in {\\mathbf S}_{n,+}$ and let\n \\begin{align*}\n D_1&:={\\operatorname{diag}}^*({\\operatorname{diag}}(\\Sigma))\\\\\n D_2&:=\\left({\\operatorname{diag}}^*({\\operatorname{diag}}(\\Sigma^{-1}))\\right)^{-1}.\n \\end{align*}\n Then the following hold,\n\\begin{align}\n&{\\operatorname{mr}}_+(\\Sigma)\\geq n_+(\\Sigma-D_1) \\label{Guttman:bound1}\\\\\n&{\\operatorname{mr}}_+(\\Sigma)\\geq n_+(\\Sigma-D_2). \\label{Guttman:bound2}\\\\\n\\nonumber\n\\end{align}\nFurther, $n_+(\\Sigma-D_1)\\leq n_+(\\Sigma-D_2)$.\n\\end{prop}\n\n\\begin{proof} The proof follows from the fact that $\\Sigma\\geq D$ implies $D\\leq D_2\\leq D_1$. See \\cite{Guttman1954} for details.\n\\end{proof}\n\nIt is also easy to see that ${\\operatorname{mr}}(\\Sigma)\\geq n_+(\\Sigma-D_1)$ which\nprovides a lower bound for the minimum rank in Shapiro's problem.\nNext we return to a bound, which we noted earlier in \\eqref{eq:rankjustify}.\n\n\\begin{prop}\\label{tracebound}\nLet $\\Sigma\\in{\\mathbf S}_{n,+}$. Then the following holds:\n\\begin{align}\\label{eq:lowerbound3}\n{\\operatorname{mr}}_+(\\Sigma)\\geq \\min_{\\Sigma\\geq D\\geq 0}{\\operatorname{trace}}(\\Sigma^{-1}(\\Sigma-D)).\n\\end{align}\n\\end{prop}\n\n\\begin{proof} The statement follows readily from \\eqref{eq:rankjustify}.\n\\end{proof}\n\nEvidently an analogous statement holds for ${\\operatorname{mr}}(\\Sigma)$.\nWe note that \\eqref{Guttman:bound1} and \\eqref{Guttman:bound2} remain invariant\nunder scaling of rows and corresponding columns, whereas \\eqref{eq:lowerbound3} does not, hence these two cannot be compared directly.\n\\end{subequations}\n\n\\section{Correspondence between decompositions}\\label{correspondence}\n\nWe now return to the decomposition of the data matrix $X=\\hat\nX+\\tilde X$ as in \\eqref{eq:decompose} and its relation to the\ncorresponding sample covariances. The decomposition of $X$ into\n``noise-free'' and ``noisy'' components implies a corresponding\ndecomposition for the sample covariance, but in the converse\ndirection, a decomposition $ \\Sigma=\\hat\\Sigma+\\tilde\\Sigma $ leads\nto a family of compatible decompositions for $X$, which corresponds\nto the boundary of a matrix-ball. This is discussed next.\n\n\\begin{prop}\\label{prop:decomposition} Let $X\\in{\\mathbb R}^{n\\times T}$, and $\\Sigma:=XX^\\prime$. If\n\\begin{equation}\\label{eq:decompose2}\n\\Sigma=\\hat \\Sigma+\\tilde \\Sigma\n\\end{equation}\nwith $\\hat \\Sigma$, $\\tilde \\Sigma$ symmetric and non-negative definite, there exists a decomposition\n\\begin{subequations}\\label{conditions}\n\\begin{equation}\\label{eq:Xdecompose}\nX=\\hat X+\\tilde X\n\\end{equation}\nfor which\n\\begin{eqnarray}\n\\label{cond2}\n&&\\hat X \\tilde X^\\prime = 0,\\\\\\label{cond3}\n&&\\hat \\Sigma = \\hat X \\hat X^\\prime,\\\\\\label{cond4}\n&&\\tilde \\Sigma = \\tilde X\\tilde X^\\prime.\n\\end{eqnarray}\n\\end{subequations}\nFurther,\nall pairs $(\\hat X,\\,\\tilde X)$ that satisfy (\\ref{eq:Xdecompose}-\\ref{cond4})\nare of the form\n\\begin{equation}\\label{parametrization}\n\\hat X=\\hat\\Sigma \\Sigma^{-1} X+R^{1\/2}V,\\;\n\\tilde X=\\tilde\\Sigma \\Sigma^{-1} X-R^{1\/2}V,\n\\end{equation}\nwith\n\\begin{subequations}\\label{Rs}\n\\begin{eqnarray}\\label{R}\nR&:=&\\hat \\Sigma - \\hat \\Sigma \\Sigma^{-1} \\hat \\Sigma\\\\\n&=&\\tilde \\Sigma - \\tilde \\Sigma \\Sigma^{-1} \\tilde \\Sigma \\label{R2}\\\\\\nonumber\n&=&\\hat \\Sigma \\Sigma^{-1}\\tilde \\Sigma\\\\\\nonumber\n&=&\\tilde \\Sigma \\Sigma^{-1}\\hat \\Sigma,\\nonumber\n\\end{eqnarray}\n\\end{subequations}\nand $V\\in{\\mathbb R}^{n\\times T}$ such that $VV'=I$, $XV'=0$.\n\\end{prop}\n\n\n\\begin{proof} The proof relies on a standard lemma (\\cite[Theorem 2]{douglas}) which states that if\n $A\\in{\\mathbb R}^{n\\times T}$, $B\\in{\\mathbb R}^{n\\times m}$ with $m\\leq T$ such that\n$A A^\\prime = B B^\\prime,$\nthen $A=BU$ for some $U\\in{\\mathbb R}^{m\\times T}$ with $U U^\\prime =I$.\nThus, we let $A:=X$,\n\\[\nS:=\\left[\\begin{matrix}\\hat \\Sigma & 0\\\\0&\\tilde \\Sigma\\end{matrix}\\right],\n\\]\nand $B:=\\left[\\begin{matrix}I&I \\end{matrix}\\right] S^{1\/2}$,\nwhere $S^{1\/2}$ is the matrix-square root of $S$.\nIt follows that there exists a matrix $U$ as above for which $A=BU$, and therefore we can take\n\\[\n\\left[\\begin{matrix}\\hat X \\\\ \\tilde X\\end{matrix}\\right]:=S^{1\/2}U.\n\\]\nThis establishes the existence of the decomposition\n\\eqref{eq:Xdecompose}.\n\nIn order to parameterize all such pairs $(\\hat X,\\,\\tilde X)$, let\n$U_o$ be an orthogonal (square) matrix such that\n\\[XU_o=[\\Sigma^{1\/2} \\; 0].\n\\]\nThen $\\hat X U_o$ and $\\tilde X U_o$ must be of the form\n\\begin{equation}\\label{UXs}\n\\hat X U_o=: \\left[\\begin{matrix}\\hat X_1&\\Delta \\end{matrix}\\right],\\;\n\\tilde X U_o=: \\left[\\begin{matrix}\\tilde X_1& -\\Delta \\end{matrix}\\right],\n\\end{equation}\nwith $\\hat X_1$, $\\tilde X_1$ square matrices. Since\n\\[\\left[\\begin{matrix}\\hat X\\\\\\tilde X \\end{matrix}\\right]\n\\left[\\begin{matrix}\\hat X^\\prime &\\tilde X^\\prime \\end{matrix}\\right]\n=\\left[\\begin{matrix}\\hat \\Sigma& 0\\\\0&\\tilde \\Sigma \\end{matrix}\\right],\n\\]\nthen\n\\begin{subequations}\n\\begin{eqnarray}\\label{first}\n&&\\hat X_1\\hat X_1^\\prime+\\Delta\\Delta^\\prime=\\hat \\Sigma\\\\\\label{second}\n&&\\hat X_1\\tilde X_1^\\prime-\\Delta\\Delta^\\prime=0\\\\\\label{third}\n&&\\tilde X_1\\tilde X_1^\\prime+\\Delta\\Delta^\\prime=\\tilde \\Sigma.\n\\end{eqnarray}\n\\end{subequations}\nSubstituting $\\hat X_1\\tilde X_1^\\prime$ for $\\Delta\\Delta^\\prime$ into (\\ref{first}) and\nusing the fact that $\\tilde X_1=X_1-\\hat X_1$ with $X_1=\\Sigma^{1\/2}$ we obtain that\n\\begin{eqnarray*}\n&&\\hat X_1=\\hat \\Sigma\\Sigma^{-1\/2}.\n\\end{eqnarray*}\nSimilarly, using (\\ref{third}) instead, we obtain that\n\\begin{eqnarray*}\n&&\\tilde X_1=\\tilde \\Sigma\\Sigma^{-1\/2}.\n\\end{eqnarray*}\nSubstituting into (\\ref{second}), (\\ref{first}) and (\\ref{third}) we obtain the following three relations\n\\begin{eqnarray*}\n\\Delta\\Delta^\\prime &=& \\hat \\Sigma \\Sigma^{-1}\\tilde \\Sigma\\\\\n&=&\\hat \\Sigma - \\hat \\Sigma \\Sigma^{-1} \\hat \\Sigma\\\\\n&=&\\tilde \\Sigma - \\tilde \\Sigma \\Sigma^{-1} \\tilde \\Sigma.\n\\end{eqnarray*}\nSince $\\Delta\\Delta^\\prime$ and the $\\Sigma$'s are all symmetric,\n\\begin{eqnarray*}\n\\Delta\\Delta^\\prime&=&\\tilde \\Sigma \\Sigma^{-1}\\hat \\Sigma\n\\end{eqnarray*}\nas well. Thus, $\\Delta=R^{1\/2}V_1$ with $V_1V_1^\\prime=I$. The proof\nis completed by substituting the expressions for $\\hat X_1$ and\n$\\Delta$ into \\eqref{UXs}.\n\\end{proof}\n\nInterestingly,\n\\[\n{\\operatorname{rank}}(R)+{\\operatorname{rank}}(\\Sigma)={\\operatorname{rank}} \\left(\\left[\n \\begin{array}{cc}\n \\hat\\Sigma & \\hat\\Sigma \\\\\n \\hat\\Sigma & \\Sigma \\\\\n \\end{array}\n \\right] \\right)={\\operatorname{rank}} \\left(\\left[\n \\begin{array}{cc}\n \\hat\\Sigma & 0 \\\\\n 0 & \\tilde\\Sigma \\\\\n \\end{array}\n \\right] \\right)={\\operatorname{rank}}(\\hat\\Sigma)+{\\operatorname{rank}}(\\tilde\\Sigma),\n\\]\nand hence, the rank of the ``uncertainty radius'' $R$ of the corresponding $\\hat X$ and $\\tilde X$-matrix spheres is\n\\[{\\operatorname{rank}}(R)= {\\operatorname{rank}}(\\hat\\Sigma)+{\\operatorname{rank}}(\\tilde\\Sigma)-{\\operatorname{rank}}(\\Sigma).\n\\]\nIn cases where identifying $\\hat X$ from the data matrix $X$,\ndifferent criteria may be used to quantify uncertainty. One such is\nthe rank of $R$ while another is its trace, which is the variance of\nestimation error in determining $\\hat X$. This topic is considered\nnext and its relation to the Frisch decomposition highlighted.\n\n\\section{Uncertainty and worst-case estimation}\\label{sec:min-max}\nThe basic premise of the decomposition (\\ref{eq:decompose2}) is\nthat, in principle, no probabilistic description of the data is\nneeded. Thus, under the assumptions of\nProposition~\\ref{prop:decomposition}, $R$ represents a deterministic\nradius of uncertainty in interpreting the data. On the other hand,\nwhen data and noise are probabilistic in nature and represent\nsamples of jointly Gaussian random vectors ${\\mathbf x},\\;{\\hat{\\mathbf x} },\\; {\\tilde{\\mathbf x}}$ as\nin (\\ref{eq:xa} - \\ref{eq:xc}), the conditional expectation of\n${\\hat{\\mathbf x} }$ given ${\\mathbf x}$ is $E\\{{\\hat{\\mathbf x} } |{\\mathbf x}\\}=\\hat\\Sigma\\Sigma^{-1} {\\mathbf x}$,\nwhile the variance of the error\n\\begin{eqnarray*}\nE\\{({\\hat{\\mathbf x} }-\\hat\\Sigma \\Sigma^{-1}{\\mathbf x})({\\hat{\\mathbf x} }-\\hat\\Sigma \\Sigma^{-1}{\\mathbf x})^\\prime\\}&=&\\hat\\Sigma - \\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma\\\\\n&=&R\n\\end{eqnarray*}\nis the radius of the deterministic uncertainty set. Either way, it is of interest to assess how this radius depends on the decomposition of $\\Sigma$.\n\n\\subsection{Uniformly optimal decomposition}\n\nSince the decomposition of $\\Sigma$ in the Frisch problem is not\nunique, it is natural to seek a uniformly optimal choice of the\nestimate $K{\\mathbf x}$ for ${\\hat{\\mathbf x} }$ over all admissible decompositions. To\nthis end, we denote the mean-squared-error loss function\n\\begin{eqnarray}\\label{eq:LossFunction}\nL(K, \\hat \\Sigma, \\tilde\\Sigma)&:=&{\\operatorname{trace}}\\left({\\mathcal E}\\left( ({\\hat{\\mathbf x} }-K{\\mathbf x})({\\hat{\\mathbf x} }-K{\\mathbf x})^\\prime\\right)\\right)\\nonumber\\\\\n&\\;=&{\\operatorname{trace}}\\left(\\hat\\Sigma-K\\hat\\Sigma-\\hat\\Sigma K'+K(\\hat\\Sigma+\\tilde\\Sigma) K' \\right),\\label{eq:loss}\n\\end{eqnarray}\nand define\n\\begin{align*}\n{\\mathcal S}(\\Sigma):= \\{(\\hat\\Sigma, \\tilde\\Sigma) : &~\\Sigma=\\hat\\Sigma+\\tilde\\Sigma,\\; \\hat\\Sigma,\\; \\tilde\\Sigma\\geq0 \\text{~and~} \\tilde\\Sigma \\text{~is diagonal} \\}\n\\end{align*}\nas the set of all admissible pairs. Thus, a uniformly-optimal\ndecomposition of $X$ into signal plus noise relates to the following\nmin-max problem:\n\\begin{align}\\label{prob:minmax}\n\\min_{K}\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} L(K, \\hat \\Sigma, \\tilde\\Sigma).\n\\end{align}\nThe minimizer of \\eqref{prob:minmax} is the uniformly optimal\nestimator gain $K$. Analogous min-max problems, over\ndifferent uncertainty sets, have been studied in the literature\n\\cite{Eldar2004competitive}. In our setting\n\\begin{subequations}\\label{eq:concave}\n\\begin{eqnarray}\n\\min_{K}\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} L(K, \\hat \\Sigma,\\tilde\\Sigma)&\\geq&\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)}\\min_{K} L(K, \\hat \\Sigma,\\tilde\\Sigma)\\label{minmaxmaxmin}\\\\\n&=&\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} {\\operatorname{trace}}\\left(\\hat\\Sigma-\\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma\\right)\\label{eq:concave1}\\\\\n&=&\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} {\\operatorname{trace}}\\left(\\tilde\\Sigma-\\tilde\\Sigma\\Sigma^{-1}\\tilde\\Sigma\\right).\\label{eq:concave2}\n\\end{eqnarray}\n\\end{subequations}\nThe functions to maximize in \\eqref{eq:concave1} and\n\\eqref{eq:concave2} are both strictly concave in $\\hat\\Sigma$ and\n$\\tilde\\Sigma$. Therefore the maximizer is unique. Thus, we denote\n\\begin{equation}\\label{optsolution}\n(K_{\\rm opt}, \\hat\\Sigma_{\\rm opt}, \\tilde\\Sigma_{\\rm opt}) :=\\arg \\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)}\\min_{K} L(K, \\hat \\Sigma,\\tilde\\Sigma),\n\\end{equation}\nwhere, clearly, $K_{\\rm opt}=\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}$.\n\nIn general, the decomposition suggested by the uniformly optimal\nestimation problem does not lead to a singular signal covariance\n$\\hat\\Sigma$. The condition for when that happens is given next.\nInterestingly, this is expressed in terms of half the candidate\nnoise covariance utilized in obtaining one of the Guttman bounds\n(Proposition \\ref{prop:Guttman}).\n\n\\begin{prop}\\label{prop:maxmin}\nLet $\\Sigma>0$, and let\n\\begin{equation}\\label{eq:D0}\nD_0:=\\frac12 {\\operatorname{diag}}^*\\left({\\operatorname{diag}}(\\Sigma^{-1})\\right)^{-1}\n\\end{equation}\n(which is equal to $\\frac12 D_2$ defined in Proposition \\ref{prop:Guttman}).\nIf $\\Sigma-D_0\\geq0$, then\n\\begin{subequations}\\label{Solution}\n\\begin{equation}\\label{InteriorSolution}\n\\tilde\\Sigma_{\\rm opt}=D_0 \\text{~and~} \\hat\\Sigma_{\\rm opt}=\\Sigma-D_0.\n\\end{equation}\nOtherwise,\n\\begin{equation}\\label{BoundarySolution}\n\\tilde\\Sigma_{\\rm opt}\\leq D_0 \\text{~and~} \\hat\\Sigma_{\\rm opt} \\text{~is singular}.\n\\end{equation}\n\\end{subequations}\n\\end{prop}\n\n\\begin{proof}\nFrom \\eqref{eq:concave2},\n\\begin{eqnarray}\nL(K_{\\rm opt}, \\hat \\Sigma_{\\rm opt}, \\tilde \\Sigma_{\\rm opt})&=&\\max \\left\\{\\tilde \\Sigma-\\tilde \\Sigma\\Sigma^{-1}\\tilde \\Sigma ~\\mid~ \\Sigma\\geq\\tilde\\Sigma\\geq0, \\tilde\\Sigma \\text{~is diagonal} \\right\\}\\nonumber\\\\\n&\\leq& \\max \\left\\{\\tilde \\Sigma-\\tilde \\Sigma\\Sigma^{-1}\\tilde \\Sigma ~\\mid~\\tilde\\Sigma \\text{~is diagonal} \\right\\}\\label{relaxedD}\\\\\n&=&\\frac12 {\\operatorname{trace}}(D_0)\\nonumber\n\\end{eqnarray}\nwith the maximum attained for $\\tilde\\Sigma=D_0$. Then\n\\eqref{InteriorSolution} follows. In order to prove\n\\eqref{BoundarySolution}, consider the Lagrangian corresponding to\n\\eqref{eq:concave2}\n\\[\n{\\mathcal L}(\\tilde\\Sigma,\\Lambda_0, \\Lambda_1) ={\\operatorname{trace}}(\\tilde\\Sigma-\\tilde\\Sigma\\Sigma^{-1}\\tilde\\Sigma+\\Lambda_0(\\Sigma-\\tilde\\Sigma)+\\Lambda_1\\tilde\\Sigma)\n\\]\nwhere $\\Lambda_0,\\;\\Lambda_1$ are Lagrange multipliers.\nThe optimal values satisfy\n\\begin{subequations}\n\\begin{eqnarray}\n&&[I-2\\Sigma^{-1}\\tilde\\Sigma_{\\rm opt}-\\Lambda_{0}+\\Lambda_{1}]_{kk}=0, \\;\\forall\\; k=1,\\ldots, n,\\label{condition1}\\\\\n&& \\Lambda_{0}\\hat\\Sigma_{\\rm opt}=0,\\; \\Lambda_{0}\\geq0,\\label{condition2}\\\\\n&& \\Lambda_{1}\\tilde\\Sigma_{\\rm opt}=0,\\; \\Lambda_{1}\\geq0 \\text{~and is diagonal}.\\label{condition3}\n\\end{eqnarray}\n\\end{subequations}\nIf $\\Sigma- D_0\\not\\geq0$ we show that $\\hat\\Sigma_{\\rm opt}$ is\nsingular. Assume the contrary, i.e., that $\\hat\\Sigma_{\\rm opt}>0$.\nFrom \\eqref{condition2}, we see that $\\Lambda_{0}=0$, while from\n\\eqref{condition1}, $ [I-2\\Sigma^{-1}\\tilde\\Sigma_{\\rm\nopt}]_{kk}\\leq 0. $ This gives that\n\\[\n[\\tilde\\Sigma_{\\rm opt}]_{kk}\\geq \\frac{1}{2[\\Sigma^{-1}]_{kk}}= [D_0]_{kk},\n\\]\nfor all $k=1, \\ldots, n$, which contradicts the fact that $\\Sigma-D_0\\not\\geq0$. Therefore $\\hat\\Sigma_{\\rm opt}$ is singular.\nWe now assume that $\\tilde\\Sigma\\not \\leq D_0$. Then there exists $k$ such that $[\\tilde\\Sigma_{\\rm opt}]_{kk}> [D_0]_{kk}$.\nFrom \\eqref{condition3} and \\eqref{condition1}, we have that\n \\[\n [\\Lambda_{1}]_{kk}=0 \\text{~and~} [I-2\\Sigma^{-1}\\tilde\\Sigma_{\\rm opt}]_{kk}\\geq0\n \\]\nwhich contradicts the assumption that $[\\tilde\\Sigma_{\\rm opt}]_{kk}>\n[D_0]_{kk}$. Therefore $\\tilde\\Sigma_{\\rm opt}\\leq D_0$ and\n\\eqref{BoundarySolution} has been established.\n\\end{proof}\n\nWe remark that while\n\\begin{eqnarray*}\n{\\mathcal E}\\left( ({\\hat{\\mathbf x} }-K{\\mathbf x})({\\hat{\\mathbf x} }-K{\\mathbf x})^\\prime\\right)&=&\\hat\\Sigma-K\\hat\\Sigma-\\hat\\Sigma K'+K\\Sigma K'\\\\\n&=&(\\hat\\Sigma\\Sigma^{-\\frac12}-K\\Sigma^{\\frac12})(\\hat\\Sigma\\Sigma^{-\\frac12}-K\\Sigma^{\\frac12})^\\prime+\\hat\\Sigma-\\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma\n\\end{eqnarray*}\nis matrix-convex in $K$ and a unique minimum for\n$K=\\hat\\Sigma\\Sigma^{-1}$, the error covariance\n$\\hat\\Sigma-\\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma $ may not have a unique\nmaximum in the positive semi-definite sense. To see this, consider\n$\\Sigma=\\left[\n \\begin{array}{cc}\n 2 & 1 \\\\\n 1 & 2 \\\\\n \\end{array}\n \\right]\n$. In this case $D_0=\\frac{3}{4}I$, $\\hat\\Sigma_{\\rm opt}=\\left[\n \\begin{array}{cc}\n 5\/4 & 1 \\\\\n 1 & 5\/4 \\\\\n \\end{array}\n \\right]$, and\n\\begin{equation}\\label{eq:Ropt}\n\\hat\\Sigma_{\\rm opt}-\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}\\hat\\Sigma_{\\rm opt}=\\left[\n \\begin{array}{cc}\n 3\/8 & 3\/16 \\\\\n 3\/16 & 3\/8 \\\\\n \\end{array}\n \\right].\n\\end{equation}\nOn the other hand, for $\\hat\\Sigma=\\left[\n \\begin{array}{cc}\n 3\/2 & 1 \\\\\n 1 & 3\/2 \\\\\n \\end{array}\n \\right]$, then\n\\[\n\\hat\\Sigma-\\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma=\\left[\n \\begin{array}{cc}\n 1\/3 & 1\/12 \\\\\n 1\/12 & 1\/3 \\\\\n \\end{array}\n \\right]\n\\]\nwhich is neither larger nor smaller than \\eqref{eq:Ropt} in the\nsense of semi-definiteness. This is a key reason for considering\nscalar loss functions of the error covariance as in\n\\eqref{eq:loss}.\n\nNext we note that there is no gap between the min-max and max-min\nvalues in the two sides of \\eqref{minmaxmaxmin}.\n\\begin{prop}\\label{prop:minmax}\nFor $\\Sigma\\in{\\mathbf S}_{n,+}$, then\n\\begin{equation}\\label{eq:equal}\n\\min_{K}\\max_{(\\hat\\Sigma, \\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} L(K, \\hat \\Sigma, \\tilde\\Sigma)=\\max_{(\\hat\\Sigma, \\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)}\\min_{K} L(K, \\hat \\Sigma, \\tilde\\Sigma).\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nWe observe that for a fixed $K$, the function $L(K, \\hat \\Sigma,\n\\tilde\\Sigma)$ is a linear function of $(\\hat\\Sigma, \\tilde\\Sigma)$.\nFor fixed $(\\hat\\Sigma, \\tilde\\Sigma)$, the function is a convex\nfunction of $K$. Under this conditions it is standard that\n\\eqref{eq:equal} holds, see e.g. \\cite[page 281]{Boyd2004convex}.\n\\end{proof}\n\n\nWe remark that when $D_0=\\frac12\n{\\operatorname{diag}}^*\\left({\\operatorname{diag}}(\\Sigma^{-1})\\right)^{-1}$ is admissible as noise\ncovariance, i.e., $\\Sigma- D_0\\geq0$, the optimal signal covariance\nis $\\hat\\Sigma_{\\rm opt}=\\Sigma-D_0$, and the gain matrix $K_{\\rm\nopt}=\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}=I-D_0\\Sigma^{-1}$ has all\ndiagonal entries equal to $\\frac{1}{2}$. Thus, with $K_{\\rm opt}$ in\n\\eqref{eq:LossFunction} the mean-square-error loss is independent of\n$\\hat\\Sigma$ and equal to ${\\operatorname{trace}}\\left(K_{\\rm opt}\\Sigma K_{\\rm\nopt}^\\prime\\right)$ for any admissible decomposition of $\\Sigma$.\n\nWe also remark that the key condition (Proposition \\ref{prop:maxmin})\n \\begin{align*}\\label{InvariantCondition}\n& \\Sigma\\geq\\frac12 {\\operatorname{diag}}^*\\left({\\operatorname{diag}}(\\Sigma^{-1}) \\right)^{-1}\\\\\n&\\Leftrightarrow 2{\\operatorname{diag}}^*\\left({\\operatorname{diag}}(\\Sigma^{-1}) \\right)\\geq \\Sigma^{-1}\n \\end{align*}\ncan be equivalently written as $\\Sigma^{-1}\\circ (2I-{\\bf 1}{\\bf\n1}')\\geq0$, and interestingly, amounts to the positive\nsemi-definitess of a matrix formed by changing the signs of all\noff-diagonal entries of $\\Sigma^{-1}$. The set of all such matrices, $\\left\\{S\n\\mid S\\geq 0,~ S\\circ (2I-{\\bf 1}{\\bf 1}')\\geq0 \\right\\}$, is\nconvex, invariant under scaling rows and corresponding columns, and\ncontains the set of diagonally dominant matrices $\\{S \\mid S\\geq 0,~\n[S]_{ii}\\geq \\sum_{j\\neq i} |[S]_{ij}| \\text{~for all ~} i\\}$.\n\nWe conclude this section by noting that ${\\operatorname{trace}}(R_{\\rm opt})$,\n with\n \\[\nR_{\\rm opt}:=\\hat\\Sigma_{\\rm opt}-\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}\\hat\\Sigma_{\\rm opt},\n\\]\nquantifies the distance between admissible decompositions of $\\Sigma$. This is stated next.\n\n\\begin{prop}\\label{lemma:radius}\nFor $\\Sigma>0$ and any pair $(\\hat\\Sigma, \\tilde\\Sigma)\\in {\\mathcal S}(\\Sigma)$,\n\\[\n{\\operatorname{trace}}\\left( (\\hat\\Sigma-\\hat\\Sigma_{\\rm opt})\\Sigma^{-1}(\\hat\\Sigma-\\hat\\Sigma_{\\rm opt})' \\right)\\leq {\\operatorname{trace}}(R_{\\rm opt}).\n\\]\n\\end{prop}\n\\begin{proof}\nClearly\n$0\\leq{\\operatorname{trace}}(\\hat\\Sigma-\\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma)$,\nwhile from Proposition \\ref{prop:minmax},\n\\begin{eqnarray}\nL(K_{\\rm opt}, \\hat\\Sigma, \\tilde\\Sigma)\n&=& {\\operatorname{trace}}(\\hat\\Sigma-2\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}\\hat\\Sigma+\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}\\hat\\Sigma_{\\rm opt}')\\label{eqB}\\\\\n&\\leq& {\\operatorname{trace}}(R_{\\rm opt}).\\nonumber\n\\end{eqnarray}\nThus, ${\\operatorname{trace}}(\\hat\\Sigma\\Sigma^{-1}\\hat\\Sigma-2\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}\\hat\\Sigma+\\hat\\Sigma_{\\rm opt}\\Sigma^{-1}\\hat\\Sigma_{\\rm opt}')\\leq {\\operatorname{trace}}(R_{\\rm opt})$.\n\\end{proof}\n\n\\subsection{Uniformly optimal estimation and trace regularization}\\label{sec:regularized}\nA decomposition of $\\Sigma$ in accordance with the min-max estimation problem of the previous section often produces an invertible signal covariance $\\hat\\Sigma$. On the other hand, it is often the case and it is the premise of factor analysis, that $\\hat\\Sigma$ is singular of low rank and, thereby, allows identifying linear relations in the data. In this section we consider combining the mean-square-error loss function with regularization term promoting a low rank for the signal covariance $\\hat\\Sigma$ \\cite{Fazel2001}. More specifically, we consider\n\\begin{equation}\\label{prob:minmaxRank}\nJ=\\min_{K}\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} \\left(L(K,\n\\hat \\Sigma, \\tilde\\Sigma)-\\lambda\\cdot {\\operatorname{trace}}(\\hat\\Sigma)\\right),\n\\end{equation}\nfor $\\lambda\\geq0$, and properties of its solutions.\n\nAs noted in Proposition \\ref{prop:minmax} (see \\cite[page 281]{Boyd2004convex}), here too there is no gap between the min-max and the max-min, which becomes\n\\begin{subequations}\n\\begin{align}\n&\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)}\\min_{K} L(K, \\hat \\Sigma, \\tilde\\Sigma)-\\lambda\\cdot {\\operatorname{trace}}(\\hat\\Sigma)\\nonumber\\\\\n&= \\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)}\\min_{K} {\\operatorname{trace}}\\left( (1-\\lambda)\\hat\\Sigma-K\\hat\\Sigma-\\hat\\Sigma K'+K(\\hat\\Sigma+\\tilde\\Sigma)K' \\right)\\nonumber\\\\\n&=\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} {\\operatorname{trace}}\\left( (1-\\lambda)\\hat\\Sigma-\\hat\\Sigma(\\hat\\Sigma+\\tilde\\Sigma)^{-1}\\hat\\Sigma \\right) \\label{eq:Kcanceled}\\\\\n&=\\max_{(\\hat\\Sigma,\\tilde\\Sigma)\\in{\\mathcal S}(\\Sigma)} {\\operatorname{trace}}\\left( -\\lambda\\Sigma+ (1+\\lambda)\\tilde\\Sigma-\\tilde\\Sigma(\\hat\\Sigma+\\tilde\\Sigma)^{-1}\\tilde\\Sigma \\right). \\label{eq:Sigtilde}\n\\end{align}\n\\end{subequations}\nSince \\eqref{eq:Kcanceled}\nand \\eqref{eq:Sigtilde} are strictly concave functions of\n$\\hat\\Sigma$ and $\\tilde\\Sigma$, respectively, there is a unique\nset of optimal values $(K_{\\lambda, \\rm opt}, \\hat\\Sigma_{\\lambda,\\rm opt}, \\tilde\\Sigma_{\\lambda,\\rm opt})$.\n\n\\begin{prop}\nLet $\\Sigma>0$, $D_0=\\frac12 \\left({\\operatorname{diag}}^*{\\operatorname{diag}}(\\Sigma^{-1})\\right)^{-1},$\n$\\lambda_{\\rm min}$ be the smallest eigenvalue of $D_0^{-\\frac12}\\Sigma D_0^{-\\frac12}$,\nand $(K_{\\lambda, \\rm opt}, \\hat\\Sigma_{\\lambda,\\rm opt}, \\tilde\\Sigma_{\\lambda,\\rm opt})$ as above, for $\\lambda\\geq0$.\nFor any\n$\\lambda\\geq\\lambda_{\\rm min}-1$,\n$\\hat\\Sigma_{\\lambda,{\\rm opt}}$ is singular.\n\\end{prop}\n\n\\begin{proof}\nThe trace of\n$( -\\lambda\\Sigma+ (1+\\lambda)\\tilde\\Sigma-\\tilde\\Sigma\\Sigma^{-1}\\tilde\\Sigma )$ is maximal for the diagonal choice $\\tilde \\Sigma = (1+\\lambda)D_0$.\nFor any $\\lambda \\geq \\lambda_{\\rm min}-1$, $\\Sigma-(1+\\lambda) D_0$ fails to be positive semidefinite. Thus, the constraint $\\Sigma-\\tilde\\Sigma\\geq 0$ in \\eqref{eq:Sigtilde} is active and $\\hat\\Sigma_{\\lambda, {\\rm opt}}$ is singular.\n\\end{proof}\n\nNote that $\\Sigma-2D_0\\not\\geq 0$ (unless $\\Sigma$ is diagonal), and therefore $\\lambda_{\\rm min}<2$. Hence, for\n$\\lambda\\geq1$, $\\hat\\Sigma_{\\lambda, {\\rm opt}}$ is singular.\nWhen $\\lambda\\to 0$ we recover the solution in \\eqref{optsolution}, whereas for $\\lambda\\to\\infty$ we recover the solution in Proposition\n\\ref{prop:mintrace}.\n\n\n\\section{Accounting for statistical errors}\\label{statisticalerrors}\n\nFrom an applications standpoint $\\Sigma$ represents an empirical\ncovariance, estimated on the basis of a finite observation record in\n$X$. Hence \\eqref{eq:diagonal} and \\eqref{eq:orthogonality} are only\napproximately valid, as already suggested in\nSection~\\ref{sec:datastrcuture}. Thus, in order to account for\nsampling errors we can introduce a penalty for the size of\n $C:=\\hat X\\tilde X^\\prime$, conditioned so that\n \\[\n \\Sigma=\\hat\\Sigma + \\tilde\\Sigma +C +C',\n \\]\nand a penalty for the distance of $\\tilde \\Sigma$ from the set $\\{D \\mid D\\mbox{ diagonal}\\}$.\n\nAlternatively, we can use the Wasserstein 2-distance\n\\cite{olkin1982,ning2011} between the respective Gaussian\nprobability density functions, which can be written in the form of a\nsemidefinite program\n\\[\nd(\\hat\\Sigma+D, \\Sigma)=\\min_{C_1}\\left({\\operatorname{trace}}(\\Sigma+\\hat\\Sigma+D+C_1+C_1') \\mid \\left[\n\\begin{array}{cc} \\hat\\Sigma+D & C_1 \\\\\n C_1' & \\Sigma \\\\\n\\end{array}\n\\right]\\geq0 \\right).\n\\]\n\nReturning to the uncertainty radius of Section \\ref{correspondence} and the\nproblem discussed in Section \\ref{sec:min-max}, we note that the problem\n\\begin{equation}\\nonumber\n\\max\\min_{K} L(K, \\hat \\Sigma,D)\\\\\n=\\max {\\operatorname{trace}}\\left(\\hat\\Sigma-\\hat\\Sigma(\\hat\\Sigma+D)^{-1}\\hat\\Sigma\\right)\n\\end{equation}\ncan be expressed as the semidefinite program\n\\begin{equation}\\nonumber\n\\max_Q \\left\\{ {\\operatorname{trace}}\\left(\\hat\\Sigma-Q \\right)\\mid\n\\left[\n \\begin{array}{cc}\n Q & \\hat\\Sigma \\\\\n \\hat\\Sigma & \\hat\\Sigma+D \\\\\n \\end{array}\n \\right]\\geq 0\n\\right\\}.\n\\end{equation}\nThus, putting the above together, a formulation that incorporates the various tradeoffs between the dimension of the signal subspace, mean-square-error loss, and statistical errors is to maximize\n\\begin{equation}\\label{eq:maxmin2}\n{\\operatorname{trace}}(\\hat\\Sigma -Q) - \\lambda_1\\, {\\operatorname{trace}}(\\hat\\Sigma) -\\lambda_2\\, {\\operatorname{trace}}(\\hat\\Sigma + D - C_1-C_1^\\prime)\n\\end{equation}\nsubject to\n\\begin{eqnarray*}\n\\left[\n \\begin{array}{cc}\n Q & \\hat\\Sigma \\\\\n \\hat\\Sigma & \\hat\\Sigma+D \\\\\n \\end{array}\n \\right]\\geq 0,\\;\\left[\n\\begin{array}{cc} \\hat\\Sigma+D & C_1 \\\\\n C_1' & \\Sigma \\\\\n\\end{array}\n\\right]\\geq0, \\mbox{ with }D\\geq 0 \\mbox{ and diagonal.}\n\\end{eqnarray*}\nThe value of the parameters $\\lambda_1$, $\\lambda_2$ dictate the relative importance that we place on the various terms and determine the tradeoffs in the problem.\n\nWe conclude with an example to highlight the potential and limitations of the techniques.\nWe generate data $X$ in the form\n\\[\nX=FV+\\tilde X\n\\]\nwhere $F\\in{\\mathbb R}^{n\\times r}$, $V\\in {\\mathbb R}^{r\\times T}$, and $\\tilde X\\in {\\mathbb R}^{n\\times T}$ with $n=50$, $r=10$, $T=100$. The elements of $F$ and $V$ are generated from normal distributions with mean zero and unit covariance. The columns of $\\tilde X$ are generated from a normal distribution with mean zero and diagonal covariance, itself having (diagonal) entries which are uniformly drawn from interval $[1, 10]$. The matrix $\\Sigma=XX'$ is subsequently scaled so that ${\\operatorname{trace}}(\\Sigma)=1$.\nWe determine\n\\[\n(\\hat\\Sigma,Q,D)={\\rm arg}\\max \\left\\{ {\\operatorname{trace}}(\\hat\\Sigma-Q)-\\lambda\\cdot {\\operatorname{trace}}(\\hat\\Sigma)\\right\\}\n\\]\nsubject to\n\\begin{eqnarray*}\n\\left[\\begin{matrix}Q &\\hat\\Sigma\\\\ \\hat\\Sigma & \\hat\\Sigma+D \\end{matrix} \\right]\\geq0, ~d(\\hat\\Sigma+D, \\Sigma)\\leq \\epsilon, \\text{~with~} \\hat\\Sigma, D\\geq0 \\text{~and~} D \\text{~diagonal},\n\\end{eqnarray*}\nand tabulate below a typical set of values for the rank of $\\hat\\Sigma$ (Table 1)\nas a function of $\\lambda$ and $\\epsilon$. We observe a ``plateau'' where the rank stabilizes at $10$ over a small range of values for $\\epsilon$ and $\\lambda$. Naturally, such a plateau may be taken as an indication of a suitable range of parameters.\nAlthough the current setting where a small perturbation in the empirical covariance $\\Sigma$ is allowed, the bounds for the rank\nin \\eqref{Guttman:bound2} and \\eqref{eq:lowerbound3} are still pertinent. In fact, for this example, in $7\/10$ instances where the ${\\operatorname{rank}}(\\hat\\Sigma)=10$ the bound in \\eqref{Guttman:bound2} (computed based on the perturbed covariance $\\hat\\Sigma+D$) has been tight and it thus a valid certificate. For the same range of parameters, the bound in \\eqref{eq:lowerbound3} has been lower than the actual rank of $\\hat\\Sigma$. In general, the bounds in \\eqref{Guttman:bound2} and \\eqref{eq:lowerbound3} are not comparable as either one may be tighter than the other.\\\\\n\\begin{center}\n\\begin{minipage}[]{.6\\textwidth}\n\\begin{tabular}[t]{|c||c|c|c|c|c|c|c|}\n \\hline\n \t\\backslashbox[.5cm]{$\\lambda$}{$\\epsilon$}\t& $0$& $0.08$ & $0.10$& $0.12$& $0.14$ & $0.16$\\\\ \\hline\\hline\n $1$ & 46 & 26 & 24 & 23 & 22 & 22 \\\\ \\hline\n $5$ & 46 & 17 & 14 & 10 & 10 & 9 \\\\ \\hline\n $10$ & 45 & 16 & 12 & 10 & 10 & 8 \\\\ \\hline\n $20$ & 45 & 15 & 12 & 10 & 10 & 8 \\\\ \\hline\n $50$ & 45 & 15 & 12 & 10 & 10 & 8 \\\\ \\hline\n $100$& 45 & 15 & 11 & 10 & 10 & 8 \\\\ \\hline\n\\end{tabular}\\\\[.05in]\n{Table 1: ${\\operatorname{rank}}(\\hat\\Sigma)$ as a function of $\\lambda$ and $\\epsilon$}\\\\[.05in]\n\\end{minipage}\n\\end{center}\n\n\\section{Conclusions} \\label{sec:conclusion}\n\nIn this paper we considered the general problem of identifying\nlinear relations among variables based on noisy measurements --a\nclassical problem of major importance in the current era of ``Big\nData.'' Novel numerical techniques and increasingly powerful\ncomputers have made it possible to successfully treat a number of\nkey issues in this topic in a unified manner. Thus, the goal of the\npaper has been to present and develop in a unified manner key ideas\nof the theory of noise-in-variables linear modeling.\n\nMore specifically, we considered two different viewpoints for the\nlinear model problem under the assumption of independent noise. From\nan estimation viewpoint, we quantify the uncertainty in estimating\n``noise-free'' data based on noise-in-variables linear models. We\nproposed a min-max estimation problem which aims at a uniformly\noptimal estimator --the solution can be obtained using convex\noptimization. From the modeling viewpoint, we also derived several\nclassical results for the Frisch problem that asks for the maximum\nnumber of simultaneous linear relations. Our results provide a\ngeometric insight to the Reiers\\o l theorem, a\n generalization to complex-valued matrices, an\niterative re-weighting trace minimization scheme for obtaining\nsolutions of low rank along with a characterization of fixed points,\nand certain computational tractable lower bounds to serve as\ncertificates for identifying the minimum rank. Finally, we consider\nregularized min-max estimation problems which integrate various\nobjectives (low-rank, minimal worst-case estimation error) and\nexplain their effectiveness in a numerical example.\n\nIn recent years, techniques such as the ones presented in this work\nare becoming increasingly important in subjects where one has very\nlarge noisy datasets including medical imaging, genomics\/proteomics,\nand finance. It is our hope that the material we presented in this\npaper will be used in these topics. It must be noted that throughout\nthe present work we emphasized independence of noise in individual\nvariables. Evidently, more general and versatile structures for the\nnoise statistics can be treated in a similar manner, and these may\nbecome important when dealing with large databases.\n\nA very important topic for future research is that of dealing with\nstatistical errors in estimating empirical statistics. It is common\nto quantify distances using standard matrix norms --as is done in\nthe present paper as well. Alternative distance measures such as the\nWasserstein distance mentioned in Section~\\ref{statisticalerrors}\nand others (see e.g., \\cite{ning2011}) may become increasingly\nimportant in quantifying statistical uncertainty.\n\nFinally, we raise the question of the asymptotic performance of certificates such as those presented in Section \\ref{sec:CertifMinRank}. It is important to know how the tightness of the certificate to the minimal rank of linear models relates to the size of the problem.\n\n\n\\section*{Acknowledgments}\n\nThis work was supported in part by grants from NSF, NIH, AFOSR, ONR,\nand MDA. This work is part of the National Alliance for Medical\nImage Computing (NA-MIC), funded by the National Institutes of\nHealth through the NIH Roadmap for Medical Research, Grant U54\nEB005149. Information on the National Centers for Biomedical\nComputing can be obtained from\nhttp:\/\/nihroadmap.nih.gov \/bioinformatics. Finally, this project was\nsupported by grants from the National Center for Research Resources\n(P41-RR-013218) and the National Institute of Biomedical Imaging\nand Bioengineering (P41-EB-015902) of the National Institutes of\nHealth.\n\n\n\\bibliographystyle{siam}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nObscuration of the nuclear \nemission in type~II AGN allows the study of soft X-ray spectral \ncomponents, which are normally outshone by the direct component \nin type~I unobscured objects. It \nhas been well known since the early day of X-ray spectroscopy that \nexcess emission above the extrapolation of the absorbed nuclear \nradiation is present in almost all bright Seyfert~2s (Turner et al. \n1997). This excess appears smooth when measured with instruments with moderate\nenergy resolutions such as CCD. \nHowever, high-resolution (grating) measurements with {\\it Chandra} and \nXMM-Newton revealed that this excess is generally due to a blending of strong \nrecombination lines from He- and H-like transitions of elements from \nCarbon to Nitrogen (Sako et al. 2000, Sambruna et al. 2001, Kinkhabwala \net al. 2002, Armentrout et al. 2007). X-ray spectral diagnostics \n(Kinkhabwala et al. 2002, Guainazzi \\& Bianchi 2006) and a close \nmorphological coincidence between the soft X-rays and the [OIII] in \nExtended Narrow Line Regions (ENLR; Bianchi et al. 2006, Bianchi et al. 2010) strongly \nindicate that the gas is photoionised by the AGN, with an important role \nplayed by resonant scattering. \n\n\nIn this context, NGC~5252 represents an extraordinary laboratory to study the \nfeedback between the AGN output and circumnuclear gas on kpc scale, thanks to \nits spectacular ionisation cones (Tadhunter \\& Tsvetanov 1989).\n\nNGC 5252 is classified as Seyfert 1.9 \n(\\cite{ost}) S0 (\\cite{dev}) nearby (z=0.023,) galaxy (N$_{H,\n Gal}$=2.14$\\times$10$^{+20}$ cm$^{-2}$, Dickey \\& Lockman, 1990). Small\nradio jets (r$\\sim$4'') have\nbeen detected and found to be aligned with the ionisation cones\n(\\cite{wilson94}). Nonetheless, the host galaxy luminosity\n(M$_{R}$$\\sim$-22. \\cite{capetti}), \nmass (M$_{bulge}$$\\sim$2.4$\\times$10$^{11}$M$_{\\sun}$, \\cite{marc}) and the\nmass of the central super-massive black-hole\n(M$_{BH}$$\\sim$10$^{9}$M$_{\\sun}$, \\cite{capetti}) are more typical of \nquasar than Seyfert galaxies. These pieces of evidence led\n\\cite{capetti} to speculate\nthat NGC 5252 is most probably to be considered a QSO relic. This view is in\nagreement with \"downsizing\" scenarios about the evolution of super-massive\nblack-hole (SMBH) in cosmic\ntimes. Accordingly with these scenarios, most massive SMBHs formed and evolved\nearlier than lower mass ones. \n\n\n\n\nIonisation cones are one of the strongest argument in favour of the Seyfert \nunification scenarios (Antonucci 1993). For this reason, NGC~5252 is also an \nimportant laboratory to test AGN geometrical models. From a diferent point of\nview, \nAGN activity has been recognized, since a \nwhile as a key component of the SMBH host galaxy co-evolution and AGN\nfeedback is likely to self-regulate or be responsible of the observed \nproperties (Menci et al. 2004). \nThe very existence of ionization cones witness that feedback\/winds\n are or were active and thus these sources are ideal laboratories\n for feedback.\n\n\nX-ray measurements allows to directly link the \nproperties of the gas emitting optical lines with the intrinsic AGN \npower, which in type~II AGN can be truly measured only at energies \nlarger than the soft photoelectric cut-off due to the AGN obscuring \nmatter. Furthermore, the morphological coincidence between X-rays and \noptical emission in ENLR (Bianchi et al. 2006) points to a fundamental \nphysical link between the two wavebands. They need to be studied \nsimultaneously in order to derive the correct energy budget in the \nionisation cones. Prompted by these motivations, we have performed deep \nX-ray observations of NGC~5252 at the highest spatial and spectral \nresolution currently available with {\\it Chandra} and XMM-Newton. The \nresults of these observations are the subject of this paper.\n\n\n\\section{The nuclear spectrum}\n\nNGC5252 was observed by XMM-Newton on 2003, July 18th, \nwith the EPIC CCDs (MOS and pn in full window, see Tab.~\\ref{tab11}) as the prime instrument\nfor a total duration of $\\sim$67 ks. The\n{\\it Observation Data Files} (ODFs) were reduced and analysed using the \nlatest Science Analysis System (SAS) software package \n(\\cite{gabriel03}) with associated latest \ncalibration files. We used the {\\tt epproc} and {\\tt emproc} tasks to \ngenerate event files and remove dead and hot pixels. \nSeveral time intervals with a high background rate were \nidentified in the\nsingle events light curve at energy $>$10 keV and were \nremoved, yielding a net exposure of $\\sim$50 ks for the MOS\nand $\\sim$38 ks for the pn. Pile-up is negligible in this source,\naccording to the {\\tt epatplot} SAS task outcome.\nPatterns $\\leq$12 and $\\leq$4 were considered for MOS and \npn, respectively. Source counts were extracted from a circular region\nwith radius 50$\\arcsec$, thus encompassing a large\nfraction of the optically-defined galaxy. Background was estimated using both\nblank-sky files and locally from a offset source-free region.\nLight curves in the soft (0.5-2 keV) and hard (2-10 keV) energy bands \nwere first investigated. We found no significant flux nor spectral \nvariations, thus considered the time averaged spectrum.\n\nThe best-fit spectrum is shown in Fig.~\\ref{fig1mc}.\n \\begin{figure}\n \\centering\n \\includegraphics[angle=-90,width=8.0cm]{spe_cont.ps}\n\\vspace{-2.0mm} \\includegraphics[angle=90,width=8.0cm]{0152940101_RGS.ps}\n \\caption{{\\it Upper panel}: XMM-Newton EPIC spectrum extracted from a\n 50$\\arcsec$ region around the NGC~5252 nucleus. For clarity, only pn\n data are presented. {\\it Lower panel:} XMM-Newton RGS spectrum of\n NGC~5252 }\n \\label{fig1mc}\n \\end{figure}\nIf the nuclear emission is modeled with a simple absorbed power-law we\nobtain an extremely flat photon index [$\\Gamma$=1.05$\\pm$0.10 and\n$N_{\\rm H}$=(2.2$\\pm$0.1)$\\times 10^{22}$~cm$^{-2}$, the reported errors\n are, here and hereafter, at 90\\% confidence level], \nplus a soft component emerging at energies below E$\\sim$1 keV. It can be\nparametrised with a scattered power-law with a steep photon index $\\Gamma$=3.0$\\pm$0.2.\nThere is also evidence for\none (or few) soft emission lines in addition to the soft\ncontinuum, a Fe K$_{\\alpha}$ line with \nE$=$6.44$\\pm$0.05 keV and EW=50$\\pm$25 eV, and an absorption edge \nat E=7.0$\\pm$0.1 keV and optical depth $\\tau$=0.31$\\pm$0.05.\n A significantly better fit \nis obtained in case of a power-law plus a thermal\n component, namely $mekal$ in $Xspec$ (\\cite{mewe85}), \n is used to model the soft X-ray band\nof NGC 5252 ($\\chi^{2}$\/d.o.f.=1436\/1410 in the first case while\n$\\chi^{2}$\/d.o.f.=1329\/1410 with $mekal$). \nIn this case, the temperature of the plasma\nis $kT$$\\sim$0.17 keV. \nThe investigation on the true nature of\nthis soft X-ray component will be the subject of the next Sections. \n Here it is important to note that, whatever the fitting of the data below $\\sim$1 keV, the best-fit\nmodel for the nuclear emission of NGC 5252 above $\\sim$1 keV\nis typical in shape, but flatter ($\\Gamma$$\\sim$1.4-1.5) than normally\nfound from Seyfert 2 galaxies ($\\Gamma$$=$1.5-2.5, Turner \\& Pounds 1989; \\cite{turner97,risaliti02,cappi06}; Dadina 2008). It is however \nconsistent with the previous ASCA measurement ($\\Gamma$$\\sim$1-1.5,\n\\cite{cappi96}), confirming the need in this source for a more complex \nabsorption \n(either multiple, ionised or both) in order to recover a steeper \ncanonical photon index.\nWith the above model we measure, for the soft (0.5-2 keV) component, a flux of \n3.5 $\\times$10$^{-13}$erg cm$^{-2}$s$^{-1}$ corresponding to a luminosity of \n4.1$\\times$10$^{41}$ erg\/s and, for the hard (2-10 keV) component, a flux of \n8.9 $\\times$10$^{-12}$erg cm$^{-2}$s$^{-1}$ corresponding to a (unabsorbed) \nluminosity of 1.2$\\times$10$^{43}$ erg\/s. This is consistent, within a few tens\npercent, with previous ASCA and BeppoSAX values (Cappi et al. 1996; Dadina\n2007). It is worth noting here that,\nfrom IR diagnostics, we expect that star forming activity should contribute to\nless than $\\sim$1\\% to the total soft X-ray emission (Cappi et al. 1996). \n\n\n\\section{High-resolution spectroscopy of the AGN environment}\n \\begin{figure*}\n \\centering\n \\includegraphics[width=8cm,angle=-90]{fig2_1.ps}\n \\caption{$Chandra$ ACIS-S images of NGC~5252 in the 0.1--1~keV (soft;\n \t\t{\\it left panel}), and 1--10~keV (hard; {\\it right\n\t\tpanel}) Images were smoothed with a\n\t\t$\\sigma = 1.25$~pixels (yielding an angular resolution of\n $\\simeq$1\" in the images)\n\t\twavelet for illustration purposes. The {\\it\n\t\tthin solid lines} represent\n\t\t9 linearly spaced contours in the range 2 to 20 counts per\n\t\tpixel. The {\\it thick dot-dashed line} indicate the position\n\t\tof the out-of-time events readout streak, removed before\n\t\tthe generation of the image. The {\\it dashed} lines represents\n\t\tthe regions, whence the spectra\n\t\tof the South and North diffuse Spots were\n\t\textracted. The {\\it solid circles} represent the regions,\n\t\twhence the spectra of the nucleus (the brightest and\n central spot) and of the South-East\n\t\tNuclear Source were extracted.\n }\n \\label{fig2}\n \\end{figure*}\n\n\nThe Reflection Grating Spectrometer (RGS; \\cite{derherder01})\non board XMM-Newton\nobserved NGC~5252 simultaneously to the EPIC cameras (Tab.~\\ref{tab11}, Fig.~\\ref{fig1mc}). It\nproduces high-resolution (first order resolution 600-1700~km~s$^{-1}$)\nspectra\nin the 6--35~\\AA\\ (0.35--2~keV) range. Its 2.5$\\arcmin$ diameter\nslit fully encompasses the ionisation cones and the host galaxy.\nThe RGS spectrum represents therefore only the average conditions of\nthe soft X-ray emitting gas across the nucleus and the cone.\n\n\\begin{footnotesize} \n\\begin{table}\n\\caption{Main characteristics of the X-ray observations presented here.} \n\\label{tab11}\n\\centering \n\\begin{tabular}{l c c l c c } \n\\hline\n&&&&&\\\\\nInstr.&Exp.&CR & Instr.&Exp.&CR \\\\\n&&&&&\\\\\n&ks&c\/s&&ks&c\/s\\\\\n&&&&&\\\\\n\\hline\\hline \n&&&&&\\\\\nEpn&38&1.10$^{a}$& RGS1&63& 0.08$^{b}$\\\\\n&&&&&\\\\\nEMOS1&49&0.37$^{a}$ &RGS2&63& 0.08$^{b}$\\\\\n&&&&&\\\\\nEMOS2&50&0.37$^{a}$&ACIS-S&60&0.32$^{a}$\\\\\n&&&&&\\\\\n\\hline\n\\end{tabular}\n\n\n$^{a}$ count-rate in the 0.2-10 keV band; $^{b}$ count-rate in the 0.4-1.2 keV\n\\end{table}\n\\end{footnotesize}\n\n\nRGS data were reduced starting from the {\\it Observation Data Files}\nwith SASv6.5 (\\cite{gabriel03}), and using the latest calibration files. \nThe SAS meta-task {\\tt rgsproc} was used\nto generate source and background spectra, assuming as a reference\ncoordinate coincident with the optical nucleus of NGC~5252. Background\nspectra were generated using both blank field maps - accumulated across\nthe whole mission - and a ``local'' background accumulated during the\nobservation. The former, based on a model of the estimated background on\nthe basis of the count rates detected in the most external of the camera\nCCDs, overestimates the intrinsic background level during the\nobservation. We have therefore employed the ``local'' background hereafter.\nA correction \nfactor to the count background spectrum has been applied to take into \naccount the size of the extraction region, which corresponds to the area \nof the RGS active CCDs outside the 98\\% percentage point of the line \nspread function in the cross-dispersion direction.\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8cm]{fig3_1.ps}\n \\caption{Radial profile ({\\it filled circles}) of the ACIS-S hard band\n\t\timage. The {\\it solid line}\n\t\trepresents the PSF for a source with the same\n\t\thard X-ray spectral energy distribution as the NGC~5252\n\t\t``nucleus normalized'' to its on-axis peak flux. When not visible,\n the error bars are within the filled circles.\n }\n \\label{fig3}\n \\end{figure}\n\n\nWe simultaneously fit the spectra of the two cameras \nfollowing the procedure outlined in Guainazzi \\& Bianchi\n(2006)\\footnote{This paper discusses a sample of 69 RGS spectra of\ntype 1.5, 1.8, 1.9 and 2 Seyfert galaxies. \nThe observation of NGC~5252 discussed in this paper belongs to this sample as well.}\nwho have performed local spectral fits around\neach of the $\\simeq$40 emission lines detected in the archetypal\nobscured Seyfert NGC~1068 (\\cite{kin02}). In these fits, both the\nbackground level and the continuum have been assumed as independent power-law\ncomponents, with photon index $\\Gamma$ set equal to 1. \nIt is worth noting that the adopted \nvalue of the power law index, here equal to the photon index of the\ncontinuum of the primary emission, does not affect the results signifcantly, given the very\nlimited band of these fits. Different choices for the \ncontinuum spectral index yield indistinguishable results.\nEach emission line has been modeled with an unresolved Gaussian profile \nfixed\nto be at the expected energies (leaving the intrinsic\nwidth of the profile free yields a negligible improvement in the quality of\nthe fit). We detect three lines (see lower panel of Fig. 1) at a confidence level larger than\n90\\% [$\\Delta \\chi^{2} = $ 10.5, 24.0, 10.8 for CV, OVII and OVIII \nlines, respectively, for one interesting parameter (Tab.~\\ref{tab1})].\n\\begin{table}\n\\caption{List of emission lines detected in the RGS spectrum of NGC~5252.\n$E_c$ is the centroid line energy; $L$ is the intrinsic line luminosity:\n$\\Delta v$ is the difference between the measured line centroid energy\nand the laboratory energy. Only statistical error on this measurements\nare quoted. $v_{{\\rm sys}}$ is the systematic error on\n$\\Delta v$ due to residual uncertainties in the RGS aspect solution\n($\\simeq$8~m\\AA)} \n\\begin{footnotesize} \n\\label{tab1} \n\\centering \n\\begin{tabular}{l c c c c} \n\\hline\\hline \nIdentification & $E_c$ & $L$ & $\\Delta v$ & $v_{sys}$ \\\\\n& (eV) & (10$^{40}$~erg~s$^{-1}$) & (km~s$^{-1}$) & (km~s$^{-1}$) \\\\\n\\hline \nC{\\sc vi} Ly-${\\alpha}$ & $367 \\pm 5$ & $3 \\pm^{11}_{2}$ & $800 \\pm 400$ & 70 \\\\\nO{\\sc vii} He-${\\alpha}$ (f) & $560.4 \\pm 0.2$ & $2.0 \\pm^{0.8}_{0.9}$ & $-370 \\pm 110$ & 110 \\\\\nO{\\sc vii} He-${\\alpha}$ (i) & $E_c (f) + 7.7$ & $<$0.9 & & \\\\\nO{\\sc vii} He-${\\alpha}$ (r) & $E_c (f) + 13.0$ & $<$1.4 & & \\\\\nO{\\sc viii} Ly-${\\alpha}$ & $654.0 \\pm^{0.7}_{1.2}$ & $0.6 \\pm 0.4$ & $300 \\pm^{300}_{600}$ & 130 \\\\\n\\hline \n\\end{tabular}\n\\end{footnotesize}\n\\end{table}\nNone of them is a Radiative Recombination Continuum (RRC). A (admittedly\nloose) constrain on the width of the Gaussian profile can be obtained on\nthe O{\\sc vii} He-$\\alpha$ triplet only: $\\sigma < 4400$~km~s$^{-1}$\n(8.2~eV).\n \\begin{figure*}\n \\centering\n \\hbox{\n \\includegraphics[width=6cm,angle=-90]{fig4a_1.ps}\n \\hspace{1.0cm}\n \\includegraphics[width=6cm,angle=-90]{fig4b_1.ps}\n }\n \\hbox{\n \\includegraphics[width=6cm,angle=-90]{fig4c_1.ps}\n \\hspace{1.0cm}\n \\includegraphics[width=6cm,angle=-90]{fig4d_1.ps}\n }\n \\caption{Spectra ({\\it upper panels}) and residuals in units of\n\t\tstandard deviations ({\\it lower panels}) for the four\n\t\tregions of NGC~5252 defined in Fig.~\\ref{fig2}.\n }\n \\label{fig4}\n \\end{figure*}\nDiagnostic parameters involving the intensity of the O{\\sc vii} He-${\\alpha}$\ntriplets can be in principle used to pinpoint the physical process responsible\nfor the bulk of the X-ray emission in high-resolution spectra. The detection\nof the forbidden ($f$) component only allows to set lower limits\non the standard triplet diagnostics (\\cite{gabriel69,porquet00}):\n$R > 1.1$, $G > 0.7$ (where R is the ratio between forbidden and\nintercobination lines and depends on the electron density, while G is the\nratio between intercombination plus forbidden lines and the resonance line, \\cite{gabriel69,porquet00}). \n\nThese limits, although fully consistent with\nphotoionised plasmas, do not rule out collisional ionisation. Guainazzi \\&\nBianchi (2007) proposed a criterion to discriminate, {\\it on a statistical\nbasis}, between AGN- and starburst-powered sources based on the location\nof the source in an empirical observable plane: integrated luminosity of the\nHe- and H-like Oxygen lines, $L_O$, against the intensity\nratio $\\eta$ between the $f$ and the\nO{\\sc viii} Ly-$\\alpha$. In NGC~5252\n$\\eta = 2.3 \\pm 0.4$, and\n$L_0 \\sim 3 \\times 10^{40}$~erg~s$^{-1}$. These values put NGC~5252 in\nthe plane locus preferentially occupied by photoionised (AGN) sources\n(\\cite{guainazzi09}).\nWe estimated also the flux density associated to the continuum, using a\nline-free energy range between 586 and 606~eV: $\\nu L_{\\nu}|_{0.6 \\ keV} =\n(7.2 \\pm^{1.7}_{2.9}) \\times 10^{40}$~erg~s$^{-1}$.\n\n\\section{X-ray imaging of the ionisation cones}\n\n{\\it Chandra} observed NGC~5252 on August 11, 2003 with the ACIS-S\ndetector in standard VFAINT configuration. Data reduction was performed\nwith CIAO version 3.3 and associated CALDB version 3.2. ``Level~1''\nevents were corrected for bad pixels, gain spatial dependency,\nand charge transfer inefficiency via {\\tt acis\\_process\\_events}.\n\nAlthough the correction for read-out streaks was applied\nas well, some out-of-time events remain in the final cleaned event\nlist, and were removed by applying a 2 pixels ($\\simeq$1$\\arcsec$)-wide\ntilted rectangular box around the streak.\n\nACIS-S images in the\n$\\sim$2$\\arcmin$ around the optical core of\nNGC~5252 are shown in Fig.~\\ref{fig2} in the 0.2--1~keV and\nin the 1--10~keV energy bands. The soft band clearly shows extended emission\nin the North-South direction on both sides of the nucleus. On the\ncontrary, the hard band image is point-like. We extracted a radial\nprofile of the latter, and compared it with the expected instrumental Point\nSpread Function (PSF) of a source\nwith the same spectral energy distribution as the\nNGC~5252 nucleus. The two profiles are perfectly consistent\nup to 30$\\arcsec$ off-axis (see Fig.~\\ref{fig3}).\n\nIn order to characterize the spectral behavior of the diffuse emission,\nwe have extracted spectra from four regions, identified in the soft image\n(Fig.~\\ref{fig2}): the nucleus (N), a S-E source about 3.2$\\arcsec$ from the\nnucleus (SENS), and the South (SS) and North (NS) diffuse Spots. Background\nspectra were generated from a large circle 57$\\arcsec$ wide around the\ngalaxy core, once a $21$$\\arcsec$ inner circle, as well as\n5$\\arcsec$ circles around each serendipitous point sources were removed. Alternative\nchoices of the background regions do not substantially change\nthe results presented in this section. Source spectra were rebinned\nin order to over-sample the intrinsic instrumental energy resolution by\na factor $\\ge$3, and to have at least 25 background-subtracted counts\nin each spectral bin. The latter criterion\nensures the applicability of the $\\chi^2$ statistics.\n\nFor all spectra, we have employed a baseline model that include \na thermal emission component from collisionally excited plasma ({\\tt mekal} in\n{\\sc Xspec}; \\cite{mewe85}). \nThis choice was done for simplicity and the only information\nobtained using {\\tt mekal} is the flux of the thermal component. \nThis is particularly true for the SS and NS regions where the complexity of \nthe {\\tt mekal} model is well above the quality of the data. \nMoreover, a photoelectrically-absorbed power-law was\nalways included in the data. \nThe physical meaning of the latter is different\ndepending on the region where the spectrum was extracted. For the nuclear\nregion, the non-thermal component represents the contribution of the active\nnucleus; for the other regions, the integrated contribution of hard galactic\nsources such as, for example, X-ray binaries, cataclysmic variables or \nsupernova remnants. We therefore\nrefrain from attributing a physical meaning to the power-law spectral\nindeces in the latter case.\nThe spectra and corresponding best-fits are shown in Fig.~\\ref{fig4}.\nA summary of the spectral results is presented\nin Tab.~\\ref{tab2}. The {\\it Chandra} data confirm that the nuclear spectrum\n\\begin{table*}\n\\label{tab2}\n\\caption{ACIS-S best-fit parameters and results for the spatially-resolved regions\nof NGC~5252. $E_c$ and $EW$ are the centroid energy and the Equivalent Width\nof a Gaussian profile at the energies of K$_{\\alpha}$ fluorescence for\nneutral or mildly ionised iron. Fluxes, ($F$), are in the observed frame.\nLuminosities, ($L$), are in the source frame, and are corrected for absorption.}\n\\begin{tabular}{lcccccccccc} \\hline \\hline\nRegion & $N_H$ & $\\Gamma$ & $kT^a$ & $E_c$ & $EW$ & $F_{0.5-2 keV}$$^b$ &\n$F_{2-10 keV}$$^b$ & $L_{0.5-2 keV}$$^c$ & $L_{2-10 keV}$$^c$ & $\\chi^2\/\\nu$ \\\\\n& $(10^{22}$~cm$^{-2}$) & & (eV) & (keV) & (eV) & & & & & \\\\ \\hline\nNucleus & $2.32 \\pm^{0.13}_{0.15}$ & $1.00 \\pm^{0.08}_{0.06}$ & $140\\pm^{60}_{40}$ & $6.37 \\pm^{0.06}_{0.05}$ & $50 \\pm 20$ & $2.66\\pm^{0.11}_{0.28}$ & $103\\pm^2_3$ & $0.82\\pm^{0.03}_{0.09}$ & $1130 \\pm^{20}_{30}$ & 176.9\/127 \\\\\nSENS & $0.47\\pm^{0.16}_{0.31}$ & $0.8 \\pm 0.4$ & $<140$ & ... & ... & $0.06 \\pm 0.05$ & $0.34 \\pm^{0.08}_{0.14}$ & $50 \\pm 40$ & $4.1 \\pm^{1.0}_{1.7}$ & 3.0\/5 \\\\\nNS & $\\equiv$$N_{H,Gal}$ & $-0.9\\pm^{0.5}_{0.6}$ & $240 \\pm^{40}_{30}$ & ... & ... & $0.066\\pm^{0.011}_{0.015}$ & $1.3 \\pm 0.3$ & $0.89 \\pm^{0.15}_{0.20}$ & $15 \\pm 3$ & 1.0\/3 \\\\\nSS & $\\equiv$$N_{H,Gal}$ & $0.0 \\pm 0.4$ & $110\\pm^{13}_{21}$ & ... & ... & $0.122 \\pm^{0.017}_{0.023}$ & $0.70 \\pm^{0.18}_{0.53}$ & $1.7 \\pm^{0.2}_{0.3}$ & $8 \\pm^2_6$ & 5.1\/7 \\\\\n& & & $510\\pm^{170}_{230}$ & & & & & & & \\\\ \\hline\n\\end{tabular}\n\n\\noindent\n$^a$derived using $mekal$ to fit the spatially resolved data\n\n\\noindent\n$^b$in units of $10^{-13}$~erg~s$^{-1}$~cm$^{-2}$\n\n\\noindent\n$^c$in units of $10^{40}$~erg~s$^{-1}$\n\n\\end{table*}\nis remarkably flat, and is\nseen through a substantial column density ($N_H \\simeq\n2.26 \\times 10^{22}$~cm$^{-2}$).\n\nThe background subtraction for the spectrum of SENS could be contaminated\nby the spilling of the nuclear emission.\nThe encircled energy fraction at a distance equal to that between\nsources N and SENS is $\\simeq$97.5\\%. However,\nsubtracting a properly rescaled nuclear spectrum to the SENS spectrum yields\nnegative counts above 2~keV. In order to have an independent estimate of the\nspectral behavior of SENS, we have extracted images\n10$\\arcsec$ around the nuclear region in narrow energy bands\n(Fig.~\\ref{fig5}): 200-400~eV,\n \\begin{figure}\n \\centering\n \\includegraphics[width=8.5cm]{fig5_1.ps}\n \\caption{Narrow-band ACIS-S images in the 10$\\arcsec$ around the\n\t\tNGC~5252 core, normalized to the peak nuclear emission.\n\t\tImages are smoothed with a 5 pixel Gaussian kernel.\n\t\tThe {\\it solid lines} in the {\\it upper left} panel\n\t\trepresent a contour plot of the 0.2--1~keV \n\t\timage, assuming the same smoothing criterion.\n }\n \\label{fig5}\n \\end{figure}\n500-600~eV, 600-700~eV and 700-1000~eV. In a line-dominated plasma, the\nabove energy ranges correspond to bands dominated by C{\\sc vi}\nand C{\\sc v} K$_{\\alpha}$, O{\\sc vii} He-$\\alpha$, O{\\sc viii} Ly-$\\alpha$,\nand Fe-L transitions, respectively.\nEach image was normalized to the peak of the\nnuclear emission in that energy band. The soft X-ray\nSENS spectrum is comparatively dominated by Oxygen transitions, with little\ncontribution in either the Carbon or the Iron band.\n\n\n\\section{Comparing soft X-ray and [OIII] morphologies}\n\nNGC 5252 was observed in the [OIII] band with the WFPC2 on-board HST on\n1995, July 23, using the linear ramp filter FR533N. The data were downloaded\nfrom MAST (multi-mission archive at STScI). The images were processed through the standard OTFR (on-the-fly reprocessing) calibration pipeline which performs analog-to-digital conversion, bad pixel masking, bias and dark subtraction, flat field correction and photometric calibration. The cosmic rays rejection was performed combining the two images that are usually taken for this scope. Geometric distortion was corrected using the {\\it multi drizzle} script (\\cite{koekemoer02}).\n\nThe relative Chandra-HST astrometry is clearly a fundamental issue for this\nwork. Chandra has a nominal position accuracy of 0.6$\\arcsec$ while the\nabsolute astrometry of HST is accurate to 1-2$\\arcsec$. Fortunately, to align\nthe two astrometric solutions, we could use a point-like source detected both\nin the WFPC2 and Chandra fields. This source was previously detected at radio\nwavelengths (\\cite{wilson94}) and is most probably associated to a background\nquasar (Tsvetanov et al. 1996). Moreover, as a second reference point, we used the brightest emission peak in the nuclear region of NGC 5252 itself.\n\n\nImages were calibrated in flux using a constant flux conversion\nfactor of $1.839 \\times 10^{-16}$, corresponding to the flux producing a\ncount rate of 1~s$^{-1}$ in the filter band. The above value is appropriate\nfor the instrumental configuration employed during the \nNGC~5252 exposure,\nas indicated by the {\\tt PHOTFLAM} keyword in the image file.\n\nThe HST image is shown in Fig.~\\ref{fig6}.\n \\begin{figure}\n \\centering\n \\includegraphics[width=8.5cm]{fig6_22.ps}\n \\caption{HST WFC~2 [OIII] image of the ionisation cones in NGC~5252.\n }\n \\label{fig6}\n \\end{figure}\nTsvetanov et al. (1996), Morse et al. (1998) \\& Capetti et al. (2005)\ndiscussed it in details.\nWe refer the reader to these papers for an extensive\ndiscussion on the properties of the optical emission. Their main\noutcomes can be summarized as follows:\n\n\\begin{itemize}\n\n\\item the surface brightness is dominated by the unresolved nucleus\n\n\\item a half-ring structure is apparent S-E of the nucleus at a\nmaximum projected distance of $\\simeq$1.5~kpc. It is probably associated\nwith the near side of an inclined gas disk, whose far side is\nobscured by the host galaxy dust (\\cite{morse98});\n\n\\item the large scale ionisation cone is traced by thin shells of\nenhanced emission at either side of the nucleus, well aligned along\na P.A.$\\simeq$110$^{\\circ}$ at distances between 5 and 11~kpc. Fainter\nco-aligned structures at scales as large as 20~kpc are \ndetected as well in the O[{\\sc iii}] images;\nhowever, we will not discuss these latter structures, as\nthey are beyond the region where X-ray emission associated with\nNGC~5252 is detected;\n\n\\item there is no evidence of radial motions. The measured velocities\nof the different structures are fully explained by the rotations of the \ntwo disks [nonetheless Acosta-Pulido et al. (1996) claimed the detection\nof radial motions describing the kinematic properties of the [OIII] emission \narcs].\n\n\\end{itemize}\n\nIn Fig.~\\ref{fig7} we present\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=8.5cm,angle=-90]{fig7.ps}\n \\caption{Iso-intensity {\\it Chandra}-ACIS 0.2-1 keV X-ray iso-intensity\n \t\tcontours superposed to the HST WFC~2 [OIII] image\n \t\tof Fig.~\\ref{fig7}. The resolution of the latter has been\n \t\tdegraded to the typical resolution of {\\it Chandra} optics\n \t\tby applying a wavelet smoothing with an 8~pixel\n \t\tkernel. The {\\it Chandra} contours represents\n \t\tnine linearly spaces count levels from 0.5 to 20 counts\n \t\tper pixel, after a wavelet smoothing with a $\\sigma$=1.25\n \t\tpixel has been applied.\n }\n \\label{fig7}\n \\end{figure}\nthe superposition between the soft X-rays iso-intensity contours to\nthe [OIII] image. The HST image spatial resolution has been\ndegraded with a 8~pixel wavelet kernel to match the resolution\nof the {\\it Chandra} optics.\nRegions of enhanced X-ray emission exhibit a remarkable coincidence with the\nmorphology of the optical narrow-band image. \n\n\n \\begin{figure}\n \\centering\n \\includegraphics[width=5.5cm,angle=-90]{fig91.ps}\n \\caption{[OIII]\/Soft-X flux ratio as a function of distance from the nucleus}\n \\label{fig8}\n \\end{figure}\n\n\nWe have calculated the ratio between the\n[OIII] band and the 0.5-2 keV flux (Fig. 8) for the regions specified, \nafter splitting region SS into two\nsub-regions divided by a E-W line at $\\delta_{J2000} = 4^{\\circ} 32\\arcmin\n21\\arcsec$ (regions ``SSNorth'' - SSN - and ``SSSouth'' - SSS - respectively).\nThe ratio exhibit a dynamical range smaller than a factor 2\nover distances ranging from less than 100 pc\nto $\\sim$1.5 kpc with a slight tendency to decrease with the distance from the \nnucleus(r). This last effect, however, is most probably an observational \nartifact due to the decreasing in surface brightness of the arcs moving away\nfrom the nucleus coupled with the sensitivity limits of $Chandra$ to extended\nsources. At a first glance, the [OIII]\/Soft-X ratio profile as a function of the \ndistance from the nucleus seems to \nsuggest that the electron density follow a r$^{-2}$ relation since the number \nof ionising photons and of the overall average ionisation state of the \nnuclear species remain almost constant.\nThis result is in \nagreement with Bianchi et al. (2006), who assumed, however,\na very simplified geometry of the emitting gas. \nA more detailed investigation on this\ntopic in NGC~5252 is hampered by the quality of the data.\n\n\n\\section{Discussion}\n\n The soft X-ray emission of NGC~5252 is clearly\n extended and ACIS images demonstrate that \nthe spectacular ionisazion cones observed in [OIII] have counterparts\nin the 0.1-1 keV band. \nThe cumulative soft X-ray spectrum observed by $XMM$--$Newton$ is described by\na soft power-law ($\\Gamma$$\\sim$3). The ACIS images suggest that this is probably due \nto a blend of emission lines that mimics such steep power-law as demonstrated \nin other type II Seyferts like NGC 1068, Circinus galaxy and Mrk 3 (\\cite{kin02}, \\cite{brink02},\n\\cite{ogle03}, \\cite{sam01}, \\cite{sako00}, \\cite{b05}, \\cite{pp05}).\nThis scenario is supported also by the detection in the RGS high resolution \nspectrum of three emission \nlines, of CV, OVII and OVIII, probably due to photoionised gas. This is consistent also by previous \noptical studies that excluded \ncollisional ionisation along the cones of NGC~5252 (Tsvetanov et al. 1996). \nMoreover, the presence of {\\it in situ} ionisation sources due to shocks formed by \nlarge scale outflows interacting with the interstellar matter has been\nexcluded (\\cite{morse98}). \nThus the source of ionising photons is most probably the nucleus. Under this\nassumption, \nwe can use the imaging of the arcs to study the physical condition\nof the gas along the ionisation cones. In particular, \nthe constant of [OIII]\/(0.5-2 keV) flux ratio along the ionisation cones \nwithin the inner\\footnote{The outer arcs and filaments (\\cite{tad}) are most probably too weak \nto be detected in X-rays. Considering the extension of the outer [OIII] arcs, the\nminimum detectable flux between 0.1-1 keV is F$_{0.1-1\n keV}$$\\sim$5$\\times$10$^{-15}$erg s$^{-1}$ cm$^{-2}$ \nwhile, assuming\na constant [OIII]\/soft X-ray ratio, the expected flux should be $\\sim$10 times\nlower. } $\\sim$1.5 kpc suggests a r$^{-2}$ law\nfor the ion density. \n\n\n\n\n\n\n\n\n\nOptical spectroscopic studies (Acosta-Pulido et al. 1996) suggest that the\nradial dependency of the ionization parameter\\footnote{U=$\\frac{L}{4 \\pi \\rho\n r^{2}}$, where L is the source's luminosity,\n $\\rho$ is the density of the ionized gas, $r$ is the distance between the\n source of ionizing photons and the ionized matter.} U follows a different law\nin the south-east ($U \\propto r^{-0.4}$) with respect to the north-east\n($U \\propto r^0$) cone. The authors speculate that the intrinsic behavior should be the one\nshown in the former, while the radial-independence of the ionization\nparameter in the latter may be due to a ``conspiracy'' introduced by the\nexistence of two counterotating disks \nof gas (\\cite{morse98}): one is coplanar \nto the stellar one, and another is inclined by $\\sim$40$^{\\circ}$. \nMorse et al. (1998) speculated that the prominence of the southeast [OIII] \ncone in the nuclear regions is due to the fact that this component is \nseen directly, while the northeast [OIII] cone is seen\nthrough the gas of the other disk. \nIf so, the absorption due to this component could alter the line ratios \npresented by \\cite{ap96} and thus the correct behavior of U should be the one derived from\nthe south-east cone. U$\\propto$r$^{-0.4}$ implies that the luminosity of the nucleus increased by a\nfactor $\\Delta$L$\\sim$3-6 in the last $\\Delta$t$\\sim$5000 years. These numbers\nbecome $\\Delta$L$\\sim$10-30 and $\\Delta$t$\\sim$30000 years if we further \nassume that\nthe U and the ion density laws are still valid up to 10 kpc from the nucleus,\ni.e. where the optical cones are still detectable in [OIII] but not in\nX-rays. \n\n\n\n\n\n\nOn the contrary, having U constant and $\\rho \\propto$ r$^{-2}$ would imply that L has remained constant during the last 5000\n(30000) years. This is consistent with the ``quasar-relic'' scenario proposed by\n\\cite{capetti}. These authors suggested that the nucleus of NGC~5252 is indeed\nthe ``relic'' of a nucleus that already experienced the activity phase in the\npast and that now persists in an almost quiescent phase. \nThis is suggested by the high mass of the SMBH (M$_{BH}$=10$^{9}$M$_{\\sun}$, \nCapetti et al. 2005) that \nindicates that the nucleus has already accreted in the past, the low \nEddington ratio (L$\\sim$10$^{-3}$L$_{Edd}$, assuming the bolometric\ncorrection from Marconi et al. (2004),\nL$_{hard-x}$$\\sim$(1\/22)$\\times$L$_{bol}$), and the early type (S0) morphology\nof the \nAGN host galaxy. In literature it is also reported that the optical emission\nline ratios in the inner 30\" are typical of LINERS \n (\\cite{gon}), thus suggesting that a low efficiency\nengine is acting at the nucleus of the source. \nIt is worth noting that also the detection of two counterotating\ndisks suggests that NGC~5252 is a \"quasar-relic\". These\ndisks are tracers of a major merging event that occurred, most probably, \nmore than 10$^8$ years ago, since the stellar disk of NGC~5252 is\nundisturbed. If the merging event triggered a phase of AGN \nactivity (see Jogge 2006, and references therein for a discussion on this\ntopic), we can expect that it lasted few\/some \n$\\sim$10$^{7}$ years (\\cite{mar}; \\cite{ste}; \\cite{jac}; \\cite{gon2}) \n after which the source has persisted in a quiescent state.\n\n\nFinally, it is worth noting that the spectrum of the nucleus hosted by\nNGC~5252 is confirmed to be quite flat (Cappi et al. 1996). \nAs shown, if modeled with a simple absorbed power-law its photon\nindex points to a very hard spectrum ($\\Gamma$$\\sim$1). \nThe low EW ($\\sim$50 eV) of the neutral (E$_{FeK\\alpha}$$\\sim$6.4 keV) iron line\nis consistent with what expected if the FeK$\\alpha$ line is\nproduced via transmission in the observed column\n(N$_{H}$$\\sim$2$\\times$10$^{22}$ cm$^{-2}$, Makishima 1986) thus excluding a reflection\ndominated spectrum.\nTo reconcile, at least marginally, \nthe hardness of the NGC~5252 nuclear spectrum, with previous results for\nSeyfert galaxies ($\\Gamma$$\\sim$1.5-2.5; Turner \\& Pounds, \n1989; Nandra \\& Pounds, 1994; Smith \\& Done, 1996; Dadina 2008), we must invoke\ncomplex absorption models involving partial covering of the source and\/or \nthe presence of ionised absorbers along the line of sight. In this\ncase the spectral index becomes $\\Gamma$$\\sim$1.4-1.5. \n It is interesting to note that the flat photon index \nmay be a further clue suggesting that the X-rays may be \nproduced in an ADAF,\nNarayan \\& Yi, 1994) as expected in a ``quasar-relic''.\n\n\n\n\n\n\\begin{acknowledgements}\n\nThis paper is based on observations obtained with XMM-Newton, an ESA\nscience mission with instruments and contributions directly funded by\nESA Member States and the USA (NASA). MD greatfully acknowledge Barbara De\nMarco for the helpful discussions. MD, MC and GM greatfully acknowledge \nASI financial support under contract I\/23\/05\/0. CV greatfully acknowledge \nASI financial support under contract I\/088\/06\/0.\n\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\subsection{Background}\n\\subsubsection*{Philosophy}\nFor us, \\emph{Real spectrum} is a loose term for a $C_2$-spectrum built\nfrom the $C_2$-spectrum $M\\R$ of Real bordism, considered by Araki,\nLandweber and Hu--Kriz \\cite{HK}. The present article shows that\nbringing together Real spectra and Gorenstein duality reveals rich and interesting structures. \n\nIt is part of our philosophy that theorems about Real spectra can\noften be shown in the same style as theorems for the underlying\ncomplex oriented spectra although the details might be more\ndifficult, and groups needed to be graded over the real representation\nring $RO(C_2)$ (indicated by $\\bigstar$) rather than over the integers\n(indicated by $*$). This extends a well known\nphenomenon: complex orientability of equivariant spectra\nmakes it easy to reduce questions to integer gradings, and we show \nthat even in the absence of complex orientability, good\nbehaviour of coefficients can be seen by grading with\nrepresentations. \n\n\\subsubsection*{Bordism with reality}\nIn studying these spectra, the real regular representation $\\rho=\\R\nC_2$ plays a special role. We write $\\sigma$ for the sign\nrepresentation on $\\R $ so that $\\rho =1+\\sigma$. \n One of the crucial features of $M\\R$ is that \n it is \\emph{strongly even} in the sense of \\cite{HM}, i.e.\n\\begin{enumerate}\n\\item \\label{item:Restr} the restriction functor $\\pi_{k\\rho}^{C_2}M\\R \\to \\pi_{2k}MU$ is an isomorphism for all $k\\in\\Z$, and\n\\item the groups $\\pi_{k\\rho-1}^{C_2}M\\R$ are zero for all $k\\in\\Z$.\n\\end{enumerate}\n\nWe now localize at 2, and (with two exceptions) all spectra and\nabelian groups will henceforth be $2$-local. The Quillen idempotent\nhas a $C_2$-equivariant refinement, and this defines the\n$C_2$-spectrum $BP\\R$ as a summand of $M\\R_{(2)}$. The isomorphism (\\ref{item:Restr}) allows us to lift the usual $v_i$ to classes $\\vb_i\\in\\pi_{(2^i-1)\\rho}^{C_2}BP\\R$. The Real spectra we are interested in are quotients of $BP\\R$ by sequences of $\\vb_i$ and localizations thereof. For example, we can follow \\cite{HK} and \\cite{Hu} and define \n$$BP\\R \\langle n \\rangle = BP\\R\/(\\vb_{n+1},\\vb_{n+2},\\dots)$$ \nand \n$$E\\R(n) = BP\\R \\langle n \\rangle[\\vb_n^{-1}].$$\nThese spectra are still strongly even, as we will show. Apart from the big literature on $K$-theory with Reality (e.g.\\ \\cite{Ati66}, \\cite{Dugger} and \\cite{B-G10}), Real spectra have been studied by Hu and Kriz, in a series of papers by Kitchloo and Wilson (see e.g.\\ \\cite{K-W15} for one of the latest installments), by Banerjee \\cite{Banerjee}, by Ricka \\cite{Ricka} and by Lorman \\cite{Lorman}. \n A crucial point is that a morphism between strongly even\n $C_2$-spectra is an equivalence if it is an equivalence of underlying\n spectra \\cite[Lemma 3.4]{HM}. \n\nWe are interested in two dualities for Real spectra: Anderson duality\nand Gorenstein duality. These are closely related \\cite{GS} but apply\nto different classes of spectra. \n\n\\subsubsection*{Anderson duality}\nThe Anderson dual $\\Z^X$ of a spectrum $X$ is an integral version of\nits Brown-Comenetz dual (in accordance with our general principle,\n$\\Z$ denotes the $2$-local integers). The homotopy groups of the\nAnderson dual lie in a short exact sequence\n\\begin{align}\\label{eq:IntroductionSES} 0 \\to \\mathrm{Ext}_{\\Z}^1(\\pi_{-*-1}X ,\\Z) \\to \\pi_*(\\Z^X) \\to\n\\mathrm{Hom}_{\\Z}(\\pi_{-*}X, \\Z) \\to 0. \\end{align}\n\nOne reason to be interested in the computation of Anderson duals is\nthat they show up in universal coefficient sequences (see\n\\cite{Anderson} or Section \\ref{subsec:Anderson}). The situation is\nnicest for spectra that are Anderson self-dual in the sense that\n$\\Z^X$ \nis a suspension of $X$. Many important spectra like $KU$, $KO$, $Tmf$ \\cite{Sto12} or\n$Tmf_1(3)$ are indeed Anderson self-dual. These examples are all unbounded as the sequence \\eqref{eq:IntroductionSES} nearly forces them to be. \n\nAnderson duality also works $C_2$-equivariantly\nas first explored by \\cite{Ricka}; the only change in the above short\nexact sequence is that equivariant homotopy groups are used. The $C_2$-spectra $K\\R$\n\\cite{H-S14} and $Tmf_1(3)$ \\cite{HM} are also $C_2$-equivariantly\nself-Anderson dual, at least if we allow suspensions by\n\\emph{representation spheres}. \n\nOne simpler example is essential background: if $\\underline{\\Z}$ denotes the constant Mackey functor (i.e., with restriction being the identity and\ninduction being multiplication by 2) then the Anderson dual of its\nEilenberg-MacLane spectrum is the Eilenberg-MacLane spectrum for the\ndual Mackey functor $\\underline{\\Z}^*=\\mathrm{Hom}_{\\Z}(\\underline{\\Z}, \\Z)$\n(i.e., with restriction being multiplication by 2 and induction being\nthe identity). It is then easy to check that in fact $H(\\underline{\\Z}^*)\\simeq \\Sigma^{2(1-\\sigma)}H\\Zu$.\n(From one point of view this is the fact that $\\R P^1=S(2\\sigma)\/\\Ctwo$ is equivalent to the circle). \nThe dualities we find are in a sense all \ndependent on this one. \n\n\\subsubsection*{Gorenstein duality}\nBy contrast with Anderson self-duality, many connective ring spectra\nare Gorenstein in the sense of \\cite{DGI}. We sketch the theory here,\n explaining it more fully in Sections \\ref{sec:kR} and \\ref{sec:dishonest}.\n\nThe starting point is a connective commutative ring $C_2$-spectrum $R$,\nwhose $0$th homotopy Mackey functor is constant at $\\Z$: \n$$\\underline{\\pi}^{\\Ctwo}_0(R)\\cong \\underline{\\Z}.$$ \nThis gives us a map $R\\longrightarrow H\\Zu$ of commutative ring spectra by killing\nhomotopy groups. We say that $R$\nis {\\em Gorenstein} of shift $a\\in RO(C_2)$ if there is an equivalence of $R$-modules\n$$\\mathrm{Hom}_R(H\\Zu ,R)\\simeq \\Sigma^aH\\Zu. $$\n\nWe are interested in the duality this often entails. \nNote that the Anderson dual $\\Z^R$ obviously has the Matlis lifting property\n$$\\mathrm{Hom}_R(H\\Zu, \\Z^R)\\simeq H\\Zu^*, $$\nwhere $\\Z^*=\\mathrm{Hom}_{\\Z}(\\underline{\\Z}, \\Z)$ as above. Thus if $R$ is Gorenstein,\nin view of the \nequivalence $H(\\underline{\\Z}^*)\\simeq \\Sigma^{2(1-\\sigma)}H\\Zu$, we have equivalences\n\\begin{align*}\\mathrm{Hom}_R(H\\Zu ,\\mathrm{Cell}_{H\\Zu} R)&\\simeq \\mathrm{Hom}_R(H\\Zu ,R)\\\\\n&\\simeq \\Sigma^aH\\Zu\\\\\n&\\simeq \\Sigma^{a-2(1-\\sigma)}H(\\underline{\\Z}^*) \\\\\n&\\simeq\n\\mathrm{Hom}_R(H\\Zu, \\Sigma^{a-2(1-\\sigma)}\\Z^R).\\end{align*}\nHere, $\\mathrm{Cell}_{H\\Zu}$ denotes the $H\\Zu$-$\\mathbb{R}$-cellularization as in Section \\ref{sec:Cell}. We would like to remove the $\\mathrm{Hom}_R(H\\Zu, \\cdot)$ from the composite equivalence above. \n\n\\begin{defn}\nWe say that $R$ has {\\em Gorenstein duality} of shift $b$ if we have\nan equivalence of $R$-modules\n$$\\mathrm{Cell}_{H\\Zu} R \\simeq \\Sigma^b \\Z^R.$$\n\\end{defn}\n\n As in the non-equivariant setting, the passage from Gorenstein to\n Gorenstein duality requires showing that\nthe above composite equivalence is compatible with the right \naction of $\\cE =\\mathrm{Hom}_R(H\\Zu, H\\Zu)$. This turns out to be considerably\nmore delicate than the non-equivariant counterpart because\nconnectivity is harder to control; but if one can lift the\n$R$-equivalence to an $\\cE$-equivalence, the conclusion is that if $R$\nis Gorenstein of shift $a$ then it has Gorenstein duality of shift $b=a-2(1-\\sigma)$. \n\n\n\\subsubsection*{Local cohomology}\nThe duality statement becomes more interesting when the cellularization can be\nconstructed algebraically. For any finitely generated ideal $J$ of the $RO(C_2)$-graded\ncoefficient ring $R_{\\bigstar}^{C_2}$, we may form the stable Koszul\ncomplex $\\Gamma_JR$, which only depends on the radical of $J$. In our\nexamples, this applies to the augmentation ideal $J=\\ker(R_{\\bigstar}^{C_2}\\longrightarrow\nH\\Zu_{\\bigstar}^{C_2})$, which may be radically generated by finitely\nmany elements $\\vb_i$ in degrees which are multiples of\n$\\rho$. Adapting the usual proof to the Real context, \nProposition \\ref{prop:cell} shows that \n$\\Gamma_JR\\longrightarrow R $ is\n$H\\Zu$-$\\R$-cellularization: \n$$\\mathrm{Cell}_{H\\Zu}R\\simeq \\Gamma_JR. $$\nThe $RO(C_2)$-graded homotopy groups of $\\Gamma_JR$ can be computed using a spectral sequence involving local cohomology. \n\n\n\n\\subsubsection*{Conclusion}\n In favourable cases the Gorenstein condition on a ring spectrum $R$\n implies Gorenstein duality for $R$; this in turn establishes a strong\n duality property on the $RO(C_2)$-graded coefficient ring, expressed using local cohomology. \n\n\n\n\\subsection{Results}\nOur main theorems establish Gorenstein duality for a large family of Real spectra. We present in this introduction the particular cases of $BP\\R \\langle n \\rangle$ and $E\\R(n)$, deferring the more general theorem to Section \\ref{sec:results}. Let again $\\sigma$ denote the non-trivial\nrepresentation of $\\Ctwo$ on the real line and $\\rho =1+\\sigma$ the \nreal regular representation. Furthermore set $D_n = 2^{n+1}-n-2$ so\nthat $D_n\\rho = |\\vb_1|+\\cdots + |\\vb_n|$. Other terms in the statement will be explained\nin Section \\ref{sec:AKG}. \n \n\\begin{thm}\nFor each $n\\geq 1$ the $\\Ctwo$-spectrum $BP\\R \\langle n \\rangle$ is Gorenstein\nof shift $-D_n\\rho -n$, and has Gorenstein duality of shift\n$-D_n\\rho -n-2(1-\\sigma)$. This means that\n$$\\Z_{(2)}^{BP\\R \\langle n \\rangle} \\simeq \\Sigma^{D_n\\rho+n+2(1-\\sigma)} \\Gamma_{\\Jb_n}BP\\R \\langle n \\rangle,$$\nwhere $\\Jb_n = (\\vb_1,\\dots, \\vb_n)$. This induces a local cohomology\nspectral sequence\n$$H^*_{\\Jb_n}(BP\\R \\langle n \\rangle^{\\Ctwo}_{\\bigstar})\\Rightarrow \\pi^{\\Ctwo}_{\\bigstar}(\\Sigma^{-D_n\\rho\n -n-2(1-\\sigma)}\\Z_{(2)}^{BP\\R \\langle n \\rangle}). $$\n\\end{thm}\n\\vspace{0.3cm}\n\\begin{thm}\\label{thm:ER(n)}\nFor each $n\\geq 1$ the $\\Ctwo$-spectrum $E\\R(n)$ has Gorenstein duality of shift\n$-D_n\\rho -(n-1)-2(1-\\sigma)$. This means that\n\\begin{align*}\n \\Z_{(2)}^{E\\R(n)} &\\simeq \\Sigma^{D_n\\rho+(n-1)+2(1-\\sigma)} \\Gamma_{\\Jb_{n-1}}E\\R(n) \\\\\n &\\simeq \\Sigma^{(n+2)(2^{2n+1}-2^{n+2})+n+3}\\Gamma_{J_{n-1}}E\\R(n) ,\n\\end{align*}\nfor $J_{n-1} = \\Jb_{n-1}\\cap\\pi_*^{C_2}E\\R(n)$. This induces likewise a local cohomology spectral sequence.\n\\end{thm}\n\nWe note that this has implications for the $\\Ctwo$-fixed point spectrum\n$(BP\\R \\langle n \\rangle)^{\\Ctwo}=BPR\\langle n\\rangle$. The graded ring\n$$\\pi_*(BPR\\langle n\\rangle)=\\pi_*^{\\Ctwo}(BP\\R \\langle n \\rangle)$$\nis the integer part of the $RO(\\Ctwo)$-graded coefficient ring\n$\\pi^{\\Ctwo}_{\\bigstar}(BP\\R \\langle n \\rangle)$. However, since the ideal\n$\\Jb_n$ is not generated in integer degrees, the statement for $BPR\\langle n\\rangle$\nis usually rather complicated, and one of our main messages is that\nworking with the equivariant spectra gives more insight. On the other hand, $ER(n) = E\\R(n)^{C_2}$ has integral Gorenstein duality because one can use the additional periodicity to move the representation suspension and the ideal $\\Jb_n$ to integral degrees. \n\nWe will discuss the general result in more detail later, but the two\nfirst cases are about familiar ring spectra. \n\n\\begin{example} (See Sections \\ref{sec:kR} and \\ref{sec:kRlcss}.) \nFor $n=1$, connective $K$-theory with Reality $k\\R$ is $2$-locally a\nform of $BP\\R\\langle 1\\rangle$.\n For this example we can work without\n2-localization, so that $\\Z$ means the integers. Our first theorem states\nthat $k\\R$ is Gorenstein of shift $-\\rho-1=-2-\\sigma$\nand that it has Gorenstein duality of shift $-4+\\sigma$. This just means that \n$$\\Z^{k\\R} \\simeq \\Sigma^{4-\\sigma} \\fib(k\\R \\to K\\R).$$\nThe local cohomology spectral sequence collapses to a short exact sequence associated to the fibre sequence just mentioned. We\nwill see in Section \\ref{sec:kRlcss} that the sequence is not split, even as abelian groups. \n\nTheorem \\ref{thm:ER(n)} recovers the main result of \\cite{H-S14},\ni.e.\\ that $\\Z^{K\\R} \\simeq \\Sigma^4K\\R$, which also implies $\\Z^{KO}\n\\simeq \\Sigma^4 KO$. It is a special feature of the case $n=1$ that we\nalso get a nice duality statement for the fixed points in the\nconnective case. Indeed, by considering the $RO(C_2)$-graded homotopy\ngroups of $k\\R$, one sees \\cite[3.4.2]{B-G10} that\n$$(k\\R \\otimes S^{-\\sigma})^{\\Ctwo}\\simeq \\Sigma^{1}ko. $$\nThis implies that connective $ko$ has untwisted Gorenstein duality of shift\n$-5$, i.e.\\ that \n$$\\Z^{ko} \\simeq \\Sigma^5\\fib(ko\\to KO).$$ \nThis admits a closely related non-equivariant proof, combining \nthe fact that $ku$ is Gorenstein (clear from coefficients) and the\nfact that complexification $ko\\longrightarrow ku$ is relatively Gorenstein\n(connective version of Wood's theorem \\cite[4.1.2]{B-G10}). \n\\end{example}\n\n\n\\begin{example}\n(See Examples \\ref{exa:forms} and \\ref{exa:tmf} or Lemma \\ref{lem:BPRnGordish} and Corollary \\ref{cor:BPRnGorDdish}.) \nThe 2-localization of the $C_2$-spectrum $tmf_1(3)$ is a form of $BP\\R\\langle\n2\\rangle$, and the theorem is closely related to results in\n\\cite{HM}. It states\nthat $tmf_1(3)$ is Gorenstein of shift $-4\\rho-2=-6-4\\sigma $\nand has Gorenstein duality of shift $-8-2\\sigma$. We show in Section\n\\ref{sec:tmfotlcss} that there are non-trivial \ndifferentials in the local cohomology spectral\nsequence.\n\nPassing to fixed points we obtain the 2-local equivalence\n$$BPR\\langle 2\\rangle=(BP\\R\\langle 2\\rangle)^{\\Ctwo}=tmf_0(3).$$\nBy contrast with the $n=1$ case, as observed in \\cite{HM}, $tmf_0(3)$ does not have untwisted\nGorenstein duality of any integer degree. \n\nA variant of Theorem \\ref{thm:ER(n)} also computes the\n$C_2$-equivariant Anderson dual of $TMF_1(3)$ and the computation of\nthe Anderson dual of $Tmf_1(3)$ from \\cite{HM} follows as well.\n\nThe results apply to $tmf_1(3)$ and $TMF_1(3)$ themselves (i.e.,\nwith just 3 inverted, and not all other odd primes). \n\\end{example}\n\nWe remark that our main theorem also recovers the main result of \\cite{Ricka} about the Anderson self-duality of integral Real Morava K-theory. \n\n\n\\subsection{Guide to the reader}\nWhile the basic structure of this paper is easily visible from the table of contents, we want to comment on a few features. \n\nThe precise statements of our main results can be found in Section \\ref{sec:results}. We will give two different proofs of them. One (Part 3) might be called `the hands on approach' which is\nelementary and explicit, and one (Part 2) uses Gorenstein techniques\ninspired by commutative algebra. \nThe intricacy and dependence \non specific calculations in the explicit approach and the make the conceptual approach\nvaluable. The subtlety of the structural requirements of the\nconceptual approach make the reassurance of the explicit approach\nvaluable. The exact results proved in Parts 2 and 3 are also\nslightly different. \n\nWhile the Gorenstein approach only relies on the knowledge of the homotopy groups of $H\\Zu$ and the reduction theorem Corollary \\ref{cor:reduction}, we need detailed information about the homotopy groups of quotients of $BP\\R$ for the hands-on approach. In Appendix \\ref{Appendix}, we give a streamlined account of the computation of $\\pi_{\\bigstar}^{C_2}BP\\R$ (which appeared first in \\cite{HK}). In Section \\ref{sec:BPRn}, we give a rather self-contained account of the homotopy groups of $BP\\R \\langle n \\rangle$ and of other quotients of $BP\\R$, which can also be read independently of the rest of the paper. While some of this is rather technical, most of the time we just have to use Corollary \\ref{Cor:crucial} whose statement (though not proof, perhaps) is easy to understand.\n\nWe give separate arguments for the computation of the Anderson dual of\n$k\\R$ so that this easier case might illustrate the more complicated\narguments of our more general theorems. Thus, if the reader is only\ninterested in $k\\R$, he or she might ignore most of this paper. More\nprecisely, under this assumption one might proceed as follows: First one looks at Section \\ref{sec:kRgroups} for a quick reminder on $\\pi_{\\bigstar}^{C_2}k\\R$, then one skims through Sections \\ref{sec:Basics} and \\ref{sec:AKG} to pick up the relevant definitions and then one proceeds directly to Section \\ref{sec:kR} or Section \\ref{sec:kRagain} to get the proof of the main result in the case of $k\\R$. Afterwards one may look at the pictures and computations in the rest of Section \\ref{sec:kRlcss} to see what happens behind the scenes of Gorenstein duality. \n\n\n\\vspace{1cm}\n\\part{Preliminaries and results}\\vspace{0.5cm}\n\\section{Basics and conventions about $C_2$-spectra}\\label{sec:Basics}\n\\subsection{Basics and conventions}\nWe will work in the homotopy category of genuine $G$-spectra (i.e., stable for suspensions by $S^V$ for any finite\ndimensional representation $V$) for $G =\\Ctwo$, the group of order $2$. We will denote by $\\otimes$ the derived smash product of spectra.\n\nWe may combine the equivariant and non-equivariant homotopy groups of\na $C_2$-spectrum into a Mackey functor, which we denote by $\\underline{\\pi}_*^{C_2}X$ and denote $C_2$-equivariant and underlying homotopy groups correspondingly by $\\pi^{C_2}_*X$ and $\\pi^e_*X$. For an abelian group $A$, we write $\\underline{A}$ for the constant Mackey functor (i.e., restriction maps are the identity),\nand $\\underline{A}^*$ for its dual (i.e., induction maps are the identity). \nWe write $HM$ for the Eilenberg-MacLane spectrum associated to a\nMackey functor $M$. \n\nAnother $C_2$-spectrum of interest to us is $k\\R$, the $C_2$-equivariant connective cover of\nAtiyah's $K$-theory with Reality \\cite{Ati66}. The term ``Real\nspectra'' derives from this example. The examples of Real bordism and the other $C_2$-spectra derived from it will be discussed in Section \\ref{sec:BPRn}.\n\nWe will usually grade our homotopy groups by the real\nrepresentation ring $RO(\\Ctwo)$, and we write $M_{\\bigstar}$ for $RO(\\Ctwo)$-graded groups. In addition to the real sign representation $\\sigma$ and the regular representation $\\rho$ the virtual\nrepresentation $\\pp =1-\\sigma$ is also significant. Examples of\n$RO(\\Ctwo)$-graded homotopy classes are the geometric Euler classes\n$a_V\\colon S^0 \\longrightarrow S^V$; in particular, $a=a_{\\sigma}$ will play a central role. In addition to $a$, we will also often have a class $u=u_{2\\sigma}$ of degree $2\\pp$. \n\nWe often want to be able to discuss gradings by certain subsets of\n$RO(\\Ctwo)$. To start with we often want to refer to gradings by multiples\nof the regular representation (where we write $M_{*\\rho}$), but we also\nneed to discuss gradings of the form $k\\rho -1$. For this, we use the notation\n$$*\\rho - =\\{ k\\rho \\; | \\; k\\in \\Z\\}\\cup \\{ k\\rho -1\\; | \\; k\\in \\Z\\}. $$\nFollowing \\cite{HM} we call an $RO(\\Ctwo)$-graded object $M$ {\\em even} if\n$M_{k\\rho -1}=0$ for all $k$. An $RO(\\Ctwo)$-graded Mackey functor is {\\em\n strongly even} if it is even and all the Mackey functors in gradings\n$k\\rho$ are constant. We call a $C_2$-spectrum (strongly) even if its homotopy groups are (strongly) even.\n\nIf $X$ is a strongly even $C_2$-spectrum and $x\\in \\pi_{2k}X$, we denote by $\\overline{x}$ its counterpart in $\\pi_{k\\rho}^{C_2}X$. If we want to stress that we consider a certain spectrum as a $C_2$-spectrum, we will also sometimes indicate this by a bar; for example, we may write $\\overline{tmf_1(3)}$ if we want to stress the $C_2$-structure of $tmf_1(3)$. \n\n\n\\subsection{Cellularity} \\label{sec:Cell}\n\\label{subsec:Rcell}\nIn a general triangulated category, it is conventional to say $M$ is \\emph{$K$-cellular} if\n$M$ is in the localizing subcategory generated by $K$ (or equivalently\nby all integer suspensions of $K$). A reference in the case of spectra is \\cite[Sec 4.1]{DGI}. We say that a $C_2$-spectrum $M$ is \\emph{$K$-$\\R$-cellular} (for a $C_2$-spectrum $K$)\nif it is in the localizing subcategory generated by the suspensions\n$S^{k\\rho}\\otimes K$ for all integers $k$. We note that this is the\nsame as the localizing subcategory generated by integer suspensions of\n$K$ and $(\\Ctwo)_+\\otimes K$ because of the cofibre sequence\n$$(C_2)_+ \\to S^0 \\to S^\\sigma.$$\nWe say that a map $N\\to M$ is a \\emph{$K$-$\\R$-cellularization} if $N$ is $K$-$\\R$-cellular and the induced map\n$$\\mathrm{Hom}(K, N) \\to \\mathrm{Hom}(K, M)$$\nis an equivalence of $C_2$-spectra. The $K$-$\\R$-cellularization is clearly unique up to equivalence. \n\nWe note that cellularity and $\\R$-cellularity are definitely\ndifferent. For example $(\\Ctwo)_+$ is not $S^0$-cellular, but it is\n$S^0$-$\\R$-cellular. \n\nIn this article, we will only use $\\R$-cellularity. \n\n\\subsection{The slice filtration}\n\\label{subsec:slice}\nRecall from \\cite[Section 4.1]{HHR} or \\cite{SlicePrimer} that the \\emph{slice cells} are the $\\Ctwo$-spectra of the form \n\\begin{itemize}\n \\item $S^{k\\rho}$ of dimension $2k$,\n \\item $S^{k\\rho-1}$ of dimension $2k-1$, and\n \\item $S^k\\otimes (\\Ctwo)_+$ of dimension $k$.\n\\end{itemize}\nA $\\Ctwo$-spectrum $X$ is $\\leq k$ if for every slice cell $W$ of dimension $\\geq k+1$ the mapping space $\\Omega^\\infty \\mathrm{Hom}_{\\mathbb{S}}(W, X)$ is equivariantly contractible. As explained in \\cite[Section 4.2]{HHR}, this leads to the definition of $X \\to P^kX$, which is the universal map into a $\\Ctwo$-spectrum that is $\\leq k$. The fibre of $$X \\to P^kX$$\nis denoted by $P_{k+1}X$. The $k$\\emph{-slice} $P_k^kX$ is defined as the fibre of $$P^kX\\to P^{k-1}X$$\nor, equivalently, as the cofibre of the map $P_{k+1}X \\to P_kX$. We have the following two useful propositions:\n\n\\begin{prop}[\\cite{SlicePrimer}, Cor 2.12, Thm 2.18]\n\\label{prop:sliceodd}\n If $X$ is an even $\\Ctwo$-spectrum, then $P^{2k-1}_{2k-1}X = 0$ for all $k\\in \\Z$.\n\\end{prop}\n\n\\begin{prop}[\\cite{SlicePrimer}, Cor 2.16, Thm 2.18]\n\\label{prop:sliceven}\n If $X$ is a $\\Ctwo$-spectrum such that the restriction map in\n $\\underline{\\pi}^{\\Ctwo}_{k\\rho}$ is injective, then $P^{2k}_{2k}X$ is\n equivalent to the Eilenberg-MacLane spectrum\n $\\underline{\\pi}^{\\Ctwo}_{k\\rho} X$.\n\\end{prop}\n\nThis allows us to give a characterization of an Eilenberg-MacLane spectrum based on regular representation degrees.\n\\begin{cor}\n\\label{cor:characterizingZ}\nAny even $\\Ctwo$-spectrum $X$ with \n$$\\underline{\\pi}^{\\Ctwo}_{k\\rho}(X)=\\begin{cases}\\underline{A} & \\text{ if } k= 0 \\\\\n0 & \\text{ else}\\end{cases}$$ \n for an abelian group $A$ is equivalent to $H\\underline{A}$.\n\\end{cor}\n\\begin{proof}\nBy the last two propositions, we have\n\\[P^k_k X\\simeq \\begin{cases} H\\underline{A} & \\text{ if } k=0 \\\\\n 0 & \\text{ else}\n \\end{cases}\n\\]\nBy the convergence of the slice spectral sequence \\cite[Theorem 4.42]{HHR}, the result follows.\n\\end{proof}\n\n\\section{Anderson duality, Koszul complexes and Gorenstein duality}\\label{sec:AKG}\n\n\\subsection{Duality for abelian groups}\\label{sec:DAb}\nIt is convenient to establish some conventions for abelian groups to\nstart with, so as to fix notation. \n\nPontrjagin duality is defined for all graded abelian groups $A$\nby \n$$A^{\\vee}=\\mathrm{Hom}_{\\Z}(A, \\Q \/\\Z). $$\nSimilarly, the rational dual is defined by \n$$A^{\\vee \\Q}=\\mathrm{Hom}_{\\Z}(A, \\Q ). $$\n\nSince $\\Q $ and $\\Q \/\\Z$ are injective abelian groups these two\ndualities are homotopy invariant and pass to the\nderived category. Finally the Anderson dual $A^*$ is defined by\napplying $\\mathrm{Hom}_{\\Z}(A, \\cdot )$ to the exact sequence\n$$0\\longrightarrow \\Z \\longrightarrow \\Q \\longrightarrow \\Q\/\\Z\\longrightarrow 0$$ \nso that we have a triangle \n$$A^*\\longrightarrow A^{\\vee \\Q }\\longrightarrow A^{\\vee }. $$\n\n\nIf $M$ is a free abelian group, then the\nAnderson dual is simply calculated by \n$$M^*=\\mathrm{Hom}_{\\Z}(M, \\Z)$$\n(since $M$ is free, the $\\mathrm{Hom}$ need not be derived). \n\nIf $M$ is a graded abelian group which is an $\\mathbb{F}_2$-vector\nspace then up to suspension the Anderson dual is the vector space dual: \n$$M^{\\vee}=\\mathrm{Hom}_{\\mathbb{F}_2}(M, \\mathbb{F}_2)\\simeq \\Sigma^{-1} M^*$$\n(since vector spaces are free, $\\mathrm{Hom}$ need not be derived). \n\n\n\\subsection{Anderson duality}\n\\label{subsec:Anderson}\nAnderson duality is the attempt to topologically realize the algebraic duality from the last subsection. It appears that it was invented by Anderson (only published in mimeographed notes \\cite{Anderson}) and Kainen \\cite{Kainen}, with similar ideas by Brown and Comenetz \\cite{B-C76}. For brevity and consistency, we will only use the term Anderson duality instead of Anderson--Kainen duality or Anderson--Brown--Comenetz duality throughout. We will work in the category of $\\Ctwo$-spectra, where Anderson duality was first explored by Ricka in \\cite{Ricka}. \n\nLet $I$ be an injective abelian group. Then we let $I^{\\mathbb{S}}$ denote the $C_2$-spectrum representing the functor \n$$X \\mapsto \\mathrm{Hom}(\\pi_{*}^{C_2}X,I).$$\nFor an arbitrary $C_2$-spectrum, we define $I^X$ as the function spectrum $F(X, I^{\\mathbb{S}})$. For a general abelian group $A$, we choose an injective resolution \n$$A \\to I \\to J$$\nand define $A^X$ as the fibre of the map $I^X \\to J^X$. For example, we get a fibre sequence\n$$\\Z^X\\longrightarrow \\Q^X\\longrightarrow (\\Q\/\\Z)^X.$$\nIn general, we get a short exact sequence of homotopy groups\n$$0 \\to \\mathrm{Ext}_{\\Z}(\\pi_{-k-1}^{C_2}(X), A) \\to \\pi_k^{C_2} (A^X) \\to \\mathrm{Hom}(\\pi_{-k}^{C_2}(X), A) \\to 0.$$\nThe analogous exact sequence is true for $RO(C_2)$-graded Mackey functor valued homotopy groups by replacing $X$ by $(C_2\/H)_+ \\wedge \\Sigma^VX$. \nOur most common choices will be $A = \\Z$ and $A=\\Z_{(2)}$.\n\nFrom time to time we we use the following property of Anderson duality: If $R$ is a strictly commutative $C_2$-ring spectrum and $M$ an $R$-module, then $\\mathrm{Hom}_R(M, A^R) \\simeq A^M$ as $R$-modules as can easily be seen by adjunction.\n\nOne of the reasons to consider Anderson duality is that it provides universal coefficient sequences. In the $C_2$-equivariant world, this takes the following form \\cite[Proposition 3.11]{Ricka}:\n\\[0 \\to \\mathrm{Ext}_\\Z^1(E_{\\alpha-1}^{\\Ctwo}(X), A) \\to (A^E)_{\\alpha}^{\\Ctwo}(X) \\to \\mathrm{Hom}_\\Z(E_{\\alpha}^{\\Ctwo}(X), A) \\to 0,\\]\nwhere $E$ and $X$ are $C_2$-spectra, $\\alpha\\in RO(C_2)$ and $A$ is an abelian group. \n\nOur first computation is the Anderson dual of the Eilenberg--MacLane\nspectrum of the constant Mackey functor $\\underline{\\Z}$.\n\n\\begin{lemma}\n\\label{lem:Zu}\nThe Anderson dual of the Eilenberg-MacLane spectrum $H\\Zu$ (as an\n$\\bbS$-module) is given by \n$$\\Z^{H\\Zu}\\simeqH\\Zu^* \\simeq \\Sigma^{2\\pp} H\\Zu, $$\nwhere $\\delta = 1-\\sigma$.\n\\end{lemma}\n\n\\begin{proof}\nThe first equivalence follows from the isomorphisms\n$$\\underline{\\pi}^{C_2}_*(\\Z^{H\\Zu}) \\cong \\mathrm{Hom}_{\\Z}(\\underline{\\pi}^{C_2}_{-*}H\\Zu, \\Z) \\cong \\underline{\\Z}^*.$$\n\nSince \n$$\\pi_*^{\\Ctwo}(S^{2-2\\sigma}\\otimes H\\Zu)=H^*_{\\Ctwo}(S^{2\\sigma -2};\n\\underline{\\Z})=H^*(S^{2\\sigma -2}\/\\Ctwo; \\Z), $$\nand $S^{2\\sigma}=S^0*S(2\\sigma)$ is the unreduced suspension of of $S(2\\sigma)$, the second equivalence is a calculation of\nthe cohomology of $\\R P^1$ . \n\\end{proof}\n\n\n\\begin{remark}\nThis proof shows that if $\\Ctwo$ is replaced by a cyclic group of any order\n we still have\n$$\\Z^{H\\Zu}=H\\Zu^* \\simeq \\Sigma^{\\lambda} H\\Zu$$\nwhere $\\lambda =\\eps -\\alpha$ (with $\\eps$ the trivial one\ndimensional complex representation and $\\alpha$ a faithful one\ndimensional representation). \n\\end{remark}\n\nAnderson duality works, of course, also for non-equivariant spectra. We learned the following proposition comparing the equivariant and non-equivariant version in a conversation with Nicolas Ricka.\n\n\\begin{prop}\\label{prop:AndersonFixed}Let $A$ be an abelian group. We have $(A^X)^{C_2} \\simeq A^{(X^{C_2})}$ for every $C_2$-spectrum $X$. \n\\end{prop}\n\\begin{proof}Let $\\infl_e^{C_2} Y$ denote the inflation of a spectrum $Y$ to a $C_2$-spectrum with `trivial action', i.e.\\ the left derived functor of first regarding it as a naive $C_2$-spectrum with trivial action and then changing the universe. This is the (derived) left adjoint for the fixed point functor \\cite[Prop 3.4]{MM02}.\n\nLet $I$ be an injective abelian group. Then there is for every spectrum $Y$ a natural isomorphism \n\\begin{align*}\n[Y, (I^X)^{C_2}] &\\cong [\\infl_e^{C_2} Y, I^X]^{C_2} \\\\\n&\\cong \\mathrm{Hom}(\\pi_0^{C_2}(\\infl_e^{C_2} Y \\otimes X), I)\\\\\n&\\cong \\mathrm{Hom}(\\pi_0(Y\\otimes X^{C_2}), I) \\\\\n&\\cong [Y, I^{(X^{C_2})}].\n\\end{align*}\nHere, we use that fixed points commute with filtered homotopy colimits and cofibre sequences and therefore also with smashing with a spectrum with trivial action. Thus, there is a canonical isomorphism in the homotopy category of spectra between $I^{(X^{C_2})}$ and $(I^X)^{C_2}$ that is also functorial in $I$ (by Yoneda). For a general abelian group $A$, we can write $A^{(X^{C_2})}$ as the fibre of $(I^0)^{X^{C_2}} \\to (I^1)^{X^{C_2}}$ (and similarly for the other side) for an injective resolution $0\\to A\\to I^0\\to I^1$. Thus, we obtain a (possibly non-canonical) equivalence between $A^{(X^{C_2})}$ and $(A^X)^{C_2}$.\n\\end{proof}\n\\begin{remark}An analogous result holds, of course, for every finite group $G$. \n\\end{remark} \n\n\\subsection{Koszul complexes and derived power torsion}\\label{sec:Koszul}\nLet $R$ be a non-equivariantly $E_{\\infty}$ $C_2$-ring spectrum and $M$ be an\n$R$-module. In this section we will recall two versions of stable\nKoszul complexes. Among their merits is that they provide models for\ncellularization or $\\R$-cellularization in cases of interest for us. A basic reference for the material in this section is \\cite{G-M95}. \n\nAs classically, the $r$-power torsion in a module $N$ can be defined as the kernel of $N \\to N[\\frac1r]$, we define the \\emph{derived $J$-power torsion} of $M$ with respect to an ideal $J = (x_1,\\dots, x_n) \\subseteq \\pi^{\\Ctwo}_{\\bigstar}(R)$ as\n\\begin{align*}\\Gamma_J M = \\fib(R \\to R[\\frac1{x_1}]) \\otimes_R \\cdots\n \\otimes_R \\fib(R \\to R[\\frac1{x_n}]) \\otimes_R M .\n\\end{align*}\nThis is also sometimes called the \\emph{stable Koszul complex} and also denoted by $K(x_1,\\dots, x_n)$. As shown in \\cite[Section 3]{G-M95}, this only depends on the ideal $J$ and not on the chosen generators. As algebraically, the derived functors of $J$-power torsion are the local cohomology groups, we might expect a spectral sequence computing the homotopy groups of $\\Gamma_J M$ in terms of local cohomology. As in \\cite[Section 3]{G-M95}, this takes the form\n\\begin{align}\\label{eqn:localcohomology} H_J^s(\\pi_{\\bigstar+V}^{C_2}M) \\Rightarrow \\pi_{V-s}^{C_2}(\\Gamma_JM).\\end{align}\n\nOur second version of the Koszul complex can be defined in the\none-generator case as\n$$\\kappa_R(x) = \\mathop{ \\mathop{\\mathrm {holim}}\\limits_\\rightarrow} \\nolimits \\Sigma^{(1-l)|x|}R\/x^l$$\nfor $x \\in \\pi^{\\Ctwo}_{\\bigstar}(R)$.\nHere, the map $R\/x^l \\to \\Sigma^{-|x|}R\/x^{l+1}$ is induced by the diagram of cofibre sequences\n\\[\n \\xymatrix{\\Sigma^{|x^l|}R \\ar[r]^{x^l}\\ar[d]^=& R \\ar[r]\\ar[d]^x & R\/x^l\\ar@{-->}[d] \\\\\n \\Sigma^{|x^l|}R \\ar[r]^{x^{l+1}} & \\Sigma^{-|x|}R \\ar[r] & \\Sigma^{-|x|}R\/x^{l+1}\n }\n\\]\n\nMore generally, we can make for a sequence $\\mathbf{x} = (x_1,\\dots, x_n)$ in $\\pi^{\\Ctwo}_{\\bigstar}(R)$ the definition\n\\begin{align*}\\kappa_R(\\mathbf{x}; M) &:= \\kappa_R(x_1)\\otimes_R\\cdots \\otimes_R \\kappa_R(x_n)\\otimes_R M\\\\\n&\\simeq \\mathop{ \\mathop{\\mathrm {holim}}\\limits_\\rightarrow} \\nolimits \\Sigma^{-((l_1-1)+\\cdots (l_n-1))|x|} M\/(x_1^{l_1},\\dots, x_n^{l_n})\\end{align*}\n\nThe spectrum $\\kappa_R(x)$ comes with an obvious filtration by $\\Sigma^{(1-l)|x|}R\/x^l$ with filtration quotients $\\Sigma^{-l|x|}R\/x$. We can smash these filtrations together to obtain a filtration of $\\kappa_R(\\mathbf{x})$\nwith filtration quotients wedges of summands of the form $\\Sigma^{-l_1|x_1| -\\cdots - l_n|x_n|}R\/(x_1,\\dots, x_n)$ (see \\cite[1.3.11-12]{tilson} or \\cite[2.8, 2.12]{tilsonArXiv}). Using the following lemma, we obtain also a corresponding filtration on $\\Gamma_JR$. \n\n\\begin{lemma}\\label{lem:Koszul} For $\\mathbf{x}$ as above, we have \n\\begin{align*}\n \\kappa_R(\\mathbf{x}) &\\simeq \\Sigma^{|x_1|+\\cdots +|x_n|+n}\\Gamma_JR.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nSee \\cite[Lemma 3.6]{G-M95}.\n\\end{proof}\n\nWe can also define $\\kappa_R(\\mathbf{x};M)$ (and likewise the other versions of Koszul complexes) for an infinite sequence of $x_i$ by just taking the filtered homotopy colimit over all finite subsequences. Usually Lemma \\ref{lem:Koszul} breaks down in the infinite case.\n\n\\begin{remark}\\label{rmk:hocolim}\n The homotopy colimit defining $\\kappa_R(\\mathbf{x};M)$ has a directed cofinal subsystem, both in the\nfinite and in the infinite case. Indeed, the colimit ranges over all sequences $(l_1,l_2,\\dots)$ with only finitely many entries nonzero. For the directed subsystem, we start with $(0,0,\\dots)$ and raise in the $n$-th step the first $n$ entries by $1$. Directed homotopy colimit are well-known to be weak colimits in the homotopy category of $R$-modules, i.e.\\ every system of compatible maps induces a (possibly non-unique) map from the homotopy colimit \\cite[Sec 3.1]{Mar83} \\cite[Sec II.5]{Sch07}.\n\\end{remark}\n\nOne of the reason for introducing $\\Gamma_JM$ is that it provides a model for the $\\R$-cellularization of $M$ with respect to $R\/J = (R\/x_1) \\otimes_R \\cdots \\otimes_R (R\/x_n)$ in the sense of Section \\ref{sec:Cell}.\n\\begin{prop}\\label{prop:cell}\n Suppose that $x_1\\dots, x_n \\in \\pi_{*\\rho}^{C_2}R$ and set $J = (x_1,\\dots, x_n)$. Then $\\Gamma_JM \\to M$ is a $\\R$-cellularization with respect to $R\/J$ in the (triangulated) category of $R$-modules. \n\\end{prop}\n\\begin{proof}\n Clearly, $\\kappa_R(x_1,\\dots, x_n; M)$ is $\\R$-cellular with respect to $M\/J$; furthermore $M\/J$ is $R\/J$-$\\R$-cellular as clearly $M$ is $R$-cellular. To finish the proof, we have to show that \n $$\\mathrm{Hom}_R(R\/J, \\Gamma_JM) \\to \\mathrm{Hom}_R(R\/J, M)$$\n is an equivalence. Note that $\\Gamma_JM = \\Gamma_{x_n}(\\Gamma_{(x_1,\\dots, x_{n-1})}M)$. Thus, it suffices by induction to show that \n $$\\mathrm{Hom}_R(A\/x, \\Gamma_x B) \\to \\mathrm{Hom}_R(A\/x, B)$$\n is an equivalence for all $R$-modules $A,B$. This is equivalent to \n $$\\mathrm{Hom}_R(A\/x, B[x^{-1}]) = 0$$\nwhich is true as multiplication by $x$ induces an equivalence\n $$\\mathrm{Hom}_R(A,B[x^{-1}]) \\xrightarrow{x^*} \\mathrm{Hom}_R(\\Sigma^{|x|}A, B[x^{-1}]).\\qedhere$$\n\\end{proof}\n\n\\begin{cor}\\label{cor:cellular}\n Let $M$ be a connective $R$-module and $A$ an abelian group. Then the Anderson dual $A^M$ is $\\R$-cellular with respect to $R\/J$ for every ideal $J\\subset \\pi_{\\bigstar}^{C_2}$ finitely generated in degrees $a+b\\sigma$ with $a\\geq 1$ and $a+b\\geq 1$. \n\\end{cor}\n\\begin{proof}\n By the last proposition, we have to show that $\\Gamma_JA^M \\simeq A^M$. For this it suffices to show that $A^M[x^{-1}]$ is contractible for every generator $x$ of $J$. As $M$ is connective, we know that $\\pi_{a+b\\sigma}M = 0$ if $a<0$ and $a+b<0$ (this follows, for example, using the cofibre sequence $(C_2)_+ \\to S^0 \\to S^\\sigma$). Thus, $\\pi_{a+b\\sigma}A^M = 0$ if $a>0$ and $a+b>0$. The result follows. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Real bordism and the spectra $BP\\mathbb{R}\\langle n \\rangle$}\\label{sec:BPRn}\n\\subsection{Basics and homotopy fixed points}\\label{sec:BPRBasics}\nThe $\\Ctwo$-spectrum $M\\R$ of \\emph{Real bordism} was originally defined by Araki and Landweber. In the naive model of $\\Ctwo$-spectra, \nwhere a $\\Ctwo$-spectrum is just given as a sequence $(X_n)$\n of pointed $\\Ctwo$-spaces together with maps \n $$\\Sigma^{\\rho}X_n \\to X_{n+1}$$\n it is just given by the Thom spaces $M\\R_n = BU(n)^{\\gamma_n}$ \n with complex conjugation as $\\Ctwo$-action. Defining it as a strictly commutative $\\Ctwo$-orthogonal spectrum requires more care and was done \n in \\cite[Example 2.14]{SchEquiv} and \\cite[Section B.12]{HHR}. An important fact is that the geometric fixed points of $M\\R$ are equivalent to \n $MO$ (first proven in \\cite{A-M} and reproven in \\cite[Proposition B.253]{HHR}).\n \nAs shown in \\cite{Ara79} and \\cite[Theorem 2.33]{HK}, there is a\nsplitting \n$$M\\R_{(2)} \\simeq \\bigoplus_{i}\\Sigma^{m_i\\rho}BP\\R,$$\nwhere the underlying spectrum of $BP\\R$ agrees with $BP$. This\nsplitting corresponds on geometric fixed points to the splitting \n$$MO \\simeq \\bigoplus_{i}\\Sigma^{m_i}H\\F_2.$$ \nAs shown in \\cite{HK} (see also\nAppendix \\ref{Appendix}), the restriction map \n$$\\pi_{*\\rho}^{\\Ctwo}BP\\R\\to \\pi_{2*}BP$$ \nis an isomorphism. Choose now arbitrary\nindecomposables $v_i \\in \\pi_{2(2^i-1)}BP$ and denote their lifts to\n$\\pi_{(2^i-1)\\rho}^{\\Ctwo}BP\\R$ and their images in $\\pi_{(2^i-1)\\rho}^{\\Ctwo}M\\R$\nby $\\vb_i$. We denote by $BP\\R \\langle n \\rangle$ the quotient \n$$BP\\R\/(\\vb_{n+1},\\vb_{n+2},\\dots)$$ \nin the homotopy category of $M\\R$-modules. At least a priori, this depends on the choice of $v_i$. \n\nWe want to understand the homotopy groups of $BP\\R \\langle n \\rangle$. This was first\ndone by Hu in \\cite{Hu} (beware though that Theorem 2.2 is not correct\nas stated there) and partially redone in \\cite{K-W13}. For the\nconvenience of the reader, we will give the computation again. Note\nthat our proofs are similar but not identical to the ones in the\nliterature. The main difference is that we do not use ascending\ninduction and prior knowledge of $H\\Z$ to compute $\\Phi^{C_2}BP\\R \\langle n \\rangle$,\nbut precise knowledge about $\\pi_{\\bigstar}^{C_2}BP\\R$ -- this is not\nsimpler than the original approach, but gives extra information about\nother quotients of $BP\\R$, which we will need later. We recommend that\nthe reader looks at Appendix A for a complete understanding of the\nresults that follow. \n\nWe will use the\nTate square \\cite{GMTate} and consider the following diagram in which\nthe rows are cofibre sequences: \n\n \\[\\xymatrix@C=0.7cm{\n BP\\R \\langle n \\rangle \\otimes (EC_2)_+ \\ar[r]\\ar[d]^\\simeq & BP\\R \\langle n \\rangle \\ar[r]\\ar[d]& BP\\R \\langle n \\rangle\\otimes \\tilde{E}C_2 \\ar[d]\\ar[r] &\\SigmaBP\\R \\langle n \\rangle\\otimes (EC_2)_+ \\ar[d]\\\\\n BP\\R \\langle n \\rangle^{(EC_2)_+} \\otimes (EC_2)_+ \\ar[r] & BP\\R \\langle n \\rangle^{(EC_2)_+} \\ar[r]& BP\\R \\langle n \\rangle^{(EC_2)_+}\\otimes \\tilde{E}C_2 \\ar[r] & \\SigmaBP\\R \\langle n \\rangle\\otimes (EC_2)_+ \n }\n \\]\n\nAfter taking fixed points this becomes\n \\[\\xymatrix{\n BP\\R \\langle n \\rangle_{h\\Ctwo} \\ar[r]\\ar[d]^= & BP\\R \\langle n \\rangle^{\\Ctwo} \\ar[r]\\ar[d]& BP\\R \\langle n \\rangle^{\\Phi \\Ctwo} \\ar[d]\\ar[r] &\\SigmaBP\\R \\langle n \\rangle_{h\\Ctwo} \\ar[d]\\\\\n BP\\R \\langle n \\rangle_{h\\Ctwo} \\ar[r] & BP\\R \\langle n \\rangle^{h\\Ctwo} \\ar[r]& BP\\R \\langle n \\rangle^{t\\Ctwo} \\ar[r] & \\SigmaBP\\R \\langle n \\rangle_{h\\Ctwo} \n }\n \\]\n\n First, we compute the homotopy groups of the homotopy fixed points. For this we use the $RO(C_2)$-graded homotopy fixed point spectral sequence, described for example in \\cite[Section 2.3]{HM}.\n \n \\begin{prop}\\label{prop:BPRnHomotopy}\n The $RO(\\Ctwo)$-graded homotopy fixed point spectral sequence\n \\[E_2=H^*(\\Ctwo; \\pi^e_{\\bigstar}BP\\R \\langle n \\rangle) \\cong \\Z_{(2)}[\\vb_1,\\dots, \\vb_n, u^{\\pm 1}, a]\/2a \\Rightarrow \\pi^{\\Ctwo}_\\bigstar (BP\\R \\langle n \\rangle^{(E\\Ctwo)_+})\\]\n has differentials generated by $d_{2^{i+1}-1}(u^{2^{i-1}}) = a^{2^{i+1}-1}\\vb_i$ for $i\\leq n$ and $E_{2^{n+1}} = E_\\infty$.\n \\end{prop}\n \\begin{proof}\n The description of $E_{2^{n+1}}$ is entirely analogous to the proof of \\ref{prop:Differentials}, using that $a^{2^{i+1}-1}\\vb_i = 0$ in $\\pi_\\bigstar^{\\Ctwo}BP\\R \\langle n \\rangle^{(E\\Ctwo)_+}$. Now we need to show that there are no further differentials: As every element in filtration $f$ is divisible by $a^f$ in $E^{2^{n+1}}$, the existence of a nonzero $d_m$ (with $m\\geq 2^{n+1}$) implies the existence of a nonzero $d_m$ with source in the $0$-line. Moreover, a nonzero $d_m$ of some element $u^{l}\\vb$ (for $\\vb$ a polynomial in the $\\vb_i$) on the $0$-line implies a nonzero $d_m$ on $u^{l}$ as $\\vb$ is a permanent cycle (in the image from $BP\\R$). The image of such a differential must be of the form $a^mu^{l'}\\vb'$, where $\\vb'$ is a polynomial in $\\vb_1,\\dots, \\vb_n$. As $a^m\\vb_i = 0$ for $1\\leq i\\leq n$ in $E^{2^{n+1}}$, the polynomial $\\vb'$ must be constant. Counting degrees, we see that \n \\[(2l-1)-2l\\sigma = |u^l|-1 = |a^mu^{l'}| = 2l' -(2l'+m)\\sigma\\]\n and thus $m = 2l-2l' = 1$. This is clearly a contradiction. \n \\end{proof}\n \n \\begin{cor}\n We have \n $$\\pi^{C_2}_{\\bigstar}(BP\\R \\langle n \\rangle^{(EC_2)_+}\\otimes \\tilde{E}C_2) \\cong \\F_2[u^{\\pm 2^n}, a^{\\pm 1}].$$\n In particular, we get $\\pi_*BP\\R \\langle n \\rangle^{tC_2} \\cong \\F_2[x^{\\pm 1}]$, where $x = u^{2^n}a^{-2^{n+1}}$ and $|x| = 2^{n+1}$. These are understood to be additive isomorphisms.\n \\end{cor}\n \\begin{proof}\n Recall that \n $$\\pi^{C_2}_{\\bigstar}(BP\\R \\langle n \\rangle^{(EC_2)_+}\\otimes \\tilde{E}C_2) = \\pi^{C_2}_{\\bigstar}(BP\\R \\langle n \\rangle^{(EC_2)_+})[a^{-1}].$$\n as $S^{\\infty\\sigma}$ is a model of $\\tilde{E}C_2$. The result follows as all $\\vb_i$ are $a$-power torsion, but $u^{2^nm}$ is not. \n \\end{proof}\n \n \\subsection{The homotopy groups of $BP\\mathbb{R}\\langle n\\rangle$}\\label{sec:BPRnC2}\n \n Computing the homotopy groups of the fixed points is more delicate\n than the computation of the homotopy fixed points. We first have to\n use our detailed knowledge about the homotopy groups of $BP\\R$.\n Given a sequence $\\underline{l} = (l_1,\\dots)$, we denote by\n $BP\\R\/\\underline{\\vb}^{\\underline{l}}$ the spectrum\n $BP\\R\/(\\vb^{l_{i_1}}_{i_1}, \\vb^{l_{i_2}}_{i_2},\\dots)$, where $i_j$\n runs over all indices such that $l_{i_j}\\neq 0$. Similarly\n $BP\\R\/\\vb_i^j$ is understood to be $BP\\R$ if $j=0$. We use the\n analogous convention when we have algebraic quotients of homotopy\n groups. \n\nWe recommend the reader skips the proof of the following result for\nfirst reading, as the technical detail is not particularly\nilluminating. \n \n \\begin{prop}\\label{prop:BPBound}\nLet $k\\geq 1$ and $\\underline{l} = (l_1,l_2, \\dots)$ be a sequence of nonnegative integers with $l_k=0$. Then the element $\\vb_k$ acts injectively on $(\\pi_{*\\rho-c}^{\\Ctwo}BP\\R)\/\\underline{\\vb}^{\\underline{l}}$ if $0\\leq c \\leq 2^{k+1}+1$, with a splitting on the level of $\\Z_{(2)}$-modules. \n\\end{prop}\n\\begin{proof}\nRecall from Appendix \\ref{Appendix} that $\\pi_\\bigstar^{\\Ctwo}BP\\R$ is isomorphic to the subalgebra of\n $$P\/(2a, \\vb_ia^{2^{i+1}-1})$$\n (where $i$ runs over all positive integers) generated by $\\vb_i(j) = u^{2^ij}\\vb_i$ (with $i,j\\in\\Z$ and $i\\geq 0$) and $a$, where $P = \\Z_{(2)}[a, \\vb_i, u^{\\pm 1}]$. The degrees of the elements are $|a| = 1-\\rho$ and $$|\\vb_i(j)| = (2^i-1)\\rho + 2^ij(4-2\\rho) = (2^i-1-2^{i+1}j)\\rho + 2^{i+2}j.$$\n We add the relations $\\vb_i^{l_i} = 0$ if $l_i\\neq 0$.\n \nWe will first show that the ideal of $\\vb_k$-torsion elements in\n$(\\pi_\\bigstar^{\\Ctwo}BP\\R)\/\\underline{\\vb}^{\\underline{l}}$ is\ncontained in the ideal generated by $a^{2^{k+1}-1}$ and\n$\\vb_s^{l_s-1}\\vb_s(j)$ for $s$ with $l_s \\neq 0$ and $j$ divisible by\n$2^{k-s}$ if $sk$. \n\nNow assume that $x$ is a $\\vb_k$-torsion element not divisible by $a^n$ for $n\\geq 2^{k+1}-1$. Then $x$ must be of the form $\\vb_s^{l_s-1}\\vb_s(j)<$ where $j$ is divisible by $2^{k-s}$ if $sk$, we see that if $|x|$ is of the form $*\\rho - c$ with $c\\geq 0$, then we have $$c \\geq 2^{k+2}-(2^{k+1}-2) = 2^{k+1}+2.$$ The statement about injectivity follows also in this case. \n\nWe still have to show the split injectivity. \nIf $\\vb_k y = 2z$, but $y$ not divisible by $2$, then $y$ must be of the form $2\\vb u^{2^kj}$ in $P$, where $\\vb$ is a polynomial in the $\\vb_i$. Thus, $|y| = 2^{k+2}j +*\\rho$, so we are fine in degree $*\\rho -c$ for $0\\leq c\\leq 2^{k+1}+1 \\leq 2^{k+2}-1$. \n\\end{proof}\n\n\\begin{remark}\n The exact bounds in the preceding proposition are not very important. The only important part for later inductive arguments is that the bound grows with $k$. \n\\end{remark}\n\n\\begin{cor}\\label{Cor:QuotientBP}\nLet $\\underline{l} = (l_1,l_2, \\dots)$ be a sequence with only finitely many nonzero entries and let $j$ be the smallest index such that $l_j \\neq 0$. Then the map \n\\[\n(\\pi_{*\\rho-c}^{\\Ctwo}BP\\R)\/\\underline{\\vb}^{\\underline{l}} \\to \\pi_{*\\rho-c}^{\\Ctwo}(BP\\R\/\\underline{\\vb}^{\\underline{l}})\n\\] \nis an isomorphism for $0\\leq c\\leq 2^{j+1}$.\n\\end{cor}\n\\begin{proof}\nWe use induction on the number $n$ of nonzero indices in $\\underline{l}$. If $n=0$ (and $j=\\infty$), the statement is clear.\n\nFor the step, define $\\underline{l}'$ to be the sequence obtained from $\\underline{l}$ by setting $l_j = 0$. Consider the short exact sequence\n\\[0 \\to (\\pi_{*\\rho-c}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}'}))\/\\vb_j^{l_j} \\to \\pi_{*\\rho-c}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}}) \\to \\left\\{\\pi_{*\\rho-(c+1)}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}'})\\right\\}_{\\vb_j^{l_j}} \\to 0.\\]\nHere, the notion $\\{X\\}_z$ denotes the elements in $X$ killed by $z$. \n\nAssume $c\\leq 2^{j+1}$. By the induction hypothesis, $\\pi_{*\\rho-c}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}'})\\cong (\\pi_{*\\rho-c}^{\\Ctwo}B)\/\\underline{\\vb}^{\\underline{l}'}$ as $c\\leq 2^{j+2}$, so that $(\\pi_{*\\rho-c}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}'}))\/\\vb_j^{l_j} \\cong (\\pi_{*\\rho-c}^{\\Ctwo}B)\/\\underline{\\vb}^{\\underline{l}}$. Furthermore,\n\\begin{align*}\n\\left\\{\\pi_{*\\rho-(c+1)}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}'})\\right\\}_{\\vb_j^{l_j}} &\\cong \\left\\{(\\pi_{*\\rho-(c+1)}^{\\Ctwo}B)\/\\underline{\\vb}^{\\underline{l}'}\\right\\}_{\\vb_j^{l_j}} \\\\\n&\\cong 0\n\\end{align*}\nas follows from $c+1 \\leq 2^{j+2}$ and $c+1 \\leq 2^{j+1}+1$ by the induction hypothesis and Proposition \\ref{prop:BPBound}. Thus, we see that $(\\pi_{*\\rho-c}^{\\Ctwo}B)\/\\underline{\\vb}^{\\underline{l}} \\to \\pi_{*\\rho-c}^{\\Ctwo}(B\/\\underline{\\vb}^{\\underline{l}})$ is an isomorphism.\n\\end{proof}\n\nThe following corollary is crucial:\n\n\\begin{cor}\\label{Cor:crucial}\n Let $I \\subset \\Z_{(2)}[\\vb_1,\\dots]$ be an ideal generated by powers of the $\\vb_i$. Then $BP\\R\/I$ is strongly even. \n\\end{cor}\n\\begin{proof}\n As being strongly even is a property closed under filtered homotopy colimits, we are reduced to the case that $I$ is finitely generated. By the last corollary, it suffices to show that $BP\\R$ itself is strongly even. That the Mackey functor $\\underline{\\pi}_{*\\rho}^{\\Ctwo}(BP\\R)$ is constant is clear from Theorem \\ref{thm:BPR}. \n \n Assume that $x$ is a nonzero element in $\\pi_{*\\rho-1}^{\\Ctwo}BP\\R$. We can assume that $x$ is represented by $a^ku^l\\vb$ in the $E_2$-term of the homotopy fixed point spectral sequence for $BP\\R$, where $\\vb$ is a monomial in the $\\vb_i$ (with $\\vb_0 =2$), $k\\geq 0$ and $l\\in\\Z$. The element $x$ is in degree $k+4l + *\\rho$. Let $j\\geq 0$ be the minimal number such that $\\vb_j|\\vb$. Then $2^j|l$ and $k\\leq 2^{j+1}-2$. This is clearly in contradiction with $k+4l = -1$. \n\\end{proof}\n\nWe recover the $C_2$-case of the reduction theorem of \\cite[Prop 4.9]{HK} and \\cite[Thm 6.5]{HHR}.\n\\begin{cor}\\label{cor:reduction}\n There is an equivalence $BP\\R\/(\\vb_1,\\vb_2,\\dots) \\simeq H\\underline{\\Z}_{(2)}$.\n\\end{cor}\n\\begin{proof}\n This follows directly from the last corollary and Corollary \\ref{cor:characterizingZ}. \n\\end{proof}\n\n\n\n\\begin{cor}\\label{cor:rho+}\n Let $I \\subset \\Z_{(2)}[\\vb_1,\\dots]$ be an ideal generated by powers of the $\\vb_i$. Then \n $$\\pi_{*\\rho+1}^{C_2}BP\\R\/I \\cong \\F_2\\{a\\}\\otimes \\Z_{(2)}[\\vb_1, \\vb_2,\n \\dots]\/I.$$\n \\end{cor}\n \\begin{proof}\n As $BP\\R\/I$ is strongly even, this follows from \\cite[Lemma 2.15]{HM}.\n \\end{proof}\n\n\nThis allows us to compute $\\pi_\\bigstar^{\\Ctwo}BP\\R \\langle n \\rangle$.\n \n \\begin{prop}\\label{prop:BPRnFixedPoints}\n The spectrum $BP\\R \\langle n \\rangle$ is the connective cover of its Borel completion $BP\\R \\langle n \\rangle^{(E\\Ctwo)_+}$. The cofibre $C$ of $BP\\R \\langle n \\rangle \\to BP\\R \\langle n \\rangle^{(E\\Ctwo)_+}$ has homotopy groups $$\\pi_\\bigstar^{\\Ctwo}C \\cong \\F_2[a^{\\pm 1}, u^{-2^n}]u^{-2^n},$$\n with the naming of the elements indicating the map $\\pi_\\bigstar^{\\Ctwo}BP\\R \\langle n \\rangle^{(E\\Ctwo)_+} \\to \\pi_\\bigstar^{\\Ctwo}C$.\n \\end{prop}\n \\begin{proof}\n This is clear on underlying homotopy groups. Thus, we have only to show that $BP\\R \\langle n \\rangle^{\\Ctwo} \\to BP\\R \\langle n \\rangle^{h\\Ctwo}$ is a connective cover. For that purpose, it is enough to show that $BP\\R \\langle n \\rangle^{\\Phi \\Ctwo}$ is connective and that the fibre of $BP\\R \\langle n \\rangle^{\\Phi \\Ctwo} \\to BP\\R \\langle n \\rangle^{t\\Ctwo}$ has homotopy groups only in negative degrees. \n \n We have $BP\\R \\langle n \\rangle^{\\Phi \\Ctwo} \\simeq BP\\R^{\\Phi \\Ctwo}\/(\\vb_{n+1},\\dots)$. As noted in the proof of Proposition \\ref{prop:vbn}, the image of $\\vb_i$ in $M\\R^{\\phi \\Ctwo}$ is $0$. As the quotient $BP\\R^{\\Phi \\Ctwo}\/\\vb_{n+1},\\dots$ can be taken in the category of $M\\R^{\\Phi \\Ctwo}$-modules, we are only quotiening out by $0$. It follows easily that $(BP\\R\/(\\vb_{n+1},\\dots, \\vb_{n+m}))^{\\Phi \\Ctwo}$ has in the homotopy groups an $\\F_2$ in all degrees of the form $\\sum_{i=n+1}^{n+m}\\varepsilon_i(|v_i|+1) = \\sum_{i=n+1}^{n+m}\\varepsilon_i2^i$ with $\\varepsilon_i = 0$ or $1$. As geometric fixed point commute with homotopy colimits, we see that $\\pi_*BP\\R \\langle n \\rangle^{\\Phi \\Ctwo} \\cong \\F_2[y]$ (additively) with $|y| = 2^{n+1}$. It remains to show that $y^k$ maps nonzero to $\\pi_*BP\\R \\langle n \\rangle^{t\\Ctwo}$ (and hence maps to $x^k)$. \n \n It is enough to show that $a^{-|y^k|-1}y^k$ maps nonzero to $\\pi_\\bigstar^{\\Ctwo} \\Sigma BP\\R \\langle n \\rangle\\otimes (E\\Ctwo)_+$ in the sequence coming from the Tate square, i.e.\\ that $a^{-|y^k|-1}y^k$ is not in the image from (the fixed points of) $BP\\R \\langle n \\rangle$. But $a^{-|y^k|-1}y^k$ is in degree $(|y^k|+1)\\rho-1$ and $\\pi_{(|y^k|+1)\\rho-1}^{\\Ctwo}BP\\R \\langle n \\rangle = 0$ by Corollary \\ref{Cor:crucial}. \n \\end{proof}\n \n Let us describe the homotopy groups of $BP\\R \\langle n \\rangle$ in more detail. We set $\\vb_0 = 2$ for convenience. Denote by $BB$ (for \\emph{basic block}) the $\\Z_{(2)}[a, \\vb_1,\\dots,\\vb_n]\/2a$-submodule of \n $$\\Z_{(2)}[\\vb_1,\\dots, \\vb_n]\/(a^{2^{k+1}-1}\\vb_k)_{0\\leq k \\leq n}$$\n generated by $1$ and by $\\vb_k(m) = u^{2^km}\\vb_k$ for $0\\leq k < n$ and $0< m <2^{n-k}$.\nBy Proposition \\ref{prop:BPRnHomotopy}, we see that \n $$\\pi_{\\bigstar}^{C_2}BP\\R \\langle n \\rangle^{(EC_2)_+} \\cong BB[U^{\\pm 1}]$$\n with $U = u^{2^n}.$ Note that this isomorphism is not claimed to be multiplicative; in general, $BP\\R \\langle n \\rangle$ is not even known to have any kind of (homotopy unital) multiplication.\n \n Define $BB'$ to be the kernel of the map $BB \\to \\F_2[a]$ given by sending all $\\vb_k$ and $\\vb_k(m)$ to zero. Set $NB = \\Sigma^{\\sigma-1}\\F_2[a]^\\vee \\oplus BB'$, where $NB$ stands for \\emph{negative block}. Then it is easy to see from the last proposition that \n $$\\pi_{\\bigstar}^{C_2}BP\\R \\langle n \\rangle \\cong BB[U] \\oplus U^{-1}NB[U^{-1}],$$ \n where this isomorphism is again only meant additively. We will be a little bit more explicit about the homotopy groups of $BP\\R \\langle n \\rangle$ in the cases $n=1$ and $2$ in Part \\ref{part:LocalCohomology}. \n\n\\subsection{Forms of $BP\\mathbb{R}\\langle n\\rangle$}Our next goal is\nto identify certain spectra as forms of $BP\\R \\langle n \\rangle$. We take the following definition from \\cite{HM}:\n\n\\begin{defn}\nLet $E$ be an even $2$-local commutative and associative $\\Ctwo$-ring spectrum up to homotopy. By \\cite[Lemma 3.3]{HM}, $E$ has a Real orientation and after choosing one, we obtain a formal group law on $\\pi_{*\\rho}^{\\Ctwo}E$. The $2$-typification of this formal group law defines a map $\\pi^e_{2*}BP \\cong \\pi_{*\\rho}^{C_2}BP\\R \\to \\pi_{*\\rho}^{C_2}E$. We call $E$ a \\emph{form of $BP\\R\\langle n\\rangle$} if the map\n\\[\\underline{\\Z_{(2)}[\\vb_1,\\dots, \\vb_n]} \\subset \\underline{\\pi}_{*\\rho}BP\\R \\to \\underline{\\pi}_{*\\rho} E\\]\nis an isomorphism of constant Mackey functors. \n\nThis depends neither on the choice of $\\vb_i$ nor on the chosen Real orientation, as can be seen using that $\\vb_i$ is well-defined modulo $(2, \\vb_1,\\dots, \\vb_{i-1})$. \n\\end{defn}\n\nEquivalently, one can say that $E$ is a form of $BP\\R \\langle n \\rangle$ if and only if\n$E$ is strongly even and its underlying spectrum is a form of\n$BP\\langle n\\rangle$. We want to show that every form of $BP\\R\\langle\nn\\rangle$ is also of the form $BP\\R \/\\vb_{n+1}, \\vb_{n+2}, \\ldots $\nfor some choice of elements $\\vb_i$. For this, we need the following lemma from \\cite[Lemma 3.4]{HM}:\n\\begin{lemma}\\label{lem:regrep}\nLet $f\\colon E\\to F$ be a map of $\\Ctwo$-spectra. Assume that f induces isomorphisms \n\\[\\pi^{\\Ctwo}_{k\\rho}E \\to \\pi^{\\Ctwo}_{k\\rho}E \\quad \\text{and} \\quad \\pi_kE \\to \\pi_kF\\]\nfor all $k\\in\\Z$. Assume furthermore that $\\pi^{\\Ctwo}_{k\\rho-1}E \\to \\pi^{\\Ctwo}_{k\\rho-1}F$ is an injection for all $k\\in\\Z$ (for example, if $\\pi^{\\Ctwo}_{k\\rho-1}E =0$). Then $f$ is an equivalence of $\\Ctwo$-spectra.\n\\end{lemma}\n\\begin{prop}\\label{prop:FormBPRn}\n Let $E$ be a form of $BP\\R\\langle n\\rangle$. Then one can choose\n indecomposables $\\vb_i\\in \\pi_{(2^i-1)\\rho}^{\\Ctwo}BP\\R$ for $i\\geq\n n+1$ such that $E \\simeq BP\\R \/(\\vb_{n+1},\\vb_{n+2},\\dots)$. \n\\end{prop}\n\\begin{proof}\n First choose any system of $\\vb_i$. Choose furthermore a Real orientation $f\\colon BP\\R \\to E$ and denote $f(\\vb_i)$ by $x_i$. Define a multiplicative section \n $$s\\colon \\pi_{*\\rho}^{\\Ctwo}E \\to \\pi_{*\\rho}^{\\Ctwo}BP\\R$$ by $s(x_i) = \\vb_i$ for $1\\leq i \\leq n$. \n \n Now define a new system of $\\vb_i$ by \n \\[\\vb_i^{\\mathrm{new}} = \\vb_i - s(f_*(\\vb_i))\\]\n for $i\\geq n+1$. As these agree with $\\vb_i$ mod $(\\vb_1,\\dots, \\vb_n)$, they are still indecomposable. Furthermore, the $\\vb_i^{\\mathrm{new}}$ are for $i\\geq n+1$ clearly in the kernel of $f_*$. Thus, we obtain a map $BP\\R \\langle n \\rangle\/(\\vb_{n+1}^{\\mathrm{new}},\\vb_{n+2}^{\\mathrm{new}},\\dots) \\to E$ that is an isomorphism on $\\pi_{*\\rho}^{\\Ctwo}$. By Corollary \\ref{Cor:crucial}, the source is strongly even. By Lemma \\ref{lem:regrep}, the map is an equivalence. \n\\end{proof}\n\n\\begin{examples}\\label{exa:forms}We consider Real versions of the classical examples $ku$ and $tmf_1(3)$.\n \\begin{enumerate}\n \\item The connective Real K-theory spectrum $k\\R_{(2)}$ is a form of $BP\\R\\langle 1\\rangle$. Indeed, the underlying spectrum $ku_{(2)}$ is well known to be a form of $BP\\langle 1\\rangle$ and $k\\R_{(2)}$ is also strongly even (as can be seen by the results from \\cite[3.7D]{B-G10} or from the computation in Section \\ref{sec:kRlcss}). \n \\item Define $\\overline{tmf_1(3)}$ as the equivariant connective cover of the spectrum $\\overline{Tmf_1(3)}$, i.e.\\ $Tmf_1(3)$ with the algebro-geometrically defined $\\Ctwo$-action (see \\cite[Section 4.1]{HM} for details). As shown in \\cite[Corollary 4.17]{HM}, $\\overline{tmf_1(3)}_{(2)}$ is a form of $BP\\R\\langle 2\\rangle$. By Proposition \\ref{prop:FormBPRn}, we can construct $\\overline{tmf_1(3)}_{(2)}$ by killing a sequence $\\vb_2, \\vb_3,\\dots$ in $BP\\R$. This construction is used in \\cite{L-OString} to define a $\\Ctwo$-equivariant version of $tmf_1(3)_{(2)}$. In particular, we see (using the discussion before Proposition 4.23 in \\cite{HM}) that $\\overline{TMF_1(3)}_{(2)}$ (with the algebro-geometrically defined $\\Ctwo$-action) agrees with the $\\mathbb{TMF}_1(3)_{(2)}$ of \\cite{L-OString}.\n \\end{enumerate}\n\\end{examples}\n\n\\section{Results and consequences}\\label{sec:results}\nIn this section, we want to discuss our main results in more detail than in the introduction and we will also derive some consequences and give some examples. Recall to that purpose the notation from Sections \\ref{sec:Koszul} and \\ref{sec:BPRBasics}. Furthermore, we will implicitly localize everything at $2$ so that $\\Z$ means $\\Z_{(2)}$ etc. Our main theorem is the following:\n\\begin{thm}\\label{thm:main}\nLet $(m_1,m_2,\\dots)$ be a sequence of nonnegative integers with only finitely many entries bigger than $1$ and let $M$ be the quotient $BP\\R\/(\\vb_1^{m_1},\\vb_2^{m_2},\\dots)$, where we only quotient by the positive powers of $\\vb_i$. Denote by $\\underline{\\vb}$ the sequence of $\\vb_i$ in $\\pi^{\\Ctwo}_{\\bigstar}M\\R$ such that $m_i = 0$, by $|\\underline{\\vb}|$ the sum of their degrees and by $m'$ the sum of all $(m_i-1)|\\vb_i|$ for $m_i> 1$. Then \n$$\\Z^M \\simeq \\Sigma^{-m'+4-2\\rho}\\kappa_{M\\R}(\\underline{\\vb}; M).$$\n\nThe most important case is that $m_{n+1} = m_{n+2} = \\cdots = 1$ so that \n$$M = BP\\R \\langle n \\rangle\/(\\vb_1^{m_1},\\dots,\\vb_n^{m_n}).$$\nIf $k$ is the number of elements in $\\underline{\\vb}$, we also get \n\\begin{align*}\n\\Z^M \\simeq \\Sigma^{-m'+k+|\\underline{\\vb}| +4-2\\rho}\\Gamma_{\\underline{\\vb}}M,\n\\end{align*}\nwhere we view $M$ as an $M\\R$-module. \n\\end{thm}\nThe first form will be proved as Theorem \\ref{Thm:QuotientDuality} and the second follows from it using Lemma \\ref{lem:Koszul}. The second form also follows from Corollary \\ref{cor:BPRnGorDdish} (using that $\\Gamma_{\\underline{\\vb}}$ preserves cofibre sequences to pass to quotients of $BP\\R \\langle n \\rangle$). \n\n\\begin{example}\\label{ex:BPRn}\n$\\Z^{BP\\R \\langle n \\rangle} \\simeq \\Sigma^{n+D_n\\rho +4-2\\rho}\\Gamma_{(\\vb_1,\\dots, \\vb_n)}BP\\R \\langle n \\rangle$ for $D_n = |v_1|+\\cdots +|v_n|$. This says that $BP\\R \\langle n \\rangle$ has Gorenstein duality with respect to $H\\Zu \\simeq BP\\R \\langle n \\rangle\/(\\vb_1,\\dots, \\vb_n)$. (The last equivalence follows from Corollary \\ref{cor:reduction}.)\n\\end{example}\n\n\\begin{example}\nSet $k\\R(n) = BP\\R \\langle n \\rangle\/(\\vb_1,\\dots, \\vb_{n-1})$ to be connective\nintegral Real Morava $K$-theory and $K\\R(n) = k\\R(n)[\\vb_n^{-1}]$ its periodic version. Then \n\\begin{align*}\\Z^{k\\R(n)} &\\simeq \\Sigma^{1+|\\vb_n|+4-2\\rho}\\Gamma_{\\vb_n}k\\R(n)\\\\\n&\\simeq \\Sigma^{(2^n-3)\\rho+4}\\cof(k\\R(n) \\to K\\R(n))\\end{align*}\nThis includes for $n=1$ the case of usual ($2$-local) connective Real K-theory. \n\\end{example}\n\\begin{example}\n To have a slightly stranger example, take $M = BP\\R\\langle 3\\rangle\/(\\vb_1^4, \\vb_3^2)$. Then \n $$\\Z^M \\simeq \\Sigma^{5-9\\rho}\\Gamma_{\\vb_2}M.$$\n\\end{example}\n\n\\vspace*{0.5cm}\n\nSo far, we have only talked about \\emph{quotients} of $BP\\R$. This does not include important Real spectra like the Real Johnson--Wilson theories $E\\R(n) = BP\\R \\langle n \\rangle[\\vb_n^{-1}]$ or the (integral) Real Morava K-theories $K\\R(n)$. For this, we have to study the behaviour of our constructions under localizations. \n\nLet $M$ be an $RO(C_2)$-graded $\\Z[v]$-module, where $v$ has some degree $|v| \\in RO(C_2)$. We say that $M$ has \\emph{bounded $v$-divisibility} if for every degree $a+b\\sigma$, there is a $k$ such that \n$$v^k\\colon M_{a+b\\sigma-|v^k|} \\to M_{a+b\\sigma}$$\nis zero. We will also apply the concept to modules that are just $\\Z|v|$-graded.\n\\begin{lemma}The class of $RO(C_2)$-graded $\\Z[v]$-modules of bounded $v$-divisibility is closed under submodules, quotients and extensions. \n\\end{lemma}\n\\begin{proof}\n This is clear for submodules and quotients. Let \n $$0 \\to K \\to M \\to N \\to 0$$\n be a short exact sequence of $\\Z[v]$-modules where $K$ and $N$ are of bounded $v$-divisibility. For a given degree $\\alpha \\in RO(C_2)$, we know that there is a $k$ such that $v^k$ maps trivially into $K_\\alpha$. Furthermore, there is an $n$ such that $v^n$ maps trivially into $N_{\\alpha-k|v|}$. Thus, multiplication by $v^{n+k}$ is the zero map $M_{\\alpha-(k+n)|v|} \\to M_\\alpha$.\n\\end{proof}\n\n\nLet $M$ be an $M\\R$-module. We say that $M$ is of \\emph{bounded $\\vb_n$-divisibility} if both $\\pi^{\\Ctwo}_{\\bigstar}M$ and $\\pi^e_*M$ are of bounded $\\vb_n$-divisibility. This is, for example, true if $M$ is connective. \n\n\\begin{lemma}\\label{lem:boundedrho}We have the following two properties of $\\vb_n$-divisiblity.\n\\begin{enumerate}\n\\item Being of bounded $\\vb_n$-divisibility is closed under cofibres and suspensions. \n\\item An $M\\R$-module $M$ is of bounded $\\vb_n$-divisibility if and only if $\\pi^{\\Ctwo}_{*\\rho}M$ and $\\pi^e_*M$ are of bounded $\\vb_n$-divisibility. \n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nBoth statements follow from the last lemma. For the second item, we additionally use the exact sequence\n$$\\pi^e_{a+b+1}M \\to \\pi^{C_2}_{a+(b+1)\\sigma}M \\to \\pi^{C_2}_{a+b\\sigma}M \\to \\pi^e_{a+b}M$$\ninduced by the cofibre sequence\n$$(C_2)_+ \\to S^0\\to S^{\\sigma}.\\qedhere$$\n\\end{proof}\n\n\\begin{lemma}\nIf $M$ has bounded $\\vb_n$-divisibility, then there is a natural equivalence \n$$M[\\vb_n^{-1}] \\simeq \\Sigma \\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits\\left( \\cdots \\to \\Sigma^{|\\vb_n|}\\Gamma_{\\vb_n}M \\xrightarrow{\\vb_n} \\Gamma_{\\vb_n} M\\right)$$\nof $M\\R$-modules.\n\\end{lemma}\n\\begin{proof}\nWe apply the endofunctor $H\\colon N\\mapsto \\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits(\\cdots \\to \\Sigma^{|\\vb_n|}N \\xrightarrow{\\vb_n} N)$ of $M\\R$-modules to the cofibre sequence\n$$\\Gamma_{\\vb_n}M \\to M\\to M[\\vb_n^{-1}].$$\nClearly $H(M[\\vb_n^{-1}])\\simeq M[\\vb_n^{-1}]$. Thus, we just have to show that $H(M)\\simeq 0$. This follows by the $\\lim^1$-sequence and bounded $\\vb_n$-divisibility. \n\\end{proof}\n\n\\begin{lemma}\nLet $B$ be a quotient of $BP\\R$ by powers of the $\\vb_i$. Then\n$B[\\vb^{-1}]$ has bounded $\\vb_n$-divisibility if $\\vb$ is a product\nof $\\vb_i$ not containing $\\vb_n$. Hence, the same is also true for\nthe stable Koszul complex $\\Gamma_{\\underline{\\vb}}B$, where $\\underline{\\vb}$ is a sequence of $\\vb_i$ not containing $\\vb_n$.\n\\end{lemma}\n\\begin{proof}\nBy Lemma \\ref{lem:boundedrho}, it is enough to check the first statement on $\\pi_{*\\rho}^{C_2}$ and on $\\pi^e_*$. On the latter, it is clear and the former is isomorphic to it by Corollary \\ref{Cor:crucial}. For the second statement we use that $\\Gamma_{\\underline{\\vb}}B$ is the fibre of $B \\to \\check{C}(\\underline{\\vb};B)$, where $\\check{C}(\\underline{\\vb};B)$ has a filtration with subquotients $M\\R$-modules of the form $\\Sigma^?B[x^{-1}]$ for some $x\\in \\pi_{\\bigstar}^{C_2}M\\R$ \\cite[Lemma 3.7]{G-M95}. Thus, the second statement follows from Lemma \\ref{lem:boundedrho}. \n\\end{proof}\n\n\\begin{thm}\nLet the notation be as in Theorem \\ref{thm:main} and assume for simplicity that only finitely many $m_i$ are zero and that $m_n = 0$. Then \n$$\\Z^{M[\\vb_n^{-1}]} \\simeq \\Sigma^{-m'+|\\underline{\\vb}|+(k-1)+4-2\\rho}\\Gamma_{\\underline{\\vb}\\setminus \\vb_n} M.$$\nHere $\\underline{\\vb} \\setminus \\vb_n$ denotes the sequence of all $\\vb_i$ such that $m_i = 0$ and $i\\neq n$. \n\\end{thm}\n\\begin{proof}\nThe preceding lemmas imply the following chain of equivalences:\n\\begin{align*}\n\\Z^{M[\\vb_n^{-1}]} &\\simeq \\Z^{\\mathop{ \\mathop{\\mathrm {holim}}\\limits_\\rightarrow} \\nolimits (M \\xrightarrow{\\vb_n} \\Sigma^{-|\\vb_n|}M \\xrightarrow{\\vb_n} \\cdots)} \\\\\n&\\simeq \\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits (\\cdots \\xrightarrow{\\vb_n} \\Z^M) \\\\\n&\\simeq \\Sigma^{-m'+|\\underline{\\vb}|+k+4-2\\rho}\\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits \\left(\\cdots \\xrightarrow{\\vb_n} \\Gamma_{\\underline{\\vb}}M\\right)\\\\\n&\\simeq \\Sigma^{-m'+|\\underline{\\vb}|+k+4-2\\rho}\\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits \\left(\\cdots \\xrightarrow{\\vb_n} \\Gamma_{\\vb_n}(\\Gamma_{\\underline{\\vb}\\setminus \\vb_n}M) \\right) \\\\\n&\\simeq \\Sigma^{-m'+|\\underline{\\vb}|+(k-1)+4-2\\rho}(\\Gamma_{\\underline{\\vb}\\setminus \\vb_n} M)[\\vb_n^{-1}] \\\\\n&\\simeq \\Sigma^{-m'+|\\underline{\\vb}|+(k-1)+4-2\\rho}\\Gamma_{\\underline{\\vb}\\setminus \\vb_n} (M[\\vb_n^{-1}])\n\\end{align*}\n\\end{proof}\n\n\\begin{example}\nWe recover the following result by Ricka \\cite{Ricka}: \n$$\\Z^{K\\R(n)} \\simeq \\Sigma^{4-2\\rho}K\\R(n).$$\nHere, $K\\R(n)$ denotes integral Morava K-theory $E\\R(n)\/(\\vb_1,\\dots, \\vb_{n-1})$. \n\\end{example}\n\n\\begin{example}\nIn the following, we will use that there are invertible classes $x,\\vb_n\\in\\pi_\\bigstar^{\\Ctwo}E\\R(n)$ of degree $-2^{2n+1}+2^{n+2}-\\rho$ and $(2^n-1)\\rho$ respectively, where $x = \\vb_n^{1-2^n}u^{2^n(1-2^{n-1})}$.\n\\begin{align*}\n\\Z^{E\\R(n)} &\\simeq \\Sigma^{D_{n-1}\\rho + (n-1)+4-2\\rho} \\Gamma_{(\\vb_1,\\dots, \\vb_{n-1})}E\\R(n) \\\\\n&\\simeq \\Sigma^{-(n+2)\\rho+(n+3)} \\Gamma_{(\\vb_1,\\dots, \\vb_{n-1})}E\\R(n) \\\\\n&\\simeq \\Sigma^{(n+2)(2^{2n+1}-2^{n+2})+n+3} \\Gamma_{(\\vb_1,\\dots, \\vb_{n-1})}E\\R(n).\n\\end{align*}\nThis says that $E\\R(n)$ has Gorenstein duality with respect to $E\\R(n)\/(\\vb_1,\\dots, \\vb_{n-1}) = K\\R(n)$. Note that we can replace the ideal $(\\vb_1,\\dots, \\vb_{n-1})$ by an ideal generated in integral degrees, namely $(\\vb_1x, \\dots, \\vb_{n-1}x^{2^{n-1}-1})$. \n\\end{example}\n\n\\begin{example}\\label{exa:tmf}\nRecall from \\cite{HM} the spectra $tmf_1(3)$, $Tmf_1(3)$ and $TMF_1(3)$ and the corresponding $C_2$-spectra $\\overline{tmf_1(3)}$, $\\overline{Tmf_1(3)}$ and $\\overline{TMF_1(3)}$. Recall that we have $\\pi_*tmf_1(3) = \\Z[a_1,a_3]$, where $a_1$ and $a_3$ can be identified with the images of the Hazewinkel generators $v_1$ and $v_2$, and that $\\overline{tmf_1(3)}$ is a form of $BP\\R\\langle 2\\rangle$ (as already discussed in Example \\ref{exa:forms}). This gives the Anderson dual of $\\overline{tmf_1(3)}$. Tweaking the last theorem a little bit, allows also to show that \n$$\\Z^{\\overline{TMF_1(3)}} \\simeq \\Sigma^{5+2\\rho}\\Gamma_{\\vb_1} \\overline{TMF_1(3)}.$$\nWe can also recover one of the main results of \\cite{HM}, namely that $\\Z^{\\overline{Tmf_1(3)}} \\simeq \\Sigma^{5+2\\rho} \\overline{Tmf_1(3)}$. Indeed, $Tmf_1(3)$ is by \\cite[Section 4.3]{HM} the cofibre of the map \n$$\\Gamma_{\\vb_1,\\vb_2}\\overline{tmf_1(3)} \\to \\overline{tmf_1(3)}.$$\nAs the source is equivalent to $\\Sigma^{-6-2\\rho}\\Z^{\\overline{tmf_1(3)}}$, applying Anderson duality shows that $\\Z^{\\overline{Tmf_1(3)}}$ is the fibre of\n$$\\Sigma^{6+2\\rho}\\overline{tmf_1(3)} \\to \\Sigma^{6+2\\rho} \\Gamma_{\\vb_1,\\vb_2} \\overline{tmf_1(3)}.$$\nThis is equivalent to $\\Sigma^{5+2\\rho}\\overline{Tmf_1(3)}$. \nThis example does not require 2-localization, only that $3$ is inverted.\n\\end{example}\n\n\\begin{remark}By Proposition \\ref{prop:AndersonFixed}, all the results in this section have direct implications for the Anderson duals of the fixed point spectra. These are easiest to understand in the case of $ER(n) = (E\\R(n))^{C_2}$, where we get\n$$\\Z^{ER(n)} \\simeq \\Sigma^{(n+2)(2^{2n+1}-2^{n+2})+n+3} \\Gamma_{(\\vb_1x,\\dots, \\vb_{n-1}x^{2^n-1})}ER(n).$$\n\\end{remark}\n\n\\vspace{0.8cm}\n\\part{The Gorenstein approach}\nIn this part, we explain the Gorenstein approach to prove Gorenstein duality, first for $k\\R$ and then for $BP\\R \\langle n \\rangle$.\n\\section{Connective $K$-theory with Reality}\n\\label{sec:kR}\nThe present section considers $K$-theory with reality, which is more\nfamiliar than $BP\\R \\langle n \\rangle$ for general $n$, and no 2-localization is\nnecessary. The arguments are especially\nsimple, firstly because $k\\R$ is a commutative ring spectrum, and\nsecondly becaue we only need to\nconsider principal ideals. Simple as the argument is, we see in\nSection \\ref{sec:kRlcss} that the consequences for coefficient rings\nare interesting. \n\n\\subsection{Gorenstein condition and Matlis lift}\nIt is well known that there is a cofibre sequence\n$$\\Sigma^{\\eps}ku\\stackrel{v}\\longrightarrow ku \\longrightarrow H\\Z. $$\nIf one knows the coefficient ring $ku_*=\\Z [v]$, this is easy \nto construct, since we can identify $ku\/v$ as the Eilenberg-MacLane\nspectrum from its homotopy groups. \n\nThere is a version with Reality \\cite{Dugger}. Indeed, we may\nconstruct the cofibre sequence\n$$\\Sigma^{\\rho} k\\R \\stackrel{\\vb}\\longrightarrow k\\R \\longrightarrow H\\Zu, $$\nwhere $k\\R \/\\vb$ is identified using Corollary \\ref{cor:characterizingZ}\n \nSince the Dugger sequence is self dual we immediately deduce that\n$k\\R$ is Gorenstein. \n\n\\begin{lemma}\n\\label{lem:kRGor}\n$$\\mathrm{Hom}_{k\\R}(H\\Zu, k\\R)=\\Sigma^{-\\rho-1}H\\Zu$$\nand $k\\R \\longrightarrow H\\Zu$ is Gorenstein. \n\\end{lemma}\n\n\\begin{proof}\nApply $\\mathrm{Hom}_{k\\R}(\\cdot , k\\R)$ to the Dugger sequence. \n\\end{proof}\n\nTo actually get Gorenstein duality we need to construct a Matlis\nlift (adapted from \\cite[Section 6]{DGI}), which is a counterpart in topology of the injective hull of the residue\nfield. \n\\begin{defn}\nIf $M$ is an $H\\Zu$-module, we say that a $k\\R$-module $\\tilde{M}$ is\na {\\em Matlis lift} of $M$ if $\\tilde{M}$ is $H\\Zu$-$\\R$-cellular and \n$$\\mathrm{Hom}_{k\\R}(T, \\tilde{M})\\simeq \\mathrm{Hom}_{H\\Zu}(T, M)$$\nfor all $H\\Zu$-modules $T$.\n\\end{defn}\n\nThe Anderson dual provides one such example. \n\n\\begin{lemma}\n\\label{lem:ML}\nThe $k\\R$-module $\\Sigma^{-2(1-\\sigma)}\\Z^{k\\R}$ is a Matlis lift of\n$H\\Zu$. Indeed, \n\n(i) $\\underline{\\Z}^{k\\R}$ is $H\\Zu$-$\\R$-cellular and \n\n(ii) There is an equivalence \n$$\\Sigma^{2\\pp} H\\Zu\\simeq H\\Zu^*=\\mathrm{Hom}_{k\\R} (H\\Zu, \\Z^{k\\R}), $$\nwhere $\\delta =1-\\sigma$. \n\\end{lemma}\n\n\\begin{proof}\nOne could prove the first part from the slice tower, but it also follows directly from Corollary \\ref{cor:cellular}. \n\nThe second statement is immediate from Lemma \\ref{lem:Zu}. \n\\end{proof}\n\n\\subsection{Gorenstein duality}\n We next want to move on to Gorenstein duality, so we write\n$$\\cE=\\mathrm{Hom}_{k\\R}(H\\Zu, H\\Zu). $$\n\nCombining Lemmas \\ref{lem:kRGor} and \\ref{lem:ML}, we have \n\\begin{eqnarray}\\label{eq:kReq}\\mathrm{Hom}_{k\\R}(H\\Zu, k\\R)\\simeq \\Sigma^{-\\rho-1}H\\Zu \\simeq\n\\mathrm{Hom}_{k\\R}(H\\Zu, \\Sigma^{-4+\\sigma}\\Z^{k\\R})\\end{eqnarray}\n\nWe now want to remove the $\\mathrm{Hom}_{k\\R}(H\\Zu , \\cdot )$ from this\nequivalence. \n\n\\begin{lemma} {\\em (Effective constructibility)}\n\\label{lem:effective}\nThe evaluation map \n$$\\mathrm{Hom}_{k\\R}(H\\Zu, M)\\otimes_{\\cE}H\\Zu \\longrightarrow M$$\nis $H\\Zu$-$\\R$-cellularization for every left $k\\R$-module $M$.\n\\end{lemma}\n\n\\begin{proof}\nSince the domain is clearly $H\\Zu$-$\\R$-cellular, it is enough to show the map is an\nequivalence for all cellular modules $M$.\n\nThis is clear for $M=H\\Zu$. The class of $M$ for which the statement is true is closed under (i) triangles, (ii)\ncoproducts (since $H\\Zu$ is small) and (iii) suspensions by\nrepresentations. This gives all $\\R$-cellular modules. \n\\end{proof}\n\nLocal cohomology gives an alternative approach to\ncellularization. Recall that we define the $\\vb$-power torsion of a $k\\R$-module $M$\nby the fibre sequence\n$$\\Gamma_{\\vb}M \\longrightarrow M \\longrightarrow M[1\/\\vb]. $$\n\nThe following lemma is a special case of Proposition \\ref{prop:cell}. \n\\begin{lemma}\n\\label{lem:Gammacell}\nThe map \n$$\\Gamma_{\\vb}M\\longrightarrow M$$ \nis $H\\Zu$-$\\R$-cellularization.\n\\end{lemma}\n\nIt remains to check that the two $\\cE$-actions on $H\\Zu$ coincide. \n\n\n\\begin{lemma}\\label{lem:UniquenesskR}\nThere is a unique right $\\cE$-module structure on $H\\Zu$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $H\\Zu'$ is a right $\\cE$-module whose underlying\n$C_2$-spectrum is equivalent to the\nEilenberg-MacLane spectrum $H\\Zu$. \nWe first claim that $H\\Zu'$ can be constructed as an\n$\\cE$-module with cells in degrees $k\\rho$ for $k\\leq 0$:\n$$H\\Zu'\\simeq_{\\cE} S^0_{\\cE}\\cup e^{-\\rho}_{\\cE}\\cup e^{-2\\rho}_{\\cE} \\cup \\cdots $$\n\nOnce that is proved, we argue as follows. If $H\\Zu''$ is another right\n$\\cE$-module with underlying $C_2$-spectrum $H\\Zu$, we may construct a\nmap $H\\Zu'\\longrightarrow H\\Zu''$ skeleton by skeleton in the usual way. \nWe start with the $\\cE$-module map $\\cE =(H\\Zu')^{(0)}\\longrightarrow H\\Zu'$ giving\nthe unit, and successively extend the map over the\ncells of $H\\Zu'$. At each stage the obstruction to the existence of an \nextension over $(H\\Zu')^{-k\\rho}$ lies in $\\pi^{\\Ctwo}_{-k\\rho-1}(H\\Zu'')$. \nThese groups are zero. We end with a map which is an isomorphism on \n0th homotopy Mackey functors and therefore an equivalence.\n\n\nFor the cell-structure, it is enough to show that for every right\n$\\cE$-module $H\\Zu'$ of the homotopy type of the Eilenberg--MacLane\nspectrum $H\\Zu$, there is a map $\\cE \\to H\\Zu'$ of right $\\cE$-modules\nwhose fibre has the homotopy type of $\\Sigma^{-\\rho-1}H\\Zu$. Indeed, suppose we have already constructed a right $\\cE$-module $(H\\Zu')^{(n)}$ with an $\\cE$-map to $H\\Zu'$ with fibre of the homotopy type $\\Sigma^{-(n+1)\\rho-1}H\\Zu$. Then it is easy to see that the cofibre $(H\\Zu')^{(n+1)}$ of the map $\\Sigma^{-(n+1)\\rho -1}\\cE \\to \\Sigma^{-(n+1)\\rho-1}H\\Zu \\to (H\\Zu')^{(n)}$ has the analogous property. Taking the homotopy colimit, we get a map $\\mathop{ \\mathop{\\mathrm {holim}}\\limits_\\rightarrow} \\nolimits (H\\Zu')^{(n)} \\to H\\Zu'$ with fibre $\\mathop{ \\mathop{\\mathrm {holim}}\\limits_\\rightarrow} \\nolimits \\Sigma^{-(n+1)\\rho-1}H\\Zu$, which is clearly zero (e.g.\\ by Lemma \\ref{lem:regrep} and the fact that $H\\Zu$ is even; we refer to \\cite[Section 3.4]{Ricka} for a table of $\\underline{\\pi}_{\\bigstar}^{C_2}H\\Zu$). \n\nWe choose the map $f\\colon \\cE \\to H\\Zu'$ representing $1\\in \\pi_0^{\\Ctwo}H\\Zu'$ and call the fibre $F$. We want to show that $f$ agrees with the canonical map $\\cE \\to H\\Zu$ on homotopy groups of the form $\\pi_{k-\\sigma}^{\\Ctwo}$ for $k\\in\\Z$. Indeed, the only nonzero class in $H\\Zu'$ in these degrees is $a\\in\\pi_{-\\sigma}^{\\Ctwo}H\\Zu'$, which has to be hit by $a\\in \\pi_{-\\sigma}^{\\Ctwo}\\cE$ as it comes from the sphere. Thus, $\\pi_{k-\\sigma}^{\\Ctwo} F \\cong \\pi_{k-\\sigma}^{\\Ctwo} \\Sigma^{-1-\\rho}H\\Zu$ for all $k$ and hence $F\\simeq \\Sigma^{-1-\\rho}H\\Zu$ as $C_2$-spectra, as we needed to show. \n\\end{proof}\n\nFrom this the required statement follows. \n\n\\begin{cor} {\\em (Gorenstein duality)} \n\\label{cor:kRGorD}\nThere is an equivalence of $k\\R$-modules\n$$\\Gamma_{\\vb} k\\R \\simeq \\Sigma^{-4+\\sigma} \\Z^{k\\R}. \\qed \\\\[1ex]$$\n\\end{cor}\n\\begin{proof}\n By \\eqref{eq:kReq} and Lemma \\ref{lem:UniquenesskR}, we know that \n $$ \\mathrm{Hom}_{k\\R}(H\\Zu, k\\R)\\otimes_{\\cE} H\\Zu\\simeq \n\\mathrm{Hom}_{k\\R}(H\\Zu, \\Sigma^{-4+\\sigma}\\Z^{k\\R})\\otimes_{\\cE}H\\Zu.$$\nBy Lemma \\ref{lem:effective}, the two sides are the cellularizations of $k\\R$ and $\\Sigma^{-4+\\sigma}\\Z^{k\\R}$ respectively. By Lemmas \\ref{lem:Gammacell} and \\ref{lem:ML}, the former is $\\Gamma_{\\vb}k\\R$ and the latter is $\\Sigma^{-4+\\sigma}\\Z^{k\\R}$ itself. \n\\end{proof}\n\n\nThe implications of this equivalence for the coefficient ring are\ninvestigated in Section \\ref{sec:kRlcss}. \n\n\\section{\\texorpdfstring{$\\protect BP\\langle n \\rangle$}{BP} with Reality}\n\\label{sec:dishonest}\n\nWe now turn to the case of $BP\\R \\langle n \\rangle$ for a general $n$. The counterpart\nof the argument of Section \\ref{sec:kR} is a little simpler when $BP\\R \\langle n \\rangle$ is a commutative ring\nspectrum. For $n=1$ and $n=2$, the spectra $k\\R$, and $tmf_1(3)$, are both known to be a\ncommutative ring spectra, and their 2-localizations give $BP\\R \\langle n \\rangle$\nwhen $n=1$ and $n=2$ respectively. However for higher\n$n$ it is not known that $BP\\R \\langle n \\rangle$ is a commutative ring spectrum. \nThis is a significant technical issue, but\none that is familiar when working with non-equivariant $BP$-related theories\nsince $BP$ is not known to be a commutative ring. The established\nmethod for getting around this is to use the fact that \n$BP$ and $BP\\langle n \\rangle$ are modules over the commutative ring $MU$.\nWe will adopt precisely the same method by working with $M\\R$-modules.\nThe only real complication is that \nwe are forced to work with spectra whose homotopy groups are bigger\nthan we might like, but if we focus on the relevant part, it causes\nno real difficulties. \n\n\n\n\n\n\\subsection{Gorenstein condition and Matlis lift}\nAs mentioned in the introduction of this section, we will work in the setting of $M\\R$-modules. More precisely, we will always (implicitly) localize at $2$ and set $S = M\\R_{(2)}$. As discussed in Section \\ref{sec:BPRBasics}, we can define $S$-modules $BP\\R \\langle n \\rangle$, once we have chosen a sequence of $\\vb_i$ (for example, the Hazewinkel or Araki generators). \n\nThe ideal \n$$\\Jb_n=(\\vbn{1}, \\ldots, \\vbn{n})$$\nplays a prominent role, and we will abuse notation by writing \n$$S\/\\overline{J}_n:=\\cof(S\\stackrel{\\vbn{1}}\\longrightarrow S)\\otimes_S\n\\cof(S\\stackrel{\\vbn{2}}\\longrightarrow S)\\otimes_S \\cdots \\otimes_S\n\\cof(S\\stackrel{\\vbn{n}}\\longrightarrow S), $$\nand then \n$$M\/\\overline{J}_n:=M\\otimes_S S\/\\overline{J}_n.$$\nIn particular, \n$$BP\\R \\langle n \\rangle\/\\overline{J}_n=BP\\R \\langle n \\rangle\/\\vbn{n}\/\\vbn{n-1}\/\\cdots \/\\vbn{1}\\simeq H\\Zu $$\nby the $C_2$-case of the reduction theorem, here proved as Corollary \\ref{cor:reduction}. \n\nIf $BP\\R \\langle n \\rangle$ is a ring spectrum\n$$\\mathrm{Hom}_{BP\\R \\langle n \\rangle}(H\\Zu, M)=\\mathrm{Hom}_{BP\\R \\langle n \\rangle}(BP\\R \\langle n \\rangle\\otimes_S S\/\\overline{J}_n,\nM)=\\mathrm{Hom}_{S}(S\/\\overline{J}_n, M), $$\nso that the right hand side gives a way for us to express the\nfact that certain $BP\\R \\langle n \\rangle$-modules (such as $BP\\R \\langle n \\rangle$ and $\\Z^{BP\\R \\langle n \\rangle}$)\nare Matlis lifts, using only module structures over $S$.\n\nApplying this when $M=BP\\R \\langle n \\rangle$, we obtain the Gorenstein condition. \n\n\\begin{lemma}\n\\label{lem:BPRnGordish}\nThe map $BP\\R \\langle n \\rangle \\longrightarrow H\\Zu$ is Gorenstein of shift $-D_n\\rho -n$ \nin the sense that \n$$\\mathrm{Hom}_S(S\/\\overline{J}_n, BP\\R \\langle n \\rangle)\\simeq \\Sigma^{-D_n\\rho -n}H\\Zu, $$\nwhere\n$$D_n\\rho =|\\vbn{n}|+|\\vbn{n-1}|+\\cdots\n+|\\vbn{1}|=\\left[2^{n+1}-n-2 \\right]\\rho . $$\n\\end{lemma}\n\n\\begin{proof}\nSince each of the maps $\\vbn{i}: \\Sigma^{|\\vbn{i}|}S\\longrightarrow S$ is self-dual, \nfor any $S$-module $M$, we have \n$$\\mathrm{Hom}_S(S\/\\overline{J}_n, M)\\simeq \\Sigma^{-D_m\\rho-n}S\/\\overline{J}_n \\otimes_S M. $$\n\\end{proof}\n\nApplying this when $M=\\Z^{BP\\R \\langle n \\rangle}$, we obtain the Anderson Matlis lift.\n\n\\begin{lemma}\n\\label{lem:MLB}\nThe Anderson dual of $BP\\R \\langle n \\rangle$ is a Matlis lift of $H\\Zu^*$ in the sense that \n\n(i) $\\Z^{BP\\R \\langle n \\rangle}$ is $H\\Zu$-$\\R$-cellular and \n\n(ii) There is an equivalence \n$$\\Sigma^{2-2\\sigma}H\\Zu \\simeq H\\Zu^* \\simeq \\mathrm{Hom}_{S} (S\/\\overline{J}_n, \\Z^{BP\\R \\langle n \\rangle}). $$\n\\end{lemma}\n\n\\begin{proof}\nOne could prove the first part from the slice tower, but it also follows directly from Corollary \\ref{cor:cellular}. \n\nFor the second statement observe that \n$$\\mathrm{Hom}_{S} (S\/\\overline{J}_n, \\Z^{BP\\R \\langle n \\rangle}) \\simeq \\mathrm{Hom}_S(S\/\\overline{J}_n\\otimes_S BP\\R \\langle n \\rangle, \\Z^S)\\simeq \\Z^{H\\Zu}.$$ \nThus, Lemma \\ref{lem:Zu} implies the statement. \n\\end{proof}\n\n\n\n\\subsection{Gorenstein duality}\nThroughout this section, we will write $R = BP\\R \\langle n \\rangle$ for brevity. \nCombining Lemmas \\ref{lem:BPRnGordish} and \\ref{lem:MLB}, we have an\nequivalence of $S$-modules\n$$\\mathrm{Hom}_{S}(S\/\\overline{J}_n, R)\\simeq \\Sigma^{-D_n \\rho-n}H\\Zu \\simeq\n\\mathrm{Hom}_{S}(S\/\\overline{J}_n, \\Sigma^{-(D_n+n+2)-(D_n-2)\\sigma}\\Z^R)$$\n\nWe now want to remove the $\\mathrm{Hom}_{S }(S\/\\overline{J}_n , \\cdot )$ from this\nequivalence. The endomorphism ring \n$$\\widetilde{\\cE}_n =\\mathrm{Hom}_S(S\/\\overline{J}_n, S\/\\overline{J}_n)$$\nof the small $S$-module $S\/\\overline{J}_n$, replaces $\\cE_n=\\mathrm{Hom}_{R}(H\\Zu,\n H\\Zu)$ from the case that $R =BP\\R \\langle n \\rangle$ is a ring spectrum. We note that \n$$\\widetilde{\\cE}_n\\otimes_S R=\\mathrm{Hom}_S(S\/\\overline{J}_n, S\/\\overline{J}_n )\\otimes_S R\\simeq \n\\mathrm{Hom}_S(S\/\\overline{J}_n, S\/\\overline{J}_n)\\otimes_S R). $$\nIf $R = BP\\R \\langle n \\rangle$ were a commutative ring, this would be a ring equivalent to\n$\\mathrm{Hom}_R(H\\Zu, H\\Zu)$. \n\n\nIn any case, the following is proved exactly like Lemma \\ref{lem:effective}\n\n\\begin{lemma} {\\em (Effective constructibility)}\\label{lem:effective2}\nThe evaluation map \n$$\\mathrm{Hom}_{S }(S\/\\overline{J}_n, M)\\otimes_{\\widetilde{\\cE}_n}S\/\\overline{J}_n \\longrightarrow M$$\nis $S\/\\overline{J}_n$-$\\R$-cellularization.\\qed \\\\[1ex] \n\\end{lemma}\n\n\n\nOf course local cohomology gives an alternative approach to\ncellularization. Recall that we define\n$$\\Gamma_{\\Jb_n}M =\\Gamma_{\\vbn{1} }S\n\\otimes_{S}\\Gamma_{\\vbn{2}}S \\otimes_{S}\\cdots \\otimes_{S}\n\\Gamma_{\\vbn{n} }S \\otimes_{S}M. $$\nThen Proposition \\ref{prop:cell} gives the following lemma. \n\n\\begin{lemma}\n$$\\Gamma_{\\Jb_n}M\\longrightarrow M$$ \nis $H\\Zu$-$\\R$-cellularization.\n\\end{lemma}\n\nIt remains to check that the two $\\widetilde{\\cE}_n$ actions on $H\\Zu$ coincide. For\n$k\\R$ (i.e., $n=1$) we showed there was a unique right $\\cE_n$-module\nstructure on $H\\Zu$. This may be true for $\\widetilde{\\cE}_n$-module structures, but we will instead\njust prove in the next subsection that the two particular $\\widetilde{\\cE}_n$-modules that\narose from the left and right hand ends of the first display of this subsection are equivalent. \n\nThe required Gorenstein duality statement follows. Its implications for the coefficient ring for\n$n=2$ are investigated explicitly in Section \\ref{sec:tmfotlcss}. \n\n\\begin{cor} \n\\label{cor:BPRnGorDdish}\n{\\em (Gorenstein duality)} There is an equivalence of $M\\R$-modules\n$$\\Gamma_{\\Jb_n} R \\simeq \\Sigma^{-(D_n+n+2)-(D_n-2)\\sigma} \\Z^R$$\nwith $R = BP\\R \\langle n \\rangle$.\n\\end{cor}\n\\begin{proof} \nWe will argue in Subsection \\ref{subsec:Eequiv} that the equivalence\n$$\\mathrm{Hom}_S(S\/\\overline{J}_n, R) \\simeq \\mathrm{Hom}_S(S\/\\overline{J}_n, \\Sigma^{-D_n\\rho\n-n-2\\delta} \\Z^R), $$\nis in fact an equivalence of right\nmodules over $\\widetilde{\\mathcal{E}}_n$. By Lemma \\ref{lem:effective2}, $R$ and \n$\\Sigma^{-(D_n+n+2)-(D_n-2)\\sigma}\n\\Z^R$ have equivalent $S\/\\overline{J}_n$ cellularizations. We have seen above that the cellularization of $R$ is $\\Gamma_{\\Jb_n} BP\\R \\langle n \\rangle $ and that $\\Sigma^{-D_n\\rho -n-2\\delta}\n\\Z^R$ itself is cellular. \n\\end{proof}\n\n\n\n\n\\subsection{The equivalence of induced and coinduced Matlis lifts of\n $\\protect H\\Zu$}\n\\label{subsec:Eequiv}\nFor brevity we will still write $R=BP\\R \\langle n \\rangle$, and note that we have a map\n$S=M\\R \\longrightarrow BP\\R \\langle n \\rangle=R$. The two $S$-modules that concern us are of a\nvery special sort, one looks as if it is obtained from an $S$-module\nby `extension of scalars from $S$ to $R$' and one\nlooks as if it is obtained by `coextension of scalars from $S$ to $R$'.\n\n\\begin{lemma}\n\\label{lem:resEtnEn}\nWe have equivalences of right $\\widetilde{\\mathcal{E}}_n$-modules\n$$\\mathrm{Hom}_S(S\/\\overline{J}_n, R)\\simeq \\mathrm{Hom}_S(S\/\\overline{J}_n, S)\\otimes_SR.$$\n$$\\mathrm{Hom}_S(S\/\\overline{J}_n, \\Z^R)=\\mathrm{Hom}_S(R, \\mathrm{Hom}_S(S\/\\overline{J}_n, \\Z^S))$$\n\\end{lemma}\n\n\\begin{proof}\nThe first equivalence is immediate from the smallness of\n$S\/\\overline{J}_n$. \n\nThe second equivalence follows from the equivalence \n$$\\Z^R\\simeq \\mathrm{Hom}_S(R, \\Z^S)$$\nof $S$-modules. \n\\end{proof}\n\n\n\nSuspending the equivalences from Lemma \\ref{lem:resEtnEn} so that we are comparing two $\\widetilde{\\mathcal{E}}_n$-modules\nequivalent to $H\\Zu$ (see Lemma \\ref{lem:MLB}) we have\n$$Y_1=\\mathrm{Hom}_S(S\/\\overline{J}_n, \\Sigma^{D_n\\rho+n}R)\\simeq \\mathrm{Hom}_S(S\/\\overline{J}_n,\\Sigma^{D_n\\rho+n}S)\\otimes_SR=X_1\\otimes_SR$$\nand \n$$Y_2=\\mathrm{Hom}_S(S\/\\overline{J}_n, \\Sigma^{2\\pp} \\Z^R)\\simeq \\mathrm{Hom}_S(S,\n\\mathrm{Hom}_S(S\/\\overline{J}_n, \\Sigma^{2\\pp} \\Z^S)) =\\mathrm{Hom}_S(R,X_2). $$\n\n \nIn Subsection \\ref{subsec:alpha} we will construct an $\\widetilde{\\mathcal{E}}_n$-map $\\alpha: X_1\\longrightarrow Y_2$ and then argue\nin Subsection \\ref{subsec:alphat} that this extends along $X_1=X_1\\otimes_SS\\longrightarrow X_1\\otimes_SR =Y_1$ to\ngive a map $\\tilde{\\alpha}: Y_1\\longrightarrow Y_2$ which is easily seen to be an\nequivalence: it is clearly a $*\\rho -$ isomorphism\nand hence an equivalence by Lemma \\ref{lem:regrep}. \n\nTo see our strategy, note that the extension problem \n$$\\diagram\nX_1\\dto \\rto^-{\\alpha} & \\mathrm{Hom}_S(S\/\\overline{J}_n, \\mathrm{Hom}_S(R,\\Z^S))\\\\\nX_1\\otimes_SR\\ar@{-->}[ur]_-{\\tilde{\\alpha}}&\n\\enddiagram$$\nin the category of $\\widetilde{\\mathcal{E}}_n$-modules is equivalent to the extension problem\n$$\\diagram\nX_1 \\otimes_{\\widetilde{\\mathcal{E}}_n} S\/\\overline{J}_n \\otimes_S R\n\\dto \\rto^-{\\alpha'} & \\Z^S\\\\\nX_1 \\otimes_{\\widetilde{\\mathcal{E}}_n} S\/\\overline{J}_n \\otimes_S R\\otimes_SR\\ar@{-->}[ur]_-{\\tilde{\\alpha}'}&\n\\enddiagram$$\nin the category of $S$-modules. The point is that by the defining property of the Anderson dual, this\nlatter extension problem can be tackled by looking in\n$\\pi^{\\Ctwo}_0$. The 0th homotopy groups of the spectra on the left are\neasily calculated from the known ring $\\pi^{\\Ctwo}_{\\bigstar}(H\\Zu)$. \n\n\\subsection{Construction of the map $\\alpha$}\n\\label{subsec:alpha}\n\nWe construct the map $\\alpha$ using a similar method as in the proof of Lemma \\ref{lem:UniquenesskR}.\n\n\\begin{lemma}\nThere is a map \n$$\\alpha : X_1 \\longrightarrow Y_2$$\nof right $\\widetilde{\\mathcal{E}}_n$-modules that takes the image of $1\\in\n\\pi^{\\Ctwo}_0(S)$ to a generator of $\\pi^{\\Ctwo}_0(H\\Zu)=\\Z$. \n\\end{lemma}\n\n\\begin{proof}\nFirst we claim that $X_1$ has a $\\widetilde{\\mathcal{E}}_n$-cell structures \nwith one 0-cell and other cells in dimensions which are negative\nmultiples of $\\rho$. More precisely, there is \n a filtration \n$$\\widetilde{\\mathcal{E}}_n\\simeq X_1^{[0]}\\to X_1^{[1]}\\to\nX_1^{[2]}\\to \\cdots \\to X_1$$\nso that $X_1\\simeq \\mathop{ \\mathop{\\mathrm {holim}}\\limits_\\rightarrow} \\nolimits_dX_1^{[d]}$ and there are cofibre sequences\n$$X_1^{[d-1]}\\longrightarrow X_1^{[d]}\\longrightarrow \\bigvee \\Sigma^{-d\\rho} \\widetilde{\\mathcal{E}}_n. $$\n\nBy definition $X_1 = \\mathrm{Hom}_S(S\/\\overline{J}_n,\\Sigma^{D_n\\rho+n}S)$. By Proposition \\ref{prop:cell} and Lemma \\ref{lem:Koszul}, this is equivalent to \n$$Hom_S(S\/\\overline{J}_n,\\Sigma^{D_n\\rho+n}\\Gamma_{\\overline{J}_n}S) \\simeq \\mathrm{Hom}_S(S\/\\overline{J}_n, \\kappa_S(\\vb_1,\\dots, \\vb_n))$$\nbecause $\\Gamma_{\\overline{J}_n}S \\to S$ is $S\/\\overline{J}_n$-$\\R$-cellularization. The usual construction of the stable Koszul complex from the unstable\nKoszul complex recalled in Subsection \\ref{sec:Koszul}, shows that \n$$\\kappa_S(\\vb_1,\\dots, \\vb_n)$$ \nhas a filtration with subquotients sums of $(-k\\rho)$-fold suspensions of $S\/\\overline{J}_n$. This induces a corresponding filtration on $X_1$. \n\n\nAs in Lemma \\ref{lem:UniquenesskR} we may construct $\\alpha$ by obstruction\ntheory. Indeed, we start\nby choosing a map $\\widetilde{\\mathcal{E}}_n=X_1^{[0]}\\longrightarrow Y_2^{[0]}$ taking the unit to\na generator. At the $d$th stage we have a\nproblem \n$$\\diagram\nX_1^{[d-1]} \\rto \\dto & Y_2\\\\\nX_1^{[d]} \\ar@{-->}[ur]&\n\\enddiagram$$\nThe obstruction to extension is in a finite product of groups\n$$[\\Sigma^{-d\\rho-1}\\widetilde{\\mathcal{E}}_n, Y_2]^{\\widetilde{\\mathcal{E}}_n}=\\pi^{\\Ctwo}_{-d\\rho -1}(H\\Zu)=0 $$\nwhere the vanishing is from the known value of $\\pi^{\\Ctwo}_{\\bigstar}(H\\Zu)$. \n\\end{proof}\n\n\\subsection{The map $\\tilde{\\alpha}$}\n\\label{subsec:alphat}\nReferring to the second extension problem diagram above, we note $S\/\\overline{J}_n\\otimes_SR\\simeq\nH\\Zu$ as $S$-modules. Thus, we have to solve the lifting problem\n$$\\diagram\nX_1 \\otimes_{\\widetilde{\\mathcal{E}}_n} H\\Zu \\otimes_S S \\dto_{1\\tensor1\\otimes \\pi} \\rto^-{\\alpha'} & \\Z^S\\\\\nX_1 \\otimes_{\\widetilde{\\mathcal{E}}_n} H\\Zu \\otimes_SR\\ar@{-->}[ur]_-{\\tilde{\\alpha}'}&\n\\enddiagram$$\nwhere $H\\Zu$ is equipped with some $\\widetilde{\\mathcal{E}}_n$-module structure. Denote the upper left corner by $T$. The map $T \\to T\\otimes_S R$ is a split inclusion on underlying $MU$-modules. Indeed, \n$$T\\simeq X_1 \\otimes_{\\widetilde{\\mathcal{E}}_n} S\/\\overline{J}_n \\otimes_S R$$\n and the map $R \\to R\\otimes_S R$ is a split inclusion on underlying spectra because $BP\\langle n\\rangle$ has the structure of a homotopy unital $MU$-algebra \\cite[V.2.6]{EKMM}. \n\nBy the definition of Anderson duals, we have a diagram of short exact sequences:\n\\[\\xymatrix{\n 0 \\ar[r]& \\mathrm{Ext}_\\Z^1(\\pi^{\\Ctwo}_{-1}(T\\otimes_SR),\\Z)\\ar[d] \\ar[r]& [T \\otimes_SR, \\Z^S]^S \\ar[d]\\ar[r]& \\mathrm{Hom}_\\Z(\\pi^{\\Ctwo}_0(T \\otimes_SR),\\Z)\\ar[d] \\ar[r]& 0 \\\\\n 0\\ar[r]&\\mathrm{Ext}_\\Z^1(\\pi^{\\Ctwo}_{-1}(T),\\Z)\\ar[r]& [T, \\Z^S]^S \\ar[r]& \\mathrm{Hom}_\\Z(\\pi^{\\Ctwo}_0(T),\\Z) \\ar[r]& 0\n }\n\\]\n\nWe want to show that the maps $\\pi_k^{C_2}T \\to \\pi_k^{C_2}T\\otimes_S R$ are split injections for $k=0,-1$, which solves the problem. For the computation of $\\pi_*^{C_2}T$ recall from the last section that $X_1$ has a filtration starting with $X_1^{[x]} = \\widetilde{\\mathcal{E}}_n$ and with subquotients sums of terms of the form $\\Sigma^{-d\\rho}\\widetilde{\\mathcal{E}}_n$. Thus, $T$ obtains a filtration starting with $T^{[1]} = H\\Zu$ and with subquotients sums of terms of the form $\\Sigma^{-d\\rho}H\\Zu$. The map $H\\Zu = T^{[1]} \\to T$ clearly induces isomorphisms on $\\underline{\\pi}_k^{\\Ctwo}$ for $k = 0,-1$ by the known homotopy groups of $H\\Zu$ (see e.g.\\ \\cite[Section 3.4]{Ricka} for a table). Thus, $\\underline{\\pi}^{C_2}_{-1}T = 0$ and $\\underline{\\pi}^{C_2}_0T = \\underline{\\Z}$. \n\nIf we have a map $\\underline{\\Z} \\to M$ from the constant Mackey functor, it is a split injection on $(C_2\/C_2)$ if it is one on $(C_2\/e)$. But we have already seen above that on underlying spectra $T\\to T\\otimes_SR$ is a split inclusion. Thus, we have shown that $\\pi_k^{C_2}T \\to \\pi_k^{C_2}(T\\otimes_S R)$ is split injective, which provides the map $\\tilde{\\alpha}'$.\n\n\\vspace{1cm}\n\\part{The hands-on approach}\nIn this part, we give a different way to compute the Anderson dual of $BP\\R \\langle n \\rangle$ by first computing the Anderson dual of $BP\\R$ itself. Again, we will first do the case of $k\\R$. \n\n\\section{The case of $k\\mathbb{R}$ again}\\label{sec:kRagain}\nTo illustrate our strategy, we give an alternative calculation of the\nAnderson dual of $k\\R$. This can also be deduced from our main theorem\nbelow, but it might be helpful to see the proof in this simpler case\nfirst. General references for the $RO(C_2)$-graded homotopy groups of $k\\R$ are \\cite[Section 3.7]{B-G10} or Section \\ref{sec:kRgroups}.\n\nWe want to show the following proposition:\n\n\\begin{prop}\nThere is an equivalence $\\kappa_{k\\R}(\\vb) \\to \\Sigma^{2\\rho -4}\\Z^{k\\R}.$\n\\end{prop}\nRecall here that $\\vb \\in \\pi^{C_2}_{\\rho}k\\R$ is the Bott element for Real K-theory and \n$$\\kappa_{k\\R}(\\vb) = \\hcolim_n \\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n.$$\n Our idea is simple: To obtain a map from the homotopy colimit, we have just to give maps \n $$\\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n \\to \\Sigma^{2\\rho -4}\\Z^{k\\R}$$\n that are compatible in the homotopy category (see Remark \\ref{rmk:hocolim}). We will show in the next lemma that these maps are essentially unique: The Mackey functor of homotopy classes of $k\\R$-linear maps $\\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n \\to \\Sigma^{2\\rho -4}\\Z^{k\\R}$ is isomorphic to $\\underline{\\Z}$ and the precomposition with the map $\\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n \\to \\Sigma^{-n\\rho}k\\R\/\\vb^{n+1}$ induces the identity on $\\underline{\\Z}$. \n\nChoosing the $C_2$-equivariant map $\\kappa_{k\\R}(\\vb) \\to \\Sigma^{2\\rho -4}\\Z^{k\\R}$ that corresponds to $1\\in \\Z$ for every $n$ induces an equivalence on underlying homotopy groups. By Lemma \\ref{lem:regrep} the result follows as soon as we have established that $\\kappa_{k\\R}(\\vb)$ is strongly even and that the Mackey functor $\\underline{\\pi}_{*\\rho}\\Sigma^{2\\rho-4}\\Z^{k\\R}$ is constant. These two facts will also be shown in the following lemma, finishing the proof of the proposition.\n\n\\begin{lemma}Denote for a $\\underline{\\Z}[\\vb]$ module $M$ by $\\{M\\}_{\\vb^n}$ the $\\vb^n$-torsion in it. Then we have:\n \\begin{enumerate}\n \\item $k\\R\/\\vb^n$ is strongly even and hence the same is true for $\\kappa_{k\\R}(\\vb)$.\n \\item $\\underline{\\pi}_{n\\rho}^{\\Ctwo}\\Sigma^{2\\rho-4}\\Z^{k\\R} \\cong \\underline{\\pi}^{\\Ctwo}_{(n-2)\\rho+4}\\Z^{k\\R}$ is constant for all $n\\in\\Z$.\n \\item $\\underline{[\\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n, \\Sigma^{2\\rho-4}\\Z^{k\\R}]}^{C_2}_{k\\R} \\cong \\left\\{\\underline{\\pi}^{\\Ctwo}_{-(n-1)\\rho}\\Sigma^{2\\rho-4}\\Z^{k\\R}\\right\\}_{\\vb^n} \\cong \\underline{\\Z}$\n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n The first part follows as \n $$\\underline{\\pi}^{\\Ctwo}_{k\\rho -i}(k\\R\/\\vb^n) =\\underline{\\pi}^{\\Ctwo}_{k\\rho -i}(k\\R)\/\\vb^n$$\n for $i=0,1$ because $\\pi_{k\\rho-i}^{C_2}k\\R = 0$ for $i=1,2$. \n \n For the second part consider the short exact sequence\n \\[0\\to \\mathrm{Ext}(\\underline{\\pi}^{C_2}_{k\\rho-5}k\\R, \\Z) \\to \\underline{\\pi}^{C_2}_{-k\\rho +4}\\Z^{k\\R} \\to \\mathrm{Hom}(\\underline{\\pi}^{C_2}_{k\\rho -4}k\\R,\\Z) \\to 0.\\]\n We have $\\underline{\\pi}^{\\Ctwo}_{k\\rho-5}k\\R = 0$ for all $k\\in\\Z$. For $k<2$, the Mackey functor $\\underline{\\pi}^{\\Ctwo}_{k\\rho-4}k\\R$ vanishes as well and for $k\\geq 2$, we have $\\underline{\\pi}^{\\Ctwo}_{k\\rho-4}k\\R \\cong \\underline{\\Z}^*$, generated by $v^{k-2}$ and $2\\vb^{k-2}u$. Thus, \n $$\\underline{\\pi}^{\\Ctwo}_{-k\\rho+4}\\Z^{k\\R} \\cong \\begin{cases} 0 & \\text{ if }k<2 \\\\\n \\underline{\\Z} & \\text{ if }k\\leq 2 \\end{cases}$$\n This shows part (2). As multiplication by $\\vb^n$ does not hit $\\underline{\\pi}^{\\Ctwo}_{(n+1)\\rho-4}k\\R$, the whole Mackey functor $\\underline{\\pi}^{\\Ctwo}_{-(n+1)\\rho+4}\\Z^{k\\R}$ is $\\vb^n$-torsion. This gives the second isomorphism of the third part. \n \n \n For the remaining isomorphism, note that the cofibre sequence\n \\[\n \\Sigma^\\rho k\\R \\xrightarrow{\\vb^n} \\Sigma^{-(n-1)\\rho}k\\R \\to \\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n \\to \\Sigma^{\\rho+1} k\\R\n \\]\ninduces a short exact sequence\n \\[0 \\to (\\underline{\\pi}^{\\Ctwo}_{\\rho +1}\\Sigma^{2\\rho-4}\\Z^{k\\R} )\/ \\vb_n \\to \\underline{[\\Sigma^{-(n-1)\\rho}k\\R\/\\vb^n, \\Sigma^{2\\rho-4}\\Z^{k\\R}]}^{C_2}_{k\\R} \\to \\left\\{\\underline{\\pi}^{\\Ctwo}_{-(n-1)\\rho}\\Sigma^{2\\rho-4}\\Z^{k\\R}\\right\\}_{\\vb^n} \\to 0\\]\n \n We have $\\underline{\\pi}_{\\rho +1}^{\\Ctwo}\\Sigma^{2\\rho-4}\\Z^{k\\R} \\cong \\underline{\\pi}_{5 -\\rho}^{\\Ctwo}\\Z^{k\\R}$, which sits in a short exact sequence\n \\[ 0 \\to \\mathrm{Ext}_\\Z(\\underline{\\pi}_{\\rho-6}^{\\Ctwo}k\\R,\\Z) \\to \\underline{\\pi}_{5 -\\rho}^{\\Ctwo}\\Z^{k\\R} \\to \\mathrm{Hom}_\\Z(\\underline{\\pi}_{\\rho-5}^{\\Ctwo}k\\R,\\Z)\\to 0.\\]\n But because of connectivity, $\\underline{\\pi}_{\\rho-c}^{\\Ctwo}k\\R = 0$ for $c\\geq 3$. \n \\end{proof}\n\n\\section{Duality for $BP\\mathbb{R}$}\nWe will use throughout the abbreviation $B = BP\\R$ and will furthermore implicitly localize everything at $2$ so that $\\Z = \\Z_{(2)}$ etc.\\ and all $\\mathrm{Hom}$ and $\\mathrm{Ext}$ groups are over $\\Z = \\Z_{(2)}$ unless marked otherwise. Denote by $\\underline{\\vb}$ a sequence of indecomposable elements $\\vb_i \\in \\pi_{(2^i-1)\\rho}^{C_2}B$. The aim of this section is to show that $\\Sigma^{2\\rho-4}\\Z^{B} \\simeq \\kappa_{M\\R}(\\underline{\\vb}; B)$. \n\nRecall that $\\kappa_{M\\R}(\\underline{\\vb}; B)$ is defined as follows: Given a sequence $\\underline{l} = (l_1,l_2,\\dots)$ with $l_i\\geq 0$, we denote by $B\/\\underline{\\vb}^{\\underline{l}}$ the spectrum $B\/(\\vb^{l_{i_1}}_{i_1}, \\vb^{l_{i_2}}_{i_2},\\dots)$, where $i_j$ runs over all indices such that $l_{i_j}> 0$. Set \n$$|\\underline{l}| = l_1|\\vb_1|+l_2|\\vb_2| + \\cdots $$\nThen \n$$\\kappa_{M\\R}(\\underline{\\vb}; B) = \\hcolim_{\\underline{l}} \\Sigma^{-|\\underline{l}-\\underline{1}|}B\/\\underline{\\vb}^{\\underline{l}},$$\nwhere $\\underline{l}$ runs over all sequences $\\underline{l}$ where all but finitely many $l_i$ are zero and $\\underline{1}$ denotes the constant sequence of ones. Furthermore, the $i$-th entry of $\\underline{l}-\\underline{1}$ is defined to be the maximum of $0$ and $l_i-1$. \n\nThus, to get a map $\\kappa_{M\\R}(\\underline{\\vb}; B) \\to \\Sigma^{2\\rho-4}\\Z^{B}$, we have to understand the homotopy classes of maps $B\/\\underline{\\vb}^{\\underline{l}} \\to \\Sigma^{2\\rho-4}\\Z^{B}$. This will be the content of the next subsection.\n\n\\subsection{Preparation}\nRecall the Mackey functor $\\underline{\\Z}^*$ defined by\n$$\\underline{\\Z}^*(\\Ctwo\/\\Ctwo) \\cong \\underline{\\Z}^*(\\Ctwo\/e)\\cong \\Z$$\nwith transfer equalling $1$ while restriction is multiplication by $2$. \n\\begin{lemma}\\label{lem:computation}\n As $\\underline{\\Z}[\\vb_1,\\vb_2,\\dots]$-modules, we have the following isomorphisms.\n \\begin{enumerate}\n \\item $\\underline{\\pi}^{C_2}_{*\\rho -4}B \\cong \\underline{\\Z}^*\\otimes_\\Z \\Z[\\vb_1,\\vb_2,\\dots]$ where $\\underline{\\Z}^*$ is generated by $1$ on underlying and by $2u^{-1}$ on $C_2$-equivariant homotopy groups.\n \\item $\\underline{\\pi}^{C_2}_{*\\rho -5}B = 0$\n \\item $\\pi^{C_2}_{*\\rho -6}B \\cong \\F_2\\{a^2\\vb_1(-1)\\} \\otimes_\\Z \\Z[\\vb_1,\\vb_2,\\dots]$. \n \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n By Theorem \\ref{thm:BPR}, the groups $\\pi_{*\\rho -c}^{C_2}B$ are additively generated by nonzero elements of the form $x = a^l\\vb$ with $\\vb$ a monomial in the $\\vb_i(j)$. Let $\\vb_i(j)$ be the one occuring with minimal $i$, where $j$ is chosen such that $\\vb = \\vb_i(j)\\vb'$ with $\\vb'$ a monomial in the $\\vb_k$ (this is possible by the third relation in Theorem \\ref{thm:BPR}). Then $|x| = *\\rho +j2^{i+2}+l$ and $0 \\leq l< 2^{i+1}-1$. \n \n For $c=4$, this implies $j=-1$, $i=0$ and $l=0$. Thus, $x$ is of the form $\\vb_0(-1)\\vb'$. As the restriction of $\\vb_0(-1)$ to $\\pi_0^eB$ equals $2$, the result follows. \n \n For $c=5$, we must have $l \\geq 2^{i+2}-5$, which implies $l\\geq 2^{i+1}-1$ or $i=0$; in the latter case $l$ must be zero, which is not possible.\n \n For $c=6$, we must have $l = -j2^{i+2}-6$, which implies $l\\geq 2^{i+1}-1$ or $i\\leq 1$ and $j=-1$. As $i=0$ is again not possible, $x = a^2\\vb_1(-1)\\vb'$ with $\\vb' \\in \\pi_{*\\rho}^{C_2}$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:AndersonQuot}\n For a sequence $\\ul = (l_1,l_2, \\dots)$, the map\n $$\\underline{\\pi}^{C_2}_{*\\rho+4} \\Z^{B\/\\uvb^{\\ul}} \\to \\mathrm{Hom}(\\underline{\\pi}^{C_2}_{-*\\rho-4}B\/\\uvb^{\\ul},\\Z) \\cong \\underline{\\Z} \\otimes_{\\Z} (\\Z[\\vb_1,\\vb_2,\\dots]\/\\uvb^{\\ul})^*$$\n is an isomorphism, where $\\Z[\\vb_1,\\vb_2,\\dots]^{*} = \\mathrm{Hom}_\\Z(\\Z[\\vb_1,\\vb_2,\\dots], \\Z)$ (so that the gradings become nonpositive). Here, the second map is the dual of the map \n $$\\underline{\\Z}^* \\otimes_{\\Z} \\Z[\\vb_1,\\vb_2,\\dots]\/\\uvb^{\\ul} \\to \\underline{\\pi}^{C_2}_{-*\\rho-4}B\/\\uvb^{\\ul}$$\n sending $1 \\in \\underline{\\Z}^*(C_2\/C_2)$ to the image of $u^{-1}$ under the map $B\\to B\/\\uvb^{\\ul}$ and $1\\in\\underline{\\Z}^*(C_2\/e)$ to $1$. \n\\end{lemma}\n\\begin{proof}\n We have a short exact sequence\n $$0 \\to \\mathrm{Ext}(\\underline{\\pi}^{C_2}_{-*\\rho-5}B\/\\uvb^{\\ul}, \\Z) \\to \\underline{\\pi}^{C_2}_{*\\rho-4} \\Z^{B\/\\uvb^{\\ul}} \\to \\mathrm{Hom}(\\underline{\\pi}^{C_2}_{-*\\rho-4}B\/\\uvb^{\\ul}, \\Z) \\to 0.$$\n If $l_1=0$, then Corollary \\ref{Cor:QuotientBP} and Lemma \\ref{lem:computation} directly imply the statement. If $l_1\\neq 0$, Corollary \\ref{Cor:QuotientBP} only allows us to identify the homotopy Mackey functor in degree $-*\\rho-4$, but not the one in degree $-*\\rho-5$. We give a separate argument in this case.\n \n If $l_1\\neq 0$, consider the sequence $\\ul' = (0,l_2,l_3,\\dots)$ and the corresponding cofibre sequence\n $$ \\Sigma^{l_1\\rho}B\/\\uvb^{\\ul'} \\xrightarrow{\\vb_1^{l_1}} B\/\\uvb^{\\ul'} \\to B\/\\uvb^{\\ul} \\to \\Sigma^{l_1\\rho+1}B\/\\uvb^{\\ul'}.$$\n This induces a short exact sequence\n $$ 0 \\to (\\underline{\\pi}^{C_2}_{*\\rho-5}B\/\\uvb^{\\ul'})\/\\vb_1^{l_1} \\to \\underline{\\pi}^{C_2}_{*\\rho-5}B\/\\uvb^{\\ul} \\to \\{\\underline{\\pi}^{C_2}_{*\\rho-6}B\/\\uvb^{\\ul'}\\}_{\\vb_1^{l_1}} \\to 0.$$\n Here the last term denotes the sub Mackey functor of $\\underline{\\pi}^{C_2}_{*\\rho-6}B\/\\uvb^{\\ul'}$ killed by $\\vb_1^{l_1}$. By Corollary \\ref{Cor:QuotientBP} and Lemma \\ref{lem:computation}, we see that $\\underline{\\pi}^{C_2}_{*\\rho-5}B\/\\uvb^{\\ul} = 0$. \n \\end{proof}\n\nAs $B = BP\\R$ is not known to have an $E_\\infty$-structure, we have to work with $M\\R$-linear maps instead, for which the following lemma is useful:\n\n\\begin{lemma}\nThe map \n\\[\\Z^B \\simeq \\mathrm{Hom}_{M\\R}(M\\R, \\Z^B) \\to \\mathrm{Hom}_{M\\R}(B, \\Z^B)\\]\nis an equivalence.\n\\end{lemma}\n\\begin{proof}\nLet $e\\colon M\\R \\to M\\R$ be the Quillen--Araki idempotent. Recall that \n\\[B = \\hcolim \\left(M\\R \\xrightarrow{e} M\\R \\xrightarrow{e} \\cdots\\right).\\]\nThus, \n\\[\\Z^B \\simeq \\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits \\left(\\cdots \\xrightarrow{e^*} \\Z^{M\\R} \\xrightarrow{e^*} \\Z^{M\\R}\\right).\\]\nHence,\n\\[\n\\mathrm{Hom}_{M\\R}(B, \\Z^B) \\simeq \\mathop{ \\mathop{\\mathrm {holim}} \\limits_\\leftarrow} \\nolimits \\left(\\cdots \\xrightarrow{e^*} \\mathrm{Hom}_{M\\R}(B,\\Z^{M\\R}) \\xrightarrow{e^*} \\mathrm{Hom}_{M\\R}(B,\\Z^{M\\R})\\right).\n\\]\nAs every $\\mathrm{Hom}_{M\\R}(B,\\Z^{M\\R})$ is equivalent to a holim over $\\mathrm{Hom}_{M\\R}(M\\R, \\Z^{M\\R})\\simeq \\Z^{M\\R}$, connected by $e^*$, we get that \n$\\mathrm{Hom}_{M\\R}(B,\\Z^B)$ is the homotopy limit $\\hlim_{\\Z^-\\times \\Z^-}\\Z^{M\\R}$, where $\\Z^-$ denotes the poset of negative numbers and all connecting maps are $e^*$. This is equivalent to the homotopy limit indexed over the diagonal, which in turn is equivalent to the homotopy limit indexed over a vertical.\n\\end{proof}\n\nRecall that we want to show that $X = \\Sigma^{2\\rho-4}\\Z^B$ is equivalent to $\\kappa_{M\\R}(\\underline{\\vb},B)$. The reason for the choice of suspension is essentially (as before) that $H\\underline{\\Z} \\simeq \\Sigma^{2\\rho-4}H\\underline{\\Z}^*$.\n\n\\begin{prop}\\label{Cor:QuotientX}\nFor a sequence $\\ul = (l_1,l_2, \\dots)$, we have an isomorphism\n\\[\n \\underline{[\\Sigma^{*\\rho}B\/\\uvb^{\\ul}, X]}^{C_2}_{M\\R} \\cong \\underline{\\Z} \\otimes_{\\Z} (\\Z[\\vb_1,\\vb_2,\\dots]\/\\uvb^{\\ul})^*,\n\\] \nnatural with respect to the maps $B\/\\uvb^{\\ul} \\to \\Sigma^{-|\\ul'-\\ul|\\rho}B\/\\uvb^{\\ul'}$ in the defining homotopy colimit for $\\kappa_{M\\R}(\\uvb; B)$ for $\\ul' = (l_1',l_2',\\dots)$ a sequence with $l_i' \\geq l_i$ for all $i\\geq 1$. \n\\end{prop}\n\\begin{proof}\nThe last lemma implies that we also have\n$$\\Z^{B\/\\uvb^{\\ul}} \\simeq \\mathrm{Hom}_{M\\R}(B\/\\uvb^{\\ul}, \\Z^B)$$\nas the functors $\\Z^{?}$ and $\\mathrm{Hom}_{M\\R}(?, \\Z^B)$ behave the same way with respect to cofibre sequences and (filtered) homotopy colimits. Then we just have to apply Lemma \\ref{lem:AndersonQuot}. \n\\end{proof}\n\n\\subsection{The theorem}\nWe first describe the homotopy groups of $X = \\Sigma^{2\\rho-4}\\Z^B$ with $B=BP\\R$ as before. \nBy Lemma \\ref{lem:AndersonQuot}, we get\n$$\\underline{\\pi}^{C_2}_{*\\rho}X \\cong \\mathrm{Hom}(\\underline{\\pi}_{(*+2)\\rho-4}^{C_2}B, \\Z) \\cong \\underline{\\Z} \\otimes_\\Z \\Z[\\vb_1,\\vb_2,\\dots]^{*}.$$ \n\nLet $\\underline{l}$ be a sequence with only finitely many nonzero entries. By Proposition \\ref{Cor:QuotientX}, the element $(\\underline{\\vb}^{\\underline{l}-\\underline{1}})^*$ induces a corresponding $M\\R$-linear map $\\Sigma^{-|\\underline{l}-\\underline{1}|}B\/\\vb^{\\underline{l}} \\to X$, which is unique up to homotopy. By this uniqueness, these maps are also compatible for comparable $\\underline{l}$. By Remark \\ref{rmk:hocolim}, this induces a map\n\\[\\kappa_{M\\R}(\\underline{\\vb},B) = \\hcolim_{\\underline{l}}\\left(\\Sigma^{-|\\underline{l}-\\underline{1}|}B\/\\vb^{\\underline{l}}\\right) \\;\\xrightarrow{h}\\, X,\\]\nwhere $\\underline{l}$ ranges over all sequences where only finitely many $l_i$ are nonzero.\n\n\\begin{thm}\\label{Thm:BPDuality}\nThis map $h\\colon \\kappa_{M\\R}(\\underline{\\vb}; B) \\to X$ is an equivalence of $\\Ctwo$-spectra. \n\\end{thm}\n\n\\begin{proof}\nBy Corollary \\ref{Cor:crucial}, we get on $\\underline{\\pi}_{*\\rho}$-level\n\\[\\clim_{\\underline{l}} \\Sigma^{-|\\underline{l}-\\underline{1}|}\\underline{\\Z}[\\vb_1,\\vb_2,\\dots]\/(\\vb_1^{l_1},\\dots) \\to \\underline{\\Z} \\otimes_\\Z \\Z[\\vb_1,\\dots]^*,\\]\nwhich is an isomorphism. The odd underlying homotopy groups of both sides are zero. To apply Lemma \\ref{lem:regrep}, it is left to show that $\\pi^{\\Ctwo}_{k\\rho-1}\\kappa_{M\\R}(\\underline{\\vb}; B) = 0$ for all $k\\in\\Z$. Again by Corollary \\ref{Cor:crucial}, it is even true that $\\pi^{\\Ctwo}_{k\\rho-1}(B\/\\vb^{\\underline{l}})$ is zero for all $k\\in\\Z$ and all sequences $\\underline{l}$.\n\\end{proof}\n\n\\section{Duality for regular quotients}\nThe goal of this section is to prove our main result Theorem \\ref{thm:main}:\n\\begin{thm}\\label{Thm:QuotientDuality}\nLet $(m_1,m_2,\\dots)$ be a sequence of nonnegative integers with only finitely many entries bigger than $1$. Denote by $\\underline{\\vb}'$ the sequence of $\\vb_i$ in $\\pi^{\\Ctwo}_{\\bigstar}M\\R$ such that $m_i = 0$ and by $m'$ the sum of all $(m_i-1)|\\vb_i|$ for $m_i> 1$. Then there is an equivalence\n$$\\Z^{B\/\\underline{\\vb}^{\\underline{m}}} \\simeq \\Sigma^{-m'+4-2\\rho}\\kappa_{M\\R}(\\underline{\\vb}'; B\/\\underline{\\vb}^{\\underline{m}}).$$\n\\end{thm}\nHere and for the rest of the section we will implicitly localize everything at $2$ again. Before we prove the theorem, we need some preparation.\n\n\\begin{lemma}\\label{Lem:AndersonQuotient}\nLet $\\underline{m} = (m_1,\\dots)$ be a sequence of nonnegative integers with a finite number $n$ of nonzero entries. Then \n\\[\\Z^{B\/\\underline{\\vb}^{\\underline{m}}} \\simeq \\Sigma^{-|\\underline{m}|-n}(\\Z^B)\/\\underline{\\vb}^{\\underline{m}}.\\]\n\\end{lemma}\n\\begin{proof}\nLet $Y$ be an arbitrary ($\\Ctwo$-)spectrum and $\\Sigma^{|v|} Y \\xrightarrow{v} Y \\to Y\/v$ be a cofibre sequence. Then we have an induced cofibre sequence\n\\[\\Z^{Y\/v} \\to \\Z^Y \\xrightarrow{v} \\Sigma^{-|v|} \\Z^Y \\to \\Sigma \\Z^{Y\/v} \\simeq \\Sigma^{-|v|}(\\Z^{Y})\/v.\\]\nThus, $\\Z^{Y\/v} \\simeq \\Sigma^{-|v|-1}(\\Z^{Y})\/v$. The claim follows by induction. \n\\end{proof}\n\n\\begin{lemma}\\label{Lem:Nilpotence}\nThe element $\\vb_i^{3k}$ acts trivially on $B\/\\vb_i^k$ for every $i\\geq 1, k\\geq 1$.\n\\end{lemma}\n\\begin{proof}\nBy the commutativity of the diagram\n\\[\\xymatrix{\n\\Sigma^{k|\\vb_i|}B \\ar[d]^{\\vb_i^k}\\ar[r]& \\Sigma^{k|\\vb_i|}B\/\\vb_i^k \\ar[d]^{\\vb_i^k} \\ar[d]\\\\\nB \\ar[r] & B\/\\vb_i^k }\n\\]\nwe see that the composite $\\Sigma^{k|\\vb_i|} B \\to \\Sigma^{k|\\vb_i|} B\/\\vb_i^k \\xrightarrow{\\vb_i^k} B\/\\vb_i^k$ is zero, so that the latter map factors over an $M\\R$-linear map $\\Sigma^{2k|\\vb_i|+1}B \\to B\/\\vb_i^k$. As $[\\Sigma^{2k|\\vb_i|+1}B, B\/\\vb_i^k]_{M\\R}$ is a retract of $[\\Sigma^{2k|\\vb_i|+1}M\\R, B\/\\vb_i^k]_{M\\R} \\cong \\pi_{2k|\\vb_i|+1}^{C_2}B\/\\vb_i^k$, we just have to show that $\\vb_i^{2k}x = 0$ for every $x\\in \\pi_{2k|\\vb_i|+1}B\/\\vb_i^k$. \n\nWe have a short exact sequence\n\\[0 \\to (\\pi_\\bigstar^{\\Ctwo}B)\/\\vb_i^k \\to \\pi_\\bigstar^{\\Ctwo}(B\/\\vb_i^k)\\to \\left\\{\\pi^{\\Ctwo}_{\\bigstar -k|\\vb_i|-1} B\\right\\}_{\\vb_i^k} \\to 0.\\]\nAs $\\vb_i^k x$ clearly maps to zero, it is the image of a $y\\in (\\pi_\\bigstar^{\\Ctwo}B)\/\\vb_i^k$. But $\\vb_i^ky = 0$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:smash}\n We have $$B\/\\vb_i^l \\otimes_{M\\R} B\/\\vb_j^m \\simeq B\/(\\vb_i^l,\\vb_j^m).$$\n Furthermore, there is an equivalence\n \\[\\hcolim_l \\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^l\\otimes_{M\\R} B\/\\vb_i^m \\simeq \\Sigma^{|\\vb_i| +1}B\/\\vb_i^m\\]\n of $M\\R$-modules if $m\\geq 1$.\n\\end{lemma}\n\\begin{proof}\nWe have $$B\\otimes_{M\\R} B \\simeq \\hcolim (B\\xrightarrow{e} B \\xrightarrow{e} \\cdots) \\simeq B,$$\nwhere $e$ denotes again the Quillen--Araki idempotent, \nand thus also\n$$B\/\\vb_i^l \\otimes_{M\\R} B\/\\vb_j^m \\simeq B\/(\\vb_i^l,\\vb_j^m).$$\n\nThus, the maps in the homotopy colimit in the lemma are induced by the following diagram of cofibre sequences:\n\\[\\xymatrix{\n\\Sigma^{|\\vb_i|}B\/\\vb_i^m \\ar[r]^-{\\vb_i^l}\\ar[d]^{\\mathrm{id}} & \\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^m \\ar[r]\\ar[d]^{\\vb_i} & \\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^l\\otimes_{M\\R} B\/\\vb_i^m \\ar[d] \n\\\\\n\\Sigma^{|\\vb_i|}B\/\\vb_i^m \\ar[r]^-{\\vb_i^{l+1}} & \\Sigma^{-l|\\vb_i|}B\/\\vb_i^m \\ar[r]& \\Sigma^{-l|\\vb_i|}B\/\\vb_i^{l+1}\\otimes_{M\\R} B\/\\vb_i^m \n}\n\\]\nWe can assume that the homotopy colimit only runs over $l\\geq 3m$ so that by the last lemma the two cofibre sequences split and we get \n\\[\\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^l\\otimes_{M\\R} B\/\\vb_i^m \\simeq \\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^m \\oplus \\Sigma^{|\\vb_i|+1}B\/\\vb_i^m.\\]\nThe corresponding map \n\\[\\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^m \\oplus \\Sigma^{|\\vb_i|+1}B\/\\vb_i^m \\to \\Sigma^{-l|\\vb_i|}B\/\\vb_i^m \\oplus \\Sigma^{|\\vb_i|+1}B\/\\vb_i^m\\]\ninduces multiplication by $\\vb_i$ on the first summand, the identity on the second plus possibly a map from the second summand to the first. \n\nUsing this decomposition, it is easy to show that\n\\[\\hcolim_{l} \\Sigma^{-(l-1)|\\vb_i|}B\/\\vb_i^l\\otimes_{M\\R} B\/\\vb_i^m\\to \\Sigma^{|\\vb_i|+1}B\/\\vb_i^m\\]\n(defined by the projection on the second summand for $l\\geq 3m$) is an equivalence. Indeed, on homotopy groups the map is clearly surjective. And if \n$$(x,y) \\in \\pi_\\bigstar^{C_2}\\Sigma^{-l|\\vb_i|}B\/\\vb_i^m \\oplus \\pi_{\\bigstar}^{C_2} \\Sigma^{|\\vb_i|+1}B\/\\vb_i^m $$\nmaps to $0 \\in \\pi_{\\bigstar}^{C_2} \\Sigma^{|\\vb_i|+1}B\/\\vb_i^m$, then $y = 0$ and $(x,0)$ represents $0$ in the colimit because $\\vb_i$ acts nilpotently. \n\\end{proof}\n\n\n\\begin{proof}[of theorem]\nAs in the theorem, let $\\underline{\\vb}'$ be the sequence of $\\vb_i$ such that $m_i = 0$ and also denote by $\\underline{\\vb}'' = (\\vb_{i_1},\\vb_{i_2},\\dots)$ the sequence of $\\vb_i$ such that $m_i \\neq 0$. \n\nWe begin with the case that $\\underline{m}$ has only finitely many\nnonzero entries (say $n$). By Lemma \\ref{Lem:AndersonQuotient} we see that \n\\[\\Z^{B\/\\underline{\\vb}^{\\underline{m}}} \\simeq \\Sigma^{-|\\underline{m}|-n}(\\Z^B)\/\\underline{\\vb}^{\\underline{m}}.\\]\nCombining this with Theorem \\ref{Thm:BPDuality}, we obtain\n\\begin{align*}\n \\Z^{B\/\\underline{\\vb}^{\\underline{m}}} &\\simeq \\Sigma^{-|\\underline{m}|-n+4-2\\rho}\\kappa_{M\\R}(\\underline{\\vb}, B)\/\\underline{\\vb}^{\\underline{m}} \\\\\n\t\t\t\t\t &\\simeq \\Sigma^{-|\\underline{m}|-n+4-2\\rho}\\kappa_{M\\R}(\\underline{\\vb}',\\kappa_{M\\R}(\\underline{\\vb}'', B\/\\underline{\\vb}^{\\underline{m}}))\n\\end{align*}\nThus, we have to show that $\\kappa_{M\\R}(\\underline{\\vb}'', B\/\\underline{\\vb}^{\\underline{m}}) \\simeq \\Sigma^{|\\vb_{i_1}|+\\cdots |\\vb_{i_n}|+n}B\/\\underline{\\vb}^{\\underline{m}}$.\n\nBy Lemma \\ref{lem:smash}, we have an equivalence\n\\[(B\/\\underline{\\vb}^{\\underline{m}})\/(\\vb_{i_1}^{l_{i_1}}, \\dots, \\vb_{i_n}^{l_{i_n}}) \\simeq (B\/\\vb_1^{l_{i_1}}\\otimes_{M\\R} B\/\\vb_1^{m_{i_1}}) \\otimes_{M\\R}\\dots\\otimes_{M\\R} (B\/\\vb_n^{l_{i_n}}\\otimes_{M\\R} B\/\\vb_n^{m_{i_n}}).\\]\nIf we let now the homotopy colimit run over the sequences $(l_{i_1},\\dots, l_{i_n})$, we can do it separately for each tensor factor. Hence, we obtain again by Lemma \\ref{lem:smash} an equivalence\n\\[\\kappa_{M\\R}(\\underline{\\vb}'', B\/\\underline{\\vb}^{\\underline{m}}) \\simeq \\Sigma^{|\\vb_{i_1}|+\\cdots |\\vb_{i_n}| +n} B\/\\underline{\\vb}^{\\underline{m}}.\\]\nThus, we have shown the theorem in the case that $\\underline{m}$ has only finitely many nonzero entries. \n\nWe prove the case that $\\underline{m}$ has possibly infinitely many nonzero entries by a colimit argument. Define $\\underline{m}_{\\leq k}$ to be the sequence obtained from $\\underline{m}$ by setting $m_{k+1}, m_{k+2}, \\dots$ to zero. Then $B\/\\underline{m} \\simeq \\hcolim_k B\/\\underline{m}_{\\leq k}$ and thus $\\Z^{B\/\\underline{m}} \\simeq \\hlim_k \\Z^{B\/\\underline{m}_{\\leq k}}$. Denote by $\\underline{\\vb}'_{\\leq k}$ the sequence of $\\vb_i$ such that $m_i = 0$ or $i>k$ and by $m'_k$ the quantity $|\\underline{m}_{\\leq k}-\\underline{1}|$; note that $m'_k = m'$ for $k$ large. \n\nWe have to show that the map\n\\[h\\colon \\Sigma^{-m'}\\kappa_{M\\R}(\\underline{\\vb}', B\/\\underline{\\vb}^{\\underline{m}}) \\to \\hlim_k \\Sigma^{-m'_k}\\kappa_{M\\R}(\\underline{\\vb}'_{\\leq k}, B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}})\\]\nis an equivalence. This map is defined as follows: We know that \n$$\\kappa_{M\\R}(\\underline{\\vb}', B\/\\underline{\\vb}^{\\underline{m}}) \\simeq \\hcolim_k \\kappa_{M\\R}(\\underline{\\vb}', B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}}).$$\nUsing this, we get a map induced from the maps $\\kappa_{M\\R}(\\underline{\\vb}', B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}}) \\to \\kappa_{M\\R}(\\underline{\\vb}'_{\\leq k}, B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}})$ for $k$ large.\n\nBy Corollary \\ref{Cor:crucial}, we can describe what happens on $\\pi_{*\\rho}^{C_2}$: The left hand side has as $\\Z$-basis monomials of the form $\\underline{\\vb}^{\\underline{n}}$ with only finitely many $n_i$ nonzero, $n_i \\leq 0$ and $n_i\\geq -m_i+1$ if $m_i\\neq 0$. Likewise, \n$$\\pi_{*\\rho}^{C_2}\\left(\\Sigma^{m'_k}\\kappa_{M\\R}(\\underline{\\vb}'_{\\leq k}, B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}})\\right)$$\nhas as $\\Z$-basis monomials of the form $\\uvb^{\\underline{n}}$ with only finitely many $n_i$ nonzero, $n_i \\leq 0$ and $n_i\\geq -m_i+1$ if $m_i\\neq 0$ and $i\\leq k$. The maps in the homotopy limit induce the obvious inclusion maps. Thus, clearly the map \n$$\\pi_{*\\rho}^{C_2}\\left(\\Sigma^{m'}\\kappa_{M\\R}(\\underline{\\vb}', B\/\\underline{\\vb}^{\\underline{m}})\\right) \\to \\lim_k \\pi_{*\\rho}^{C_2}\\left(\\Sigma^{m'_k}\\kappa_{M\\R}(\\underline{\\vb}'_{\\leq k}, B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}})\\right)$$\nis an isomorphism. \n\nIt remains to show that $\\lim^1_k\\pi_{*\\rho+1}^{C_2}\\left(\\Sigma^{m'_k}\\kappa_{M\\R}(\\underline{\\vb}'_{\\leq k}, B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}})\\right)$ vanishes. By Corollary \\ref{cor:rho+}, every term has as $\\F_2$-basis monomials of the form $a\\uvb^{\\underline{n}}$ with only finitely many $n_i$ nonzero, $n_i \\leq 0$ and $n_i\\geq -m_i+1$ if $m_i\\neq 0$ and $i\\leq k$. The system becomes stationary in every degree, more precisely if $\\ast > -2^{k+1}$. Thus, the $\\lim^1$-term vanishes. A similar $\\lim^1$-argument also shows that the odd underlying homotopy groups of $\\hlim_k \\Sigma^{-m'_k}\\kappa_{M\\R}(\\underline{\\vb}'_{\\leq k}, B\/\\underline{\\vb}^{\\underline{m}_{\\leq k}})$ vanish.\n\nAs the source of $h$ is strongly even by Corollary \\ref{Cor:crucial} and by the arguments we just gave the morphism $h$ induces an isomorphism on $\\underline{\\pi}_{*\\rho}^{C_2}$ and on (odd) underlying homotopy groups, Lemma \\ref{lem:regrep} implies that $h$ is an equivalence. \n\\end{proof}\n\n\\vspace{1cm} \n\\part{Local cohomology computations}\\label{part:LocalCohomology}\n\nIn Part 4, we will describe the local cohomology spectral sequence in\nsome detail, and use it to understand the structure of the\n$H\\Zu$-cellularization of $BP\\R \\langle n \\rangle$. The calculation is not difficult,\nbut on the other hand it is quite hard to follow because it is made up\nof a large number of easy calculations which interact a little, and\nbecause one needs to find a helpful way to follow the $RO(\\Ctwo )$-graded\ncalculations. \n\nIn contrast the case of $k\\R$ is simple enough to be explained fully without\nfurther scaffolding, and it introduces many of the structures that we will want to\nhighlight. Since it may also be of wider interest than the general\ncase of $BP\\R \\langle n \\rangle$ we devote Section \\ref{sec:kRlcss} to it before\nreturning to the general case in Section \\ref{sec:BPRnlcss}. Section \\ref{sec:tmfotlcss} will then give a more detailed account in the interesting case $n=2$. \n\nLet us also recall some notation used throughout this part. As in the\nrest of the paper we work 2-locally, except when speaking about $k\\R$\nor $tmf_1(3)$ when fewer primes need be inverted. We often\nwrite $\\delta = 1-\\sigma \\in RO(C_2)$. We also recall the duality\nconventions from Section \\ref{sec:DAb}; in particular, for an\n$\\F_2$-vector space $V^{\\vee}$ equals the dual vector space\n$\\mathrm{Hom}_{\\F_2}(V,\\F_2)$ and for a torsionfree $\\Z$-module $M$, we set\n$M^*=\\mathrm{Hom}(M, \\Z)$. \n\n\nIf $R$ is a $C_2$-spectrum, we will use the notation $R^{C_2}_\\bigstar$ for its $RO(C_2)$-graded homotopy groups. We will also write $R^{hC_2}_{\\bigstar}=\\pi_{\\bigstar}^{C_2}(R^{(EC_2)_+})$ and similarly for geometric fixed points and the Tate construction. \n\n\\section{The local cohomology spectral sequence for $\\protect k\\R$}\n\\label{sec:kRlcss}\n\nThis section focuses entirely on the classical case of $k\\R$, where\nthere are already a number of features of interest. This gives a\nchance to introduce some of the structures we will use for the general\ncase. \n\n\\subsection{The local cohomology spectral sequence}\n\nGorenstein duality for $k\\R$ (Corollary \\ref{cor:kRGorD}) \nhas interesting implications for the coefficient ring, both\ncomputationally and structurally. \nWriting $\\bigstar$ for $RO(\\Ctwo)$-grading as usual, the local cohomology spectral\nsequence \\cite[Section 3]{G-M95} takes the following form. \n\n\\begin{prop} \n\\label{prop:kRlcss}\n There is a spectral sequence of $k\\R^{\\Ctwo}_{\\bigstar}$-modules\n$$H^*_{(\\vb)}(k\\R^{\\Ctwo}_{\\bigstar}) \\Rightarrow \\Sigma^{-4+\\sigma} \\pi^{\\Ctwo}_{\\bigstar}(\\Z^{k\\R}).$$\nThe homotopy of the Anderson dual in an arbitrary degree $\\alpha \\in\nRO(C_2)$ lies in an exact sequence\n$$0\\longrightarrow \\mathrm{Ext}_{\\Z}(k\\R^{\\Ctwo}_{-\\alpha -1}, \\Z)\\longrightarrow \n\\pi^{\\Ctwo}_{\\alpha}(\\Z^{k\\R}) \\longrightarrow \\mathrm{Hom}_{\\Z}(k\\R^{\\Ctwo}_{-\\alpha}, \\Z)\n\\longrightarrow 0. $$\nSince local cohomology is entirely in cohomological degrees 0 and 1,\nthe spectral sequence collapses to a short exact sequence \n$$0\\longrightarrow \\Sigma^{-1} H^1_{(\\vb)}(k\\R^{\\Ctwo}_{\\bigstar}) \\longrightarrow \\Sigma^{-4+\\sigma}\n\\pi^{\\Ctwo}_{\\bigstar}(\\Z^{k\\R}) \\longrightarrow H^0_{(\\vb)}(k\\R^{\\Ctwo}_{\\bigstar}) \\longrightarrow\n0. $$\nThis sequence is not split, even as abelian groups. \n\\end{prop}\n\nOne should not view Proposition \\ref{prop:kRlcss} as an algebraic\nformality: it embodies the fact that $k\\R^{\\Ctwo}_{\\bigstar}$ is a very special\nring. To illustrate this, we recall the calculation of \n$k\\R^{\\Ctwo}_{\\bigstar}$ in Subsection \\ref{sec:kRgroups}. In Subsection\n\\ref{subsec:kRloccoh} we calculate its local cohomology, and how\nthe Gorenstein duality isomorphism with the known homotopy of the Anderson \ndual works. \n\n\\subsection{The ring $\\protect k\\R^{\\Ctwo}_{\\bigstar}$}\\label{sec:kRgroups}\n\nOne may easily calculate $k\\R^{\\Ctwo}_{\\bigstar}$. This has already been done in \\cite{B-G10}, but we sketch a slightly different method. We will first calculate $k\\R^{h\\Ctwo}_{\\bigstar} $ and then use the Tate square \\cite{GMTate}. \n\nIn the homotopy fixed point spectral sequence\n$$\\Z[\\vb, a, u^{\\pm 1}]\/2a \\Rightarrow k\\R^{hC_2}_{\\bigstar}$$\nall differentials are generated by $d_3(u)=\\vb a^3$. Indeed, this differential is forced by $\\eta^4 = 0$ and there is no room for further ones. \nIt follows that $U=u^2$ is an infinite cycle, and so the\nwhole ring is $U$-periodic: \n$$k\\R^{h\\Ctwo}_{\\bigstar}=BB [U,U^{-1}], $$\nwhere $BB$ is a certain `basic block'. This basic block is a sum\n$$BB=BR\\oplus (2u)\\cdot \\Z [\\vb]$$\nas $BR$-modules, where \n$$ BR=\\Z [\\vb,a]\/(2a, \\vb a^3).$$\n\nIt is worth illustrating $BB$ in the plane (with $BB_{a+b\\sigma}$\nplaced at the point $(a,b)$). The squares and circles represent copies of $\\Z$, and\nthe dots represent copies of $\\mathbb{F}_2$. The left hand vertical column\nconsists of 1 (at the origin, $(0,0)$) and the powers of $a$, but the feature to concentrate on\nis the diagonal lines representing $\\Z [\\vb]$ submodules. These are\neither copies of $\\Z [\\vb]$ or of $\\mathbb{F}_2 [\\vb]$ or simply copies of $\\mathbb{F}_2$. \n\n$$\\begin{tikzpicture}[scale =1]\n\\draw[step=0.5, gray, very thin] (-3,-3) grid (3, 3);\n\\draw (-1.5,1.5 ) node[anchor=east, draw=orange]{\\Large{BB}};\n\\foreach \\y in {1,2,3,4,5,6}\n\\draw (0,-\\y\/2) node[anchor=east] {$a^{\\y}$};\n\\foreach \\y in {1,2,3,4,5,6}\n\\node at (0,-\\y\/2) [fill=red, inner sep=1pt, shape=circle, draw] {};\n\n\\foreach \\y in {0, 1,2,3,4,5}\n\\node at (\\y\/2+1\/2,\\y\/2) [fill=red, inner sep=1pt, shape=circle, draw]\n{};\n\\foreach \\y in {0, 1,2,3,4,5}\n\\node at (\\y\/2+1\/2,\\y\/2-1\/2) [fill=red, inner sep=1pt, shape=circle, draw] {};\n\n\\draw [->] (0,0)-- (3,3);\n\\node at (0,0) [shape = rectangle, draw]{};\n\\draw (0,0) node[anchor=east]{1};\n\\foreach \\y in {1,2,3,4,5,6}\n\\draw (\\y\/2,\\y\/2) node[anchor=east] {$\\vb^{\\y}$};\n\\foreach \\y in {1,2,3,4,5,6}\n\\node at (\\y\/2,\\y\/2) [shape=rectangle, draw] {};\n\n\\draw[red] (0,-0.5)--(3,2.5);\n\\draw[red] (0,-1.0)--(3,2.0);\n\n\\draw [->](1,-1)-- (3,1);\n\\node at (1,-1) [shape=circle, draw] {};\n\\draw (1,-1) node[anchor=east]{$2u$};\n\n\\foreach \\y in {1,2,3,4}\n\\node at (1+\\y\/2,-1+\\y\/2) [shape=circle, draw] {};\n\\end{tikzpicture}$$\n\nProceeding with the calculation, we may invert $a$ to find the\nhomotopy of the Tate spectrum $k\\R^t=F(E(C_2)_+, k\\R ) \\wedge S^{\\infty\n \\sigma}$: \n$$k\\R^{t\\Ctwo}_{\\bigstar}=\\mathbb{F}_2 [a,a^{-1}][U,U^{-1}]. $$\nOne also sees that the homotopy of the geometric fixed points (the\nequivariant homotopy of $k\\R^{\\Phi}=k\\R \\wedge S^{\\infty\\sigma}$) is \n$$k\\R^{\\Phi \\Ctwo}_{\\bigstar}=\\mathbb{F}_2 [a,a^{-1}][U] $$\nusing the following lemma:\n\\begin{lemma}\n\\label{lem:Phiconn}\n Let $X$ be a $\\Ctwo$-spectrum which is non-equivariantly connective\n and such that $X^{\\Ctwo} \\to X^{h\\Ctwo}$ is a connective cover. Then $X^{\\Phi \\Ctwo} \\to X^{t\\Ctwo}$ is a connective cover as well.\n\\end{lemma}\n\n\n\\begin{proof}\n This follows from the diagram of long exact sequences\n \\[\\xymatrix{\n \\pi_kX_{h\\Ctwo} \\ar[r]\\ar[d] & \\pi_kX^{\\Ctwo} \\ar[r]\\ar[d]& \\pi_kX^{\\Phi \\Ctwo} \\ar[r]\\ar[d] & \\pi_{k-1}X_{h\\Ctwo} \\ar[r]\\ar[d] & \\pi_{k-1}X^{\\Ctwo}\\ar[d] \\\\\n \\pi_kX_{h\\Ctwo} \\ar[r] & \\pi_kX^{h\\Ctwo} \\ar[r]& \\pi_kX^{t\\Ctwo} \\ar[r] & \\pi_{k-1}X_{h\\Ctwo} \\ar[r] & \\pi_{k-1}X^{h\\Ctwo},\n }\n \\]\n the fact that $X_{h\\Ctwo}$ is connective and the $5$-lemma. \n\\end{proof}\n\n\nNow the Tate square \n$$\\diagram \nk\\R \\rto \\dto & k\\R \\wedge S^{\\infty \\sigma} \\dto\\\\\nk\\R^{(E\\Ctwo)_+} \\rto & k\\R^{(E\\Ctwo)_+} \\wedge S^{\\infty \\sigma} \n\\enddiagram$$\ngives $k\\R^{\\Ctwo}_{\\bigstar}$.\n\nIt is convenient to observe that the two\nrows are of the form $M\\longrightarrow M[1\/a]$, so that the fibre is $\\Gamma_a\nM$. Since the two rows have equivalent fibres, we calculate the\nhomotopy of the second and obtain \n$$k\\R^{\\bigstar}_{hC_2}=NB [U, U^{-1}], $$\nwhere $NB$ is quickly calculated as the $(a)$-local cohomology\n$H^*_{(a)}(BB)$ (and named $NB$ for `negative block'). The element\n$a$ acts vertically and we can immediately read off the answer: the\ntower $\\Z [a]\/(2a)$ gives some $H^1$, and the rest is $a$-power torsion:\n$$NB=BB'\\oplus \\Sigma^{-\\pp} \\mathbb{F}_2 [a]^{\\vee}, $$\nwhere $BB' \\subset BB$ is the sub-$BR$-module\ngenerated by $2, \\vb, 2u$ (Informally, we may say that $BB'$ omits from $BB$ all monomials $a^k$ for\n$k\\geq 1$ and the generator 1). \nNote that $NB$ is placed so that its element $2$ is in degree 0 for\nease of comparison to $BB$; all occurrences of $NB$ in $k\\R^{\\Ctwo}_{\\bigstar}$\ninvolve nontrivial suspensions. \n\n\n\n\n\nAgain, it is helpful to display the negative block. This differs from\n$BB$ in that the powers of $a$ have been deleted, and replaced by a\nnew left hand column $\\Sigma^{-\\pp}\\mathbb{F}_2 [a]^{\\vee}$. The other new\nfeature is that the copy of $\\Z [\\vb]$ generated by $1$ has been\nreplaced by the kernel $(2,\\vb)$ of $\\Z\n[\\vb]\\longrightarrow \\mathbb{F}_2$, as indicated by the circle at the origin, labelled by its generator 2.\n\n$$\\begin{tikzpicture}[scale =1]\n\\draw[step=0.5, gray, very thin] (-3,-3) grid (3, 3);\n\\draw (-1.5,1.5 ) node[anchor=east, draw=orange]{\\Large{NB}};\n\\foreach \\y in {1,2,3,4,5,6}\n\\node at (-0.5,\\y\/2) [fill=red, inner sep=1pt, shape=circle, draw] {};\n\\draw[red] (-0.5,0.5)--(-0.5,3);\n\n\\foreach \\y in {0, 1,2,3,4,5}\n\\node at (\\y\/2+1\/2,\\y\/2) [fill=red, inner sep=1pt, shape=circle, draw]\n{};\n\\foreach \\y in {0, 1,2,3,4,5}\n\\node at (\\y\/2+1\/2,\\y\/2-1\/2) [fill=red, inner sep=1pt, shape=circle, draw] {};\n\n\n\\draw [->] (0,0)-- (3,3);\n\\node at (0,0) [shape = circle, draw]{};\n\\draw (0,0) node[anchor=east]{2};\n\\foreach \\y in {1,2,3,4,5,6}\n\\draw (\\y\/2,\\y\/2) node[anchor=east] {$\\vb^{\\y}$};\n\\foreach \\y in {1,2,3,4,5,6}\n\\node at (\\y\/2,\\y\/2) [shape=rectangle, draw] {};\n\\draw[red] (0.5,0)--(3,2.5);\n\\node at (0.5,0) [fill=red, inner sep=1pt, shape=circle, draw] {};\n\\draw[red] (0.5,-0.5)--(3,2.0);\n\\node at (0.5,-0.5) [fill=red, inner sep=1pt, shape=circle, draw] {};\n\n\\draw [->](1,-1)-- (3,1);\n\\node at (1,-1) [shape=circle, draw] {};\n\\draw (1,-1) node[anchor=east]{$2u$};\n\n\\foreach \\y in {1,2,3,4}\n\\node at (1+\\y\/2,-1+\\y\/2) [shape=circle, draw] {};\n\n\\foreach \\y in {1,2,3,4,5,6}\n\\draw[red] (\\y\/2,\\y\/2) -- (\\y\/2, \\y\/2-1);\n\\end{tikzpicture}$$\n\\vspace{0.2cm}\n\nThe Tate square then lets us read off \n$$k\\R^{\\Ctwo}_{\\bigstar}=\\bigoplus_{k\\leq -1} NB\\cdot \\{ U^{k}\\} \\oplus\n\\bigoplus_{k\\geq 0} BB\\cdot \\{U^{k}\\} =(U^{-1} \\cdot NB [U^{-1}] )\\oplus BB[U] $$\nThe $\\Z [U]$ module structure is given by letting $U$ act in the\nobvious way on the $NB$ and $BB$ parts, and by the maps\n$$NB \\longrightarrow BB'\\longrightarrow BB $$\nin passage from the $U^{-1}$ factor of $NB$ to the $U^0$ factor of $BB$.\n\nPerhaps it is helpful to note that with the exception of the towers\n$U^{-k}\\Sigma^{-\\pp}\\mathbb{F}_2 [a]^{\\vee}$, we have a subring of\n$BB[U,U^{-1}]$, which consists of blocks $BB\\cdot U^i$ for \n$i \\geq 0$ and blocks $BB'\\cdot U^i$ for $i<0$.\n\n\\subsection{Local cohomology}\n\\label{subsec:kRloccoh}\nRecall that we are calculating local cohomology with respect to the principal\nideal $(\\vb)$ so that we only need to consider $k\\R^{\\Ctwo}_{\\bigstar}$ as a\n$\\Z [\\vb]$-module. As such it is a sum of suspensions of the blocks\n$BB$ and $NB$, so we just need to calculate the local cohomology of\nthese. \n\nMore significantly, $\\Z [\\vb]$ is graded over multiples of the regular\nrepresentation, so local cohomology calculations may be performed on one diagonal\nat a time (i.e., we fix $n$ and consider gradings $n+*\\rho$). The only modules that occur are\n$$\\Z [\\vb], \\mathbb{F}_2 [\\vb], \\mathbb{F}_2 \\mbox{ and the ideal } (2,\n\\vb)\\subseteq \\Z[\\vb], $$\neach of which has local cohomology that is very easily calculated. \n\n\n\n\n\\begin{lemma}\nThe local cohomology of the basic block $BB$ is as follows. \n$$H^0_{(\\vb)}(BB)=a^3 \\mathbb{F}_2 [a]$$\n$$H^1_{(\\vb)}(BB)=\\Sigma^{-\\rho} \\Z [\\vb]^{*}\\oplus \\Sigma^{-\\rho+2\\delta} \\Z [\\vb]^{*}\\oplus\n\\Sigma^{-\\rho-\\sigma} \\mathbb{F}_2 [\\vb]^{\\vee}\\oplus \\Sigma^{-\\rho-2\\sigma}\n\\mathbb{F}_2 [\\vb]^{\\vee}. $$\n\\end{lemma}\n\n\\begin{proof}\nThe local cohomology is the cohomology of the complex\n$$BB\\longrightarrow BB[1\/\\vb]. $$ \nIt is clear that \n$$BB[1\/\\vb]=\\Z [\\vb,\\vb^{-1}]\\oplus u\\cdot \\Z [\\vb,\\vb^{-1}]\\oplus \na\\cdot \\mathbb{F}_2 [\\vb,\\vb^{-1}]\\oplus a^2\\cdot \\mathbb{F}_2 [\\vb,\\vb^{-1}]\\qedhere$$\n\\end{proof}\n\n\nTurning to $NB$, we recall that $NB=BB'\\oplus \\Sigma^{-\\delta} \\mathbb{F}_2\n[a]^{\\vee}$, and we have a short exact sequence\n$$0\\longrightarrow BB'\\longrightarrow BB\\longrightarrow \\mathbb{F}_2 [a]\\longrightarrow 0. $$ \nThe local cohomology is thus easily deduced from that of $BB$.\n\n\n\\begin{lemma}\nThe local cohomology of the negative block $NB$ is as follows. \n$$H^0_{(\\vb)}(NB)=\\Sigma^{-\\pp}\\mathbb{F}_2 [a]^{\\vee}$$\n$$H^1_{(\\vb)}(NB)=\\Sigma^{-\\rho} \\Z [\\vb]^{*}\\oplus \\mathbb{F}_2 \\oplus \\Sigma^{-\\rho+2\\delta} \\Z [\\vb]^{*}\\oplus\n\\Sigma^{-\\sigma} \\mathbb{F}_2 [\\vb]^{\\vee}\\oplus \\Sigma^{-2\\sigma} \\mathbb{F}_2 [\\vb]^{\\vee}$$\nMore properly, the $\\Z [\\vb]$-module structure of the sum of the first\ntwo terms is \n$$\\Sigma^{-\\rho} \\Z [\\vb]^{*}\\oplus \\mathbb{F}_2 \\cong \\Z [\\vb]^*\/(2\\cdot 1^*).$$\n\\end{lemma}\n\n\\begin{proof}\nThe local cohomology is the cohomology of the complex\n$$NB\\longrightarrow NB[1\/\\vb]. $$\n\nIt is clear that $NB[1\/\\vb]=BB[1\/\\vb]$, which makes the part coming\nfrom the $2$-torsion clear. For the $\\Z$-torsion free part, it is\nhelpful to consider the exact sequence\n$$0\\longrightarrow (2, \\vb) \\longrightarrow \\Z [\\vb]\\longrightarrow \\mathbb{F}_2 \\longrightarrow 0$$\nand then consider the long exact sequence in local cohomology.\n\\end{proof}\n\nImmediately from the defining cofibre sequence $\\Gamma_{\\vb}k\\R \\longrightarrow\nk\\R \\longrightarrow k\\R [1\/\\vb]$ we see that there is a short exact sequence\n$$0\\longrightarrow H^1_{(\\vb)}(\\Sigma^{-1}k\\R^{\\Ctwo}_{\\bigstar}) \\longrightarrow\n\\pi^{\\Ctwo}_{\\bigstar}(\\Gamma_{(\\vb)}k\\R)\\longrightarrow\nH^0_{(\\vb)}(k\\R^{\\Ctwo}_{\\bigstar}) \\longrightarrow 0. $$\nThis gives $\\pi^{\\Ctwo}_{\\bigstar}(\\Gamma_{(\\vb)}k\\R)$ up to\nextension. The Gorenstein duality isomorphism can be used to resolve the remaining extension\nissues, and the answer is recorded in the proposition below. \n\nThe diagram Figure \\ref{fig:GBBkR} should help the reader interpret the statement and proof of\nthe calculation of the homotopy of $\\Gamma_{(\\vb)}k\\R$. We have\nomitted dots, circles and boxes except at the ends of diagonals or\nwhere an additional generator is required. The vertical lines denote\nmultiplication by $a$ and the dashed vertical line is an exotic\nmultiplication by $a$ that is not visible on the level of local\ncohomology. The green diamond does not denote a class, but marks the\npoint one has to reflect (non-torsion classes) at to see Anderson\nduality. Torsion classes are shifted by $-1$ after reflection (i.e., shifted\none step horizontally to the left). \n\n\\begin{center}\n\\begin{figure}\\includegraphics{GBBkR}\n\\caption{Gorenstein duality for $k\\R$ \\label{fig:GBBkR}}\n\\end{figure}\n\\end{center}\n\n\n\\begin{prop}\nThe homotopy of the derived $\\vb$-power torsion is given by \n$$\\pi_{\\bigstar}^{\\Ctwo} (\\Gamma_{(\\vb)}k\\R)\\cong (U^{-1}\\cdot GNB [U^{-1}]) \\oplus GBB [U]$$\nwhere $GBB$ and $GNB$ are based on the local cohomology of $BB$ and\n$NB$ respectively, and described as follows. We have \n$$GBB= \\Sigma^{-2-\\sigma} \\left[ \n\\Z [\\vb]^{*}\\oplus a\\cdot \\mathbb{F}_2 [\\vb]^{\\vee}\n\\oplus a^2\\cdot \\mathbb{F}_2 [\\vb]^{\\vee}\n\\oplus u\\cdot N \\right] $$\nwhere $N$ (with top in degree 0) is given by an exact sequence\n$$0\\longrightarrow \\Z [\\vb]^* \\longrightarrow N \\longrightarrow \\mathbb{F}_2 [a]\\longrightarrow 0, $$\nnon-split in degree 0.\n\nSimilarly, \n$$GNB= \\Sigma^{-1} \n\\left[ \\Z [\\vb]^{*}\/(2 \\cdot (1^*) ) \\oplus \na\\cdot \\mathbb{F}_2 [\\vb]^{\\vee}\n\\oplus a^2\\cdot \\mathbb{F}_2 [\\vb]^{\\vee}\n\\oplus \\Sigma^{1-3\\sigma}\\Z [\\vb]^* \\oplus \\Sigma^{\\sigma}\\mathbb{F}_2\n[a]^{\\vee} \\right] $$\nwhere the action of $a$ is as suggested by the sum decomposition\nexcept that multiplication by $a$ is non-trivial wherever possible\n(i.e., when one dot is vertically above another, or where a box is\nvertically above a dot). \n\\end{prop}\n\n\\begin{proof}\nWe first note that the contributions from the different blocks do not\ninteract. Indeed, the only time that different blocks give\ncontributions in the same degree come from the $\\mathbb{F}_2 [a]$ towers of\n$BB$: one class in that degree is $\\vb$-divisible (and not killed\nby $\\vb$) and the other class is annihilated by $\\vb$. We may therefore \nconsider the blocks entirely separately.\n\nThe block $GBB$ comes from the local cohomology of $BB$ and therefore\nlives in a short exact sequence \n$$0\\longrightarrow H^1_{(\\vb)}(\\Sigma^{-1}BB) \\longrightarrow\nGBB \\longrightarrow H^0_{(\\vb)}(BB) \\longrightarrow 0$$\n\nThe block $GNB$ comes from the local cohomology of $NB$\nand therefore lives in a short exact sequence \n$$0\\longrightarrow H^1_{(\\vb)}(\\Sigma^{-1}NB) \\longrightarrow\nGNB \\longrightarrow H^0_{(\\vb)}(NB) \\longrightarrow 0$$\n\nMost questions about module structure over $BB[U]$ are resolved by\ndegree, but there are two which remain. These can be resolved Gorenstein duality \\ref{cor:kRGorD} and the known module structure in $\\Z^{k\\R}$. \n\nIn $GBB$, the additive\nextension in $\\pi^{\\Ctwo}_{-3\\sigma}$ is non-trivial: \n$$\\pi_{-3\\sigma}^{\\Ctwo} (\\Gamma_{(\\vb)}k\\R)\\cong \\Z. $$\nAlso the multiplication by $a$ \n$$\\mathbb{F}_2 \\cong GNB_{-1+\\sigma} \\to GNB_{-1}\\cong \\mathbb{F}_2$$\nis nonzero (where $GNB_{-1+\\sigma}$ corresponds to $\\pi^{C_2}_{-5+5\\sigma}(\\Gamma_{(\\vb)}k\\R)$ in the $U^{-1}$-shift). \n\\end{proof}\n\n\\begin{remark}\n It is striking that the duality relates the top $BB$ to the bottom\n$NB$ (i.e., Anderson duality takes the part of $\\Gamma_{\\vb}k\\R$\ncoming from the local cohomology of $BB$ to $NB$), and it takes the\nbottom $NB$ to the top $BB$ (i.e., Anderson duality takes the part of $\\Gamma_{\\vb}k\\R$\ncoming from the local cohomology of $NB$ to $BB$). \nIndeed, as commented after Lemma \\ref{lem:Phiconn}, since $NB=\\Gamma_{(a)}BB$, we have\n$$\\Sigma^{2+\\sigma}\\Gamma_{(\\vb)}BB\\simeq (\\Gamma_{(a)} BB)^*$$\nand \n$$\\Gamma_{(\\vb, a)} BB\\simeq \\Sigma^{-2-\\sigma} BB^*, $$\nwith the second stating that $BB$ is Gorenstein of shift $-2-\\sigma$\nfor the ideal $(a, \\vb )$. \n\n\nBy extension, Anderson duality takes the part of $\\Gamma_{\\vb}k\\R$\ncoming from the local cohomology of all copies of $BB$ to all copies\nof $NB$ and vice versa. This might suggest separating $k\\R$ into a\npart with homotopy $BB[U]$, giving a cofibre sequence \n$$\\langle BB[U]\\rangle\\longrightarrow k\\R \\longrightarrow \\langle U^{-1}NB[U^{-1}]\\rangle , $$\nwhere the angle brackets refer to a spectrum with the indicated\nhomotopy. However one may see that there is no $C_2$-spectrum with homotopy the Mackey functor corresponding to $BB[U]$\n(considering the $b\\sigma$ and $(b+1)\\sigma$ rows one sees that the\nnon-equivariant homotopy of the spectrum would be zero up to about \ndegree $2b$; taking all rows together it would have to be\nnon-equivariantly contractible and hence $a$-periodic). Similarly,\nthere is no spectrum with homotopy $U^{-1}NB[U^{-1}]$, so these dualities are purely\nalgebraic. \n\\end{remark}\n\n\n\n\\section{The local cohomology spectral sequence for $\\protect BP\\R \\langle n \\rangle$}\n\\label{sec:BPRnlcss}\n\n\n\nGorenstein duality for $BP\\R \\langle n \\rangle$ (Example \\ref{ex:BPRn}) \nhas interesting implications for the coefficient ring, both\ncomputationally and structurally. \nWriting $\\bigstar$ for $RO(\\Ctwo)$-grading as usual, the local cohomology spectral\nsequence \\cite[Section 3]{G-M95} takes the form described in the\nfollowing proposition.\n We now revert to our standard assumption of\nworking 2-locally, so that $\\Z$ means the 2-local integers. \n\n\n\\begin{prop} \n\\label{prop:BPRnlcss}\nThere is a spectral sequence of $BP\\R \\langle n \\rangle^{\\Ctwo}_{\\bigstar}$-modules\n$$H^*_{\\Jb_n}(BP\\R \\langle n \\rangle^{\\Ctwo}_{\\bigstar}) \\Rightarrow \\Sigma^{-(D_n+n+2)-(D_n-2)\\sigma} \\pi^{\\Ctwo}_{\\bigstar}(\\Z^{BP\\R \\langle n \\rangle})$$\nfor $\\Jb_n = (\\vb_1,\\dots, \\vb_n)$. \nThe homotopy of the Anderson dual in an arbitrary degree $\\alpha \\in\nRO(C_2)$ is easily calculated\n$$0\\longrightarrow \\mathrm{Ext}_{\\Z}(BP\\R \\langle n \\rangle^{\\Ctwo}_{-\\alpha -1}, \\Z)\\longrightarrow \n\\pi^{\\Ctwo}_{\\alpha}\\Z^{BP\\R \\langle n \\rangle} \\longrightarrow \\mathrm{Hom}_{\\Z}(BP\\R \\langle n \\rangle^{\\Ctwo}_{-\\alpha}, \\Z) \\longrightarrow 0. $$\nFor $n\\geq 2$ the local cohomology spectral sequence has some non-trivial\ndifferentials. \n\\end{prop}\n\nOne should not view Proposition \\ref{prop:BPRnlcss} as an algebraic\nformality: it embodies the fact that $BP\\R \\langle n \\rangle^{\\Ctwo}_{\\bigstar}$ is a very special\nring. \n\nIn the present section we will discuss the implications of this for\nthe coefficient ring for general $n$. The perspective is a bit distant\nso the reader is encouraged to refer back to $k\\R$ (i.e., the case\n$n=1$) in Section \\ref{sec:kRlcss} to anchor the generalities. \n\nHowever the case $n=1$ is too simple to show some of what happens, so \nwe will also illustrate the case $tmf_1(3)$ (i.e.,\nthe case $n=2$) in Section \\ref{sec:tmfotlcss}.\n\n\\subsection{Reduction to diagonals}\nFor brevity we write $R_{\\bigstar}=BP\\R \\langle n \\rangle_{\\bigstar}^{\\Ctwo}$. Because the ideal\n$\\Jb_n=(\\vbn{1}, \\ldots , \\vbn{n})$\n is generated by elements whose degrees are a multiple of\n$\\rho$, we can\ndo $\\Jb_n$-local cohomology calculations over the subring $R_{*\\rho}$\nof elements in degrees which are multiples of $\\rho$. \n\nThus, for an $R_{\\bigstar}$-module $M_{\\bigstar}$ we have a direct sum decomposition\n$$M_{\\bigstar}=\\bigoplus_{d} M_{d+*\\rho}$$\nas $R_{*\\rho}$-modules, where we refer to the gradings $d+*\\rho$ as the\n{\\em $d$-diagonal}. Hence, we also have\n$$H^i_{\\Jb_n}(M_{\\bigstar})=\\bigoplus_d H^i_{\\Jb_n}(M_{d+*\\rho}). $$\n(We have abused notation by also writing $\\overline{J}_n$ for the ideal of\n$R_{*\\rho}$ generated by $\\vbn{1}, \\ldots , \\vbn{n}$.)\n\n\\subsection{The general shape of $BP\\R \\langle n \\rangle^{\\Ctwo}_{\\bigstar}$}\nBy the description at the end of Section \\ref{sec:BPRnC2}, we have an isomorphism\n$$R_{\\bigstar}= U^{-1}\\cdot NB[U^{-1}] \\oplus BB [U]$$\nwith $BB$ and $NB$ as described there. It is easy to see that $BB$ and $NB$ decompose as $R_{*\\rho}$-modules into modules of a certain form we will describe now. We will implicitly $2$-localize everywhere. \n\nThe modules $BB$ and $NB$ decompose into are\n$$P=R_{*\\rho}=\\Z[\\vbn{1}, \\ldots, \\vbn{n}] \\mbox{ and } \\Pb{s}=P\/(\\vbn{0}, \\ldots,\n\\vbn{s})=\\mathbb{F}_2 [\\vbn{s+1},\\ldots , \\vbn{n}] $$\nfor $s\\geq 0$\nand the ideals expressed by the exact sequences \n\\begin{align*} 0\\longrightarrow (2, \\vbn{1}, \\ldots , \\vbn{t})\\longrightarrow P \\longrightarrow\n \\Pb{t}\\longrightarrow 0 \\end{align*}\nor \n\\begin{align*}0\\longrightarrow (\\vbn{s+1}, \\ldots , \\vbn{t})\\longrightarrow \\Pb{s}\\longrightarrow \\Pb{t}\\longrightarrow 0\\end{align*}\nwith $s\\geq 0$.\n\nTheir local cohomology is easily calculated. In the first two cases, the modules only have local cohomology in a\nsingle degree\n\\begin{align*}H_{\\Jb_n}^*(P)&=H_{\\Jb_n}^n(P)=P^*(-D_n\\rho) \\\\\nH_{\\Jb_n}^*(\\Pb{s})&=H_{\\Jb_n}^{n-s}(\\Pb{s})=\\Pb{s}^{\\vee}((D_s-D_n)\\rho). \\end{align*}\nThe top non-zero degree of $P^*$ is zero, so that $1^* \\in\nP^*(-D_n\\rho)$ is in degree $-D_n\\rho = -|\\vb_1|-\\cdots -\n|\\vb_n|$. We alert the reader to the fact that star is used in two ways: occasionally in\n$H^*$ to mean cohomological grading and rather frequently here in\n$P^*$ to mean the $\\Z$-dual of $P$. \n\nNow we turn to the ideal $(\\vbn{s+1}, \\ldots , \\vbn{t})$. If $t=s+1$ the ideal is principal\nand $(\\vbn{s+1})\\cong \\Pb{s}((s+1)\\rho)$; thus we get a single local cohomology group\n$$H^{n-s}_{\\Jb_n} ((\\vbn{s+1}) \\Pb{s})=\\Pb{s}^{\\vee}((D_s-D_n+s+1)\\rho)$$\nas can be seen from the long exact sequence of local cohomology. \n\nOtherwise we get two local cohomology groups\n$$H^{n-s}_{\\Jb_n} ((\\vbn{s+1}, \\ldots , \\vbn{t})\\Pb{s})=\\Pb{s}^{\\vee}((D_n-D_s)\\rho)\n\\mbox{ and } H^{n-t+1}_{\\Jb_n} ((\\vbn{s+1}, \\ldots , \\vbn{t})\n\\Pb{s})=\\Pb{t}^{\\vee}((D_n-D_t)\\rho).$$\n\nThe case of $(2, \\vbn{1}, \\ldots , \\vbn{t})$ is similar but with an extra case. The case $t=0$ is easy since then\n$(2)\\cong P$ so the local cohomology is all in cohomological degree $n$ where it is\n$P^*(-D_n\\rho)$. If $t=1$ we again get a single local cohomology group\n$$H^{n}_{\\Jb_n} ((2,\\vbn{1}) P)=P^*(-D_n\\rho)\\oplus \\Pb{1}^{\\vee}((D_{1}-D_n)\\rho).$$\nOtherwise we get two local cohomology groups\n$$H^{n}_{\\Jb_n} ((2, \\ldots , \\vbn{t})P)=P^*(-D_n\\rho)\n\\mbox{ and } H^{n-t+1}_{\\Jb_n} ((2, \\ldots , \\vbn{t})P) =\\Pb{t}^{\\vee}((D_t-D_n)\\rho).$$\n\n\n\\subsection{The special case $n=1$}\nThe best way to make the patterns apparent is to look at the simplest\ncases. In this section we begin with $k\\R^{C_2}_{\\bigstar}$ as treated in\nSection \\ref{sec:kRlcss} above, and we encourage the reader to relate\nthe calculations here to the diagrams in Section \\ref{sec:kRlcss}. In that case, \n$$P=k\\R^{C_2}_{*\\rho}=\\Z [\\vbn{1}], \\Pb{0}=\\mathbb{F}_2 [\\vbn{1}] \\mbox{ and } \\Pb{1}=\\mathbb{F}_2.$$\n\nDisplaying $BB$ by $d$-diagonal, we have\n\n$$\\begin{array}{c|cc}\n&BB&(n=1)\\\\\n\\hline\nd&1&u\\\\\n\\hline\n0&P&\\\\\n1&\\Pb{0}&\\\\\n2&\\Pb{0}&\\\\\n3&\\Pb{1}&\\\\\n4&\\Pb{1}&(2)P\\\\\n5&\\Pb{1}&\\\\\n6&\\Pb{1}&\\\\\n7&\\Pb{1}&\\\\\n8&\\Pb{1}&\\\\\n\\end{array}$$\n\\vspace{0.2cm}\n\nThe position of the modules along the $d$-diagonal can be inferred\nfrom the label at the top of the column. Thus the first column has\ngenerators in degree $-d\\sigma$, and the second column similarly, but\n in the column of $u$ (namely the 2-column). Noting that $u$ is on the\n 4-diagonal, the $d$th row has generators in $|u| -(d-4)\\sigma = 2-(d-2)\\sigma$. For example, along the 4-diagonal we have $a^4\\Pb{1}\n\\oplus (2u)P$.\n\n\n\nTaking local cohomology, and shifting $H^s_{\\Jb_n}$ down by $s$ (as in the local cohomology spectral sequence), we have\n$$\\begin{array}{c|cc}\n&H^*_{(\\vb_1)}(BB)&(n=1)\\\\\n\\hline\nd&1&u\\\\\n\\hline\n-1&{\\color{brown}P^*(-2\\rho)}&\\\\\n0&{\\color{brown} \\Pb{0}^{\\vee}(-2\\rho)}&\\\\\n1&{\\color{brown}\\Pb{0}^{\\vee}(-2\\rho)}&\\\\\n2&&\\\\\n3&\\Pb{1}&{\\color{brown}P^*(-2\\rho)}\\\\\n4&\\Pb{1}&\\\\\n5&\\Pb{1}&\\\\\n6&\\Pb{1}&\\\\\n7&\\Pb{1}&\\\\\n8&\\Pb{1}&\\\\\n\\end{array}$$\n\\vspace{0.2cm}\n\nHere, we colored $H^1$-groups brown. Note that shifting down by $s$ both lowers $d$ by $s$ and adds a shift by $-s\\rho$. For example, considering the 3-diagonal of this table, the $\\Pb{1}$ comes directly\nfrom the 3-diagonal of $BB$, whilst the $P^*(-2\\rho)$ comes from the\n$(2)P$ on the 4-diagonal of $BB$; the local cohomology is\n$P^*(-\\rho)$, but its diagonal is shifted by $-1$ since it is a first\nlocal cohomology, and because it is by reference to the 2-column the\nshift is $-\\rho$. The top of this module is calculated by\nreference to the column of $|u|$ (i.e., the 2-column), and has top in degree\n $2-(3-2)\\sigma-2\\rho=-3\\sigma$.\n\nWe saw in Section \\ref{sec:kRlcss} that the two modules on the\n3-diagonal give a non-trivial additive extension (in degree\n$-3\\sigma$) after running the spectral sequence. \n\n\n\\subsection{The special case $n=2$}\n\\label{subsec:tmfotloccoh}\nContinuing our effort to make patterns visible, we consider\n$tmf_1(3)^{C_2}_{\\bigstar}$ in this subsection (i.e., the case $n=2$). With\n$\\Z$ denoting the integers with 3 inverted here, this has\n$$P=tmf_1(3)^{C_2}_{*\\rho}=\\Z [\\vbn{1}, \\vbn{2}], \\Pb{0}=\\mathbb{F}_2 [\\vbn{1},\n\\vbn{2}], \\Pb{1}=\\mathbb{F}_2 [\\vbn{2}] \\mbox{ and } \\Pb{2}=\\mathbb{F}_2.$$\n\nThus for $n=2$ we have\n$$\\begin{array}{c|cccc}\n&BB&(n=2)&&\\\\\n\\hline\nd&1&u&u^2&u^3\\\\\n\\hline\n0&P&&&\\\\\n1&\\Pb{0}&&&\\\\\n2&\\Pb{0}&&&\\\\\n3&\\Pb{1}&&&\\\\\n4&\\Pb{1}&(2)&&\\\\\n5&\\Pb{1}&&&\\\\\n6&\\Pb{1}&&&\\\\\n7&\\Pb{2}&&&\\\\\n8&\\Pb{2}&&(2,\\vbn{1})P&\\\\\n9&\\Pb{2}&&(\\vbn{1})\\Pb{0}&\\\\\n10&\\Pb{2}&&(\\vbn{1})\\Pb{0}&\\\\\n11&\\Pb{2}&&&\\\\\n12&\\Pb{2}&&&(2)\\\\\n13&\\Pb{2}&&&\\\\\n\\end{array}$$\nOnce again, the column labelled $u^i$ is the $2i$th column, and shifts\nalong the diagonal have as reference point where this column meets the\nrelevant diagonal. \n\nWe take local cohomology, again remembering that $H^s_{\\Jb_n}$ is\nshifted down by $s$, which changes the diagonal by $s$. For example, on the 7-diagonal, $\\Pb{2}$ comes from the 7-diagonal in\n$BB$, whereas the $\\Pb{0}^{\\vee}(-5\\rho)$ comes from the 2nd local\ncohomology of the entry $(\\vbn{1})\\Pb{0}$ on the $9$-diagonal; the local\ncohomology of $\\Pb{0}$ is $\\Pb{0}^{\\vee}(-4\\rho)$, this is shifted by\na further $-2\\rho$ from the change of diagonal, and $+\\rho$ because of\nthe $\\vbn{1}$. \n$$\\begin{array}{c|cccc}\n&H^*_{(\\vb_1,\\vb_2)}(BB)&(n=2)&&\\\\\n\\hline\nd&1&u&u^2&u^3\\\\\n\\hline\n-2&{\\color{teal}P^*(-6\\rho)}&&&\\\\\n-1&{\\color{teal}\\Pb{0}^{\\vee}(-6\\rho)}&&&\\\\\n0&{\\color{teal}\\Pb{0}^{\\vee}(-6\\rho)}&&&\\\\\n1&&&&\\\\\n2&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&{\\color{teal}P^*(-6\\rho)}&&\\\\\n3&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&&&\\\\\n4&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&&&\\\\\n5&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&&&\\\\\n6&&& {\\color{brown}\\Pb{1}^{\\vee}(-5\\rho)}\\,\\oplus\\, {\\color{teal}P^*(-6\\rho)}&\\\\\n7&\\Pb{2}&&{\\color{teal}\\Pb{0}^{\\vee}(-5\\rho)}&\\\\\n8&\\Pb{2}&&{\\color{teal}\\Pb{0}^{\\vee}(-5\\rho)}&\\\\\n9&\\Pb{2}&&&\\\\\n10&\\Pb{2}&&&{\\color{teal}P^*(-6\\rho)}\\\\\n11&\\Pb{2}&&&\\\\\n12&\\Pb{2}&&&\\\\\n13&\\Pb{2}&&&\\\\\n\\end{array}$$\n\nWe have colored again $H^1$-groups in brown and now also $H^2$-groups in teal. We will see below that there are non-trivial extensions on the 2- and 10-diagonals, and that\nthere are differentials in the local cohomology spectral sequence from\nthe 7-, 8- and 9-diagonals (differentials go from the $d$-diagonal to\nthe $(d-1)$-diagonal). \n\n\\subsection{Moving from the basic block $BB$ to the negative block $NB$}\nMoving from $BB$ to $NB$ only affects the $0$ column, where in each\ncase $M$ is replaced by $\\ker (M \\longrightarrow \\mathbb{F}_2)=(2)M$. In effect this\nreplaces $\\Pb{n}$ by $0$. It also adds on a new $(-1)$-column \nof $\\Pb{n}=\\mathbb{F}_2$ going up from the $\\sigma$ row. We resist the temptation\nto display a table for $NB$ explicitly, but note that\n$NB=\\Gamma_{(a)}BB$ as for $k\\R$. \n\n\\subsection{Gorenstein duality}\n\\label{subsec:GorDBPRn}\nWith the above data in mind, we may consider the $d$-diagonal $BB_d$,\nwhere the lowest value of $d$ is 0 and the highest is $N=4(2^n-1)$. If\nwe ignore the difference\nbetween $BB$ and $NB$ (which is at most $\\mathbb{F}_2$ in any degree) we find\napproximately that $BB_d$ has a relationship to $BB_{N-d}$, namely \nsomething like an equality\n$$H^n_{\\Jb_n}(BB_d)^*=BB_{N-d}. $$\nThere are various ways in which this is inaccurate and needs to be modified. \nFirstly, if the local cohomology of $BB_d$ is entirely in cohomological \ndegree $n-\\eps $ with $\\eps \\neq 0$, there will be a shift of $\\eps$\n(if it is in several degrees there is a further\ncomplication). Secondly, Anderson duality introduces a shift of 1 diagonal if\napplied to torsion modules. Thirdly, we have seen that there may be\nextensions between these local cohomology groups, sometimes removing\n$\\Z$-torsion. Finally, there may be differentials. \n\nIn fact all of these effects are `small' in the sense that the growth\nrate along a diagonal is bounded by a polynomial of degree $n-1$. \nEncouraged by this, if we ignore all of these effects we see \nthat $BB$ is a Gorenstein module in the sense that the reverse-graded \nversion is equivalent to \nthe dual of its local cohomology.\n$$H^n_{\\Jb_n}(BB)^*=\\mathrm{rev}(BB). $$ \n\nThis is rather as if there is a cofibre sequence \n$$S\\longrightarrow BP\\R \\langle n \\rangle\\longrightarrow Q$$\nwith $S$ Gorenstein and $Q$ a Poincar\\'e duality algebra of formal dimension\n$N=2(1-\\sigma)(2^n-1)$. \n\n\n\n\n\n\\section{The local cohmology spectral sequence for $tmf_1(3)$}\n\\label{sec:tmfotlcss}\nWe examine the local cohomology spectral sequence and Gorenstein duality\nin more detail for $tmf_1(3)$. Actually, our calculations are equally valid for all forms of $BP\\R\\langle 2\\rangle$, but we prefer the more evocative name $tmf_1(3)$ of the most prominent example. More of the general features are visible for $tmf_1(3)$\nthan for $k\\R$.\n\n As usual we will implicitly localize everywhere at $2$ (although for $tmf_1(3)$ itself it would actually suffice to just invert $3$).\n\n\\subsection{The local cohomology spectral sequence}\nWe make explicit the implications for the coefficient ring, both\ncomputationally and structurally. Writing $\\bigstar$ for\n$RO(\\Ctwo)$-grading as usual, the spectral sequence takes the\nfollowing form. \n\n\\begin{prop} \n\\label{prop:tmfotlcss}\nThere is a spectral sequence of $tmf_1(3)^{\\Ctwo}_{\\bigstar}$-modules\n$$H^*_{\\Jb_n}(tmf_1(3)^{\\Ctwo}_{\\bigstar}) \\Rightarrow \\Sigma^{-8-2\\sigma} \\pi^{\\Ctwo}_{\\bigstar}(\\Z^{tmf_1(3)}).$$\nThe homotopy of the Anderson dual is easily calculated\n$$0\\longrightarrow \\mathrm{Ext}_{\\Z}(tmf_1(3)^{\\Ctwo}_{-\\alpha -1}, \\Z)\\longrightarrow \n\\pi^{\\Ctwo}_{\\alpha}\\Z^{tmf_1(3)} \\longrightarrow \\mathrm{Hom}_{\\Z}(tmf_1(3)^{\\Ctwo}_{-\\alpha}, \\Z) \\longrightarrow 0. $$\nThe local cohomology spectral sequence has some non-trivial differentials. \n\\end{prop}\n\n\\subsection{The ring $\\protect tmf_1(3)^{\\Ctwo}_{\\bigstar}$}\\label{sec:tmfgroups}\n\nThe ring $tmf_1(3)^{\\Ctwo}_{\\bigstar}$ is approximately calculated in \\cite{HM} and is more precisely desribed as \n$$BB[U] \\oplus U^{-1}NB[U^{-1}]$$\nas at the end of Section \\ref{sec:BPRnC2} with $n=2$. We already tabulated $BB$ in Section \\ref{subsec:tmfotloccoh}, but we want also want to display a bigger chart of $\\pi_{\\bigstar}^{C_2}tmf_1(3)$ as Figure \\ref{fig:tmf13} to give the reader a feeling of how the blocks piece together. \n\nA black diagonal line means a copy of $P$ when it starts in a box, a\ncopy of $(2)P$ when it starts in a small circle, a copy of\n$(2,\\vb_1)P$ when it starts in a dot and a copy of $(2,\\vb_1,\\vb_2)$\nwhen it starts in a big circle. A red diagonal line means a copy of\n$\\overline{P}_0$ and a green diagonal line a copy of $\\overline{P}_1$. A red dot is a copy of $\\F_2 = \\overline{P}_2$. \n\n\n\\begin{center}\n\\begin{figure}\\includegraphics{tmf13}\n\\caption{The homotopy of $tmf_1(3)$ \\label{fig:tmf13}}\n\\end{figure}\n\\end{center}\n\n\\subsection{Local cohomology}\nWe are calculating local cohomology with respect to the \nideal $\\Jb_2=(\\vbn{1}, \\vbn{2})$ so that we only need to consider $tmf_1(3)^{\\Ctwo}_{\\bigstar}$ as a\n$\\Z [\\vbn{1}, \\vbn{2}]$-module. As such it is a sum of suspensions of the blocks\n$BB$ and $NB$, so we just need to calculate the local cohomology of\nthese. This was described in Section \\ref{sec:BPRnlcss} above. Here we will\nsimply describe the extensions and the behaviour of the local\ncohomology spectral sequence. \n\nThe basis of this discussion are the tables of $BB$ and $GBB$ from Subsection\n\\ref{subsec:tmfotloccoh} together with the analogues for $NB$ and\n$GNB$. Although these are organized by diagonal, Figure \\ref{fig:GBBtmf13}\ndisplaying $BB, GBB, U^{-1} NB$ and $U^{-1}GNB$ may help visualize the \nway the modules are distributed along each diagonal. The vertical lines denote\nmultiplication by $a$ and the dashed vertical line is an exotic\nmultiplication by $a$ that is not visible on the level of local\ncohomology. The green diamond does not denote a class, but marks the\npoint one has to reflect (non-torsion classes) at to see Anderson\nduality. Torsion classes are shifted after reflection by $-1$ (i.e.,\none step horizontally to the left). \n\n\\begin{center}\n\\begin{figure}\\includegraphics{GBBtmf13}\n\\caption{Gorenstein duality for $tmf_1(3)$ \\label{fig:GBBtmf13}}\n\\end{figure}\n\\end{center}\n\n\n\nThe strategy is to take the known subquotients from the local\ncohomology calculation, and resolve the extension problems using Gorenstein\nduality. \n\n\n\n\n\\begin{prop}\nWe have an isomorphism \n$$\\pi_{\\bigstar}^{C_2}\\Gamma_{\\Jb_2}tmf_1(3) \\cong GBB[U]\\oplus U^{-1}GNB[U^{-1}],$$\nwhere $GBB$ and $GNB$ are described in the following. We will simultaneously describe what differentials and extensions in the local cohomology spectral sequence caused the passage from $H^*_{\\Jb_2}(BB)$ and $H^*_{\\Jb_2}(NB)$ to $GBB$ and $GNB$ respectively. \n\n(i) The $\\Z [\\vbn{1}, \\vbn{2}]$-modules along the diagonals in $GBB$ are as\nfollows. \n$$\\begin{array}{c|cl}\n&GBB&(n=2)\\\\\n\\hline\ni&\\mbox{Module}&\\mbox{Top degree}\\\\\n\\hline\n-2&P^*&-6-4\\sigma\\\\\n-1&\\Pb{0}^{\\vee}&-6-5\\sigma\\\\\n0&\\Pb{0}^{\\vee}&-6-6\\sigma\\\\\n1&0&\\\\\n2&[(2,\\vbn{1})P]^*&-4-6\\sigma\\\\\n3&\\Pb{1}^{\\vee}&-4-7\\sigma\\\\\n4&\\Pb{1}^{\\vee}&-4-8\\sigma\\\\\n5&\\Pb{1}^{\\vee}&-4-9\\sigma\\\\\n6&[(2,\\vbn{1})P]^*&-2-8\\sigma\\\\\n7&(\\vbn{1}, \\vbn{2})\\Pb{0}&-2-9\\sigma\\\\\n8&(\\vbn{1}, \\vbn{2})\\Pb{0}&-2-10\\sigma\\\\\n9&0&\\\\\n10&[(2, \\vbn{1}, \\vbn{2})P]^*&0-10\\sigma\\\\\n10+k\\geq 11&\\mathbb{F}_2&0-(10+k)\\sigma\\\\\n\\end{array}$$\nThere are three non-trivial differentials\n$$d_2: H^0_{\\Jb_2}(BB)\\longrightarrow H^2_{\\Jb_2}(BB)$$\nfrom the groups at $-7\\sigma, -8\\sigma, -9\\sigma$ to the groups at \n$-7\\sigma-1, -8\\sigma -1, -9\\sigma-1$, which have affected the values\non the 6-, 7-, 8- and 9-diagonals in the table. \n\nThe extensions \n$$0\\longrightarrow P^* \\longrightarrow [(2,\\vbn{1})P]^* \\longrightarrow \\mathbb{F}_2 [\\vb_2]^{\\vee}\\longrightarrow 0$$\non the 2-diagonal and the 6-diagonal are Anderson dual to the defining short exact sequence\n$$0\\longrightarrow (2,\\vbn{1})P\\longrightarrow P \\longrightarrow \\mathbb{F}_2 [\\vb_2]\\longrightarrow 0$$\nin the following sense: The Anderson dual of the latter exact sequence is a triangle\n$$\\mathbb{F}_2[\\vb_2]^* \\to P^*\\to [(2,\\vbn{1})P]^* \\to \\Sigma\\mathbb{F}_2[\\vb_2]^* \\cong \\mathbb{F}_2[\\vb_2]^{\\vee},$$\nwhich induces (on homology) the extensions above. \nThe extension \n$$0\\longrightarrow P^* \\longrightarrow [(2,\\vbn{1}, \\vbn{2} )P]^* \\longrightarrow \\mathbb{F}_2 \\longrightarrow 0$$\non the 10-diagonal is Anderson dual to the short exact sequence\n$$0\\longrightarrow (2,\\vbn{1}, \\vbn{2})P\\longrightarrow P \\longrightarrow \\mathbb{F}_2 \\longrightarrow 0.$$\n\n(ii) The $\\Z [\\vbn{1}, \\vbn{2}]$-modules along the diagonals in $GNB$ are as\nfollows (take the direct sum of the two entries for the $(-2)$-,\n$(-1)$-, $0$- $1$- and $2$-diagonals) \n$$\\begin{array}{c|cl}\n&GNB&(n=2)\\\\\n\\hline\ni&\\mbox{Module}&\\mbox{Top degree}\\\\\n\\hline\n-k\\leq -3&\\mathbb{F}_2&-1-k\\sigma\\\\\n-2&P^*, \\mathbb{F}_2 &-6-4\\sigma, -1+\\sigma\\\\\n-1&\\Pb{0}^{\\vee}, \\mathbb{F}_2&-6-5\\sigma, -1+0\\sigma\\\\\n0&\\Pb{0}^{\\vee}, \\mathbb{F}_2 &-6-6\\sigma, -1-\\sigma\\\\\n1&\\mathbb{F}_2 &-1-2\\sigma\\\\\n2&P^*, \\Pb{1}^{\\vee}&-4-6\\sigma, -1-3\\sigma\\\\\n3&\\Pb{1}^{\\vee}&-1-4\\sigma\\\\\n4&\\Pb{1}^{\\vee}&-1-5\\sigma\\\\\n5&\\Pb{1}^{\\vee}&-1-6\\sigma\\\\\n6&[(2,\\vbn{1})P]^*&-1-7\\sigma\\\\\n7&\\Pb{0}^{\\vee}&-1-8\\sigma\\\\\n8&\\Pb{0}^{\\vee}&-1-9\\sigma\\\\\n9&0&\\\\\n10&P^*&0-10\\sigma\\\\\n\\end{array}$$\n\nThe extension\n$$0\\longrightarrow P^* \\longrightarrow [(2,\\vbn{1})P]^* \\longrightarrow \\mathbb{F}_2 [v_2]^{\\vee}\\longrightarrow 0$$\non the 6-diagonal is Anderson dual to the short exact sequence\n$$0\\longrightarrow (2,\\vbn{1})P\\longrightarrow P \\longrightarrow \\mathbb{F}_2 [v_2]\\longrightarrow 0.$$\n\\end{prop}\n\n\n\n\\begin{proof}\nWe first note that the contributions from the different blocks do not\ninteract. Indeed, the only time that different blocks give\ncontributions in the same degree comes from the $\\mathbb{F}_2 [a]$ towers of\n$BB$, and one class in that degree is divisible by $\\vbn{1}$ or $\\vbn{2}$ and not killed\nby both $\\vbn{1}$ and $\\vbn{2}$. We may therefore \nconsider the blocks entirely separately.\n \nThe block $GBB$ comes from the local cohomology of $BB$ in the sense\nthat there is a spectral sequence\n$$H^*_{\\Jb_2}(BB)\\Rightarrow GBB .$$\nThus there is a filtration \n$$GBB=GBB^0\\supseteq GBB^1\\supseteq GBB^2\\supseteq GBB^3=0$$\nwith \n$$0\\longrightarrow GBB^0\/GBB^1 \\longrightarrow H^0_{\\Jb_2}(BB) \\stackrel{d_2}\\longrightarrow\n\\Sigma^{-1}H^2_{\\Jb_2}(BB) \\longrightarrow \\Sigma^1 GBB^2\\longrightarrow 0$$\nand \n$$GBB^1\/GBB^2\\cong \\Sigma^{-1} H^1_{\\Jb_2}(BB). $$ \n\nThe block $GNB$ comes from the local cohomology of $NB$ in a precisely\nanalogous way.\n\nMost questions about module structure over $BB[U]$ are resolved by\ndegree. The remaining issues are resolved by\nusing Gorenstein duality. \n\n\nReferring to the table for $H^*_{\\Jb_2}(BB)$ in Subsection\n\\ref{subsec:tmfotloccoh}, the first potential extension is on the\n2-diagonal. Using Gorenstein duality to compare with $NB_{\\delta=8}$ \nwe see that the actual extension on the 2-diagonal of $GBB$ is \n$$0\\longrightarrow P^*\\longrightarrow [(2,\\vb_1)P]^*\\longrightarrow \\Pb{1}^{\\vee} \\longrightarrow\n0, $$\nwhere we have shifted the modules so they all have top degree 0. \nThere is an additive extension on the 10-diagonal by\nreference to the Anderson dual. \nFinally the three non-zero $d_2$ differentials from $-1-k\\sigma$ for\n$k=7,8$ and $9$ are necessary for connectivity (this removes the \nneed to discuss the possible extensions on the 7- and 8-diagonals). \n\n\nThe situation is rather similar for $GNB$. We will not explicitly\ndisplay $NB$ since the only effect (apart from the addition of \n$\\mathbb{F}_2 [a]^{\\vee}$) is on the first column, where a\nmodule is replaced by the kernel of a surjection to $\\mathbb{F}_2$. It is\nperhaps worth displaying $H^2_{\\Jb_2}(NB)$, where we leave out the big $\\mathbb{F}_2[a]^{\\vee}$-tower in $H^0_{\\Jb_2}NB$. We will color again $H^1$-groups in brown and $H^2$-groups in teal.\n$$\\begin{array}{c|cccc}\n&H^*_{\\Jb_2}(NB)&(n=2)&&\\\\\n\\hline\ni&1&u^2&u^4&u^6\\\\\n\\hline\n-2&{\\color{teal}P^*(-6\\rho)}&&&\\\\\n-1&{\\color{teal}\\Pb{0}^{\\vee}(-6\\rho)}\\oplus {\\color{brown}\\Pb{2}}&&&\\\\\n0&{\\color{teal}\\Pb{0}^{\\vee}(-6\\rho)}\\oplus {\\color{brown}\\Pb{2}}&&&\\\\\n1&{\\color{brown}\\Pb{2}}&&&\\\\\n2&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&{\\color{teal}P^*(-6\\rho)}&&\\\\\n3&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&&&\\\\\n4&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&&&\\\\\n5&{\\color{brown}\\Pb{1}^{\\vee}(-4\\rho)}&&&\\\\\n6&&& {\\color{brown}\\Pb{1}^{\\vee}(-5\\rho)}\\oplus {\\color{teal}P^*(-6\\rho)}&\\\\\n7&&&{\\color{teal}\\Pb{0}^{\\vee}(-5\\rho)}&\\\\\n8&&&{\\color{teal}\\Pb{0}^{\\vee}(-5\\rho)}&\\\\\n9&&&&\\\\\n10&&&&{\\color{teal}P^*(-6\\rho)}\\\\\n11&&&&\\\\\n12&&&&\\\\\n13&&&&\\\\\n\\end{array}$$\n In this case all extensions are split, except for the one on the\n6-diagonal and there are no differentials. The $a$ multiplications\nin the $\\mathbb{F}_2 [a]^{\\vee}$ tower are clear from Gorenstein duality and\nthe $a$-tower $\\mathbb{F}_2 [a]$ in $BB$.\n\\end{proof}\n\n\\begin{remark}\n(i) Summarizing the way a diagonal $BB_{\\delta}$ contributes to $NB$ as in \n$$H^*_{\\Jb_2}(BB_{\\delta})^*\\sim NB_{\\delta'}$$ \nas sketched in Subsection \\ref{subsec:GorDBPRn}. We have \n\n$$\\begin{array}{|cc||cc|}\n\\delta &\\delta' s.t. H^*_{\\Jb_2}(BB_{\\delta})^*\\sim\nNB_{\\delta'}&\\delta &\\delta' s.t. H^*_{\\Jb_2}(NB_{\\delta})^*\\sim BB_{\\delta'}\\\\\n\\hline\n0&12&0&12\\\\\n1&10&1&10\\\\\n2&9&2&9\\\\\n3&8&3&8\\\\\n4&8,6&4&8,6\\\\\n5&5&5&5\\\\\n6&4&6&4\\\\\n7&2&7&.\\\\\n8&4,3&8&4\\\\\n9&2&9&2\\\\\n10&1,0&10&1\\\\\n11&0&11&.\\\\\n12&0&12&0\\\\\n\\hline\n\\end{array}$$\n\nBecause most of the modules are $2$-torsion the most common pairing is\nbetween $\\delta$ and $11-\\delta$ rather than between \n$\\delta$ and $12-\\delta$ as happens for the main $U$-power diagonals. \n\n(ii) We also note as before that since $NB=\\Gamma_{(a)}BB$, we have\n$$\\Sigma^{6+4\\sigma}\\Gamma_{(\\vb_1, \\vb_2)}BB\\sim (\\Gamma_{(a)} BB)^*$$\n(where we have written $\\sim$ rather than $\\simeq$ in recognition of\nthe differentials) and \n$$\\Sigma^{6+4\\sigma}\\Gamma_{(\\vb_1, \\vb_2, a)} BB\\simeq BB^*, $$\nwith the second stating that $BB$ is Gorenstein of shift $-6-4\\sigma$\nfor the ideal $(\\vb_1, \\vb_2, a )$. \n\\end{remark}\n\n\n\\vspace{1cm}\n\\addtocontents{toc}{\\vspace{\\normalbaselineskip}}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:Intro}\nEfficient telescope scheduling is essential to maximize the scientific output of survey telescopes. \nOptimizing a survey's scientific merit function requires scheduling decisions that consider slew rate, sky brightness, source location, transparency and extinction, and many other factors. For surveys such as LSST that will scan the entire accessible sky \\citep{Ivezic2014}, a determination of the sky brightness as a function of azimuth, elevation and passband is an important factor in crafting an optimal sequence of observations. Contributions to the night sky include unresolved or diffuse celestial sources, emission from OH molecules in the upper atmosphere, zodiacal light, man-made light pollution, and moonlight that scatters from clouds and from the constituents of the atmosphere. Some of these contributions to the night sky are stable over time and do not impact the order in which we observe fields. Other \ncontributions to the night sky are time-variable but deterministic. The monthly \nvariation due to moonlight falls in this category, for cloud-free conditions. Other \ncontributions to sky brightness, such as variable OH emission \\citep{High2010} and moonlight scattering from clouds, are more stochastic in nature. \n\nUnderstanding the brightness of the cloudless moonlit sky in the LSST bands is one key component in scheduling decisions. To make predictions of potential future performance, LSST has developed an operations simulator to study different scheduling algorithms \\citep{OpSim}. \nThus far, the LSST operations simulator has used historical weather records and measures of the atmospheric conditions for the Cerro Pachon site (taken over a 10-year time period). \nThe operations simulator generates a sky model that predicts sky brightness based on the Krisciunas and Schaefer model \\citep{KrSc1991}, which is further discussed below. It also simulates atmospheric seeing and cloud coverage as a function of time. This information is used to estimate the efficiency of the survey for different candidate scheduling algorithms. As LSST advances into the construction phase, and eventually into full operation, we need a higher fidelity determination of the brightness of the cloudless \nnight sky. This will be augmented with all-sky-camera data \\citep{AllSkyCamera} to make real-time, condition-dependent adjustments to the sequencing of LSST observations. \n\nWe define the lunar contribution to sky brightness as the difference between the observed sky brightness (in units of magnitudes per square arc sec) with the Moon above the horizon and the moonless sky brightness, for a given the phase of the lunar cycle. From the standpoint of scheduling decisions, what matters most is the {\\it relative} lunar brightness variation across the accessible sky, and so our primary goal in this paper is to determine this spatial structure, using the sun as a proxy for the moon. This allows us to obtain high signal-to-noise data, without complications from other contributions to sky brightness. \n\nAn {\\it ab initio} computation of the lunar sky illumination is complicated due to multiple scattering effects. Sunlight reflects off the moon, and a portion of this light is scattered towards the Earth. This light impinging on the top of the atmosphere is then scattered by molecules and aerosols, and some is absorbed. The moonlight can be scattered multiple times, including off of the ground, before it reaches the telescope pupil. An empirical measurement is arguably more secure than a radiative transfer calculation that must make assumptions about the size, shape, and vertical distribution of aerosols. \n\nWalker developed \\citep{Walker1987} a scattered moonlight model that included a table of sky brightness in five photometric bands, at five different moon phases. It did not account for the positions of the Moon or observation target, and was measured during solar minimum. Because of these shortcomings, it is not accurate enough for current and future telescope operations. Later, Krisciunas and Schaefer used an empirical fit to 33 observations taken in the V-band taken at the 2800\\,m level of Mauna Kea, resulting in an accuracy between 8\\% and 23\\% if not near full Moon \\citep{KrSc1991}. This model predicted the moonlight as a function of the Moon's phase, the zenith distance of the Moon, the zenith distance of the sky position, the angular separation of the Moon and sky position, and the band's atmospheric extinction coefficient. More recently, a spectroscopic extension of this model was used to fit sky brightness data from Cerro Paranal \\citep{Noll2012}. This treatment includes all relevant components, such as scattered moonlight and starlight, zodiacal light, airglow line emission and continuum, scattering and absorption within the Earth's atmosphere, and thermal emission from the atmosphere and telescope. This model was recently updated with an observed solar spectrum, a lunar albedo fit, and scattering and absorption calculations \\citep{Jones2013}. Winkler et al. characterized the nighttime sky brightness profile under a variety of atmospheric conditions using measurements from the South African Astronomical Observatory soon after the Mount Pinatubo volcanic eruption in 1991 \\citep{WiWy2013}.\nOur goals in this paper are more limited, as we are primarily interested in the \nspatial structure and spectrum of the scattered moonlight component of the night sky. \n\nBased on this discussion, it is clear that models for scattered moonlight are very complicated. This motivates our attempt to empirically determine the relative sky brightness as a function of lunar phase, and its dependence on the positions of the target and the moon. We measured the solar sky brightness as a function of angle between sky location and the sun, as well as zenith angles of the sun and the telescope. We argue that this is useful because up to wavelength-dependent lunar albedo factors, the sky illumination pattern that the moon casts has the same functional form as from the sun in the daytime. \n\nWe have made measurements of the daytime sky brightness at Cerro Pachon, the LSST site in Chile, with an array of six photodiodes with filters in \nthe {\\it u, g, r, i, z,} and {\\it y} bands. \nThere is an extensive history of daytime sky brightness measurements. In particular, the \nangular and wavelength dependence of the observed solar scattering can be used to deduce properties of atmospheric aerosols and precipitable water vapor. \nWe use a similar measurement scheme to the AERONET remote sensing aerosol monitoring network \\citep{Holben1998}. The AERONET sky brightness data are taken in optical passbands that differ from those that LSST will use. Rather than invoke a set of \ncolor transformations to convert from AERONET into LSST bands, here we make a \ndirect measurement of sky brightness in the LSST passbands. \n\nWe describe the apparatus in section \\ref{sec:apparatus}. Measurements and analysis are presented in section \\ref{sec:results}. We conclude with a discussion of topics for further study in section \\ref{sec:Conclusion}. \n\n\\section{Apparatus}\n\\label{sec:apparatus}\n\n\\begin{figure}[t]\n \\includegraphics[width=6.0in]{plots\/photodiode_drawing.pdf}\n \\caption{Sketch of the photodiode portion of the apparatus. The photodiode mounts include a photodiode, an iris, a filter, and a baffle tube.\n \\label{fig:apparatus}\n\\end{figure}\n\nFig.~\\ref{fig:apparatus} shows a sketch of the photodiode mount, which is identical \nfor all six channels except for the interference filters that define the passbands. \nLight enters a 50\\,mm inner diameter cylinder, 152.4 mm long, that serves to block off-axis stray light. The 50\\,mm diameter filters are placed at the base of this baffle tube. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=6.0in]{plots\/filter.pdf}\n\\caption{Photon sensitivity function curves for the photodiodes (upper) used in this experiment, and \nthe expected photon sensitivity function for LSST (lower curves, due to more complex optical system). From left to right the bands are $u,g,r,i,z$ and $y$. We use this information\nto make a throughput correction when predicting scattered moonlight backgrounds for LSST. The Astrodon filters we used for the photodiodes are designed to avoid the water band at 940 nm. }\n\\label{fig:QE}\n\\end{center}\n\\end{figure}\n\nWe used ``Generation 2 Sloan Digital Sky Survey (SDSS)'' {\\it u,g,r,i,z,y} filters from Astrodon \\citep{Astrodon}. \nFigure~\\ref{fig:QE} shows their transmission spectra as well as the current-design LSST filters \\citep{Ivezic2014}, for comparison. The Astrodon filters we used are essentially flat-topped, with minimal leakage or in-band ripple. An adjustable iris (Thor Labs SMD12C) sits behind the interference filter. We found we could operate with these irises set to their maximum opening diameter of 12\\,mm, for all six passbands. \nThe only other transmissive optical element that lies between the Si and the sky is a quartz window in front of the photodiode. \n\nThe photodiodes are SM1PD2A cathode-grounded Si UV-enhanced photodiodes, obtained from Thor Labs. The photodiodes have a 10\\,mm x 10\\,mm active area behind a 9\\,mm diameter input aperture. The only other transmissive optical element that lies between the Si and the sky is a quartz window in front of the photodiode. The etendue of the system is established by the combination of the 12\\,mm diameter iris and the 9\\,mm circular photodiode input aperture. These two circular apertures are separated by a distance of 60\\,mm. \n\n\\subsection{Etendue of the photodiode plus tube system, in comparison to an LSST pixel}\n\nAs seen from the plane of the diode aperture, the full-angle subtended by the adjustable iris is then $2\\,\\textrm{arctan}(\\frac{6}{60})=11.4^\\circ$, which subtends a solid angle of \n$\\Omega_\\textrm{diode}=2\\,\\pi\\,(1-\\cos(\\frac{11.4}{2}))=3.11\\times10^{-2}$ steradians. For comparison, \na pixel on LSST subtends 0.2 arcsec on a side, for a solid angle of $7.4\\times10^{-13}$\\,steradians\/pixel.\n\nIf we consider the iris as establishing the field of view of the photodiode system, then the \naperture in front of the diode determines the sensor's unvignetted collecting area, where $A_\\textrm{photodiode}=\\pi\\,(4.5 \\times 10^{-3} \\textrm{m})^2 = 6.36\\times10^{-5}\\,\\textrm{m}^2$. This amounts to computing the overlap of the sensor and iris apertures. For comparison, the effective collection area of LSST is equivalent to a diameter of 6.5\\,m, for a collection area of $A_\\textrm{LSST}=\\pi\\,(\\frac{6.5\\textrm{m}}{2})^2=33.2\\,\\textrm{m}^2$. The ratio of the etendue of an LSST pixel to the photodiode is then $R=\\frac{A_\\textrm{LSST} \\Omega_\\textrm{LSST pixel}}{A_\\textrm{photodiode} \\Omega_\\textrm{photodiode}}=1.23\\times 10^{-5}$.\n\n\n\nThe interpretation of the data will benefit from knowing the ratio between the instrumental response function of the photodiode system and LSST. Table~\\ref{tab:throughputs} compares the band-integrated system throughput figure for the photodiode system (the Thor labs QE times the Astrodon filter response) with two versions of the LSST throughput. LSST is considering using CCDs from two vendors, e2v and ITL, and these have somewhat different quantum efficiency curves. We have, therefore, provided in Table~\\ref{tab:throughputs} the results from integrating over the response functions (including in the LSST case the three reflections, the obscuration, the filter and corrector transmissions, and the detector QE) at a spacing of one nm. The units in Table~\\ref{tab:throughputs} are nm, and can be interpreted as the sensitivity-weighted equivalent width of the respective filters. Taking the ratio of these numbers, passband by passband, allows us to scale the photodiode measurements to anticipated values on the LSST focal plane. \n\n\\begin{table}[htdp]\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\nband & diodes & LSST with ITL & LSST with e2v & $\\left [ \\frac{T_{ITL}} {T_{diodes}} \\right ]$ & $\\left [ \\frac{T_{e2v}} {T_{diodes}} \\right ]$ \\\\\n\\hline\nu & 33.8 & 20.6 & 15.3 & 0.61 & 0.45 \\\\\ng & 99.0 & 61.3 & 65.4 & 0.62 & 0.66 \\\\\nr & 93.2 & 60.3 & 62.9 &0.65 & 0.67 \\\\\ni & 106.7 & 53.7 & 53.2 & 0.50 & 0.50 \\\\\nz & 155.2 & - & - & - & - \\\\\nzs & 65.8 & 44.3 & 43.3 & 0.67 & 0.66 \\\\\ny & 69.5 & 27.9 & 27.2 & 0.40 & 0.40 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{System throughput values. The first three columns are the integral of the system response function at 1 nm spacings, for each passband in the different systems. \nThe last two columns show ratios of the LSST\nthroughput to that of the diodes. The diode instrument has no reflective optics \nand a minimum of air-glass interfaces, whereas LSST has three reflections from \naluminum as well as a three element corrector. Also, the photodiodes are considerably thicker than the LSST CCDs, and have enhanced UV sensitivity. This accounts for the \nincreased diode throughput compared to the LSST system.}\n\\label{tab:throughputs}\n\\end{table}%\n\n \n\nThe photodiodes are connected via coaxial cable to a set of manual switches that feed a selected one of the six signals to a Thor Labs model PDA200C photocurrent amplifier, which produces a $\\pm$ 10\\,V signal proportional to the current from the selected photodiode. This signal was connected to an Arduino Uno, which digitizes this signal with a 10 bit A\/D converter. The Arduino was connected to a serial port on the data collection computer.\n\nThe six photodiode tubes are mounted on a Celestron model CG-5 equatorial telescope mount, which is controlled by external connection to a laptop computer. Stellarium (\\cite{Stellarium}) is used to control the telescope mount pointing. The mount's RA and DEC motors have a precision of 0.05$^\\circ$. The computer registered the right ascension ($\\alpha$) and declination ($\\delta$) for each brightness measurement, along with the photocurrent from each of the six photodiodes. \nThese $\\alpha$ and $\\delta$ measurements were converted to alt-az coordinates for data analysis, since as shown below this is the most natural angular coordinate system for this problem. \n\n\n\\subsection{Sky Scanning Strategy and Angular Coordinates}\n\nAssuming that the scattering properties of the atmosphere are axisymmetric about local vertical, the normalized sky brightness (scaled to the brightness of the illuminating source) depends on three angles: the zenith angle $z_\\textrm{source}$ of the source (sun or moon), the zenith angle $z_\\textrm{tel}$ of the telescope boresight, and the azimuthal angle $\\Delta \\phi$ between the source and the boresight.\n \nAn ``almucantar'' is the line on the sky at the elevation angle of the Sun, at some\ngiven time. An advantage to making sky brightness measurements along an almucantar \nis that the boresight and source elevation angles are constant, and equal. Only their\nazimuthal separation is varied. \nDuring an almucantar measurement, observations are made at the solar elevation angle through 360$^\\circ$ of azimuth. The almucantar sweep is a special case of a constant-zenith-angle scan, which is our favored data collection method. The range of scattering angles along an almucantar decreases as the solar zenith angle decreases; thus almucantar sequences made at airmass of 2 or more achieve maximum scattering angles of 120$^\\circ$ or larger.\n\nWe elected to obtain our sky brightness data in a succession of constant-zenith angle scans, taking a data point every 45$^\\circ$ of azimuth, except near the zenith. This gives us 8 data points in azimuth at each telescope zenith angle. We obtained data at zenith angles of 0, 30, 45, 60, and 75$^\\circ$, and along the almucantar, over the course of the day, in each passband. The resulting daytime sky brightness (DSB) data by passband, $DSB(z_\\textrm{source}, z_\\textrm{tel}, \\phi, \\textrm{filter})$, comprise our measurement. We generate an all-sky map of sky brightness. The data are processed as follows. For each solar zenith angle, we generate an all-sky map of brightness vs. position. We then repeat for different values of solar zenith angle. We make a polynomial fit to brightness vs. altitude and delta-azimuth. This is a map of relative night sky brightness if the moon were at the location of the sun, up to an overall scale factor per passband. We can use the geometry of the photodiode tube to compute number of photons per square arcsec per square meter and then scale the value by about 14 magnitudes to get the lunar contribution. The scaling factors are computed in the next section. We can then compute an estimate for other lunar phases, based on the lunar phase function. Of course, the actual sky brightness is a combination of the lunar contribution, which we compute, plus other factors, which depend on the instrument, plate scale, integration time, etc. as well as solar cycle and site characteristics. \n \n\\subsection{Scaling from Solar to Lunar Illumination}\n\nKeiffer and Stone (\\cite{KiSt2005}, hereafter K\\&S)) describe how to scale between solar \nillumination and lunar\nillumination at the top of the atmosphere, depending on both reflection geometry\nand wavelength. The ratio $R$ of the lunar to solar irradiance at the top of the atmosphere \nis given by\n$R(\\lambda,g)=A(\\lambda, g) \\left [ \\Omega_M\/\\pi \\right ] $, where $\\Omega_M$ \nis the solid angle subtended by the moon, and $A(\\lambda,g)$ is a wavelength \ndependent scattering function that depends on the angle, g, between the vectors \nfrom the moon to the Earth and the moon to the sun. We\nhave assumed nominal values for the sun-moon and moon-earth distances. \nThe geometrical dilution factor is 6.42\/$\\pi \\times 10^{-5} = 2.04 \\times 10^{-5}$, or 11.72 magnitudes. \n\nTo correct for the wavelength-dependent and phase-angle-dependent lunar scattering function, we took the parametric description of $A(\\lambda,g)$ provided in K\\&S, integrated across our passbands (note: truncated u band at 350, no data bluer), to determine the scattering-dependent magnitude differences between sunlight and moonlight at the top of the atmosphere, for different lunar phases. In order to avoid the sharp peak in reflection due\nto the ``opposition effect'', we limited the range of phase angles to $|g| > $ 2 degrees.\n\nThe additional attenuation from lunar scattering as a function of passband at \nfull moon (taken here to be our minimum scattering\nangle of 2 degrees) is listed in Table \\ref{tab:fullmoon}. We obtained these values by numerically computing \n\n\\begin{equation}\n\\Delta \\textrm{mag}_i (g) = -2.5 \\textrm{log}_{10} \\left( \\frac{\\int A(\\lambda,g) T_i(\\lambda) {\\rm d \\lambda}} {\\int T(\\lambda) {\\rm d \\lambda}} \\right)\n\\label{eq:fullmoon}\n\\end{equation}\n\n\\noindent\nwhere $T_i (\\lambda)$ is a top-hat approximation of filter $i$. \n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\nBand & $\\Delta \\textrm{mag}(g=2^\\circ)$ & $\\Delta \\textrm{mag}(g=10^\\circ)$ & $\\Delta \\textrm{mag}(g=45^\\circ)$ & $\\Delta \\textrm{mag}(g=90^\\circ)$ \\\\\n\\hline\n$u$ & 2.60 & 3.05 & 4.06 & 5.52\\\\\n$g$ & 2.36 & 2.78 & 3.77 & 5.19 \\\\\n$r$ & 2.10 & 2.50 & 3.45 & 4.84 \\\\\n$i$ & 1.92 & 2.31 & 3.23 & 4.59 \\\\\n$z$ & 2.17 & 2.57 & 3.53 & 4.92 \\\\\n$y$ & 1.72 & 2.12 & 2.91 & 4.31 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Irradiance attenuation due to lunar scattering, in the LSST bands, at various lunar phase angles. The illumination from the moon is slightly redder than sunlight, in general. \nThis reddening effect increases as the phase angle increases.}\n\\label{tab:fullmoon}\n\\end{table*}%\n\nBased on these values, since the sky brightness scales linearly with the irradiance provided at the top of the atmosphere, we expect the $r$ band full-moon lunar sky brightness to be a factor of 11.72+2.10=13.82 magnitudes fainter than what we observe in the daytime, if the moon were placed in the same alt-az position as the sun. \n\n\nThis allows us to generate, up to a single passband-dependent overall scale factor that depends on the \neffective etendue of the photodiode tube, the equivalent full-moon sky brightness map\nfor the case where the moon is in the same location as the sun. \nWe simply take the solar-illuminated sky brightness data, and scale all values to the\nr band brightness at the zenith. Then we make a passband-dependent adjustment based\non the color of the reflected sunlight as shown in Table \\ref{tab:fullmoon}.\n\n\\section{Results}\n\\label{sec:results}\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\nFilter & Dark Current [nA] & $1 \\sigma$\\,Uncertainty [nA] \\\\\\hline\\hline\nu & 4.3 & 0.2 \\\\\\hline\ng & 2.1 & 0.3 \\\\\\hline\nr & 5.3 & 0.2 \\\\\\hline\ni & 4.2 & 0.2 \\\\\\hline\nz & 4.6 & 0.2 \\\\\\hline\ny & 2.0 & 0.3 \\\\\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Dark current measurement for all six filters. Measurements were taken periodically during data taking to check if there was a significant temperature dependence. Dark current values were found to be approximately the same over the run, and these values are about 100 times smaller than the currents in the sky brightness analysis.\n}\n\\label{tab:DarkCurrent}\n\\end{table}\n\nWe begin by measuring dark current values for each photodiode channel, and the results are displayed in Table~\\ref{tab:DarkCurrent}. The photodiodes have dark current values ranging from 2.0 to 5.3\\,nA with statistical uncertainties between 0.2 to 0.3\\,nA. These dark currents are well under 1\\% of the signal levels from the sky. Because it is a negligible contribution, we ignore the dark current contribution in the analysis that follows.\n\n\\subsection{Observations}\n\nWe obtained sky brightness data from the roof of the ALO building on Cerro\nPachon, (located at S 30:15:06, W 70:44:18) during the daytime on \n2014 Sept 4, 5, 6 and 7. The conditions on Sept 5 were less favorable, \nwith high cirrus clouds in the sky. \nWe cycled through the sky sampling strategy described above, \ntaking 2000 data points in each passband, running through the 6 bands\nin succession. Each data collection period at a fixed pointing lasted about \ntwo minutes, and a full cycle across the sky lasted about an hour. In all, we collected 10 sequences, spanning a range of solar elevations from 20$^\\circ$ to 55$^\\circ$. \n\n\\subsection{Spatial structure of scattered light}\n\nNight sky structure was investigated by \\cite{ChHa1996}, in the context of flat-fielding. The authors are unaware of a comprehensive program to map (and visualize) the sky brightness under variable lunar illumination conditions. In the following, we show the sky brightness dependence on the zenith angle $z_\\textrm{source}$ of the source (sun or moon), the zenith angle $z_\\textrm{tel}$ of the telescope boresight, and the azimuthal angle $\\phi$ between the source and the boresight. We perform fits of sky brightness to the three independent variables and compute the effect on 5 sigma point source detection magnitude for a survey such as LSST. The data allows us to study the color across the sky. We also note how to scale overall brightness in each band as a function of lunar phase.\n\n\\subsection{Spatial Structure of Lunar Sky Brightness}\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\nBand & a ($\\times 10^{11}$) & b ($\\times 10^{11}$) & c ($\\times 10^{11}$) & d ($\\times 10^{11}$) & Median Residual ($\\times 10^{11}$) \\\\\n\\hline\nu & 88.5 (6.2) & -0.5 (0.1) & -0.5 (0.1) & 0.4 (0.1) & 5\\\\\ng & 386.5 (34.0) & -2.2 (0.2) & -2.4 (0.2) & 0.8 (0.5) & 13\\\\\nr & 189.0 (32.7) & -1.4 (0.2) & -1.1 (0.2) & 0.8 (0.5) & 11\\\\\ni & 164.8 (33.1) & -1.5 (0.2) & -0.7 (0.2) & 0.6 (0.5) & 12\\\\\nz & 231.2 (62.3) & -2.8 (0.3) & -0.7 (0.4) & 1.4 (0.9) & 21\\\\\nzs & 131.1 (45.6) & -1.4 (0.2) & -0.5 (0.3) & 0.2 (0.6) & 10\\\\\ny & 92.0 (32.7) & -1.3 (0.2) & -0.2 (0.2) & 0.9 (0.5) & 20\\\\\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Daytime sky brightness values as function of angle between the point on sky and sun, altitude of the point on the sky, and altitude of sun fit to a plane of the form $a + b x + c y + d y$ in electrons\/s. The zs band values, which are given to approximate the LSST z filter, are computed using Astrodon z minus y. Here, x corresponds to the angle between the point on the sky and the sun, y corresponds to the altitude of the point on the sky, and z corresponds to the altitude of the sun. The median of the absolute value of the residuals is given by the final column. }\n\\label{tab:results}\n\\end{table*}\n\nThe first result we present is sky brightness as a function of angle between the point on sky and the sun, the altitude of the point on the sky, and the altitude of the sun. The measurements describe a three-dimensional surface corresponding to these parameters. We convert the photodiode output from $\\mu$A to electrons\/s. We fit resulting data to a plane of the form $a + b x + c y + d y$ for each of the six bands. Here, x corresponds to the angle between the point on sky and the sun, y corresponds to the altitude of the point on the sky, and z corresponds to the altitude of the sun. The coefficients and the standard errors, as well as the median of the residuals for the fits are shown in Table~\\ref{tab:results}. We find the residuals of the fits to be small; these are generally an order of magnitude smaller than the overall scale factor (a). There are a number of notable features. The first is that as $\\phi$ increases, the sky brightness decreases. This in and of itself is not surprising, but the rates of decrease are significant. For example, in the g band, for every 10 degrees that $\\phi$ decreases, the number of photons by more than a factor of 2. There is a similar effect for the altitude of the point on the sky. As the point moves to the horizon, the sky brightness increases. This effect is more pronounced for the bands near the blue. Finally, the sky brightness increases as the altitude of the sun increases. Perhaps more interesting is the significant color dependence of the results. Although the trends described above hold true regardless of color, the magnitude of their effect is very different. The difference between the g and y bands is about a factor of 2 difference in the point on the sky dependence and more than a factor of 10 difference in the altitude of the point on the sky. The effect of the altitude of the sun, $z_\\textrm{source}$, is more constant across color.\n\nIt is straightforward to apply the measured planar fit coefficients to an individual observation. Table~\\ref{tab:fullmoon} provides the appropriate scale factor for different moon phases and passbands. If the passband of interest varies significantly from the filters in this study, one can compute the appropriate factor from equation~\\ref{eq:fullmoon}. After this, one finds the altitude and azimuth at the site of interest at the time of the observation as well as the altitude and azimuth of the target. Three angles are then computed from these quantities: the angle between the point on the sky and the moon, the altitude of the point on the sky, and the altitude of the moon. One then computes $a + b x + c y + d y$ for the appropriate passband, where a, b, c, and d are given in Table~\\ref{tab:results}, and x corresponds to the angle between the point on the sky and the moon, y corresponds to the altitude of the point on the sky, and z corresponds to the altitude of the moon.\nThis table allows for a straightforward comparison between the ratio of sky brightnesses for different colors. The ratio of fluxes in u to g, for example, are fairly flat across the sky due to the similar ratios between coefficients. On the other hand, the ratio of fluxes in u to i, depends heavily on angle to the source, with virtually no dependence on zenith angle. This indicates that the sky is much redder close to the moon than far away.\nA code that performs these steps is available at https:\/\/github.com\/mcoughlin\/skybrightness for public download. Hopefully, this will allow other researchers to easily use the data product. Required inputs are the latitude, longitude, and elevation of the site, right ascension and declination of the source, the passband of interest, and the times of observation.\n\n\n\n\\subsection{From Relative Sky Brightness to $m_5$ Variations to Optimal Scheduling}\n\nWe can use the data we have generated of {\\it relative} daytime sky brightness to generate a sky map of degradation in the point source magnitude that can be detected in the case where scattered moonlight dominates the Poisson noise. We define the {\\it sky brightness factor}, SBF, to be the ratio of the local sky to the darkest attainable sky surface brightness at that moment. In order to achieve the same SNR with varying sky backgrounds, the source must be brighter by a factor of $\\Delta m_5 = \\frac{1}{2} \\times -2.5\\,\\textrm{log}_{\\textrm{10}}(SBF)$. Because the sky brightness structure is linear in the illumination level, the sun-illuminated measurements are perfectly valid for making these $\\Delta m_5$ maps. If one region of the sky is twice as bright as another in the daytime, then for the moonlight-dominated case, replacing the sun with the moon will not change that fact. An example is shown in Figure \\ref{fig:dm5}, for the u-band. For the scheduling of LSST observations over the course of a night, a week, or a month, this\nis the format in which we think the sky brightness data is most useful. We stress that this comes directly from the daytime measurements of relative brightness, with no \nconversion needed, as long as scattered moonlight dominates the sky background. \n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=5.5in]{plots\/a1_alt_angle_meshgrid.pdf}\n\\caption{Variation $\\Delta m_5$ in the point source magnitude that can be detected at 5$\\sigma$ in the u-band, against spatially varying sky brightness. This contour plot shows the change in point \nsource detection threshold as a function of altitude and angular \nseparation from the moon.\nThe color bar indicates the change in $m_5$ for a fixed exposure time, in magnitudes. Maps\nsuch as this \ncan be used to optimize the sequence of LSST observations. }\n\\label{fig:dm5}\n\\end{center}\n\\end{figure}\n \n\\subsection{Prediction of Scattered Moonlight Contribution to LSST backgrounds}\n\nTable~\\ref{tab:zenithresults} presents the data analysis sequence, for sky brightness obtained at zenith with a source elevation angle of 45$^\\circ$. The table shows, for each passband, the measured photocurrent, the dark current value, the number of photoelectrons per second \nproduced in the photodiode, the geometrical factor $GF$ for scaling from solar to lunar irradiance, the \nattenuation due to the lunar phase function at $PF$ full moon ($g=2^\\circ$), the ratio $R$ of LSST pixel to photodiode etendues, the ratio of throughput times etendue for the two systems, and the\nnumber of LSST photoelectrons per pixel per second. We compute\n\n\\begin{equation}\n\\Phi_\\textrm{LSST}=\\left [ \\frac{I_\\textrm{meas}-I_\\textrm{dark}}{1.60 \\times 10^{-19} Coul} \\right ] * GF * PF * \\left [ \\frac{T_{LSST}} {T_{diodes}} \\right ] * \\left [ \\frac{(A \\Omega)_{LSST~pixel}} {(A \\Omega)_{diodes}} \\right ]\n\\end{equation}\nto obtain the expected number of photoelectrons per pixel per second we expect on the LSST focal plane, for full moon conditions, with the telescope pointed to the zenith, and a \nlunar zenith angle of 45$^\\circ$. Note that we do not include any factor for atmospheric attenuation since we wish to use the photon arrival rate on the photodiode to predict the lunar \nbackground flux on the LSST focal plane. \n\n\\begin{table}[htdp]\n\\begin{center}\n\\begin{tabular}{|l|l|l|l|l|l|l|l|l|}\n\\hline\nBand & I$_\\textrm{meas}$ & $\\Phi_\\textrm{diode}$ & Geometry & Phase & $\\frac{T(ITL,e2v)\\,A\\,\\Omega(LSST)}{TA\\,\\Omega(Photodiode)}$ & $\\Phi_\\textrm{ITL}$ & $\\Phi_\\textrm{e2v}$ & ETC \\\\\n~ & $\\mu$A & e\/s & Factor & Factor & ~ & e\/pix\/s & e\/pix\/s &e\/pix\/s \\\\\n\\hline\nu & 1.0 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.091 & (0.61,0.45)*$1.23\\times 10^{-5}$ & 86 & 64 & 106 \\\\\ng & 3.5 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.114 & (0.62,0.66)*$1.23\\times 10^{-5}$ & 307 & 327 & 451 \\\\\nr & 1.8 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.14 & (0.65,0.67)*$1.23\\times 10^{-5}$ & 168 & 173 & 186 \\\\\ni & 1.5 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.17 & (0.50,0.50)*$1.23\\times 10^{-5}$ & 106 & 106 & 116 \\\\\nz & 2.2 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.13 & (0.67,0.66)*$1.23\\times 10^{-5}$ & 210 & 207 & - \\\\ \nzs & 0.9 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.13 & (0.67,0.66)*$1.23\\times 10^{-5}$ & 84 & 83 & 89 \\\\\ny & 1.1 & $6.25\\times 10^{12}$ & $2.04\\times 10^{-5}$ & 0.20 & (0.40,0.40)*$1.23\\times 10^{-5}$ & 60 & 60 & 23 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Zenith daytime sky brightness values, converted to expected LSST \nlunar sky backgrounds at zenith at full moon. These are all adjusted to a common angle between the point on sky and sun, altitude of the point on the sky, and altitude of sun of 45$^\\circ$. The zs band values, which are given to approximate the LSST z filter, are computed using Astrodon z minus y. There is qualitative agreement between the values calculated from the daytime measurement and the LSST exposure time calculator. The exposure time calculator uses the full sky spectrum, not just lunar part, and thus we expect the photodiode measurements to generally underestimate the exposure time calculator numbers.}\n\\label{tab:zenithresults}\n\\end{table}%\n \n\n\n \n\n\\section{Conclusion}\n\\label{sec:Conclusion}\n\nMeasurements of sky brightness are important for efficient telescope scheduling and predictions for LSST. While daytime measurements are approximations to Lunar measurements, it provides high signal to noise ratio measurements in the LSST bands. We measured the fall-off in sky brightness with angle from the sun and zenith angle.\n\nThere are a number of important conclusions to draw from the measurements. The first is that there are substantial gradients in scattered sky brightness, as much as 2 magnitudes. The second is that the scattered sky brightness increases closer to the horizon, perhaps due to \nmore column density winning over extinction. The third is that when observing in bright time, there is significant benefit to point away from the moon.\n \n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=5.5in]{plots\/delta_mag.pdf}\n\\caption{u, g, and r-band flux as a function of time for a single point on the sky using data from Table~\\ref{tab:zenithresults}. Code to produce this plot is available at https:\/\/github.com\/mcoughlin\/skybrightness for public download. This shows in general the differences in flux for a single observation throughout the night.}\n\\label{fig:deltam}\n\\end{center}\n\\end{figure} \n \nAs motivation for future work, figure~\\ref{fig:deltam} shows u, g, and r-band flux as a function of time for a single point on the sky using the measurements described in this paper.\nWe can use these measurements to prioritize observations throughout a given night.\nIn the future, we intend to improve on these measurements by designing the apparatus to take simultaneous measurements. \nWith such an apparatus, we will be able to, for example, measure the color of clouds. \nWe will also be able to take significantly more observations, allowing for refinement of this model, continuing to make it more useful for observers and those exploring scheduling strategies.\n\n\\section{Acknowledgments}\nMC was supported by the National Science Foundation Graduate Research Fellowship\nProgram, under NSF grant number DGE 1144152. CWS is grateful to the DOE Office \nof Science for their support under award DE-SC0007881. Thanks also to Prof. Gary Swensen of Univ. of Illinois for hospitality in the ALO building at Pachon.\n\n\\bibliographystyle{plainnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{sec:intro}\nLet $(X_1, X_2,\\dots, X_n)$ be $n$ independent random points uniformly distributed on the square $\\cc01^2$. \nThe semi-discrete random matching problem concerns the study of the properties of the optimal \ncoupling (with respect to a certain cost) of these $n$ points with the Lebesgue measure \n$\\restricts{\\Leb^2}{\\cc01^2}$. \n\nMore precisely, denoting $\\mu^n \\defeq \\frac1n \\sum_{i=1}^n\\delta_{X_i}$ the empirical measure and \n$\\m\\defeq\\restricts{\\Leb^2}{\\cc01^2}$, we want to investigate the optimal transport from \n$\\m$ to $\\mu^n$.\n\nThe ultimate goal is understanding both the distribution of the random variable associated to the optimal \ntransport cost and the properties of the (random) optimal map. In the present paper we will show that\nthe optimal transport map can be well-approximated by the identity plus the gradient of the solution \nof a Poisson problem. In the large literature devoted to the matching problem,\nwe believe that (except for the 1-dimensional case) this is one of the few results describing\nthe behavior of the optimal map, and not only of the transport cost, see also \n\\cite{Goldman2018} in connection with the behavior of the optimal transport map in the Lebesgue-to-Poisson problem \non large scales. \n\nBefore going on, let us briefly recall the definitions of optimal transport and Wasserstein distance. \nWe suggest the monographs \\cite{Villani08,Santambrogio15} for an introduction to the topic.\n\\begin{definition}[Wasserstein distance]\n Let $(X, d)$ be a compact metric space and let $\\mu,\\nu\\in\\prob(X)$ be probability measures.\n Given $p\\in\\co{1}{\\infty}$, we define the $p$-Wasserstein distance between $\\mu$ and $\\nu$ as\n \\begin{equation}\\label{eq:defWp}\n W_p^p(\\mu, \\nu) \\defeq \\inf_{\\gamma\\in\\Gamma(\\mu, \\nu)} \\int_{X\\times X} d^p(x, y) \\de\\gamma(x, y) \\comma\n \\end{equation}\n where $\\Gamma(\\mu, \\nu)$ is the set of all $\\gamma\\in\\prob(X\\times X)$ such that the projections $\\pi_i$, $i=1,2$, on the \n two factors are $\\mu$ and $\\nu$, that is $(\\pi_1)_\\#\\gamma = \\mu$ and $(\\pi_2)_\\#\\gamma=\\nu$.\n\\end{definition}\n\\begin{remark}\n The infimum in the previous definition is always attained (\\cite[Theorem 1.4]{Santambrogio15}).\n \n Moreover, if $(X, d)$ is a Riemannian manifold and $\\mu\\ll\\m$, where $\\m$ is the volume measure of \n the manifold, the Wasserstein distance is realized by a map (\\cite{McCann01}). \n Namely, the infimum \\cref{eq:defWp} is attained and the unique minimizer is induced by a Borel map $T:M\\to M$, \n so that $T_\\#\\mu = \\nu$ and \n \\begin{equation*}\n W_p^p(\\mu, \\nu) = \\int_M d^p(x, T(x)) \\de\\mu(x) \\fullstop\n \\end{equation*}\n\\end{remark}\n\nEven though the square is a fundamental example, the random matching problem makes perfect sense even in \nmore general spaces (changing the reference measure $\\m$ accordingly).\nHistorically, in the combinatorial literature\\footnote{In the combinatorial literature the problem considered\nwas the bipartite matching problem, in which two independent random point clouds have to be matched. The \nsemi-discrete matching and the bipartite matching are tightly linked and, given that we will consider only\nthe former, we are going to talk about the combinatorial literature as if it were considering the semi-discrete\nmatching.}, the most common ambient space was $\\cc01^d$ for some $d\\ge 1$ and the aspect of the problem that \nattracted more attention was estimating the expected value of the $W_1$ cost. \nIn the papers \\cite{Ajtai1984,Talagrand1992,Dobric1995,Ledoux2017} (and possibly in other ones) \nthe problem was solved in all dimensions and for all $1\\le p < \\infty$, obtaining the growth \nestimates\\footnote{The notation $f(n)\\approx g(n)$ means that there exists a positive constant $C>0$ such \nthat $C^{-1}g(n) \\le f(n)\\le C g(n)$ for every $n$.}\n\\begin{equation*}\n \\E{W_p^p(\\m, \\mu^n)} \\approx \n \\begin{cases}\n n^{-\\frac p2} &\\text{ if $d=1$,}\\\\\n \\left(\\frac{\\log(n)}n\\right)^{\\frac p2} &\\text{ if $d=2$,}\\\\\n n^{-\\frac pd} &\\text{ if $d\\ge 3$.}\n \\end{cases} \n\\end{equation*}\nAs might be clear from the presence of a logarithm, the matching problem exhibits some unexpected behavior\nin dimension $2$. \n\nSee the introductions of \\cite{ambrosio-glaudo2018,Ledoux2017} or \\cite[Chapter 4, 14, 15]{Talagrand14}\nfor a more in-depth description of the history of the problem.\n\nNowadays the topic is active again (\\cite{holden2018,Talagrand2018,Goldman2018,Ledoux2017,ambrosio-glaudo2018,Ledoux18,Ledoux19}),\nalso as a consequence of \\cite{Ambrosio-Stra-Trevisan2018}, in which the authors, \nfollowing an ansatz suggested in \\cite{CaraccioloEtAl2014}, manage to obtain the leading term of the \nasymptotic expansion of the expected matching cost in dimension $2$ with respect to the quadratic \ndistance\\footnote{The notation $f(n)\\sim g(n)$ means that $\\frac{f(n)}{g(n)}\\to 1$ when $n\\to\\infty$.}:\n\\begin{equation}\\label{eq:limit_value}\n \\E{W_2^2(\\m, \\mu^n)} \\sim \\frac{\\log(n)}{4\\pi n} \\fullstop\n\\end{equation}\nThe approach is far from being combinatorial, indeed it relies on a first-order approximation of the \nWasserstein distance with the $H^{-1}$ negative Sobolev norm. Their proof works on any closed compact \n$2$-dimensional manifold.\n\nGiven that we will build upon it, let us give a brief sketch of the approach. \nWhat we are going to describe is simpler than the original approach of \\cite{Ambrosio-Stra-Trevisan2018} and\ncan be found in full details in \\cite{ambrosio-glaudo2018}. For simplicity we will assume to work on the \nsquare.\n\nLet $T^n$ be the optimal map from $\\m$ to $\\mu^n$, whose existence is ensured by Brenier's Theorem \n(see \\cite{Brenier91}). Still by Brenier's Theorem, we know that $T^n = \\id + \\nabla \\tilde f^n$, where\n$\\id$ is the identity map and $\\tilde f^n:\\cc01^2\\to M$ is a convex function. \nWith high probability $\\mu^n$ is well-spread on the square, thus we expect $\\nabla \\tilde f^n$ to be \n\\emph{very small}.\nWe know $(T^n)_\\#\\m=\\mu^n$ and we would like to apply the change of variable formula to deduce something \non the Hessian of $\\tilde f^n$. \nThe issue is that the singularity of $\\mu^n$ prevents a direct application of the change of variable \nformula. \nAnyhow, proceeding formally we obtain $\\det(\\id+\\nabla^2 \\tilde f^n)^{-1}=\\mu^n$. \nGoing on with the formal computation, if we consider only the first order term of \nthe left hand side, the previous identity simplifies to\n\\begin{equation*}\n -\\lapl \\tilde f^n \\approx \\mu^n-1 \\fullstop\n\\end{equation*}\nSomewhat unexpectedly, this last equation makes perfect sense. \nTherefore we might claim that if we define $f^n:\\cc01^2\\to\\R$ as the solution of $-\\lapl f^n=\\mu^n-1$\n(with null Neumann boundary condition), then $T^n$ is well-approximated by $\\id+\\nabla f^n$ and \nfurthermore the transport cost is well-approximated by $\\int \\abs{\\nabla f^n}^2\\de\\m$.\n\nThis conjecture is appealing, but false, if taken literally. Indeed, it is very easy to check that the integral\n$\\int \\abs{\\nabla f^n}^2\\de\\m$ diverges. \n\nThe ingredient that fixes this issue is a regularization argument. \nMore precisely, let $\\mu^{n,t}\\defeq P_t^*\\mu^n$ be the evolution at a certain small time $t>0$ of the \nempirical measure through the heat semigroup (see \\cite[Chapter 6]{Chavel84}).\nIf we repeat the ansatz with $\\mu^n$ replaced by $\\mu^{n,t}$ we obtain a function $f^{n,t}:\\cc01^2\\to\\R$ \nthat solves \n\\begin{equation*}\n -\\lapl f^{n,t} = \\mu^{n,t}-1\n\\end{equation*}\nwith null Neumann boundary conditions. Let us remark that in fact $f^{n,t}=P_t f^n$.\n\nOnce again, we can hope that $\\id+\\nabla f^{n,t}$ approximates very well $T^n$ and furthermore that the transport\ncost from $\\m$ to $\\mu^n$ is well-approximated by $\\int \\abs{\\nabla f^{n,t}}^2\\de\\m$.\n\nThis time the predictions are sound.\nChoosing carefully the time $t=t(n)$, we can show that, with high probability, the map $\\id+\\nabla f^{n,t}$ is\noptimal from $\\m$ to $\\left(\\id+\\nabla f^{n,t}\\right)_\\#\\m$ and the Dirichlet energy of $f^{n,t}$ approximate\nvery well $W_2^2(\\m, \\mu^n)$.\nOnly one part of the conjecture is left unproven by \\cite{ambrosio-glaudo2018}: is it true that \n$\\id+\\nabla f^{n,t}$ approximates, in some adequate sense, the optimal map $T^n$?\nThe goal of the present paper is to answer positively this question.\n\nWe are going to prove the following.\n\\begin{theorem}\\label{thm:main_theorem}\n Let $(M,\\metric)$ be a $2$-dimensional closed compact Riemannian manifold (or the square $\\cc01^2$) whose\n volume measure $\\m$ is a probability. We will denote with $d:M\\times M\\to\\co0\\infty$ the Riemannian \n distance on $M$.\n \n Given $n\\in\\N$, let $X_1, X_2,\\dots, X_n$ be $n$ independent random points $\\m$-uniformly \n distributed on $M$. \n Let us denote $\\mu^n\\defeq \\frac1n \\sum_i \\delta_{X_i}$ the empirical measure associated to the \n random point cloud and let $T^n$ be the optimal transport map from $\\m$ to $\\mu^n$.\n \n For a fixed time $t>0$, let $\\mu^{n,t}\\defeq P_t^*\\mu^n\\in\\prob(M)$ and let $f^{n,t}:M\\to\\R$ be the unique\n null-mean solution\\footnote{If $M=\\cc01^2$ we ask also that $f$ satisfies the null Neumann boundary\n conditions.} of the Poisson problem $-\\lapl f^{n,t}=\\mu^{n,t}-1$.\n \n If we set $t=t(n)=\\frac{\\log(n)^4}n$, on average $T^n$ is very close to $\\exp(\\nabla f^{n,t})$ in the \n $L^2$-norm, that is\n \\begin{equation}\\label{eq:main-quantitative}\n \\frac{\\E{\\int_M d^2(T^n, \\exp(\\nabla f^{n,t}))\\de\\m}}{\\frac{\\log(n)}{n}} \\ll \\sqrt{\\frac{\\log \\left(\\log (n)\\right)}{ \\log(n) }} \\fullstop\n \\end{equation}\n In particular,\n \\begin{equation*}\n \\lim_{n\\to\\infty}\\frac{\\E{\\int_M d^2(T^n, \\exp(\\nabla f^{n,t}))\\de\\m}}\n {\\E{\\int_M d^2(T^n, \\id)\\de\\m}} \n = 0 \\fullstop\n \\end{equation*}\n\\end{theorem}\n\\begin{remark}\n To handle the case of the square $M=\\cc01^2$ some care is required. Indeed the presence of boundary makes\n things more delicate. This is the reason why only the square is considered in the theorem and not any\n $2$-dimensional compact manifold with boundary. \n \n See \\cite[Subsection 2.1 and Remark 3.10]{ambrosio-glaudo2018} for some further details on this matter.\n\\end{remark}\n\n\\begin{remark}\\label{rem:distance-tangent}\nBy McCann's Theorem \\cite{McCann01} we can write $T^n = \\exp(\\nabla f^n)$, hence a natural \nquestion is if \\cref{eq:main-quantitative} holds with $|\\nabla (f^n - f^{n,t})|$ in place of \n$d(T^n, \\exp(\\nabla f^{n,t}))$. Using the fact that the exponential map restricted to a \nsufficiently small neighbourhood of the null vector field is a global diffeomorphism with its \nimage, it would be sufficient to show that, for every \n$\\varepsilon>0$, $\\P{\\|d(T^{n},\\id)\\|_\\infty > \\varepsilon} \\ll \\log(n)\/n$, as $n \\to \\infty$. \nWe will prove this estimate in \\cref{prop:linf_is_small}, that provides the desired approximation at\nthe level of the gradients\n\\begin{equation}\\label{eq:main_gradients}\n \\lim_{n\\to\\infty} \n \\frac{\\E{\\norm{\\nabla f^n-\\nabla f^{n,t}}_{L^2(M)}^2}}\n {\\E{\\norm{\\nabla f^n}_{L^2(M)}^2}} = 0\\fullstop \n\\end{equation}\n\\end{remark}\n\nThe strategy of the proof is to show that the information that we already have on $\\exp(\\nabla f^{n,t})$ \n(namely that it is an optimal map between $\\m$ and some measure $\\hat\\mu^{n,t}$ that is very close to \n$\\mu^{n,t}$) is enough to deduce that it must be near to the optimal map $T^n$. \n\nAs part of the strategy of proof, we obtain, in \\cref{sec:stability}, a new stability result for the optimal transport \nmap on a general compact Riemannian manifold (not only of dimension $2$). \nThis is the natural generalization to Riemannian manifolds of \\cite{gigli2011}.\nThe said stability result follows rather easily from the study of the short-time behavior of the Hopf-Lax \nsemigroup we perform in \\cref{sec:hopflax}. \nThe Hopf-Lax semigroup comes up in our investigation as, when $t=1$, it becomes the operator of\n$c$-conjugation and thus produce the second Kantorovich potential once the first is known (see \n\\cite[Section 1.2]{Santambrogio15} for the theory of Kantorovich potentials and $c$-conjugation).\n\nThe main theorem is established in \\cref{sec:random_matching}.\n\\vspace{2mm}\n\n\\noindent {\\textit{Acknowledgments. } F. Glaudo has received funding from the European Research Council under the \nGrant Agreement No 721675. L. Ambrosio acknowledges the support of the MIUR PRIN 2015 project.\n\n\\subsection{Notation for constants}\nWe will use the letters $c$ and $C$ to denote constants, whose dependencies are denoted by $c=c(A, B,\\dots)$. \nThe value of such constants can change from one time to the other.\n\nMoreover we will frequently use the notation $A\\lesssim B$ to hide a constant that depends only on the\nambient manifold $M$. This expression means that there exists a constant $C=C(M)$ such that $A\\le C\\cdot B$.\n\n\n\n\n\\section{Short-time behavior of the Hopf-Lax semigroup with datum in \\texorpdfstring{$C^{1,1}$}{C(1,1)}}\n\\label{sec:hopflax}\nLet us begin recalling the definition of the Hopf-Lax semigroup (also called Hamilton-Jacobi semigroup).\n\\begin{definition}[Hopf-Lax semigroup]\n Let $(X, d)$ be a compact length space\\footnote{A metric space is a length space if the distance between\n any two points is the infimum of the length of the curves between the two points. Let us remark that for\n the definition we need neither the compactness nor the length property of $X$, but without these \n assumptions many of the properties of the Hopf-Lax semigroup fail (first of all the fact that it is \n a semigroup).}.\n For any function $f\\in C(X)$ and any $t\\geq 0$, let $Q_t f:X\\to\\R$ be defined by\n \\begin{equation*}\n Q_t f(y) = \\min_{x\\in X} \\frac1{2t} d^2(x,y) + f(x) \\quad (t>0),\\qquad Q_0f=f\\fullstop\n \\end{equation*}\n\\end{definition}\nWithout additional assumptions on $X$ or $f$ it is already possible to deduce many properties of the Hopf-Lax\nsemigroup. Let us give a very short summary of the most important ones.\n\\begin{itemize}\n \\item When $t\\to 0$ the functions $Q_t f$ converge uniformly to $f$.\n \\item The Hopf-Lax semigroup is indeed a semigroup, that is $Q_{s+t}f = Q_sQ_t f$ for any $s,\\, t \\geq 0$.\n \\item In a \\emph{suitable weak sense},the Hamilton-Jacobi equation\n \\begin{equation*}\n \\frac{\\de}{\\de t} Q_t f + \\frac12\\abs{\\nabla Q_t f}^2 = 0 \n \\end{equation*}\n holds.\n Let us emphasize that the mentioned equation does not make sense if we don't give an appropriate definition\n of norm of the gradient as we are working in a metric setting.\n\\end{itemize}\nSee \\cite{Lott-Villani07}, in particular Theorem~2.5, for a detailed proof of the mentioned\nproperties. \n\nThere is a vast literature investigating the regularity properties of the Hopf-Lax semigroup and its \nconnection with the Hamilton-Jacobi equation, in particular that it is the unique solution in the viscosity\nsense (see for instance \\cite{lions1982,benton1977,bardi2008}). \nNonetheless we could not find a complete reference for the short-time behavior of the Hopf-Lax semigroup on \na Riemannian manifold (as the majority of the results are stated on the Euclidean space) with a relatively \nregular initial datum (namely $C^{1,1}$). This is exactly the topic of this section. \n\nWhat we are going to show, apart from \\cref{it:hopflax_convexity}, is not new. For instance,\nin \\cite[Section 5]{fathi2003}, the author proves the validity of the method of characteristics in \na way very similar to ours. In that paper more general Lagrangians are considered and as a consequence \nthe proofs are more involved and require much more geometric tools and notation.\n\nFor us, the ambient space is a compact Riemannian manifold $(M, \\metric)$ and the function \n$f\\in C^{1,1}(M)$ is differentiable with Lipschitz continuous gradient. Moreover, either $M$ is closed\nor it is the square $\\cc01^2$. For a general\nmanifold with boundary the results are false, the square is special because its boundary is piecewise geodesic.\nHandling all manifolds with totally geodesic boundary would be possible, but would require some additional \ncare. In order to simplify the exposition we decided to state the results only for the square.\nThroughout this section we will often use implicitly that a Lipschitz continuous function is differentiable\nalmost everywhere (see \\cite[Theorem 3.2]{Evans-Gariepy}).\n\nWe will show that, up to a small time that depends on the $C^{1,1}$-norm of $f$, the Hopf-Lax semigroup \nis \\emph{as good as one might hope}.\nWe will describe explicitly the minimizer $x=x_t(y)$ of the variational problem that \ndefines $Q_t f(y)$ deducing some \\emph{explicit} formulas for $Q_tf$ and its gradient and we will\nshow that $Q_t f$ solves the Hamilton-Jacobi equation in the classical sense.\nFinally we will be able to control the $C^{1,1}$-norm of $Q_t f$ and the $C^{0,1}$-norm of $Q_t f-f$. \n\nHow can we achieve these results for short times when $f\\in C^{1,1}$? \nThe main ingredient is the possibility to identify the minimizer $x=x_t(y)$ in the definition of $Q_t f(y)$.\nGiven $x\\in M$, let $\\gamma:\\co{0}{\\infty}\\to M$ be the unique geodesic with $\\gamma(0) = 0$ and \n$\\gamma'(0)=\\nabla f(x)$. If $y=\\gamma(t)$, then the minimizer in the definition of $Q_t f(y)$ is exactly $x$.\nThis approach is exactly the method of characteristics when applied on a Riemannian manifold (\\emph{straight \nlines on a manifold are geodesics}).\n\nLet us begin with a technical lemma. \n\\begin{lemma}\\label{lem:exp_is_diffeo}\n Let $(M, \\metric)$ be a closed compact Riemannian manifold (or the square $\\cc01^2$).\n\n There exists a constant $c=c(M)$ such that the following statement holds.\n Let $X\\in\\chi(M)$ be a Lipschitz continuous vector \n field\\footnote{If $M=\\cc01^2$ we ask also that $X$ is tangent to the boundary.}\n with $\\norm{X}_\\infty\\le c$ and $\\norm{\\nabla X}_\\infty\\le c$ and, for any $0\\le t\\le 1$, let \n $\\varphi_t:M\\to M$ be the map defined as $\\varphi_t(x) \\defeq \\exp(tX(x))$, where $\\exp:TM\\to M$ denotes the \n exponential map.\n For any $0\\le t\\le 1$, the map $\\varphi_t$ is a homeomorphism \n such that $\\Lip(\\varphi_t)$, $\\Lip(\\varphi_t^{-1}) \\le 2$ and the vector field $X_t\\in\\chi(M)$ defined\n as\n \\begin{equation*}\n X_t \\defeq \\frac{\\partial \\varphi_s}{\\partial s}\\Big|_{s=t}\n \\end{equation*}\n is Lipschitz continuous with $\\norm{\\nabla X_t}_{\\infty}\\lesssim\\norm{\\nabla X}_{\\infty}$.\n\\end{lemma}\n\\begin{proof}\n We will give only a sketch of the proof of the first part of the statement as the argument is well-known.\n\n Let us begin by proving the result when $M$ is closed (in particular we exclude only $M=\\cc01^2$).\n \n We can deduce the first part of the statement from the fact that $\\varphi=\\varphi_1$ is injective and \n locally (i.e. on sufficiently small balls) it is a bi-Lipschitz transformation with its image.\n \n Working in a suitably chosen finite atlas (whose existence follows from the compactness of $M$), the fact \n that $\\varphi$ is a bi-Lipschitz diffeomorphism is a consequence of the following very \n well-known lemma about perturbations of the identity (see \\cite[Theorem 9.24]{rudin1976} or \n \\cite[Theorem 5.3]{fathi2003}).\n If $T:\\Omega\\subseteq \\R^d\\to\\R^d$ is such that $T-\\id$ is $L$-Lipschitz with $L<1$, then $T$ is locally\n invertible and $\\Lip(T) \\le 1+L,\\ \\Lip(T^{-1}) \\le (1-L)^{-1}$.\n \n The global injectivity follows directly from the fact that it is locally bi-Lipschitz. Indeed if \n $\\varphi(x_1)=\\varphi(x_2)$ then $d(x_1, x_2) \\le 2\\norm{X}_\\infty$ and therefore we can exploit the local\n injectivity of $\\varphi$.\n \n When $M=\\cc01^2$ we need only a simple additional remark. Given that $X$ is tangent to the boundary, the map \n $\\varphi$ is a homeomorphism of the boundary. As a consequence of this fact, it is not\n difficult to prove (by injectivity) that the image of the interior of the square is mapped \n by $\\varphi$ in itself. \n From here on we can simply mimic the proof described above for closed manifolds and achieve the\n result also for the case of the square.\n \n We move our attention to the second part of the statement.\n By a simple homogeneity argument, it is sufficient to prove that\n $\\norm{\\nabla X_t}_{\\infty}\\lesssim 1$.\n \n Once again we work in chart. Let $\\Omega\\subseteq\\R^d$ be the domain of the chart. \n As usual, $X_t$ can be understood as a vector field on $\\Omega$ and $\\varphi_t$ as a map from \n $\\Omega'\\Subset\\Omega$ into $\\Omega$.\n Choosing the chart appropriately, we can assume that the Euclidean distance is bi-Lipschitz equivalent\n to the distance induced by the metric $\\metric$.\n \n The Lipschitz continuity of $X_t$ with respect to the metric $\\metric$ is equivalent to proving that, for \n any $x,y\\in\\Omega$, it holds\n \\begin{equation*}\n \\abs{X_t(x)-X_t(y)} \\lesssim \\abs{x-y} \\comma\n \\end{equation*}\n where all the absolute values are with respect to the standard Euclidean norm. \n Since $\\varphi_t$ is surjective, it is sufficient to prove that, for any $x,y\\in\\Omega'$, it holds\n \\begin{equation}\\label{eq:exp_diffeotmp1}\n \\abs{X_t(\\varphi_t(x))-X_t(\\varphi_t(y))} \\lesssim \\abs{\\varphi_t(x)-\\varphi_t(y)} \\fullstop\n \\end{equation}\n \n Given that $\\varphi_t^{-1}$ is Lipschitz, we already know\n \\begin{equation}\\label{eq:exp_diffeotmp2}\n \\abs{x-y} \\lesssim \\abs{\\varphi_t(x)-\\varphi_t(y)} \n \\quad\\text{and}\\quad\n \\abs{X(x)-X(y)} \\lesssim \\abs{\\varphi_t(x)-\\varphi_t(y)} \\fullstop\n \\end{equation}\n Let $\\gamma_x:\\cc01\\to\\Omega$ be the unique geodesic, with respect to $\\metric$, such that $\\gamma_x(0)=x$\n and $\\gamma_x'(0)=X(x)$. Let $\\gamma_y:\\cc01\\to\\Omega$ be defined analogously. By definition, it holds\n \\begin{equation}\\label{eq:exp_diffeotmp3}\n X_t(\\varphi_t(x)) = \\gamma_x'(t)\n \\quad\\text{and}\\quad\n X_t(\\varphi_t(y)) = \\gamma_y'(t) \\fullstop\n \\end{equation}\n Taking into account \\cref{eq:exp_diffeotmp1}, \\cref{eq:exp_diffeotmp2} and \\cref{eq:exp_diffeotmp3}, the \n Lipschitz continuity of $X_t$ would follow from the inequality\n \\begin{equation}\\label{eq:exp_diffeotmp4}\n \\abs{\\gamma_x'(t)-\\gamma_y'(t)}\n \\lesssim \\abs{\\gamma_x(0)-\\gamma_y(0)} + \\abs{\\gamma_x'(0)-\\gamma_y'(0)}\n \\fullstop\n \\end{equation}\n The curves $\\gamma_x,\\gamma_y$ are geodesics, hence the vectors $(\\gamma_x, \\gamma_x')$ and \n $(\\gamma_y,\\gamma_y')$ solve the same autonomous ordinary differential equation with different initial data.\n Hence \\cref{eq:exp_diffeotmp4} follows from the well-known Lipschitz dependence of the solution \n from the initial data (see \\cite[Theorem 2.6]{teschl2012}) and therefore the proof is concluded.\n\\end{proof}\n\nWe can now state and prove the main theorem of this section. \nThe technically demanding part of these notes is entirely enclosed in the following theorem.\n\\begin{theorem}\\label{thm:hopflax_properties}\n Let $(M, \\metric)$ be a closed compact Riemannian manifold (or the square $\\cc01^2$).\n \n Let $f\\in C^{1,1}(M)$ be a scalar function\\footnote{If $M=\\cc01^2$ we ask also that $f$ satisfies the \n null Neumann boundary conditions.} and, for any positive time $t>0$, let us define the map \n $\\varphi_t:M\\to M$ as $\\varphi_t(x) \\defeq \\exp(t\\nabla f(x))$.\n \n There exists a constant $c=c(M)$ such that the following properties hold for any time \n $0\\le t\\le c \\left(\\norm{\\nabla f}_\\infty + \\norm{\\nabla^2 f}_\\infty\\right)^{-1}$:\n \\begin{enumerate}[ref={(\\arabic*)}]\n \\item \\label{it:varphi_t_diffeo}\n The map $\\varphi_t$ is a bi-Lipschitz homeomorphism such that \n $\\Lip(\\varphi_t),\\, \\Lip(\\varphi_t^{-1}) \\leq 2$.\n \\item \\label{it:hopflax_explicit} For any $y\\in M$, it holds\n \\begin{equation*}\n Q_t f(y) = \\frac1{2t} d^2(\\varphi_t^{-1}(y), y) + f(\\varphi_t^{-1}(y)) \\fullstop\n \\end{equation*}\n \\item \\label{it:hopflax_convexity}\n For any $y,\\,y'\\in M$, one has the (strict-convexity-like) estimate\n \\begin{equation*}\n \\frac{d^2(y, y')}t \\lesssim Q_tf(y)-Q_tf(y')\n +\\frac1{2t}\\left[d^2(\\varphi_t^{-1}(y), y') - d^2(\\varphi_t^{-1}(y), y)\\right]\\fullstop\n \\end{equation*}\n \\item \\label{it:hopflax_regularity}\n The function $Q_tf$ is Lipschitz continuous in time and $C^{1,1}(M)$ in space. In particular we have\n $\\norm{\\partial_t Q_t f}_{\\infty}\\le \\norm{\\nabla f}_{\\infty}$ and\n $\\norm{\\nabla^2Q_tf}_{\\infty} \\lesssim \\norm{\\nabla^2 f}_{\\infty}$.\n \\item \\label{it:hamilton_jacobi} \n The function $Q_tf$ is a classical solution of the Hamilton-Jacobi equation\n \\begin{equation*}\n \\frac{\\de}{\\de t} Q_t f + \\frac12 \\abs{\\nabla Q_t f}^2 = 0 \\fullstop\n \\end{equation*}\n \\item \\label{it:gradient_conservation} \n For any $x\\in M$, if $\\gamma:\\cc01\\to M$ is the geodesic such that $\\gamma(0)=x$ and \n $\\gamma'(0) = \\nabla f(x)$, then it holds\n \\begin{equation*}\n Q_tf(\\gamma(t)) = f(x) + \\frac t2 \\abs{\\nabla f}^2(x) \\quad\\text{ and }\\quad\n \\nabla Q_t f(\\gamma(t)) = \\gamma'(t) \\fullstop\n \\end{equation*}\n \\item \\label{it:hopflax_lip}\n One has\n \\begin{equation*}\n \\Lip(Q_tf-f) \\le t\\norm{\\nabla f}_\\infty\\cdot\\norm{\\nabla^2 f}_\\infty \\fullstop\n \\end{equation*}\n \\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n Thanks to the following homogeneity, for any $t>0$ and $\\lambda>0$, of the Hopf-Lax semigroup\n \\begin{equation*}\n Q_t(\\lambda f)(y) = \\lambda Q_{\\lambda t} f(y) \\comma\n \\end{equation*}\n we can assume without loss of generality that $\\norm{\\nabla f}_\\infty + \\norm{\\nabla^2 f}_\\infty \\le c$ \n and prove that the statements hold up to time $1$.\n Thus, we will implicitly assume that the time variable satisfies $0\\le t\\le 1$.\n We will choose the value of the constant $c$ during the proof, it should be clear that all constraints \n we impose depend only on the manifold $M$ and not on the function $f$.\n\n The statement of \\cref{it:varphi_t_diffeo} follows from \\cref{lem:exp_is_diffeo}.\n \n To prove \\cref{it:hopflax_explicit} we need some preliminary observations. \n If $c=c(M)$ is sufficiently small (so that the constraint on $f$ is sufficiently strong), thanks to \n the compactness of $M$ we can find a radius $r=r(M)>0$ such that:\n \\begin{enumerate}[label=(\\alph*)]\n \\item If $p,\\,q\\in M$ satisfy $d(p,q)\\le r$ then\n \\begin{equation*}\n \\nabla^2 d^2(\\emptyparam,p)(q) \\ge \\frac12 \\metric \\fullstop\n \\end{equation*}\n \\item For any $y\\in M$, to compute $Q_tf(y)$ it is sufficient to minimize on $B(y,r)$:\n \\begin{equation*}\n Q_tf(y) = \\inf_{x\\in B(y, r)} \\frac1{2t}d^2(x, y)+f(x) \\fullstop\n \\end{equation*}\n \\item For any $y\\in M$ it holds the inequality $d(y, \\varphi_t^{-1}(y)) \\le r$. \n In particular we can assume that $\\varphi_t^{-1}(y)$ is not in the cut-locus of $y$.\n \\item For any $y\\in M$ it holds the identity\n \\begin{equation*}\n \\nabla \\left(\\frac1{2t}d^2(\\emptyparam, y) + f(\\emptyparam)\\right)(\\varphi_t^{-1}(y)) = 0 \\fullstop\n \\end{equation*}\n This identity can be shown computing the gradient of the distance from $y$ squared, since we know\n that $y=\\exp(t\\nabla f(x))$ where $x=\\varphi_t^{-1}(y)$. Indeed, given that $x$ does not belong to the \n cut-locus of $y$, we know\n \\begin{equation*}\n \\nabla \\left(\\frac12 d^2(\\emptyparam, y)\\right)(x) = -t\\nabla f(x)\n \\end{equation*}\n and the desired identity follows.\n \\end{enumerate}\n With these observations at our disposal, the proof of \\cref{it:hopflax_explicit} is straight-forward.\n Given a time $0\\le t\\le 1$ and a point $y\\in M$, let us consider the function $w_{t,y}:M\\to\\R$ defined\n as\n \\begin{equation*}\n w_{t,y}(x) = \\frac1{2t}d^2(x,y) + f(x) \\fullstop\n \\end{equation*}\n We know that $Q_tf(y) = \\min_{x\\in B(y, r)} w_{t,y}(x)$. Moreover $\\nabla w_{t,y}(\\varphi_t^{-1}(y))=0$ and,\n if the constraint on $\\norm{\\nabla^2 f}_\\infty$ is sufficiently small, we also know \n $\\nabla^2 w_{t,y}\\ge \\frac1{3t}\\metric$ in $B(y, r)$. \n Hence, by convexity, we deduce that $\\varphi_t^{-1}(y)$ is the global minimum point of \n $w_{t,y}$ and \\cref{it:hopflax_explicit} follows.\n \n Let us now move to the proof of \\cref{it:hopflax_convexity}. \n Let $x,\\, x'\\in M$ be such that $\\varphi_t(x) = y$ and $\\varphi_t(x')=y'$. \n Applying \\cref{it:hopflax_explicit} and recalling that $\\varphi_t$ is a bi-Lipschitz \n diffeomorphism, we can see that the inequality we want to prove is equivalent to\n \\begin{equation*}\n \\frac1t d^2(x, x') \\lesssim f(x)-f(x') + \\frac 1{2t}\\left(d^2(x, y')-d^2(x', y') \\right)\n \\end{equation*}\n and, using the same notation as above, this becomes\n \\begin{equation*}\n \\frac1t d^2(x, x') \\lesssim w_{t, y'}(x) - w_{t, y'}(x') \\fullstop\n \\end{equation*}\n The latter inequality follows from the strict convexity of $w_{t,y'}$ that we have already shown \n while proving \\cref{it:hopflax_explicit}.\n\n Showing from scratch that $Q_tf$ solves the Hamilton-Jacobi equation would not be hard, but for this \n we refer to \\cite[Theorem 2.5, viii]{Lott-Villani07}, where the authors show that $Q_tf$ is \n a \\emph{suitably weak} solution of the Hamilton-Jacobi equation. \n From their statement, we can deduce that if $Q_tf$ is differentiable at $x\\in M$, then\n \\begin{equation}\\label{eq:hamilton_jacobi_ae}\n \\frac{\\de}{\\de t}Q_tf(x) + \\abs{\\nabla Q_tf(x)}^2 = 0 \\fullstop\n \\end{equation}\n Since we will show that $Q_tf$ is $C^{1,1}(M)$, the validity of \n \\cref{it:hopflax_regularity,it:hamilton_jacobi} is a consequence of \\cref{eq:hamilton_jacobi_ae}.\n \n The first part of \\cref{it:gradient_conservation}, namely \n $Q_tf(\\gamma(t)) = f(x) + \\frac t2\\abs{\\nabla f}^2(x)$, is implied by \\cref{it:hopflax_explicit}. \n To obtain the identity involving the gradient, let us differentiate the previous equality with respect to the\n time variable. If $Q_t f$ is differentiable at $\\gamma(t)$, it holds\n \\begin{equation}\\label{eq:almost_gradient_conservation}\n \\frac{\\de}{\\de t} (Q_t f)(\\gamma(t)) + \\scalprod{\\nabla Q_t f(\\gamma(t))}{\\gamma'(t)} = \n \\frac{\\de}{\\de t} (Q_tf(\\gamma(t))) = \\frac12\\abs{\\nabla f}^2(x)\\fullstop\n \\end{equation}\n Applying \\cref{it:hamilton_jacobi} and the fact that $\\abs{\\gamma'(t)} = \\abs{\\nabla f}(x)$, from\n \\cref{eq:almost_gradient_conservation} we can deduce\n \\begin{equation}\\label{eq:hopflax_tmp}\n -\\frac12 \\abs{\\nabla Q_tf}^2(\\gamma(t)) + \\scalprod{\\nabla Q_t f(\\gamma(t))}{\\gamma'(t)} \n = \\frac12\\abs{\\gamma'}^2(x) \\iff \\abs{\\nabla Q_tf(\\gamma(t)) - \\gamma'(t)}^2 = 0 \\fullstop\n \\end{equation}\n This does not imply directly \\cref{it:gradient_conservation} since we have shown the identity only if\n $Q_t f$ is differentiable at $\\gamma(t)$.\n As a byproduct of \\cref{it:hopflax_explicit}, we know that $Q_t f$ is Lipschitz continuous and \n therefore, from \\cref{eq:hopflax_tmp}, we can deduce that, fixed $t$, for almost every $x\\in M$ it holds\n \\begin{equation*}\n \\nabla Q_t f(\\varphi_t(x)) = \\frac{\\partial \\varphi_s(x)}{\\partial s}\\Big|_{s=t} \\fullstop\n \\end{equation*}\n Since the right-hand side is Lipschitz continuous (see \\cref{lem:exp_is_diffeo}) it follows that \n $Q_tf\\in C^{1,1}(M)$ and, as anticipated, this concludes the proofs of \n \\cref{it:hopflax_regularity},\\cref{it:hamilton_jacobi} and \\cref{it:gradient_conservation}.\n \n Finally let us tackle \\cref{it:hopflax_lip}.\n Given $y\\in M$, let $x=\\varphi_t^{-1}(y)$. Thanks to \\cref{it:gradient_conservation}, if we consider \n the geodesic $\\gamma:\\cc01\\to M$ such that $\\gamma(0)=x$ and $\\gamma'(0)=\\nabla f(x)$, we know that \n $\\gamma(t)=y$ and $\\gamma'(t)=\\nabla Q_tf(y)$.\n \n Thus we have\n \\begin{equation*}\n \\abs{\\nabla f(y) -\\nabla Q_tf(y)} \n \\le \\int_0^t \\abs{\\nabla_{\\gamma'}\\left(\\nabla f(\\gamma)-\\gamma'\\right)}\\de s\n \\le t\\abs{\\nabla f(x)}\\cdot\\norm{\\nabla^2 f}_\\infty\n \\end{equation*}\n and this is the desired statement.\n\\end{proof}\n\n\\begin{remark}\n Let us emphasize that the only statement contained in \\cref{thm:hopflax_properties} that we are going to\n use is \\cref{it:hopflax_convexity}. Indeed it will be crucial when studying the stability of optimal\n maps. Furthermore, such a statement should be seen more like as a property of the $c$-conjugate \n (see \\cite[Section 1.2]{Santambrogio15}) than as a property of the Hopf-Lax semigroup.\n \n We have proven all other statements in order to give a complete reference on the short-time behavior of\n the Hopf-Lax semigroup when the initial datum is in $C^{1,1}(M)$.\n\\end{remark}\n\n\n\n\\section{Quantitative Stability of the Optimal Map}\\label{sec:stability}\nIn this section we will always refer to the optimal transport with respect to the quadratic\ncost between two probability measures in $\\prob(M)$ that are absolutely continuous with respect to the volume\nmeasure $\\m$ of a compact Riemannian manifold $(M, \\metric)$.\n\nThe duality theory of optimal transport can be seen as a tool to bound from above and from below \nthe optimal transport cost. Indeed, simply producing a transport map we can bound the cost from above, whereas\nwith a pair of potentials we can bound it from below. Estimating the optimal cost is the best one can \ndesire for a generic convex problem, but for the optimal transport problem we know that the optimal \nmap is unique (see \\cite{McCann01}) and thence we would like to be able to approximate it.\n\nIn details, we want to investigate the following problem.\n\\begin{problem}\n Let $\\nu, \\,\\mu_1,\\, \\mu_2\\in \\prob(M)$ be probability measures with $\\nu\\ll \\m$. Let $S,\\, T$ be \n the optimal transport maps from $\\nu$ to $\\mu_1$ and $\\mu_2$ respectively. \n Estimate the $L^2(\\nu)$-distance $\\norm{d(S, T)}^2_{L^2(\\nu)}$ between the two maps.\n\\end{problem}\nThe approach we are going to adopt builds upon the method, suggested to N.Gigli by the first author, who\nused it in \\cite[Proposition 3.3 and Corollary 3.4]{gigli2011}. \nIn the proof of the mentioned results, the author obtains (even if not stated in this way) exactly\nthe same inequality we are going to obtain. The substantial difference is that those results (and their proofs)\nwork only when the ambient is the Euclidean space. \n\nTransporting the proofs from the flat to the curved setting is not straight-forward. The proof of \nProposition~3.3 of the mentioned paper does not work on a Riemannian manifold, \nbecause curvature comes into play when comparing tangent vectors at different points.\nTo overcome this difficulty we have\ncome up with \\cref{it:hopflax_convexity} of \\cref{thm:hopflax_properties}. On the contrary, the proof of \nCorollary 3.4 is easily adapted on a compact Riemannian manifold.\n\nLet us also mention the recent result \\cite[Theorem 4.1]{berman2018}. In the said theorem the author\nobtain a quantitative stability of the optimal map when, instead of changing the target measure as we are\ndoing, the source measure is changed. The proof is totally different from ours and is mainly based on\ncomplex analytic tools. Also in that paper only the Euclidean setting (and the flat torus) is considered.\n\nWe will attack the stability problem only in the \\emph{perturbative setting}, namely when the optimal\nmap from $\\nu$ to $\\mu_1$ is the identity up to the first order.\nWorking only in the perturbative setting might look like an extremely strong assumption that would yield\nno applications at all. This is not the case, indeed what we call \\emph{perturbative setting} is \nmore or less equivalent to requiring only that the optimal transport map $T$ is local (meaning that \n$T-\\id$ is uniformly small) and well-behaved.\nFor example, and this is the whole point of \\cite{ambrosio-glaudo2018}, the optimal map from the reference \nmeasure to a random point cloud is (with high probability) a perturbation of the identity.\n\nWe don't need any hypothesis on the optimal map between $\\nu$ and $\\mu_2$.\n\n\\begin{theorem}\\label{thm:optimal_map_stability}\n Let $(M,\\metric)$ be a closed compact Riemannian manifold (or the square $\\cc01^2$) and let us \n denote by $\\m$ its volume measure.\n\n Let $\\nu, \\mu_1, \\mu_2\\in\\prob(M)$ be three probability measures with $\\nu\\ll\\m$ and let $S, T:M\\to M$ \n be the optimal transport maps respectively for the pairs of measures $(\\nu, \\mu_1)$ and $(\\nu, \\mu_2)$. \n We assume that $S=\\exp(\\nabla f)$ where $f:M\\to\\R$ is a \n $C^{1,1}$-function\\footnote{If $M=\\cc01^2$ we ask also that $f$ satisfies the null Neumann boundary\n conditions.} such that $\\norm{\\nabla f}_\\infty + \\norm{\\nabla^2 f}_\\infty \\le c$ where $c=c(M)$ is \n the constant considered in the statement of \\cref{thm:hopflax_properties}.\n \n Then it holds\n \\begin{equation*}\n \\int_M d^2(S, T)\\de\\nu \\lesssim W_2^2(\\mu_1, \\mu_2) + W_2(\\mu_1, \\mu_2)W_2(\\nu, \\mu_1) \\fullstop\n \\end{equation*}\n\\end{theorem}\n\\begin{proof}\n Let us consider a generic transport map $S':M\\to M$ from $\\nu$ to $\\mu_1$ and recall that,\n according to \\cite{Glaudo19}, if $c(M)$ is small enough, then the map $S$ is optimal.\n\n Given $x\\in M$, let us apply \\cref{it:hopflax_convexity} of \\cref{thm:hopflax_properties} with \n $y=S(x)$ and $y'=S'(x)$ and $t=1$\n \\begin{equation*}\n d^2(S(x),S'(x)) \n \\lesssim Q_1f(S(x))-Q_1f(S'(x)) + \\frac12\\left(d^2(x, S'(x))-d^2(x, S(x))\\right) \\fullstop\n \\end{equation*}\n Integrating this inequality with respect to $\\nu$ we obtain\n \\begin{equation}\\label{eq:stability_same_measures}\n \\int_M d^2(S, S')\\de\\nu \\lesssim \\norm{d(S', \\id)}_{L^2(\\nu)}^2 - W_2^2(\\nu, \\mu_1)\n \\end{equation}\n as the first two terms cancel thanks to the fact that both $S$ and $S'$ sends $\\nu$ into $\\mu_1$.\n \n We can now prove the main statement under the additional assumption that there exists an optimal map \n $R:M\\to M$ from $\\mu_2$ to $\\mu_1$. Applying \\cref{eq:stability_same_measures} with $S' = R\\circ T$ we get\n \\begin{equation}\\label{eq:temporary_ineq}\n \\int_M d^2(S, R\\circ T)\\de\\nu \\lesssim \\norm{d(R\\circ T, \\id)}_{L^2(\\nu)}^2 - W_2^2(\\nu, \\mu_1) \\fullstop\n \\end{equation}\n Thanks to the triangle inequality, it holds\n \\begin{align*}\n \\norm{d(R\\circ T, \\id)}_{L^2(\\nu)} \n &\\le \\norm{d(R\\circ T, T)}_{L^2(\\nu)} + \\norm{d(T, \\id)}_{L^2(\\nu)}\n = \\norm{d(R, \\id)}_{L^2(\\mu_2)} + W_2(\\nu, \\mu_2) \\\\\n &\\le 2 W_2(\\mu_1, \\mu_2) + W_2(\\nu, \\mu_1) \\fullstop\n \\end{align*}\n Applying this last inequality into \\cref{eq:temporary_ineq} yields\n \\begin{align*}\n \\int_M d^2(S, R\\circ T)\\de\\nu \n &\\lesssim \\left[2 W_2(\\mu_1, \\mu_2) + W_2(\\nu, \\mu_1)\\right]^2 - W_2^2(\\nu, \\mu_1) \\\\\n &\\lesssim W_2^2(\\mu_1, \\mu_2) + W_2(\\mu_1, \\mu_2)W_2(\\nu, \\mu_1)\n \\end{align*}\n and the desired statement follows from the triangle inequality\n \\begin{align*}\n \\int_M d^2(S, T) &\\lesssim \\int_M d^2(S, R\\circ T)\\de\\nu + \\int_M d^2(R\\circ T, T)\\de\\nu \\\\\n &\\lesssim W_2^2(\\mu_1, \\mu_2) + W_2(\\mu_1, \\mu_2)W_2(\\nu, \\mu_1) + \\int_M d^2(R, \\id)\\de\\mu_2 \\\\\n &= 2 W_2^2(\\mu_1, \\mu_2) + W_2(\\mu_1, \\mu_2)W_2(\\nu, \\mu_1) \\fullstop\n \\end{align*}\n\n \n It remains to drop the assumption on the existence of the optimal map $R$. Given that our ambient manifold \n is compact, we can apply the nonquantitative strong stability (see \\cite[Corollary 5.23]{Villani08}). \n Let us take a sequence of absolutely continuous probability measures $\\mu_2^n$ that weakly converges \n to $\\mu_2$. \n Thanks to McCann's Theorem (see \\cite{McCann01}) the optimal map $R^n$ from $\\mu_2^n$ to $\\mu_1$ exists \n and thanks to the strong stability we know that the optimal maps $T^n$ from $\\nu$ to $\\mu_1^n$ converge \n strongly in $L^2(\\nu)$ to $T$. \n Hence it is readily seen that the result for $\\mu_2$ can be obtained by passing to the limit the \n result for $\\mu_2^n$. \n\\end{proof}\n\n\\begin{remark}\n The first part of the proof of \\cref{thm:optimal_map_stability} might seem a bit magical. Let us describe\n what is happening under the hood. \n \n The function $f$ is the Kantorovich potential of the couple $(\\nu, \\mu_1)$ and\n hence, by standard theory in optimal transport, it must be $c$-concave. \n \n Our hypotheses ensure us that it is not only $c$-concave, but even \\emph{strictly} $c$-concave. \n Furthermore, the theory we have developed on the Hopf-Lax semigroup tells us that even the other \n potential $f^c=Q_1 f$ is strictly $c$-concave (this is exactly \\cref{it:hopflax_convexity}).\n \n The result follows integrating the strict $c$-concavity inequality with respect to the measure\n $\\nu$.\n\\end{remark}\n\n\n\\begin{remark}\n The main use of \\cref{thm:optimal_map_stability} is the following one. \n Assume that the optimal map from $\\nu$ to $\\mu_1$ is local and well-behaved (this ensures the validity \n of the hypotheses of the theorem) and furthermore that $\\mu_2$ is much closer to $\\mu_1$ than to $\\nu$. \n In this situation, the theorem tells us\n \\begin{equation*}\n \\int_M d^2(S, T)\\de\\nu \\ll \\int_M d^2(S, \\id)\\de\\nu \\comma\n \\end{equation*}\n and this conveys exactly the information that $S$ approximates very well $T$. Notice also\n that the improvement from $C^{0,1\/2}$ dependence of \\cite{gigli2011} to the kind of Lipschitz dependence \n is due to the fact that we are working in a perturbative regime, close to the reference measure.\n\\end{remark}\n\n\\section{Optimal map in the random matching problem}\\label{sec:random_matching}\nWe want to apply our result on the stability of the optimal map in the perturbative setting to the \nsemi-discrete random matching problem. In this section we will work on a compact closed Riemannian manifold \n$(M, \\metric)$ of dimension $2$ (or the square $\\cc01^2$). \nWe will denote with $\\m$ the volume measure, with the implicit assumption that it is a probability.\n\nIn this setting, the semi-discrete random matching problem can be formulated as follows.\nFor a fixed $n\\in\\N$, consider $n$ independent random points $X_1, X_2, \\dots, X_n$ $\\m$-uniformly distributed\non $M$. Study the optimal transport map $T^n$ (with respect to the quadratic cost) from $\\m$ to the empirical \nmeasure $\\mu^n = \\frac1n\\sum_{i} \\delta_{X_i}$.\n\nSince we want to attack the problem applying \\cref{thm:optimal_map_stability}, first of all we have to choose \n$\\nu,\\, \\mu_1$ and $\\mu_2$.\nThe choices of $\\nu$ and $\\mu_2$ are very natural, indeed we set $\\nu=\\m$ and $\\mu_2=\\mu^n$. This way the map\n$T$ is $T^n$ .\n\nFar less obvious is the choice of $\\mu_1$, $S$ and $f$. As one might expect from the statement \nof \\cref{thm:main_theorem} and from the ansatz described in the introduction, our choice is $f=f^{n,t}$.\nThus $S=\\exp(\\nabla f^{n,t})$ (for some appropriate $t=t(n)$). \nFurthermore, keeping the same notation of \\cite{ambrosio-glaudo2018}, the measure $\\mu_1=S_\\#\\m$ will \nbe denoted by $\\hat\\mu^{n,t}$.\n\nFirst of all it is crucial to understand whether we are in position to apply \\cref{thm:optimal_map_stability}.\nIndeed we need to check if $\\nabla^2 f^{n,t}$ and $\\nabla f^{n,t}$ are sufficiently small. Moreover we have\nto obtain a strong estimate on $W_2^2(\\mu_1, \\mu_2)$. \nBoth this facts are among the main results obtained in \\cite{ambrosio-glaudo2018}. \nHence let us state them in the following proposition.\n\n\\begin{proposition}[Summary of results from \\texorpdfstring{\\cite{ambrosio-glaudo2018}}{[AG18]}]\\label{prop:ag18}\n Let $(M, \\metric)$ be a closed compact $2$-dimensional Riemannian manifold (or the square $\\cc01^2$) whose\n volume measure $\\m$ is a probability.\n Given $n\\in\\N$, let $X_1,\\dots, X_n$ be $n$ independent random points $\\m$-uniformly distributed on $M$\n and denote $\\mu^n=\\frac1n\\sum_i \\delta_{X_i}$ the associated empirical measure.\n \n For a choice of the time $t>0$, let $\\mu^{n,t}=P^*_t(\\mu^n)$ be the evolution through the heat flow of\n the empirical measure and let $f^{n,t}:M\\to\\R$ be the unique null-mean solution\\footnote{If $M=\\cc01^2$\n we ask also that $f$ satisfies the null Neumann boundary conditions.} to the Poisson equation\n $-\\lapl f^{n,t} = \\mu^{n,t}-1$.\n Finally, let us define the probability measure $\\hat\\mu^{n,t}$ as the push-forward of $\\m$ through the map\n $\\exp(\\nabla f^{n,t})$.\n \n For any $\\xi>0$, let $A^{n,t}_\\xi$ be the probabilistic event $\\{\\norm{\\nabla^2 f^{n,t}}_\\infty < \\xi\\}$.\n \n If $t=t(n)=\\frac{\\log^4(n)}{n}$ and $\\xi=\\xi(n) = \\frac1{\\log(n)}$, the following statements\\footnote{In \n \\cite{ambrosio-glaudo2018} the time $t(n)$ is chosen as $t(n)=\\gamma\\frac{\\log^3(n)}{n}$, where \n $\\gamma$ is a constant. \n As we clarify in \\cref{rem:time_is_flexible}, the choice of the exponent of the logarithm in the \n definition of $t(n)$ is not rigid. \n We choose the exponent $4$ instead of $3$ since it lets us get some estimates in a cleaner form and \n makes it possible to avoid inserting a constant in the definition of $t(n)$.}\n hold\n \\begin{itemize}\n \\item We know the asymptotic behavior of the expected matching cost\n \\begin{equation}\\label{eq:matching_cost_limit}\n \\lim_{n\\to\\infty} \\E{W_2^2(\\m, \\mu^n)}\\left(\\frac1{4\\pi}\\frac{\\log(n)}n\\right)^{-1} = 1 \\fullstop\n \\end{equation}\n \\item The probability of the complement of $A^{n,t}_\\xi$ decays faster than any power.\n In formulas, for any $k>0$ there exists a constant $C=C(M, k)$ such that \n \\begin{equation}\\label{eq:exceptional_set_rare}\n \\P{\\left(A^{n,t}_\\xi\\right)^\\complement} \n \\le C(M, k) n^{-k} \\fullstop\n \\end{equation}\n \\item One has the refined contractivity estimate\\footnote{This does not follow from the well-known \n contractivity property for the heat semigroup. Indeed the standard contractivity would yield \n an estimate of order $t=\\gamma\\frac{\\log^4(n)}{n}\\gg \\frac{\\log(n)}{n}$\n and such magnitude is too \n large for our purposes.}\n \\begin{equation}\\label{eq:diffusion_error}\n \\E{W_2^2(\\mu^n, \\mu^{n,t})} \\lesssim \\frac{\\log(\\log(n))}n \n \\left(\\ll \\frac{\\log(n)}{n}\\right)\\fullstop\n \\end{equation}\n \\item We are able to control the perturbation error with\n \\begin{equation}\\label{eq:perturbation_error}\n \\E{W_2^2(\\mu^{n,t}, \\hat\\mu^{n,t})} \\lesssim \\frac{1}{n\\log(n)} \n \\left(\\ll \\frac{\\log(n)}{n}\\right)\\fullstop\n \\end{equation}\n \\item \\label{it:exp_is_optimal}\n When $n$ is sufficiently large, in the event $A^{n,t}_\\xi$ the map $\\exp(\\nabla f^{n,t})$ is \n optimal from $\\m$ to $\\hat\\mu^{n,t}$.\n \\end{itemize}\n\\end{proposition}\n\\begin{proof}\n All of these results are contained in \\cite{ambrosio-glaudo2018} and thus we will only\n give a precise reference for them. All references are to propositions contained\n in \\cite{ambrosio-glaudo2018}.\n \n The validity of \\cref{eq:matching_cost_limit} is contained in Theorem 1.2.\n The fact that the event $A^{n,t}_\\xi$ has overwhelming probability follows from Theorem 3.3.\n The refined contractivity estimate \\cref{eq:diffusion_error} is Theorem 5.2.\n \n The estimate \\cref{eq:perturbation_error} follows from Equation 6.2 and Lemma 3.14. \n More specifically Equation 6.2 tells us that in the event $A^{n,t}_\\xi$ it holds\n \\begin{equation*}\n \\E{W_2^2(\\mu^{n,t}, \\hat\\mu^{n,t})} \\lesssim \\xi^2 \\int_M\\abs{\\nabla f^{n,t}}^2\\de\\m\n \\end{equation*}\n and Lemma 3.14 gives us the expected value of the Dirichlet energy of $f^{n,t}$. The behavior in the \n complementary of $A^{n,t}_\\xi$ can be ignored thanks to \\cref{eq:exceptional_set_rare}.\n \n It remains to show that in the event $A^{n,t}_\\xi$, the map $\\exp(\\nabla f^{n,t})$ is optimal. \n This follows directly from \\cite[Theorem 1.1]{Glaudo19}.\n\\end{proof}\n\n\\begin{remark}\\label{rem:repeat_old_remark}\n Let us repeat the elementary observation made in \\cite[Remark 5.3]{ambrosio-glaudo2018}, as it will \n be useful.\n \n Let $X, Y$ be two random variables such that, in an event $E$, it holds $X\\le Y$. Then\n \\begin{equation*}\n \\E{X} \\le \\E{Y} + (\\norm{X}_\\infty + \\norm{Y}_\\infty)\\P{E^\\complement} \\fullstop\n \\end{equation*}\n In particular, if the infinity norm of $X, Y$ is suitably controlled and the probability of $E^\\complement$\n is exceedingly small, we can assume $\\E{X}\\le \\E{Y}$ up to a small error.\n \n This observation allows us to restrict our study to the \\emph{good} event $A^{n,t}_\\xi$. Indeed all \n quantities involved in our computations have at most polynomial growth, whereas \n $\\P{(A^{n,t}_\\xi)^\\complement}$ decays faster than any power.\n\\end{remark}\n\n\nOnce we have these results in our hands, the proof of the main theorem follows rather easily. Indeed\nwe just have to check that all assumptions of our stability result are satisfied.\n\n\\begin{proof}[Proof of \\cref{thm:main_theorem}]\n Let us assume to be in the event $A^{n,t}_\\xi$ with $\\xi = \\frac1{\\log(n)}$.\n Hence, thanks to \\cref{it:exp_is_optimal}, we can apply \\cref{thm:optimal_map_stability} to the triple\n of measures $\\nu=\\m$, $\\mu_1=\\hat\\mu^{n,t}$ and $\\mu_2=\\mu^n$ (with \n $S=\\exp(\\nabla f^{n,t})$ and $T=T^n$). \n We obtain\n \\begin{align*}\n \\int_M d^2(\\exp(\\nabla f^{n,t}), T^n)\\de\\m &\\lesssim \n W_2^2(\\mu^n, \\hat\\mu^{n,t}) + W_2(\\mu^n, \\hat\\mu^{n,t})W_2(\\m, \\hat\\mu^{n,t}) \\\\\n &\\lesssim \n W_2^2(\\mu^n, \\hat\\mu^{n,t}) + W_2(\\mu^n, \\hat\\mu^{n,t})W_2(\\m, \\mu^n) \n \\fullstop\n \\end{align*}\n \n Recalling \\cref{rem:repeat_old_remark} and \\cref{eq:exceptional_set_rare}, if we consider the expected \n value we can apply the latter inequality as if it were true unconditionally and not only in the \n event $A^{n,t}_\\xi$. Thus, taking the expected value and applying Cauchy-Schwarz's inequality, we get\n \\begin{equation*}\n \\E{\\int_M d^2(\\exp(\\nabla f^{n,t}), T^n)\\de\\m} \\lesssim \n \\E{W_2^2(\\mu^n, \\hat\\mu^{n,t})} \n + \\sqrt{\\E{W^2_2(\\mu^n, \\hat\\mu^{n,t})}\\cdot \\E{W^2_2(\\m, \\mu^n)}}\n \\fullstop\n \\end{equation*}\n The desired statement follows directly applying \n \\cref{eq:matching_cost_limit,eq:diffusion_error,eq:perturbation_error}.\n\\end{proof}\n\n\\begin{remark}\\label{rem:time_is_flexible}\n It might seem that our choice of the time $t=\\log^4(n)\/n$ is a little arbitrary, and indeed it is. Any time\n $t=t(n)$ of order $\\log^\\alpha(n)\/n$, for some $\\alpha > 3$, would have worked flawlessly.\n\\end{remark}\n\nIt remains to justify \\cref{rem:distance-tangent}. As already said, the desired estimate boils down to\nthe validity of \n\\begin{equation}\\label{eq:linf_distance_bounded}\n \\P{\\|d(T^{n},\\id)\\|_\\infty > \\varepsilon} \\ll \\frac{\\log(n)}n\n\\end{equation}\nfor any fixed $\\eps>0$.\nThe strategy of the proof is as follows. \nWith \\cref{lem:linf_l2_bound} (see also \\cite[Lemma 4.1]{goldman2017}) we reduce the hard task of \ncontrolling the $L^\\infty$-distance between $T^n$ and $\\id$ to the easier task of controlling \n$W_2^2(\\m,\\mu^n)$. \nThis latter estimate is then shown to be a consequence of \\cref{eq:exceptional_set_rare}.\n\n\\begin{lemma}\\label{lem:linf_l2_bound}\n Let $(M,\\metric)$ be a $d$-dimensional compact Riemannian manifold (possibly with Lipschitz boundary) and \n let $\\m$ be the volume measure on $M$.\n \n If $T:M\\to M$ is the optimal map with respect to the quadratic cost from $\\m$ to $T_\\#\\m$,\n then one has\n \\begin{equation*}\n \\norm{d(\\id, T)}_{L^\\infty(M)}\n \\lesssim \\left(\\int_M d^2(\\id, T)\\de\\m\\right)^{\\frac1{d+2}} \\fullstop\n \\end{equation*}\n\\end{lemma}\n\\begin{proof}\n Since the map $T$ is optimal, its graph is essentially contained in $c$-cyclically monotone set \n (see~\\cite[Theorem 1.38]{Santambrogio15}). \n More precisely, there exists a Borel set $C\\subseteq M$ such that $\\{(x,T(x)):\\ x\\in C\\}$ is $c$-cyclically \n monotone and $M\\setminus C$ is $\\m$-negligible. \n We will reduce our considerations to points in $C$ in order to exploit the $c$-cyclical monotonicity.\n\n Let us fix a point $x_0\\in C$ and let us define $\\alpha \\defeq \\frac12 d(x_0, T(x_0))$.\n Let us define the point $p\\in M$ as the middle point between $x_0$ and $T(x_0)$, that is \n $d(x_0, p) = d(p, T(x_0)) = \\alpha$. \n Let us consider a point $x\\in B(p, \\eps\\alpha)\\cap C$ where $\\eps>0$ is a small constant that will \n be chosen a posteriori. Finally let us define $\\beta\\defeq d(x, T(x))$. We want to show that \n $\\beta$ cannot be much smaller than $\\alpha$.\n \\begin{figure}[htb]\n \\centering\n \\input{linf_l2_bound.tikz}\n \\caption{The points considered in the proof of of \\cref{lem:linf_l2_bound}.}\n \\end{figure}\n \n \n Thanks to the $c$-cyclical monotonicity of $C$, it holds\n \\begin{equation*}\n d^2(x_0, T(x_0)) + d^2(x, T(x)) \\le d^2(x, T(x_0)) + d^2(x_0, T(x))\n \\end{equation*}\n and thus, applying repeatedly the triangle inequality, we deduce\n \\begin{align*}\n 4\\alpha^2 + \\beta^2 \n &\\le \\left(d(x, p) + d(p, T(x_0))\\right)^2 + \\left(d(x_0, p) + d(p, x) + d(x, T(x))\\right)^2 \\\\\n &\\le (\\eps\\alpha + \\alpha)^2 + (\\alpha + \\eps\\alpha + \\beta)^2 \n = 2(1+\\eps)^2\\alpha^2 + \\beta^2 + 2(1+\\eps)\\alpha\\beta \\\\\n &\\hphantom{asdasdasdasdasdasdasd}\\Updownarrow \\\\\n &\\hphantom{asdasdasd}(2-(1+\\eps)^2) \\alpha \\le (1+\\eps)\\beta \\fullstop\n \\end{align*}\n If $\\eps$ is chosen sufficiently small (i.e. $\\eps = 1\/3$), the desired estimate\n $\\alpha\\lesssim\\beta$ follows.\n \n Since $x$ can be chosen arbitrarily in $B(p, \\eps\\alpha)\\cap C$, the estimate \n $\\alpha\\lesssim \\beta$ implies\n \\begin{align*}\n \\int_M d^2(x, T(x))\\de\\m(x) \n &\\ge \\int_{B(p, \\eps\\alpha)} d^2(x, T(x))\\de\\m(x)\n \\gtrsim \\m(B(p, \\eps\\alpha)) d^2(x_0, T(x_0)) \\\\\n &\\gtrsim \\eps\\alpha^d d^2(x_0, T(x_0)\n \\gtrsim \\bigl(d(x_0, T(x_0))\\bigr)^{d+2} \\comma\n \\end{align*}\n where we have used that a ball with radius $r$ not larger than the diameter of \n $M$ has measure comparable to $r^d$ (follows from the Ahlfors-regularity of compact \n Riemannian manifolds with Lipschitz boundary).\n This completes the proof since $x_0$ can be chosen arbitrarily in a set with full measure.\n\\end{proof}\n\\begin{remark}\n The previous lemma holds, with the same proof, on any Ahlfors-regular metric measure space \n that is also a length space.\n\\end{remark}\n\\begin{remark}\n If we apply \\cref{lem:linf_l2_bound} on a $2$-dimensional manifold with $T^n$ being the optimal map\n (with respect to the quadratic cost) from $\\m$ to the empirical measure $\\mu^n$, we obtain\n \\begin{equation*}\n \\norm{d(\\id,T^n}_{L^{\\infty}(M)} \n \\lesssim W_2(\\m, \\mu^n)^{\\frac{1}{2}} \\fullstop\n \\end{equation*}\n Since we know (as a consequence of \\cref{eq:limit_value}) that with high probability\n $W^2_2(\\m, \\mu^n) \\lesssim n^{-1}\\log(n)$, we deduce that with high probability\n it holds\n \\begin{equation*}\n \\norm{d(\\id,T^n}_{L^{\\infty}(M)} \\lesssim \\left(\\frac{\\log(n)}{n}\\right)^{\\frac14} \\fullstop\n \\end{equation*}\n This estimate does not match the asymptotic behavior of the $\\infty$-Wasserstein distance between $\\m$\n and $\\mu^n$. In fact, as proven in \\cite{leighton1989,shor1991,trillos2015}, with high probability \n it holds\n \\begin{equation*}\n W_{\\infty}(\\m, \\mu^n) \\approx \\frac{\\log(n)^{\\frac34}}{n^{\\frac12}} \\fullstop\n \\end{equation*}\n\\end{remark}\n\nWe are now ready to show \\cref{eq:linf_distance_bounded} (to be precise we prove a much stronger \nestimate).\n\\begin{proposition}\\label{prop:linf_is_small}\n Using the same notation and definitions of the statement of \\cref{thm:main_theorem}, for any \n $\\eps>0$ and any $k>0$ there exists a constant $C=C(M, \\eps, k)$ such that\n \\begin{equation}\n \\P{\\|d(T^{n},\\id)\\|_\\infty > \\varepsilon} \\le C(M, \\eps, k)n^{-k} \\fullstop\n \\end{equation}\n\\end{proposition}\n\\begin{proof}\n We show that for any $\\eps>0$ and any $k>0$ there exists a constant $C=C(M, \\eps, k)$\n such that\n \\begin{equation}\\label{eq:wasserstein_is_small}\n \\P{W_2(\\m, \\mu^n) > \\varepsilon} \\le C(M, \\eps, k)n^{-k} \\fullstop\n \\end{equation}\n In fact, if we are able to prove \\cref{eq:wasserstein_is_small}, then the statement of the proposition \n follows applying \\cref{lem:linf_l2_bound} with $T=T^n$ (changing adequately $\\eps,k$ and the value\n of the constant $C$).\n \n The triangle inequality gives us\n \\begin{equation}\\label{eq:linf_tmp1}\n W_2(\\m, \\mu^n) \\le W_2(\\mu^{n,t}, \\mu^n) + W_2(\\m, \\mu^{n,t}) \\fullstop\n \\end{equation}\n The first term can be bounded using the contractivity property of the heat semigroup, obtaining\n \\begin{equation}\\label{eq:linf_tmp2}\n W_2(\\mu^{n,t}, \\mu^n) \\lesssim \\sqrt{t} \\fullstop\n \\end{equation}\n For the second term we employ the transport inequality \\cite[(4.1)]{ambrosio-glaudo2018} and get\n \\begin{equation}\\label{eq:linf_tmp3}\n W^2_2(\\m, \\mu^{n,t}) \\lesssim \\int_M \\abs{\\nabla f^{n,t}}^2\\de\\m \\fullstop\n \\end{equation}\n If we assume to be in the event $A^{n,t}_\\xi$ (that is defined in the statement of \\cref{prop:ag18})\n with $\\xi=\\xi(n)=\\frac1{\\log(n)}$, we have\n \\begin{equation}\\label{eq:linf_tmp4}\n \\int_M \\abs{\\nabla f^{n,t}}^2\\de\\m \\lesssim \\norm{\\nabla f}_{L^{\\infty}(M)}^2\n \\lesssim \\norm{\\nabla^2 f}_{L^{\\infty}(M)}^2 \\le \\xi^2 \\fullstop\n \\end{equation}\n Joining \\cref{eq:linf_tmp1,eq:linf_tmp2,eq:linf_tmp3,eq:linf_tmp4} we deduce that in the event \n $A^{n,t}_\\xi$ it holds\n \\begin{equation*}\n W_2(\\m, \\mu^n) \\lesssim \\sqrt{t} + \\xi \\fullstop\n \\end{equation*}\n Since $t(n)\\to 0$ and $\\xi(n)\\to 0$ as $n\\to\\infty$, this implies (for $n$ sufficiently large) that\n in the event $A^{n,t}_\\xi$ it holds $W_2(\\m, \\mu^n) \\le \\eps$. Hence \\cref{eq:wasserstein_is_small}\n is a consequence of \\cref{eq:exceptional_set_rare} and this concludes of the proof.\n\\end{proof}\n\n\n\n\\printbibliography\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzfymy b/data_all_eng_slimpj/shuffled/split2/finalzzfymy new file mode 100644 index 0000000000000000000000000000000000000000..0c3a9929859a591ebc3d557e30c4bf954b840b26 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzfymy @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nA \\emph{homomorphism} from a directed graph $G$ to a directed graph $H$ is a map from the vertices of $G$ to the vertices of $H$ which maps each edge of $G$ to an edge of $H$. Two directed graphs $G$ and $H$ are called \\emph{homomorphically equivalent} if there is a homomorphism from $G$ to $H$ and from $H$ to $G$.\nThe study of the \\emph{homomorphism order}\non the class of all finite directed graphs (or short: \\emph{digraphs}), factored by homomorphic equivalence, has a long history in graph theory. It is known to have a quite complicated structure; we refer to Ne\\v{s}et\\v{r}il and Tardif~\\cite{NesetrilTardif} and the references therein. \n\nA classical topic in graph homomorphisms is the $H$-coloring problem, which is the computational problem of deciding whether a given finite digraph $G$ maps homomorphically to $H$. The computational complexity of this problem has been classified for finite undirected graphs $H$ by Hell and Ne\\v{s}et\\v{r}il~\\cite{HellNesetril} in 1990: they are either in L or NP-complete. Feder and Vardi~\\cite{FederVardi} proved that every finite-domain CSP is polynomial-time equivalent to an $H$-coloring problem for a finite \\emph{directed} graph $H$\\footnote{This result has been sharpened in~\\cite{BulinDelicJacksonNiven}.}, and they conjectured\nthat each of these problems are either in P or NP-complete. \nThis conjecture was eventually solved in 2017 by Bulatov and, independently, by Zhuk~\\cite{BulatovFVConjecture,ZhukFVConjecture}. \nHowever, other long-standing open problems about the complexity of $H$-coloring for finite digraphs $H$ remain open, for example the characterisation of when this problem is in the complexity class L, or in NL~\\cite{LinearDatalog,EgriLaroseTessonLogspace,Kazda18}. \n\nThe border between polynomial-time tractable and NP-complete $H$-colouring problems can be described in terms of \\emph{primitive positive (pp) constructions}, which is a concept that has been introduced by Barto, Opr\\v{s}al, and Pinsker~\\cite{wonderland} in the setting of general relational structures. The idea is that if $G$ has a pp construction in $H$, then, intuitively, \\emph{`$H$ can simulate $G$'}, and the $G$-coloring problem reduces (in logarithmic space) to the $H$-coloring problem. \nIn particular, $H$-coloring is NP-hard if $K_3$ has a pp construction in $H$, where $K_3$ is the clique with three vertices, by reduction from the NP-hard three-colorability problem. It follows from the proof of the dichotomy conjecture that \n otherwise $H$-coloring is in P. Note that pp constructability can also be used to\nstudy the question of which $H$-coloring problems are in L or in NL. The surprising power of pp constructions is the motivation for studying pp constructions on finite digraphs more systematically. \n\nFor digraphs $G$ and $H$ that have at least one edge, the definition of pp constructions takes the following elegant combinatorial form: \n$G$ pp constructs $H$ if there exists a digraph $K$ and $a,b \\in V(K)^d$ for some $d \\in {\\mathbb N}$ \n such that $G$ is homomorphically equivalent to the digraph with vertices $V(H)^d$ \nand where $(u,v)$ forms an edge if there is a homomorphism from $K$ to $H$ that maps $(a,b)$ to $(u,v)$. \nWe write $H \\leq G$ if $G$ has a pp construction in $H$. It can be shown that $\\leq$ is transitive (Corollary 3.10 in~\\cite{wonderland}) and so it gives rise to a partial order $\\mathfrak P_{\\Digraphs}$ on the class of all finite digraphs (where we take the liberty to identify two digraphs $G$ and $H$ if they pp construct each other). \nSince all finite digraphs have a pp construction in $K_3$ (see, e.g. [11]), it is the smallest element of the poset $\\mathfrak P_{\\Digraphs}$.\nThe poset also has a greatest element, namely the digraph\n$P_1$ with just one vertex and no edges. \nThe digraph $P_1$ has a unique greatest lower bound in $\\mathfrak P_{\\Digraphs}$, namely the digraph $P_2$ consisting of two vertices and one directed edge; this is not hard to see and will be shown in Section~\\ref{sect:result}. \n\nIn this article, we present a complete description of the greatest lower bounds of $P_2$ in $\\mathfrak P_{\\Digraphs}$; we call these digraphs \\emph{submaximal}. We also prove that every finite digraph which does not pp constructs $P_2$ is smaller than one of the submaximal digraphs (Theorem~\\ref{thm:submaximalGraphs}; also see Figure~\\ref{fig:main}). \nThe submaximal digraphs are:\n\\begin{itemize}\n \\item The directed cycles $C_p$ for $p$ prime.\n (For $k \\in {\\mathbb N}^+$, the directed cycle $C_k$ \n is defined to be the digraph \n $({\\mathbb Z}_{k};\\{(u,v) \\mid u+1=v \\mod k\\})$.) \n \\item $T_3 \\coloneqq (\\{0,1,2\\},<)$, the transitive tournament with three vertices.\n\\end{itemize}\n\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=1.3]\n \\node (0) at (2,2) {$\\P_{1} \\equiv C_{1}$};\n \\node (1) at (2,1) {$\\P_{2}$};\n \\node (20) at (0,0) {${T}_{3}$};\n \\node (21) at (1,0) {${C}_{2}$};\n \\node (22) at (2,0) {${C}_{3}$};\n \\node (23) at (3,0) {${C}_{5}$};\n \\node (24) at (4,0) {$\\dots$};\n \n \n \\node (3) at (2,-0.7) {$\\vdots$};\n \\node[rotate = 45] (3) at (0.8,-0.7) {$\\vdots$};\n \\node[rotate = -45] (3) at (3.2,-0.7) {$\\vdots$};\n \n \n \\node (4) at (2,-1.5) {$K_3$};\n \n \\path\n (0) edge (1)\n (1) edge (20)\n (1) edge (21)\n (1) edge (22)\n (1) edge (23)\n (1) edge (24)\n ;\n \n \\end{tikzpicture}\n \\caption{The pp constructability poset on finite digraphs.}\n \\label{fig:main}\n\\end{figure}\n\n\\subsection*{Related work}\nThe pp constructability poset for smooth digraphs, i.e., digraphs where every vertex has indegree at least one and outdegree at least one (digraphs without sources and sinks), has been described in~\\cite{smooth-digraphs}. \nThe pp constructability poset on general relational structures over a two-element set has been described in~\\cite{PPPoset}.\n\n\n\\section{Minor conditions} \nPrimitive positive constructability has a universal algebraic characterisation; this characterisation plays a role in our proof, so we present it here. \nIf $H = (V,E)$ is a digraph then $H^k$ denotes the $k$-th direct power of $H$, which is the digraph with vertex set $V^k$ and edges set $$\\{((u_1,\\dots,u_k),(v_1,\\dots,v_k)) \\mid (u_1,v_1) \\in E,\\dots,(u_k,v_k) \\in E\\}.$$\nA \\emph{polymorphism} of $H$ is a homomorphism $f$ from $H^k$ to $H$, for some $k \\in {\\mathbb N}$, which is called the \\emph{arity} of $f$. We write $\\ensuremath{\\operatorname{Pol}}(H)$ for the set of all polymorphisms of $H$. This set contains the projections and is closed under composition.\\footnote{Sets of operations with these properties are called \\emph{clones} in universal algebra.} An operation $f$ is called \\emph{idempotent} if $f(x,\\dots,x) = x$ for all $x \\in V$. \n\nA central topic in universal algebra are \\emph{minor conditions}. If $f \\colon V^k \\to V$ is an operation and $\\sigma \\colon \\{1,\\dots,k\\} \\to \\{1,\\dots,n\\}$ is a function,\nthen $f_\\sigma$ denotes the operation \n$$(x_1,\\dots,x_n) \\mapsto f(x_{\\sigma(1)},\\dots,x_{\\sigma(k)}),$$\nand $f_\\sigma$ is called a \\emph{minor} of $f$. \nA \\emph{minor condition} is a set $\\Sigma$ of expressions of the form $f_{\\sigma} = g_{\\tau}$ where $f$ and $g$ are function symbols ($f$ and $g$ might be the same symbol). \n\n\\begin{example}\nAn operation $f \\colon V^n \\to V$ is called \\emph{cyclic}\nif for all $x_1,\\dots,x_n \\in V$\n$$f(x_1,x_2,\\dots,x_n) = f(x_2,\\dots,x_n,x_1).$$ This condition \ncan be expressed by the minor condition \n$$\\Sigma_n \\coloneqq \\{f_{{\\operatorname{id}}} = f_{\\tau}\\}$$ where ${\\operatorname{id}}$ denotes the identity function on $\\{1,2,\\dots,n\\}$ and $\\tau$ denotes the cyclic permutation $(1,2,\\dots,n)$ on $\\{1,\\dots,n\\}$. \n\\end{example} \n\nIf a minor condition $\\Sigma$ contains several expressions, then \ndifferent expressions in $\\Sigma$ might share the same function symbols.\n\n\\begin{example}\nAn idempotent operation $f$ is called a \\emph{Maltsev operation} \nif for all $x,y \\in V$ \n\\[f(y,y,x) = f(x,x,x) = f(x,y,y) .\\] \nThis condition can be expressed by the minor condition \n\\[\\Sigma_M \\coloneqq \\{f_{\\sigma} = f_\\tau, f_{\\tau} = f_{\\rho}\\}\\]\nwhere $\\sigma, \\tau, \\rho \\colon \\{1,2,3\\} \\to \\{1,2\\}$ are given by $\\sigma(1,2,3) = (2,2,1)$, $\\tau(1,2,3) = (1,1,1)$, and \n$\\rho(1,2,3) = (1,2,2)$. \n\\end{example}\n\nA set of operations $F$ \\emph{satisfies} a minor condition $\\Sigma$ if the function symbols in $\\Sigma$ can be replaced by operations from $F$ so that all the expressions in $\\Sigma$ hold; in this case we write $F \\models \\Sigma$. \nIf $H$ is a digraph, then $\\Sigma(H)$ denotes the class of all minor conditions that are satisfied in $\\ensuremath{\\operatorname{Pol}}(H)$. \n\n\\begin{theorem}[Barto, Opr\\v{s}al, and Pinsker~\\cite{wonderland}]\\label{thm:wonderland} \nLet $G$ and $H$ be finite digraphs. Then \n\\begin{align*}\n H \\text{ pp constructs } G &&\\text{if and only if} && \\Sigma(H)\\subseteq \\Sigma(G). \n\\end{align*}\n\\end{theorem}\n\n\\section{The pp construction poset}\\label{sect:result}\nWe have already defined pp constructability for digraphs in the introduction, but present an equivalent description here which is convenient when specifying pp constructions, and which is also closer to the presentation of Barto, Opr\\v{s}al, and Pinsker~\\cite{wonderland}. \nA \\emph{primitive positive formula} is a formula $\\phi(x_1,\\dots,x_k)$ of the form\n$$ \\exists y_1,\\dots,y_n (\\psi_1 \\wedge \\cdots \\wedge \\psi_m)$$\nwhere each of the formulas $\\psi_1,\\dots,\\psi_m$ is of the form $\\bot$ (for \\emph{false}), of the form $z_1=z_2$, or of the form $E(z_1,z_2)$ where\n$z_1,z_2$ are variables from $\\{x_1,\\dots,x_k,y_1,\\dots,y_n\\}$. \n\n\n\n\\begin{definition}\nLet $H = (V,E)$ be a digraph. A digraph $G$ with vertex set $V^d$ is called a \\emph{pp power of $H$ of dimension $d$} if there exists a primitive positive formula $\\phi(x_1,\\dots,x_d,y_1,\\dots,y_d)$\nsuch that \nthe edge set of $G$ equals \n$$\\{((u_1,\\dots,u_d),(v_1,\\dots,v_d)) \\mid \\phi(u_1,\\dots,u_d,v_1,\\dots,v_d) \\text{ holds in } H\\}.$$\n\\end{definition}\nIt follows from the definitions that $H \\leq G$ if and only if $G$ is homomorphically equivalent to a pp power of $H$. \nWe write \n\\begin{itemize}\n\\item $H \\equiv G$ if $H \\leq G$ and $G \\leq H$;\n\\item \n$H < G$ if $H \\leq G$ and not $G \\leq H$. \n\\end{itemize}\n\n\\begin{lemma}\n$\\P_1$ is the greatest element of $\\mathfrak P_{\\Digraphs}$. Moreover, $\\P_1 \\equiv {C}_1$. \n\\end{lemma}\n\\begin{proof}\nLet ${ G}$ be a finite digraph. Consider the pp power of ${ G}$ of dimension one given by the formula $\\phi(x,y) \\coloneqq \\;\\perp$. The resulting digraph has no edges and is therefore homomorphically equivalent to $\\P_1$.\nNow consider the pp power of ${ G}$ of dimension one given by the formula $\\phi(x,y) \\coloneqq (x=y)$. \nThe resulting digraph is homomorphically equivalent to the digraph ${C}_1$ with one vertex and a loop, which implies the statement. \n\\end{proof}\n\nIn the proof of the following lemma we need the fundamental concept of \\emph{cores} from the theory of graph homomorphisms (see, e.g.,~\\cite{HNBook}). A digraph $H = (V,E)$ is called a \\emph{core} if every \\emph{endomorphism} of $H$ (i.e., every homomorphism from $H$ to $H$) is an \\emph{embedding} (i.e., an\nisomorphism between $H$ and an induced subgraph of $H$). It is easy to see that every finite digraph $H$ is homomorphically equivalent to a core digraph, and that all core digraphs $G$ that are homomorphically equivalent to $H$ are isomorphic to each other; we therefore call $G$ \\emph{the} core of $H$. \nWhen studying $\\mathfrak P_{\\Digraphs}$ we may therefore restrict our attention to core digraphs; the big advantage of cores is the following useful lemma.\n\n\\begin{lemma}[follows from Lemma 3.9 in~\\cite{wonderland}]\\label{lem:constants}\nLet $H = (V,E)$ be a finite core digraph. Then $H \\leq G$ if and only if $G$ is homomorphically equivalent to a pp power of $H$ where the primitive positive formula might additionally contain conjuncts of the form $x = c$ where $x$ is a variable and $c \\in V$ is a constant. \n\\end{lemma}\n\n\\begin{lemma}\nWe have $\\P_2 < \\P_1$. Moreover, \n$\\P_2$ is the only coatom of $\\mathfrak P_{\\Digraphs}$, \ni.e., $\\P_2$ is the unique greatest lower bound of $\\P_1$ in $\\mathfrak P_{\\Digraphs}$. \n\\end{lemma}\n\\begin{proof}\nWe have already seen that $\\P_2 \\leq \\P_1$. \nTo prove that $\\P_2 \\not \\leq P_1$, \nfirst observe that $\\P_1$ has constant polymorphisms, while $\\P_2$ does not. Let $\\Sigma_c \\coloneqq \\{f_{\\rho} = f_{\\sigma} \\}$ \nwhere $f$ is a unary function symbol, \n$\\rho \\colon \\{1\\} \\to \\{1,2\\}$ is constant 1, and $\\sigma \\colon \\{1\\} \\to \\{1,2\\}$ is constant 2. \nThen $\\ensuremath{\\operatorname{Pol}}(\\P_1) \\models \\Sigma_c$, \nbut $\\ensuremath{\\operatorname{Pol}}(\\P_2) \\not \\models \\Sigma_c$. Then (the easy direction of) \nTheorem~\\ref{thm:wonderland} implies that $\\P_1 \\leq \\P_2$ does not hold. \n\nFor the second statement, let ${ G}$ be a finite digraph such that ${ G}<\\P_1$. \nWe have to show that ${ G} \\leq \\P_2$. \nWithout loss of generality we may assume that ${ G}$ is a core. Hence, by Lemma~\\ref{lem:constants}, we can use constants in pp constructions. Note that $G$ must have at least two different vertices $u$ and $v$. The pp power of ${ G}$ of dimension one given by the formula $\\phi(x,y) \\coloneqq (x=u) \\wedge (y=v)$ is a digraph that has exactly one edge that is not a loop and that is therefore homomorphically equivalent to $\\P_2$.\n\\end{proof}\n\nThe following theorem is our main result and will be shown in the remainder of the article; see Figure~\\ref{fig:main}. \n\n\\begin{theorem}\\label{thm:submaximalGraphs}\nThe submaximal elements of $\\mathfrak P_{\\Digraphs}$ are precisely ${T}_3$, ${C}_2$, ${C}_3$, ${C}_5$, $\\dots$ \nIf ${ G}$ is a finite digraph that does not have a pp construction in $P_2$, then ${ G}\\leq {T}_3$ or ${ G}\\leq{C}_p$ for some prime $p$.\n\\end{theorem}\n\n\\section{Submaximal digraphs and minor conditions}\nWe first discuss which of \nthe minor conditions that we have encountered \nare satisfied by the polymorphisms of \nthe digraphs that appear in Theorem~\\ref{thm:submaximalGraphs}. \nThe following facts are well-known; we present the proof for the convenience of the reader. \n\n\\begin{lemma}\\label{lem:CpConditions1}\nLet $p$ and $q$ be primes. Then\n$\\ensuremath{\\operatorname{Pol}}(C_p) \\models \\Sigma_q$ if and only if $q \\neq p$. \n\\end{lemma}\n\\begin{proof}\n If $p\\neq q$, then there is an $n\\in \\mathbb{N}^+\\!$ such that $p\\cdot n=1\\pmod q$. The map\n\\[(x_1,\\dots,x_p) \\mapsto n\\cdot(x_1 +\\ldots + x_p) \\pmod q\\]\nis a polymorphism of ${C}_q$ satisfying $\\Sigma_p$.\n\nNow suppose that $p=q$. We assume for contradiction that $f$ is a polymorphism of ${C}_q$ satisfying $\\Sigma_p$. Then\n\\[f(0,\\dots,q-2,q-1) = a = f(1,\\dots,q-1,0)\\]\nand hence $(a,a) \\in E$, which is impossible since $C_p$ has a loop only if $p=1$. \n\\end{proof}\n\n\\begin{lemma}\\label{lem:CpConditions2}\n$\\ensuremath{\\operatorname{Pol}}(C_n) \\models \\Sigma_M$ for every $n \\in {\\mathbb N}$. \n\\end{lemma}\n\\begin{proof}\nThe ternary operation $(x_1,x_2,x_3) \\mapsto x_1 - x_2 + x_3 \\pmod n$ is a Maltsev polymorphism of $C_n$. \n\\end{proof}\n\n\nA finite digraph $H$ is called \n\\emph{$k$-rectangular} if whenever $H$ contains directed paths of length $k$ from $a$ to $b$, from $c$ to $b$, \nand from $c$ to $d$, then also from $a$ to $d$. See Figure~\\ref{fig:rect}. \nA digraph $H$ is called \\emph{totally rectangular} if it is $k$-rectangular for all $k \\geq 1$. \n\n\\begin{lemma}\\label{lem:maltIffRect}\nA finite digraph $H$ is totally rectangular if and only if\nit has a Maltsev polymorphism. \nA finite core digraph $H$ has a Maltsev polymorphism if and only if \n$\\ensuremath{\\operatorname{Pol}}(H) \\models \\Sigma_M$. \n\\end{lemma}\n\\begin{proof}\nThe first part of the statement is Corollary 4.11 in~\\cite{CarvalhoEgriJacksonNiven}. For the second statement, \nlet $H = (V,E)$ be a core digraph which has a polymorphism $f$ that satisfies $f(x,y,y)=f(x,x,x)=f(y,y,x)$ for all $x,y \\in V$; we have to find a polymorphism that is additionally idempotent. \nNote that the function $x \\mapsto f(x,x,x)$ is an endomorphism; since $H$ is a core, the endomorphism is injective. Since $H$ is finite the endomorphism must in fact be an automorphism, and has an inverse $i$ which is an endomorphism as well. Then the operation $(x_1,x_2,x_3) \\mapsto i(f(x_1,x_2,x_3))$ is idempotent and a Maltsev operation. \n\\end{proof}\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=1.3]\n \\node (a) at (0,0) {$a$};\n \\node (b) at (0,1) {$b$};\n \\node (c) at (1,1) {$c$};\n \\node (d) at (1,0) {$d$};\n \\path[->]\n (a) edge (b)\n \n (c) edge (b)\n (c) edge (d)\n (a) edge[dashed] (d)\n ;\n \\end{tikzpicture}\n \\caption{Rectangularity in digraphs.}\n \\label{fig:rect}\n\\end{figure}\n\n\n\n\\begin{lemma}\\label{lem:T3Conditions}\n$\\ensuremath{\\operatorname{Pol}}(T_3) \\models \\Sigma_n$ for every $n \\in {\\mathbb N}$, but $\\ensuremath{\\operatorname{Pol}}(T_3) \\not \\models \\Sigma_M$. \n\\end{lemma}\n\\begin{proof}\nThe operation $(x_1,\\dots,x_n) \\mapsto \\max(x_1,\\dots,x_n)$ is a polymorphism of $T_3$ that satisfies $\\Sigma_n$. \nOn the other hand, $T_3 = (\\{0,1,2\\},E)$ is not 1-rectangular, witnessed by $(1,2),(0,2),(0,1) \\in E$ but $(1,1) \\notin E$; the second statement therefore follows from \nLemma~\\ref{lem:maltIffRect}. \n\\end{proof}\n\n\nThe following theorem states that \nthe digraph $\\P_2$ is the unique smallest element of \n$\\mathfrak P_{\\Digraphs}$ that satisfies $\\Sigma_M$ and $\\Sigma_p$ for all $p$ prime. \n\n\\begin{theorem}\\label{thm:maltAndCyclImplyIdempotent}\nLet ${ G}$ be a finite digraph\nthat satisfies $\\Sigma_M$\nand $\\Sigma_p$ for all primes $p$. Then \n$\\P_2 \\leq { G}$. \n\\end{theorem}\n\n\nIn the proof of Theorem~\\ref{thm:maltAndCyclImplyIdempotent} we make use the following result of Carvalho, Egri, Jackson, and Niven~\\cite{CarvalhoEgriJacksonNiven}, which guides us in our further proof steps. \n\n\\begin{theorem}[Lemma 3.10 in~\\cite{CarvalhoEgriJacksonNiven}]\\label{thm:maltImpliesPathOrDuoc}\nIf ${ G}$ is totally rectangular, then ${ G}$ is homomorphically equivalent to either a directed path or a disjoint union of directed cycles.\n\\end{theorem}\n\nWe write $\\P_k$ for the directed path with the $k$ vertices $\\{0,\\dots,k-1\\}$. \n\n\\begin{lemma}\\label{lem:IdempConstructPaths}\nThe digraph $\\P_2$ pp constructs $\\P_{k}$ for all $k\\in\\mathbb{N}^+$\\!.\n\\end{lemma}\n\\begin{proof}\nClearly, $\\P_2 \\leq \\P_1$ and $\\P_2 \\leq \\P_2$.\nLet $k\\geq3$ and consider the pp power ${ G}$ of $\\P_2$ of dimension $k-1$ given by the following formula \n$\\phi(x_1,\\dots x_{k-1},y_1,\\dots,y_{k-1})$ \n\\begin{align*}\n (x_1 = y_2) \\wedge (x_2=y_3) \\wedge \\dots \\wedge (x_{k-2}=y_{k-1}) \\wedge E(x_{k-1},y_1). \n\\end{align*}\nThen ${ G}$ contains the following path of length $k$\n\\[(0,0,\\dots,0)\\to (1,0,\\dots,0) \\to (1,1,\\dots,0) \\to \\dots \\to (1,1,\\dots,1).\\]\nwhich shows that there exists a homomorphism from $\\P_k$ to ${ G}$. Note that whenever there is an edge from $u$ to $v$ in ${ G}$, then the tuple $v$ contains exactly one 1 more than the tuple $u$.\nTherefore, the function $V({ G}) \\to \\{1,\\dots,k\\}$ that maps $v$ to the number of 1's in $v$ is a homomorphism from ${ G}$ to $\\P_k$. Hence $\\P_2\\leq\\P_k$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:maltAndCyclImplyIdempotent}]\nLet ${ G}$ be a finite digraph satisfying $\\Sigma_M$ and $\\Sigma_p$ for every prime $p$. By Lemma~\\ref{lem:maltIffRect} and Theorem~\\ref{thm:maltImpliesPathOrDuoc} there are two cases to consider:\nthe first is that ${ G}$ is homomorphically equivalent to $\\P_k$ for some $k$. Then $\\P_2\\leq { G}$ by Lemma~\\ref{lem:IdempConstructPaths}. \n\nThe second case is that ${ G}$ is homomorphically equivalent to a disjoint union of directed cycles.\nWithout loss of generality we may assume that ${ G}$ is a disjoint union of directed cycles. \nLet $(a_0,\\dots,a_{\\ell-1})$ be a shortest cycle in ${ G}$. Let $p$ be a prime and $k\\in\\mathbb{N}^+\\!$ such that $p\\cdot k=\\ell$, and\nlet $f\\in\\ensuremath{\\operatorname{Pol}}({ G})$ be a function that witnesses that $\\ensuremath{\\operatorname{Pol}}({ G})\\models\\Sigma_p$. \nThen\n\\begin{align*}\n f(a_0,a_k,\\dots,a_{(p-1)\\cdot k})=a=f(a_k,a_{2 k},\\dots,a_{0}).\n\\end{align*}\nSince $f$ is a polymorphism of ${ G}$ there is a directed path of length $k$ from $a$ to $a$. Thus, ${ G}$ contains a directed cycle whose length divides $k$, which contradicts the assumption that $\\ell$ is the length of the shortest directed cycle in ${ G}$. Therefore, $\\ell$ has no prime divisors, and $\\ell=1$. So ${ G}$ contains a loop and hence is homomorphically equivalent to $C_1$; it follows that $\\P_2 \\leq { G}$. \n\\end{proof}\n\n\\section{Proof of the main result}\n\n\nWe use the following general result about when a finite digraph can pp construct a finite disjoint union of cycles.\n\\begin{lemma}[Lemma 6.8 in~\\cite{smooth-digraphs}]\nLet ${C}$ be a finite disjoint union of cycles and let ${ G}$ be a finite digraph. Then \n\\begin{align*}\n{ G}\\leq {C} && \\text{iff} && \\ensuremath{\\operatorname{Pol}}({ G})\\models\\Sigma_{C\\dotdiv c} \\text{ implies } \\ensuremath{\\operatorname{Pol}}({C})\\models\\Sigma_{C\\dotdiv c}\\text{ for all $c\\in\\mathbb{N}^+$\\!.}\n\\end{align*}\n\\end{lemma}\n\nFor the special case that $C=C_p$, there are only two conditions of the form $\\Sigma_{C\\dotdiv c}$, namely $\\Sigma_1$, which is trivial, and $\\Sigma_p$, which is not satisfied by $C_p$. Hence, we obtain the following result. \n\\begin{theorem}\n\\label{thm:notSatSigma}\nIf ${ G}$ is a finite digraph. If $p$ is a prime number such that $\\ensuremath{\\operatorname{Pol}}({ G}) \\not\\models\\Sigma_p$, then ${ G} \\leq{C}_p$.\n\\end{theorem}\n\n\nWe also need a similar results for $\\Sigma_M$ instead of $\\Sigma_p$. \n\n\\begin{lemma}\\label{lem:notSatMalt}\nLet ${ G}$ be a finite digraph. \nIf $\\ensuremath{\\operatorname{Pol}}({ G}) \\not \\models \\Sigma_M$, then ${ G} \\leq {T}_3$. \n\\end{lemma}\n\n\n\\begin{proof\nSince $\\leq$ is transitive we may assume without loss of generality that $\\H = (V,E)$ is a core. By Lemma~\\ref{lem:maltIffRect}, $\\H$ is not totally rectangular. Hence, there are vertices $a,b,c,d \\in V$ such that in ${ G}$ there are directed paths of length $k$ from $a$ to $b$, from $c$ to $b$, from $c$ to $d$, and there is no directed path of length $k$ from $a$ to $d$. Note that by Lemma~\\ref{lem:constants} we are allowed to use constants in pp constructions. \nWe write $x\\stackrel{k}\\to y$\nas a shortcut for the primitive positive formula \n$\\exists u_1,\\dots,u_{k-2} (E(x,u_1) \\wedge E(u_1,u_2) \\wedge \\cdots \\wedge E(u_{k-2},y))$. \nConsider the pp power of ${ G}$ of dimension two given by the formula\n\\begin{align*}\n \\phi(x_1,x_2,y_1,y_2)&\\coloneqq x_1\\stackrel{k}\\to y_2 \\wedge (x_2=d) \\wedge (y_1=a).\n\\end{align*}\n\nLet $\\H$ be the resulting digraph. Consider the vertices $v_0=(c,d)$, $v_1=(a,d)$, and $v_2=(a,b)$ of $\\H$. Note that the only vertex of $\\H$ that can have incoming and outgoing edges is $v_1$. Since there is no path of length $k$ from $a$ to $d$ the vertex $v_1$ does not have a loop. Furthermore, $\\H$ has the edges $(v_0,v_1), (v_1,v_2),$ and $(v_0,v_2)$ (see Figure~\\ref{fig:T3constr}). \nHence, $i\\mapsto v_i$ is an embedding of ${T}_3$ into $\\H$. \nLet $V_0$ be the set of all vertices in $H$ that have outgoing edges and $V_2$ be the set of all vertices in $H$ that have incoming edges. Note that $V_0\\mathbin\\Delta V_2$ consists of $v_1$ and all isolated vertices. Clearly, $V_0\\setminus V_2$, $V_0\\mathbin\\Delta V_2$, and $V_2\\setminus V_0$ form a partition of $V(H)$ and\nthe map\n\\begin{align*}\n v\\mapsto\n \\begin{cases}\n v_2&\\text{if }v\\in V_2\\setminus V_0\\\\\n v_1&\\text{if }v\\in V_0\\mathbin\\Delta V_2\\\\\n v_0&\\text{if }v\\in V_0\\setminus V_2\n \\end{cases}\n\\end{align*}\nis a homomorphism from $\\H$ to ${T}_3$. Hence ${ G} \\leq {T}_3$.\n\\end{proof}\n\n\n\\begin{figure}\n \\centering\n \\begin{tikzpicture}\n \\node (0) at (0,0) {$(c,d)$};\n \\node (1) at (0,1.5) {$(a,d)$};\n \\node (2) at (0,3) {$(a,b)$};\n \n \\path[->]\n (0) edge (1)\n (1) edge (2)\n (0) edge[bend left=42] (2)\n ;\n \\end{tikzpicture}\n \\caption{The primitive positive construction of $T_3$ in the proof of Lemma~\\ref{lem:notSatMalt}.}\n \\label{fig:T3constr}\n\\end{figure}\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:submaximalGraphs}]\nLet ${ G}$ be a digraph such that $\\P_2\\not\\leq { G}$. \nTheorem~\\ref{thm:maltAndCyclImplyIdempotent} implies that either $\\ensuremath{\\operatorname{Pol}}({ G})$ does not satisfy $\\Sigma_M$ or that it does not satisfy $\\Sigma_p$ for some prime $p$. In the first case ${ G} \\leq {T}_3$, by Lemma~\\ref{lem:notSatMalt}. \nIn the second case ${ G}\\leq{C}_p$, by Theorem~\\ref{thm:notSatSigma}.\nHence, all submaximal elements of $\\mathfrak P_{\\Digraphs}$ are contained in \n$\\{{T}_3, {C}_2 ,{C}_3, {C}_5, \\dots\\}$. \nLemma~\\ref{lem:CpConditions1}, Lemma~\\ref{lem:CpConditions2}, and Lemma~\\ref{lem:T3Conditions} imply that these digraphs form an antichain in $\\mathfrak P_{\\Digraphs}$, and hence \neach of these digraphs is submaximal. \n\\end{proof}\n\n\n\nNote that our result implies the following. \n\n\\begin{corollary}\\label{cor:maltAndCyclImplyIdempotent}\nIf a finite digraph ${ G}$ satisfies $\\Sigma_M$, $\\Sigma_2$, $\\Sigma_3$, $\\Sigma_5$, $\\dots$, then any minor condition satisfied by $\\ensuremath{\\operatorname{Pol}}(\\P_2)$ is also satisfied by $\\ensuremath{\\operatorname{Pol}}({ G})$.\n\\end{corollary}\n\nThe statement of Corollary~\\ref{cor:maltAndCyclImplyIdempotent} may also be phrased as\n$$\\{\\Sigma_M, \\Sigma_2, \\Sigma_3, \\Sigma_5, \\dots\\}\\subseteq \\Sigma({ G}) \\quad \\Rightarrow \\quad \\Sigma(\\P_2)\\subseteq \\Sigma({ G}).$$\n\n\\begin{remark}\nWe do not know whether Corollary~\\ref{cor:maltAndCyclImplyIdempotent} holds for arbitrary clones of operations on a finite set, instead of just clones of the form $\\ensuremath{\\operatorname{Pol}}({ G})$ for a finite digraph $G$. However, the statement is false for clones of operations on an infinite set, as illustrated by the clone of operations on ${\\mathbb Q}$ of the form\n$(x_1,\\dots,x_n) \\mapsto a_1x_1+\\cdots+a_nx_n$ for \n$a_1,\\dots,a_n \\in {\\mathbb Q}$ such that $a_1 + \\cdots + a_n = 1$. \nThis clone satisfies $\\Sigma_n$ for every $n \\in \\mathbb N$,\nand contains the function $(x_1,x_2,x_3) \\mapsto x_1 - x_2 + x_3$, so it also satisfies $\\Sigma_M$. \nHowever, it is easy to see that it does not contain an operation $f$ that satisfies\n$$f(x,x,y) = f(y,y,x) = f(x,y,y) = f(y,x,x)$$\nfor all $x,y \\in {\\mathbb Q}$;\nhowever, this minor condition is satisfied by $\\ensuremath{\\operatorname{Pol}}(\\P_2)$ (for example by $f = \\max$). \n\\end{remark}\n\n\\begin{remark}\nMany, but not all the statements that we have shown also apply to \\emph{infinite} digraphs. In Theorem~\\ref{thm:wonderland}, only the forward direction holds if $G$ and $H$ are infinite; however, in this text we only used the forward direction of this theorem. \n\nEvery digraph with a Maltsev polymorphism is totally rectangular even if the digraph\nis infinite. \nThe proof of Theorem~\\ref{thm:maltImpliesPathOrDuoc} of Carvalho, Egri, Jackson, and Niven can be generalised to show that \nevery infinite digraph which is totally rectangular \nhomomorphically equivalent to an infinite disjoint unions of cycles or one of the infinite paths $P^\\infty\\coloneqq(\\mathbb{N},\\{(u,u+1)\\mid u\\in\\mathbb{N}\\})$, \n$P_\\infty\\coloneqq(\\mathbb{N},\\{(u+1,u)\\mid u\\in\\mathbb{N}\\})$, \nthe disjoint union $P_\\infty + P^\\infty$ of $P_\\infty$ and $P^\\infty$, \nand $P^\\infty_\\infty\\coloneqq(\\mathbb{Z},\\{(u,u+1)\\mid u\\in\\mathbb{Z}\\})$. All of these graphs have a Maltsev polymorphism. \n\nInfinite disjoint unions of cycles are clearly not submaximal. \nClearly, $P_2$ cannot pp construct the core digraphs $P_{\\infty}$, $P^{\\infty}$, $P_\\infty + P^\\infty$, and $P^\\infty_\\infty$, \nand these graphs can pp construct $P_2$. Clearly $P_{\\infty}$ and $P^{\\infty}$ pp-construct each other. \nWe do not know whether these graphs are submaximal in the class of all digraphs. \n\\end{remark}\n\n\n\\section{Concluding remarks}\nPrimitive positive constructability orders finite digraphs $H$ by their `strength' with respect to the $H$-coloring problem. Many deep combinatorial statements about graphs and digraphs can be phrased in terms of this order. We showed that at least the top region of the resulting poset can be described completely.\nA full description of the entire poset $\\mathfrak P_{\\Digraphs}$ would be highly desirable.\nWe state three concrete open problems.\n\\begin{enumerate}\n \\item Is $\\mathfrak P_{\\Digraphs}$ a lattice? (Primitive positive constructability is known to form a join meet-lattice on the class of all finite relational structures factored by homomorphic equivalence, but it is not clear to the authors whether the clone product construction for the meet used there can be carried out in the category of digraphs.) \n \\item Does $\\mathfrak P_{\\Digraphs}$ contain infinite ascending chains? (We have seen an infinite antichain in this article; an infinite descending chain of digraphs with a Maltsev polymorphism can be found in~\\cite{smooth-digraphs} and \n the existence of infinite descending chains of digraphs without a Maltsev polymorphism follows from results of~\\cite{BMOOPW}, and also from results in~\\cite{JacksonKN16}.) \n \n \n \n \\item What are the greatest lower bounds of $T_3$ in $\\mathfrak P_{\\Digraphs}$? \n\\end{enumerate}\n\nWe already mentioned that the pp constructability poset on disjoint unions of cycles has been described in~\\cite{smooth-digraphs}; in particular, it contains no infinite ascending chains and is a lattice. \nNote that this result combined with the result of the present paper shows that when exploring $\\mathfrak P_{\\Digraphs}$ it only remains to explore the interval between $K_3$ and $T_3$: \nif a digraph $\\H$ does not have a Maltsev polymorphism, then we proved that it is below $T_3$ (and above $K_3$);\notherwise, it is homomorphically equivalent to a directed path or a disjoint union of cycles and hence falls into the region that has already been completely described. \n\n\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \nLet $p$ be an odd prime number and let $\\mathbb{F}_q$ be the finite field with $q=p^M$ elements. Denote by $\\text{W}(\\mathbb{F}_q)$ the ring Witt vectors of $\\mathbb{F}_q$. For each prime number $l$, choose an embedding $\\iota_l:\\bar{\\mathbb{Q}}\\hookrightarrow \\bar{\\mathbb{Q}}_l$. Denote by $\\operatorname{G}_l$ the absolute Galois group $\\operatorname{Gal}(\\bar{\\mathbb{Q}}_l\/\\mathbb{Q}_l)$. Let $S$ be a finite set of prime numbers containing $p$. Denote by $\\mathbb{Q}_S$ the maximal algebraic extension of $\\mathbb{Q}$ which is unramified at each prime $l\\notin S$ and set $\\operatorname{G}_{\\mathbb{Q},S}=\\operatorname{Gal}(\\mathbb{Q}_S\/\\mathbb{Q})$. The weak form of Serre's conjecture states that an odd and irreducible two-dimensional Galois representation \n\\[\\bar{\\varrho}:\\operatorname{G}_{\\mathbb{Q},S}\\rightarrow \\text{GL}_2(\\mathbb{F}_q)\\]is modular, i.e., lifts to a characteristic zero representation $\\varrho$ attached to an eigencuspform. The strong form asserts that the eigencuspform may be chosen to have optimal level equal to the prime to $p$ part of the Artin conductor of $\\bar{\\varrho}$. Ribet \\cite{ribetLL} proved via a level lowering argument that the weak form implies the strong form. Khare-Wintenberger \\cite{KW2} went on to prove the full statement, building on Ribet's work.\n\\par \nIn \\cite{hamblenramakrishna}, Hamblen and Ramakrishna prove a generalization of the weak form of Serre's conjecture for reducible two-dimensional Galois representations. They impose some conditions on a two-dimensional representation\n$\\bar{\\varrho}:\\operatorname{G}_{\\mathbb{Q},S}\\rightarrow \\op{GL}_2(\\mathbb{F}_q)$, namely,\n\\begin{enumerate}\n\\item $\\bar{\\varrho}$ is reducible of the form \n\\[\\bar{\\varrho}=\\mtx{\\varphi}{\\ast}{}{1},\\] where $\\varphi:\\operatorname{G}_{\\mathbb{Q},S}\\rightarrow \\mathbb{F}_q^{\\times}$ is a Galois character.\n\\item The representation $\\bar{\\varrho}$ is odd, i.e. if $c\\in \\operatorname{G}_{\\mathbb{Q}}$ denotes complex conjugation, \\[\\det \\bar{\\varrho}(c)=-1.\\]\n\\item The representation $\\bar{\\varrho}$ is indecomposable, i.e. the cohomology class \\[\\ast\\in H^1(\\operatorname{G}_{\\mathbb{Q},S}, \\mathbb{F}_q(\\varphi))\\] is non-zero.\n\\item \nThe Galois character $\\varphi$ is stipulated to satisfy some conditions, for instance, $\\varphi\\neq \\bar{\\chi},\\bar{\\chi}^{-1}$ where $\\bar{\\chi}$ is the mod $p$ cyclotomic character and $\\varphi^2\\neq 1$. Further, the image of $\\varphi$ is stipulated to span $\\mathbb{F}_q$ over $\\mathbb{F}_p$.\n\\item \nThere are further conditions on the restriction $\\bar{\\varrho}_{\\restriction \\operatorname{G}_p}$. The reader may refer to condition 5 of Theorem 2 in \\cite{hamblenramakrishna} for further details.\n\\end{enumerate} \nHamblen and Ramakrishna show that if $\\bar{\\varrho}$ satisfies the above mentioned conditions, then on enlarging the set of ramification $S$ by a finite set of primes $X$, $\\bar{\\varrho}$ has an odd, irreducible, $p$-ordinary lift $\\varrho$ which is unramified outside $S\\cup X$\n \\[\\begin{tikzpicture}[node distance = 2.0cm, auto]\n \\node (GSX) {$\\operatorname{G}_{\\mathbb{Q},S\\cup X}$};\n \\node (GS) [right of=GSX] {$\\operatorname{G}_{\\mathbb{Q},S}$};\n \\node (GL2) [right of=GS]{$\\text{GL}_2(\\mathbb{F}_q).$};\n \\node (GL2W) [above of= GL2]{$\\text{GL}_2(\\text{W}(\\mathbb{F}_q))$};\n \\draw[->] (GSX) to node {} (GS);\n \\draw[->] (GS) to node {$\\bar{\\varrho}$} (GL2);\n \\draw[->] (GL2W) to node {} (GL2);\n \\draw[dashed,->] (GSX) to node {$\\varrho$} (GL2W);\n \\end{tikzpicture}\\]This lift is \\textit{geometric} in the sense of Fontaine and Mazur \\cite{fontainemazur}. By the result of Skinner and Wiles in \\cite{skinnerwiles}, the representation $\\varrho$ arises from a $p$-ordinary eigencuspform. This settles the weak form of Serre's conjecture for such $\\bar{\\varrho}$.\n\\par\nThe prospect of generalizing this lifting result leads us to examine higher dimensional Galois representations with image in a smooth group-scheme $\\operatorname{G}$ over $\\text{W}(\\mathbb{F}_q)$. Assume that $\\op{G}_{\\restriction \\mathbb{F}_q}$ is split and reductive and choose a split Borel $\\operatorname{B}_{\/\\mathbb{F}_q}\\subset \\op{G}_{\\restriction \\mathbb{F}_q}$. Let $\\bar{\\rho}$ be a homomorphism \\[\\bar{\\rho}:\\operatorname{G}_{\\mathbb{Q}}\\rightarrow \\operatorname{G}(\\mathbb{F}_q).\\] Let $\\mathfrak{g}$ denote the Lie-algebra of the adjoint group $\\operatorname{G}_{\\restriction \\mathbb{F}_q}^{ad}$ and $\\Phi(\\operatorname{G}_{\\restriction \\mathbb{F}_q}^{ad})$ be a root system compatible with the choice of Borel. Denote by $\\mathfrak{n}\\subset \\mathfrak{g}$ the span of the positive roots. The $\\mathbb{F}_q$-vector space $\\mathfrak{g}$ acquires an adjoint Galois action\n\\[\\operatorname{Ad}^0\\bar{\\rho}:\\operatorname{G}_{\\mathbb{Q}}\\rightarrow \\operatorname{Aut}_{\\mathbb{F}_q}(\\mathfrak{g}).\\] Denote by $\\operatorname{Ad}^0\\bar{\\rho}$ the Galois module with underlying vector space $\\mathfrak{g}$. It is imperative that $\\bar{\\rho}$ is \\textit{odd}. For an involution $\\tau\\in \\op{Aut}(\\operatorname{G}_{\\restriction \\mathbb{F}_q})$, let $(\\operatorname{Ad}^0\\bar{\\rho})^{\\tau}$ denote the subspace of $\\operatorname{Ad}^0\\bar{\\rho}$ fixed by $\\tau$. It was shown by E. Cartan that\n\\begin{equation*}\n \\dim (\\operatorname{Ad}^0\\bar{\\rho})^{\\tau}\\geq\\dim \\mathfrak{n}\n\\end{equation*}\n(see \\cite[Proposition 2.2]{Yun} for further details).\nThe representation $\\bar{\\rho}$ is \\textit{odd} if equality is achieved for the involution $\\op{ad} \\bar{\\rho}(c)$, i.e.\n\\begin{equation}\\label{oddness}\n \\dim (\\operatorname{Ad}^0\\bar{\\rho})^{\\op{ad}\\bar{\\rho}(c)}=\\dim \\mathfrak{n}.\n\\end{equation}\nIn particular, the group $\\operatorname{G}$ must contain an element $h=\\bar{\\rho}(c)$ for which equality $\\ref{oddness}$ is achieved. Such an element is said to induce a \\textit{Chevalley involution}. When $n>2$, the general linear group $\\operatorname{GL}_n(\\mathbb{F}_q)$ contains no such element. Hence there are no odd representations for the group $\\operatorname{GL}_n(\\mathbb{F}_q)$ when $n>2$. On the other hand, the general symplectic group $\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$ does contain elements which induce Chevalley involutions. \n\\par Ramakrishna in \\cite{RamLGR} and \\cite{RamFM} showed that odd, irreducible representations $\\bar{\\rho}:\\op{G}_{\\mathbb{Q}}\\rightarrow \\op{GL}_2(\\bar{\\mathbb{F}}_p)$ satisfying some additional hypotheses exhibit characteristic zero lifts which are \\textit{geometric} in the sense of Fontaine and Mazur. These results provided evidence for Serre's conjecture, before it was proved by Khare and Wintenberger. Taylor in \\cite{taylor} introduced a reformulation of Ramakrishna's method, by showing that the vanishing of a certain \\textit{dual Selmer group} is sufficient in asserting the existence of global Galois deformations with fixed local conditions. This new formulation paved the way for higher dimensional generalizations. In \\cite{partikisthesis}, Patrikis generalized Ramakrishna's lifting theorem to odd representations with \\textit{big image} in $\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$. Fakhruddin, Khare and Patrikis studied more general odd, irreducible representations in \\cite{FKP1} and \\cite{FKP2}.\n\\par We assume that our representation has image in $\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$ for $n\\geq 2$. Associate to a commutative $\\text{W}(\\mathbb{F}_q)$-algebra $R$, a non-degenerate alternating form on $R^{2n}$ prescribed by the matrix\\[J:=\\mtx{}{\\operatorname{Id}_n}{-\\operatorname{Id}_n}{}.\\] The group of general symplectic matrices $\\operatorname{GSp}_{2n}(R)$ consists of matrices $X$ which preserve this form up to a scalar i.e. satisfy\n$X^t J X\\in R^{\\times} \\cdot J$. The similitude character $\\nu:\\operatorname{GSp}_{2n}(R)\\rightarrow R^{\\times}$ is defined by the relation $X^t J X=\\nu(X)\\cdot J$. The space $\\operatorname{Ad}^0\\bar{\\rho}$ is an $\\mathbb{F}_q[\\operatorname{G}_{\\mathbb{Q},S}]$-module with underlying space $\\operatorname{sp}_{2n}(\\mathbb{F}_q)$. The Galois action is prescribed by\n \\[g\\cdot X=\\bar{\\rho}(g) X \\bar{\\rho}(g)^{-1}\\]where $g\\in \\operatorname{G}_{\\mathbb{Q},S}$ and $X\\in \\operatorname{sp}_{2n}(\\mathbb{F}_q)$. Let $\\operatorname{B}(R)$ be the Borel subgroup consisting of matrices\n\\[M=\\mtx{C}{CD}{}{\\xi (C^t)^{-1}}\\]where $C\\in \\operatorname{GL}_n(R)$ is upper triangular, $D\\in \\operatorname{GL}_n(R)$ is symmetric and $\\xi\\in R^{\\times}$. Note that in this setting, $\\op{B}$ is defined over $\\text{W}(\\mathbb{F}_q)$. Denote by $U_1\\subset \\op{B}$ the unipotent subgroup.\n\\par\nLet $\\bar{\\rho}:\\operatorname{G}_{\\mathbb{Q},S}\\rightarrow \\operatorname{GSp}_{2n}(\\mathbb{F}_q)$ be a continuous Galois representation with image in $\\op{B}(\\mathbb{F}_q)$. Composing $\\bar{\\rho}$ with the similitude-character $\\bar{\\nu}$ defines a Galois character denoted by $\\bar{\\kappa}$. Denote by $\\mathcal{T}\\subseteq \\operatorname{GSp}_{2n}$ the diagonal torus and $e_{i,j}\\in \\op{GL}_{2n}(\\mathbb{F}_q)$ the matrix with $1$ in the $(i,j)$-position and $0$ in all other positions. Set $\\mathfrak{t}$ for the $\\mathbb{F}_q$-span of $H_1,\\dots, H_n$, where $H_i:=e_{i,i}-e_{n+i,n+i}$. Let $L_1, \\dots, L_n\\in \\mathfrak{t}^*$ be the dual basis of $H_1,\\dots, H_n$. An integer linear combination $\\lambda$ of $L_1, \\dots, L_n$ is viewed as character on the torus $\\mathcal{T}(\\mathbb{F}_q)$, which is trivial on the center of $\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$. Via the natural quotient map $\\op{B}\\rightarrow \\mathcal{T}$, a character on $\\mathcal{T}$ induces a character on $\\op{B}$. The character on $\\op{B}$ induced by $\\lambda$ is denoted by \\[\\omega_{\\lambda}:\\op{B}(\\mathbb{F}_q)\\rightarrow \\mathbb{F}_q^{\\times}.\\] Associated to $\\omega_{\\lambda}$ is the Galois character\n\\[\\sigma_{\\lambda}=\\omega_{\\lambda}\\circ \\bar{\\rho}:\\operatorname{G}_{\\mathbb{Q},S}\\rightarrow \\mathbb{F}_q^{\\times}.\\] Let \"$1$\" be a formal symbol for the trivial linear combination of $L_1,\\dots, L_n$ and set $\\sigma_1$ equal to the trivial character. The roots $\\Phi=\\Phi(\\operatorname{Ad}^0\\bar{\\rho}, \\mathfrak{t})$ are specified by\n\\[\\begin{split}\n\\Phi=& \\{\\pm 2L_1, \\dots, \\pm 2 L_n\\} \\\\\\cup &\\{\\pm(L_i+L_j)\\mid 1\\leq i2n$,\n\\item\\label{thc2}$\\bar{\\rho}$ is odd, i.e.\n$\\dim (\\operatorname{Ad}^0\\bar{\\rho})^{\\op{ad}\\bar{\\rho}(c)}=\\dim \\mathfrak{n}$.\n\\item\\label{thc3}The image of $\\bar{\\rho}$ contains the unipotent subgroup $\\operatorname{U}_1(\\mathbb{F}_q)$.\n\\item\\label{thc4}Both the following conditions on the distinctness of the characters $\\{\\sigma_{\\lambda}\\}$ are satisfied:\n\\begin{enumerate}\n \\item For $\\lambda, \\lambda'\\in \\Phi\\cup \\{1\\}$ such that $\\lambda\\neq \\lambda'$, $\\sigma_{\\lambda}$ is not a $\\op{Gal}(\\mathbb{F}_q\/\\mathbb{F}_p)$-twist of $\\sigma_{\\lambda'}$.\n \\item Moreover for $\\lambda, \\lambda'\\in \\Phi\\cup \\{1\\}$ not necessarily distinct, $\\sigma_{\\lambda}$ is not a $\\op{Gal}(\\mathbb{F}_q\/\\mathbb{F}_p)$-twist of $\\bar{\\chi}\\sigma_{\\lambda'}$.\n\\end{enumerate}\n\\item\\label{thc5}\nFor each of the roots $\\lambda\\in \\Phi$, the $\\mathbb{F}_p$-linear span of the image of $\\sigma_{\\lambda}$ in $\\mathbb{F}_q$ is $\\mathbb{F}_q$.\n\n\\item\\label{thc7}\nAt each prime $v\\in S$ such that $v\\neq p$, there is a liftable local deformation condition $\\mathcal{C}_v$ with tangent space $\\mathcal{N}_v$ of dimension \n\\[\\dim \\mathcal{N}_v=h^0(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho}).\\]\\item\\label{thc8}Tilouine's regularity conditions $(\\operatorname{REG})$ and $(\\operatorname{REG})^*$ are satisfied, i.e. \n\\[H^0(\\op{G}_p, \\operatorname{Ad}^0\\bar{\\rho}\/\\mathfrak{b})=0\\text{ and }H^0(\\op{G}_p, (\\operatorname{Ad}^0\\bar{\\rho}\/\\mathfrak{b})(\\bar{\\chi}))=0.\\]\n\\end{enumerate} Let $\\kappa$ be a fixed choice of a lift of the character $\\bar{\\kappa}$ such that $\\kappa=\\kappa_0\\chi^k$, where $k$ is a positive integer divisible by $p(p-1)$ and $\\kappa_0$ is the Teichm\\\"uller lift of $\\bar{\\kappa}$. Then $\\exists$ a finite set of auxiliary primes $X$ disjoint from $S$ and a lift $\\rho$ \\[\\begin{tikzpicture}[node distance = 2.0cm, auto]\n \\node (GSX) {$\\operatorname{G}_{\\mathbb{Q},S\\cup X}$};\n \\node (GS) [right of=GSX] {$\\operatorname{G}_{\\mathbb{Q},S}$};\n \\node (GL2) [right of=GS]{$\\operatorname{GSp}_{2n}(\\mathbb{F}_q).$};\n \\node (GL2W) [above of= GL2]{$\\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q))$};\n \\draw[->] (GSX) to node {} (GS);\n \\draw[->] (GS) to node {$\\bar{\\rho}$} (GL2);\n \\draw[->] (GL2W) to node {} (GL2);\n \\draw[dashed,->] (GSX) to node {$\\rho$} (GL2W);\n \\end{tikzpicture}\\] for which \n\\begin{enumerate}\n\\item $\\rho$ is irreducible,\n\\item $\\rho$ is $p$-ordinary (in the sense of \\cite[section 4.1]{patrikisexceptional}),\n\\item $\\nu\\circ \\rho= \\kappa$,\n\\item for $v\\in S\\backslash \\{p\\}$, the restriction to the decomposition group $\\rho_{\\restriction \\operatorname{G}_v}\\in \\mathcal{C}_v$.\n\\end{enumerate}\n\\end{Th}\n\nThe lift $\\rho$ is geometric in the sense of Fontaine and Mazur. For $\\lambda\\in \\Phi$, setting $\\lambda=-\\lambda'$ in condition $\\eqref{thc4}$, we have that $\\sigma_{\\lambda}^2\\neq 1$. Note that the conditions also imply that $\\sigma_{\\lambda}\\neq \\bar{\\chi},\\bar{\\chi}^{-1}$. This is reminiscent of the condition $\\varphi^2\\neq 1$ of Hamblen-Ramakrishna. In particular, condition $\\eqref{thc4}$ implies that $p>2$. The requirement $p>2n$ is primarily made so that we may suitably work with the exponential map in various places.\n\\par It is a consequence of Tilouine's regularity conditions that the ordinary deformations of $\\bar{\\rho}_{\\restriction \\op{G}_p}$ constitute a liftable deformation condition $\\mathcal{C}_p$ for which the tangent space $\\mathcal{N}_p$ has dimension:\n\\[\\dim \\mathcal{N}_p=h^0(\\op{G}_p,\\operatorname{Ad}^0\\bar{\\rho})+\\dim \\mathfrak{n}.\\] For a discussion on the ordinary deformation condition and Tilioune's regularity conditions, the reader may refer to \\cite[section 4]{patrikisexceptional}. With reference to condition $\\eqref{thc7}$, the reader may consult \\cite[sections 4.3 and 4.4]{patrikisexceptional} for examples of such deformation conditions.\n\\par When $n=1$, the unipotent group is abelian and examples of such Galois representations $\\bar{\\rho}$ are constructed via class field theory. The reader may, for instance, refer to \\cite{ribetclassgroups}, \\cite[section 7]{hamblenramakrishna} and \\cite{ray} for more details. In section $\\ref{examples}$, examples of Galois representations $\\bar{\\rho}:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{F}_p)$ satisfying the conditions of Theorem $\\ref{main}$ are constructed. It is shown that if $p\\geq 23$ is a regular prime, there exists a Galois representation\n \\[\\bar{\\rho}=\\left( {\\begin{array}{cccc}\n \\bar{\\chi}^3 &\\ast & \\ast & \\ast \\\\\n & 1 & \\ast & \\ast \\\\\n & & \\bar{\\chi}^6 & \\\\\n & & \\ast & \\bar{\\chi}^9\n \\end{array} } \\right):\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{F}_p)\\] which satisfies the conditions of Theorem $\\ref{main}$.\n\\subsection*{Acknowledgements}\nThe author is very grateful to his advisor Ravi Ramakrishna for introducing him to the fascinating subject of Galois deformations. He thanks Brian Hwang and Nicolas Templier for fruitful conversations. The author also appreciates the suggestions made by the anonymous referee which have led to significant improvement of the article.\n\\section{Notation}\\label{notationsection}\n\\begin{itemize}\n\\item For an $\\mathbb{F}_q$-vector space $M$, set $\\dim M:=\\dim_{\\mathbb{F}_q} M$.\n\\item At every prime $v$, choose an embedding $\\iota_v:\\bar{\\mathbb{Q}}\\hookrightarrow\\bar{\\mathbb{Q}}_v$. The absolute Galois group $\\operatorname{G}_v=\\operatorname{Gal}(\\bar{\\mathbb{Q}}_v\/\\mathbb{Q}_v)$ is identified with the decomposition group of the prime dividing $v$ determined by $\\iota_v$. \n\\item \n Let $e_{i,j}$ denote the $2n\\times 2n$ square matrix with coefficients in $\\mathbb{F}_q$ with $1$ in the $(i,j)$-th position and $0$ at all other positions. \\item The space $\\operatorname{Ad}^0\\bar{\\rho}$ is an $\\mathbb{F}_q[\\operatorname{G}_{\\mathbb{Q},S}]$-module with underlying space $\\operatorname{sp}_{2n}(\\mathbb{F}_q)$. The Galois action is prescribed by\n \\[g\\cdot X=\\bar{\\rho}(g) X \\bar{\\rho}(g)^{-1}\\]where $g\\in \\operatorname{G}_{\\mathbb{Q},S}$ and $X\\in \\operatorname{sp}_{2n}(\\mathbb{F}_q)$.\n \\item The space of diagonal matrices in $\\operatorname{Ad}^0\\bar{\\rho}$ is denoted by $\\mathfrak{t}$. Let $H_1,\\dots, H_n$ be the basis of $\\mathfrak{t}$ defined by $H_i:=e_{i,i}-e_{n+i,n+i}$. Let $L_1, \\dots, L_n\\in \\mathfrak{t}^*$ be the dual basis.\n \\item Let $\\Phi$ be the set of roots of $\\operatorname{sp}_{2n}(\\mathbb{F}_q)$ and $\\lambda_1,\\dots, \\lambda_n\\in \\Phi$ be the simple roots defined as follows\n \\[\\lambda_i:=\\begin{cases}\n L_i-L_{i+1}\\text{ for }i2n$. The group $U_1$ is the unipotent subgroup of $\\op{B}(\\mathbb{F}_q)$.\n\\item Throughout, $h^i$ will be an abbreviation for $\\dim H^i$. For instance, $h^i(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ is an abbreviation for $\\dim H^i(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$.\n\\item Let $M$ be an $\\mathbb{F}_q[\\op{G}_S]$-module, let $\\Sh^i_S(M)$ consist of cohomology classes $f\\in H^i(\\operatorname{G}_{\\mathbb{Q},S}, M)$ such that $f_{\\restriction \\op{G}_v}=0$ for all $v\\in S$.\n\\end{itemize}\n\\section{The General Lifting Strategy}\\label{section3}\n\\par Let $\\bar{\\varrho}$ be a Galois representation $\\bar{\\varrho}:\\operatorname{G}_{\\mathbb{Q}}\\rightarrow \\operatorname{GL}_2(\\bar{\\mathbb{F}}_p)$ which is irreducible, odd and unramified outside finitely many primes. Ramakrishna in \\cite{RamFM} and \\cite{RamLGR} showed that if $\\bar{\\varrho}$ satisfies additional conditions, it lifts to a continuous Galois representation $\\varrho$ which is geometric in the sense of Fontaine and Mazur. In other words, $\\varrho$ is odd, unramified outside finitely many primes and $\\varrho_{\\restriction \\operatorname{G}_p}$ is de Rham. This geometric lifting theorem provided evidence for the weak version of Serre's conjecture before it was proved by Khare and Wintenberger. The geometric lifting construction was adapted to the reducible setting in \\cite{hamblenramakrishna}. The main result of this manuscript is a higher dimensional generalization of the lifting theorem of Hamblen-Ramakrishna. The basic strategy involves successively lifting $\\bar{\\rho}$ to a characteristic zero irreducible geometric representation $\\rho$ by successively lifting $\\rho_m$ to $\\rho_{m+1}$\n\\[\\begin{tikzpicture}[node distance = 2.0 cm, auto]\n \\node (GSX) at (0,0){$\\operatorname{G}_{\\mathbb{Q},S\\cup X}$};\n \\node (GL2) at (5,0){$\\operatorname{GSp}_{2n}(\\mathbb{F}_q).$};\n \\node (GL2Wn) at (3,2)[above of= GL2]{$\\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^{m})$};\n \\node (GL2Wnplus1) at (5,4){$\\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^{m+1})$};\n \\draw[->] (GSX) to node [swap]{$\\bar{\\rho}$} (GL2);\n \\draw[->] (GL2Wn) to node {} (GL2);\n \\draw[->] (GSX) to node [swap]{$\\rho_m$} (GL2Wn);\n \\draw[->] (GL2Wnplus1) to node {} (GL2Wn);\n \\draw[dashed,->] (GSX) to node {$\\rho_{m+1}$} (GL2Wnplus1);\n \\end{tikzpicture}\\] \n \n \\begin{Def} Let $\\mathcal{C}$ be the category of coefficient rings over $\\text{W}(\\mathbb{F}_q)$ with residue field $\\mathbb{F}_q$. The objects of this category consist of local $\\text{W}(\\mathbb{F}_q)$-algebras $(R,\\mathfrak{m})$ for which\n \\begin{itemize}\n \\item $R$ is complete and Noetherian,\n \\item $R\/\\mathfrak{m}$ is isomorphic to $\\mathbb{F}_q$ as a $\\text{W}(\\mathbb{F}_q)$-algebra. The residual map \\[\\phi:R\\rightarrow \\mathbb{F}_q\\]is the composite of the quotient map $R\\rightarrow R\/\\mathfrak{m}$ with the unique isomorphism of $W(\\mathbb{F}_q)$-algebras $R\/\\mathfrak{m}\\xrightarrow{\\sim}\\mathbb{F}_q$.\n \\end{itemize} A morphism $F:(R_1,\\mathfrak{m}_1)\\rightarrow (R_2,\\mathfrak{m}_2)$ is a homorphism of local rings which is also a $\\text{W}(\\mathbb{F}_q)$-algebra homorphism. Recall that $\\kappa$ is a fixed choice of lift of $\\bar{\\kappa}$. Let $\\kappa_v$ denote the restriction of $\\kappa$ to $\\operatorname{G}_v$.\n \\end{Def}\n \\par Let $v$ be a prime and $R\\in \\mathcal{C}$. Denote by $\\phi^*: \\operatorname{GSp}_{2n}(R)\\rightarrow \\operatorname{GSp}_{2n}(\\mathbb{F}_q)$ the group homomorphism induced by the residual homomorphism $\\phi: R\\rightarrow \\mathbb{F}_q$. We say that $\\rho_R:\\operatorname{G}_v\\rightarrow \\operatorname{GSp}_{2n}(R)$ is an $R$-lift of $\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$ if $\\phi^*\\circ \\rho_R=\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$, i.e. the following diagram commutes\n \\[ \\begin{tikzpicture}[node distance = 2.2 cm, auto]\n \\node(G) at (0,0){$\\operatorname{G}_{v}$};\n \\node (A) at (3,0) {$\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$.};\n \\node (B) at (3,2) {$\\operatorname{GSp}_{2n}(R)$};\n \\draw[->] (G) to node [swap]{$\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$} (A);\n \\draw[->] (B) to node{$\\phi^*$} (A);\n \\draw[->] (G) to node {$\\rho_R$} (B);\n \\end{tikzpicture}\\]Further, we shall require that the similitude character of $\\rho_R$ coincides with the composite of $\\kappa_v$ with the homomorphism $W(\\mathbb{F}_q)^{\\times}\\rightarrow R^{\\times}$ induced by the structure map.\n \\par Two lifts $\\rho_R$ and $\\rho_R'$ are said to be strictly-equivalent if there is\n \\[A\\in \\text{ker}\\lbrace \\operatorname{GSp}_{2n}(R)\\xrightarrow{\\phi^*} \\operatorname{GSp}_{2n}(\\mathbb{F}_q)\\rbrace \\]\n such that \n $\\rho_R=A\\rho_R' A^{-1}$. A deformation is a strict equivalence class of lifts. Let $\\operatorname{Def}_v(R)$ be the set of $R$-deformations of $\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$. The association $R\\mapsto \\operatorname{Def}_v(R)$ defines a covariant functor \\[\\operatorname{Def}_v:\\mathcal{C}\\rightarrow \\operatorname{Sets}. \\]\n The tangent space $\\operatorname{Def}_v(\\mathbb{F}_q[\\epsilon]\/(\\epsilon^2))$ naturally acquires the structure of an $\\mathbb{F}_q$-vector space and is isomorphic to $H^1(\\operatorname{G}_v,\\operatorname{Ad}^0\\bar{\\rho})$. Under this association, a cohomology class $f$ is identified with the deformation $(\\operatorname{Id}+\\epsilon f)\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$.\n \\par For $m\\in \\mathbb{Z}_{\\geq 2}$, the deformations $\\operatorname{Def}_v(\\text{W}(\\mathbb{F}_q)\/p^{m})$ are equipped with action of the cohomology group $H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$. For $\\varrho_m\\in \\operatorname{Def}_v(\\text{W}(\\mathbb{F}_q)\/p^{m})$ and $f\\in H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$, the twist of $\\varrho_m$ by $f$ is defined by the formula $(\\op{Id}+p^{m-1}f)\\varrho_m$. The set of deformations $\\varrho_m$ of a fixed $\\varrho_{m-1}\\in \\operatorname{Def}_v(\\text{W}(\\mathbb{F}_q)\/p^{m-1})$ is either empty or in bijection with $H^1(\\operatorname{G}_v,\\operatorname{Ad}^0\\bar{\\rho})$.\n \\begin{Def}\\label{defconditiondef} (see \\cite{taylor}) We say that a sub-functor $\\mathcal{C}_v$ of $\\operatorname{Def}_v$ is a deformation condition if (1) to (3) below are satisfied. If condition (4) is satisfied, $\\mathcal{C}_v$ is said to be liftable.\n \\begin{enumerate}\n \\item First, we require that $\\mathcal{C}_v(\\mathbb{F}_q)=\\{\\bar{\\rho}_{\\restriction \\operatorname{G}_v}\\}.$\n \\item For $R_1$ and $R_2$ be $\\mathcal{C}$, let $\\rho_1\\in \\mathcal{C}_v(R_1)$ and $\\rho_2\\in \\mathcal{C}_v(R_2)$. Let $I_1$ be an ideal in $R_1$ and $I_2$ an ideal in $R_2$ such that there is an isomorphism $\\alpha:R_1\/I_1\\xrightarrow{\\sim} R_2\/I_2$ satisfying \\[\\alpha(\\rho_1 \\;\\text{mod}{I_1})=\\rho_2 \\;\\text{mod}{I_2}.\\] Let $R_3$ be the fibred product \\[R_3=\\lbrace(r_1,r_2)\\mid \\alpha(r_1\\text{mod} I_1)=r_2 \\text{mod} I_2\\rbrace\\] and $\\rho_3$ the $R_3$-deformation induced from $\\rho_1$ and $\\rho_2$. Then $\\rho_3$ satisfies $\\mathcal{C}_v(R_3)$.\n \\item Let $R\\in \\mathcal{C}$ with maximal ideal $\\mathfrak{m}_R$. If $\\rho\\in \\operatorname{Def}_v(R)$ is such that $\\rho\\mod{\\mathfrak{m}_R^n}$ satisfies $\\mathcal{C}_v$ for all $n\\in \\mathbb{Z}_{\\geq 1}$, then $\\rho$ also satisfies $\\mathcal{C}_v$.\n \\item Let $R\\in \\mathcal{C}$ and $I$ an ideal such that $I.\\mathfrak{m}_R=0$. For $\\rho\\in \\mathcal{C}_v(R\/I)$, there exists $\\tilde{\\rho}\\in \\mathcal{C}_v(R)$ such that $\\rho=\\tilde{\\rho}\\mod{I}$.\n \\end{enumerate}\n \\end{Def}\n Let $\\mathcal{C}_v$ be a local deformation condition at the prime $v$. The tangent space $\\mathcal{N}_v$ consists of $f\\in H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$, such that $(\\op{Id}+\\epsilon f) \\bar{\\rho}_{\\restriction \\op{G}_v}\\in \\mathcal{C}_v(\\mathbb{F}_q[\\epsilon]\/(\\epsilon^2))$. The action of $\\mathcal{N}_v$ on $\\operatorname{Def}_v(\\text{W}(\\mathbb{F}_q)\/p^m)$ stabilizes $\\mathcal{C}_v(\\text{W}(\\mathbb{F}_q)\/p^m)$. In other words, if $\\varrho_m\\in \\mathcal{C}_v(\\text{W}(\\mathbb{F}_q)\/p^m)$ and $f\\in \\mathcal{N}_v$, then \n \\[(\\op{Id}+p^{m-1}f)\\varrho_{m}\\in \\mathcal{C}_v(\\text{W}(\\mathbb{F}_q)\/p^m). \\]It is assumed that each prime $v\\in S\\backslash \\{p\\}$ is equipped with a liftable local deformation condition $\\mathcal{C}_v$ such that \n\\[\\dim \\mathcal{N}_v=h^0(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho}).\\] The reader may consult \\cite[sections 4.3 and 4.4]{patrikisexceptional} for examples of such deformation conditions. The deformation condition $\\mathcal{C}_p$ is the ordinary deformation condition. Since we have assumed that Tilouine's regularity conditions are satisfied (cf. \\cite[section 4.1]{patrikisexceptional}), $\\mathcal{C}_p$ is liftable and the tangent space $\\mathcal{N}_p$ has dimension equal to\n\\[\\dim \\mathcal{N}_p=h^0(\\operatorname{G}_p, \\operatorname{Ad}^0\\bar{\\rho})+\\dim \\mathfrak{n},\\]see \\cite[Proposition 4.4]{patrikisexceptional}. We allow the successive lifts $\\rho_m$ to be ramified at a set of primes $S\\cup X$. Each auxiliary prime $v\\in X$ is equipped with a liftable subfunctor $\\mathcal{C}_v$ of $\\operatorname{Def}_v$. These auxiliary primes are referred to as trivial primes and were introduced by Hamblen and Ramakrishna in the two-dimensional setting \\cite[section 4]{hamblenramakrishna}. These are primes $v\\equiv 1\\mod{p}$, not contained in $S$, at which $\\bar{\\rho}_{\\restriction \\op{G}_v}$ the trivial representation and $v\\not\\equiv 1\\mod{p^2}$. We use a higher dimensional generalization of the deformation functor $\\mathcal{C}_v$ at a trivial prime $v$, due to Fakhruddin, Khare and Patrikis \\cite[Definition 3.1]{FKP1}. At each trivial prime $v$ there is a subspace $\\mathcal{N}_v$ of $H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ of dimension $h^0(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ which behaves like a versal tangent space. For $m\\geq 3$, the action of $\\mathcal{N}_v$ on $\\operatorname{Def}(\\text{W}(\\mathbb{F}_q)\/p^m)$ stabilizes $\\mathcal{C}_v(\\text{W}(\\mathbb{F}_q)\/p^m)$. This is proved in the $\\op{GL}_2$-case by Hamblen-Ramakrishna, see \\cite[Corollory 25, 29]{hamblenramakrishna}. For more general groups, we refer to Fakhruddin-Khare-Patrikis \\cite[Lemma 3.6,3.10]{FKP1} for the precise statement. However, this is not the case for $m=2$.\n\\par Let $X$ be a finite set of trivial primes disjoint from $S$. For $v\\in S\\cup X$, set $\\mathcal{N}_v^{\\perp}\\subseteq H^1(\\operatorname{G}_v,\\operatorname{Ad}^0\\bar{\\rho}^*)$ to be the orthogonal complement of $\\mathcal{N}_v$ with respect to the non-degenerate Tate pairing \n\\[H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})\\times H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho}^*)\\rightarrow H^2(\\operatorname{G}_v, \\mathbb{F}_q(\\bar{\\chi}))\\xrightarrow{\\sim}\\mathbb{F}_q.\\] Set $\\mathcal{N}_{\\infty}=0$ and $\\mathcal{N}_{\\infty}^{\\perp}=0$. The Selmer-condition $\\mathcal{N}$ is the tuple $\\{\\mathcal{N}_v\\}_{v\\in S\\cup X\\cup \\{\\infty\\}}$ and the dual Selmer condition $\\mathcal{N}^{\\perp}$ is $\\{\\mathcal{N}_v^{\\perp}\\}_{v\\in S\\cup X\\cup\\{\\infty\\}}$. Attached to $\\mathcal{N}$ and $\\mathcal{N}^{\\perp}$ are the Selmer and dual-Selmer groups defined as follows:\n \\[H^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho}):=\\text{ker}\\left\\{ H^1(\\operatorname{G}_{\\mathbb{Q}, S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho})\\xrightarrow{\\operatorname{res}_{S\\cup X}} \\bigoplus_{v\\in S\\cup X} \\frac{H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_v}\\right\\}\\]\n and\n \\[H^1_{\\mathcal{N}^{\\perp}}(\\op{G}_{\\mathbb{Q},S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho}^*):=\\text{ker}\\left\\{ H^1(\\operatorname{G}_{\\mathbb{Q}, S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho}^{*})\\xrightarrow{\\op{res}_{S\\cup X}'} \\bigoplus_{v\\in S\\cup X} \\frac{H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho}^*)}{\\mathcal{N}_v^{\\perp}}\\right\\}\\]\n respectively. The following formula is due to Wiles (see \\cite[Theorem 8.7.9]{NW}):\n \\begin{equation}\\label{wilesformula}\\begin{split}h^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho})-h^1_{\\mathcal{N}^{\\perp}}(\\operatorname{G}_{\\mathbb{Q}, S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho}^{*})&=h^0(\\op{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho})-h^0(\\op{G}_{\\mathbb{Q}},\\operatorname{Ad}^0\\bar{\\rho}^*)\\\\ &+\\sum_{v\\in S\\cup X\\cup \\{\\infty\\}} \\left(\\dim \\mathcal{N}_v-h^0(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})\\right).\\\\\\end{split}\\end{equation} Since $\\bar{\\rho}$ is odd, one has that $h^0(\\op{G}_{\\infty}, \\operatorname{Ad}^0\\bar{\\rho})=\\dim \\mathfrak{n}$. It follows from the above formula that the dimensions of the Selmer group and dual Selmer group coincide, i.e.\n \\[h^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho})=h^1_{\\mathcal{N}^{\\perp}}(\\operatorname{G}_{\\mathbb{Q}, S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho}^{*}).\\]The Selmer and dual Selmer groups fit into a long exact sequence called the Poitou-Tate sequence. We only point out that the cokernel of $\\op{res}_{S\\cup X}$ injects into $H^1_{\\mathcal{N}^{\\perp}}(\\op{S}_{S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho}^*)^{\\vee}$. In particular, if the Selmer group is zero, then so is the dual Selmer group, in which case the restriction map $\\operatorname{res}_{S\\cup X}$ is an isomorphism. Since the spaces $\\mathcal{N}_v$ at a trivial prime $v$ stabilize lifts only past mod $p^3$, it becomes necessary to produce a mod $p^3$ lift $\\rho_3$ of $\\bar{\\rho}$ before applying the general lifting-strategy. All deformations $\\rho_m$ discussed in this paper will have similitude character equal to $\\kappa\\mod{p^m}$.\n \\par The three main steps are as follows:\n \\begin{enumerate}\n \\item first it is shown that there is a finite set of trivial primes $X_1$ disjoint from $S$ such that the representation $\\bar{\\rho}$ lifts to a mod $p^2$ representation $\\rho_2$ which is unramified outside $S\\cup X_1$.\n \\item It is shown in \\cite[section 5]{hamblenramakrishna} that there is a finite set of trivial primes $X_2\\supset X_1$ disjoint from $S$ and a mod $p^3$ lift $\\rho_3$ of $\\rho_2$ which satisfies the following conditions\n \\begin{itemize}\n \\item $\\rho_3$ is irreducible, i.e. does not contain a free rank one Galois stable $\\text{W}(\\mathbb{F}_q)\/p^3$-submodule.\n \\item It is unramified outside $S\\cup X_2$.\n \\item The lift $\\rho_3$ is also arranged to satisfy conditions $\\mathcal{C}_v$ at each prime $v\\in S\\cup X_2$.\n \\end{itemize}\n This strategy for getting past mod-$p^2$ is based on the methods developed by Khare, Larsen and Ramakrishna in \\cite{KLR}.\n \\item At this stage, all that remains to be shown is that the set of primes $X_2$ may be further enlarged to a set of trivial primes $X$ which is disjoint from $S$ such that the Selmer group $H^1_{\\mathcal{N}}(\\operatorname{G}_{\\mathbb{Q},S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho})$ is equal to zero.\n \\end{enumerate}\n \\par The rest of the argument warrants some explanation. Since the Selmer group is zero, the map $\\op{res}_{S\\cup X}$ is an isomorphism. Suppose for $m\\geq 3$ and $\\rho_m$ is a mod $p^m$ lift of $\\rho_3$ which is unramified outside $S\\cup X$ and satisfies the conditions $\\mathcal{C}_v$ at each prime $v\\in S\\cup X$. We show that $\\rho_m$ may be lifted to $\\rho_{m+1}$ which satisfies the same conditions. Since the dual Selmer group is zero, so is $\\Sh^1_{S\\cup X}(\\operatorname{Ad}^0\\bar{\\rho}^*)$, and it follows from global-duality that $\\Sh^2_{S\\cup X}(\\operatorname{Ad}^0\\bar{\\rho})$ is zero. Since local condition $\\mathcal{C}_v$ is liftable, there are no local obstructions to lifting ${\\rho_m}$. The cohomological obstruction to lifting $\\rho_{m}$ to $\\rho_{m+1}$ is a class in $\\Sh^2_{S\\cup X}(\\operatorname{Ad}^0\\bar{\\rho})$ and hence is zero. As a result, $\\rho_m$ does lift one more step to $\\rho_{m+1}$. In order to complete the inductive argument, it is shown that $\\rho_{m+1}$ can be chosen to satisfy the conditions $\\mathcal{C}_v$. After picking a suitable global cohomology class $z\\in H^1(\\operatorname{G}_{\\mathbb{Q}, S\\cup X}, \\operatorname{Ad}^0\\bar{\\rho})$ and replacing $\\rho_{m+1}$ by its twist $(\\operatorname{Id}+p^{m}z)\\rho_{m+1}$, this may be arranged. At each prime $v\\in S\\cup X$, there is a cohomology class $z_v\\in H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ such that the twist $(\\operatorname{Id}+p^mz_v){\\rho_{m+1}}_{\\restriction \\operatorname{G}_v}$ satisfies $\\mathcal{C}_v$. Since we assume that $m\\geq 3$, we have that $\\mathcal{N}_v$ stabilizes $\\mathcal{C}_v$. For $v\\in S\\cup X$, the elements $z_v$ are defined modulo $\\mathcal{N}_v$. Since $\\operatorname{res}_{S\\cup X}$ is an isomorphism, the tuple\n $(z_v)\\in \\bigoplus_{v\\in S\\cup X} H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})\/\\mathcal{N}_v$ arises from a unique global cohomology class $z$ which is unramified outside $S\\cup X$. After replacing $\\rho_{m+1}$ by $(\\op{Id}+p^m z) \\rho_{m+1}$, it satisfies the conditions $\\mathcal{C}_v$ at each prime $v\\in S\\cup X$. This completes the inductive lifting argument.\n\\section{Preliminaries}\n\\par In this section, we prove a number of Galois theoretic results which will be applied in later sections. Let $M$ be a finite abelian group with $\\op{G}_{\\mathbb{Q}}$-action and $E$ be a number field. Denote by $E(M)$ the extension of $E$ \\textit{cut out} by $M$. In other words, it is the Galois extension of $E$ which is fixed by the kernel of the action of $\\op{G}_{E}$ on $M$. Let $M_1,\\dots, M_k$ be finite abelian groups on which $\\op{G}_{\\mathbb{Q}}$ acts. Denote by $E(M_1,\\dots, M_k)$ the composite of the fields $E(M_1)\\cdots E(M_k)$. Let $K:=\\mathbb{Q}(\\bar{\\rho}, \\mu_p)$ and $L:=\\mathbb{Q}(\\bar{\\rho})$ and set $\\op{G}':=\\op{Gal}(K\/\\mathbb{Q})$ and $\\op{G}:=\\op{Gal}(L\/\\mathbb{Q})$. Let $F$ be the subfield $\\mathbb{Q}(\\varphi_1,\\dots, \\varphi_n, \\bar{\\kappa})$ of $L$, where we recall that the characters $\\varphi_1,\\dots, \\varphi_n$ are as in $\\eqref{introducingbarrho}$. Denote by $N':=\\operatorname{Gal}(K\/F(\\mu_p))$ and $N:=\\operatorname{Gal}(L\/F)$. The groups $\\op{G},\\op{G}', N$ and $N'$ are depicted in the following field diagram\n\\begin{equation*}\n\\begin{tikzpicture}[node distance = 1.5cm, auto]\n \\node(Q) {$\\mathbb{Q}.$};\n \\node (L) [above of =Q]{$F$};\n \\node (E) [above of=L, right of=L] {$F(\\mu_p)$};\n \\node (F) [above of=L, left of =L] {$\\mathbb{Q}(\\bar{\\rho})$};\n \\node (K) [above of=E, left of=E] {$\\mathbb{Q}(\\bar{\\rho},\\mu_p)$};\n \\draw[-] (Q) to node {} (L);\n \\draw[-] (L) to node {} (E);\n \\draw[-] (L) to node {\\scriptsize$N$} (F);\n \\draw[-] (F) to node {} (K);\n \\draw[-] (E) to node [swap]{\\scriptsize$N'$} (K);\n \\end{tikzpicture}\n \\end{equation*}\n Condition $\\eqref{thc3}$ of Theorem $\\ref{main}$ asserts that the image of $\\bar{\\rho}$ contains $U_1(\\mathbb{F}_q)$. Therefore $N$ may be identified with $\\bar{\\rho}(N)=U_1(\\mathbb{F}_q)$. In particular the abelianization $N^{ab}$ may be identified with $U_1(\\mathbb{F}_q)\/U_2(\\mathbb{F}_q)$. Since $N$ is a $p$-group and $[F(\\mu_p):F]$ is coprime to $p$, it follows that $\\mathbb{Q}(\\bar{\\rho})$ and $F(\\mu_p)$ are linearly disjoint over $F$. It follows that $N$ is canonically isomorphic to $N'$. The inclusion of $\\mathcal{T}$ into $\\op{B}$ is a section of the quotient map $\\op{B}\\rightarrow \\mathcal{T}$. This induces a semi-direct product decomposition $\\op{B}=U_1\\rtimes \\mathcal{T}$. Let $\\mathcal{T}'$ be the intersection of the image of $\\bar{\\rho}$ with $\\mathcal{T}$. The group $\\op{G}$ may be identified with the image of $\\bar{\\rho}$. It is easy to see that $\\op{G}$ has a semi-direct product decomposition $\\op{G}\\simeq \\bar{\\rho}(\\op{G})=U_1(\\mathbb{F}_q)\\rtimes \\mathcal{T}'$.\\begin{Lemma}\\label{l1}\nSuppose $0<|k|\\leq 2n-1$, there is an isomorphism of $\\mathbb{F}_q[\\operatorname{G}_{\\mathbb{Q},S}]$-modules\n\\[(\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1}\\simeq \\bigoplus_{ht \\lambda=k} \\mathbb{F}_q(\\sigma_{\\lambda}).\\]On the other hand, \\[(\\operatorname{Ad}^0\\bar{\\rho})_0\/(\\operatorname{Ad}^0\\bar{\\rho})_1=\\mathfrak{b}\/\\mathfrak{n}\\simeq \\mathfrak{t}.\\]\n\\end{Lemma}\n\\begin{proof}\nLet $\\lambda$ be of height $k$. Let $X\\in (\\operatorname{Ad}^0\\bar{\\rho})_{\\lambda}$, we observe that\n\\[\\bar{\\rho}(g)\\cdot X \\cdot \\bar{\\rho}(g)^{-1}\\equiv \\sigma_{\\lambda}(g) X \\mod{(\\operatorname{Ad}^0\\bar{\\rho})_{k+1}}.\\] Likewise, for $X\\in \\mathfrak{b}$, the conjugation action on $X$ modulo $\\mathfrak{n}$ is trivial.\n\\end{proof}\n\n\\par Let $\\zeta$ be a non-zero element of $\\mathbb{F}_q(\\bar{\\chi})$. For $i=1,\\dots, n$, set $\\delta_{i,j}=\\zeta$ if $i=j$ and $0$ otherwise. Likewise, for roots $\\lambda$ and $\\gamma$, set $\\delta_{\\lambda, \\gamma}$ to equal $\\zeta$ if $\\lambda=\\gamma$ and $0$ otherwise. Denote by $X_{\\lambda}^*$ and $H_i^*$ the elements of $\\operatorname{Ad}^0\\bar{\\rho}^*$ which are defined by the following relations: \\begin{equation}\\label{XHdual}\\begin{split}& X_{\\lambda}^*(X_{\\gamma})=\\delta_{\\lambda,\\gamma}\\text{ and }X_{\\lambda}^*(H_i)=0,\\\\\n & H_i^*(X_{\\lambda})=0 \\text{ and }H_i^*(H_j)=\\delta_{i,j}.\\end{split}\\end{equation} The element $H_i^*\\in \\operatorname{Ad}^0\\bar{\\rho}^*$ should not be confused with $L_i\\in \\mathfrak{t}^*$. Let $(\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}}$ be the span of $H_1^*,\\dots, H_n^*$ and $(\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{\\lambda}}$ the span of $X_{-\\lambda}^*$. Let $P$ be a Galois-stable subgroup of $\\operatorname{Ad}^0\\bar{\\rho}^*$. Associated to $P$ are its eigenspaces for the action of $\\bar{\\rho}^{-1}(\\mathcal{T})$. For $\\lambda\\in \\Phi\\cup \\{1\\}$, set $P_{\\bar{\\chi}\\sigma_{\\lambda}}$ to be the intersection of $P$ with $(\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{\\lambda}}$. Likewise, associate to a Galois stable subgroup $Q\\subseteq \\operatorname{Ad}^0\\bar{\\rho}$, an eigenspace $Q_{\\sigma_{\\lambda}}$. Define $Q_1$ to be the intersection $Q\\cap \\mathfrak{t}$. For $\\lambda \\in \\Phi$, denote by $Q_{\\sigma_{\\lambda}}$ the intersection $Q\\cap (\\operatorname{Ad}^0\\bar{\\rho})_{\\lambda}$.\n \\par The representation $\\bar{\\rho}$ factors through $\\op{G}$. Let $\\mathbb{T}$ be the subgroup of $\\op{G}'$ consisting of $g$ such that $\\bar{\\rho}(g)\\in \\mathcal{T}$. For $\\lambda\\in \\Phi\\cup \\{1\\}$, $\\mathbb{T}$ acts on $P_{\\bar{\\chi}\\sigma_{\\lambda}}$ by the character $\\bar{\\chi}\\sigma_{\\lambda}$ and on $Q_{\\sigma_{\\lambda}}$ by the character $\\sigma_{\\lambda}$. Since the characters $\\sigma_{\\lambda}$ are assumed to be distinct, it is easy to see that\n \\[P_{\\bar{\\chi}\\sigma_{\\lambda}}=\\{p\\in P| t\\cdot p=\\bar{\\chi}\\sigma_{\\lambda}(t)p\\text{ for }t\\in \\mathbb{T}\\}\\]\n \\[Q_{\\sigma_{\\lambda}}=\\{q\\in Q| t\\cdot q=\\sigma_{\\lambda}(t)q\\text{ for }t\\in \\mathbb{T}\\}.\\] The order of $\\mathbb{T}$ is coprime to $p$, hence Maschke's theorem asserts that any finite dimensional $\\mathbb{F}_p[\\op{G}']$-module $M$ decomposes into a direct sum $M=\\bigoplus_{\\tau} M_{\\tau}$, where $\\tau$ is a character of $\\mathbb{T}$ and $M_{\\tau}$ is the $\\tau$-eigenspace $M_{\\tau}:=\\{m\\in M| g\\cdot m=\\tau(g) m\\}$. Thus, we have the next Lemma, which follows from the discussion above.\n \\begin{Lemma}\\label{Pdecomposition} Let $P\\subseteq \\operatorname{Ad}^0\\bar{\\rho}^*$ and $Q\\subseteq \\operatorname{Ad}^0\\bar{\\rho}$ be Galois-stable subgroups.\n \\begin{enumerate}\n \\item As a $\\mathbb{T}$-module, $P$ decomposes into a direct sum of eigenspaces: \\[P=\\bigoplus_{\\lambda\\in \\Phi\\cup\\{1\\}} P_{\\bar{\\chi}\\sigma_{\\lambda}}.\\]\n \\item As a $\\mathbb{T}$-module, $Q$ decomposes into a direct sum of eigenspaces:\n \\[Q=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}} Q_{\\sigma_{\\lambda}}.\\]\n \\end{enumerate}\n \\end{Lemma}\n \n \\begin{comment} We prove the statement for $P\\subseteq \\operatorname{Ad}^0\\bar{\\rho}^*$. The proof is identical for $Q\\subseteq \\operatorname{Ad}^0\\bar{\\rho}$. Let $\\Phi'$ denote $\\Phi\\cup \\{1\\}$. As $\\lambda$ ranges over $\\Phi'$, the characters $\\sigma_{\\lambda}$ are all assumed to be distinct. For $p\\in P$, represent \\[p=\\sum_{\\lambda\\in \\Phi'} p_{\\lambda}\\] where $p_{\\lambda}$ is the projection of $p$ to $(\\operatorname{Ad}^0\\bar{\\rho})_{\\bar{\\chi} \\sigma_{\\lambda}}$. For $\\lambda\\in \\Phi'$, let $P_{\\lambda}$ denote the projection of $P$ to $(\\operatorname{Ad}^0\\bar{\\rho})_{\\bar{\\chi}\\sigma_{\\lambda}}$. Let $\\Phi_0\\subset \\Phi'$ consist of $\\lambda$ such that the intersection $P_{\\bar{\\chi}\\sigma_{\\lambda}}$ is not equal $P_{\\lambda}$. We show that $\\Phi_0$ is the empty set. Assume by way of contradiction that $\\Phi_0$ is not empty. Let $\\mathcal{P}$ consist of $p\\in P$ for which $p_{\\lambda}\\notin P$ for some $\\lambda\\in \\Phi_0$. Since $\\Phi_0$ is nonempty, so is $\\mathcal{P}$. For $p\\in \\mathcal{P}$, set $\\lambda_0(p)$ and $\\lambda_1(p)$ to be the minimal and maximal elements of $\\Omega_p:=\\{\\lambda\\in \\Phi'\\mid p_{\\lambda}\\notin P\\}$ respectively. By assumption, $p_{\\lambda}\\in P$ for $p\\notin \\Omega_p$. If $\\lambda_1(p)=\\lambda_0(p)$ then $p_{\\lambda_0}=p-\\sum_{\\lambda\\notin\\Omega_p}p_{\\lambda}$ is contained in $P$, which is clearly not the case as $\\lambda_0(p)$ is in $\\Omega_p$. Therefore, $\\lambda_0(p)$ is not equal to $\\lambda_1(p)$. Let $q\\in \\mathcal{P}$ be an element for which $\\lambda_0(p)\\leq \\lambda_0(q)$ for $p\\in \\mathcal{P}$. Set $\\lambda_i:=\\lambda_i(q)$ for $i=0,1$. It follows from conditions $\\eqref{thc3}$ and $\\eqref{thc4}$ of Theorem $\\ref{main}$, that there exists $g$ such that $\\bar{\\rho}(g)\\in \\mathcal{T}$ and $\\sigma_{\\lambda_0}(g)\\neq \\sigma_{\\lambda_1}(g)$. The element $q'$ defined by\n \\[q':=g\\cdot q-(\\bar{\\chi}\\sigma_{\\lambda_0})(g)q=\\sum_{\\lambda} \\left((\\bar{\\chi}\\sigma_{\\lambda})(g)-(\\bar{\\chi}\\sigma_{\\lambda_0})(g)\\right)q_{\\lambda}\\]\n is contained in $\\mathcal{P}$ and $\\lambda_1(q')=\\lambda_1$. Furthermore, it has been arranged that $q'_{\\lambda_0}=0$ and $q'_{\\lambda}$ is a scalar multiple of $q_{\\lambda}$ for every root $\\lambda$. Hence, we deduce that $\\lambda_0(q')>\\lambda_0$, which is a contradiction to the maximality of $\\lambda_0=\\lambda_0(q)$. Therefore $\\Phi_0$ is empty and the proof is complete.\\end{comment}\n Set the height of the formal symbol \"$1$\" to be equal to zero. Fix a total order on $\\Phi\\cup \\{1\\}$ such that $\\op{ht}(\\lambda)\\leq \\op{ht}(\\gamma)$ if $\\lambda\\leq \\gamma$. \n\\begin{Lemma}\\label{mainin}\n\\begin{enumerate} Let $P$ be a non-zero Galois stable subgroup of $\\operatorname{Ad}^0\\bar{\\rho}^*$. Let $Q$ be a non-zero Galois stable subgroup of $\\operatorname{Ad}^0\\bar{\\rho}$. The following statements hold:\n \\item \\label{43c1} $P_{\\bar{\\chi}\\sigma_{2L_1}}$ is equal to $(\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{2L_1}}$,\n \\item \\label{43c2} $Q_{\\sigma_{2L_1}}$ is equal to $(\\operatorname{Ad}^0\\bar{\\rho})_{\\sigma_{2L_1}}$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nIt follows from Lemma $\\ref{Pdecomposition}$ that $P$ decomposes into $\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}} P_{\\bar{\\chi}\\sigma_{\\lambda}}$. By condition $\\eqref{thc5}$ of Theorem $\\ref{main}$, the image of $\\sigma_{2L_1}$ spans $\\mathbb{F}_q$. Since $\\bar{\\chi}$, takes values in $\\mathbb{F}_p^{\\times}$, the same is true for the image of $\\bar{\\chi}\\sigma_{2L_1}$. Therefore, if $P_{\\bar{\\chi}\\sigma_{2L_1}}$ is not zero, then $P_{\\bar{\\chi}\\sigma_{2L_1}}=(\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{2L_1}}$. Suppose by way of contradiction that $P_{\\bar{\\chi}\\sigma_{2L_1}}=0$. Then we may choose $\\gamma\\in \\Phi\\cup \\{1\\}$ such that:\n\\begin{itemize}\n \\item $P_{\\bar{\\chi}\\sigma_{\\gamma}}\\neq 0$,\n \\item $P_{\\bar{\\chi}\\sigma_{\\lambda}}=0$ for all $\\lambda>\\gamma$.\n\\end{itemize} By assumption, $\\gamma$ is not the maximal root $2L_1$. There exists $\\gamma_1\\in \\Phi$ such that the difference $\\mu:=\\gamma_1-\\gamma$ is in $ \\Phi^{+}$. This can be shown by considering all possibilities for $\\gamma$:\n\\begin{enumerate}\n\\item $\\gamma=1$, then let $\\mu=\\gamma_1$ be any positive root,\n \\item $\\gamma=2L_i$ for $i> 1$, then $\\mu=L_{i-1}-L_i$ and $\\gamma_1=L_{i-1}+L_i$,\n \\item $\\gamma=-2L_i$ for $i>1$, then $\\mu=L_{i-1}+L_i$ and $\\gamma_1=L_{i-1}-L_i$,\n \\item $\\gamma=-2L_1$, then $\\mu=L_{1}+L_2$ and $\\gamma_1=-L_{1}+L_2$,\n \\item $\\gamma=L_i+L_j$ for $i2n$. Note that $\\mu$ is a positive root and hence, \n\\[g^{-1}\\cdot Y-Y=[X,Y]\\mod{(\\operatorname{Ad}^0\\bar{\\rho})_{\\op{ht}(\\mu)+k+1}}.\\]Note that since $-\\gamma_1+\\mu=-\\gamma$ is a root, \\[[(\\operatorname{Ad}^0\\bar{\\rho})_{\\mu},(\\operatorname{Ad}^0\\bar{\\rho})_{-\\gamma_1}]=(\\operatorname{Ad}^0\\bar{\\rho})_{-\\gamma}\\] (cf. \\cite[p. 39]{humphreys}). Letting $Y$ run through an appropriate basis of $\\operatorname{Ad}^0\\bar{\\rho}$, it follows from the above identity that $g\\cdot p-p$ can be expressed as a sum $a+b$ where $a\\neq 0$ is in $P_{\\bar{\\chi}\\sigma_{\\gamma_1}}$ and $b\\in \\bigoplus _{\\lambda>\\gamma_1}P_{\\bar{\\chi}\\sigma_{\\lambda}}$. In particular, this shows that the projection of $g\\cdot p$ to $P_{\\bar{\\chi}\\sigma_{\\gamma_1}}$ is non-zero.\n\\par Since $\\gamma_1=\\mu+\\gamma$ and $\\mu\\in \\Phi^+$, the height of $\\gamma_1$ is strictly larger than the height of $\\gamma$. As a result, $\\gamma_1>\\gamma$. Therefore, the subgroup $P_{\\bar{\\chi}\\sigma_{\\gamma_1}}=0$. This contradiction shows that $\\gamma=2L_1$ and $P_{\\bar{\\chi}\\sigma_{2L_1}}\\neq 0$. This concludes part $\\eqref{43c1}$. The proof of part $\\eqref{43c2}$ is similar and is left to the reader.\n\\end{proof}\nFor $\\lambda\\in \\Phi\\cup \\{1\\}$, set \\[N_{\\lambda}=\\begin{cases}\n1\\text{ if }\\lambda\\in \\Delta,\\\\\n0\\text{ otherwise.}\n\\end{cases}\\]\n\\begin{Lemma}\\label{l2}\nLet $\\lambda\\in \\Phi\\cup \\{1\\}$ and $\\sigma_{\\lambda}$ the associated character. The following assertions are satisfied:\n\\begin{enumerate}\n\\item\\label{l2c1}\n$\\dim \\rm{Hom}( N, \\mathbb{F}_q(\\sigma_{\\lambda}))^{\\op{G}\/N}=N_{\\lambda}$.\n\\item\\label{l2c2}\n$\\dim \\rm{Hom}(N', \\mathbb{F}_q(\\sigma_{\\lambda})^*)^{\\op{G}'\/N'}=0$.\n\\item\\label{l2c3} For $k\\neq 1$, \\[ H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})=0.\\]\nOn the other hand,\n\\[h^1 (\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_2)= \\dim \\mathfrak{t}.\\]\n\\item\\label{l2c4} For all $k$,\n $h^1(\\op{G}', ((\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})^*)=0$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nBy condition $\\ref{thc3}$ of Theorem $\\ref{main}$, $N$ may be identified with $U_1(\\mathbb{F}_q)$. The abelianization $N^{ab}$ is \\[U_1\/U_2(\\mathbb{F}_q)\\simeq \\bigoplus_{\\gamma\\in \\Delta} \\mathbb{F}_q(\\sigma_{\\gamma}).\\]\nBy condition $\\eqref{thc5}$ of Theorem $\\ref{main}$, any $\\op{G}\/N$ equivariant map $F:\\mathbb{F}_q(\\sigma_{\\gamma})\\rightarrow \\mathbb{F}_q(\\sigma_{\\lambda})$ is determined by the image of any nonzero element, hence \\[\\dim \\op{Hom}(\\mathbb{F}_q(\\sigma_{\\gamma}),\\mathbb{F}_q(\\sigma_{\\lambda}))^{\\op{G}\/N}\\leq 1.\\] We have that \\[F(\\sigma_{\\gamma}(g_1)\\sigma_{\\gamma}(g_2))=\\sigma_{\\lambda}(g_1)\\sigma_{\\lambda}(g_2)F(1)=F(\\sigma_{\\gamma}(g_1))F(\\sigma_{\\gamma}(g_2))F(1).\\] Since the image of $\\sigma_{\\lambda}$ spans $\\mathbb{F}_q$, it follows that $F$ is a scalar multiple of an element of $\\op{Gal}(\\mathbb{F}_q\/\\mathbb{F}_p)$. By assumption, if $\\lambda\\neq \\gamma$, the characters $\\sigma_{\\lambda}$ and $\\sigma_{\\gamma}$ are not equal up to a twist of $\\op{Gal}(\\mathbb{F}_q\/\\mathbb{F}_p)$. Therefore,\n\\[\\rm{Hom}( \\mathbb{F}_q(\\sigma_{\\gamma}), \\mathbb{F}_q(\\sigma_{\\lambda}))^{\\op{G}\/N}=\\begin{cases} \\mathbb{F}_q \\mbox{ if $\\sigma_{\\lambda}=\\sigma_{\\gamma}$,}\\\\\n0 \\mbox{ otherwise,}\n\\end{cases}\\]and part $\\eqref{l2c1}$ follows this.\n\nObserve that $N'$ is isomorphic to $N$ and $\\op{G}'\/N'$ is the Galois group $\\op{Gal}(\\mathbb{Q}(\\{\\varphi_i\\}, \\bar{\\kappa},\\bar{\\chi})\/\\mathbb{Q})$. By condition $\\eqref{thc4}$, the characters $\\sigma_{\\gamma}$ and $\\sigma_{\\lambda}^*=\\bar{\\chi}\\sigma_{-\\lambda}$ are not equivalent up to a twist of $\\op{Gal}(\\mathbb{F}_q\/\\mathbb{F}_p)$. Part $\\eqref{l2c2}$ follows via the same reasoning as part $\\eqref{l2c1}$.\n\\par The order of $\\op{G}\/N$ is coprime to $p$. By inflation-restriction and part $(\\ref{l2c1})$\n\\begin{equation*}\n\\begin{split}\n& \\dim H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})\\\\\n= & \\dim \\rm{Hom}(N, (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})^{\\op{G}\/N}\\\\\n= & \\sum_{\\lambda\\in\\Phi, ht(\\lambda)=k} \\dim \\rm{Hom}(N, \\mathbb{F}_q(\\sigma_{\\lambda}))^{\\op{G}\/N}\\\\\n= & \\sum_{\\lambda\\in \\Phi, ht(\\lambda)=k} N_{\\lambda}.\\\\\n\\end{split}\n\\end{equation*}\n\n\nIt follows that if $k\\neq 1$, \n\\[ H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})=0\\]and that\n\\[h^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_{2})=\\# \\Delta =n=\\dim \\mathfrak{t}.\\] This concludes part $\\eqref{l2c3}$.\n\\par The order of $\\op{G}'\/N'$ is coprime to $p$. Therefore, by inflation-restriction,\n\\begin{equation*}\n\\begin{split}\n& \\dim H^1(\\op{G}', (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})\\\\\n= & \\dim \\rm{Hom}(N', (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})^{\\op{G}'\/N'}\\\\\n= & \\sum_{\\lambda\\in\\Phi, ht(\\lambda)=k} \\dim \\rm{Hom}(N', \\mathbb{F}_q(\\sigma_{\\lambda})^*)^{\\op{G}'\/N'}\\\\\n=& 0.\n\\end{split}\n\\end{equation*} This concludes the proof of part $\\eqref{l2c4}$.\n\\end{proof}\n\\begin{Def}\\label{perpdef}\nLet $(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}\\subset \\operatorname{Ad}^0\\bar{\\rho}^*$ be the subspace of $\\operatorname{Ad}^0\\bar{\\rho}^*$ consisting of $f\\in \\operatorname{Ad}^0\\bar{\\rho}^*$ for which $f_{\\restriction (\\operatorname{Ad}^0\\bar{\\rho})_k}=0$.\n\\end{Def}\n\\begin{Remark}\nFor $k>-2n+1$, the submodule $(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}\\neq 0$ and by Lemma $\\ref{mainin}$, \\[(\\operatorname{Ad}^0\\bar{\\rho})_{k,\\bar{\\chi}\\sigma_{2L_1}}^{\\perp}\\simeq(\\operatorname{Ad}^0\\bar{\\rho})_{\\bar{\\chi}\\sigma_{2L_1}}^*.\\] \n\\end{Remark}\n\\begin{Lemma}\\label{l3}\nLet $k$ be an integer,\n\\begin{enumerate}\n\\item\\label{l3c1}\n$H^1(\\op{G},\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0$ and $H^1(\\op{G}',\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0$,\n\\item\\label{l3c2}\n$H^1(\\op{G}',(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})=0$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nWe begin with the proof of part $\\eqref{l3c1}$. Consider the case when $k\\leq 1$. By part $\\eqref{l2c3}$ of Lemma $\\ref{l2}$, for $i\\leq 1$, \n\\begin{equation*}\nH^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_{i-1}\/(\\operatorname{Ad}^0\\bar{\\rho})_{i})=0.\n\\end{equation*}\nand hence there is an injection\n\\begin{equation*}H^1(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_i)\\hookrightarrow H^1(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_{i-1}).\\end{equation*}\nWe deduce that $H^1(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0$.\n\\par Next consider the case $k>1$. Associated to \n\\begin{equation*}\n0\\rightarrow (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_k\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1\\rightarrow 0 \\end{equation*}\nis the long exact sequence in cohomology. It follows from $\\ref{mainin}$ that any non-zero submodule of $\\operatorname{Ad}^0\\bar{\\rho}$ has a non-trivial $\\sigma_{2L_1}$ eigenspace for the $\\mathbb{T}$-action. As a consequence, $H^0(\\operatorname{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho})=0$. Since Lemma $\\ref{l2}$ asserts that \\[H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_k\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})=0,\\] we have that $H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_{k+1})$ surjects onto $H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_{k})$ for $k>1$. As a result, $H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho})$ surjects onto $H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_{k})$, and therefore, \\[H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_{k})=0.\\]\n\\par Since it has been shown that $H^1(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1)=0$, we have a short exact sequence\n\\[0\\rightarrow H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1)\\rightarrow H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_k)\\rightarrow H^1(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)\\rightarrow 0.\\]It suffices to show that\n\\[\\dim H^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1)\\geq \\dim H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_k).\\]\nCondition $\\eqref{thc4}$ of Theorem $\\ref{main}$ stipulates that for $\\lambda \\in \\Phi$, $\\sigma_{\\lambda}$ is not equal to $\\sigma_1=1$. Therefore for $i\\leq 0$,\n\\begin{equation*}\nH^0(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_{i-1}\/(\\operatorname{Ad}^0\\bar{\\rho})_{i})=\\bigoplus_{ \\text{ht} \\gamma=i-1} H^0(\\op{G}, \\mathbb{F}_q(\\sigma_{\\gamma}))=0.\n\\end{equation*}\nFor $i\\leq 0$, we deduce that\n\\begin{equation*}\nH^0(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_i\/(\\operatorname{Ad}^0\\bar{\\rho})_1)\\xrightarrow{\\sim} H^0(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_{i-1}\/(\\operatorname{Ad}^0\\bar{\\rho})_1).\n\\end{equation*}\nComposing these isomorphisms we have that\n\\begin{equation*}\n(\\operatorname{Ad}^0\\bar{\\rho})_0\/(\\operatorname{Ad}^0\\bar{\\rho})_1=H^0(\\op{G},(\\operatorname{Ad}^0\\bar{\\rho})_0\/(\\operatorname{Ad}^0\\bar{\\rho})_1)\\xrightarrow{\\sim} H^0(\\op{G},\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1).\n\\end{equation*}\nWe have deduced that\n\\begin{equation*}\nh^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1)=\\dim (\\operatorname{Ad}^0\\bar{\\rho})_0\/(\\operatorname{Ad}^0\\bar{\\rho})_1=\\dim \\mathfrak{t}.\n\\end{equation*}\nBy Lemma $\\ref{l2}$ part $\\eqref{l2c3}$,\n\\begin{equation*}\nh^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_2)=\\dim \\mathfrak{t}=h^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_1).\n\\end{equation*}\nBy Lemma $\\ref{l2}$, for $i\\geq 2$, we have that \\begin{equation*}H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_i\/(\\operatorname{Ad}^0\\bar{\\rho})_{i+1})=0.\\end{equation*} and it follows that $H^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_2\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0$.\nHence it follows that\n\\begin{equation}\\label{deltaiso}h^1(\\op{G},(\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_k)\\leq h^1(\\op{G}, (\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_2).\\end{equation}We conclude that\n\\[h^1(\\op{G},(\\operatorname{Ad}^0\\bar{\\rho})_1\/(\\operatorname{Ad}^0\\bar{\\rho})_k)\\leq h^0(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/ (\\operatorname{Ad}^0\\bar{\\rho})_1).\\]Therefore we conclude that $H^1(\\op{G}, \\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0$.\n\\par Since $[L:K]$ is coprime to $p$, from a direct application of inflation-restriction it follows that $H^1(\\op{G}',\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0$.\nWe have proved part $(\\ref{l3c1})$.\n\\par Consider the short exact sequence \\[0\\rightarrow (\\operatorname{Ad}^0\\bar{\\rho})_{i-1}^{\\perp}\\rightarrow (\\operatorname{Ad}^0\\bar{\\rho})_i^{\\perp} \\rightarrow ((\\operatorname{Ad}^0\\bar{\\rho})_{i-1}\/(\\operatorname{Ad}^0\\bar{\\rho})_{i})^*\\rightarrow 0\\] and the associated sequence in cohomology. By Lemma $\\ref{l2}$, \n\\begin{equation*}H^1(\\op{G}', ((\\operatorname{Ad}^0\\bar{\\rho})_{i-1}\/(\\operatorname{Ad}^0\\bar{\\rho})_{i})^*)=0\n\\end{equation*}\nfrom which we deduce that \n\\begin{equation*}\nH^1(\\op{G}',(\\operatorname{Ad}^0\\bar{\\rho})_{i-1}^{\\perp})\\rightarrow H^1(\\op{G}', (\\operatorname{Ad}^0\\bar{\\rho})_i^{\\perp})\n\\end{equation*}\nis a surjection for all $i$. As \\[(\\operatorname{Ad}^0\\bar{\\rho})_{-2n+1}^{\\perp}\\simeq (\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_{-2n+1})^*=0\\] we deduce that $H^1(\\op{G}', (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})=0$.\n\\end{proof}\n\\par For $\\psi$ in $H^1(\\operatorname{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho}^*)$, the restriction $\\psi_{\\restriction \\op{G}_K}:\\op{G}_K\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}^*$ is a homomorphism since the action of $\\op{G}_K$ on $\\operatorname{Ad}^0\\bar{\\rho}^*$ is trivial. Let $K_{\\psi}\\supseteq K$ be the extension \\textit{cut out} by $\\psi$, i.e. $K_{\\psi}$ is the smallest extension of $K$ which is fixed by the kernel of $\\psi_{\\restriction \\op{G}_K}$. Identify $\\op{Gal}(K_{\\psi}\/K)$ with $\\psi(\\op{G}_K)\\subseteq \\operatorname{Ad}^0\\bar{\\rho}^*$ and let $J_{\\psi}\\subseteq K_{\\psi}$ be the subfield for which $\\op{Gal}(K_{\\psi}\/J_{\\psi})\\simeq \\psi(\\op{G}_K)_{\\bar{\\chi}\\sigma_{2L_1}}$. Likewise for $f$ in $H^1(\\operatorname{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho})$ denote by $L_f$, the extension of $L$ cut out by $f$. Set $K_f$ to be the composite of $L_f$ with $K$. Since $p\\nmid [K:L]$, we have that $\\op{Gal}(K_f\/K)\\simeq \\op{Gal}(L_f\/L)$.\n\\begin{Lemma}\\label{l4}Let $\\mathcal{J}\\supseteq S$ be a finite set of primes. Let $f\\in H^1(\\op{G}_{\\mathbb{Q},\\mathcal{J}}, \\operatorname{Ad}^0\\bar{\\rho})$ and $\\psi\\in H^1(\\op{G}_{\\mathbb{Q},\\mathcal{J}}, \\operatorname{Ad}^0\\bar{\\rho})$ be a non-zero cohomology classes. Then the following assertions are satisfied:\n\\begin{enumerate}\n\\item\\label{l4p1} $L_{f}\\supsetneq L$ (equivalently, $K_f\\supsetneq K$),\n\\item\\label{l4p2}\n$K_{\\psi}\\supsetneq J_{\\psi}$, in particular, $K_{\\psi}\\supsetneq K$.\n\\end{enumerate}\n\\begin{proof}\n\\par For part $\\eqref{l4p2}$, recall that Lemma $\\ref{l3}$ asserts that\n$H^1(\\op{G}', \\operatorname{Ad}^0\\bar{\\rho}^*)=0$. Therefore, the restriction $\\psi_{\\restriction \\op{G}_K}$ is not zero. This shows that $K_{\\psi}\\supsetneq K$. That $K_{\\psi}$ strictly contains $J_{\\psi}$ is a direct consequence of Lemma $\\ref{mainin}$. Part $\\eqref{l4p1}$ also follows from Lemma $\\ref{l3}$.\n\\end{proof}\n\\end{Lemma}\n\n\\begin{Lemma}Let $P\\subseteq \\operatorname{Ad}^0\\bar{\\rho}^*$ (resp. $Q\\subseteq \\operatorname{Ad}^0\\bar{\\rho}$) be a nonzero Galois-stable subgroup and $\\iota_P:P\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}^*$ (resp. $\\iota_Q:Q\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}$) denote the inclusion. Then, \\label{y1}\n\\begin{enumerate}\n \\item\\label{48c1} \n$\\rm{Hom}_{\\mathbb{F}_p}(P,\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}=\\mathbb{F}_q\\cdot \\iota_P$,\n\\item\\label{48c2} \n$\\rm{Hom}_{\\mathbb{F}_p}(Q,\\operatorname{Ad}^0\\bar{\\rho}^*)^{G}=\\mathbb{F}_q\\cdot \\iota_Q$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nWe prove part $\\eqref{48c1}$, the proof of $\\eqref{48c2}$ is similar. Let $\\Phi'=\\Phi\\cup \\{1\\}$ and $m:=\\# \\Phi'$. Enumerate $\\Phi'=\\{\\gamma_1,\\dots, \\gamma_m\\}$, so that $\\gamma_i>\\gamma_j$ if $ii$ be such that $\\varphi(p)\\in W_j$ and $\\varphi(p)\\notin W_{j-1}$. It follows from conditions $\\eqref{thc3}$ and $\\eqref{thc4}$ of Theorem $\\ref{main}$, that there exists $g\\in \\mathbb{T}$ such that $\\sigma_{\\gamma_i}(g)\\neq \\sigma_{\\gamma_j}(g)$. Express $\\varphi(p)=x_j+x_{j-1}$, where $x_{j}\\in (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi} \\sigma_{\\gamma_j}}$ and $x_{j-1}\\in W_{j-1}$. We have that \n\\[g\\varphi(p)=\\bar{\\chi}(g)\\sigma_{\\gamma_j}(g)x_{j}+g\\cdot x_{j-1}=\\varphi(gp)=\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g)(x_j+x_{j-1}).\\]\nWe have that\n\\[\\bar{\\chi}(g)(\\sigma_{\\gamma_j}(g)-\\sigma_{\\gamma_i}(g))x_{j}=\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g)x_{j-1}-g\\cdot x_{j-1}\\]is contained in $W_{j-1}$. This is contradiction, we deduce that $\\varphi(P_i)\\subseteq W_i$. We show that $\\varphi(p)=\\beta p$ for $p\\in P_{\\bar{\\chi}\\sigma_{\\gamma_i}}$ and this shall complete the inductive step. Write $\\varphi(p)=z_i+z_{i-1}$ where $z_i\\in (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{\\gamma_i}}$ and $z_{i-1}\\in W_{i-1}$. Let $g\\in \\mathbb{T}$, we have that \\[\\varphi(g p)=\\varphi(\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g) p)=\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g)\\varphi( p)=\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g)z_i+\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g) z_{i-1}.\\] We have that\n\\[g\\varphi(p)=g\\cdot z_i+g \\cdot z_{i-1}=\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g)z_i+g\\cdot z_{i-1}.\\]\nTherefore, we deduce that \n\\[g\\cdot z_{i-1}=\\bar{\\chi}(g)\\sigma_{\\gamma_i}(g) z_{i-1}.\\] We show that $z_{i-1}=0$. If $z_{i-1}\\neq 0$, there exists $k1$, we have that ${P'}_{\\bar{\\chi}\\sigma_{2L_1}}=0$ and therefore Lemma $\\ref{mainin}$, asserts that $P'=0$. We conclude that $\\varphi(p)=z_i=\\beta p$. This concludes the induction step and the proof of the result.\n\\end{comment}\n\\end{proof}\n\\begin{Cor}\\label{Coradd}\nThe following statements hold:\n\\begin{enumerate}\n \\item\\label{coradd1}\n let $P_1$ and $P_2$ be Galois-stable subgroups of $\\operatorname{Ad}^0\\bar{\\rho}^*$ such that there is an isomorphism $\\phi:P_1\\xrightarrow{\\sim} P_2$ of Galois modules. Then $P_1=P_2$ and $\\phi$ is multiplication by a scalar.\n \\item \\label{coradd2}\n Let $Q_1$ and $Q_2$ be Galois-stable subgroups of $\\operatorname{Ad}^0\\bar{\\rho}$ such that there is an isomorphism $\\phi:Q_1\\xrightarrow{\\sim} Q_2$ of Galois modules. Then $Q_1=Q_2$ and $\\phi$ is multiplication by a scalar.\n\\end{enumerate}\n\\end{Cor}\n\\begin{proof}\nWe prove part $\\eqref{coradd1}$, part $\\eqref{coradd2}$ is identical. Let $\\iota_{P_i}:P_i\\hookrightarrow \\operatorname{Ad}^0\\bar{\\rho}^*$ be the inclusion. By Proposition $\\ref{y1}$, the two inclusions $\\iota_{P_1}$ and $\\iota_{P_2}\\circ \\phi$ are the same upto a scalar. The assertion follows.\n\\end{proof}\nLet $Q$ be a $\\op{G}$-submodule of $\\operatorname{Ad}^0\\bar{\\rho}$, by Lemma $\\ref{Pdecomposition}$, the projection of $Q$ to $(\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}$ equals $Q_{\\sigma_{-2L_1}}$. For convenience of notation, let $Q_{-2L_1}$ denote $Q_{\\sigma_{-2L_1}}$.\n\\begin{Lemma}\\label{fullrankLemma}\nLet $Q$ be a Galois-stable submodule of $\\operatorname{Ad}^0\\bar{\\rho}$ for which $ Q_{-2L_1}\\neq 0$, then $Q=\\operatorname{Ad}^0\\bar{\\rho}$.\n\\end{Lemma}\n\\begin{proof}\nLet $P:=\\{\\gamma\\in \\operatorname{Ad}^0\\bar{\\rho}^*\\mid \\gamma(x)=0 \\text{ for }x\\in Q\\}$. The assumption on $Q$ implies that $P_{\\bar{\\chi}\\sigma_{2L_1}}\\neq (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{2L_1}}$. Since the image of $\\bar{\\chi}\\sigma_{2L_1}$ spans $\\mathbb{F}_q$, $P_{\\bar{\\chi}\\sigma_{2L_1}}\\neq (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{2L_1}}$ implies that $P_{\\bar{\\chi}\\sigma_{2L_1}}=0$ By Lemma $\\ref{mainin}$, $P=0$, and therefore, $Q=\\operatorname{Ad}^0\\bar{\\rho}$.\n\\end{proof}\n\n\\begin{Lemma}\\label{22Dec5} The following statements hold:\n\\begin{enumerate}\n \\item\\label{411c1} the fields $K=\\mathbb{Q}(\\bar{\\rho},\\mu_p)$ and $\\mathbb{Q}(\\mu_{p^2})$ are linearly disjoint over $\\mathbb{Q}(\\mu_p)$.\n \\item\\label{411c2} Let $\\mathcal{J}\\supseteq S$ be a finite set of prime numbers, $\\psi_1,\\dots, \\psi_t\\in H^1(\\operatorname{G}_{\\mathbb{Q},\\mathcal{J}},\\operatorname{Ad}^0\\bar{\\rho}^*)$ and set $K_j:=K_{\\psi_j}$ for $j=1,\\dots, t$. Then the composite $K_1\\cdots K_t$ and $\\mathbb{Q}(\\mu_{p^2})$ are linearly disjoint over $\\mathbb{Q}(\\mu_p)$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nSuppose by way of contradiction that $\\mathbb{Q}(\\mu_{p^2})\\subseteq K$. Set $V:=\\op{Gal}(K\/F(\\mu_{p^2}))$ and $\\mathcal{A}:=\\op{G}'\/N'=\\op{Gal}(F(\\mu_p)\/\\mathbb{Q})$. For $n\\in N'^{ab}$ and $g\\in \\mathcal{A}$, let $\\tilde{n}$ and $\\tilde{g}$ be lifts of $n$ and $g$ to $N'$ and $\\op{G}'$ respectively. The action of $\\mathcal{A}$ on $N'^{ab}$ is induced by conjugation, defined by $g\\cdot n:= \\tilde{g}\\tilde{n}\\tilde{g}^{-1}\\mod{[N',N']}$. The groups $N$ and $N'$ are isomorphic and the image of $\\bar{\\rho}$ is assumed to contain $U_1(\\mathbb{F}_q)$ (condition $\\ref{thc3}$ of Theorem $\\ref{main}$). The quotient $N'\/V=\\op{Gal}(F(\\mu_{p^2})\/F(\\mu_p))\\simeq \\mathbb{F}_p$. Let $\\pi: \\op{N}'^{ab}\\rightarrow \\mathbb{F}_p$ denote the map induced by the mod-$V$ quotient. Being the composite of Galois extensions, $F(\\mu_{p^2})$ is Galois over $\\mathbb{Q}$. As a result, $\\pi$ is $\\mathcal{A}$-equivariant. Furthermore, since $F(\\mu_{p^2})$ is an abelian extension of $\\mathbb{Q}$, the $\\mathcal{A}$-action on $N'\/V$ is trivial. On the other hand, as an $\\mathcal{A}$-module, $N'^{ab}\\simeq \\bigoplus_{\\lambda\\in \\Delta} \\mathbb{F}_q(\\sigma_{\\lambda})$. It follows from condition $\\ref{thc4}$ of Theorem $\\ref{main}$ that $\\sigma_{\\lambda}\\neq 1$ for $\\lambda\\in \\Phi$. As a result, \n\\[\\op{Hom}(N'^{ab}, \\mathbb{F}_p)^{\\op{G}'}\\simeq \\bigoplus_{\\lambda\\in \\Delta}\\op{Hom}(\\mathbb{F}_q(\\sigma_{\\lambda}), \\mathbb{F}_p)^{\\op{G}'}=0.\\] This is a contradiction which concludes the proof of the first part.\n\\par \nSet $\\mathcal{K}_j$ to be $K_1\\dots K_j$ and $\\mathcal{K}_0:=K$. Setting $E:=\\mathbb{Q}(\\mu_{p^2})$, it suffices to show that $\\mathcal{K}_j\\cap E=\\mathcal{K}_{j-1}\\cap E$. We begin with the case $j=1$. For $\\psi\\in H^1(\\op{G}_{\\mathbb{Q},\\mathcal{J}}, \\operatorname{Ad}^0\\bar{\\rho}^*)$, regard $\\operatorname{Gal}(K_{\\psi}\/K)$ as an $\\mathbb{F}_q[\\op{G}']$-module, where the Galois action is induced via conjugation. The $\\op{G}'$-module $P_1:=\\operatorname{Gal}(K_{1}\/K)$ is identified with $\\psi_1(\\op{G}_K)$. Let $Q_1\\subseteq P_1$ be the $\\op{G}'$-stable subgroup defined by $Q_1:=\\op{Gal}(K_1\/(K_1\\cap E)\\cdot K)$. The action of $\\op{G}'$ on $P_1\/Q_1=\\op{Gal}((K_1\\cap E)\\cdot K\/K)$ is trivial. By Lemma $\\ref{Pdecomposition}$, the quotient $P_1\/Q_1$ decomposes into subgroups \n\\[P_1\/Q_1=\\bigoplus_{\\lambda\\in \\Phi\\cup\\{1\\}} (P_1)_{\\bar{\\chi}\\sigma_{\\lambda}}\/(Q_1)_{\\bar{\\chi}\\sigma_{\\lambda}}.\\]\nThe characters $\\sigma_{\\lambda}\\neq \\bar{\\chi}^{-1}$ and hence $P_1=Q_1$. We have thus shown that $K_1\\cap E= K\\cap E$.\n\\par Let $P_j$ be defined by $P_j:=\\operatorname{Gal}(\\mathcal{K}_j\/\\mathcal{K}_{j-1})$. The $\\op{G}'$-module $P_j$ is isomorphic to \\[ \\operatorname{Gal}(K_j\/K_j\\cap \\mathcal{K}_{j-1})\\subseteq \\psi_j(\\op{G}_K)\\subseteq \\operatorname{Ad}^0\\bar{\\rho}^*.\\] Let $Q_j$ be the $\\op{G}'$-stable subgroup $\\op{Gal}(\\mathcal{K}_j\/(\\mathcal{K}_j\\cap E)\\cdot \\mathcal{K}_{j-1})$ and note that the $\\op{G}'$ action on $P_j\/Q_j$ is trivial. Invoking the same argument as in the case when $j=1$, we have that $P_j=Q_j$ and hence $\\mathcal{K}_j\\cap E=\\mathcal{K}_{j-1}\\cap E$. This completes the proof. \n\\end{proof}\n\n\n\n\\begin{Def}\n\n\\begin{enumerate}\n \\item Let $M_1$ and $M_2$ be $\\mathbb{F}_p[\\op{G}']$-modules. We say that $M_1$ is unrelated to $M_2$ if for every $\\mathbb{F}_p[\\op{G}']$-submodule $N$ of $M_1$, \n\\[\\op{Hom}(N,M_2)^{\\op{G}'}=0.\\]\n\n\\item Let $E$ be a finite extension of $K$ such that $E$ is Galois over $\\mathbb{Q}$ and $\\op{Gal}(E\/K)$ is an $\\mathbb{F}_p$-vector space. Let $M$ be an $\\mathbb{F}_p[\\op{G}']$-module. We say that $E$ is unrelated to $M$ if $\\op{Gal}(E\/K)$ is $\\op{G}'$-unrelated to $M$. Here, the $\\op{G}'$-action on $\\op{Gal}(E\/K)$ is induced via conjugation (let $x\\in \\op{Gal}(E\/K)$ and $g\\in \\op{G}'$, pick a lift $\\tilde{g}$ of $g$, set $g\\cdot x:=\\tilde{g}x\\tilde{g}^{-1}$).\n\\end{enumerate}\n\n\\end{Def}\n\\begin{Prop}\\label{414}\nLet $\\mathcal{J}\\supseteq S$ be a finite set of prime numbers and \\[\\theta_0,\\dots, \\theta_t\\in H^1(\\op{G}_{\\mathbb{Q},\\mathcal{J}},\\operatorname{Ad}^0\\bar{\\rho}^*)\\] be linearly independent over $\\mathbb{F}_q$. Set $K_i:=K_{\\theta_i}$ and let $\\mathbb{L}_1,\\dots, \\mathbb{L}_k$ be a (possibly empty) set of Galois extensions of $\\mathbb{Q}$. Assume that $\\mathbb{L}_i$ contains $K$ and $\\op{Gal}(\\mathbb{L}_i\/K)$ is an $\\mathbb{F}_p$-vector space for $i=1,\\dots, k$. Suppose that $\\mathbb{L}_i$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$ for $i=1,\\dots, k$. Denote by $\\mathcal{L}$ the composite $\\mathbb{L}_1\\cdots \\mathbb{L}_k$. If the set $\\{\\mathbb{L}_1,\\dots, \\mathbb{L}_k\\}$ is empty, set $\\mathcal{L}=K$. The field $K_0$ is not contained in the composite of the fields $K_1\\cdots K_t \\cdot \\mathcal{L}$.\n\\end{Prop}\n\\begin{proof}\nLet $\\mathcal{K}$ denote the composite of the fields $K_1,\\dots, K_t$. If $K_0$ is contained in $\\mathcal{K}\\cdot \\mathcal{L}$, then $\\theta_0,\\dots, \\theta_t\\in H^1(\\op{Gal}(\\mathcal{K}\\cdot \\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)$ and hence\n\\[h^1(\\op{Gal}(\\mathcal{K}\\cdot \\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\\geq t+1.\\] Hence it suffices to show that \n\\[h^1(\\op{Gal}(\\mathcal{K}\\cdot \\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\\leq t.\\]First we show that \n\\[h^1(\\op{Gal}( \\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)=0.\\] Denote by $\\mathcal{L}_i$ the composite of the fields $\\mathbb{L}_1\\cdots \\mathbb{L}_i$ and set $\\mathcal{L}_0:=K$. Note that $\\op{Gal}(\\mathcal{L}_i\/\\mathcal{L}_{i-1})$ is isomorphic to $\\op{Gal}(\\mathbb{L}_i\/\\mathbb{L}_i\\cap \\mathcal{L}_{i-1})$, which is an $\\mathbb{F}_p[\\op{G}']$-submodule of $\\op{Gal}(\\mathbb{L}_i\/K)$. Since $\\mathbb{L}_i$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$,\n\\[\\rm{Hom}(\\op{Gal}(\\mathcal{L}_i\/\\mathcal{L}_{i-1}),\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}=0.\\]Hence the inflation map\n\\[H^1(\\op{Gal}(\\mathcal{L}_{i-1}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\\xrightarrow{\\op{inf}} H^1(\\op{Gal}(\\mathcal{L}_{i}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\\]is an isomorphism. We deduce that $H^1(\\op{Gal}(\\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)$ is isomorphic to $H^1(\\op{G}',\\operatorname{Ad}^0\\bar{\\rho}^*)$ and hence, is zero.\n\\par Let $\\mathcal{K}_i$ denote the composite $K_1\\cdots K_i$ and $\\mathcal{K}_0$ denote $K$. Note that $\\op{Gal}(\\mathcal{K}_i\\cdot \\mathcal{L}\/\\mathcal{K}_{i-1}\\cdot\\mathcal{L})$ is an $\\mathbb{F}_p[\\op{G}']$-submodule of $\\op{Gal}(K_i\/K)$, and hence, of $\\operatorname{Ad}^0\\bar{\\rho}^*$. Lemma $\\ref{y1}$ asserts that\n\\[\\dim \\rm{Hom}(\\op{Gal}(\\mathcal{K}_i\\cdot \\mathcal{L}\/\\mathcal{K}_{i-1}\\cdot\\mathcal{L}),\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}\\leq 1.\\]Therefore, by inflation-restriction,\n\\[h^1(\\op{Gal}(\\mathcal{K}_i\\cdot \\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\\leq h^1(\\op{Gal}(\\mathcal{K}_{i-1}\\cdot\\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*) +1.\\]Consequently, we deduce that\n$h^1(\\op{Gal}(\\mathcal{K}\\cdot \\mathcal{L}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\\leq t$ and the proof is complete.\n\\end{proof}\n\\begin{Lemma}\\label{415} Let $\\mathcal{J}\\supseteq S$ be a finite set of primes.\n \\begin{enumerate}\n \n \\item\\label{415c1} Let $M$ be a nontrivial quotient of $\\operatorname{Ad}^0\\bar{\\rho}^*$ and $\\eta\\in H^1(\\op{G}_{\\mathbb{Q},\\mathcal{J}},M)$ be non-zero. Let $K_{\\eta}$ be the field extension of $K$ cut out by $\\eta$. The field $K_{\\eta}$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n \\item \\label{415c2} The field $K(\\mu_{p^2})$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n \\item\\label{415c3} Let $f\\in H^1(\\op{G}_{\\mathbb{Q},\\mathcal{J}}, \\operatorname{Ad}^0\\bar{\\rho})$, then the extension $K_f$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n \\item \\label{415c4} Suppose that we are given a lift $\\zeta_2:\\op{G}_{\\mathbb{Q}, \\mathcal{J}}\\rightarrow \\op{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)$ of $\\bar{\\rho}$ with similitude character $\\kappa\\mod{p^2}$. The field extension $K(\\zeta_2)$ cut out by the kernel of $\\zeta_2$, is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n \\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\n\\par For part $\\eqref{415c1}$, it suffices to show that $M$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$. Since $M$ is a non-trivial quotient of $\\operatorname{Ad}^0\\bar{\\rho}^*$, it follows from Lemma $\\ref{mainin}$ that the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $M$ is zero. Let $N\\subseteq M$ be a $\\op{G}'$-submodule and $f:N\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}^*$ be a homomorphism. The $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $f(N)$ is zero, hence by Lemma $\\ref{mainin}$, the map $f=0$.\n\\par Since $\\mathbb{Q}(\\mu_{p^2})$ is an abelian extension of $\\mathbb{Q}$, the $G'$-action on $\\op{Gal}(K(\\mu_{p^2})\/K)$ is trivial. On the other hand, $\\operatorname{Ad}^0\\bar{\\rho}^*$ has no trivial $\\mathbb{T}$-eigenspace. It follows that $K(\\mu_{p^2})$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$ and part $\\eqref{415c2}$ follows.\n\\par \nLet $Q$ be a $\\op{G}'$-submodule of $\\operatorname{Ad}^0\\bar{\\rho}$, Lemma $\\ref{Pdecomposition}$ asserts that $Q=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}} Q_{\\sigma_{\\lambda}}$. On the other hand, $\\operatorname{Ad}^0\\bar{\\rho}^*=\\bigoplus_{\\gamma\\in \\Phi\\cup \\{1\\}} (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{\\lambda}}$. It follows from condition $\\eqref{thc4}$ of Theorem $\\ref{main}$ that \n\\[\\op{Hom}(Q_{\\sigma_{\\lambda}}, (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{\\gamma}})^{\\mathbb{T}}=0.\\] As a result, $\\op{Hom}(Q,\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}=0$ and hence part $\\eqref{415c3}$ follows.\n\\par For part \\eqref{415c4}, identify $\\operatorname{Ad}^0\\bar{\\rho}$ with the kernel of the mod-$p$ reduction map \\[\\op{Sp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)\\rightarrow \\op{Sp}_{2n}(\\mathbb{F}_q),\\]by identifying $X\\in \\operatorname{Ad}^0\\bar{\\rho}$ with $\\op{Id}+pX$. Recall that $\\kappa=\\kappa_0\\chi^k$, where $k$ is a positive integer which is divisible by $p(p-1)$. Setting $\\kappa_2:=\\kappa\\mod{p^2}$, we see that the restriction $\\kappa_{2\\restriction \\op{G}_K}$ is trivial. Therefore, $\\op{Gal}(K(\\zeta_2)\/K)$ may be identified with a Galois submodule of $\\operatorname{Ad}^0\\bar{\\rho}$. Here, $g\\in \\op{Gal}(K(\\zeta_2)\/K)$ is identified with \n\\[\\zeta_2(g)\\in \\op{ker}\\left(\\op{Sp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)\\rightarrow \\op{Sp}_{2n}(\\mathbb{F}_q)\\right)\\simeq \\operatorname{Ad}^0\\bar{\\rho}.\\] The same reasoning as in the previous case shows that $K(\\zeta_2)$ is indeed unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n\\end{proof}\n\\begin{Lemma}\\label{lemma416}\nLet $L_1,\\dots, L_k$ and $K_1,\\dots , K_l$ be Galois extensions of $\\mathbb{Q}$ which contain $K$. Assume that:\n\\begin{itemize}\n \\item $\\op{Gal}(L_i\/K)$ and $\\op{Gal}(K_i\/K)$ are finite dimensional $\\mathbb{F}_p$-vector spaces.\n \\item As a $\\op{G}'$-module, $\\op{Gal}(L_i\/K)$ is isomorphic to a subquotient of $\\operatorname{Ad}^0\\bar{\\rho}$ for $i=1,\\dots, k$.\n \\item As a $\\op{G}'$-module, $\\op{Gal}(K_i\/K)$ is isomorphic to a subquotient of $\\operatorname{Ad}^0\\bar{\\rho}^*$ for $i=1,\\dots, l$.\n\\end{itemize} Then the composite $L_1\\cdots L_k$ is linearly disjoint from $K_1,\\dots, K_l$.\n\\end{Lemma}\n\\begin{proof}\nThe order of $\\mathbb{T}$ is coprime to $p$, hence Maschke's theorem asserts that any finite dimensional $\\mathbb{F}_p[\\op{G}']$-module $M$ decomposes into a direct sum \n\\[M=\\oplus_{\\tau} M_{\\tau}.\\]Here, $\\tau$ is a character of $\\mathbb{T}$ and $M_{\\tau}$ is the $\\tau$-eigenspace \n\\[M_{\\tau}:=\\{m\\in M| g\\cdot m=\\tau(g) m\\}.\\]The action of $\\op{G}'$ on $\\op{Gal}(L_i\/K)$ and $\\op{Gal}(K_i\/K)$ is induced by conjugation. By assumption, $\\op{Gal}(L_i\/K)$ is isomorphic to a subquotient of $\\operatorname{Ad}^0\\bar{\\rho}$, i.e. there exist $\\op{G}'$-submodules $Q_1\\subseteq Q_2$ of $\\operatorname{Ad}^0\\bar{\\rho}$ such that $\\op{Gal}(L_i\/K)\\simeq Q_2\/Q_1$. By Lemma $\\ref{Pdecomposition}$, the module $Q_i$ decomposes into $\\mathbb{T}$-eigenspaces\n\\[Q_i=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}}(Q_i)_{\\sigma_{\\lambda}}\\]for $i=1,2$. Therefore, the quotient $\\op{Gal}(L_i\/K)$ decomposes into \\[\\op{Gal}(L_i\/K)=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}}(\\op{Gal}(L_i\/K))_{\\sigma_{\\lambda}}\\] where $(\\op{Gal}(L_i\/K))_{\\sigma_{\\lambda}}:=(Q_2)_{\\sigma_{\\lambda}}\/(Q_1)_{\\sigma_{\\lambda}}$ is the $\\sigma_{\\lambda}$-eigenspace for the action of $\\mathbb{T}$ on $\\op{Gal}(L_i\/K)$. Likewise, $\\op{Gal}(K_i\/K)$ decomposes into \\[\\op{Gal}(K_i\/K)=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}}(\\op{Gal}(K_i\/K))_{\\bar{\\chi}\\sigma_{\\lambda}}.\\]\n\\par Let $\\mathcal{L}$ be the composite $L_1\\cdots L_k$ and $\\mathcal{K}$ be the composite $K_1\\cdots K_l$. Letting $\\mathcal{L}_i$ be the composite $L_1\\cdots L_i$, filter $\\mathcal{L}$ by\n \\[\\mathcal{L}\\supseteq \\mathcal{L}_{k-1}\\cdots \\supseteq \\mathcal{L}_1\\supseteq K.\\] The Galois group \\[\\op{Gal}(\\mathcal{L}_i\/\\mathcal{L}_{i-1})\\simeq \\op{Gal}(L_i\/L_i\\cap \\mathcal{L}_{i-1})\\] is a $\\op{G}'$-submodule of $\\op{Gal}(L_i\/K)$. Hence the characters for the action of $\\mathbb{T}$ on $\\op{Gal}(\\mathcal{L}_i\/\\mathcal{L}_{i-1})$ are each of the form $\\sigma_{\\lambda}$. Similar reasoning shows that the characters for the action of $\\mathbb{T}$ on $\\op{Gal}(\\mathcal{K}\/K)$ are each of the form $\\bar{\\chi}\\sigma_{\\lambda}$. Set $E=\\mathcal{K}\\cap \\mathcal{L}$ and $M=\\op{Gal}(E\/K)$. Being a quotient of $\\op{Gal}(\\mathcal{L}\/K)$, $M$ decomposes into eigenspaces for the action of the torus \n\\[M=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}} M_{\\sigma_{\\lambda}}.\\]Since $M$ is a quotient of $\\op{Gal}(\\mathcal{K}\/K)$, \\[M=\\bigoplus_{\\gamma\\in \\Phi\\cup \\{1\\}} M_{\\bar{\\chi}\\sigma_{\\gamma}}.\\] It is assumed that the image of $\\sigma_{\\lambda}$ spans $\\mathbb{F}_q$ and that $\\sigma_{\\lambda}$ is not a $\\op{Gal}(\\mathbb{F}_q\/\\mathbb{F}_p)$ twist of $\\bar{\\chi}\\sigma_{\\gamma}$. Hence, it follows that\n\\[\\op{Hom}(\\mathbb{F}_q(\\sigma_{\\lambda}),\\mathbb{F}_q(\\bar{\\chi}\\sigma_{\\gamma}))^{\\mathbb{T}}=0.\\]Therefore, $\\rm{Hom}(M,M)^{\\op{G}'}=0$ and in particular, the identity map is zero. This implies that $\\mathcal{K}\\cap \\mathcal{L}=K$.\n\\end{proof}\n\\section{Deformation conditions at Auxiliary Primes}\n\nWe introduce the auxiliary primes $v$ and the liftable deformation problem $\\mathcal{C}_v$ at $v$.\n\\begin{Def} \nA prime number $v$ is a trivial prime if the following splitting conditions are satisfied:\n\\begin{itemize}\n\\item $\\operatorname{G}_v\\subseteq \\ker\\bar{\\rho}$,\n\n\\item $v\\equiv 1 \\mod{p}$ and $v \\not\\equiv 1 \\mod{p^2}$.\\end{itemize} \n\\end{Def}\n In other words, a prime number $v$ is trivial if it splits in $\\mathbb{Q}(\\bar{\\rho},\\mu_p)$ and does not split in $\\mathbb{Q}(\\mu_{p^2})$. By Lemma $\\ref{22Dec5}$, $\\mathbb{Q}(\\bar{\\rho},\\mu_p)$ does not contain $\\mathbb{Q}(\\mu_{p^2})$. This is a Chebotarev condition, i.e. defined by a finite union of sets that are defined by applying the Chebotarev density theorem. Therefore, the set of trivial primes has positive Dirichlet density, in particular, it is infinite.\\par Let $v$ be a trivial prime. The deformations of the trivial representation $\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$ are tamely ramified. The Galois group of the maximal pro-p extension of $\\mathbb{Q}_{v}$ is generated by a Frobenius $\\sigma_v$ and a generator of tame pro-$p$ inertia $\\tau_v$. These satisfy the relation \n$\\sigma_v\\tau_v\\sigma_v^{-1}=\\tau_l^{v}$. We define the deformation functor $\\mathcal{C}_v$. The functor $\\mathcal{C}_v$ will be liftable, however, it will not be a deformation condition. Let $\\alpha$ be a root which shall be specified later. The root-subgroup $\\text{U}_{\\alpha}\\subset \\operatorname{GSp}_{2n}$ is the subgroup generated by the image of the root-subspace $(\\op{sp}_{2n})_{\\alpha}$ under the exponential map. We let $\\text{Z}(\\text{U}_{\\alpha})$ be the subgroup of $\\operatorname{GSp}_{2n}$ consisting of elements which commute with $\\text{U}_{\\alpha}$.\n\n\n\\begin{Def}\\label{ramtriv}\\cite[Definition 3.1]{FKP1}\n Let $\\mathcal{D}_v^{\\alpha}$ consist of the deformation classes of lifts such that some representative $\\varrho$ satisfies:\n \\begin{enumerate}\n \\item $\\varrho(\\sigma_v)\\in \\mathcal{T}\\cdot \\text{Z}(\\text{U}_{\\alpha})$ and $\\varrho(\\tau_v)\\in \\text{U}_{\\alpha}$,\n \\item under the composite \n \\[\\mathcal{T}\\cdot \\text{Z}(\\text{U}_{\\alpha})\\rightarrow \\mathcal{T}\/(\\mathcal{T}\\cap \\text{Z}(\\text{U}_{\\alpha}))\\xrightarrow{\\alpha} \\text{GL}_1\\] $\\varrho(\\sigma_v)$ maps to $v$.\n \\end{enumerate}\n \\begin{Remark}\n When $n=1$ and $\\alpha$ is the positive root of $\\operatorname{sl}_2$, the deformation functor $\\mathcal{D}_v^{\\alpha}$ consists of $\\varrho$ such that there exists $x$ and $y$ such that \\[\\varrho(\\sigma_v)=c\\mtx{v}{x}{0}{1}\\text{ and } \\varrho(\\tau_v)=\\mtx{1}{y}{0}{1}.\\]Here $c$ is equal to $(\\kappa(\\sigma_v)\/v)^{\\frac{1}{2}}$.\n \\end{Remark}\n\\end{Def}\nWe shall denote by the kernel of $\\alpha$ restricted to $\\mathfrak{t}$ by $\\mathfrak{t}_{\\alpha}$. Since the action of $\\operatorname{G}_v$ on $\\operatorname{Ad}^0\\bar{\\rho}$ is trivial, \n\\[H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})=\\text{Hom}(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho}).\\] Let $\\mathcal{P}_v^{\\alpha}$ be the subspace of $H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ consisting of $\\phi$ such that\n\\[\\phi(\\sigma_v)\\in \\mathfrak{t}_{\\alpha}+\\text{Cent}((\\operatorname{Ad}^0\\bar{\\rho})_{\\alpha})\\]\n\\[\\phi(\\tau_v)\\in (\\operatorname{Ad}^0\\bar{\\rho})_{\\alpha}.\\]\nLet $\\Phi^{\\alpha}$ denote the subset of roots $\\beta\\in \\Phi$ such that $[(\\operatorname{Ad}^0\\bar{\\rho})_{\\alpha}, (\\operatorname{Ad}^0\\bar{\\rho})_{\\beta}]\\neq 0$. Recall that $X_{\\alpha}$ is a choice of root vector for $\\alpha$.\n\n \n\\begin{Def}\\label{defconditions}\n\\begin{enumerate}\n \\item Let $v$ be a trivial prime which is unramified mod $p^2$ in our lifting argument. Set $\\alpha=2L_1$ and $\\mathcal{C}_v=\\mathcal{C}_v^{nr}$ consist of deformations with a representative \\[\\varrho'=(\\operatorname{Id}+X_{-\\alpha})\\varrho (\\operatorname{Id}+X_{-\\alpha})^{-1}\\] where $\\varrho$ is a representative for a deformation in $\\mathcal{D}_v^{\\alpha}$ which satisfies further conditions. In accordance with \\cite[Definition 3.5]{FKP1}, we assume that the mod-$p^2$ reduction $\\varrho_2:=\\varrho\\mod{p^2}$ satisfies the following conditions:\n \\begin{enumerate}\n \\item $\\varrho_2$ is unramified, with $\\varrho_2(\\sigma_v)\\in \\mathcal{T}(\\text{W}(\\mathbb{F}_q)\/p^2)$,\n \\item for all $\\beta\\in \\Phi^{\\alpha}$, \n \\[\\beta(\\varrho_2(\\sigma_v))\\neq 1\\mod{p^2}.\\]\n \\end{enumerate} Let $\\mathcal{S}_{v}^{\\alpha}$ consist of $\\phi\\in H^1(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ such that $\\phi(\\sigma_v)\\in \\bigoplus_{\\beta\\in \\Phi^{\\alpha}} (\\operatorname{Ad}^0\\bar{\\rho})_{\\beta}$ and $\\phi(\\tau_v)=0$. Let $\\mathcal{N}_v$ be specified by \\[\\mathcal{N}_v=\\mathcal{N}_v^{nr}:=(\\operatorname{Id}+X_{-\\alpha})(\\mathcal{P}_v^{\\alpha}+\\mathcal{S}_v^{\\alpha})(\\operatorname{Id}+X_{-\\alpha})^{-1}.\\]\n \\item \\label{defconditions2}Let $v$ be a trivial prime which will be ramified mod $p^2$ in our lifting argument. Let $\\alpha=-2L_1$ and $\\mathcal{C}_v=\\mathcal{C}_v^{ram}$ consist of deformations in $\\mathcal{D}_v^{\\alpha}$ with representative $\\varrho$ satisfying some additional conditions, which we specify. In accordance with \\cite[Definition 3.9]{FKP1}, assume that the mod-$p^2$ reduction $\\varrho_2$ satisfies the following conditions:\n \\begin{enumerate}\n \\item $\\varrho_2(\\tau_v)\\in u_{\\alpha}(py)$ where $y\\in \\text{W}(\\mathbb{F}_q)^{\\times}$, and $u_{\\alpha}: (\\operatorname{Ad}^0\\bar{\\rho})_{\\alpha}\\rightarrow \\op{GSp}_{2n}$ is the root group homomorphism over $\\op{W}(\\mathbb{F}_q)$.\n \\item For all $\\beta\\in \\Phi^{\\alpha}$, \n \\[\\beta(\\varrho_2(\\sigma_v))\\neq 1\\mod{p^2}.\\]\n \\end{enumerate}Let $\\mathcal{S}_v^{\\alpha}$ denote the space of cohomology classes specified in the proof of \\cite[Lemma 3.10]{FKP1}. Let $\\mathcal{N}_v=\\mathcal{N}_v^{ram}$ be defined by \\[\\mathcal{N}_v:=\\mathcal{P}_v^{\\alpha}+\\mathcal{S}_v^{\\alpha}.\\] \n\\end{enumerate}\n\\end{Def}\nThe following gives us a criterion for an element $f\\in H^1(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ to not be contained in $\\mathcal{N}_v^{nr}$. This criterion is used in the proof of Proposition $\\ref{lastchebotarev}$.\n\\begin{Lemma}\\label{lemma55}\nLet $v$ be a trivial prime and $\\mathcal{C}_v=\\mathcal{C}_v^{nr}$. Let $f\\in \\mathcal{N}_v$, express $f(\\sigma_v)=\\sum_{\\lambda\\in \\Phi} {a_{\\lambda}} X_{\\lambda}+\\sum_{i=1}^n a_i H_i$. Write $X_{-2L_1}=c e_{n+1,1}$ and $X_{2L_1}=d e_{1,n+1}$. We have that $a_{2L_1}= -(cd)^{-1} a_{1}$.\n\\end{Lemma}\n\\begin{proof}\nSet $g:=(\\op{Id} + X_{-2L_1})^{-1} f(\\op{Id} + X_{-2L_1})$ and express $g(\\sigma_v)=\\sum_{\\lambda\\in \\Phi} {b_{\\lambda}} X_{\\lambda}+\\sum_{i=1}^n b_i H_i$. Note that for $\\phi\\in \\mathcal{P}_v^{2L_1}$, \\[\\phi(\\sigma_v)\\in \\mathfrak{t}_{2L_1}+\\text{Cent}((\\operatorname{Ad}^0\\bar{\\rho})_{2L_1})\\] and hence has zero $H_1$-component. For $\\phi\\in \\mathcal{S}_v^{2L_1}$, we have that\n\\[\\phi(\\sigma_v)\\in \\bigoplus_{\\beta\\in \\Phi^{2L_1}} (\\operatorname{Ad}^0\\bar{\\rho})_{\\beta}.\\] We deduce that the $H_1$-component $b_1$ is equal to zero. We show that the $H_1$-component of $g(\\sigma_v)$ is equal to $a_1+cd a_{2L_1}$ from the relation $g(\\sigma_v)=(\\op{Id}+X_{-2L_1})^{-1}f(\\sigma_v) (\\op{Id}+X_{-2L_1})$. Note that $X_{-2L_1}^2=0$ and thus, $(\\op{Id}+X_{-2L_1})^{-1}=(\\op{Id}-X_{-2L_1})$. One has that\n\\[\\begin{split}g(\\sigma_v)=&(\\op{Id}-X_{-2L_1})f(\\sigma_v) (\\op{Id}+X_{-2L_1})\\\\=&(\\op{Id}-c e_{n+1,1})f(\\sigma_v) (\\op{Id}+c e_{n+1,1})\\\\\n=&f(\\sigma_v)+ c[f(\\sigma_v),e_{n+1,1}]-c^2e_{n+1,1}f(\\sigma_v) e_{n+1,1}.\\end{split}\\]\nNote that \n\\[c^2e_{n+1,1}f(\\sigma_v) e_{n+1,1}=a_{2L_1}c^2 e_{n+1,1}X_{2L_1} e_{n+1,1}= a_{2L_1}c^2d e_{n+1,1}= a_{2L_1}cd X_{-2L_1},\\]and thus does not contribute to the $H_1$-component of $g(\\sigma_v)$. The contribution of \\[c[f(\\sigma_v), e_{n+1,1}]=\\sum_{\\lambda\\in \\Phi} {ca_{\\lambda}} [X_{\\lambda},e_{n+1,1}]+\\sum_{i=1}^n ca_i [H_i,e_{n+1,1}]\\] to the $H_1$-component of $g(\\sigma_v)$ is from the term\n\\[ca_{2L_1} [X_{2L_1},e_{n+1,1}]=ca_{2L_1} [d e_{1,n+1},e_{n+1,1}]=cd a_{2L_1} H_1.\\] Thus, we have shown that $b_1=a_1+cd a_{2L_1}$. Since, $b_1=0$, we deduce that $a_1= -cd a_{2L_1}$.\n\\end{proof}\n\\begin{Lemma}\\cite[Lemma 3.2, 3.6,3.10]{FKP1}\nLet $v$ be a trivial prime (for which either $\\mathcal{C}_v=\\mathcal{C}_v^{ram}$ or $\\mathcal{C}_v^{nr}$ is the chosen deformation condition) and $X\\in \\mathcal{N}_v$,\n\\begin{enumerate}\n \\item $\\dim \\mathcal{N}_v=\\dim\\operatorname{Ad}^0\\bar{\\rho}=h^0(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$.\n \\item Let $m\\geq 3$ and $\\rho_m\\in \\mathcal{C}_v(\\text{W}(\\mathbb{F}_q)\/p^{m})$, then\n\\[(\\operatorname{Id}+p^{n-1}X)\\rho_m\\in \\mathcal{C}_v(\\text{W}(\\mathbb{F}_q)\/p^{m}).\\]\n\\item The deformation functor $\\mathcal{C}_v$ is liftable.\n\\end{enumerate}\n\\end{Lemma}\n\\par Prior to lifting $\\bar{\\rho}$ to characteristic zero, we show that $\\bar{\\rho}$ lifts to $\\rho_2$ after increasing the set of ramification from $S$ to $S\\cup X_1$. One may choose a continuous lift $\\tau$ of $\\bar{\\rho}$ as depicted\n\\[ \\begin{tikzpicture}[node distance = 2.6 cm, auto]\n \\node(G) at (0,0) {$\\operatorname{G}_{\\mathbb{Q},S\\cup X_1}$};\n \\node (A) at (3,0) {$\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$};\n \\node (B) at (3,2){$\\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)$};\n \\draw[->] (G) to node [swap]{$\\bar{\\rho}$} (A);\n \\draw[->] (B) to node{} (A);\n \\draw[->] (G) to node {$\\tau$} (B);\n \\end{tikzpicture}\\]such that the composite $\\nu\\circ \\tau=\\psi \\mod{p^2}$.\n The obstruction class \\[\\mathcal{O}(\\bar{\\rho})_{\\restriction S\\cup X_1}\\in H^2(\\op{G}_{S\\cup X_1}, \\operatorname{Ad}^0\\bar{\\rho})\\] is represented by the $2$-cocycle\n \\[(g_1,g_2)\\mapsto \\tau(g_1 g_2)\\tau(g_2)^{-1}\\tau(g_1)^{-1}.\\]\n \n The residual representation $\\bar{\\rho}$ lifts to a representation $\\rho_2$ ramified only at primes in $S\\cup X_1$ if and only if this obstruction is zero. For $v\\in S$, the local representation $\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$ satisfies $\\mathcal{C}_v$ which is a liftable deformation condition (by assumption) and thus lifts to mod $p^2$. The residual representation $\\bar{\\rho}$ is unramified at each prime $v\\in X_1$ and thus it is easy to see that $\\bar{\\rho}_{\\restriction \\operatorname{G}_v}$ lifts to mod $p^2$ for $v\\in X_1$. As a consequence, $\\mathcal{O}(\\bar{\\rho})_{\\restriction S\\cup X_1}$ is contained in $\\Sh^2_{S\\cup X_1}(\\operatorname{Ad}^0\\bar{\\rho})$. We will show that a set of finitely many trivial primes $X_1$ can be chosen so that \\[\\Sh^2_{S\\cup X_1}(\\operatorname{Ad}^0\\bar{\\rho})=0.\\]For such a choice of $X_1$, there is a deformation $\\rho_2$\n \\[ \\begin{tikzpicture}[node distance = 2.8 cm, auto]\n \\node(G) at (0,0) {$\\operatorname{G}_{\\mathbb{Q},S\\cup X_1}$};\n \\node (A) at (3,0) {$\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$.};\n \\node (B) at (3,2) {$\\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)$};\n \\draw[->] (G) to node [swap]{$\\bar{\\rho}$} (A);\n \\draw[->] (B) to node{} (A);\n \\draw[->] (G) to node {$\\rho_2$} (B);\n \\end{tikzpicture}\\]\n \n\\begin{Prop}\\label{Shavanishing}\nLet $\\mathscr{M}$ denote the finite set of $\\operatorname{G}_{\\mathbb{Q}}$-modules defined by \\[\\begin{split}\\mathscr{M}:=&\\{(\\operatorname{Ad}^0\\bar{\\rho})\/(\\operatorname{Ad}^0\\bar{\\rho})_k, \\mid -2n+1\\leq k \\leq 2n\\}\\\\&\\cup \\{(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}, \\mid -2n+1\\leq k \\leq 2n\\}.\\\\\n\\end{split}\\]\nThere is a finite set $T\\supset S$ such that $T\\backslash S$ consists of only trivial primes such that for all $M\\in \\mathscr{M}$,\n\\begin{equation}\\label{equationTminusS}\n\\ker \\{H^1(\\op{G}_{\\mathbb{Q},T},M)\\rightarrow \\bigoplus_{w\\in T\\backslash S} H^1(\\op{G}_w, M)\\}=0\n\\end{equation}\nand so in particular,\n\\begin{equation*}\n\\Sh_T^1(M)=0.\n\\end{equation*}\n\\end{Prop}\n\\begin{proof}\nWe show that $T$ can be chosen for which \n\\begin{equation*}\n\\Sh_T^1(\\operatorname{Ad}^0\\bar{\\rho}^*)=0,\n\\end{equation*}\nthe argument for any $M\\in \\mathscr{M}$ is identical. For $0\\neq \\psi\\in H^1(\\operatorname{G}_{\\mathbb{Q},S}, \\operatorname{Ad}^0\\bar{\\rho}^*)$, let $K_{\\psi}\\supset \\mathbb{Q}(\\operatorname{Ad}^0\\bar{\\rho}^*)$ be the field extension cut out by $\\psi$. By Lemma $\\ref{l4}$, the extension $K_{\\psi}$ is not equal to $\\mathbb{Q}(\\operatorname{Ad}^0\\bar{\\rho}^*)$. The extension $K(\\mu_{p^2})$ is linearly disjoint with $K_{\\psi}$ over $K$. By Lemma $\\ref{22Dec5}$, $K(\\mu_{p^2})$ is not contained in $K$ and $K(\\mu_{p^2})\\cap K_{\\psi}=K$. As a result, there is a nonempty Chebotarev class of primes which split in $K$ and are non-split in $K_{\\psi}$ and $K(\\mu_{p^2})$. If $v$ is such a prime, it must be a trivial prime since it splits in $K$ and is non-split in $\\mathbb{Q}(\\mu_{p^2})$. On the other hand, since $v$ is non-split in $K_{\\psi}$, deduce that $\\psi_{\\restriction \\op{G}_v}\\neq 0$. We may therefore choose a finite set of primes $T$ such that \n\\begin{itemize}\n\\item $T$ is finite,\n\\item $T\\backslash S$ consists of only trivial primes,\n\\item $\\ker \\{H^1(\\op{G}_{\\mathbb{Q},T},\\operatorname{Ad}^0\\bar{\\rho}^*)\\rightarrow \\bigoplus_{w\\in T\\backslash S} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}^*)\\}=0$.\n\\end{itemize}\n\\end{proof}\nThe set of trivial primes $X_1$ is taken to be $T\\backslash S$.\n \n\\section{Lifting to mod $p^3$}\n\n\\par By Proposition $\\ref{Shavanishing}$, there is a finite set of primes $T$ containing $S$ such that $T\\backslash S$ consists of trivial primes and $\\Sh_{ T}^1(\\operatorname{Ad}^0\\bar{\\rho}^*)=0$. Let $X_1$ be the set of trivial primes $T\\backslash S$. At each prime $v\\in X_1$, let $\\mathcal{C}_v$ be the liftable deformation problem $\\mathcal{C}_v^{nr}$. By global duality, $\\Sh_{T}^2(\\operatorname{Ad}^0\\bar{\\rho})=0$ and thus the cohomological obstruction to lifting $\\bar{\\rho}$ to a representation $\\zeta_2$\n\\begin{equation}\\label{zeta2} \\begin{tikzpicture}[node distance = 2.2 cm, auto]\n \\node(G) at (0,0){$\\operatorname{G}_{\\mathbb{Q},T}$};\n \\node (A) at (3,0) {$\\operatorname{GSp}_{2n}(\\mathbb{F}_q)$};\n \\node (B) at (3,2) {$\\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)$};\n \\draw[->] (G) to node [swap]{$\\bar{\\rho}$} (A);\n \\draw[->] (B) to node{} (A);\n \\draw[->] (G) to node {$\\zeta_2$} (B);\n \\end{tikzpicture}\\end{equation}\n vanishes. Here, $\\zeta_2$ is stipulated to have similitude character $\\kappa\\mod{p^2}$. Let $v\\in T$, recall that the set of $W(\\mathbb{F}_q)\/p^2$ lifts of $\\bar{\\rho}_{\\restriction \\op{G}_v}$ is an $H^1(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$-torsor. Therefore there exists $z_v\\in H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho})$ such that the twist $(\\operatorname{Id}+z_v p) {\\zeta_2}_{\\restriction \\operatorname{G}_v}$ satisfies $\\mathcal{C}_v$. Further, for $v\\in X_1$, the class $z_v$ may is chosen so that this twist is unramified. We show that there is a set $W$ of at most two trivial primes such that on increasing the set $T$ to $Z=T\\cup W$ there exists a global cohomology class $h\\in H^1(\\op{G}_{\\mathbb{Q}, Z}, \\operatorname{Ad}^0\\bar{\\rho})$ such that\n \\begin{itemize}\n \n \\item $h_{\\restriction \\operatorname{G}_v}=z_v$ for $v\\in T$,\n \\item $(1+ph)\\zeta_{2}|_{\\op{G}_v}\\in \\mathcal{C}_v^{ram}$ for $v\\in W$.\n \\end{itemize} Further, letting $\\rho_2$ be the twist $\\rho_2=(\\operatorname{Id}+ph)\\zeta_2$, each local representation ${\\rho_2}_{\\restriction \\operatorname{G}_v}$ satisfies $\\mathcal{C}_v$ for $v\\in Z$. As a consequence, the obstruction class $\\mathcal{O}(\\rho_2)$ is in $ \\Sh_{Z}^2(\\operatorname{Ad}^0\\bar{\\rho})$. Since $Z$ contains $T$, the group $\\Sh_{Z}^2(\\operatorname{Ad}^0\\bar{\\rho})$ is zero. As a result, $\\rho_2$ must lift to $W(\\mathbb{F})\/p^3$. Assume that there is no such class $h$ for a set $W$ such that $\\# W\\leq 1$. It is shown that there is a pair of trivial primes $v_1,v_2\\notin T$ such that $W$ can be chosen to be equal to $\\{v_1,v_2\\}$. The set of trivial primes $X_2$ is then chosen to be $Z\\backslash S$. For $v\\in W$, choose $\\mathcal{C}_v$ to be equal to $\\mathcal{C}_v^{ram}$. In what follows, a Chebotarev class refers to a nonempty collection of primes defined by the application of the Chebotarev density theorem. Note that a Chebotarev class has positive Dirichlet density, and is in particular, infinite.\n \\begin{Prop}\\label{P1}\nLet $T$ be as in Proposition $\\ref{Shavanishing}$ and $\\psi$ be a nonzero element in $H^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*)$ and let $W\\subset H^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*)$ be a subspace not containing $\\psi$. Then, there exists a Chebotarev class of trivial primes $v$ such that \\[\\begin{split} &{\\psi}_{\\restriction \\operatorname{G}_v}\\neq 0\\\\\n&{\\beta}_{\\restriction \\operatorname{G}_v}=0 \\text{ for all } \\beta\\in W.\n\\end{split}\\]Moreover we may choose $v$ so that $v$ does not split completely in the $\\bar{\\chi} \\sigma_{2L_1}$-eigenspace of $\\operatorname{Gal}(K_{\\psi}\/K)$ when viewed as a Galois submodule of $\\operatorname{Ad}^0\\bar{\\rho}^*$. \n\\end{Prop}\n\\begin{proof}\nLet $\\{\\psi_1,\\dots, \\psi_m\\}$ be a basis of $W$. Since $\\psi$ is not contained in the span of $W$, the classes $\\psi,\\psi_1,\\dots, \\psi_m$ are linearly independent. Extend $\\psi_1,\\dots, \\psi_m$ to $\\psi_1,\\dots, \\psi_r$, so that $\\psi,\\psi_1,\\dots, \\psi_r$ is a basis of $H^1(\\op{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*)$. Let $\\widetilde{W}$ be the span of $\\{\\psi_1,\\dots, \\psi_r\\}$. It suffices to prove the statement for $\\widetilde{W}$ in place of $W$, since $W$ is contained in $\\widetilde{W}$. Let $\\mathfrak{F}$ denote the composite $K_{\\psi_1} \\cdots K_{\\psi_r}$. Set $P:=\\operatorname{Gal}(K_{\\psi}\/K)$ and recall that $J_{\\psi}\\subset K_{\\psi}$ is the field fixed by $P_{\\bar{\\chi} \\sigma_{2L_1}}$. Lemma $\\ref{l4}$ asserts that $J_{\\psi}\\neq K_{\\psi}$. We will show that $\\mathfrak{F}\\cap K_{\\psi}\\subseteq J_{\\psi}$. First, we show how the result follows from this.\n\\par Set $\\mathfrak{L}:=\\mathfrak{F}\\cdot K_{\\psi}=K_{\\psi_1}\\cdots K_{\\psi_r}\\cdot K_{\\psi}$. We consider the following field diagram,\n\\begin{equation*}\n\\begin{tikzpicture}[node distance = 1.8cm, auto]\n \\node (Qmu) {$\\mathbb{Q}(\\mu_p).$};\n \\node (FK) [above of=Qmu, node distance= 1.25cm] {$\\mathfrak{F}\\cap K_{\\psi}$};\n \\node (J) [above of=FK, right of= FK, node distance= 0.9 cm] {$J_{\\psi}$};\n \\node (Kpsi) [above of=J, right of= J, node distance= 0.9 cm] {$K_{\\psi}$};\n \\node (F) [above of=FK, left of= FK] {\\small $\\mathfrak{F}$};\n \\node(P) [above of= Qmu, right of= Qmu] {$\\mathbb{Q}(\\mu_{p^2})$};\n \\node(FdotK)[above of= F, right of= F, node distance= 1.8cm] {$\\mathfrak{L}$};\n \\draw[-] (Qmu) to node {} (FK);\n \\draw[-] (Qmu) to node {} (P);\n \\draw[-] (FK) to node {} (J);\n \\draw[-] (FK) to node {} (F);\n \\draw[-] (F) to node {} (FdotK);\n \\draw[-] (J) to node {} (Kpsi);\n \\draw[-] (Kpsi) to node {} (FdotK);\n \\end{tikzpicture}\n \\end{equation*}\nBy Lemma $\\ref{22Dec5}$, the intersection $ K\\cap \\mathbb{Q}(\\mu_{p^2})=\\mathbb{Q}(\\mu_p) $. In fact, Lemma $\\ref{22Dec5}$ asserts that $\\mathfrak{F}\\cap\\mathbb{Q}(\\mu_{p^2})=\\mathbb{Q}(\\mu_p)$. Therefore there is a prime $v$ which is\n\\begin{enumerate}\n \\item split in $\\op{Gal}(\\mathfrak{F}\/\\mathbb{Q})$,\n \\item nonsplit in $\\op{Gal}(\\mathbb{Q}(\\mu_{p^2})\/\\mathbb{Q}(\\mu_p))$,\n \\item nonsplit in $\\op{Gal}(K_{\\psi}\/J_{\\psi})$.\n\\end{enumerate} Since $K=\\mathbb{Q}(\\bar{\\rho},\\mu_p)$ is contained in $\\mathfrak{F}$, the prime $v$ is a trivial prime. Since $v$ splits in $\\op{Gal}(\\mathfrak{F}\/\\mathbb{Q})$, we have that $\\psi_{i\\restriction \\op{G}_v}=0$ for $i=1,\\dots, r$. Since $v$ does not split in $\\op{Gal}(K_{\\psi}\/K)$, we have that $\\psi_{\\restriction \\op{G}_v}\\neq 0$.\n\\par We begin by showing that $K_{\\psi}$ is not contained in $\\mathfrak{F}$. This is equivalent to the assertion that $\\mathfrak{L}$ is not equal to $\\mathfrak{F}$. Each of the classes $\\psi, \\psi_1,\\dots, \\psi_r$ is in the image of the inflation map \n\\[H^1(\\operatorname{Gal}(\\mathfrak{L}\/\\mathbb{Q}), \\operatorname{Ad}^0\\bar{\\rho}^*)\\xrightarrow{\\op{inf}}H^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*),\\]and hence the above map is an isomorphism. It follows that\n\\begin{equation*}\nh^1(\\operatorname{Gal}(\\mathfrak{L}\/\\mathbb{Q}), \\operatorname{Ad}^0\\bar{\\rho}^*)=h^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*)\\geq r+1.\\end{equation*}\nIt suffices to show that\n$h^1(\\operatorname{Gal}(\\mathfrak{F}\/\\mathbb{Q}), \\operatorname{Ad}^0\\bar{\\rho}^*)\\leq r$. We show by induction on $i$ that\n\\begin{equation*}\nh^1(\\operatorname{Gal}( K_{\\psi_1}\\cdots K_{\\psi_i}\/\\mathbb{Q}), \\operatorname{Ad}^0\\bar{\\rho}^*)\\leq i.\\end{equation*} Lemma $\\ref{l3}$ asserts that $H^1(\\op{G}',\\operatorname{Ad}^0\\bar{\\rho}^*)=0$ and hence by inflation-restriction,\n\\begin{equation*}\nH^1(\\operatorname{Gal}(K_{\\psi_1}\/\\mathbb{Q}), \\operatorname{Ad}^0\\bar{\\rho}^*)\\simeq \\op{Hom}(P_1, \\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}.\n\\end{equation*}\nLemma $\\ref{y1}$ asserts that\n\\[\\dim \\op{Hom}(P_1,\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}\\leq 1\\]and hence the case $i=1$ follows.\n\nFor the induction step, set $\\mathfrak{F}_i:=K_{\\psi_1}\\cdots K_{\\psi_{i}}$ and \\[P_{i}:=\\op{Gal}(\\mathfrak{F}_{i}\/\\mathfrak{F}_{i-1})\\simeq \\op{Gal}(K_{\\psi_{i}}\/K_{\\psi_{i}}\\cap \\mathfrak{F}_{i-1}).\\] Lemma $\\ref{y1}$ asserts that \\[\\dim \\op{Hom}(P_{i},\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\op{G}'}\\leq 1\\]\nfrom which we see from inflation-restriction\n\\begin{equation*}\nh^1(\\operatorname{Gal}(\\mathfrak{F}_i\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)\n \\leq h^1(\\operatorname{Gal}(\\mathfrak{F}_{i-1}\/\\mathbb{Q}),\\operatorname{Ad}^0\\bar{\\rho}^*)+1.\n\\end{equation*}\n We conclude that $\\mathfrak{L}\\neq \\mathfrak{F}$ and thus we have deduced that $K_{\\psi}\\cap \\mathfrak{F}\\neq K_{\\psi}$. Set $Q:=\\op{Gal}(K_{\\psi}\/K_{\\psi}\\cap \\mathfrak{F})$, by Lemma $\\ref{mainin}$, \n \\begin{equation*}\n Q_{\\bar{\\chi}\\sigma_{2L_1}}\\simeq (\\operatorname{Ad}^0\\bar{\\rho}^*)_{\\bar{\\chi}\\sigma_{2L_1}}\\simeq P_{\\bar{\\chi}\\sigma_{2L_1}}.\n \\end{equation*}\nWe deduce that $K_{\\psi}\\cap \\mathfrak{F}$ is contained in $J_{\\psi}$. This completes the proof.\n\\end{proof}\n\\begin{Def}\nLet $\\mathcal{J}$ be a set of trivial primes that contains the set $S$ and $v\\notin \\mathcal{J}$ be a trivial prime. Denote by $\\Psi_{\\mathcal{J}}^k$ and $\\Psi_{\\mathcal{J},v}^k$ the maps defined by\n\\begin{equation*}\n\\Psi_{\\mathcal{J}}^k:H^1(\\operatorname{G}_{\\mathbb{Q},\\mathcal{J}},(\\operatorname{Ad}^0\\bar{\\rho})_k)\\xrightarrow{res_{\\mathcal{J}}}\\bigoplus_{w\\in \\mathcal{J}} H^1(\\op{G}_w, (\\operatorname{Ad}^0\\bar{\\rho})_k)\n\\end{equation*}\nand\n\\begin{equation*}\n\\Psi_{\\mathcal{J},v}^k: H^1(\\operatorname{G}_{\\mathbb{Q},\\mathcal{J}\\cup\\{v\\}},(\\operatorname{Ad}^0\\bar{\\rho})_k)\\xrightarrow{res_\\mathcal{J}}\\bigoplus_{w\\in \\mathcal{J}} H^1(\\op{G}_w, (\\operatorname{Ad}^0\\bar{\\rho})_k).\n\\end{equation*}\nLet $\\tau_v$ be a generator of the maximal pro-$p$ quotient of the tame inertia at $v$, denote by\n\\begin{equation*}\n\\pi_{\\mathcal{J},v}^k: H^1(\\operatorname{G}_{\\mathbb{Q},\\mathcal{J}\\cup\\{v\\}}, (\\operatorname{Ad}^0\\bar{\\rho})_k)\\rightarrow (\\operatorname{Ad}^0\\bar{\\rho})_k\n\\end{equation*}\nthe evaluation map defined by\n\\[\\pi_{\\mathcal{J},v}^k(f):=f(\\tau_v).\\]\n\\end{Def}\n\\begin{Lemma}\\label{lemmaDec26}\nLet $T$ be a set of primes as in Proposition $\\ref{Shavanishing}$ that contains the set $S$ and $k$ an integer. Suppose $v\\notin T$ is a trivial prime with the property that for all $\\beta\\in H^1(\\op{G}_{\\mathbb{Q},T},(\\operatorname{Ad}^0\\bar{\\rho})_k^*)$, the restriction $\\beta_{\\restriction \\operatorname{G}_v}=0$. The following are exact:\n\\begin{equation}\\label{shortexact1}\n0\\rightarrow \\op{ker}\\Psi_{T}^k\\xrightarrow{inf} \\op{ker}\\Psi_{T,v}^k\\xrightarrow{\\pi_v^k} (\\operatorname{Ad}^0\\bar{\\rho})_k\\rightarrow 0,\n\\end{equation} \n\\begin{equation}\\label{shortexact2}\n0\\rightarrow H^1(\\op{G}_{\\mathbb{Q},T},(\\operatorname{Ad}^0\\bar{\\rho})_k)\\xrightarrow{inf} H^1(\\op{G}_{\\mathbb{Q},T\\cup\\{v\\}},(\\operatorname{Ad}^0\\bar{\\rho})_k)\\xrightarrow{\\pi_v^k} (\\operatorname{Ad}^0\\bar{\\rho})_k\\rightarrow 0.\n\\end{equation} \nFurther, the image of $\\Psi_T$ is equal to the image of $\\Psi_{T,v}$.\n\\end{Lemma}\n\n\\begin{proof}\nClearly the composite of the maps is zero and $\\eqref{shortexact1}$ is exact in the middle. Denote by $\\op{res}_v$ the restriction map: \\[\\op{res}_v:H^1(\\op{G}_{\\mathbb{Q}, T\\cup\\{v\\}},(\\operatorname{Ad}^0\\bar{\\rho})_k^*)\\rightarrow H^1(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k^*).\\]By assumption, $H^1(\\op{G}_{\\mathbb{Q},T},(\\operatorname{Ad}^0\\bar{\\rho})_k^*)$ and $\\op{ker}\\op{res}_v$ are equal. By the local Euler characteristic formula and local duality, \\[h^1(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k)-h^0(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k)\\]\\[=h^2(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k)=h^0(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k^*)=\\dim (\\operatorname{Ad}^0\\bar{\\rho})_k.\\] By Wiles' Formula $\\eqref{wilesformula}$, \n\\[\\begin{split}\\dim \\op{ker}\\Psi_{T,v}^k=&\\dim \\op{ker}\\Psi_{T}^k+\\dim \\op{ker}\\op{res}_v-h^1(\\op{G}_{\\mathbb{Q},T},(\\operatorname{Ad}^0\\bar{\\rho})_k^*)\\\\+&h^1(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k^*)-h^0(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k^*)\\\\\n=&\\dim \\op{ker}\\Psi_{T}^k+\\dim (\\operatorname{Ad}^0\\bar{\\rho})_k\\\\\n\\end{split}\\]and the exactness of $\\eqref{shortexact1}$ follows. The exactness of $\\eqref{shortexact2}$ follows by the same arguments. Therefore, \n\\[\\begin{split}\\dim \\op{im} \\Psi_{T,v}=&h^1(\\op{G}_{\\mathbb{Q},T\\cup \\{v\\}}, (\\operatorname{Ad}^0\\bar{\\rho})_k)-\\dim \\op{ker} \\Psi_{T,v}\\\\=&h^1(\\op{G}_{\\mathbb{Q},T}, (\\operatorname{Ad}^0\\bar{\\rho})_k)-\\dim \\op{ker} \\Psi_{T}=\\dim \\op{im} \\Psi_{T}.\\\\\\end{split}\\]\n\\end{proof}\nLet $M$ be an $\\mathbb{F}_q[\\operatorname{G}_{w}]$-module which is a finite dimensional $\\mathbb{F}_q$-vector space. The cup product induces the map\n\\[H^1(\\operatorname{G}_w, M)\\times H^1(\\operatorname{G}_w, M^*)\\rightarrow H^2(\\operatorname{G}_w, \\mathbb{F}_q(\\bar{\\chi}))\\xrightarrow{\\sim} \\mathbb{F}_q\\] taking $f_1\\in H^1(\\operatorname{G}_w, M)$ and $f_2\\in H^1(\\operatorname{G}_w, M^*)$ to $\\op{inv}_w(f_1\\cup f_2)\\in \\mathbb{F}_q$. Define the non-degenerate pairing \n\\[\\left(\\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho})\\right)\\times \\left(\\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}^*)\\right) \\rightarrow \\mathbb{F}_q\\]defined by\n$a\\cup b=\\sum_{w\\in T} \\op{inv}_w (a_w\\cup b_w)$. Denote by $\\op{Ann}((z_w)_{w\\in T})$ the annihilator of the tuple $(z_w)_{w\\in T}$. Recall that we assume that $(z_w)_{w\\in T}$ does not arise from a global class unramified outside $T$. In particular, the tuple $(z_w)_{w\\in T}$ is not zero, and as a result, $\\op{Ann}((z_w)_{w\\in T})$ is a codimension one subspace of $\\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}^*)$. Let $\\Psi_T$ and $\\Psi_T^*$ denote the restriction maps\n\\begin{equation*}\n\\begin{split}&\\Psi_{T}:H^1(\\operatorname{G}_{\\mathbb{Q},T},\\operatorname{Ad}^0\\bar{\\rho})\\rightarrow\\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}),\\\\&\\Psi_{T}^*:H^1(\\operatorname{G}_{\\mathbb{Q},T},\\operatorname{Ad}^0\\bar{\\rho}^*)\\rightarrow\\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}^*).\n\\end{split}\n\\end{equation*}\nFrom the exactness of the Poitou-Tate sequence \\cite[Theorem 8.6.14]{NW}, it follows that the images of $\\Psi_T$ and $\\Psi_T^*$ are exact annihilators of one another. Since it is assumed that $(z_w)_{w\\in T}$ is not in the image $\\Psi_T^*$, it follows that the image of $\\Psi_T$ is not contained in $\\op{Ann}((z_w)_{w\\in T})$. As a result, ${\\Psi_T^*}^{-1}(\\op{Ann}(z_w)_{w\\in T})$ has codimension one in $H^1(\\op{G}_{\\mathbb{Q},T},\\operatorname{Ad}^0\\bar{\\rho}^*)$. Set $(\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}$ for the $\\mathbb{F}_q$ span of the root vector $X_{-2L_1}$.\n\\begin{Prop}\\label{P2}\nLet $T$ be as in Proposition $\\ref{Shavanishing}$. There exists a Chebotarev class $\\mathfrak{l}$ of trivial primes $v$ such that\n\\begin{enumerate}\n\\item\\label{22Decc1} $\\beta_{\\restriction \\operatorname{G}_v}=0$ for all $\\beta \\in H^1(\\operatorname{G}_{\\mathbb{Q},T}, (\\operatorname{Ad}^0\\bar{\\rho})_d^*)$ for $d\\geq -2n+2$,\n\\item\\label{22Decc2} there exists an $\\mathbb{F}_q$ basis $\\{\\psi, \\psi_1,\\dots, \\psi_r\\}$ of $H^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*)$ such that \n\\begin{itemize}\n\\item\\label{12}\n$\\{\\psi_1,\\dots, \\psi_r\\}$ is a basis of ${\\Psi_T^*}^{-1}(Ann(z_{w})_{w\\in T})$ \n\\item\n$\\psi_{\\restriction \\operatorname{G}_v}\\neq 0$ and ${\\psi_j}_{\\restriction \\operatorname{G}_v}=0$ for all $j\\geq 1$. \n\\end{itemize}\n\\end{enumerate}\nFurthermore, there is, for each $v\\in \\mathfrak{l}$, an element $h^{(v)}\\in H^1(\\operatorname{G}_{T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})$ such that \n\\[h^{(v)}|_{\\op{G}_w}=z_w\\] for all $w\\in T$ and \n\\begin{equation}\\label{equation64}\nh^{(v)}(\\tau_v)\\in (\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}\\backslash \\{0\\}.\\end{equation}\n\\end{Prop}\n\\begin{proof}\nFirst, we analyze condition $\\eqref{22Decc1}$.\nRecall that $(\\operatorname{Ad}^0\\bar{\\rho})_d^*$ is the quotient of $\\operatorname{Ad}^0\\bar{\\rho}^*$ by the Galois stable subspace $(\\operatorname{Ad}^0\\bar{\\rho})_d^{\\perp}$, see Definition $\\ref{perpdef}$. Its $\\mathbb{T}$-eigenspaces consist of $(\\operatorname{Ad}^0\\bar{\\rho})_{d,\\bar{\\chi}\\sigma_{\\lambda}^{-1}}^*$, where $\\lambda$ ranges through $\\Phi\\cup \\{1\\}$ with $\\op{ht}(\\lambda)\\geq d$. Condition $\\eqref{thc4}$ of Theorem $\\ref{main}$ asserts that $\\bar{\\chi}\\sigma_{-\\lambda}\\neq \\sigma_{1}$, i.e., $\\sigma_{\\lambda}\\neq \\bar{\\chi}$. Therefore, $(\\operatorname{Ad}^0\\bar{\\rho})_d^*$ contains no trivial eigenspace. Hence, the splitting conditions imposed by $\\eqref{22Decc1}$ are independent of the non-splitting condition in $\\mathbb{Q}(\\mu_{p^2})$ imposed by the fact that trivial primes are not $1\\mod{p^2}$. On the other hand, by Proposition $\\ref{P1}$, condition $\\eqref{22Decc2}$ can be satisfied by a Chebotarev class of trivial primes.\n\\par Next, we show that conditions $\\eqref{22Decc1}$ and $\\eqref{22Decc2}$ can be satisfied simultaneously. To show this, note that the condition requiring $\\psi_{\\restriction{\\op{G}_v}}\\neq 0$ is a non-splitting condition of $v$ in $\\op{Gal}(K_{\\psi}\/K)$. By Lemma $\\ref{l4}$, the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace for the $\\mathbb{T}$-action on $\\op{Gal}(K_{\\psi}\/K)$ is nontrivial. We shall require that $v$ does not split in the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $\\op{Gal}(K_{\\psi}\/K)$. On the other hand, \\[(\\operatorname{Ad}^0\\bar{\\rho}^*)_d=\\bigoplus_{ \\op{ht}(\\lambda)\\geq d}(\\operatorname{Ad}^0\\bar{\\rho}^*)_{d,\\bar{\\chi}\\sigma_{\\lambda}^{-1}},\\]does not contain the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $\\operatorname{Ad}^0\\bar{\\rho}^*$. Note that the character $\\bar{\\chi}\\sigma_{2L_1}$ is not twist equivalent to any of the characters occurring in the $\\mathbb{T}$-eigenspace decomposition of $(\\operatorname{Ad}^0\\bar{\\rho}^*)_d$. As a result, it follows via an argument identical to that in proof of Lemma $\\ref{lemma416}$, that the non-splitting condition of $v$ in $\\op{Gal}(K_{\\psi}\/K)_{\\bar{\\chi}\\sigma_{2L_1}}$ may be simultaneously satisfied along with the rest of the splitting conditions.\n\n\n\\par Let $v$ be a trivial prime which satisfies conditions $\\eqref{22Decc1}$ and $\\eqref{22Decc2}$ and moreover is non-split in $\\op{Gal}(K_{\\psi}\/K)_{\\bar{\\chi}\\sigma_{2L_1}}$. Let $d\\geq -2n+2$, Lemma $\\ref{lemmaDec26}$ asserts that the image of\n\\begin{equation*}\n\\Psi_T^d:H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, (\\operatorname{Ad}^0\\bar{\\rho})_d)\\rightarrow \\bigoplus_{w\\in T} H^1(\\op{G}_w, (\\operatorname{Ad}^0\\bar{\\rho})_d)\n\\end{equation*}\nis the same as the image of\n\\begin{equation*}\n\\Psi_{T,v}^d:H^1(\\operatorname{G}_{\\mathbb{Q},T}, (\\operatorname{Ad}^0\\bar{\\rho})_d)\\rightarrow \\bigoplus_{w\\in T} H^1(\\op{G}_w, (\\operatorname{Ad}^0\\bar{\\rho})_d).\n\\end{equation*}\nFor a trivial prime $v$ for which condition $\\eqref{22Decc2}$ is satisfied, it follows from an application of Wiles' formula $\\eqref{wilesformula}$ that the image of the map \n\\begin{equation*}\n\\Psi_{T,v}:H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})\\rightarrow \\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho})\n\\end{equation*}\nis greater than that of the map\n\\begin{equation*}\n\\Psi_T:H^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho})\\rightarrow \\bigoplus_{w\\in T} H^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}).\n\\end{equation*} We next deduce the existence of $h^{(v)}\\in H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})$ satisfying the specified properties. Since the image of $\\Psi_{T,v}$ is greater than the image of $\\Psi_T$, there is a class $g$ in $ H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})$ such that $\\Psi_{T,v}(g)\\notin \\text{Image}(\\Psi_T)$. Let \\[W_1:=\\text{Image}(\\Psi_T)+\\mathbb{F}_q\\cdot \\Psi_{T,v}(g)\\]\nand \n\\[W_2:=\\text{Image}(\\Psi_T)+\\mathbb{F}_q\\cdot (z_w)_{w\\in T}.\\]\nThe argument in \\cite[Proposition 34]{hamblenramakrishna} applies verbatim to imply that $W_1=W_2$ and so we deduce the existence of $h^{(v)}\\in H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})$ for which \\[h^{(v)}_{\\restriction \\op{G}_w}={z_w}_{\\restriction \\op{G}_w}\\] for all $w\\in T$. As we have observed,\n\\[\\text{Image}(\\Psi_{T}^{-2n+2})=\\text{Image}(\\Psi_{T,v}^{-2n+2})\\] since $h^{(v)}\\notin \\text{Image}(\\Psi_T)$ it follows that $h^{(v)}(\\tau_v)$ is not contained in $(\\operatorname{Ad}^0\\bar{\\rho})_{-2n+2}$. Invoking Lemma $\\ref{lemmaDec26}$, we deduce that on adding a suitable linear combination of elements to $h^{(v)}$ from $\\ker \\Psi_{T,v}^d$ for $d> -2n+1$, we modify the class $h^{(v)}$ so that \\[h^{(v)}(\\tau_v)\\in (\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}\\backslash \\{0\\}\\] as required. \n\\end{proof}\nFor $d\\in \\mathbb{Z}$, the natural inclusion $(\\operatorname{Ad}^0\\bar{\\rho})_d^{\\perp}\\hookrightarrow \\operatorname{Ad}^0\\bar{\\rho}^*$ induces a natural map of cohomology groups $H^1(\\op{G}_{\\mathbb{Q}}, (\\operatorname{Ad}^0\\bar{\\rho})_d^{\\perp})\\rightarrow H^1(\\op{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho}^*)$.\n\\begin{Lemma}\nLet $\\mathfrak{l}$ be the Chebotarev class of trivial primes in the Proposition $\\ref{P2}$. Let $\\{X_{\\lambda}^*\\}_{\\lambda\\in \\Phi}$ and $\\{H_1^*,\\dots, H_n^*\\}$ be as in $\\eqref{XHdual}$. There exists an $\\mathbb{F}_q$-independent set \\[\\{\\eta_{\\lambda}^{(v)}\\mid \\lambda \\in \\Phi\\}\\cup\\{\\ \\eta_{1}^{(v)}, \\dots,\\eta_{n}^{(v)}\\}\\] contained in $H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho}^*)$, satisfying the following properties:\n\\begin{enumerate}\n\\item\n$\\eta_{\\lambda}^{(v)}$ is in the image of the natural map \\[H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, (\\operatorname{Ad}^0\\bar{\\rho})_{h+1}^{\\perp})\\rightarrow H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho}^*),\\] where $h=\\op{ht}(\\lambda)$.\n\\item For $i=1,\\dots, n$, the cohomology class $\\eta_{i}^{(v)}$ \nis in the image of the natural map \\[H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, (\\operatorname{Ad}^0\\bar{\\rho})_{1}^{\\perp})\\rightarrow H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho}^*).\\]\n\\item \nFor $\\lambda \\in \\Phi$, we have that $\\eta_{\\lambda}^{(v)}(\\tau_v)= X_{\\lambda}^*$.\n\\item \nFor $i=1,\\dots, n$, we have that $\\eta_{i}^{(v)}(\\tau_v)=H_i^*$.\n\\item \nThe images of the elements $\\eta_{\\lambda}^{(v)}$ are a basis for the cokernel of the inflation map\n\\begin{equation*}\nH^1(\\operatorname{G}_{\\mathbb{Q},T}, \\operatorname{Ad}^0\\bar{\\rho}^*)\\rightarrow H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho}^*).\\end{equation*}\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\n The dual to $(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}$ is $\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k$. Proposition $\\ref{Shavanishing}$ asserts that \\[\\Sh_T^1((\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp*})=0\\] for all $k\\in \\mathbb{Z}$. Wiles' formula \\eqref{wilesformula} asserts that\n\\[\\begin{split}&h^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}},(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})-\\dim \\Sh_{T\\cup\\{v\\}}^1((\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp*})\\\\=& h^1(\\operatorname{G}_{\\mathbb{Q},T},(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})-\\dim \\Sh_{T}^1((\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp*})\\\\+&h^1(\\op{G}_v, (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})-h^0(\\op{G}_v, (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}).\\end{split}.\\] Also, by Proposition $\\ref{Shavanishing}$, we have that \\[\\Sh_T^1(\\operatorname{Ad}^0\\bar{\\rho}\/(\\operatorname{Ad}^0\\bar{\\rho})_k)=0.\\] On applying the local Euler characteristic formula and Tate duality we have that \n\\[h^1(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k)^{\\perp}-h^0(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k)^{\\perp})=h^0(\\op{G}_v,(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp*})=\\dim (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}.\\]For the last equality, note that $\\bar{\\chi}_{\\restriction \\op{G}_v}=1$ since $v\\equiv 1\\mod{p}$ and that the action on $(\\operatorname{Ad}^0\\bar{\\rho})_k)^{\\perp}$ is trivial. It follows that \\[h^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}},(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})=h^1(\\operatorname{G}_{\\mathbb{Q},T},(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})+\\dim (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}\\] and the evaluation map at $\\tau_v$\n\\[H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}}, (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})\\rightarrow (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}\\] induces a short exact sequence\n\\[0\\rightarrow H^1(\\operatorname{G}_{\\mathbb{Q},T}, (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})\\rightarrow H^1(\\operatorname{G}_{\\mathbb{Q},T\\cup\\{v\\}},(\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp})\\rightarrow (\\operatorname{Ad}^0\\bar{\\rho})_k^{\\perp}\\rightarrow 0.\\] The assertion of the Lemma follows.\n\\end{proof}\nLet $v$ a trivial prime in the Chebotarev class $\\mathfrak{l}$ of Proposition $\\ref{P2}$. For $\\lambda\\in \\Phi$, denote by $K_{\\lambda}^{(v)}:=K_{\\eta_{\\lambda}^{(v)}}$ and for $i=1,\\dots, n$, set $K_i^{(v)}:=K_{\\eta_{i}^{(v)}}$. Let $J_i^{(v)}\\subsetneq K_i^{(v)}$ and $J_{\\lambda}^{(v)}\\subsetneq K_{\\lambda}^{(v)}$ denote $J_{\\eta_{i}^{(v)}}$ and $J_{\\eta_{\\lambda}^{(v)}}$ respectively. If $E=K_i^{(v)}$ (resp. $K_{\\lambda}^{(v)}$), denote by $J_E$ the sub-extension $J_{i}^{(v)}$ (resp. $J_{\\lambda}^{(v)}$). Let $\\mathcal{F}^{(v)}$ denote the collection of fields consisting of $K_i^{(v)}$ for $i=1,\\dots, n$ and $K_{\\lambda}^{(v)}$ for $\\lambda\\in \\Phi$. The Chebotarev class $\\mathfrak{l}$ from Proposition $\\ref{P2}$ is defined by Chebotarev classes in a collection of fields $\\mathcal{F}_{\\mathfrak{l}}$. More specifically, $\\mathcal{F}_{\\mathfrak{l}}$ is the collection of fields:\n\\begin{itemize}\n \\item $K_{\\psi}, K_{\\psi_1},\\dots, K_{\\psi_r}$ from Proposition $\\ref{P2}$,\n \\item $K_{\\beta}$ as $\\beta$ runs through all cohomology classes $H^1(\\op{G}_{\\mathbb{Q},T}, (\\operatorname{Ad}^0\\bar{\\rho}^*)_d)$, where $d\\geq -2n+2$,\n \\item $K(\\zeta_2)$ (with $\\zeta_2$ defined at the start of the section),\n \\item $K(\\mu_{p^2})$.\n\\end{itemize}For $v\\in \\mathfrak{l}$ from Proposition $\\ref{P2}$, recall that $L_{h^{(v)}}$ is the field extension of $L$ cut out by \\[h^{(v)}_{\\restriction \\op{G}_L}:\\op{G}_L\\rightarrow \\operatorname{Ad}^0\\bar{\\rho}.\\] Associate to a set of trivial primes $A=\\{v_1,\\dots, v_k\\}$ in $\\mathfrak{l}$, \\[\\mathcal{F}_A:=\\cup_{i=1}^k \\mathcal{F}^{(v_i)}\\text{, and } \\mathcal{L}_A:=\\{L_{h^{(v_1)}},\\dots, L_{h^{(v_k)}}\\}.\\]\n\\begin{Lemma}\\label{eigenspaceeta}\nLet $A=\\{v_1,\\dots, v_k\\}\\subset \\mathfrak{l}$.\n\\begin{enumerate}\n \\item\\label{66c1} Let $F_1$ be a field in the collection $\\mathcal{F}_A$ and $F_2$ be the composite of all the other fields in $ \\mathcal{F}_A\\cup\\mathcal{L}_A\\cup \\mathcal{F}_{\\mathfrak{l}}$. Then $F_1$ is not contained in $F_2$. Moreover, the intersection $F_1\\cap F_2$ is contained in $J_{F_1}$.\n \\item\\label{66c2} Let $M_1$ be a field in the collection $\\mathcal{L}_A$ and $M_2$ denote the composite of all the other fields in $\\mathcal{F}_A\\cup \\mathcal{L}_A\\cup \\mathcal{F}_{\\mathfrak{l}}$. The intersection $M_1\\cap M_2=L$.\n\\end{enumerate}\n\\end{Lemma}\n\\begin{proof}\nPart $\\eqref{66c1}$ is obtained from an application of Proposition $\\ref{414}$, as we now explain. In accordance with the statement of Proposition $\\ref{414}$, we define a sequence of linearly independent classes \\[\\theta_0,\\dots, \\theta_t\\in H^1(\\op{G}_{\\mathbb{Q},T\\cup A},\\operatorname{Ad}^0\\bar{\\rho}^*),\\] and a sequence of fields $\\mathbb{L}_1, \\dots, \\mathbb{L}_b$, each of which is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n\\par Consider the classes $\\eta_i^{(v_j)}$ and $\\eta_{\\lambda}^{(v_j)}$ as $i=1,\\dots, n$, $\\lambda\\in \\Phi$ and $j=1,\\dots, k$. Enumerate these classes by $\\theta_0,\\dots, \\theta_a$, so that $\\theta_0$ is the cohomology class specified in the description of $F_1$, i.e. $F_1=K_{\\theta_0}$. The number $a$ is equal to $(nk\\# \\Phi)-1$ and $\\mathcal{F}_A=\\{K_{\\theta_0}, K_{\\theta_1},\\dots, K_{\\theta_a}\\}$. Let $\\theta_{a+1},\\dots, \\theta_{t}$ be a basis of $H^1(\\op{G}_{\\mathbb{Q},T},\\operatorname{Ad}^0\\bar{\\rho}^*)$. Recall that for $\\lambda \\in \\Phi$, we have that $\\eta_{\\lambda}^{(v_j)}(\\tau_{v_j})= X_{\\lambda}^*$, and for $i=1,\\dots, n$, we have that $\\eta_{i}^{(v_j)}(\\tau_{v_j})=H_i^*$. The classes $\\theta_{a+1},\\dots, \\theta_{t}$ are unramified at each of the primes $v_j\\in A$. It is thus, easy to see that $\\theta_0,\\dots, \\theta_{t}$ are linearly independent. Let $\\mathbb{L}_1,\\dots, \\mathbb{L}_l$ be an enumeration for the fields $K_{\\beta}$, as $\\beta$ runs through all cohomology classes $H^1(\\op{G}_{\\mathbb{Q},T}, (\\operatorname{Ad}^0\\bar{\\rho}^*)_d)$ for $d\\geq -2n+2$. Let $\\mathbb{L}_{l+1}$ be the field $K(\\zeta_2)$ and $\\mathbb{L}_{l+2}$ the field $K(\\mu_{p^2})$. The collection of fields $\\mathcal{F}_{\\mathfrak{l}}$ consists of $K_{\\theta_{i}}$ for $i=a+1,\\dots, t$ and $\\mathbb{L}_i$ for $i=1,\\dots, l+2$. Next, we have to account for the fields in $\\mathcal{L}_A$. Let $\\mathbb{L}_{l+3},\\dots, \\mathbb{L}_b$ be an enumeration of the fields $L_{h^{(v_1)}}, \\dots, L_{h^{(v_k)}}$. Thus, the collection of fields $\\mathcal{L}_A$ is $\\{\\mathbb{L}_{l+3},\\dots, \\mathbb{L}_b\\}$. In order to apply Proposition $\\ref{414}$, it suffices to show that each of the fields $\\mathbb{L}_1,\\dots, \\mathbb{L}_{b}$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$. Note that:\n\\begin{itemize}\n \\item by Lemma $\\ref{415}$, part $\\eqref{415c1}$, each of the fields $\\mathbb{L}_{1},\\dots, \\mathbb{L}_l$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$,\n \\item by part $\\eqref{415c2}$, $\\mathbb{L}_{l+2}$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$,\n \\item by part $\\eqref{415c3}$, $\\mathbb{L}_{l+3},\\dots, \\mathbb{L}_{b}$ are unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$,\n \\item and by part $\\eqref{415c4}$, $\\mathbb{L}_{l+1}$ is unrelated to $\\operatorname{Ad}^0\\bar{\\rho}^*$.\n\\end{itemize}\n\n By Proposition $\\ref{414}$, $F_1$ is not contained in $F_2$ and it follows from Lemma $\\ref{mainin}$ that $F_1\\cap F_2\\subseteq J_{F_1}$.\n\\par Assume without loss of generality that $M_1=L_{h^{(v_1)}}$. Recall that by \\eqref{equation64}, we have that \\[h^{(v_1)}(\\tau_{v_1})\\in (\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}\\backslash\\{0\\}.\\]As a result, $v_1$ is ramified in the $\\sigma_{-2L_1}$-eigenspace of $\\op{Gal}(M_1\/L)$. On the other hand, $v_1$ is unramified in each of the field extensions in $\\mathcal{F}_{\\mathfrak{l}}$ and $\\mathcal{L}_A\\backslash \\{M_1\\}$. Since the classes $\\eta_i^{(v_j)}$ and $\\eta_{\\lambda}^{(v_j)}$ are valued in $\\operatorname{Ad}^0\\bar{\\rho}^*$, there is no $\\sigma_{-2L_1}$-eigenspace for the action of $\\mathbb{T}$ on $\\op{Gal}(K_{\\theta_i}\/K)$ for $K_{\\theta_i}\\in \\mathcal{F}_A$. As a result, $v_1$ is unramified in the $\\sigma_{-2L_1}$-eigenspace of $\\op{Gal}(M_2\/L)$. Therefore, $M_1\\not\\subseteq M_2$. Identify $Q:=\\operatorname{Gal}(M_1\/M_1\\cap M_2)$ with a subgroup of $h^{(v_1)}(\\op{G}_L)\\subseteq \\operatorname{Ad}^0\\bar{\\rho}$. By Lemma $\\ref{fullrankLemma}$ it suffices to show that $Q_{-2L_1} \\neq 0$. Since $v_1$ is unramified in $M_2$, the image of $\\tau_{v_1}$ in $\\op{Gal}(M_1\/L)$ lies in $\\op{Gal}(M_1\/M_1\\cap M_2)$. Since $h^{(v_1)}(\\tau_{v_1})$ is in $(\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}\\backslash\\{0\\}$, we deduce that $Q_{-2L_1} \\neq 0$. The assertion $\\eqref{66c2}$ follows.\n\\end{proof}\n\\begin{Lemma}\nLet $v\\in \\mathfrak{l}$ and $h^{(v)}$ be as in Proposition $\\ref{P2}$. Then the $\\operatorname{Gal}(L_{h^{(v)}}\/L)\\simeq \\operatorname{Ad}^0\\bar{\\rho}$.\n\\end{Lemma}\\label{hvLemma}\n\\begin{proof}\nLet $Q:=\\operatorname{Gal}(L_{h^{(v)}}\/L)\\subseteq \\operatorname{Ad}^0\\bar{\\rho}$. Since $Q_{-2L_1} \\neq 0$, the assertion follows from Lemma $\\ref{fullrankLemma}$.\n\\end{proof}\n\\begin{Prop}\\label{lifttorho3} For a pair $(v_1,v_2)$ of trivial primes in $\\mathfrak{l}$ in Proposition $\\ref{P2}$ set $h=-h^{(v_1)}+2h^{(v_2)}$ and $\\rho_2:=(I+ph)\\zeta_2$. There is a pair $(v_1,v_2)$ such that $\\rho_{2\\restriction \\op{G}_w}\\in \\mathcal{C}_w$ for all $w\\in T$ and $\\rho_{2\\restriction \\op{G}_{v_i}}\\in \\mathcal{C}_{v_i}^{ram}$ for $i=1,2$.\n\\end{Prop}\n\\begin{proof}\nFor $i=1,2,$ we set $\\mathcal{C}_{v_i}:=\\mathcal{C}_{v_i}^{ram}$. Note that $h_{\\restriction \\op{G}_w}=z_w$\nfor all $w\\in T$ and hence $\\rho_{2\\restriction \\op{G}_w}\\in \\mathcal{C}_w$ for all $w\\in T$. This is not the case at the primes $v_1$ and $v_2$. We show that one may indeed find a pair $(v_1,v_2)\\in \\mathfrak{l}\\times \\mathfrak{l}$ so that $(I+pz_{v_i})\\zeta_2\\in \\mathcal{C}_{v_i}^{ram}$ for $i=1,2$. Consider for $v\\in \\mathfrak{l}$, the pair of elements $(\\zeta_2(\\sigma_v),h^{(v)}(\\sigma_v))$, and let $A=(A_1,A_2)$ be the pair of matrices which occurs most frequently, that is, with maximal upper density. The choice of $A$ is not necessarily unique. Let $\\mathfrak{l}_1=\\{v\\in \\mathfrak{l}\\mid \\zeta_2(\\sigma_v)=A_1,h^{(v)}(\\sigma_v)=A_2\\}$. Since there are finitely many choices for $A$, the set of primes $\\mathfrak{l}_1$ has positive upper-density. Since $h(\\tau_{v_i})\\in(\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}$ and $\\zeta_2$ is unramified at $v_i$, we have that \\[(\\operatorname{Id}+ph(\\tau_{v_i}))\\zeta_2(\\tau_{v_i})=(\\operatorname{Id}+ph(\\tau_{v_i}))\\in \\op{U}_{-2L_1}.\\] Furthermore, since $h(\\tau_{v_i})\\neq 0$, the additional condition on $(\\operatorname{Id}+ph(\\tau_{v_i}))\\zeta_2(\\tau_{v_i})$ (see Definition $\\ref{defconditions}$ part $\\eqref{defconditions2}$) is satisfied. Since $\\zeta_2(\\sigma_v)$ is fixed throughout $\\mathfrak{l}_1$, there are (not necessarily unique) matrices $C_i$ such that if $h(\\sigma_{v_i})=C_i$, we will have \n$(\\operatorname{Id}+ph){\\zeta_2}_{\\restriction \\operatorname{G}_{v_i}} \\in \\mathcal{C}_{v_i}$ for $i=1,2$. The values $h^{(v_i)}(\\sigma_{v_j})$ are represented in the table below:\n\\begin{center}\n\\begin{tabular}{c|c|c } \n & $\\sigma_{v_1}$ & $\\sigma_{v_2}$ \\\\ [0.5 ex]\n \\hline\n $h^{(v_1)}$ & $A_2$ & $R$ \\\\\n \\hline\n $h^{(v_2)}$ & $E$ & $A_2$. \\\\ \n\\end{tabular}\n\\end{center}\nWe need $E=(A_2+C_1)\/2$ and $R=2A_2-C_2$. Note that for an arbitrary pair $(v_1,v_2)\\in \\mathfrak{l}_1\\times \\mathfrak{l}_1$, this need not be the case. What follows is a recipe for producing a pair $(v_1,v_2)$ such that $E=(A_2+C_1)\/2$ and $R=2A_2-C_2$.\n\\par For $v\\in \\mathfrak{l}_1$, let $\\delta^{(v)}\\in H^1(\\operatorname{G}_v, \\operatorname{Ad}^0\\bar{\\rho}^*)$ be the cohomology class given by $\\delta^{(v)}(\\sigma_v)=X_{-2L_1}^*$ and $\\delta^{(v)}(\\tau_v)=0$. Let $y$\nbe the element that occurs most frequently among the elements $\\operatorname{inv}_v (\\delta^{(v)} \\cup h^{(v)})$ among primes $v$ of $\\mathfrak{l}_1$. Set \\[\\mathfrak{l}_2 =\\{\nv\\in \\mathfrak{l}_1 \\mid \\operatorname{inv}_v (\\delta^{(v)} \\cup h^{(v)})= y\\},\\] $\\mathfrak{l}_2$ has positive\nupper density. Suppose we first choose $v_1\\in \\mathfrak{l}_2$. Recall that $h^{(v_1)}(\\tau_{v_1})\\in (\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}$. By Lemma $\\ref{fullrankLemma}$, the class $h^{(v_1)}$ has full rank, i.e. $h^{(v_1)}(\\op{G}_K)=\\operatorname{Ad}^0\\bar{\\rho}$. In particular, $2A_2-C_2$ is contained in $h^{(v_1)}(\\op{G}_K)$. Choosing $v_2$ such that $h^{(v_1)}(\\sigma_{v_2})=2A_2-C_2$ is a Chebotarev condition on the splitting of $v_2$ in $L_{h^{(v_1)}}$. We show that $h^{(v_2)}(\\sigma_{v_1})$ is determined by how $v_2$ splits in the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace each of the fields in $\\mathcal{F}^{(v_1)}$. Since $h^{(v_2)}$ is unramified at $v_1$, the values $\\eta_{\\lambda}^{(v_1)}(\\tau_{v_1})$ and $h^{(v_2)}(\\sigma_{v_1})$ determine $(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})_{\\restriction \\op{G}_{v_1}}$. Express $h^{(v_2)}(\\sigma_{v_1})=\\sum_{\\lambda} a_{\\lambda} X_{\\lambda}+\\sum_{i=1}^n a_{i} H_{i}$. As $\\eta_{\\lambda}^{(v_1)}(\\tau_{v_1})= X_{\\lambda}^*$, we see that $\\op{inv}_{v_1}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})$ determines $a_{\\lambda}$. Likewise, $\\op{inv}_{v_1}(\\eta_{i}^{(v_1)}\\cup h^{(v_2)})$ determines $a_{i}$. For $v\\in \\mathfrak{l}$ and $\\lambda\\in \\Phi$, set $z_{\\lambda}^{(v)}$ to be equal to $\\op{inv}_v(\\eta_{\\lambda}^{(v)}\\cup h^{(v)})$. The global reciprocity law asserts that\n\\[\\sum_{w\\in T\\cup \\{v_1,v_2\\}} \\op{inv}_w(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})=0,\\text{ and }\\sum_{w\\in T\\cup \\{v_1\\}} \\op{inv}_w(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_1)})=0.\\]Since $h^{(v_2)}_{\\restriction \\op{G}_w}=z_w=h^{(v_1)}_{\\restriction \\op{G}_w}$ for $w\\in T$, we deduce that\n\\begin{equation*}\n\\begin{split}\n\\op{inv}_{v_1}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)}) &=-\\sum_{w\\in T} \\op{inv}_{w}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})-\\op{inv}_{v_2}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})\\\\\n&=-\\sum_{w\\in T} \\op{inv}_{w}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_1)})-\\op{inv}_{v_2}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})\\\\\n&=\\op{inv}_{v_1}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_1)})-\\op{inv}_{v_2}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})\\\\\n&=z_{\\lambda}^{(v_1)}-\\op{inv}_{v_2}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)}).\n\\end{split}\n\\end{equation*}\nSince $z_{\\lambda}^{(v_1)}$ depends on $v_1$ which is fixed, the variance of the right hand side of the equation comes from the term $\\op{inv}_{v_2}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})$. The specification of $h^{(v_2)}(\\sigma_{v_1})$ amounts to the specification of $\\op{inv}_{v_1}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})$ for $\\lambda\\in \\Phi$ and $\\op{inv}_{v_1}(\\eta_{i}^{(v_1)}\\cup h^{(v_2)})$ for $i=1,\\dots,n$. Set $u_{\\lambda}$ to be $\\eta_{\\lambda}^{(v_1)}(\\sigma_{v_2})(X_{-2L_1})$ for $\\lambda \\in \\Phi$ and set $u_i$ to be $\\eta_{i}^{(v_1)}(\\sigma_{v_2})(X_{-2L_1})$ for $i=1,\\dots, n$. Since $h^{(v_2)}(\\tau_{v_2})$ is a multiple of $X_{-2L_1}$, we see that \n\\[\\begin{split}&\\op{inv}_{v_2}(\\eta_{\\lambda}^{(v_1)}\\cup h^{(v_2)})=\\op{inv}_{v_2}(u_{\\lambda}\\delta^{(v_2)}\\cup h^{(v_2)})\\\\\n&\\op{inv}_{v_2}(\\eta_{i}^{(v_1)}\\cup h^{(v_2)})=\\op{inv}_{v_2}(u_{i}\\delta^{(v_2)}\\cup h^{(v_2)}).\\\\\n\\end{split}\\] Moreover since $h^{(v_2)}(\\tau_{v_2})$ is non-zero, we can choose $b\\in \\mathbb{F}_q$ such that $\\op{inv}_{v_2}(b\\delta^{(v_2)}\\cup h^{(v_2)})$ takes on any desired value. Note that $\\op{inv}_{v_2}(\\delta^{(v_2)}\\cup h^{(v_2)})$ is set to equal $y$ for all $v_2\\in \\mathfrak{l}_2$, i.e., does not depend on the choice of $v_2\\in \\mathfrak{l}_2$. As a result, for $v_1\\in \\mathfrak{l}_2$, there exist values $\\{b_{\\lambda}\\}_{\\lambda\\in \\Phi}$ and $\\{b_i\\}_{i=1,\\dots, n}$ depending only on $v_1$ such that if $u_{\\lambda}=b_{\\lambda}$ for $\\lambda \\in \\Phi$ and $u_{i}=b_{i}$ for $i=1,\\dots, n$, then, \\[h^{(v_2)}(\\sigma_{v_1})=(A_2+C_1)\/2.\\] The condition requiring $h^{(v_2)}(\\sigma_{v_1})=(A_2+C_1)\/2$, is determined by $\\eta_{\\lambda}^{(v_1)}(\\sigma_{v_2})(X_{-2L_1})$ for $\\lambda \\in \\Phi$ and by $\\eta_{i}^{(v_1)}(\\sigma_{v_2})(X_{-2L_1})$ for $i=1,\\dots, n$. Note that $v_2$ is unramified in $K_{\\eta_{\\lambda}^{(v_1)}}$ and the value of $\\eta_{\\lambda}^{(v_1)}(\\sigma_{v_2})(X_{-2L_1})$ is determined by the projection of $\\sigma_{v_2}$ to the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $\\op{Gal}(K_{\\eta_{\\lambda}^{(v_1)}}\/K)$, when viewed as a $\\mathbb{T}$-module. Recall that $J_{\\eta_{\\lambda}^{(v_1)}}$ is the subextension $K\\subseteq J_{\\eta_{\\lambda}^{(v_1)}}\\subsetneq K_{\\eta_{\\lambda}^{(v_1)}}$ such that \\[\\op{Gal}(K_{\\eta_{\\lambda}^{(v_1)}}\/K)_{\\bar{\\chi}\\sigma_{2L_1}}\\simeq \\op{Gal}(K_{\\eta_{\\lambda}^{(v_1)}}\/J_{\\eta_{\\lambda}^{(v_1)}}).\\] Since $v_2$ is a trivial prime, it is split in $K$. One may indeed insist that $v_2$ is split in $J_{\\eta_{\\lambda}^{(v_1)}}$ and $\\sigma_{v_2}$ takes on the appropriate value in $\\op{Gal}(K_{\\eta_{\\lambda}^{(v_1)}}\/J_{\\eta_{\\lambda}^{(v_1)}})$ so that $u_{\\lambda}=b_{\\lambda}$. Hence, the condition $u_{\\lambda}=b_{\\lambda}$ is simply a condition on the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $K_{\\eta_{\\lambda}^{(v_1)}}\/K$. Likewise, the condition $u_{i}=b_{i}$ is a condition on the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace of $K_{\\eta_{i}^{(v_1)}}\/K$. To summarize, the condition requiring $h^{(v_2)}(\\sigma_{v_1})=(A_2+C_1)\/2$, is equivalent to Chebotarev conditions on the splitting of $v_2$ in the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspaces of the fields in $\\mathcal{F}^{(v_1)}$, in the sense made precise in the preceding discussion.\n\\par Suppose that for the choice of $v_1\\in \\mathfrak{l}_2$, there is a $v_2\\in \\mathfrak{l}_2$ for which the required conditions are satisfied:\n\\begin{enumerate}\n \\item the condition on the splitting of $v_2$ in $L_{h^{(v_1)}}$ which amounts to specifying $h^{(v_1)}(\\sigma_{v_2})$,\n \\item the condition on the splitting of $v_2$ in the fields $\\mathcal{F}^{(v_1)}$ which amounts to specifying $h^{(v_2)}(\\sigma_{v_1})$.\n\\end{enumerate}Then we are done. Note that $\\mathfrak{l}_2$ is not a Chebotarev condition, it has only been observed that $\\mathfrak{l}_2$ has positive upper density. Consider the case when there is no choice of $v_2\\in \\mathfrak{l}_2$ for which the above conditions are satisfied. Let $\\mathfrak{l}_{v_1}$ be the subset of $\\mathfrak{l}$ for which $(R, E)\\neq (2A_2-C_2, \\frac{(A_2 + C_1)}{2})$ for the choice of $v_1$. We have thus assumed that $\\mathfrak{l}_2\\subseteq \\mathfrak{l}_{v_1}$, it follows that the upper density $\\delta(\\mathfrak{l}_2)$ is less than or equal to the upper density $\\delta(\\mathfrak{l}_{v_1})$. \n\\par Set $\\mathcal{E}^{(v_1)}$ to be the composite of the field $L_{h^{(v_1)}}$ with the fields in $\\mathcal{F}^{(v_1)}$ and let $\\mathfrak{F}_{\\mathfrak{l}}$ be the composite of fields in $\\mathcal{F}_{\\mathfrak{l}}$. We show that there is an element $x\\in\\op{Gal}(\\mathcal{E}^{(v_1)}\\cdot \\mathfrak{F}_{\\mathfrak{l}}\/K)$ such that if $v_2$ is trivial prime such that the Frobenius at $v_2$ maps to $x$, then $v_2\\in \\mathfrak{l}$ and the conditions on $v_2$ are satisfied. Said differently, if $\\sigma_{v_2}=x$, then $v_2\\in \\mathfrak{l}\\backslash \\mathfrak{l}_{v_1}$. If $F_1$ is any of the fields in $\\mathcal{F}^{(v_1)}$ and $F_2$ is the composite of the other fields in $\\mathcal{F}^{(v_1)}\\cup \\mathcal{F}_{\\mathfrak{l}}$, Lemma $\\ref{eigenspaceeta}$ asserts that $F_1\\cap F_2\\subseteq J_{F_1}$. Lemma $\\ref{eigenspaceeta}$ asserts that $L_{h^{(v_1)}}$ is linearly disjoint over $L$ from the composite of all fields in $\\mathcal{F}^{(v_1)}\\cup \\mathcal{F}_{\\mathfrak{l}}$. To construct such an element $x$, enumerate the fields in $\\mathcal{F}^{(v_1)}=\\{E_1,\\dots, E_{k-1}\\}$ and set $E_{k}:=F_{h^{(v_1)}}$. Set $E_0:=\\mathfrak{F}_{\\mathfrak{l}}$ and let $\\mathcal{E}_j$ be the composite $E_0\\cdots E_j$, note that $\\mathcal{E}_{k}=\\mathcal{E}^{(v_1)}\\cdot \\mathfrak{F}_{\\mathfrak{l}}$. Consider the filtration\n\\[\\mathcal{E}_{k}\\supset \\mathcal{E}_{k-1}\\supset \\dots \\supset \\mathcal{E}_1\\supset \\mathcal{E}_0\\supset K.\\] Let $x_0\\in \\op{Gal}(\\mathcal{E}_0\/K)$ be an element defining $\\mathfrak{l}$. Note that $\\op{Gal}(\\mathcal{E}_1\/\\mathcal{E}_0)\\simeq \\op{Gal}(E_1\/E_1\\cap \\mathcal{E}_0)$ and the intersection $E_1\\cap \\mathcal{E}_0$ is contained in $J_{E_1}$. The condition on $E_1\/K$ is on the $\\bar{\\chi}\\sigma_{2L_1}$-eigenspace $\\op{Gal}(E_1\/J_{E_1})$. Hence $x_0$ lifts to a suitable $x_1\\in \\op{Gal}(\\mathcal{E}_1\/K)$. Repeating the process, we see that $x_1$ lifts to $x_{k-1}\\in \\op{Gal}(\\mathcal{E}_{k-1}\/K)$ such that if $\\sigma_{v_2}=x_{k-1}$, then $v_2\\in \\mathfrak{l}$ and $h^{(v_2)}(\\sigma_{v_1})=(A_2+C_1)\/2$. Since $E_k\\cap \\mathcal{E}_{k-1}=K$, it follows that $x_{k-1}$ can be lifted to $x_{k}\\in \\op{Gal}(\\mathcal{E}_{k}\/K)$ such that if $\\sigma_{v_2}=x$, then all conditions on $v_2$ are satisfied.\n\nAs a result, $\\delta(\\mathfrak{l}\\backslash \\mathfrak{l}_{v_1})\\geq \\frac{1}{[\\mathcal{E}^{(v_1)}\\cdot \\mathfrak{F}_{\\mathfrak{l}}:K]}$, and hence,\n\\[\\delta(\\mathfrak{l}_{v_1})\\leq \\left(1-\\frac{1}{[\\mathcal{E}^{(v_1)}\\cdot \\mathfrak{F}_{\\mathfrak{l}}:K]}\\right).\\]For $F\\in \\mathcal{F}^{(v_1)}$, the Galois group $\\op{Gal}(F\/K)$ may be identified with a Galois submodule of $\\operatorname{Ad}^0\\bar{\\rho}^*$. Hence $[F:K]\\leq q^{\\dim(\\operatorname{Ad}^0\\bar{\\rho})}$ for $F\\in \\mathcal{F}^{(v_1)}$ is a uniform bound independent of $v_1$. Similar reasoning shows that $[L^{h^{(v_1)}}:L]\\leq q^{\\dim (\\operatorname{Ad}^0\\bar{\\rho})} $. Setting $N:=(\\#\\Phi +n+1)\\cdot \\dim \\operatorname{Ad}^0\\bar{\\rho}$, deduce that \\[\\delta(\\mathfrak{l}_{v_1})\\leq 1-q^{-N}[\\mathfrak{F}_{\\mathfrak{l}}:K]^{-1}.\\]\n\\par Suppose that there is a sequence of $m$ primes $v_{1}^{(1)},\\dots, v_{1}^{(m)}\\in \\mathfrak{l}_2$, such that it is not possible to find a second prime $v_2$ for any of the primes $v_1^{(j)}$. In other words, $\\mathfrak{l}_2\\subseteq \\cap_{j=1}^m \\mathfrak{l}_{v_1^{(j)}}$. We show that the density of $\\cap_{j=1}^m \\mathfrak{l}_{v_1^{(j)}}$ approaches zero as $m$ approaches infinity. Since the upper density of $\\mathfrak{l}_2$ is positive, we will eventually find a pair $(v_1,v_2)$. For convenience of notation, set $w_j:=v_1^{(j)}$ and set $A=\\{w_1,\\dots, w_m\\}$. Fix $1\\leq j\\leq m$ and enumerate the fields $\\mathcal{F}^{(w_j)}=\\{E_1,\\dots, E_{k-1}\\}$ and set $E_k=F_{h^{(w_j)}}$. Denote by $\\mathfrak{E}_j:=\\mathfrak{F}_{\\mathfrak{l}}\\cdot \\mathcal{E}^{(w_1)}\\cdots\\mathcal{E}^{(w_j)}$ and let $C_j$ be the subset of $\\op{Gal}(\\mathfrak{E}_j\/K)$ defining the set $\\cap_{i=1}^j \\mathfrak{l}_{w_i}$. This means that $v_2\\in \\cap_{i=1}^j \\mathfrak{l}_{w_i}$ if and only if $\\sigma_{v_2}\\in C_j$. We show that any element $w\\in \\op{Gal}(\\mathfrak{E}_{j-1}\/K)$ lifts to an element $\\tilde{w}\\in\\op{Gal}(\\mathfrak{E}_{j}\/K)$ which is not in $C_j$. This is shown by filtering $\\mathfrak{E}_j\/\\mathfrak{E}_{j-1}$ by \n\\[\\mathfrak{E}_{j}=\\mathcal{E}_k\\supset \\mathcal{E}_{k-1}\\supset \\cdots \\mathcal{E}_1\\supset \\mathcal{E}_0=\\mathfrak{E}_{j-1},\\]\nwhere $\\mathcal{E}_l:=\\mathfrak{E}_{j-1}E_1\\cdots E_l$. The argument is identical to that provided before.\n\\par\nAs a result, \\[\\# C_j\\leq ([\\mathfrak{E}_j:\\mathfrak{E}_{j-1}] -1)\\# C_{j-1}.\\] Therefore,\n\\[\\begin{split}\\delta(\\cap_{i=1}^j \\mathfrak{l}_{w_i})=\\frac{\\# C_j}{[\\mathfrak{E}_j:K]}\\leq & \\left(1-\\frac{1}{[\\mathfrak{E}_j:\\mathfrak{E}_{j-1}]}\\right)\\frac{\\# C_{j-1}}{[\\mathfrak{E}_{j-1}:K]},\\\\\n\\leq & \\left(1-\\frac{1}{[\\mathcal{E}^{(w_j)}:K]}\\right)\\frac{\\# C_{j-1}}{[\\mathfrak{E}_{j-1}:K]},\\\\\n\\leq & (1-q^{-N}) \\delta(\\cap_{i=1}^{j-1} \\mathfrak{l}_{w_i}).\n\\end{split}\\]Therefore, $\\delta(\\cap_{i=1}^m \\mathfrak{l}_{w_i})\\leq (1-q^{-N})^{m-1} (1-q^{-N}[\\mathfrak{F}_{\\mathfrak{l}}:K]^{-1})$. Since $\\mathfrak{l}_2$ has positive upper density there is a large value of $m$ such that $\\mathfrak{l}_2$ is not contained in $\\cap_{i=1}^m \\mathfrak{l}_{w_i}$. This shows that a pair $(v_1,v_2)$ satisfying the required conditions does exist.\n\\end{proof}\n\n\\begin{Prop}\\label{bigimageprop}\nLet $\\rho_2$ be as in Proposition $\\ref{lifttorho3}$. The image of $\\rho_2$ is the principal congruence subgroup of $\\op{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)$ of similitude character $1$.\n\\end{Prop}\n\\begin{proof}\nRecall that the similitude character $\\kappa$ is prescribed to equal $\\kappa_0\\chi^k$ where $\\kappa_0$ is the Teichm\\\"uller lift of $\\bar{\\kappa}$ and $k$ is divisible by $p(p-1)$. Therefore, we have that $\\kappa\\equiv \\kappa_0\\mod{p^2}$, and as a result, elements in the principal congruence subgroup in the image of $\\rho_2$ necessarily have similitude character $1$. Therefore, $\\rho_2(\\op{G}_L)$ may be identified with a subspace of $\\operatorname{Ad}^0\\bar{\\rho}$. In greater detail, $\\rho_2(g)$ is identified with $\\frac{1}{p}(\\rho_2(g)-\\op{Id})$, for $g\\in \\op{G}_L=\\op{ker}\\bar{\\rho}$. It may be checked that $\\rho_2(\\op{G}_L)$ is a $\\op{G}$-submodule of $\\operatorname{Ad}^0\\bar{\\rho}$ and that the natural $\\op{G}$-action on $\\op{Gal}(\\mathbb{Q}(\\rho_2)\/L)$ (induced by conjugation) coincides with the $\\op{G}$-action on $\\rho_2(\\op{G}_L)$ viewed as a submodule of $\\operatorname{Ad}^0\\bar{\\rho}$. Recall that $\\rho_2=(\\op{Id}+ph)\\zeta_2$, where $h$ is the cohomology class given by $-h^{(v_1)}+2h^{(v_2)}$. Since $\\bar{\\rho}$ is unramified at $v_1$, we have that $\\tau_{v_1}\\in \\op{G}_L$. The cohomology class $h^{(v_2)}$ is unramified at $v_1$, as is $\\zeta_2$. Therefore, we have that\n\\[\\rho_2(\\tau_{v_1})=(\\op{Id}+ph(\\tau_{v_1}))\\zeta_2(\\tau_{v_1})=(\\op{Id}-ph^{(v_1)}(\\tau_{v_1})).\\]\nRecall that by \\eqref{equation64}, we have that \\[h^{(v_1)}(\\tau_{v_1})\\in (\\operatorname{Ad}^0\\bar{\\rho})_{-2L_1}\\backslash\\{0\\}.\\]Therefore, $\\rho_2(\\op{G}_L)$ is identified with a Galois-submodule of $\\operatorname{Ad}^0\\bar{\\rho}$ which contains an element with non-zero $-2L_1$-component. From Lemma $\\ref{fullrankLemma}$, it is deduced that this module must be all of $\\operatorname{Ad}^0\\bar{\\rho}$. This completes the proof.\n\\end{proof}\n\\section{Annihiliating the dual-Selmer Group}\nLet $\\rho_3:\\operatorname{G}_{\\mathbb{Q},T\\cup \\{v_1,v_2\\}}\\rightarrow \\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^3)$ be the lift of $\\bar{\\rho}$ obtained from the application of Propositions $\\ref{lifttorho3}$ and $\\ref{bigimageprop}$. Recall that the Galois group $\\op{Gal}(K(\\rho_2)\/K)$ is identified with $\\operatorname{Ad}^0\\bar{\\rho}$. As a result, once it is shown that $\\rho_3$ lifts to a characteristic zero representation $\\rho$, it shall follow that $\\rho$ is irreducible. In showing that $\\rho_3$ can be lifted to characteristic zero, we enlarge the set of primes $Z=T\\cup \\{v_1,v_2\\}$ to a finite set of primes $Y$ such that $X:=Y\\backslash S$ consists only of trivial primes. For $i=1,2$ set $\\mathcal{C}_{v_i}=\\mathcal{C}_{v_i}^{ram}$ and for primes $v\\in X\\backslash \\{v_1,v_2\\}$, set $ \\mathcal{C}_v=\\mathcal{C}_v^{nr}$. We show that the dual-Selmer group $H^1_{\\mathcal{N}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y},\\operatorname{Ad}^0\\bar{\\rho}^*)$ vanishes for a suitably chosen set of primes $Y$. For convenience of notation, denote by $\\mathscr{W}$ the Galois submodule $(\\operatorname{Ad}^0\\bar{\\rho})_{-2n+2}$ of $\\operatorname{Ad}^0\\bar{\\rho}$ spanned by root spaces $(\\operatorname{Ad}^0\\bar{\\rho})_{\\beta}$ for $\\beta\\neq -2L_1$. \n\\begin{Prop}\\label{lastchebotarev}\nLet $\\rho_3:\\operatorname{G}_{\\mathbb{Q},T\\cup \\{v_1,v_2\\}}\\rightarrow \\operatorname{GSp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^3)$ be the lift of $\\bar{\\rho}$ obtained from the application of Propositions $\\ref{lifttorho3}$ and $\\ref{bigimageprop}$. Let $Y$ be a finite set of primes which contains $Z=T\\cup \\{v_1,v_2\\}$ such that $Y\\backslash S$ consists of trivial primes. Suppose $f\\in H^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho})$ and $\\psi\\in H^1_{\\mathcal{N}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho}^*)$ are nonzero classes. Then there exists a prime $v\\notin Y$ such that\n\\begin{enumerate}\n\\item \\label{71one} $v$ is a trivial prime,\n\\item \\label{71two}$\\rho_{3\\restriction \\op{G}_v}$ satisfies $\\mathcal{C}_v=\\mathcal{C}_v^{nr}$, \n\\item \\label{71three} $f$ does not satisfy $\\mathcal{N}_v=\\mathcal{N}_v^{nr}$,\n\\item\\label{71four}$\\beta_{\\restriction \\operatorname{G}_v}=0$ for all $\\beta\\in H^1(\\op{G}_{\\mathbb{Q},Y}, \\mathscr{W}^*)$,\n\\item \\label{71five} $\\psi_{\\restriction \\operatorname{G}_v}\\neq 0$ and one can extend $\\{\\psi\\}$ to a basis $\\psi_1=\\psi,\\psi_2,\\dots, \\psi_k$ of $H^1(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho}^*)$ such that $\\psi_{i \\restriction \\operatorname{G}_v}=0$ for $i>1$.\n\\end{enumerate}\n\\end{Prop}\n\\begin{proof}\nEach condition is a union of Chebotarev conditions on a number of finite extensions $J$ of $K$. Each of the extensions $J$ are Galois over $\\mathbb{Q}$ with $\\op{Gal}(J\/K)$ an $\\mathbb{F}_p$-vector space. Let $g\\in \\op{G}'$ and $x\\in \\op{Gal}(J\/K)$, define, $g\\cdot x:=\\tilde{g} x \\tilde{g}^{-1}$ where $\\tilde{g}$ is a lift of $g$ to $\\op{Gal}(J\/\\mathbb{Q})$. This gives $\\op{Gal}(J\/K)$ the structure of an $\\mathbb{F}_p[\\op{G}']$-module. For each condition, we list the choices for $J$ below as well as characters for the $\\mathbb{T}$-action on $\\op{Gal}(J\/K)$:\n\\begin{center}\n\\begin{tabular}{c|c|c } \n \\text{Condition} & $J$ & $\\text{Eigenspaces of } \\operatorname{Gal}(J\/K)$ \\\\ [1 ex]\n \\hline\n $(1)$ & $K(\\mu_{p^2})$ & $1$ \\\\\n \\hline\n $(2)$ & $K(\\rho_2)$ & $1,\\{\\sigma_{\\lambda}\\}_{\\lambda\\in \\Phi}$ \\\\\n \\hline\n $(3)$ & $K_f$ & $1,\\{\\sigma_{\\lambda}\\}_{\\lambda\\in \\Phi}$ \\\\\n \\hline\n $(4)$ & $K_{\\beta}$ \\text{ for } $\\beta \\in H^1(\\op{G}_{\\mathbb{Q},Y},\\mathscr{W}^*)$& $\\bar{\\chi},\\{\\bar{\\chi}\\sigma_{\\lambda}^{-1}|\\lambda\\neq -2L_1\\}$\\\\\n \\hline\n $(5)$ & $K_{\\psi_i} $ & $\\bar{\\chi},\\{\\bar{\\chi}\\sigma_{\\lambda}^{-1}\\}$.\\\\\n\\end{tabular}\n\\end{center}\nWe show that these conditions may be simultaneously satisfied. First, we show that each of the conditions is a nonempty Chebotarev condition (or a union of finitely many Chebotarev conditions). It is clear that condition $\\eqref{71one}$ and $\\eqref{71two}$ are nonempty Chebotarev conditions. Lemma $\\ref{lemma55}$ gives a criterion for the element $f$ to not be in the space $\\mathcal{N}_v$. In accordance with Lemma $\\ref{lemma55}$, write $X_{-2L_1}=c e_{n+1,1}$ and $X_{2L_1}=d e_{1,n+1}$. Since $f$ is non-zero, $f(\\op{G}_L)$ is a non-zero Galois-stable submodule of $\\operatorname{Ad}^0\\bar{\\rho}$. Hence, by Lemma $\\ref{mainin}$, contains $(\\operatorname{Ad}^0\\bar{\\rho})_{\\sigma_{2L_1}}$. Therefore, the image of $f_{\\restriction \\op{G}_L}$ contains an element \n\\[f(g)=\\sum_{\\lambda\\in \\Phi} a_{\\lambda} X_{\\lambda} +\\sum_{i=1}^n a_i H_i\\]such that $a_{2L_1}\\neq -(cd)^{-1} a_1$. As a result, condition $\\eqref{71three}$ is a union of finitely many nonempty Chebotarev conditions. Condition $\\eqref{71four}$ requires that the prime splits in the composite of the fields $K_{\\beta}$. That condition $\\eqref{71five}$ is a nonempty Chebotarev condition follows from Proposition $\\ref{P2}$.\n\nNext we examine the independence of these conditions. It follows from Lemma $\\ref{lemma416}$ that the composite of the fields defining the first three conditions is linearly disjoint over $K$ from the composite of the fields defining the last two conditions. As a result, the conditions may be treated separately from the last two. It follows from Proposition $\\ref{P2}$ that the conditions $\\eqref{71four}$ and $\\eqref{71five}$ are compatible with each other. Therefore, it remains to show that $\\eqref{71one}$,$\\eqref{71two}$ and $\\eqref{71three}$ may be simultaneously satisfied. We begin with the independence of $\\eqref{71one}$ and $\\eqref{71two}$. Proposition $\\ref{bigimageprop}$ asserts that $\\operatorname{Gal}(K(\\rho_2)\/K)= \\operatorname{Ad}^0\\bar{\\rho}$. Suppose that $Q$ is a proper $\\op{G}'$-stable subgroup of $\\operatorname{Ad}^0\\bar{\\rho}$. Lemma $\\ref{Pdecomposition}$ asserts that $Q$ decomposes into $\\mathbb{T}$-eigenspaces $Q=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}} Q_{\\sigma_{\\lambda}}$ and Lemma $\\ref{fullrankLemma}$ asserts that the eigenspace $Q_{-2L_1}:=Q_{\\sigma_{-2L_1}}$ must be trivial. Hence the quotient $\\operatorname{Ad}^0\\bar{\\rho}\/Q$ must have a non-zero $\\sigma_{-2L_1}$-eigenspace. It follows that there is no proper Galois stable subgroup $Q$ of $\\operatorname{Ad}^0\\bar{\\rho}$ such that $\\operatorname{Ad}^0\\bar{\\rho}\/Q$ is has trivial Galois action. Since $\\op{G}'$ acts trivially\non $\\operatorname{Gal}(K(\\mu_{p^2} )\/K)$ it follows that $K(\\rho_2) \\cap K(\\mu_{p^2}) = K$. Thus conditions $\\eqref{71one}$ and $\\eqref{71two}$ are independent.\n\\par We show that the first three conditions may be simultaneously satisfied by considering the cases $K(\\rho_2)\\supseteq K_f$ and $K(\\rho_2)\\not \\supseteq K_f$ separately. First consider the case when $K(\\rho_2)\\supseteq K_f$. Let $r:=\\dim_{\\mathbb{F}_p} f(\\op{G}_K)$. Since $\\op{Gal}(K(\\rho_2)\/K)\\simeq \\operatorname{Ad}^0\\bar{\\rho}$, if $r< \\dim_{\\mathbb{F}_p} \\operatorname{Ad}^0\\bar{\\rho}$ the containment $K(\\rho_2)\\supset K_f$ is proper. Since $f$ is non-zero, Lemma $\\ref{l4}$ asserts that $K_f\\neq K$. Let $Q\\subset \\op{Gal}(K(\\rho_2)\/K)$ be the proper subgroup such that $\\op{Gal}(K(\\rho_2)\/K)\/Q\\simeq \\op{Gal}(K_f\/K)$. Lemma $\\ref{Pdecomposition}$ asserts that $Q$ decomposes into $\\mathbb{T}$-eigenspaces $Q=\\bigoplus_{\\lambda\\in \\Phi\\cup \\{1\\}} Q_{\\sigma_{\\lambda}}$ and Lemma $\\ref{fullrankLemma}$ asserts that the eigenspace $Q_{-2L_1}:=Q_{\\sigma_{-2L_1}}$ must be trivial. Hence the quotient $\\op{Gal}(K_f\/K)$ must have a non-zero $\\sigma_{-2L_1}$-eigenspace. Identify $\\op{Gal}(K_f\/K)$ with $f(\\op{G}_K)\\subset \\operatorname{Ad}^0\\bar{\\rho}$. Since $r<\\dim_{\\mathbb{F}_p} \\operatorname{Ad}^0\\bar{\\rho}$, Lemma $\\ref{fullrankLemma}$ asserts that $f(\\op{G}_K)_{-2L_1}=0$, a contradiction. Hence, $K(\\rho_2)\\supseteq K_f$ forces equality $K(\\rho_2)= K_f$. Let \\[\\alpha_1:=f_{\\restriction \\op{G}_K}:\\op{Gal}(K_f\/K)\\xrightarrow{\\sim} \\operatorname{Ad}^0\\bar{\\rho}\\]\nand \n\\[\\alpha_2:=\\rho_{2\\restriction \\op{G}_K}:\\op{Gal}(K_f\/K)\\xrightarrow{\\sim} \\operatorname{Ad}^0\\bar{\\rho}. \\] The composite $\\alpha_1\\alpha_2^{-1}$ is a $\\op{G}'$-automorphism of $\\operatorname{Ad}^0\\bar{\\rho}$. It follows from Corollary $\\ref{Coradd}$ that $\\alpha_1\\alpha_2^{-1}$ is a scalar $a\\in \\mathbb{F}_q^{\\times}$ and hence $\\alpha_1=a\\alpha_2$. Let $v$ satisfy $\\eqref{71one}$, $\\eqref{71two}$, $\\eqref{71four}$ and $\\eqref{71five}$ such that \n\\[(\\operatorname{Id}+X_{-2L_1})^{-1}\\rho_2(\\sigma_v)(\\operatorname{Id}+X_{-2L_1})\\in \\mathcal{T}\\]has non-trivial $H_1$ component. Since $v$ is a trivial prime, $\\sigma_v$ lies in $\\op{G}_K$. Identifying $\\op{ker}\\{\\operatorname{GSp}(\\text{W}(\\mathbb{F}_q)\/p^2)\\rightarrow \\operatorname{GSp}(\\mathbb{F}_q)\\}$ with $\\operatorname{Ad}^0\\bar{\\rho}$, we view $\\rho_2(\\sigma_{v})$ as an element in $\\operatorname{Ad}^0\\bar{\\rho}$. Since $f(\\sigma_v)=a\\rho_2(\\sigma_v)$, we see that $(\\operatorname{Id}+X_{-2L_1})^{-1}f(\\sigma_v)(\\operatorname{Id}+X_{-2L_1})$ has non-zero $H_1$ component and hence is not contained in $\\mathfrak{t}_{2L_1}+\\op{Cent}((\\operatorname{Ad}^0\\bar{\\rho})_{2L_1})$. As a result, $f$ is not in $(\\operatorname{Id}+X_{-2L_1})\\mathcal{P}_v^{2L_1}(\\operatorname{Id}+X_{-2L_1})^{-1}$. It is easy to see that $f$ is not in $\\mathcal{N}_v$ and hence $\\eqref{71three}$ is also satisfied.\n\n\\par We consider the case when $K_f\\not\\subseteq K(\\rho_2)$. Since \\[[K(\\rho_2):K]=\\# \\operatorname{Ad}^0\\bar{\\rho} \\geq [K_f:K],\\]$K(\\rho_2)$ is not contained in $K_f$. It follows that $K(\\rho_2)\\supsetneq K(\\rho_2)\\cap K_f$ and $K_f\\supsetneq K(\\rho_2)\\cap K_f$ and thus by Lemma $\\ref{mainin}$, the images of \\[\\op{Gal}(K(\\rho_2)\/K(\\rho_2)\\cap K_f)\\hookrightarrow \\operatorname{Ad}^0\\bar{\\rho}\\text{ and }\\op{Gal}(K_f\/K(\\rho_2)\\cap K_f)\\hookrightarrow \\operatorname{Ad}^0\\bar{\\rho}\\] contain $(\\operatorname{Ad}^0\\bar{\\rho})_{\\sigma_{2L_1}}$. If $K_f\\subseteq K(\\rho_2,\\mu_{p^2})$, then it follows that\n\\[\\dim_{\\mathbb{F}_p} (\\op{Gal}(K(\\rho_2,\\mu_{p^2})\/K)_{\\sigma_{2L_1}})\\geq 2\\dim_{\\mathbb{F}_p} (\\operatorname{Ad}^0\\bar{\\rho})_{\\sigma_{2L_1}}=2[\\mathbb{F}_q:\\mathbb{F}_p].\\]However, $\\op{Gal}(K(\\rho_2,\\mu_{p^2})\/K)_{\\sigma_{2L_1}}$ may be identified with \\[\\op{Gal}(K(\\rho_2)\/K)_{\\sigma_{2L_1}}\\simeq (\\operatorname{Ad}^0\\bar{\\rho})_{\\sigma_{2L_1}}\\] since $K( \\mu_{p^2})$ contributes to the trivial eigenspace.\nHence, $K_f \\not \\subseteq K(\\rho_2, \\mu_{p^2} )$. Let $v$ be a prime satisfying conditions $\\eqref{71one}$, $\\eqref{71two}$, $\\eqref{71four}$ and $\\eqref{71five}$. Lemma $\\ref{mainin}$ asserts that the image of \\[\\op{Gal}(K_f\/K_f\\cap K(\\rho_2,\\mu_{p^2}))\\hookrightarrow \\operatorname{Ad}^0\\bar{\\rho}\\] contains $(\\operatorname{Ad}^0\\bar{\\rho})_{\\sigma_{2L_1}}$ and thus, we have the freedom to stipulate that the $X_{2L_1}$-component of $f(\\sigma_v)$ be anything we like. Lemma $\\ref{lemma55}$ asserts that if $f\\in \\mathcal{N}_v$, an explicit relationship must be satisfied between the $X_{2L_1}$-component and the $H_1$-component of $f(\\sigma_v)$. It follows that we may alter the $X_{2L_1}$-component of $f(\\sigma_v)$ so that $f\\notin \\mathcal{N}_v$. Therefore all conditions may be satisfied and the proof is complete.\n\\end{proof}\n\\begin{Prop}\nThere is a finite set $Y\\supseteq Z$ such that $Y\\backslash S$ consists of trivial primes and $H^1_{\\mathcal{N^{\\perp}}}(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho}^*)=0$.\n\\end{Prop}\n\\begin{proof}\n\\par Let $Y$ be a finite set of primes containing $Z$ such that $Y\\backslash S$ consists of trivial primes. If $H^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho})\\neq 0$, we exhibit a trivial prime $v$ not contained in $Y$ such that \n\\[h^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})h^1_{\\mathcal{M}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho}^*).\n\\end{equation}\n\\par Consider the restriction maps\n\\begin{equation*}\n\\Phi_1:H^1(\\op{G}_{\\mathbb{Q},Y}, \\mathscr{W})\\rightarrow \\bigoplus_{w\\in Y}H^1(\\op{G}_w, \\mathscr{W})\n\\end{equation*}\nand\n\\begin{equation*}\n\\Phi_2:H^1(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\mathscr{W})\\rightarrow \\bigoplus_{w\\in Y} H^1(\\op{G}_w, \\mathscr{W}).\n\\end{equation*} We show that the maps $\\Phi_1$ and $\\Phi_2$ have the same image. By the Poitou-Tate sequence,\n\\[0\\rightarrow \\op{image}(\\Phi_1)\\rightarrow \\bigoplus_{w\\in Y} H^1(\\op{G}_w, \\mathscr{W})\\rightarrow H^1(\\op{G}_{\\mathbb{Q},Y}, \\mathscr{W}^*)^{\\vee},\\] i.e.,\nthe image of $\\Phi_1$ is the exact annihiliator of the image of the restriction map\n\\[H^1(\\op{G}_{\\mathbb{Q},Y}, \\mathscr{W}^*)\\rightarrow \\bigoplus_{w\\in Y} H^1(\\op{G}_w, \\mathscr{W}^*).\\]\nLet $\\mathfrak{M}$ be the Selmer condition \n\\[\\mathfrak{M}_w:=\\begin{cases}0\\text{ if }w\\in Y\\\\\nH^1(\\operatorname{G}_v, \\mathscr{W})\\text{ if }w=v\\\\\nH^1_{nr}(\\op{G}_w, \\mathscr{W})\\text{ if }w\\notin Y\\cup\\{v\\},\n\\end{cases}\\] with dual Selmer condition \n\\[\\mathfrak{M}_w^{\\perp}=\\begin{cases}H^1(\\operatorname{G}_v, \\mathscr{W}^*)\\text{ if }w\\in Y\\\\\n0\\text{ if }w=v\\\\\nH^1_{nr}(\\op{G}_w, \\mathscr{W}^*)\\text{ if }w\\notin Y\\cup\\{v\\}.\n\\end{cases}\\]\nBy the Poitou-Tate sequence, the image of $\\Phi_2$ is the exact annihilator of the restriction map \n\\[H^1_{\\mathfrak{M}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\mathscr{W}^*)\\rightarrow \\bigoplus_{w\\in Y} H^1(\\op{G}_w, \\mathscr{W}^*).\\] By Proposition $\\ref{lastchebotarev}$ condition $\\eqref{71four}$, \n\\[H^1_{\\mathfrak{M}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\mathscr{W}^*)=H^1(\\op{G}_{\\mathbb{Q},Y}, \\mathscr{W}^*)\\] and therefore, the image of $\\Phi_1$ is equal to the image of $\\Phi_2$.\n\\par We deduce that\n\\begin{equation}\\label{phi3phi4}\n\\begin{split}\n\\dim \\ker \\Phi_2-\\dim \\ker \\Phi_1 &=h^1(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}},\\mathscr{W})-h^1(\\op{G}_{\\mathbb{Q},Y},\\mathscr{W})\\\\\n&=h^1(\\operatorname{G}_v, \\mathscr{W})-h^0(\\operatorname{G}_v, \\mathscr{W})\\\\\n&=h^1(\\operatorname{G}_v, \\mathscr{W})-h^1_{nr}(\\operatorname{G}_v, \\mathscr{W}).\\\\\n\\end{split}\n\\end{equation}\nBy $\\ref{phi3phi4}$, we deduce that the sequence\n\\begin{equation}\\label{lastequation}0\\rightarrow \\ker \\Phi_1\\rightarrow \\ker\\Phi_2\\rightarrow \\frac{H^1(\\operatorname{G}_v, \\mathscr{W})}{H^1_{nr}(\\operatorname{G}_v, \\mathscr{W})}\\rightarrow 0\\end{equation}\nis a short exact sequence.\n\n\\par Define the maps\n\\begin{equation*}\n\\Phi_3: H^1(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}},\\operatorname{Ad}^0\\bar{\\rho}) \\rightarrow \\bigoplus_{w\\in Y} \\frac{H^1(\\op{G}_w,\\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_w}\n\\end{equation*}\nand \n\\begin{equation*}\n\\Phi_4: H^1(\\op{G}_{\\mathbb{Q}, Y},\\operatorname{Ad}^0\\bar{\\rho}) \\rightarrow \\bigoplus_{w\\in Y} \\frac{H^1(\\op{G}_w,\\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_w}.\n\\end{equation*} From the Cassels-Poitou-Tate long exact sequence and the vanishing of $\\Sh^2_Y(\\operatorname{Ad}^0\\bar{\\rho})$, we deduce that the following sequences are exact\n\\begin{equation*}\n H^1(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}},\\operatorname{Ad}^0\\bar{\\rho}) \\xrightarrow{\\Phi_3} \\bigoplus_{w\\in Y} \\frac{H^1(\\op{G}_w,\\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_w}\\rightarrow H^1_{\\mathcal{M}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}},\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\vee}\\rightarrow 0\n\\end{equation*}\n\\begin{equation*}\n H^1(\\op{G}_{\\mathbb{Q},Y},\\operatorname{Ad}^0\\bar{\\rho}) \\xrightarrow{\\Phi_4} \\bigoplus_{w\\in Y} \\frac{H^1(\\op{G}_w,\\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_w}\\rightarrow H^1_{\\mathcal{N}^{\\perp}}(\\op{G}_{\\mathbb{Q},Y},\\operatorname{Ad}^0\\bar{\\rho}^*)^{\\vee}\\rightarrow 0.\n\\end{equation*}\nSet $t'$ to denote the difference $\\dim \\op{image} \\Phi_3-\\dim \\op{image} \\Phi_4$. From the assertion made in $\\eqref{dimgreaterone}$ we conclude that $t'\\geq 1$.\n\\par We claim that it suffices to find $\\dim \\operatorname{Ad}^0\\bar{\\rho}-t'+1$ elements in $\\ker\\Phi_3$, no linear combination of which lies in $\\mathcal{N}_v$. It follows then that the image of \\[\\ker\\Phi_3\\rightarrow \\frac{H^1(\\operatorname{G}_v,\\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_v}\\] has dimension strictly greater than $\\dim \\operatorname{Ad}^0\\bar{\\rho}-t'$. From the exactness of \n\\[0\\rightarrow H^1_{\\mathcal{N}}(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})\\rightarrow \\ker\\Phi_3\\rightarrow \\frac{H^1(\\operatorname{G}_v,\\operatorname{Ad}^0\\bar{\\rho})}{\\mathcal{N}_v}\\]one may deduce that\n\\[\\begin{split}h_{\\mathcal{N}}^1(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})<&\\dim \\ker \\Phi_3-\\dim \\operatorname{Ad}^0\\bar{\\rho}+t'.\\\\\n= & h^1(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})-\\dim \\operatorname{Ad}^0\\bar{\\rho} -\\dim \\op{im} \\Phi_4.\\\\\\end{split}\\]\nNote that $\\Sh^1_{Y}(\\operatorname{Ad}^0\\bar{\\rho})=0$ and thus an application of Wiles' formula \\eqref{wilesformula} shows that\n\\[\\begin{split}h^1(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho})=& h^0(\\op{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho})-h^0(\\op{G}_{\\mathbb{Q}},\\operatorname{Ad}^0\\bar{\\rho}^*)\\\\\n+&\\sum_{w\\in Y\\cup \\{\\infty\\}} (h^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho})-h^0(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho}))\\end{split}\\]\nand \n\\[\\begin{split}h^1(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})=& h^0(\\op{G}_{\\mathbb{Q}}, \\operatorname{Ad}^0\\bar{\\rho})-h^0(\\op{G}_{\\mathbb{Q}},\\operatorname{Ad}^0\\bar{\\rho}^*)\\\\\n+&\\sum_{w\\in Y\\cup \\{v\\}\\cup \\{\\infty\\}} (h^1(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho})-h^0(\\op{G}_w, \\operatorname{Ad}^0\\bar{\\rho})).\\end{split}\\]\nTherefore, \n\\[\\begin{split}\n h^1(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})=& h^1(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho})+h^1(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})-h^0(\\op{G}_v, \\operatorname{Ad}^0\\bar{\\rho})\\\\\n =&h^1(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho})+\\dim \\operatorname{Ad}^0\\bar{\\rho}.\n\\end{split}\\]\nTherefore, we have that \n\\[\\begin{split}h_{\\mathcal{N}}^1(\\op{G}_{\\mathbb{Q},Y\\cup\\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})<& h^1(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho})-\\dim \\op{im} \\Phi_4\\\\\n=&\\dim \\op{ker} \\Phi_4\\\\\n=& h_{\\mathcal{N}}^1(\\op{G}_{\\mathbb{Q},Y}, \\operatorname{Ad}^0\\bar{\\rho}).\n\\end{split}\\]\nTherefore in order to complete the proof we proceed to construct $\\dim \\operatorname{Ad}^0\\bar{\\rho}-t'+1$ elements in $\\ker\\Phi_3$ no linear combination of which lies in $\\mathcal{N}_v$. We are in fact able to construct $\\dim \\operatorname{Ad}^0\\bar{\\rho}$ elements, which suffices since $t'\\geq 1$.\n\\par Note that $\\operatorname{Ad}^0\\bar{\\rho}\/\\mathscr{W}$ is isomorphic to $\\mathbb{F}_q(\\sigma_{-2L_1})$ and hence $H^0(\\op{G}_{\\mathbb{Q}},\\operatorname{Ad}^0\\bar{\\rho}\/\\mathscr{W})$ is zero. We find that $H^1(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}}, \\mathscr{W})$ injects into $H^1(\\op{G}_{\\mathbb{Q},Y\\cup \\{v\\}}, \\operatorname{Ad}^0\\bar{\\rho})$ and thereby it follows that $\\op{ker}\\Phi_2$ is contained in $\\op{ker}\\Phi_3$. Let $Z_1,\\dots, Z_s$ be a basis of $\\mathscr{W}$. By the exactness of $\\ref{lastequation}$ there exist $\\omega_i\\in \\text{ker}\\Phi_2$ such that $\\omega_i(\\tau_v)=Z_i$ for $i=1,\\dots, s$. We show that no linear combination of $\\{f, \\omega_1,\\dots, \\omega_s\\}$ lies in $\\mathcal{N}_v$. Let $Q=c_0 f+\\sum_{i=1}^s c_i \\omega_i$ be in $\\mathcal{N}_v$. Since $f$ is unramified at $v$, $f(\\tau_v)=0$. On the other hand,\n$Q(\\tau_v)=\\sum_{i=1}^s c_i Z_i$ is contained in $\\mathscr{W}$. Since $Q\\in \\mathcal{N}_v$, \\[Q(\\tau_v)=c'(\\operatorname{Id}+X_{-\\alpha})X_{\\alpha}(\\operatorname{Id}+X_{-\\alpha})^{-1}\\] for $\\alpha=2L_1$ and some constant $c'$. The root vectors $X_{\\alpha}$ and $X_{-\\alpha}$ are constant multiples of $e_{1,n+1}$ and $e_{n+1,1}$ respectively. Assume without loss of generality that $X_{\\alpha}=e_{1,n+1}$ and $X_{-\\alpha}=e_{n+1,1}$. Clearly, $X_{-\\alpha}^2=0$ and hence $(1+X_{-\\alpha})^{-1}=(1-X_{-\\alpha})$. We see that\n\\[\\begin{split}Q(\\tau_v)=&c'(\\operatorname{Id}+X_{-\\alpha})X_{\\alpha}(\\operatorname{Id}-X_{-\\alpha}) \\\\\n=& c'\\left(X_{\\alpha} +[X_{-\\alpha},X_{\\alpha}]-X_{-\\alpha}X_{\\alpha}X_{-\\alpha}\\right)\\\\\n=& c'\\left(e_{1,n+1}-H_1-e_{n+1,1}\\right). \\end{split}\\]We deduce that $Q(\\tau_v)=0$ since $e_{n+1,1}\\notin \\mathcal{W}$. Therefore, $c_i=0$ for all $i=1,\\dots, s$. As a consequence, $Q=c_0 f$. However, $f$ is not contained in $\\mathcal{N}_v$. It follows that $c_0=0$ and therefore, $Q=0$. Therefore no linear combination of $\\{f, \\omega_1,\\dots, \\omega_s\\}$ lies in $\\mathcal{N}_v$ and this completes the proof.\n\\end{proof}\nTo conclude the proof of Theorem $\\ref{main}$, we observe that on choosing an appropriately large choice of trivial primes the dual Selmer group vanishes and hence, by the lifting construction outlined in section $\\ref{section3}$, $\\rho_3$ lifts to a characteristic zero representation $\\rho$. Furthermore, $\\rho$ can be arranged to have similitude character $\\kappa$, and satisfy the local conditions $\\mathcal{C}_v$ at the primes $v\\in S$. Proposition $\\ref{bigimageprop}$ asserts that the image of $\\rho_2$ contains \\[\\widehat{\\op{Sp}}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2):=\\left\\{\\op{Sp}_{2n}(\\text{W}(\\mathbb{F}_q)\/p^2)\\rightarrow\\op{Sp}_{2n}(\\mathbb{F}_q) \\right\\}\\]and it follows that $\\rho$ is irreducible.\n\\section{Examples}\\label{examples}\n\\par Let $p$ be an odd prime. Under certain hypotheses on $p$, we show that there are examples of reducible Galois representations \n$\\bar{\\rho}:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{F}_p)$ which satisfy the conditions of Theorem $\\ref{main}$. First, we sketch the strategy used. The reader may refer to section $\\ref{notationsection}$ for some of the notation used in this section. Recall that $\\bar{\\chi}$ denotes the mod-$p$ cyclotomic character. Let $\\varphi_1,\\varphi_2$ and $\\bar{\\kappa}:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GL}_1(\\mathbb{F}_q)$ be characters to be specificied later and $\\bar{r}$ the diagonal representation specified by \\[\\bar{r}:=\\left( {\\begin{array}{cccc}\n \\varphi_1 & & & \\\\\n & \\varphi_2 & & \\\\\n & & \\varphi_1^{-1} \\bar{\\kappa} & \\\\\n & & & \\varphi_2^{-1} \\bar{\\kappa} \n \\end{array} } \\right):\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{F}_p).\\] The characters, $\\varphi_i$ and $\\bar{\\kappa}$ will be powers of $\\bar{\\chi}$.\n Let $D\\in \\op{GSp}_4(\\mathbb{Q}_p)$ be the diagonal matrix \\[D:=\\left( {\\begin{array}{cccc}\n p & & & \\\\\n & 1& & \\\\\n & & p^{-2} & \\\\\n & & & p^{-1}\n \\end{array} } \\right)\\] and set $H$ to denote $(D\\op{GSp}_4(\\mathbb{Z}_p)D^{-1})\\cap (D^{-1}\\op{GSp}_4(\\mathbb{Z}_p)D)$. Note that $H$ is the subgroup of $\\op{GSp}_4(\\mathbb{Z}_p)$ consisting of matrices \\[X=\\left( {\\begin{array}{cccc}\n a_{1,1} & p a_{1,2}& p^3 a_{1,3}& p^2 a_{1,4} \\\\\n p a_{2,1} & a_{2,2}& p^2 a_{2,3}& p a_{2,4}\\\\\n p^3 a_{3,1} & p^2 a_{3,2} & a_{3,3} & p a_{3,4}\\\\\n p^2 a_{4,1}& p a_{4,2} & p a_{4,3} & a_{4,4}\n \\end{array} } \\right).\\] Note that $D^{-1} X D$ is equal to \\[\\left( {\\begin{array}{cccc}\n a_{1,1} & a_{1,2}& a_{1,3}& a_{1,4} \\\\\n p^2 a_{2,1} & a_{2,2}& a_{2,3}& a_{2,4}\\\\\n p^6 a_{3,1} & p^4 a_{3,2} & a_{3,3} & p^2 a_{3,4}\\\\\n p^4 a_{4,1}& p^2 a_{4,2} & a_{4,3} & a_{4,4}\n \\end{array} } \\right)\\] and thus reduces to the Borel $\\op{B}(\\mathbb{F}_q)$ modulo $p$. Let $H_0$ denote the intersection of $H$ with $\\op{Sp}_4(\\mathbb{Z}_p)$. Recall that for $k\\geq 1$, $\\op{U}_k(\\mathbb{F}_p)\\subset \\op{B}(\\mathbb{F}_q)$ is the exponential subgroup generated by $\\operatorname{exp}((\\operatorname{Ad}^0\\bar{\\rho})_k)$. The strategy we adopt is as follows:\n \\begin{enumerate}\n \\item Under some conditions on $p$, we may choose $\\varphi_1,\\varphi_2$ and $\\bar{\\kappa}$ such that $H^2(\\op{G}_{\\mathbb{Q},\\{p\\}},\\op{Ad}^0\\bar{r})$ is zero. Thus the global deformation problem (unramified outside $\\{p\\}$) is unobstructed.\n \\item We show that there is a lift\n \\[r:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{Z}_p)\\] of $\\bar{r}$ with image in $H$. Letting $\\bar{\\rho}$ denote the mod-$p$ reduction of $D^{-1} r D$, we note that the image of $\\bar{\\rho}$ is contained in $\\op{B}(\\mathbb{F}_p)$.\n \\item Let $\\Pi$ denote the intersection of the image of $\\bar{\\rho}$ with $\\op{U}_1(\\mathbb{F}_p)$. It is shown that after the mod-$p^2$ lift $r_2$ of $\\bar{r}$ may be carefully chosen so that $\\Pi$ surjects onto $\\op{U}_1(\\mathbb{F}_p)\/\\op{U}_2(\\mathbb{F}_p)$. Lemma $\\ref{bigimagelemmaU1}$ shows that the image of $\\bar{\\rho}$ contains $\\op{U}_1(\\mathbb{F}_p)$. Moreover, the characters $\\varphi_1, \\varphi_2$ and $\\bar{\\kappa}$ are suitably chosen so that all the conditions of Theorem $\\ref{main}$ are satisfied.\n \\end{enumerate}\n Recall that $\\Phi^+$ consists of roots $\\{2L_1,2L_2, (L_1-L_2), (L_1+L_2)\\}$ and the simple roots are $\\lambda_1=L_1-L_2$ and $\\lambda_2=2L_2$. The root vectors are as follows \\[\n{\\small X_{2L_1}}:={\\tiny\\left( {\\begin{array}{cccc}\n 0 & & 1 & \\\\\n & 0 & & \\\\\n & & 0 & \\\\\n & & & 0\n \\end{array} }\\right)}, {\\small X_{2L_2}}:={\\tiny\\left( {\\begin{array}{cccc}\n 0 & & & \\\\\n & 0 & & 1 \\\\\n & & 0 & \\\\\n & & & 0\n \\end{array} }\\right)},\\]\\[{\\small X_{L_1+L_2}}:={\\tiny\\left( {\\begin{array}{cccc}\n 0 & & & 1 \\\\\n & 0 & 1 & \\\\\n & & 0 & \\\\\n & & & 0\n \\end{array} }\\right)}\\text{ and }{\\small X_{L_1-L_2}}:={\\tiny\\left( {\\begin{array}{cccc}\n 0 & 1 & & \\\\\n & 0 & & \\\\\n & & 0 & \\\\\n & & -1 & 0\n \\end{array} }\\right).}\n\\] For $m\\geq 1$, set $H_0(\\mathbb{Z}\/p^m)$ (resp. $H(\\mathbb{Z}\/p^m)$) to denote the image of $H_0$ (resp. $H$) in $\\op{Sp}_4(\\mathbb{Z}\/p^m)$. Let $\\mathfrak{h}_m$ denote the kernel of the mod-$p^m$ reduction map $H_0(\\mathbb{Z}\/p^{m+1})\\rightarrow H_0(\\mathbb{Z}\/p^{m})$. Identify $\\mathfrak{h}_m$ with a subspace of $\\op{Ad}^0\\bar{r}$, so that $\\op{Id}+p^m X$ is identified with $X$ in $\\op{Ad}^0\\bar{r}$. It is easy to see that \n\\[\\mathfrak{h}_m=\\begin{cases}\n\\mathbb{F}_p\\langle H_1, H_2, X_{\\pm(L_1-L_2)}, X_{\\pm{2L_2}}\\rangle\\text{, for }m=1,\\\\\n\\mathbb{F}_p\\langle H_1, H_2, X_{\\pm(L_1-L_2)}, X_{\\pm{2L_2}}, X_{\\pm(L_1+L_2)}\\rangle\\text{, for }m=2,\\\\\n\\op{Ad}^0\\bar{r}\\text{, for }m\\geq 3.\n\n\\end{cases}\\]\n \\par Let $A$ be the Class group of $\\mathbb{Q}(\\mu_p)$ and let $\\mathcal{C}$ denote the mod-$p$ class group $\\mathcal{C}:=A\\otimes \\mathbb{F}_p$. The Galois group $\\op{Gal}(\\mathbb{Q}(\\mu_p)\/\\mathbb{Q})$ acts on $\\mathcal{C}$ via the natural action. Since the order of $\\op{Gal}(\\mathbb{Q}(\\mu_p)\/\\mathbb{Q})$ is prime to $p$, it follows that $\\mathcal{C}$ decomposes into eigenspaces \n \\[\\mathcal{C}=\\bigoplus_{i=0}^{p-2} \\mathcal{C}(\\bar{\\chi}^i),\\]\n where $\\mathcal{C}(\\bar{\\chi}^i)=\\{x\\in \\mathcal{C} \\mid g\\cdot x=\\bar{\\chi}^{i}(g) x \\}$.\n \\begin{Lemma}\\label{lemma31}\n For $0\\leq i\\leq p-2$,\n \\begin{enumerate}\n \\item\\label{lemma31p1} the group $\\Sh^1_{\\{p\\}}(\\mathbb{F}_p(\\bar{\\chi}^i))$ injects into $\\op{Hom}(\\mathcal{C}(\\bar{\\chi}^i),\\mathbb{F}_p)$,\n \\item\\label{lemma31p2} the group $\\Sh^2_{\\{p\\}}(\\mathbb{F}_p(\\bar{\\chi}^i))$ equals zero if $\\mathcal{C}(\\bar{\\chi}^{p-i})$ equals zero.\n \\end{enumerate}\n \\end{Lemma}\n \\begin{proof}\n Since the order of $\\op{Gal}(\\mathbb{Q}(\\mu_p)\/\\mathbb{Q})$ is prime to $p$, it follows that \\[H^1(\\op{Gal}(\\mathbb{Q}(\\mu_p)\/\\mathbb{Q}), \\mathbb{F}_p(\\bar{\\chi}^i))=0.\\] As a result, part $\\eqref{lemma31p1}$ follows from the inflation-restriction sequence. Part $\\eqref{lemma31p2}$ follows from part $\\eqref{lemma31p1}$ and Poitou-Tate duality for $\\Sh$-groups \\cite[Theorem 8.6.7]{NW}.\n \\end{proof}\n\n \\begin{Lemma}\\label{bigimagelemmaU1}\nSuppose that $p>2$ is a prime number and let $\\Pi$ be the a subgroup of $\\op{U}_1(\\mathbb{F}_p)$ such that the quotient map $\\Pi\\rightarrow \\op{U}_1(\\mathbb{F}_p)\/\\op{U}_2(\\mathbb{F}_p)$ is surjective. Then $\\Pi$ is equal to $\\op{U}_1(\\mathbb{F}_p)$.\n \\end{Lemma}\n \\begin{proof}\n For $x,y\\in \\op{U}_1(\\mathbb{F}_p)$, set $\\{x,y\\}$ to denote the commutator $xyx^{-1}y^{-1}$. As $\\mathbb{F}_p$-vector spaces, we have that\n \\[\\op{U}_k(\\mathbb{F}_p)\/\\op{U}_{k+1}(\\mathbb{F}_p)=\\begin{cases}\n \\mathbb{F}_p\\langle \\op{exp}(X_{L_1-L_2}), \\op{exp}(X_{2L_2})\\rangle \\text{ if }k=1,\\\\\n \n \\mathbb{F}_p \\langle \\op{exp}(X_{L_1+L_2})\\rangle \\text{ if }k=2,\\\\\n \\mathbb{F}_p\\langle \\op{exp}(X_{2L_1}) \\rangle\\text{ if }k=3,\\\\\n 0\\text{ if }k>3.\n \n \\end{cases}\\]We check that if $x=\\op{exp}(X_{\\lambda})$ and $y=\\op{exp}(X_{\\mu})$, for roots $\\mu$ and $\\lambda$ in $\\Phi^+$, with height $k$ and $l$ respectively, then\n \\[\\{x,y\\}=\\op{exp}([X_{\\lambda},X_{\\mu}])\\mod{\\op{U}_{k+l+1}(\\mathbb{F}_p)}.\\]We have the relations\n \\begin{equation}\\label{relationsrootvectors}[X_{L_1-L_2}, X_{2L_2}]=X_{L_1+L_2}\\text{ and } [X_{L_1-L_2}, X_{L_1+L_2}]=2X_{2L_1}\\end{equation} and that $X_{\\lambda}^2=0$ for $\\lambda\\in \\Phi^+$. We have therefore,\n \\[\\begin{split}\\{x,y\\}=&(\\op{Id}+X_{\\lambda})(\\op{Id}+X_{\\mu})(\\op{Id}-X_{\\lambda})(\\op{Id}-X_{\\mu})\\\\\n =&\\op{Id}+[X_{\\lambda}, X_{\\mu}]+(X_{\\mu}X_{\\lambda} X_{\\mu}-X_{\\lambda}X_{\\mu} X_{\\lambda})+(X_{\\lambda}X_{\\mu})^2.\\end{split}\\]Since $X_{\\lambda}^2$ and $X_{\\mu}^2$ are both equal to zero, we have that \n \\[X_{\\lambda} X_{\\mu} X_{\\lambda}=\\frac{1}{2}[[X_{\\lambda}, X_{\\mu}],X_{\\lambda}]\\] and thus $X_{\\lambda} X_{\\mu} X_{\\lambda}\\in (\\operatorname{Ad}^0\\bar{\\rho})_{2k+l}$. Likewise, the same reasoning shows that \\[X_{\\mu} X_{\\lambda} X_{\\mu}=\\frac{1}{2}[[X_{\\mu}, X_{\\lambda}],X_{\\mu}]\\] and therefore, $X_{\\mu} X_{\\lambda} X_{\\mu}\\in (\\operatorname{Ad}^0\\bar{\\rho})_{k+2l}$. Next, observe that $(X_{\\lambda} X_{\\mu})^2$ is equal to $\\frac{1}{2}([X_{\\lambda}, X_{\\mu}])^2$. This too follows from the relations $X_{\\lambda}^2=X_{\\mu}^2=0$. From the relations $\\eqref{relationsrootvectors}$, we have that if $[X_{\\lambda}, X_{\\mu}]$ is nonzero, then, $\\lambda+\\mu$ is a root and there is a constant $c$ such that $[X_{\\lambda}, X_{\\mu}]=c X_{\\lambda+\\mu}$. Since, $X_{\\lambda+\\mu}^2=0$\n, it follows that $(X_{\\lambda} X_{\\mu})^2=0$. Since $X_{\\lambda} X_{\\mu} X_{\\lambda}\\in (\\operatorname{Ad}^0\\bar{\\rho})_{2k+l}$, and the maximal height of any root is $3$, it follows that either $X_{\\lambda} X_{\\mu} X_{\\lambda}$ is zero, or a constant multiple of $X_{2L_1}$. It may be checked that $X_{2L_1} X_{\\lambda}=X_{\\lambda} X_{2L_1}=0$ for all $\\lambda\\in \\Phi^+$. As a consequence, we arrive at the following relation:\n\\[\\{x,y\\}=\\op{exp}([X_{\\lambda},X_{\\mu}])\\op{exp}(-\\frac{1}{2}[[X_{\\lambda}, X_{\\mu}],X_{\\lambda}])\\op{exp}(\\frac{1}{2}[[X_{\\mu}, X_{\\lambda}],X_{\\mu}]).\\] Note that $\\op{exp}(-\\frac{1}{2}[[X_{\\lambda}, X_{\\mu}],X_{\\lambda}])$ and $\\op{exp}(\\frac{1}{2}[[X_{\\mu}, X_{\\lambda}],X_{\\mu}])$ are in $\\op{U}_{k+l+1}$. Therefore, we deduce that\n \\[\\{x,y\\}=\\op{exp}([X_{\\lambda},X_{\\mu}])\\mod{\\op{U}_{k+l+1}(\\mathbb{F}_p)}.\\]\nWe deduce from the relations $\\eqref{relationsrootvectors}$ that the commutator $(x,y)\\mapsto\\{x,y\\}$ induces a surjective map:\n \\[\\op{U}_1(\\mathbb{F}_p)\/\\op{U}_2(\\mathbb{F}_p)\\times \\op{U}_k(\\mathbb{F}_p)\/\\op{U}_{k+1}(\\mathbb{F}_p)\\rightarrow \\op{U}_{k+1}(\\mathbb{F}_p)\/\\op{U}_{k+2}(\\mathbb{F}_p).\\]\n It follows by ascending induction on $k$, that the quotient map \\[\\Pi\\cap \\op{U}_k(\\mathbb{F}_p)\\rightarrow \\op{U}_k(\\mathbb{F}_p)\/\\op{U}_{k+1}(\\mathbb{F}_p)\\] is surjective for $k\\geq 1$. By descending induction on $k$, we deduce that $\\Pi\\cap \\op{U}_k(\\mathbb{F}_p)=\\op{U}_k(\\mathbb{F}_p)$ for $k\\geq 1$. In particular, $\\Pi$ is equal to $\\op{U}_1(\\mathbb{F}_p)$ and the proof is complete.\n \\end{proof}\n \\begin{Prop}\n Let $p\\geq 23$ be a prime such that $\\mathcal{C}(\\bar{\\chi}^{p-i})=0$ for $i\\in \\{\\pm 3, \\pm 6, \\pm 9\\}$. There exists a Galois representation\n \\[\\bar{\\rho}:=\\left( {\\begin{array}{cccc}\n \\bar{\\chi}^3 &\\ast & \\ast & \\ast \\\\\n & 1 & \\ast & \\ast \\\\\n & & \\bar{\\chi}^6 & \\\\\n & & \\ast & \\bar{\\chi}^9\n \\end{array} } \\right):\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{B}(\\mathbb{F}_p)\\] which satisfies the conditions of Theorem $\\ref{main}$. The similitude character of $\\bar{\\rho}$ is the odd character $\\bar{\\chi}^9$. Let $\\kappa$ be a fixed choice of a lift of $\\bar{\\kappa}$ such that $\\kappa=\\kappa_0\\chi^k$, where $k$ is a positive integer divisible by $p(p-1)$ and $\\kappa_0$ is the Teichm\\\"uller lift of $\\bar{\\kappa}$. There exists a finite set of auxiliary primes $X$ such that $p\\notin X$ and a lift $\\rho$ \\[\\begin{tikzpicture}[node distance = 2.0cm, auto]\n \\node (GSX) {$\\operatorname{G}_{\\mathbb{Q},\\{p\\}\\cup X}$};\n \\node (GS) [right of=GSX] {$\\operatorname{G}_{\\mathbb{Q},\\{p\\}}$};\n \\node (GL2) [right of=GS]{$\\operatorname{GSp}_{4}(\\mathbb{F}_p).$};\n \\node (GL2W) [above of= GL2]{$\\operatorname{GSp}_{4}(\\mathbb{Z}_p)$};\n \\draw[->] (GSX) to node {} (GS);\n \\draw[->] (GS) to node {$\\bar{\\rho}$} (GL2);\n \\draw[->] (GL2W) to node {} (GL2);\n \\draw[dashed,->] (GSX) to node {$\\rho$} (GL2W);\n \\end{tikzpicture}\\] for which \n\\begin{enumerate}\n\\item\\label{83p1} $\\rho$ is irreducible,\n\\item\\label{83p2} $\\rho$ is $p$-ordinary (in the sense of \\cite[section 4.1]{patrikisexceptional}),\n\\item\\label{83p3} $\\nu\\circ \\rho= \\kappa$.\n\\end{enumerate}\n \\end{Prop}\n \\begin{proof}\n We show a representation $\\bar{\\rho}$ satisfying the conditions of Theorem $\\ref{main}$ exists. It shall then follow from Theorem $\\ref{main}$ that there exists a lift $\\rho$ which satisfies the conditions $\\eqref{83p1}$, $\\eqref{83p2}$ and $\\eqref{83p3}$ above. Let $\\bar{r}$ be the representation with image in the diagonal torus:\n \\[\\bar{r}:=\\left( {\\begin{array}{cccc}\n \\bar{\\chi}^3 & & & \\\\\n & 1 & & \\\\\n & & \\bar{\\chi}^6 & \\\\\n & & & \\bar{\\chi}^9 \n \\end{array} } \\right):\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{F}_p).\\]\n \n The following matrix aids (in an informal way) in describing the eigenspace decomposition of $\\op{Ad}^0\\bar{r}$:\n \\[\\left( {\\begin{array}{cccc}\n 1 & \\bar{\\chi}^3 & \\bar{\\chi}^{-3} & \\bar{\\chi}^{-6}\\\\\n \\bar{\\chi}^{-3} & 1 & \\bar{\\chi}^{-6} & \\bar{\\chi}^{-9} \\\\\n \\bar{\\chi}^{3}& \\bar{\\chi}^{6}& 1 & \\bar{\\chi}^{-3} \\\\\n \\bar{\\chi}^{6}& \\bar{\\chi}^{9}& \\bar{\\chi}^3 & 1\n \\end{array} } \\right).\\]More precisely, $\\op{Ad}^0\\bar{r}$ is an $11$-dimensional space which decomposes into one-dimensional eigenspaces, and we have that\n \\[\\sigma_{\\pm 2L_1} = \\bar{\\chi}^{\\mp 3},\\sigma_{\\pm 2L_2} = \\bar{\\chi}^{\\mp 9},\\sigma_{\\pm(L_1+L_2)} = \\bar{\\chi}^{\\mp 6}\\text{ and }\\sigma_{\\pm (L_1-L_2)} = \\bar{\\chi}^{\\pm 3}. \\]\n We show that for $m\\geq 1$, the global cohomology group $H^2(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathfrak{h}_m)$ is zero. As a Galois module, $\\mathfrak{h}_m$ decomposes into one-dimensional eigenspaces $\\mathbb{F}_p(\\sigma)$, where $\\sigma$ ranges through some of the characters $1,\\bar{\\chi}^{\\pm 3}, \\bar{\\chi}^{\\pm 6}, \\bar{\\chi}^{\\pm 9}$. It suffices to show that for the above choices of $\\sigma$, the cohomology group $H^2(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathbb{F}_p(\\sigma))$ is zero. We show that $H^2(\\op{G}_p, \\mathbb{F}_p(\\sigma))$ is zero and $\\Sh^2_{\\{p\\}}(\\mathbb{F}_p(\\sigma))$ is zero. The dual $\\mathbb{F}_p(\\sigma)^*:=\\op{Hom}(\\mathbb{F}_p(\\sigma), \\mu_p)$ is isomorphic to $\\mathbb{F}_p(\\bar{\\chi}\\sigma^{-1})$. Since $p\\geq 13$, the character $\\bar{\\chi}\\sigma^{-1}_{\\restriction \\op{G}_p}\\neq 1$. It follows from local duality that $H^2(\\op{G}_p, \\mathbb{F}_p(\\sigma))$ is zero. It is a standard fact that $\\mathcal{C}(\\bar{\\chi})$ is zero, see \\cite[Proposition 6.16]{washington}. It follows from Lemma $\\ref{lemma31}$ and the assumptions on $p$ that $\\Sh^2_{\\{p\\}}(\\op{Ad}^0\\bar{r})$ is zero. We have thus shown that $H^2(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathfrak{h}_m)$ is zero for all $m\\geq 1$.\n \\par We stipulate that all deformations of $\\bar{r}$ have similitude character equal to $\\chi^9$, where we recall that $\\chi$ denotes the cyclotomic character. For $m\\geq 1$, let $\\chi_m$ denote $\\chi$ modulo $p^m$. Recall that $\\mathfrak{h}_1$ is spanned by $H_1,H_2$ and $\\sigma_{\\pm \\lambda_i}$ for $i=1,2$. Since the characters $\\sigma_{\\lambda_i}$ for $i=1,2$ are both odd and \\[H^2(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathbb{F}_p(\\sigma_{\\lambda_i}))=0,\\] it follows from the global Euler characteristic formula \\cite[Theorem 8.7.4]{NW} that \\[\\op{dim}H^1(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathbb{F}_p(\\sigma))=1.\\] Let $f_i$ be a generator for $H^1(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathbb{F}_p(\\sigma_{\\lambda_i}))$ for $i=1,2$. Let $r_2'$ be the mod-$p^2$ lift \\[r_2':=\\left( {\\begin{array}{cccc}\n {\\chi}_2^3 & & & \\\\\n & 1 & & \\\\\n & & {\\chi}_2^6 & \\\\\n & & & {\\chi}_2^9 \n \\end{array} } \\right):\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{GSp}_4(\\mathbb{F}_p)\\] and $r_2$ be the twist $(\\op{Id}+p(f_1+f_2))r_2'$. Note that the image of $r_2$ is in $H(\\mathbb{Z}\/p^2)$. The obstruction to lifting $r_2$ to $r_3:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow H(\\mathbb{Z}\/p^3)$ is in $H^2(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathfrak{h}_2)$, hence, is zero. Hence, $r_2$ lifts to $r_3$. Since $H^2(\\op{G}_{\\mathbb{Q},\\{p\\}}, \\mathfrak{h}_m)=0$ for all $m\\geq 1$, it follows that if $r_m:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow H(\\mathbb{Z}\/p^m)$ is a lift of $r_2$ (with similitude character $\\chi_m^9$), then $r_m$ lifts one more step to $r_{m+1}:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow H(\\mathbb{Z}\/p^{m+1})$. Furthermore, the lift $r_{m+1}$ can be prescribed to have similitude character $\\chi_{m+1}^9$. The key ingredient here is that $H$ is a subgroup of $\\op{GSp}_4(\\mathbb{Z}_p)$. Since $H$ is a closed subgroup, it follows that $r_2$ lifts to a continuous characteristic zero representation $r:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow H(\\mathbb{Z}_p)$ with similitude character $\\chi^9$. Let $\\bar{\\rho}:\\op{G}_{\\mathbb{Q},\\{p\\}}\\rightarrow \\op{B}(\\mathbb{F}_p)$ be the mod-$p$ reduction of $D^{-1} r D$ and let $\\Pi$ be $\\bar{\\rho}(\\op{G}_{\\mathbb{Q}(\\mu_p)})$. Lemma $\\ref{bigimagelemmaU1}$ asserts that if $\\Pi\\rightarrow \\op{U}_1(\\mathbb{F}_p)\/\\op{U}_2(\\mathbb{F}_p)$ is surjective, then the image of $\\bar{\\rho}$ contains $\\op{U}_1(\\mathbb{F}_p)$. Let $\\Phi(r_2)\\subseteq \\mathfrak{h}_1$ be $r_2(\\op{ker} \\bar{r})$. In fact, $\\Phi(r_2)$ is contained in $\\mathbb{F}_p\\langle H_1, H_2, X_{L_1-L_2}, X_{2L_2}\\rangle $. Since the characters $1, \\sigma_{\\lambda_1}=\\bar{\\chi}^3, \\sigma_{\\lambda_2}=\\bar{\\chi}^{-9} $ are distinct, it follows that $\\Phi(r_2)$ decomposes into distinct eigenspaces\n \\[\\Phi(r_2)=\\Phi(r_2)^{\\op{Gal}(\\mathbb{Q}(\\mu_p)\/\\mathbb{Q})}\\oplus\\Phi(r_2)_{\\bar{\\chi}^3}\\oplus \\Phi(r_2)_{\\bar{\\chi}^{-9}} .\\]\n Since $\\op{Gal}(\\mathbb{Q}(\\mu_p)\/\\mathbb{Q})$ is prime to $p$, it follows from a straightforward application of the inflation restriction sequence that $f_{i\\restriction \\op{G}_{\\mathbb{Q}(\\mu_p)}}$ is nonzero. As a result, $\\Phi(r_2)_{\\bar{\\chi}^3}$ and $\\Phi(r_2)_{\\bar{\\chi}^{-9}}$ are nonzero, and hence, $X_{\\lambda_i}\\in \\Phi(r_2)$. Since $\\op{exp}(X_{\\lambda_1})$ and $\\op{exp}(X_{\\lambda_2})$ are generators of $\\op{U}_1(\\mathbb{F}_p)\/\\op{U}_2(\\mathbb{F}_p)$, it follows that $\\Pi$ surjects onto $\\op{U}_1(\\mathbb{F}_p)\/\\op{U}_2(\\mathbb{F}_p)$. Thus, the image of $\\bar{\\rho}$ contains $\\op{U}_1(\\mathbb{F}_p)$.\n \\par We show that the conditions of Theorem $\\ref{main}$ are satisfied.\n \\begin{itemize}\n \\item Condition $\\eqref{thc1}$ asserts that $p>4$, we have assumed that $p\\geq 23$.\n \\item Condition $\\eqref{thc2}$ asserts that $\\dim (\\operatorname{Ad}^0\\bar{\\rho})^{\\op{ad}\\bar{\\rho}(c)}=\\dim \\mathfrak{n}$. Since $p>2$, up to conjugation, $\\bar{\\rho}(c)$ is equal to $\\left( {\\begin{array}{cccc}\n -1 & & & \\\\\n & 1 & & \\\\\n & & 1 & \\\\\n & & & -1\n \\end{array} } \\right)$. Explicit computation shows that w.r.t this basis, \n \\[(\\operatorname{Ad}^0\\bar{\\rho})^{\\op{ad}\\bar{\\rho}(c)}=\\mathbb{F}_p\\langle H_1, H_2, (L_1+L_2), -(L_2+L_2) \\rangle, \\]and hence, $\\dim (\\operatorname{Ad}^0\\bar{\\rho})^{\\op{ad}\\bar{\\rho}(c)}$ is equal to $4$. On the other hand, there are $4$ positive roots and the dimension of $\\mathfrak{n}$ is $4$.\n \\item Condition $\\eqref{thc3}$ asserts that the image of $\\bar{\\rho}$ contains the unipotent group $\\op{U}_1(\\mathbb{F}_p)$. This has been shown to be the case.\n \\item For condition $\\eqref{thc4}$, consider $\\sigma_{\\lambda}=\\bar{\\chi}^i$ and $\\sigma_{\\lambda'}=\\bar{\\chi}^j$. Since $i,j\\in\\{1, \\pm 3, \\pm 6, \\pm 9\\}$, we see that $|i-j|\\leq 182,$ this classification remains valid for homogeneous right coideal \nsubalgebras of the multiparameter version, see \\cite{Luz1, Tow}, of the Lusztig quantum group \n$u_q(\\frak{ sl}_{n+1}).$ We are reminded that any Hopf algebra generated by group-like and skew-\nprimitive elements is pointed, while in a pointed Hopf algebra the group-like elements span\nthe coradical, see \\cite[Definition 5.1.5]{Mon}.\n\nIn the second section we introduce main concepts and provide the general results \non the structure of the character Hopf algebras that are of use for classification. \nIn Lemma \\ref{qsim} we note that \nif the given character Hopf algebra $H=A\\# {\\bf k}[G]$ is a bosonisation\nof a quantum symmetric algebra $A,$ then each invariant \ndifferential subspace $U$ of $A$ defines a right coideal $U\\# {\\bf k}[G].$\nThis statement allows one to use noncommutative differential calculus, \n\\cite[p.6]{Luz2}, \\cite{MS}, \\cite{Kh1},\ndue to P. Schauenburg's characterization of quantum Borel subalgebras \\cite{Scha}.\nThe key point of the section is the construction of a PBW-basis\nover the coradical for a right coideal \nsubalgebra by means of \\cite{KhT, KhQ}. This basis, in particular, provides some \ninvariants for right coideal subalgebras (Definition \\ref{root}). \n\nIn the third section we define the multiparameter quantification of a Kac-Moody algebra as a character \nHopf algebra. This approach \\cite{Kh4} combines and generalizes all known quantifications. We do \nnot put unnecessary restrictions on the characteristic and on the quantification parameters. This allows \none, for example, to define a new class of finite Frobenius algebras as the Lusztig quantum groups \nover a finite field. All their right coideal subalgebras are also Frobenius \\cite{Skr} (finite Frobenius \nalgebras, in turn, have a significant r\\^{o}le in the coding theory \\cite{GMO}). \nIn Proposition \\ref{rtri} we provide \na short proof of the so called ``triangular decomposition\" in a quite general form.\n\nIn the fourth section (Proposition \\ref{phi}) we show that each homogeneous right coideal subalgebra\nin the quantum Borel algebra $U_q^+(\\frak{ sl}_{n+1})$ has PBW-generators over {\\bf k}$\\, [G]$\nof the following form\n\\begin{equation} \\Psi ^{\\hbox{\\bf s}}(k,m)=\n \\hbox{\\Large [[}\\ldots \\hbox{\\Large [}u[1+s_r,m], u[1+s_{r-1},s_r]\\hbox{\\Large ]}, \\ldots \\ ,\nu[1+s_1,s_2]\\hbox{\\Large ]},\\, u[k,s_1]\\hbox{\\Large ]},\n\\label{cbr1int}\n\\end{equation}\nwhere the brackets are defined by the structure \nof a character Hopf algebra, $[u,v]=uv-\\chi ^u(g_v)vu;$ \n$u[i,j]=[\\ldots [x_i,x_{i+1}],\\ldots ,x_j];$ $k\\leq s_12,$ then this is the case for homogeneous\nright coideal subalgebras of $u_q^{\\pm }(\\frak{ sl}_{n+1}).$ (If $q$ is not a root of 1 then\nall right coideal subalgebras that contain $G$ are homogeneous, Corollary \\ref{odn1}). \n\nIn Section 6 we consider right coideal subalgebras\nin the quantum Borel algebra that do not contain the coradical. \nNote that for every submonoid \n$\\Omega \\subseteq G$ the set of all linear combinations {\\bf k}$\\, [\\Omega]$\nis a right coideal subalgebra. We show that if the intersection $\\Omega $ of a homogeneous \nright coideal subalgebra $U$ with $G$ is a subgroup, then\n$U=\\, ${\\bf U}$_{\\theta }^{1}\\, ${\\bf k }$[\\Omega ].$ Here {\\bf U}$_{\\theta }^{1}$\nis a subalgebra generated by $g^{-1}_aa$ when $a$ runs through the described above \ngenerators of {\\bf U}$_{\\theta }.$\n\nIn Section 7 we characterize ad$_r$-invariant right coideal subalgebras that have \ntrivial intersection with the coradical in terms of K\\'eb\\'e's construction \\cite{Keb, Keb1}. \n\nWe see that the construction of {\\bf U}$_{\\theta }$ is completely\nconstructive, although it is not straightforward. \nHence by means of computer calculations one may find \nall necessary invariants of the coideal subalgebras and relations between them.\nIn the eighth section we provide a tableaux of the coideal subalgebras and their main\ncharacteristics for $n=3$ that was found by means of computer calculations.\n\nIn Sections 9-11 we consider the whole of $U_q(\\frak{ sl}_{n+1}).$\nThe triangular decomposition, \n\\begin{equation}\nU_q(\\frak{ sl}_{n+1})= U_q^-(\\frak{ sl}_{n+1})\\otimes _{{\\bf k}[F]} {\\bf k}[H]\n\\otimes _{{\\bf k}[G]}U_q^+(\\frak{ sl}_{n+1}),\n\\label{trint}\n\\end{equation}\nprovides a hope that any (homogeneous) right coideal subalgebra that contains the coradical\nhas the triangular decomposition as well, and for any two right coideal subalgebras \n$U_{\\theta }\\subseteq U_q^+ (\\frak{ sl}_{n+1}),$ $U_{\\theta ^{\\prime }}\\subseteq U_q^-(\\frak{ sl}_{n+1})$ \nthe tensor product \n\\begin{equation}\n \\hbox{\\bf U}\\, =U_{\\theta ^{\\prime }}\\otimes _{{\\bf k}[F]} {\\bf k}[H]\n\\otimes _{{\\bf k}[G]}U_{\\theta }\n\\label{truint}\n\\end{equation}\nis a right coideal subalgebra. In this hypothesis just one statement fails, \nthe tensor product indeed is a right coideal but not always a subalgebra.\n\n\nTo describe conditions when (\\ref{truint}) is a subalgebra\nwe display the element $\\Psi ^{\\hbox{\\bf s}}(k,m)$\nschematically as a sequence of black and white points labeled by the numbers\n$k-1,$ $k,$ $k+1, \\ldots $ $m-1,$ $m,$ where the first point is always white, and\nthe last one is always black, while an intermediate point labeled by $i$ is black if and only if \n$i\\in {\\bf S}:$ \n$$\n \\stackrel{k-1}{\\circ } \\ \\ \\stackrel{k}{\\circ } \\ \\ \\stackrel{k+1}{\\circ } \n\\ \\ \\stackrel{k+2}{\\bullet }\\ \\ \\ \\stackrel{k+3}{\\circ }\\ \\cdots\n\\ \\ \\stackrel{m-2}{\\bullet } \\ \\ \\stackrel{m-1}{\\circ }\\ \\ \\stackrel{m}{\\bullet }\n$$\nConsider two elements \n$\\Psi ^{T_k}(k,\\tilde{\\theta }_k)$ and $\\Psi ^{T_i^{\\prime }}(i,\\tilde{\\theta }_i^{\\prime }),$\nwhere $T_k,$ $T_i^{\\prime }$ are defined as above by $\\theta $ and $\\theta ^{\\prime},$\nrespectively. Let us display these elements graphically\n\\begin{equation}\n\\begin{matrix}\n\\stackrel{k-1}{\\circ } \\ & \\cdots \\ & \\stackrel{i-1}{\\bullet } \n\\ & \\stackrel{i}{\\bullet }\\ \\ & \\stackrel{i+1}{\\circ }\\ & \\cdots &\n\\ & \\stackrel{\\tilde{\\theta }_k}{\\bullet } \\ & \\ & \\stackrel{\\tilde{\\theta }_i^{\\prime }}{\\cdot } \\cr\n\\ \\ & \\ \\ & \\circ \n\\ & \\circ \\ \\ & \\bullet \\ & \\cdots &\n\\ & \\bullet \\ & \\cdots \\ & \\bullet\n\\end{matrix}\n\\label{gra1int}\n\\end{equation}\nIn Theorem \\ref{osn5} we prove that (\\ref{truint}) is a subalgebra if and only if for each pair\n$(k, i),$ $1\\leq k,i\\leq n$ one of the following two options is fulfilled:\n\na) Representation (\\ref{gra1int}) has no fragments of the form \n$$\n\\begin{matrix}\n\\stackrel{t}{\\circ } \\ & \\cdots & \\stackrel{l}{\\bullet } \\cr\n\\circ \n\\ & \\cdots & \\bullet \n\\end{matrix}\n$$\n\nb) Representation (\\ref{gra1int}) has the form \n$$\n\\begin{matrix}\n\\stackrel{k-1}{\\circ } \\ & \\cdots & \\circ & \\cdots & \\bullet & \\cdots & \\stackrel{m}{\\bullet } \\cr\n\\circ \n\\ & \\cdots & \\bullet & \\cdots & \\circ & \\cdots & \\bullet \n\\end{matrix}\n$$\nwhere no one of the intermediate columns has points of the same color.\n\nThe obtained criterion allows use of the computer in order to find the total number $C_n$ of right coideal subalgebras which contain the coradical: \n$$\nC_2=26; \\ C_3=252; \\ C_4=3,368; \\ C_5=58,810; \\ C_6=1,290,930; \\ C_7=34,604,844.\n$$\n\n\\noindent\n{\\bf Remark}. If a Hopf algebra $H$ has a Hopf algebra pairing \n$\\langle , \\rangle : M\\times H\\rightarrow {\\bf k}$ with a Hopf algebra $M,$\nthen $M$ acts on $H$ via $m\\rightharpoonup h=\\sum h^{(1)}\\langle m, h^{(2)}\\rangle .$\nCertainly, in this case each right coideal is $M$-invariant. Conversely, if the pairing is left faithful\n(that is, $\\langle M, h\\rangle =0$ implies $h=0$) then each $M$-invariant subspace is a right coideal.\nFor $H=U_q(\\frak{sl}_{n+1})$ (or for $H=u_q(\\frak{sl}_{n+1})$ if $q^t=1$) there exists a Hopf algebra pairing with $M=GL_q(n),$ see \\cite{RTF, CM, Tow}.\nHence, alternatively, our main result provides a classification of $GL_q(n)$-invariant subalgebras that contain the coradical.\n\n\\smallskip\n\nThe computer part of this work has been done by the second author, \nwhile the proofs are due to the first one.\n\n\n\\section{Preliminaries}\n\\noindent\n{\\bf PBW-generators}.\nLet $S$ be an algebra over a field {\\bf k} and $K$ its subalgebra\nwith a fixed basis $\\{ g_j\\, |\\, j\\in J\\} .$ A linearly ordered subset $W\\subseteq S$ is said to be\na {\\it set of PBW-generators of $S$ over $K$} if there exists \na function $h:W\\rightarrow {\\bf Z}^+\\cup {\\infty },$\ncalled the {\\it height function}, such that the set of all products\n\\begin{equation}\ng_jw_1^{n_1}w_2^{n_2}\\cdots w_k^{n_k}, \n\\label{pbge}\n\\end{equation}\nwhere $j\\in J,\\ \\ w_11.$\nIn this case all elements $[x_i,x_j]=x_ix_j-p_{ij}x_jx_i,$ $|i-j|>1$ are skew primitive.\nTherefore the ideal of $G\\langle X\\rangle $ generated by these elements is a Hopf ideal.\nWe denote by ${{\\mathfrak A}}_n$ the quotient character \nHopf algebra, \n$$\n{{\\mathfrak A}}_n=G\\langle X\\, ||\\, [x_i,x_j]=0,\\ j-i>1 \\rangle \\stackrel{df}{=} \nG\\langle X\\rangle \/ {\\rm Id}\\, \\langle [x_i,x_j],\\ j-i>1\\rangle .\n$$\n\n\\begin{definition} \\rm\nThe elements $u,v$ are said to be \n{\\it separated} if there exists an index $j,$ $1\\leq j\\leq n,$\nsuch that either $u\\in {\\bf k}\\langle x_i\\ |\\ ij\\rangle $ or vise versa \n$u\\in {\\bf k}\\langle x_i\\ |\\ i>j\\rangle ,$\n$v\\in {\\bf k}\\langle x_i\\ |\\ i\nm_1^{\\prime }x_{i_1}+m_2^{\\prime }x_{i_2}+\\ldots +m_k^{\\prime }x_{i_k}\n\\label{ord}\n\\end{equation}\nif the first from the left nonzero number in\n$(m_1-m_1^{\\prime}, m_2-m_2^{\\prime}, \\ldots , m_k-m_k^{\\prime})$\nis positive, where $x_{i_1}>x_{i_2}>\\ldots >x_{i_k}$ in $X.$\nWe associate a formal degree $D(u)=\\sum _{x\\in X}m_xx\\in \\Gamma ^+$\nto a word $u$ in $G\\cup X,$ where $\\{ m_x\\, | \\, x\\in X\\}$ is the constitution of $u$\n(in \\cite[\\S 2.1]{FG} the formal sum $D(u)$ is called the {\\it weight} of $u$).\nRespectively, if $f=\\sum \\alpha _i u_i\\in G\\langle X\\rangle ,$ $0\\neq \\alpha _i\\in {\\bf k}$\nthen \n\\begin{equation}\nD(f)={\\rm max}_i\\{ D(u_i)\\} .\n\\label{degr}\n\\end{equation}\n\nOn the set of all words in $X$ we fix the lexicographical order\nwith the priority from the left to the right,\nwhere a proper beginning of a word is considered to \nbe greater than the word itself.\n\nA non-empty word $u$\nis called a {\\it standard} word (or {\\it Lyndon} word, or \n{\\it Lyndon-Shirshov} word) if $vw>wv$\nfor each decomposition $u=vw$ with non-empty $v,w.$\nA {\\it nonassociative} word is a word where brackets \n$[, ]$ somehow arranged to show how multiplication applies.\nIf $[u]$ denotes a nonassociative word then by $u$ we denote \nan associative word obtained from $[u]$ by removing the brackets\n(of course, $[u]$ is not uniquely defined by $u$ in general, however Lemma \\ref{ind}\nsays that the value of $[u]$ in ${{\\mathfrak A}}_n$ is uniquely defined provided that $u=u(k,m)$).\nThe set of {\\it standard nonassociative} words is the biggest set $SL$\nthat contains all variables $x_i$\nand satisfies the following properties.\n\n1)\\ If $[u]=[[v][w]]\\in SL$\nthen $[v],[w]\\in SL,$\nand $v>w$\nare standard.\n\n2)\\ If $[u]=[\\, [[v_1][v_2]]\\, [w]\\, ]\\in SL$ then $v_2\\leq w.$\n\n\\noindent\nEvery standard word has\nonly one alignment of brackets such that the appeared\nnonassociative word is standard (Shirshov theorem \\cite{pSh1}).\nIn order to find this alignment one may use the following\nprocedure: The factors $v, w$\nof the nonassociative decomposition $[u]=[[v][w]]$\nare the standard words such that $u=vw$\nand $v$ has the minimal length (\\cite{pSh2}, see also \\cite{Lot}).\n\\begin{definition} \\rm A {\\it super-letter}\nis a polynomial that equals a nonassociative standard word\nwhere the brackets mean (\\ref{sqo}).\nA {\\it super-word} is a word in super-letters. \nA $G$-{\\it super-word} is a super-word multiplied from the left by a group-like element. \n\\label{sup1}\n\\end{definition}\n\nBy Shirshov's theorem every standard word $u$\ndefines only one super-letter, in what follows we shall denote it by $[u].$\nThe order on the super-letters is defined in the natural way:\n$[u]>[v]\\iff u>v.$\n\\begin{definition} \\rm\nA super-letter $[u]$\nis called {\\it hard in }$H$\nprovided that its value in $H$\nis not a linear combination\nof values of super-words of the same degree (\\ref{degr})\nin smaller than $[u]$ super-letters, \n\\underline{and $G$-super-words of smaller degrees}.\n\\label{tv1}\n\\end{definition}\n\\begin{definition} \\rm\nWe say that a {\\it height} of a hard in $H$ super-letter $[u]$\nequals $h=h([u])$ if $h$\nis the smallest number such that: first, $p_{uu}$\nis a primitive $t$-th root of 1 and either $h=t$\nor $h=tl^r,$ where $l=$char({\\bf k}); and then the value in $H$\nof $[u]^h$\nis a linear combination of super-words of the same degree (\\ref{degr})\nin less than $[u]$ super-letters,\n\\underline{and $G$-super-words of smaller degrees}.\nIf there exists no such number then the height equals infinity.\n\\label{h1}\n\\end{definition}\nCertainly, if the algebra $H$ is homogeneous in each $a_i$ then one may omit\nthe underlined parts of the definitions.\n\\begin{theorem} $(${\\rm \\cite[{Theorem 2}]{Kh3}}$).$\nThe values of all hard in $H$ super-letters with the above defined height function\nform a set of PBW-generators for $H$ over {\\bf k}$[G].$\n\\label{BW}\n\\end{theorem}\n \n\\noindent \n{\\bf PBW-basis of a coideal subalgebra}. According to \\cite[Theorem 1.1]{KhT, KhQ}\nevery right coideal subalgebra {\\bf U} that contains all group-like elements has a PBW-basis\nover {\\bf k}$[G]$ which can be extended up to a PBW-basis of $H.$\n\nThe PBW-generators $T$ for {\\bf U} \ncan be obtained from the PBW-basis of $H$ given in Theorem \\ref{BW}\nin the following way. \n\nSuppose that for a given hard super-letter $[u]$ there exists an element $c\\in ${\\bf U}\nwith the leading term $[u]^s$ in the PBW-decomposition given in Theorem \\ref{BW}: \n\\begin{equation}\nc=[u]^s+\\sum \\alpha _iW_i+\\ldots \\in \\hbox{\\bf U},\n\\label{vad1}\n\\end{equation}\nwhere $W_i$ are the basis super-words starting with less than $[u]$ super-letters, \n$D(W_i)=sD(u),$ and by the dots we denote a linear combination of $G$-super-words \nof $D$-degree less than $sD(u).$ We fix one of the elements with the minimal $s,$ and denote it by $c_u.$ Thus, for every hard in $H$ super-letter $[u]$ we have at most one element $c_u.$\nWe define the height function by means of the following lemma.\n\n\\begin{lemma}$\\!\\!\\! ($\\cite[{\\rm Lemma 4.3}]{KhT, KhQ}$).$ \nIn the representation $(\\ref{vad1})$ of the chosen element $c_u$\neither $s=1,$ or $p(u,u)$ is a primitive $t$-th root of $1$ and $s=t$ or \n$($in the case of positive characteristic$)$ $s=t({\\rm char}\\, {\\bf k})^r.$\n\\label{nco1}\n\\end{lemma}\nIf the height of $[u]$ in $H$ is infinite, then the height of $c_u$ in {\\bf U}\nis defined to be infinite as well. If the height of $[u]$ in $H$ equals $t,$ then, due to \nthe above lemma, $s=1$ (in the PBW-decomposition (\\ref{vad1}) the exponent \n$s$ must be less than\nthe height of $[u]$). In this case the height of $c_u$ in {\\bf U} is supposed to be $t$ as well.\nIf the characteristic $l$ is positive, and the height of $[u]$ in $H$ equals\n$tl^r,$ then we define the height of $c_u$ in {\\bf U} to be equal to $tl^r\/s$\n(thus, in characteristic zero the height of $c_u$ in {\\bf U} always\nequals the height of $[u]$ in $H$).\n\n\\begin{proposition}\nThe set of all chosen $c_u$ with the above defined height function\nforms a set of PBW-generators for {\\bf U} over {\\bf k}$[G].$ \n\\label{pro}\n\\end{proposition}\n\\begin{proof} See, \\cite[Proposition 4.4]{KhT, KhQ}. \\end{proof}\n\nWe note that there is an essential freedom in construction of the PBW-generators \nfor a right coideal subalgebra. In particular the PBW-basis is not uniquely defined in the above process. Nevertheless the set of leading terms of the PBW-generators indeed is uniquely defined.\n\n\\begin{definition} \\rm A hard super-letter $[u]$ is called {\\bf U}-{\\it effective} if there exists\n$c\\in \\, ${\\bf U} of the form (\\ref{vad1}). The degree $sD(u)\\in \\Gamma ^+ $ of $c$ with minimal $s$\nis said to be an {\\bf U}-{\\it root}. An {\\bf U}-root $\\gamma \\in \\Gamma ^+ $\n is called a {\\it simple} {\\bf U}-{\\it root} if it is not a sum of two or more other {\\bf U}-roots.\n\\label{root}\n\\end{definition}\nThus, the set of {\\bf U}-effective super-letters, the set of {\\bf U}-roots, and the set of \nsimple {\\bf U}-roots are invariants of any right coideal subalgebra {\\bf U}.\n\n{\\bf Remark}. There is already a fundamental for Lie theory notion of roots associated\nto semisimple Lie algebras. Certainly, the set of PBW-generators for \nthe universal enveloping algebra $U({\\frak g})$ \ncoincides with a basis of the Lie algebra ${\\frak g}.$ If we apply our definition to $U({\\frak g})$ then $U({\\frak g})$-roots are the formal degrees of basis elements related to a fixed set of generators\n$x_i, i\\in I.$ At the same time the formal degrees of basis elements for the Borel subalgebra \nare in one-to-one correspondence with positive roots: to each root \n$\\alpha _{i_1}+\\alpha _{i_2}+\\cdots +\\alpha _{i_k}$\ncorresponds a basis element $[\\ldots [x_{i_1},x_{i_2}],\\ldots ,x_{i_k}],$ \nsee \\cite[Chapter IV, \\S 3, Statement XVII]{Jac}. Therefore our definition of a root\nis a natural generalization of the classical notion. Probably the analogy would be \nmore clear if in our definition of the formal degree \nwe will replace the symbols $x_i$ with the characters $\\chi ^i$\nand identify the generators $g_i$ of the group $G$ with (exponents of the) basis elements $h_i$\nof the Cartan subalgebra since the classical roots are elements of the dual space\n$(\\sum_i {\\bf k}h_i)^*.$ Lemma \\ref{odn} below shows that this replacement is admissible.\nWe belive that by this very reason in \\cite{FG} the formal degree is referred to as {\\it weight},\nthe notion already well defined in the Lie theory. \n\n\\smallskip\n\\noindent \n{\\bf Differential calculi}. \nThe free algebra ${\\bf k}\\langle X\\rangle $ has a coordinate differential calculus\n\\begin{equation}\n\\partial_i(x_j)=\\delta _i^j,\\ \\ \\partial _i (uv)=\\partial _i(u)\\cdot v+\\chi ^u(g_i)u\\cdot \\partial _i(v).\n\\label{defdif}\n\\end{equation}\nThe partial derivatives connect the calculus with the coproduct on $G\\langle X\\rangle$ via\n\\begin{equation}\n\\Delta (u)\\equiv u\\otimes 1+\\sum _ig_i\\partial_i(u)\\otimes x_i\\ \\ \\ \n(\\hbox{mod }G\\langle X\\rangle \\otimes \\hbox{\\bf k}\\langle X\\rangle ^{(2)}),\n\\label{calc}\n\\end{equation}\nwhere ${\\bf k}\\langle X\\rangle ^{(2)}$ is the set (an ideal)\nof noncommutative polynomials without free and linear terms.\nSymetrically the equation\n\\begin{equation}\n\\Delta (u)\\equiv g_u\\otimes u+\\sum _ig_ug_i^{-1}x_i\\otimes\\partial _i^*(u)\\ \\ \\ \n(\\hbox{mod }G\\langle X\\rangle ^{(2)}\\otimes \\hbox{\\bf k}\\langle X\\rangle )\n\\label{calcdu}\n\\end{equation}\ndefines a dual differential calculus on ${\\bf k}\\langle X\\rangle $\nwhere the partial derivatives satisfy\n\\begin{equation}\n\\partial _j^*( x_i)=\\delta _i^j,\\ \\ \\partial _i^*(uv)=\n\\chi ^{i}(g_v) \\partial _i^*(u)\\cdot v+u\\cdot \\partial _i^*(v).\n\\label{difdu}\n\\end{equation}\nHere $G\\langle X\\rangle ^{(2)}{\\bf k}$ is an ideal of $G\\langle X\\rangle $\nof elements without free and linear terms.\nIf the kernel of $\\xi $ defined in (\\ref{gom}) is contained in $G\\langle X\\rangle ^{(2)}$ then formule\n(\\ref{calc}), (\\ref{calcdu}) with the $a$'s in place of the $x$'s define coordinate differential calculi\non the subalgebra $A$ of $H$ generated by the $a$'s. In this case restriction of $\\xi $\non $\\hbox{\\bf k}\\langle X\\rangle$ is a differential homomorphism, while\n (\\ref{calc}) and (\\ref{calcdu}) imply that each skew-primitive element $u$\nfrom $A^{(2)}=\\xi ({\\bf k}\\langle X\\rangle ^{(2)})$ is a constant with respect to both calculi,\n$\\partial _i(u)=\\partial _i^*(u)=0,$ $1\\leq i\\leq n.$\nMore details one can find in \\cite{MS, Kh1, Kh5}.\n\n\\smallskip\n\\noindent \n{\\bf Shuffle representation}. Let Ker$\\, \\xi \\subseteq G\\langle X\\rangle ^{(2)}.$ \nIn this case there exists a Hopf algebra projection $\\pi : H\\rightarrow {\\bf k}[G],$\n$a_i\\rightarrow 0, $ $g_i\\rightarrow g_i .$ Hence by the Radford theorem\n\\cite{Rad} we have a decomposition in a biproduct,\n$H=A\\# {\\bf k}[G],$ by means of the isomorphism\n$u\\rightarrow \\vartheta (u^{(1)})\\# \\pi(u^{(2)})$ with $\\vartheta (u)=\\sum _{(u)}u^{(1)}\\pi (S(u^{(2)})),$\nsee details in \\cite[\\S 1.5, \\S 1.7]{AS}.\n\nIf Ker$\\, \\xi $ is the biggest Hopf ideal\nin $G\\langle X\\rangle ^{(2)},$ or, equivalently, if $H$ is a Hopf algebra \nof type one in the sense of Nichols \\cite{Nic}, or, equivalently, \nif $A$ is a {\\it quantum symmetric algebra} (a Nichols algebra \\cite[\\S 1.3, Section 2]{AS}),\nthen $A$ has a shuffle representation as follows.\n\nThe algebra $A$ has a structure of a {\\it braided Hopf algebra}, \\cite{Tak1}, \nwith a braiding $\\tau (u\\otimes v)=p(v,u)^{-1}v\\otimes u.$\nThe braided coproduct \n$\\Delta ^b$ is connected with the coproduct on $H$ in the following way,\nsee \\cite[p. 93, (3.18)]{Kh5},\n\\begin{equation}\n\\Delta ^b(u)=\\sum _{(u)}u^{(1)}\\hbox{gr}(u^{(2)})^{-1}\\underline{\\otimes} u^{(2)}.\n\\label{copro}\n\\end{equation}\nAt the same time the tensor space $T(V),$ $V=\\sum_i {\\bf k}x_i$ also has a structure of a braided Hopf algebra.\nThis is the {\\it quantum shuffle algebra} $Sh_{\\tau }(V)$ with the coproduct \n\\begin{equation}\n\\Delta ^b(u)=\\sum _{i=0}^m(z_1\\ldots z_i)\\underline{\\otimes} (z_{i+1}\\ldots z_m),\n\\label{bcopro}\n\\end{equation}\nwhere $z_i\\in X,$ and $u=(z_1z_2\\ldots z_{m-1}z_m)$ is the tensor\n$z_1\\otimes z_2\\otimes \\ldots \\otimes z_{m-1}\\otimes z_m$ \nconsidered as an element of $Sh_{\\tau }(V).$\nThe map $a_i\\rightarrow (x_i)$ defines an embedding of the braided Hopf algebra\n$A$ into the the braided Hopf algebra $Sh_{\\tau }(V).$\nMore details can be find in \\cite{Nic, Wor, Scha, Ros, AG, FG1,Tak1, Flo, Kh5, Kh1}.\n\n\\smallskip\n\\noindent \n{\\bf Differential subalgebras}. \nIf {\\bf U} is a right coideal subalgebra of $H$ and {\\bf k}$[G]\\subseteq \\,${\\bf U}, then \n$\\vartheta (\\hbox{\\bf U})\\subseteq \\, ${\\bf U}, hence \n{\\bf U}$=U_A\\# {\\bf k}[G]$\nwith $U_A=\\vartheta (\\hbox{\\bf U})=\\hbox{\\bf U}\\cap A.$ Formula\n(\\ref{calc}) implies that $U_A$ is a differential subalgebra of $A,$\nthat certainly satisfies $gU_Ag^{-1}\\subseteq U_A,$ $g\\in G.$\nThe converse statement is valid if Ker$\\, \\xi $ is the biggest Hopf ideal.\n\\begin{lemma} Suppose that Ker$\\, \\xi $ is the biggest Hopf ideal\n in $G\\langle X\\rangle ^{(2)}.$\nIf $U$ is a differential subspace of $A={\\bf k}\\langle a_i\\rangle $ $=\\vartheta (H),$\nand $gUg^{-1}\\subseteq U,$ $g\\in G,$\nthen $U\\# {\\bf k}[G]$ is a right coideal of $H.$\n\\label{qsim}\n\\end{lemma}\n\\begin{proof}\nThe braided coproduct (\\ref{copro}) also defines a differential calculus \n\\begin{equation}\n\\Delta ^b(u)\\equiv u\\underline{\\otimes} 1+\n\\sum _i\\frac{\\partial ^bu}{\\partial x_i}\\underline{\\otimes} x_i\\ \\ \\ \n(\\hbox{mod }A \\underline{\\otimes} A^{(2)}).\n\\label{calc1}\n\\end{equation}\nIn \\cite[Theorem 4.8]{Kh1} this calculus is denoted by $d^*.$\nFormulae (\\ref{calc}), (\\ref{copro}), and (\\ref{calc1}) imply\n$\\partial ^bu\/\\partial x_i=g_i\\partial _i(u)g_i^{-1}.$\n\nSince $A$ has a representation as a subalgebra of the \nquantum shuffle algebra $Sh_{\\tau }(V),$ \nby \\cite[Theorem 5.1]{Kh1} applied to the calculus $d^*$\nthe restriction $\\Omega =\\xi |_{\\hbox{\\bf k}\\langle X\\rangle}$ has the following\ndifferential form\n\\begin{equation}\n\\Omega (u)=\\sum _{i_1,i_2,\\cdots ,i_n}\\frac{(\\partial^b) ^nu}{\\partial x_{i_1}\\partial x_{i_2}\\cdots \n\\partial x_{i_n}}(x_{i_n}x_{i_{n-1}}\\cdots x_{i_1}),\\ \\ \\ u\\in V^{\\otimes n},\n\\label{oderc3}\n\\end{equation} \nwhere as above $(x_{i_n}x_{i_{n-1}}\\cdots x_{i_1})$ is the tensor\n$x_{i_n}{\\otimes } x_{i_{n-1}}{\\otimes } \\cdots {\\otimes } x_{i_1}$\nconsidered as an element of $Sh_{\\tau }(V).$ By means of (\\ref{bcopro}) we have \n$$\n\\Delta ^b(\\Omega (u))=\\sum _{i_1,i_2,\\cdots ,i_n}\n\\frac{(\\partial ^b)^nu}{\\partial x_{i_1}\\partial x_{i_2}\\cdots \n\\partial x_{i_n}} \\sum _{k=1}^{n+1}(x_{i_n}\\cdots x_{i_k})\n\\underline{\\otimes }(x_{i_{k-1}}\\cdots x_{i_1})\n$$\n\n$$\n=\\sum _{k=1}^{n+1}\\sum _{i_1,i_2,\\cdots ,i_{k-1}}\\left(\\sum _{i_k,i_{k+1},\\cdots ,i_n}\n\\frac{(\\partial ^b)^{n-k+1}\\left[ \\frac{(\\partial ^b)^{k-1}u}{\\partial x_{i_1} \\cdots \n\\partial x_{i_{k-1}}}\\right]}{\\partial x_{i_k} \\cdots \n\\partial x_{i_n}} (x_{i_n}\\cdots x_{i_k})\\right)\n\\underline {\\otimes }\n (x_{i_{k-1}}\\cdots x_{i_1})\n$$\n\\begin{equation}\n=\\sum _{k=1}^{n+1}\\sum _{i_1,i_2,\\cdots ,i_{k-1}}\n\\Omega \\left( \\frac{(\\partial ^b)^{k-1} u}{\\partial x_{i_1} \\cdots \n\\partial x_{i_{k-1}}}\\right) \n\\underline {\\otimes }\n(x_{i_{k-1}}\\cdots x_{i_1}).\n\\label{oderc1}\n\\end{equation}\nSince $\\Omega $ is a $d^*$-differential map, we have got \n\\begin{equation}\n\\Delta ^b(w)=\\sum _{k=1}^{n+1}\n\\sum _{i_1,i_2,\\cdots ,i_{k-1}}\n\\frac{(\\partial ^b)^{k-1} w}{\\partial x_{i_1}\\cdots \\partial x_{i_k}}\n\\underline {\\otimes }\n(x_{i_{k-1}}\\cdots x_{i_1}), \\ \\ w=\\Omega (u).\n\\label{oderc4}\n\\end{equation} \nThis formula implies that each differential subspace $W\\subseteq A$ with respect to $d^*$\n is a right coideal with respect to $\\Delta ^b.$\nIndeed, $\\Delta ^b(W)\\subseteq (A\\underline{\\otimes } A)\\cap \n(W\\underline{\\otimes } Sh_{\\tau})$ $=W\\underline {\\otimes } A.$\nSince\n$\\partial ^bu\/\\partial x_i=g_i\\partial_i(u)g_i^{-1},$\nthe space $U$ given in the lemma is a right coideal\nwith respect to the coproduct $\\Delta ^b.$ Now (\\ref{copro})\nshows that $U{\\bf k}[G]$ is a right coideal of $H.$ \nThe lemma is proved.\\end{proof}\n\n\\section{Multiparameter quantification of Kac-Moody algebras\\\\ as character Hopf algebras}\n\n\\noindent\n{\\bf Quantification of Borel subalgebras}.\nLet $C=||a_{ij}||$ be a symmetrizable by $D={\\rm diag }(d_1, \\ldots d_n)$ generalized Cartan matrix, $d_ia_{ij}=d_ja_{ji}.$\nDenote by $\\mathfrak g$ a Kac-Moody algebra defined by $C,$ see \\cite{Kac}.\nSuppose that the quantification parameters $p_{ij}=p(x_i,x_j)=\\chi ^i(g_j)$ are related by\n\\begin{equation}\np_{ii}=q^{d_i}, \\ \\ p_{ij}p_{ji}=q^{d_ia_{ij}},\\ \\ \\ 1\\leq i,j\\leq n. \n\\label{KM1}\n\\end{equation}\nIn this case the multiparameter quantization $U^+_q ({\\mathfrak g})$ of the \nBorel subalgebra ${\\mathfrak g}^+$\nis a character Hopf algebra defined by Serre relations \nwith the skew brackets in place of the Lie operation:\n\\begin{equation}\n[\\ldots [[x_i,\\underbrace{x_j],x_j], \\ldots ,x_j]}_{1-a_{ji} \\hbox{ times}}=0, \\ \\ 1\\leq i\\neq j\\leq n.\n\\label{KM2}\n\\end{equation}\nBy \\cite[Theorem 6.1]{Khar} the left hand sides of these relations are skew-primitive elements \nin $G\\langle X\\rangle .$ Therefore the ideal generated by these elements is a Hopf ideal,\nwhile $U^+_q ({\\mathfrak g})$ indeed has a natural character Hopf algebra structure.\n\n\\begin{lemma} If $C$ is a Cartan matrix of finite type $($in particular the symmetric matrix $DC$ is positively defined $),$ and $q$ is not a root of $1$ then the grading of $U^+_q ({\\mathfrak g})$ by \ncharacters $(\\ref{grad})$ coincides with the grading by $\\Gamma ^+.$ \n\\label{odn}\n\\end{lemma}\n\\begin{proof} Since every homogeneous in $\\Gamma ^+$ element is homogeneous with respect to \n(\\ref{grad}), it suffices to show that the characters $\\chi ^i=\\chi ^{x_i},$ $1\\leq i\\leq n$\ngenerate a free Abelian group. Suppose in contrary that\n\\begin{equation}\n\\chi_1 \\stackrel{df}{=}(\\chi ^1)^{k_1}\\cdots (\\chi ^n)^{k_n}=(\\chi ^1)^{m_1}\\cdots (\\chi ^n)^{m_n}\n\\stackrel{df}{=} \\chi_2,\n\\label{fch}\n\\end{equation}\nwhere $k_i, m_i\\geq 0, \\ k_im_i=0,$ $1\\leq i\\leq n,$ and one of the $k_i$'s is nonzero.\nLet $g=g_1^{k_1}\\cdots g_n^{k_n},$ $h=g_1^{m_1}\\cdots g_n^{m_n}.$\nBy means of (\\ref{KM1}) we have\n$$\n\\chi _1(g)=\\prod _{1\\leq i,j\\leq n} p_{ij}^{k_jk_i}= \\prod _{i0.\n$$\nIn the same way $\\chi _2(h)=q^M,$ $M\\geq 0.$\nRelations (\\ref{KM1}) imply \n$$\n\\chi _2(g)\\chi _1(h)=\\prod _{1\\leq i,j\\leq n}p_{ij}^{m_ik_j}\\cdot \\prod _{1\\leq i,j\\leq n}p_{ij}^{m_jk_i}\n=\\prod _{1\\leq i, j\\leq n}(p_{ij}p_{ji})^{m_ik_j}=q^{L},\n$$\nwith $L=\\sum _{i,j}d_ia_{ij}m_ik_j \\leq 0$ since in the Catran matrix $a_{ij}\\leq 0$ for $i\\neq j,$\nwhile $k_im_i=0.$\n\nWe have $\\chi _1(g)=\\chi _2(g),$ and $\\chi_1(h)=\\chi _2(h).$ Therefore \n$q^{M+N}$ $=\\chi _1(g)\\chi _2(h)$ $=\\chi _2(g)\\chi _1(h)$ $=q^{L}.$ A contradiction.\n\\end{proof}\n\n\\smallskip\n\\noindent\n{\\bf Remark}. Of course, if the characters $\\chi_i,$ $1\\leq i\\leq n$ generate a \nfree Abelian group then $g_i,$ $1\\leq i\\leq n$ generate a free Abelian group as well.\nIn particular relations (\\ref{KM1}) imply that $G$ is a free Abelian group with the free generators\n$g_i,$ $1\\leq i\\leq n$ provided that $q$ is not a root of 1 and $C$ is of finite type.\n\\begin{corollary} \nIf $q$ is not a root of 1 and $C$ is of fifnite type, \nthen every subalgebra $U$ of $U_q^+({\\frak g})$ containing $G$\nis homogeneous with respect to each of the variables $x_i$.\n\\label{odn1}\n\\end{corollary}\n\\begin{proof} By the above lemma it suffices to note that $U$ is homogeneous \nwith respect to the grading by characters (\\ref{grad}). If $c=\\sum_ic_i\\in U$\nwith $c_i\\in H^{\\chi _i}$ and different $\\chi_i\\in G^*,$ then \n\\begin{equation}\ng^{-1}cg=\\sum _i \\chi_i(g)c_i\\in U, \\ \\ g\\in G.\n\\label{eqw}\n\\end{equation}\nAccording to the Dedekind Lemma there exist\nelements $h_i\\in G,$ such that the matrix $M=||\\chi_i(h_j)||$ is invertible.\nHence we may solve the system of equations (\\ref{eqw}) considering $c_i$\nas variables. In particular $c_i\\in U.$\n\\end{proof}\n\n\n\\smallskip\nIf the multiplicative order $t$ of $q$ is finite, \nthen we define $u^+_q ({\\mathfrak g})$ as $G\\langle X\\rangle\/{\\bf \\Lambda },$\nwhere ${\\bf \\Lambda }$ is the biggest Hopf ideal in $G\\langle X\\rangle ^{(2)}.$\nThis is a ${\\Gamma }^+$-homogeneous ideal, see \\cite[Lemma 2.2]{KA}. \nCertainly ${\\bf \\Lambda }$ contains all skew-primitive elements of\n$G\\langle X\\rangle ^{(2)}$ (each of them generates a Hopf ideal). Hence\nby \\cite[Theorem 6.1]{Khar} relations (\\ref{KM2}) are still valid in $u^+_q ({\\mathfrak g}).$\n\n\\smallskip\n\\noindent \n{\\bf Quantification of Kac-Moody algebras}.\nConsider a new set of variables $X^-=$ $\\{ x^-_1, x^-_2,\\ldots, x^-_n\\} .$ Suppose that an Abelian group $F$ generated by the elements $f_1, f_2, \\ldots ,f_n$ acts on the linear space spanned by \n$X^-$ so that $(x_i^-)^{f_j}=p_{ji}^{-1}x_i^-,$ where $p_{ij}$ are the same parameters, \nsee (\\ref{KM1}), that define $U_q^+(\\frak{g}).$ Relations (\\ref{KM1}) are invariant\nunder the substitutions $p_{ij}\\leftarrow p_{ji}^{-1},$ $q\\leftarrow q^{-1}.$ This allows us to define the\ncharacter Hopf algebra $U_q^-(\\frak{g})$ as $U_{q^{-1}}^+(\\frak{g})$ with the characters\n$\\chi ^i_-,$ $1\\leq i\\leq n$ such that $\\chi ^i_-(f_j)=p_{ji}^{-1}.$\n\nWe may extend the characters $\\chi ^i $ on $G\\times F$ in the following way\n\\begin{equation}\n\\chi ^i(f_j)\\stackrel{df}{=}p_{ji}=\\chi ^j(g_i).\n\\label{shar1}\n\\end{equation}\nIndeed, if $\\prod_k f_k^{m_k}=1$ in $F,$ then application to $x_i^-$\nimplies $\\prod_k p_{ki}^{-m_k}=1,$ hence $\\chi ^i(\\prod _k f_k^{m_k})=\\prod p_{ki}^{m_k}$\nequals 1 as well. In the same way we may extend the characters $\\chi ^i_-$ on $G\\times F$\nso that \n\\begin{equation}\n\\chi ^i_-=(\\chi ^i)^{-1} \\ \\ \\hbox{as characters of } G\\times F.\n\\label{shar2}\n\\end{equation}\n\n\nIn what follows we denote by $H$ a quotient group $(G\\times F)\/N,$ where \n$N$ is an arbitrary subgroup with $\\chi ^{i}(N)=1,$ $1\\leq i\\leq n.$ For example, if the quantification parameters satisfy additional symmetry conditions $p_{ij}=p_{ji},$ $1\\leq i,j\\leq n,$\nas this is a case for the original Drinfeld-Jimbo and Lusztig quantifications, then \n$\\chi ^i(g_k^{-1}f_k)=p_{ik}^{-1}p_{ki}=1,$ and we may take $N$ to be the subgroup generated by \n$g_k^{-1}f_k,$ $1\\leq k\\leq n. $ In this particular case the groups $H,$ $G,$ $F$ may be identified. \n\nIn the general case without loss of generality we may suppose that $G,F\\subseteq H.$\nCertainly $\\chi ^i, 1\\leq i\\leq n$ are characters of $H$ and $H$ still \nacts on the space spanned by $X\\cup X^-$ by means of these characters and their inverses. \n Consider the skew group algebra $H\\langle X\\cup X^-\\rangle $ as a character Hopf algebra:\n\\begin{equation}\n\\Delta (x_i)=x_i\\otimes 1+g_i\\otimes x_i,\\ \\ \\ \\Delta (x_i^-)=x_i^-\\otimes 1+f_i\\otimes x_i^-,\n\\label{AIcm}\n\\end{equation}\n\\begin{equation}\ng^{-1}x_ig=\\chi ^i(g)\\cdot x_i, \\ \\ g^{-1}x_i^-g=(\\chi ^i)^{-1}(g)\\cdot x_i^-, \\ \\ g\\in H.\n\\label{AIcm2}\n\\end{equation}\nWe define the algebra $U_q(\\frak{g})$ as a quotient\nof $H\\langle X\\cup X^-\\rangle $ by the following relations:\n\\begin{equation}\n[\\ldots [[x_i,\\underbrace{x_j],x_j], \\ldots ,x_j]}_{1-a_{ji} \\hbox{ times}}=0, \\ \\ 1\\leq i\\neq j\\leq n;\n\\label{rela1}\n\\end{equation}\n\\begin{equation}\n[\\ldots [[x_i^-,\\underbrace{x_j^-],x_j^-], \\ldots ,x_j^-]}_{1-a_{ji} \\hbox{ times}}=0, \\ \\ 1\\leq i\\neq j\\leq n;\n\\label{rela2}\n\\end{equation}\n\\begin{equation}\n[x_i, x_j^-]=\\delta_i^j(1-g_if_i), \\ \\ \\ \\ \\ 1\\leq i,j\\leq n\n\\label{rela3}\n\\end{equation}\nwhere the brackets are defined on $H\\langle X\\cup X^-\\rangle $ by the structure of character Hopf algebra, see (\\ref{sqo}). Since due to (\\ref{KM1}) and \\cite[Theorem 6.1]{Khar} all polynomials in the above relations are skew primitive in $H\\langle X\\cup X^-\\rangle ,$ they define a Hopf ideal of \n$H\\langle X\\cup X^-\\rangle $; that is, the natural homomorphism \n\\begin{equation}\nH\\langle X\\cup X^-\\rangle \\rightarrow U_q(\\frak{g})\n\\label{gom2}\n\\end{equation}\ndefines a Hopf algebra structure on $U_q(\\frak{g}).$\n\nIf $q$ has a finite multiplicative order then $u_q(\\frak{g})$ is defined by relations (\\ref{rela3})\nand $u=0,$ $u\\in {\\bf \\Lambda },$ $u^-=0,$ $u^-\\in {\\bf \\Lambda }^-,$ where ${\\bf \\Lambda },$\n${\\bf \\Lambda }^-$ are the biggest Hopf ideals respectively in $G\\langle X\\rangle ^{(2)}$\nand $F\\langle X^-\\rangle ^{(2)}.$\n\nBoth algebras $U_q(\\mathfrak{g}),$ and $u_q(\\mathfrak{g})$ have a grading by the additive \ngroup $\\Gamma $ generated by $\\Gamma ^+,$\nsee p.\\pageref{Gamma}, provided that we put $D(x_i^-)=-D(x_i)=-x_i,$ $D(H)=0$ \nsince in this way relations (\\ref{rela3}) became homogeneous. \n\n\\begin{corollary} \nIf $q$ is not a root of 1 and the Cartan matrix $C=||a_{ij}||$ is of finite type then every subalgebra \n$U$ of $U_q(\\frak{g})$ containing $H$ is $\\Gamma $-homogeneous.\n\\label{odn11}\n\\end{corollary}\n\\begin{proof}\nBy Lemma \\ref{odn} \nand definition (\\ref{shar2}) grading by $\\Gamma $ coincides with the grading\n(\\ref{grad}) by the group of characters (freely) generated by $\\chi ^i,$ $1\\leq i\\leq n.$\nHence every subspace invariant under the conjugations by $H$ is \n$\\Gamma $-homogeneous. \n\\end{proof}\n\n\\smallskip\nThe defined quantification reduces to known ones under a suitable choice of \n$x_i, x_i^-$ depending up the particular definition of $U_q (\\frak{g}).$ For example \nfor classical case of one parameter quantification we have $G=F=H,$ and in the \nnotations of \\cite{Luz2} we may identify\n$$\nx_i= E_i,\\ g_i= K_i,\\ x_i^-= \nF_iK_i(v^{-d_i}-v^{d_i})^{-1}, \\ p_{ij}= v^{-d_ia_{ij}}, \n$$\nwhile in the notations of \\cite{Luz1, Mul} we may take\n$$\nx_i= E_i,\\ g_i=\\tilde{K}_i, \\ x_i^-= F_i\\tilde{K}_i(v_i^{-1}-v_i)^{-1}, \\\np_{i\\mu }= v^{-\\langle \\mu ,i^{\\prime }\\rangle }.\n$$\nFor two-parameter quantizations, say in the notations of \\cite{BGH}, we may put\n$$\nx_i\\leftarrow e_i, \\ g_i\\leftarrow \\omega _i, \\ x_i^-\\leftarrow f_i(\\omega _i^{\\prime })^{-1}(r_i-s_i)^{-1},\\ \nf_i\\leftarrow (\\omega _i^{\\prime })^{-1},\n$$\nand find values of parameters $p_{ij}$ by means of \\cite[(B2), (C2), (D2)]{BGH}.\nFor the multiparameter case of Reshetikhin or DeConcini-Kac-Procesi \nin the notations of \\cite{CV}, we may take \n$$\nx_i\\leftarrow E_iL_{\\beta _i}, \\ g_i\\leftarrow L_{\\beta _i-\\alpha _i+\\gamma _i}, \\ x_i^-\\leftarrow F_iL_{\\alpha _i+\\beta _i}^{-1}(q_i-q_i^{-1})^{-1},\\ f_i\\leftarrow L_{\\gamma _i+\\alpha _i+\\beta _i}^{-1}.\n$$\n\n\\smallskip\n\\noindent\n{\\bf Triangular decomposition}.\nOne may prove that the subalgebra of \n$U_q (\\frak{g})$ generated by $G$ and values of $x_i,$ $1\\leq i\\leq n$ is isomorphic to\n $U_q^+ (\\frak{g})$ \nwhile the subalgebra generated by $F$ and values of $x_i^-,$ \n$1\\leq i\\leq n$ is isomorphic to $U_q^- (\\frak{g}).$ \nMoreover, one has the following so called ``triangular decomposition'' for both algebras:\n\\begin{equation}\nU_q(\\frak{g})= U_q^-(\\frak{g})\\otimes _{{\\bf k}[F]} {\\bf k}[H]\n\\otimes _{{\\bf k}[G]}U_q^+(\\frak{g}),\n\\label{tr}\n\\end{equation}\n\\begin{equation}\nu_q(\\frak{g})= u_q^-(\\frak{g})\\otimes _{{\\bf k}[F]} {\\bf k}[H]\n\\otimes _{{\\bf k}[G]}u_q^+(\\frak{g}).\n\\label{tr1}\n\\end{equation}\nActually this is not so evident (see \\cite{Luz1, Luz2} for standard one parameter version, \n\\cite{CV} for the multiparameter version with Cartan matrix of finite type,\n \\cite{BGH} for two-parameter version with particular Cartan matrices only). \nWe shall provide here a relatively short proof in the general setting that uses\na lemma on tensor decomposition\nfor character Hopf algebras, \\cite[Lemma 6.2]{Kh4}, and (in case $q^t=1$) the \nHeyneman--Radford theorem.\n\\begin{proposition}\nLet $J\\subseteq G\\langle X\\rangle ^{(2)} ,$ $J^-\\subseteq F\\langle X^-\\rangle ^{(2)}$\nbe constitution homogeneous Hopf ideals of \n$G\\langle X\\rangle $ and $F\\langle X^-\\rangle $ respectively.\nDenote by ${\\mathfrak A}$ the algebra generated over $H$ by $X\\cup X^-$ and defined by the relations $(\\ref{rela3})$ and\n$u_s=0,$ $s\\in S,$ $u^-_t=0,$ $t\\in T,$ where $\\{ u_s,$ $s\\in S\\} $ \n$($respectively $\\{ u_t^-,$ $t\\in T\\} )$ \nis a set of homogeneous generators of the ideal $J$ $($respectively $J^-).$ We have\n\\begin{equation}\n{\\mathfrak A}= (F\\langle X^-\\rangle\/J^-)\\otimes _{{\\bf k}[F]} {\\bf k}[H]\n\\otimes _{{\\bf k}[G]}(G\\langle X\\rangle\/J).\n\\label{tria}\n\\end{equation}\n\\label{rtri}\n\\end{proposition}\n\\begin{proof}\nWe note first that the algebra ${\\mathfrak A}_1$ generated over $H$ by $X$ and defined by the relations\n$u_s=0,$ $s\\in S$ has the form ${\\bf k}[H]\\otimes _{{\\bf k}[G]}(G\\langle X\\rangle\/J),$\nwhile the algebra ${\\mathfrak A}_2$ generated over $H$ by $X^-$ and defined by the relations\n$u^-_t=0,$ $t\\in T$ has the form $(F\\langle X^-\\rangle\/J^-)\\otimes _{{\\bf k}[F]}{\\bf k}[H].$\nHence it suffices to show that ${\\mathfrak A}={\\mathfrak A}_2\\otimes _{{\\bf k}[H]}{\\mathfrak A}_1.$ \n\nDenote by $D_i,$ $D_i^- ,$ $1\\leq i\\leq n$ the linear maps \n$$D_i:{\\bf k}\\langle X^-\\rangle \\rightarrow H\\langle X^-\\rangle ,\\ \\ \\ \nD_i^-:{\\bf k}\\langle X\\rangle \\rightarrow H\\langle X\\rangle $$\nthat satisfy the initial conditions \n\\begin{equation}\nD_i(x_j^-)=D_j^-(x_i)=\\delta_i^j(1-g_if_i),\n\\label{inc}\n\\end{equation}\nand the skew differential Leibniz rules\n\\begin{equation}\nD_i(v^-\\cdot w^-)=D_i(v^-)\\cdot w^-+p(x_i, v^-)v\\cdot D_i(w^-),\\ \\ \\ v^-, w^-\\in {\\bf k}\\langle X^-\\rangle ;\n\\end{equation}\n\\begin{equation}\nD_i^-(u\\cdot v)=p(v,x_i^-)D_i^-(u)\\cdot v+u\\cdot D_i^-(v), \\ \\ \\ u, v\\in {\\bf k}\\langle X\\rangle .\n\\label{sqdf2}\n\\end{equation}\nLemma 6.2 \\cite{Kh4} (under the substitutions \n$k\\leftarrow n,$ $n\\leftarrow 2n,$ $x_{n+i}\\leftarrow x_i^-,$ $G\\leftarrow H,$ \n$H\\leftarrow {\\mathfrak A})$ \ngives the required decomposition provided that there exist homogeneous defining\nrelations $\\{ \\varphi _s=0,$ $s\\in S\\} ,$ and $\\{ \\psi _t=0,$ $t\\in T\\} $\n for ${\\mathfrak A}_1$ and ${\\mathfrak A}_2$ respectively, such that \n\\begin{equation}\nD_i(\\psi_t)\\in H\\cdot J^-,\\ \\ D_i^-(\\varphi _s)\\in H\\cdot J, \\ \\ \\ 1\\leq i\\leq n,\\ s\\in S,\\ t\\in T.\n\\label{inw}\n\\end{equation}\nConsider the linear maps\n\\begin{equation}\n\\tilde{D}_i^-: u\\rightarrow \\partial_ i^*(u)\n-p_{ii}^{-1}p(u,x_i)\\partial_i(u)g_if_i, \\ \\ \\ u\\in{\\bf k}\\langle X\\rangle ,\n\\label{sqi1}\n\\end{equation}\nwhere the partial derivatives are defined in (\\ref{calc}) and (\\ref{calcdu}). We have \n$\\tilde{D}_i^-(x_j)=\\delta _i^j(1-g_if_i),$ while relations (\\ref{defdif}) and (\\ref{difdu}) imply\nthe differential Leibniz rule \n$\\tilde{D}_i^-(u\\cdot v)=p(x_i,v)\\tilde{D}_i^-(u)\\cdot v+u\\cdot \\tilde{D}_i^-(v).$ Since \naccording to (\\ref{shar1}) we have $p(x_i,v)=p(v, x_i^-),$ the Leibniz rule and initial values for \n$D_i^-$ coincides with that for $\\tilde{D}_i^-.$ Hence $D_i^-=\\tilde{D}_i^-.$ In perfect analogy we have\n\\begin{equation}\nD_i(u^-)=\\partial_{-i}^*(u^-)p(x_i,u^-)p_{ii}^{-1}-g_if_i \\partial_{-i}(u^-),\n \\ \\ \\ u^-\\in{\\bf k}\\langle X^-\\rangle ,\n\\label{sqi2}\n\\end{equation}\nwhere $\\partial _{-i},$ $\\partial _{-i}^* $ are left and right partial derivatives on\n ${\\bf k}\\langle X^-\\rangle $ with respect to $x_i^-.$\n\nNow if $u_s,$ $s\\in S$ and $u_t^-,$ $t\\in T$ are skew primitive elements\n(as this is the case for ${\\mathfrak A}=U_q(\\frak{g})$) then they are constants \nfor $\\partial _i,$ $\\partial _i^*,$ and $\\partial _{-i},$ $\\partial _{-i}^*,$\nrespectively.\nHence (\\ref{sqi1}), (\\ref{sqi2}) imply $D_i^-(u_s)=D_i(u_t^-)=0$ and \\cite[Lemma 6.2]{Kh4} applies.\n\nIn the general case by the Heyneman-Radford theorem (see \\cite[Corollary 5.4.7]{Mon} or very ``skew \nprimitive\" version \\cite[Corollary 5.3]{Kh2}) the Hopf ideal $J$ has a nonzero\nskew primitive element provided that $J\\neq 0.$ Denote by $J_1$ an ideal generated\nby all skew primitive elements of $J.$ Clearly $J_1$ is a Hopf ideal. \nSince all homogeneous components of a skew primitive \nelement are skew primitive, the Hopf ideal $J_1$ is homogeneous. Moreover, we have\n$D_i^-(u_s)=0,$ $s\\in S_1$ where $u_s$ run through the set of all homogeneous skew primitive \nelements of $J.$ Now consider the Hopf ideal $J\/J_1$ of the quotient Hopf algebra\n$G\\langle X\\rangle \/J_1.$ If $J\\neq J_1$ then this ideal also has nonzero skew primitive elements.\nDenote by $J_2\/J_1$ the ideal generated by all skew primitive elements of $J\/J_1,$\nwhere $J_2$ is its preimage with respect to the natural homomorphism\n$G\\langle X\\rangle \\rightarrow G\\langle X\\rangle \/J_1.$ Again we have \n$D_i^-(\\bar{u}_s)=0,$ $s\\in S_2$ in $G\\langle X\\rangle \/J_1,$ where $\\bar{u}_s$ \nrun through the set of all homogeneous skew primitive elements of $J\/J_1.$\nIn particular the ideal $J_2$ has a set of generators $u_s,$ $s\\in S_1\\cup S_2$\nsuch that $D_i^-(u_s)\\in J_1.$ Continuing the process we shall find a set of generators \n$u_s,$ $s\\in S_1\\cup S_2\\cup S_3\\cup \\ldots $ for $J$ such that $D_i^-(u_s)\\in J,$ all $s.$\n\nIn perfect analogy we find a set of generators $u_t^-,$ \nfor $J^-$ such that $D_i(u_t^-)\\in J^-,$ all $t.$ Hence \\cite[Lemma 6.2]{Kh4} applies. \n\\end{proof}\n\n\\noindent\n{\\bf Remark}. In the proof we do not use relations (\\ref{KM1}) on the quantification parameters, while relations (\\ref{shar1}), (\\ref{shar2}) on the characters are essential. Also we are reminded that originally the maps $D_i,$ $D_i^-$ were defined so that $D_i(u^-)=[x_i,u^-],$ $D_i^-(u)=[u,x_i^-]$ in the algebra \n$H\\langle X\\cup X^-|| [x_i,x_j^-]=\\delta _i^j(1-g_if_i)\\rangle .$ Hence equalities $D_i^-=\\tilde{D}_i^-$\nand (\\ref{sqi2}) imply a differential representation:\n\\begin{equation}\n[x_i,u^-]=\\partial_{-i}^*(u^-)p(x_i,u^-)p_{ii}^{-1}-g_if_i \\partial_{-i}(u^-),\\ \\ \\ u^-\\in{\\bf k}\\langle X^-\\rangle ,\n\\label{sqi3}\n\\end{equation}\n\\begin{equation}\n[u, x_i^-]= \\partial_ i^*(u)-p_{ii}^{-1}p(u,x_i)\\partial_i(u)g_if_i, \\ \\ \\ u\\in{\\bf k}\\langle X\\rangle .\n\\label{sqi4}\n\\end{equation}\n\n\n\\section{PBW-generators for coideal subalgebras in \n$U_q^+(\\frak{sl}_{n+1}),$ $u_q^+(\\frak{sl}_{n+1})$}\nSuppose that the quantification parameters $p_{ij}=p(x_i,x_j)=\\chi ^{i}(g_{j})$\nare connected by the following relations \n\\begin{equation}\np_{ii}=q; \\ \\ p_{i\\, i+1}p_{i+1\\, i}=q^{-1}; \\ \\ \np_{ij}p_{ji}=1,\\ |i-j|>1,\n\\label{a1rel}\n\\end{equation}\nwhere $q\\not= \\pm 1.$ By definition \n $U_q^+(\\frak{ sl}_{n+1})$ as a character Hopf algebra is set up by the relations\n\\begin{equation}\n[[x_i,x_{i+1}],x_{i+1}]=[x_i,[x_i,x_{i+1}]]=[x_i,x_j]=0, \\ \\ |i-j|>1.\n\\label{rela}\n\\end{equation}\nThe structure of this algebra is defined by the following theorem (see, \\cite[Theorem $A_n$]{Kh4},\nor in other terms \\cite[Lemmas pp. 176, 184]{CM}, \\cite{CLMT}). \nRecall that $u(k,m)$ $=x_kx_{k+1}\\ldots x_m,$ $k\\leq m.$\nThe standard word $u(k,m)$ defines a super-letter\n$u[k,m]\\stackrel{df}{=}$ $[x_k[x_{k+1}[\\ldots[x_{m-1}x_m]\\ldots ]]],$\nwhile by definition $u[k,k]=x_k.$ Of course the value of $u[k,m]$ in\n$U_q^+(\\frak{ sl}_{n+1})$ is independent of the alignment of brackets (Lemma \\ref{ind}). \nBy $U_q^+(\\frak{ sl}_{n+1})^{(2)}$\nwe denote the ideal $\\xi (G\\langle X\\rangle ^{(2)})$ \ngenerated by the values of $x_ix_j,$ $1\\leq i,j\\leq n.$\n\\begin{theorem}\n$1.$ The values of the super-letters $u[k,m],$ $1\\leq k\\leq m\\leq n$ in $U_q^+(\\frak{ sl}_{n+1})$\nform the set of PBW-generators of $U_q^+(\\frak{ sl}_{n+1})$ over ${\\bf k}[G].$ All heights are infinite.\n\n$2.$ If $q$ is not a root of unity, then the ideal $U_q^+(\\frak{ sl}_{n+1})^{(2)}$ has no nonzero skew-primitive elements.\n\n$3.$ If $u$ is a standard word then either $u=u(k,m)$ or $[u]=0$ in $U_q^+(\\frak{ sl}_{n+1}).$\n\\label{strA}\n\\end{theorem}\nAccording to the Heyneman-Radford theorem (see \\cite{HR}, or \\cite[Corollary 5.4.7]{Mon}) every non-zero bi-ideal of a character Hopf algebra always has nonzero skew primitive elements. By this reason the second statement \nimplies that Ker$\\, \\xi ,$ $\\ \\xi :G\\langle X\\rangle \\rightarrow U_q^+(\\frak{ sl}_{n+1}),$ is the biggest\nHopf ideal in $G\\langle X\\rangle ^{(2)}.$ In particular one can apply Lemma \\ref{qsim}\nto $U_q^+(\\frak{ sl}_{n+1})$ provided $q$ is not a root of 1.\n\nIf $q$ is a root of 1 ($q\\neq \\pm 1$), then by definition\n$u_q^+(\\frak{ sl}_{n+1})$ is a quotient $U_q^+(\\frak{ sl}_{n+1})\/{ \\Lambda },$ where\n$\\Lambda $ is the biggest Hopf subideal of $U_q^+(\\frak{ sl}_{n+1})^{(2)}.$\nHence we may apply Lemma \\ref{qsim} to $u_q^+(\\frak{ sl}_{n+1})$ as well.\n\n\\smallskip\nIf {\\bf U} is a right coideal subalgebra of $U_q^+(\\frak{ sl}_{n+1})$\nthat contains {\\bf k}$\\, [G],$ then by Proposition \\ref{pro} and Theorem \\ref{strA} it has \nPBW-generators of the form (\\ref{vad1}):\n\\begin{equation}\nc_u=u^s+\\sum \\alpha _iW_i+\\ldots \\in \\hbox{\\bf U}, \\ \\ \\ u=u[k,m].\n\\label{vad25}\n\\end{equation}\nBy means of relations (\\ref{a1rel}) we have\n$p_{uu}=p(x_kx_{k+1}\\cdots x_m,$ $ x_kx_{k+1}\\cdots x_m)=q.$\nThus, if $q$ is not a root of 1, Lemma \\ref{nco1}\nshows that in (\\ref{vad25}) the exponent $s$ equals 1, \nwhile all heights of the $c_u$'s in {\\bf U} are infinite.\n\nIf $q$ has a finite multiplicative order $t>2,$ then $u[k,m]^t=0$ in $u_q^+(\\frak{ sl}_{n+1}),$\nsee for example \\cite[Theorem 3.2]{KA}; that is, by \\cite[Lemma 3.3]{KA}\nthe values of $u[k,m]$ are still the \nPBW-generators of $u_q^+(\\frak{ sl}_{n+1}),$ but all of them have the finite height $t.$\nBy Lemma \\ref{nco1} in (\\ref{vad25}) we have $s\\in \\{ 1, t, tl^r\\}.$ \nSince $u[k,m]^t=u[k,m]^{tl^r}=0,$ the exponent $s$\nin (\\ref{vad25}) equals 1, while all heights of the $c_u$'s in {\\bf U} equal $t.$\n\nHence in both cases the PBW-generators of {\\bf U} have the following form\n\\begin{equation}\nc_u=u[k,m]+\\sum \\alpha _iW_i+\\sum_j \\beta_jV_j \\in \\hbox{\\bf U}.\n\\label{vad22}\n\\end{equation}\nwhere $W_i$ are the basis super-words starting with less than $u[k,m]$ super-letters, \n$D(W_i)=$ $D(u[k,m])$ $=x_k+x_{k+1}+\\ldots +x_m,$ and $V_j$ are $G$-super-words \nof $D$-degree less than $x_k+x_{k+1}+\\ldots +x_m.$\n\nNow, in order to reduce the freedom in construction of the PBW-generators, \nwe are going to show that (\\ref{vad22}) with homogeneous $c_u$ implies that \n{\\bf U} has an element of the same form that belongs to\na special finite set of elements (\\ref{cbr1}).\n\n\\begin{proposition} \nIf a right coideal subalgebra {\\bf U}$\\supseteq {\\bf k}[G]$ of $U_q^+(\\frak{ sl}_{n+1})$\nor $u_q^+(\\frak{ sl}_{n+1})$ contains a homogeneous element $c$\nwith the leading term $u[k,m],$ $k\\leq m,$ then for a suitable subset $\\bf S$\nof the interval $[k,m-1]$ the value of the below defined element\n$\\Psi ^{\\hbox{\\bf s}}(k,m)$ belongs to {\\bf U}.\n\\label{phi}\n\\end{proposition}\n\n\\begin{definition} \\rm\nLet {\\bf S} be a set of integers from the interval $[1,n].$ We define a\n{\\it piecewise continuous word related to} {\\bf S} as follows\n\\begin{equation}\nu^{\\hbox{\\bf s}}(k,m) \n\\stackrel{df}{=}u(1+s_r,m)u(1+s_{r-1},s_r)\\cdots u(1+s_1,s_2)u(k,s_1),\n\\label{pdw}\n\\end{equation}\nwhere {\\bf S}$\\, \\cap [k,m-1]=\\{ s_1,s_2,\\ldots s_r\\}, \\ $\n$k\\leq s_11.$\nThus by Lemma \\ref{ind} applied to (\\ref{cbr1}) the value in $U_q^+(\\frak{ sl}_{n+1})$\nor $u_q^+(\\frak{ sl}_{n+1})$ of the bracketing is independent\nof the alignment of the big brackets. In particular we have the following decomposition\n\\begin{equation}\n \\Psi ^{\\hbox{\\bf s}}(k,m)=\\hbox{\\Large [} \\Psi ^{\\hbox{\\bf s}}(1+s_i,m),\n\\Psi ^{\\hbox{\\bf s}}(k,s_i) \\hbox{\\Large ]}.\n\\label{cbrr}\n\\end{equation}\nOf course the word $u(k,m)$ is a piecewise continuous word with empty {\\bf S}, \nor more generally, with {\\bf S}$\\, \\cap [k,m-1]=\\emptyset .$\n\nLet us choose an arbitrary $s_{r+1},$ $s_r1+s_i; \\hfill \\cr\n0,\\hfill &\\hbox{if }s_{i+1}=1+s_i,\\hfill \\end{matrix} \\right.\n\\label{der7}\n\\end{equation}\nwhere \n$\\mu =(1-q^{-1})^2p(u(1+s_{i+1},m),x_{1+s_i}).$ \n\nFormulae (\\ref{der35} -- \\ref{der7}) show that products of pairwise separated\nelements from $W^{\\hbox{\\bf s}}(k,m)$ span a differential subspace, that contains\nall first derivatives of $\\Psi ^{\\hbox{\\bf s}}(k,m).$ Hence\nby induction it contains the derivatives of higher order as well. \n\nTo see that any product of pairwise separated\nelements from $W^{\\hbox{\\bf s}}(k,m)$ is proportional to some derivative\nof $\\Psi ^{\\hbox{\\bf s}}(k,m)$ we shall prove the following relation. \n\\begin{equation}\n\\Psi ^{\\hbox{\\bf s}}(k,m)\\cdot D_w=\\alpha \\in {\\bf k},\\, \\alpha \\neq 0,\\\n\\hbox{ if } \\ w=u^{\\hbox{\\bf s}}(k,m).\n\\label{der9}\n\\end{equation}\nIf {\\bf S}$\\, \\cap [k,m-1]=\\emptyset $ then $w=x_kx_{k+1}\\ldots x_m$ and relation follows from (\\ref{der1}).\nLet {\\bf S}$\\, \\cap [k,m-1]\\neq \\emptyset .$ By definition (\\ref{pdw}) we have $w=v\\cdot w^{\\prime }, $\n where $v=x_{1+s_r}x_{2+s_r}\\ldots x_m,$ $w^{\\prime }=u^{\\hbox{\\bf s}}(k,s_r).$\nHence \n$$\n\\Psi ^{\\hbox{\\bf s}}(k,m)\\cdot D_w=\\partial_{1+s_r}(\\Psi ^{\\hbox{\\bf s}}(k,m))\n\\cdot D_{v^{\\prime }}D_{w^{\\prime }},\n$$\nwhere $v^{\\prime }=x_{2+s_r}\\ldots x_m.$ By (\\ref{der8})\nthe element $\\partial_{1+s_r}(\\Psi ^{\\hbox{\\bf s}}(k,m))$ is proportional\nto $u[2+s_r]\\cdot \\Psi ^{\\hbox{\\bf s}}(k,s_r).$ Since $\\Psi ^{\\hbox{\\bf s}}(k,s_r)$\nis independent of $x_j,$ $2+s_r\\leq j\\leq m,$ skew differential Leibniz rule (\\ref{defdif}) implies\n$$\n(u[2+s_r]\\cdot \\Psi ^{\\hbox{\\bf s}}(k,s_r))\\cdot D_{v^{\\prime }}=\n(u[2+s_r]\\cdot D_{v^{\\prime }})\\Psi ^{\\hbox{\\bf s}}(k,s_r).\n$$\nBy means of the multiple application of (\\ref{der1}) we see that \n$\\Psi ^{\\hbox{\\bf s}}(k,m)\\cdot D_w$ is proportional to \n$\\Psi ^{\\hbox{\\bf s}}(k,s_r)\\cdot D_{w^{\\prime }}.$ By induction on $m-k$ \nwe get (\\ref{der9}).\n\n\\smallskip\nNow consider a product of separated elements from $W^{\\hbox{\\bf s}}(k,m),$\n\\begin{equation}\n\\Psi ^{\\hbox{\\bf s}}(a_1,b_1)\\cdot \\Psi ^{\\hbox{\\bf s}}(a_2,b_2)\\ldots \\Psi ^{\\hbox{\\bf s}}(a_l,b_l),\n\\ \\ k\\leq b_ik.$ In this case by definition \n$a-1\\notin \\,${\\bf S}, say $s_i1+s_i.$ Hence by (\\ref{der7}),\nor by (\\ref{der4}) provided $i=0,$ or by (\\ref{der8}) provided $i=r$,\nthe element $\\partial _{1+s_i}(\\Psi ^{\\hbox{\\bf s}}(k,m))$ is proportional to \n$\\Psi ^{\\hbox{\\bf s}}(2+s_i,m)\\cdot \\Psi ^{\\hbox{\\bf s}}(k,s_i),$ where formally \n$\\Psi ^{\\hbox{\\bf s}}(k,s_0)=1.$ Since \n$\\Psi ^{\\hbox{\\bf s}}(2+s_i,m)$ is independent of $x_j,$ $k\\leq j\\leq s_i,$ formula\n(\\ref{der9}) shows that $\\Psi ^{\\hbox{\\bf s}}(k,m)\\cdot D_u,$ $u=x_{1+s_i}u^{\\hbox{\\bf s}}(k,s_i)$\nis proportional to $\\Psi ^{\\hbox{\\bf s}}(2+s_i,m).$ If $2+s_i=a,$ the required representation \nof $\\Psi ^{\\hbox{\\bf s}}(a,m)$ is found. If $2+s_i1.$ By definition $1+b_1s_{t+1}.$ Hence all terms in the sum (\\ref{s}),\nexcept one that corresponds \nto $i=s_{t+1},$ are zero. \nMoreover (\\ref{xy}) implies that, due to the choice of $T,$\nthe element $u^{[T]}(1+s_{t+1}, m)\\cdot D_w$ is a nonzero scalar,\nwhile $u^{[L]}(1+s_{t+1}, m)\\cdot D_w=0$ for any other super-word \n$u^{[L]}(1+s_{t+1}, m)$ that appears in the decomposition of $A_{s_{t+1}}$\nwith a nonzero coefficient.\nHence $A_{s_{t+1}}\\cdot D_w$ is a nonzero scalar $\\mu .$ Finally, we have\n\\begin{equation}\n\\Psi ^{\\hbox{\\bf s}_t}(k,s_{t+1})=\\mu ^{-1} B\\cdot D_v \\in \\hbox{\\bf U}.\n\\label{pit2}\n\\end{equation}\n\n\n\\smallskip\n2. Let us derivate (\\ref{pit1}) by $x_{1+s_t}.$ By formulae \n(\\ref{der8}) and (\\ref{defdif}) we have\n$$\n\\mu u[(2+s_t, m)] \\cdot\n\\Psi ^{\\hbox{\\bf s}_{t-1}}(k,s_t)\n$$\n\\begin{equation}\n+\\sum _{i=s_{t+1}}^{m-1}\\mu_iA_iu[(2+s_{t},i)]\\cdot \\Psi ^{\\hbox{\\bf s}_{t-1}}(k,s_t)\\in \\, \\hbox{\\bf U},\n\\label{pit3}\n\\end{equation}\nwhere $\\mu =(1-q^{-1})^2,$ $\\mu_i=(1-q^{-1})^2p(A_i,x_{1+s_t})$ with the only possible \nexception\n$\\mu _{s_{t+1}}=(1-q^{-1})p(A_i,x_{1+s_t})$ provided $s_{t+1}=1+s_t.$\nHere we may apply (\\ref{der8}) \nsince the number $r$ related to {\\bf S}$_t$ \nequals $t.$\n\nDenote by $z$ the piecewise continuous word $u^{\\hbox{\\bf s}_{t-1}}(k,s_t).$\nLet us apply $\\cdot D_z$ to (\\ref{pit3}). Formula (\\ref{der9}) shows that\n$\\Psi ^{\\hbox{\\bf s}_{t-1}}(k,s_t)\\cdot D_z$ is a nonzero scalar. Hence we get\n$$\n\\mu u[2+s_t,m]+\\sum _{i=s_{t+1}}^{m-1}\\mu_iA_iu[2+s_t,i]\\in \\hbox{\\bf U}.\n$$\nLet us apply $\\cdot D_w$ with $w=u(2+s_t,s_{t+1})$ to this sum.\nFormulae (\\ref{defdif}) and (\\ref{der4}) imply\n\\begin{equation}\n(1-q^{-1})u[1+s_{t+1},m]+\\beta _1A_{s_{t+1}}+\n(1-q^{-1})\\sum _{i>s_{t+1}}\\beta _iA_iu[1+s_{t+1},i]\\in \\hbox{\\bf U}.\n\\label{pit4}\n\\end{equation}\nwhere $\\beta _1=p(A_{s_{t+1}},x_{1+s_t}w)$ $=p_{vu},$ \n$\\beta _i$ $=p(A_i, x_{1+s_t}w)$ $=p_{v_{i}\\, u}$\nwith \n$$\nu=u(1+s_t,s_{t+1}), \\ v=u(1+s_{t+1},m),\\ v_i=u(1+i,m).\n$$\nLet us multiply the element (\\ref{pit4}) from the right by \n$\\Psi ^{\\hbox{\\bf s}_t}(k,s_{t+1}) \\in \\, ${\\bf U}, see (\\ref{pit2}), and subtract the result from\n(\\ref{pit1}) multiplied by $\\beta _1.$ \nWe get\n$$\n\\beta _1\\Psi ^{\\hbox{\\bf s}_t}(k,m)+(q^{-1}-1)u[1+s_{t+1},m]\\cdot \n\\Psi ^{\\hbox{\\bf s}_t}(k,s_{t+1})\n$$\n\\begin{equation}\n+\\sum _{i=s_{t+2}}^{m-1}A_i\\{\\beta _1 \\Psi ^{\\hbox{\\bf s}_t}(k,i)\n+(q^{-1}-1)\\beta_iu[1+s_{t+1},i]\\cdot \\Psi ^{\\hbox{\\bf s}_t}(k,s_{t+1}) \\} \\in \\hbox{\\bf U}.\n\\label{pit6}\n\\end{equation}\nBy the recurrence formula (\\ref{cbr3}) the first line of the above formula\nequals $-\\Psi ^{\\hbox{\\bf s}_{t+1}}(k,m),$ while the expression in the braces\nequals $-\\beta_i\\Psi^{\\hbox{\\bf s}_{t+1}}(k,i).$\nThus we get the required relation\n\\begin{equation}\n\\Psi ^{\\hbox{\\bf s}_{t+1}}(k,m)+\\sum _{i=s_{t+2}}^{m-1} \n\\beta_iA_i\\Psi^{\\hbox{\\bf s}_{t+1}}(k,i)\\in \\hbox{\\bf U}.\n\\label{pit7}\n\\end{equation}\nProposition \\ref{phi} is proved.\n\n\\begin{corollary} \nIf $q$ is not a root of $1,$ then $U_q^+(\\frak{ sl}_{n+1})$ has\njust a finite number of right coideal subalgebras that include the coradical.\nIf the multiplicative order of $q$ equals $t>2,$ then \n$u_q^+(\\frak{ sl}_{n+1})$ has\njust a finite number of homogeneous right coideal subalgebras \nthat include the coradical.\n\\label{fin1}\n\\end{corollary}\n\\begin{proof}\nThis follows from Lemma \\ref{odn} and Propositions \\ref{pro}, \\ref{phi}.\n Indeed, one has $n(n-1)\/2$ options for possible value of an {\\bf U}-root\n(Definition \\ref{root}). There exists $2^{n(n-1)\/2}$ variants for sets of {\\bf U}-roots. For any given \nroot $\\gamma =x_k+x_{k+1}+\\ldots +x_m$ there exists not more than $2^{m-k}<2^n$\noptions for {\\bf S} to define a PBW-generator $\\Psi ^{\\hbox{\\bf s}}(k,m).$ Hence the total number of possible sets of PBW-generators is less than $n^{(2^n)}\\cdot 2^{n(n-1)\/2}.$ \\end{proof}\n\n\\section{Root sequence}\n\nOur next goal is to show that the exact number of (homogeneous) right coideal \nsubalgebras in $U_q^+(\\frak{ sl}_{n+1})$ (in $u_q^+(\\frak{ sl}_{n+1})$)\nthat contain {\\bf k}$\\, [G]$\nequals $(n+1)!.$ In what follows for short we shall denote by $[k:m]$ the element \n$x_k+x_{k+1}+\\ldots +x_m\\in \\Gamma ^+$ considered as an $U_q^+(\\frak{ sl}_{n+1})$-root.\n\n\\begin{definition} \\rm\nLet $\\gamma _k$ be a simple {\\bf U}-root of the form $[k:m]$ with the maximal $m.$\nDenote by $\\theta _k$ the number $m-k+1,$ the length (weight) of $\\gamma _k.$\nIf there are no simple {\\bf U}-roots of the form $[k:m],$ we put $\\theta _k=0.$\nThe sequence $r({\\bf U})=(\\theta_1, \\theta_2, \\ldots ,\\theta_n)$\nsatisfies $0\\leq \\theta_k\\leq n-k+1$ and it is uniquely defined by {\\bf U}.\nWe shall call $r({\\bf U})$ a {\\it root sequence of } {\\bf U}, or just an $r$-{\\it sequence of} {\\bf U}.\nBy $\\tilde{\\theta }_k$ we denote $k+\\theta_k -1,$ the maximal value of $m$ for the simple\n {\\bf U}-roots of the form $[k:m]$ with fixed $k.$\n\\label{tet}\n\\end{definition}\n\n\\begin{theorem} \nFor each sequence \n$\\theta=(\\theta_1, \\theta_2, \\ldots ,\\theta_n),$ such that $0\\leq \\theta_k\\leq n-k+1,$\n$1\\leq k\\leq n$ there exists one and only one $($homogeneous$)$ right coideal subalgebra\n{\\bf U}$\\, \\supseteq G$ of $U_q^+(\\frak{ sl}_{n+1})$ $($respectively, of $u_q^+(\\frak{ sl}_{n+1}))$\nwith $r({\\bf U})=\\theta .$ In what follows we shall denote this subalgebra by {\\bf U}$_{\\theta }.$\n\\label{teor}\n\\end{theorem}\nThe proof will result from the following lemmas.\n\n\\begin{lemma} \nIf $[k:m]$ is an {\\bf U}-root, then for each $r,$ $k\\leq r< m$\neither $[k:r]$ or $[r+1:m]$ is an {\\bf U}-root.\n\\label{su}\n\\end{lemma}\n\\begin{proof} By Proposition \\ref{phi} we have $\\Psi ^{\\hbox{\\bf s}}(k,m)\\in \\, ${\\bf U}\nfor a suitable {\\bf S}. If $r\\in \\, ${\\bf S}, then Theorem \\ref{26} and definition (\\ref{pbse}) imply\n$\\Psi ^{\\hbox{\\bf s}}(k,r)\\in \\, ${\\bf U}, hence $[k:r]$ is an {\\bf U}-root.\nIf $r\\notin \\, ${\\bf S}, then again Theorem \\ref{26} and (\\ref{pbse}) with $a=r+1$\nimply $\\Psi ^{\\hbox{\\bf s}}(r+1,m)\\in \\, ${\\bf U}, hence $[r+1:m]$ is an {\\bf U}-root.\n\\end{proof}\n\n\\begin{lemma} \nIf $[k:m]$ is a simple {\\bf U}-root, then \nthere exists only one subset {\\bf S} of the interval $[k,m-1],$\nsuch that $\\Psi ^{\\hbox{\\bf s}}(k,m)\\in \\, ${\\bf U}.\nMoreover the set {\\bf S} is uniquely defined by the set of all {\\bf U}-roots.\n\\label{su1}\n\\end{lemma}\n\\begin{proof} \nLet $\\Psi ^{\\hbox{\\bf s}}(k,m)\\in \\, ${\\bf U}.\nBy the definition of a simple root for each $r,$ $k\\leq r\\tilde{\\theta }_k$, while $[k,\\tilde{\\theta }_k]$ is a simple {\\bf U}-root. \n\nIf $m<\\tilde{\\theta }_k,$\nthen $[m+1:\\tilde{\\theta }_k]$ is an {\\bf U}-root if and only if it is a sum of simple {\\bf U}-roots\nstarting with a number greater than $k.$ Hence by induction the $r$-sequence\ndefines all roots of the form $[m+1:\\tilde{\\theta }_k],$ $k\\leq m<\\tilde{\\theta }_k.$ \n\nBy Lemma \\ref{su}\nthe weight $[k:m]$ is an {\\bf U}-root if and only if $[m+1: \\tilde{\\theta }_k]$ is not an {\\bf U}-root\n(recall that $[k:\\tilde{\\theta }_k]$ is simple). Hence the $r$-sequence also defines the set of\nall {\\bf U}-roots of the form $[k:m],$ $m<\\tilde{\\theta }_k.$ An {\\bf U}-root $[k:m],$\n$m<\\tilde{\\theta }_k$\nis simple if and only if there does not exist $r,$ $k\\leq rk$ implies $m\\in T_{k_1}.$\nAt the same time $k_1-1\\in R_k,$ hence by definition $m\\in T_k.$\n\n\\smallskip\n\\noindent\n{\\sc Claim} 2. {\\it If $s\\in T_k,$ $m\\in T_{s+1},$ then $m\\in T_k.$} \n\n\\smallskip\n\\noindent\nBy means of Claim 1 applied to $s$ we find a sequence \n$k_0=kk.$ Conditions (\\ref{pet1}$a$) and (\\ref{pet1}$b$) are valid for $m\\leftarrow s.$\nSuppose that (\\ref{pet1}$c$) fails. In this case we may find a number $t,$ $k\\leq t1.$ Again by the first\nclaim we have $m\\in T_{k_1}.$\nSince $k_1-1$ belongs to $R_k,$ it satisfies condition (\\ref{pet1}$b$), \n$\\tilde{\\theta }_k\\notin T_{k_1}.$ However Claim 2 shows that the conditions \n $m\\in T_{k_1},$ $\\tilde{\\theta }_k\\in T_{m+1} $\n imply $\\tilde{\\theta }_k\\in T_{k_1}.$\nA contradiction, that proves the claim.\n\n\\smallskip\n\\noindent\n{\\sc Claim 5}. {\\it The subalgebra $U^{\\prime }$ generated by $\\Psi ^{\\hbox{\\bf s}}(k,m),$\n$1\\leq k\\leq m\\leq n,$ $m\\in T_k,$ {\\bf S}$\\, =T_k$ is a differential subalgebra.}\n\n\\smallskip\n\\noindent\nIt suffices to show that all partial derivatives of $\\Psi ^{\\hbox{\\bf s}}(k,m)$ belong to $U.$\nBy Theorem \\ref{26} we have to check that $\\Psi ^{T_k}(a,b)\\in U$ provided that\n$b\\in T_k,$ $a-1\\notin T_k,$ $k\\leq a\\leq b\\leq m.$ By definition\n$\\Psi ^{T_a}(a,b)\\in U$ since due to the third claim $b\\in T_a.$ If \n\\begin{equation}\nT_k\\cap [a, b-1]=T_a\\cap [a,b-1],\n\\label{tor}\n\\end{equation}\nthen we have nothing to prove. In general, however, just the inclusion\n$T_k\\cap [a, b-1]\\subseteq T_a\\cap [a,b-1]$ holds: if $t\\in T_k,$ $a\\leq t,$\nthen Claim 3 with $s\\leftarrow a-1$ says $t\\in T_a$ (since $a-1\\notin T_k$).\n\nWe shall prove $\\Psi ^{T_k}(a,b)\\in U$ by induction on $b-a.$ If $b=a,$\nthen (\\ref{tor}) certainly holds.\n\nLet us choose the minimal $s\\in T_a,$ $s\\notin T_k.$ Then $T_k\\cap [a, s-1]=T_a\\cap [a,s-1].$\nHence $\\Psi ^{T_k}(a,s)=\\Psi ^{T_a}(a,s)\\in U.$\nBy the inductive supposition applied to the interval $[s+1,b]$\nwe get $\\Psi ^{T_k}(s+1,b)\\in U.$\nBy decomposition (\\ref{cbrr}) we have \n$$\n\\Psi ^{{T_k}\\cup \\{ s \\}}(a,b)=[\\Psi ^{T_k}(s+1,b), \\Psi ^{T_k}(a,s)]\\in U.\n$$\nAt the same time (\\ref{cby}) implies\n$$\n\\Psi ^{{T_k}\\cup \\{ s \\}}(a,b)-(1-q^{-1})\\Psi ^{T_k}(s+1,b)\\cdot \\Psi ^{T_k}(a,s)=\n-p_{vu}\\Psi ^{T_k}(a,b).\n$$\nTherefore $\\Psi ^{T_k}(a,b)\\in U,$ which is required.\n\n\\smallskip\n\\noindent\n{\\sc Claim 6}. {\\it {\\bf U}$\\, =U\\#{\\bf k}[G]$ is a right coideal subalgebra.}\n\n\\smallskip\n\\noindent\nSince $U$ is homogeneous in each variable, we have \n$g^{-1}Ug\\subseteq U,$ $g\\in G.$\nIt remains to apply Lemma \\ref{qsim}.\n\n\\smallskip\n\\noindent\n{\\sc Claim 7}. {\\it The set of all \\, {\\bf U}-roots is $\\{ [k:m]\\, |\\, m\\in T_k\\}.$ In particular\n$\\{ \\Psi ^{T_k}(k,m)\\, |$ $m\\in T_k\\} $ is a set of PBW-generators of {\\bf U} over {\\bf k}$[G].$}\n\n\\smallskip\n\\noindent\nIf $\\gamma =[a:b]$ is an {\\bf U}-root, then, by definition, in {\\bf U} there exists\na homogeneous element (\\ref{vad22}) of degree $\\gamma .$ Since by definition \n$U$ is generated by $\\{ \\Psi ^{T_k}(k,m)\\, |\\, m\\in T_k\\} ,$ the degree $\\gamma $\nis a sum of degrees of the generators: \n$\\gamma =[k_1:k_2-1]+[k_2:k_3-1]+\\ldots +[k_{r-1}:k_r-1],$ $k_{i+1}-1\\in T_{k_i},$ $1\\leq i2$) that do not contain the coradical. First of all we note that for every submonoid \n$\\Omega \\subseteq G$ the set of all linear combinations {\\bf k}$\\, [\\Omega]$\nis a right coideal subalgebra.\nConversely if $U_0\\subseteq \\,${\\bf k}$\\, [G]$ is a right coideal subalgebra then\n$U_0=\\, ${\\bf k}$\\, [\\Omega]$ for $\\Omega =U_0\\cap G$\nsince $a=\\sum_i \\alpha_i g_i\\in U_0$ implies $\\Delta (a)=\n\\sum_i \\alpha_i g_i\\otimes g_i\\in U_0\\otimes \\, ${\\bf k}$\\, [G];$ that is, $\\alpha_i g_i\\in U_0.$\n\\begin{definition} \\rm\nFor a sequence $\\theta=(\\theta_1, \\theta_2, \\ldots ,\\theta_n),$ such that $0\\leq \\theta_k\\leq n-k+1,$\n$1\\leq k\\leq n$ we denote by {\\bf U}$^1_{\\theta }$ a subalgebra with 1 generated by \n$g^{-1}\\Psi ^{\\hbox{\\bf s}}(k,m),$ where $g=g_kg_{k+1}\\ldots g_m,$ and $\\Psi ^{\\hbox{\\bf s}}(k,m)$\nruns through the set of PBW-generators of {\\bf U}$_{\\theta },$ see Theorem \\ref{teor}\nand Claim 7, Section 5.\n\\label{1te}\n\\end{definition}\n\\begin{lemma} The subalgebra\n{\\bf U}$^1_{\\theta }$ is a homogeneous right coideal, and \n{\\bf U}$^1_{\\theta }\\cap G=\\{ 1\\} .$ \n\\label{1su}\n\\end{lemma}\n\\begin{proof}\nThe subalgebra\n{\\bf U}$^1_{\\theta }$ is homogeneous since it is generated by homogeneous elements.\nIts zero homogeneous component equals {\\bf k} since among the generators just one,\nthe unity, has zero degree. \n\nDenote by $A_{\\theta }$ a {\\bf k}-subalgebra generated by \nthe PBW-generators $\\Psi ^{\\hbox{\\bf s}}(k,m)$ of {\\bf U}$_{\\theta }.$\nThe algebra {\\bf U}$^1_{\\theta }$ is spanned by all elements of the form \n$g_a^{-1}a,$ $a\\in A_{\\theta }.$ Since {\\bf U}$_{\\theta }$ is a right coideal, \nfor any homogeneous $a\\in A_{\\theta }$ we have\n$\\Delta (a)=\\sum g(a^{(2)})a^{(1)}\\otimes a^{(2)}$ where $a^{(1)}\\in A_{\\theta },$\n$g_a=g(a^{(1)})g(a^{(2)}).$ Therefore \n$\\Delta (g_a^{-1}a)=\\sum g(a^{(1)})^{-1}a^{(1)}\\otimes g_a^{-1}a^{(2)}$ with \n$g(a^{(1)})^{-1}a^{(1)}\\in \\, ${\\bf U}$_{\\theta }^1.$\n\\end{proof}\n\\begin{theorem} If $U$ is a homogeneous right coideal subalgebra \nof $U^{+}_q(\\mathfrak{sl}_{n+1})\\, ($resp. of $u^{+}_q(\\mathfrak{sl}_{n+1}))$\nsuch that $\\Omega \\stackrel{df}{=} U\\cap G$ is a group,\nthen $U=\\, ${\\bf U}$_{\\theta }^{1}\\, ${\\bf k }$[\\Omega ]$\nfor a suitable $ \\theta .$\n\\label{orc}\n\\end{theorem}\n\\begin{proof}\nLet $u=\\sum h_ia_i\\in U$ be a homogeneous element of degree $\\gamma \\in \\Gamma ^{+}$\nwith different $h_i\\in G,$ and $a_i\\in A,$ where by $A$ we denote the {\\bf k}-subalgebra\ngenerated by $x_i,$ $1\\leq i\\leq n.$ Denote by ${\\pi }_{\\gamma }$ the natural\nprojection on the homogeneous component of degree $\\gamma .$\nRespectively $\\pi _g,$ $g\\in G$ is a projection on the subspace {\\bf k}$\\, g.$\nWe have $\\Delta (u)\\cdot (\\pi _{\\gamma }\\otimes \\pi _{h_i})=h_ia_i\\otimes h_i.$\nThus $h_ia_i\\in U.$\n\nBy Theorem \\ref{teor} we have {\\bf k}$\\, [G]U=\\,${\\bf U}$_{\\theta }$ for a suitable $\\theta .$\nIf $u=ha\\in U,$ $h\\in G,$ $a\\in A,$ then $\\Delta (u)\\cdot (\\pi _{hg_a}\\otimes \\pi _{\\gamma })=\nhg_a\\otimes ha.$ Therefore $hg_a\\in U\\cap G=\\Omega ;$ that is, $u=\\omega g_a^{-1}a,$\n$\\omega \\in \\Omega .$ Since $\\Omega $ is a subgroup we get $g_a^{-1}a\\in U.$\nIt remains to note that all elements $g_a^{-1}a,$ such that $ha\\in U$ span the algebra \n{\\bf U}$_{\\theta }^1.$\n\\end{proof}\nIf $U\\cap G$ is not a group then $U$ may have a more complicated structure.\n\\begin{example} \\rm \nLet $\\Omega $ be a submonoid of $G.$ Denote by $\\overline{\\Omega }$ an arbitrary family of sets \n$\\{ \\Omega _{\\gamma }, \\gamma \\in \\Gamma ^{+}\\} $ that satisfies \nthe following conditions\n$$\n\\Omega _0=\\Omega, \\ \\ \\ \\ \n\\Omega _{\\gamma }\\cdot \\Omega _{\\gamma ^{\\prime }}\\subseteq \n\\Omega _{\\gamma +\\gamma ^{\\prime }}\\subseteq\n \\Omega _{\\gamma }\\cap \\Omega _{\\gamma ^{\\prime }}.\n$$\nIn this case the linear space {\\bf U}$_{\\theta }^{\\overline{\\Omega }}$ spanned\nby the elements $\\omega _{\\gamma }a,$ \n$\\omega _{\\gamma }\\in \\Omega _{\\gamma },$ $a\\in \\, ${\\bf U}$_{\\theta }^1,$\ndeg$\\, (a)=\\gamma $ is a right coideal subalgebra such that \n{\\bf U}$_{\\theta }^{\\overline{\\Omega }}\\cap G=\\Omega .$ The $\\gamma $-homogeneous component\nof this algebra equals $\\Omega _{\\gamma }\\, (${\\bf U}$_{\\theta }^1)_{\\gamma }.$\n Hence different $\\overline{\\Omega }$ define different homogeneous right coideal subalgebras.\n\\label{oxc1}\n\\end{example}\nFinally we point out a simplest one-parameter family of inhomogeneous \nright coideal subalgebras that have trivial intersection with the coradical.\n\\begin{example} \\rm \nLet $a=g_1^{-1}(x_1+\\alpha ),$ $\\alpha \\in \\,${\\bf k}.\nWe have \n$$\n\\Delta (a)=g_1^{-1}x_1\\otimes g_1^{-1}+ 1\\otimes g_1^{-1}x_1+\\alpha g_1^{-1}\\otimes g_1^{-1}\n=a\\otimes g_1^{-1}+1\\otimes g_1^{-1}x_1.\n$$\nTherefore the two-dimensional space spanned by $a$ and $1$ is a right coideal. \nThus the algebra {\\bf k}$\\, [a]$ with 1 generated by $a$ is a right coideal subalgebra,\nin which case {\\bf k}$\\, [a]\\cap G=\\{ 1\\} .$\n\\label{oxc2}\n\\end{example}\n\n\\section{K\\'eb\\'e construction and ${\\rm ad}_r$-invariant subalgebras}\n\nIn this section we characterize ad$_r$-invariant right coideal subalgebras that have trivial intersection \nwith the coradical in terms of K\\'eb\\'e's construction \\cite{Keb, Keb1}. Recall that the right \nadjoint action of a Hopf algebra $H$ on itself is defined by the formula\n$$\n({\\rm ad}_ra)b=\\sum \\sigma (a^{(1)})ba^{(2)},\n$$\nwhere $\\sigma $ is the antipode. The map $a\\rightarrow \\, $ad$_ra$ is a homomorphism \nof algebras ad$_r:H\\rightarrow \\hbox{End}H.$ In particular a subspace is invariant under \nthe action of all operators ad$_rH$ if and only if it is invariant under the actions of ad$_rh_i$\nfor some set of generators $\\{ h_i\\}.$ For $H=U_q^+(\\mathfrak{sl}_{n+1})$ or for\n$H=u_q^+(\\mathfrak{sl}_{n+1})$ we have\n$$\n(\\hbox{ad}_rg)b=g^{-1}bg, \\ \\ g\\in G; \\ \\ \\ (\\hbox{ad}_rx_i)b=g_i^{-1}(bx_i-x_ib).\n$$\nThe latter equality would be more familiar if we take $b=g_a^{-1}a$ with\n$a\\in A\\stackrel{df}{=}\\hbox{\\bf k}\\langle x_1,\\ldots ,x_n\\rangle :$\n\\begin{equation}\n(\\hbox{ad}_rx_i)(g_a^{-1}a)=g_i^{-1}(g_a^{-1}ax_i-x_ig_a^{-1}a)=-g_i^{-1}g_a^{-1}[x_i,a].\n\\label{keb1}\n\\end{equation}\nIn particular the subalgebra $H^1$ generated by $g_a^{-1}a,$ $a\\in A$ \n(in our terms this is {\\bf U}$^1_{\\theta }$ for $\\theta =(1,1,\\ldots, 1)$) is ad$_r$-invariant.\n\n\nThe following construction of ad$_r$-invariant right coideal subalgebras appeared in \n\\cite{Keb, Keb1}, see also \\cite[Section 6]{Let}. Let $\\pi $ be a subset of $[1,k].$\nDenote by $K(\\pi )$ a subalgebra generated by elements of the form\n$$\n\\hbox{ad}_r(x_{i_1}x_{i_2}\\ldots x_{i_k})\\, g_j^{-1}x_j, \\ \\ \\ j\\in \\pi,\\ \\ i_r\\notin \\pi, 1\\leq r\\leq k.\n$$\nThe algebra $K(\\pi )$ is ad$_r$-invariant right coideal (see, \\cite[Lemma 1.2]{Let} up to a left-right symmetry). This is homogeneous, and $K(\\pi )\\cap G=\\{ 1\\}$\nsince due to (\\ref{keb1}) the inclusion $K(\\pi )\\subseteq H^1$ is valid. Thus by Theorem \\ref{orc} \nwe have $K(\\pi )={\\bf U}^1_{\\theta }$ for a suitable $\\theta .$\n \n\\begin{theorem}\nThe following conditions on $U=\\,${\\bf U}$^1_{\\theta }$ are equivalent\n\ni. $U$ is {\\rm ad}$_r$-invariant.\n\nii. The sets $T_k,$ see Definition $\\ref{tski},$ have the form $T_k=[j(k),n],$ where \n$$\nj(k)\\stackrel{df}{=}\\, {\\rm min}\\, \\{ j\\, |\\, k\\leq j,\\ \\ j\\in T_j\\} .\n$$\n\niii. $U=K(\\pi )$ for a suitable $\\pi \\subseteq [1,n].$ \n\\label{Korc}\n\\end{theorem}\n\\begin{proof}\n{\\it iii}$\\, \\Rightarrow \\,${\\it i}. We have mentioned above.\n \n{\\it i}$\\, \\Rightarrow \\,${\\it ii}. By Claim 7 we have $m\\in T_k$ if and only if \n$\\Psi ^{T_k}(k,m)\\in \\hbox{\\bf U}_{\\theta }.$ In particular $j\\in T_j$ if and only if \n$x_j\\in \\hbox{\\bf U}_{\\theta },$ or, equivalently, $g_j^{-1}x_j\\in U.$\nIf $j\\in T_j$ and $k\\leq j,$ then by (\\ref{keb1}) we have\n$$\n{\\rm ad}_r(x_{j-1}x_{j-2}\\ldots x_k)\\ g_j^{-1}x_j=g_{u[k,j]}^{-1}u[k,j]\\in U.\n$$\nHence, by definition $[k:j]$ is an {\\bf U}$_{\\theta }$-root; that is, \naccording to Claim 7, we get $j\\in T_k.$ Moreover, if $i>j$ then by\n(\\ref{cin}) we have \n$$\n{\\rm ad}_r(x_{j+1}x_{j+2}\\ldots x_i)\\ g_{u[k,j]}^{-1}u[k,j]=g^{-1}[x_i,[x_{i-1},\\ldots [x_{j+1}, u[k,j]]\\ldots]]\n$$\n$$\n\\sim g^{-1}\\Psi ^{\\{ j+1,j+2,\\ldots ,i\\}}(k,i),\n$$\nwhere $g=g_kg_{k+1}\\ldots g_i.$ In particular $[k:i]$ is an {\\bf U}$_{\\theta }$-root; that is, \nagain according to Claim 7, we get $i\\in T_k.$ This proves $[j(k),n]\\subseteq T_k.$\n\nIf $m$ is the smallest element from $T_k$ then $u[k,m]=\\Psi ^{T_k}(k,m)\\in \\hbox{\\bf U}_{\\theta },$\nhence by multiple use of (\\ref{der1}) we get $x_m\\in \\hbox{\\bf U}_{\\theta };$ that is, $m\\in T_m,$\nand $m=j(k).$\n\n{\\it ii}$\\, \\Rightarrow \\,${\\it iii}. Let $\\pi =\\{ j\\, |\\, j\\in T_j\\} .$ For all $k,m,j$ such that \n$m\\in T_k,$ $j=j(k)$ we have \n$$\n{\\rm ad}_r(x_{j-1}x_{j-2}\\ldots x_kx_{j+1}x_{j+2}\\ldots x_m)\\ g_j^{-1}x_j\n$$\n$$=g^{-1}[x_m,[x_{m-1},\\ldots [x_{j+1}, u[k,j]]\\ldots]]=g^{-1}\\Psi ^{T_k}(k,m),\n$$\nwhere $g=g_mg_{m-1}\\ldots g_k.$ Since $K(\\pi )$ is ad$_r$-invariant, we get\n$g^{-1}\\Psi ^{T_k}(k,m)\\in K(\\pi ).$ Now Definition \\ref{1te} implies $U\\subseteq K(\\pi ).$\n\nSince $g_j^{-1}x_j\\in U,$ to check $K(\\pi )\\subseteq U$ it remains to show that\nad$_r(x_i)U\\subseteq U$ for $i\\notin \\pi .$ By (\\ref{br1}) and (\\ref{keb1}) it suffices to prove that \n$[x_i , \\Psi ^{T_k}(k,m)]\\in {\\bf U}_{\\theta }$ for $i\\notin \\pi ,$ $m\\in T_k.$\n\nLet $i=k-1.$ Since $k-1=i\\notin \\pi ,$ we have $k-1\\notin T_{k-1},$ and hence \n$j(k-1)=j(k).$ Eq. (\\ref{cbry})\nimplies $[x_{k-1}, \\Psi ^{T_k}(k,m)]\\sim \\Psi ^{T_{k-1}}(k-1,m)\\in {\\bf U}_{\\theta },$\nwhere $m\\in T_{k-1}$ follows from $T_{k-1}=[j(k),n]=T_k.$\n\nIf $ij(k),$ and $m+1\\in T_k.$ Now formula (\\ref{cin}) yields \n$$[x_{m+1}, \\Psi ^{T_k}(k,m)]\\sim \\Psi ^{T_k}(k,m+1)\\in {\\bf U}_{\\theta }.$$\n\nIf $i>m+1,$ then $[x_i , \\Psi ^{T_k}(k,m)]=0$ since $x_i$ and $\\Psi ^{T_k}(k,m)$\nare separated.\n\nWe shall show by induction on $m-k$ that in all remaining cases $[x_i , \\Psi ^{T_k}(k,m)]=0.$\nMore precisely, we prove $[x_i, \\Psi ^{\\hbox{\\bf s}}(k,m)]=0$ provided that \n{\\bf S} has the form $[j,n],$ and $k\\leq i\\leq m,$ $i\\neq \\hbox{\\rm min}\\, \\{ j,m \\}.$ \n\nIf $m-k=1,$ then for $j\\geq m$ we have just one option $i=k.$ The required relation \ntakes the form $[x_k,[x_k,x_{k+1}]]=0$ which is one of the defining relations (\\ref{rela}).\nFor $j=k$ we also have just one option $i=m=k+1.$ The required relation is \n$[x_m,[x_m,x_{m-1}]]=0.$ This relation is valid in $U_q^{+}(\\mathfrak{sl}_{n+1})$\nsince (\\ref{a1rel}) imply $[x_m,[x_m,x_{m-1}]]\\sim [[x_{m-1},x_m],x_m],$\nsee, for example, \\cite[Corollary 4.10]{Kh4}.\n\nIf $m-k>2$ then either $ik+1.$ In the former case we have $[x_i,x_m]=0,$\nand by the inductive supposition $[x_i , \\Psi ^{\\hbox{\\bf s}}(k,m-1)]=0.$ Hence representation\n(\\ref{cin}) implies the required equality. In the latter case we have $[x_i,x_k]=0,$\nand by the inductive supposition $[x_i , \\Psi ^{\\hbox{\\bf s}}(k+1,m)]=0.$ In this case representation\n(\\ref{cin1}) implies the required equality.\n\nFinally, suppose that $m-k=2.$ To simplify the notations we put $k=1,$ $m=3.$\n\nIf $j\\geq 3$ then $\\Psi ^{\\hbox{\\bf s}}(1,3)=[[x_1,x_2],x_3],$ \nand we have two options $i=1,$ $i=2.$ If $i=1,$ we have to show\n$[x_1,[[x_1,x_2],x_3]]=0.$ This relation is valid since $x_1$ (skew)commutes both with \n$[x_1,x_2]$ and $x_3$ (but not vise versa: $[x_1,x_2]$ does not (skew)commute with $x_1$\nsince $[[x_1,x_2],x_1]\\neq 0!$) Let $i=2.$ We may apply (\\ref{bri}) since \n$p_{21}p_{22}p_{23}\\cdot p_{12}p_{22}p_{32}=1.$ Thus \nby (\\ref{bri}) and (\\ref{jak3}) we have \n$$\n[x_2,[[x_1,x_2],x_3]]\\sim [[[x_1,x_2],x_3],x_2]=[[x_1,[x_2,x_3]],x_2].\n$$\nThe word $x_1x_2x_3x_2$ is standard, and the standard alignment of brackets is precisely\n$[[x_1,[x_2,x_3]],x_2].$ Hence by the third statement of Theorem \\ref{strA} this is zero in \n$U_q^{+}(\\mathfrak{sl}_{n+1}).$\n\nIf $j=2,$ then $\\Psi ^{\\hbox{\\bf s}}(1,3)=[x_3, [x_1,x_2]],$ and \nwe have two options $i=1,$ $i=3.$ If $i=1$ then\n$[x_1,[x_3,[x_1,x_2]]]=0$ since $x_1$ (skew)commutes both with \n$[x_1,x_2]$ and $x_3.$ Let $i=3.$ By (\\ref{jak4}) we have $[x_3,[x_1,x_2]]\\sim [x_1,[x_3,x_2]]].$\nSince $x_3$ (skew)commutes both with $x_1$ and $[x_3,x_2],$ we get \n$[x_3,[x_1,[x_3,x_2]]]=0.$\n\nIf $j=1$ then $\\Psi ^{\\hbox{\\bf s}}(1,3)=[[x_3,x_2],x_1],$\n and we have two options $i=2,$ $i=3.$ If $i=3$ then\n$[x_3,[[x_3,x_2],x_1]]=0$ since $x_3$ (skew)commutes both with \n$[x_3,x_2]$ and $x_1.$ Let $i=2.$ \nWe may use (\\ref{bri}) since \n$p_{23}p_{22}p_{21}\\cdot p_{32}p_{22}p_{21}=1.$\nThus by (\\ref{bri}) and (\\ref{jak3}) we have \n$$\n[x_2,[[x_3,x_2],x_1]]\\sim [[[x_3,x_2],x_1],x_2]=[[x_3,[x_2,x_1]],x_2].\n$$\nThis element in new variables $y_1=x_3,$ $y_2=x_2,$ $y_3=x_3$ takes up the form\n$[[y_1,[y_2,y_3]],y_2].$ By the third statement of Theorem \\ref{strA}\nthis is zero in $U_q^{+}(\\mathfrak{sl}_{n+1}).$\n\\end{proof}\n\n\\section{Examples}\nIn this section by means of Theorem \\ref{teor} we provide some examples of right coideal subalgebras in $U_q^+(sl_n)$ or $u_q^+(sl_n)$ with their main characteristics:\nPBW-generators, the root sequence $r({\\bf U}),$ \nthe sets $T_i,$ $R_i,$ right coideal subalgebra generators, and maximal Hopf subalgebras.\nWe start with a characterization of \n$2^n$ ``trivial\" examples --- Hopf subalgebras.\n\n\\begin{proposition} \nA right coideal subalgebra {\\bf U}$={\\bf U}_{\\theta }$\nis a Hopf subalgebra if and only if \nfor every $k,$ $1\\leq k\\leq n$ either $\\theta _k=0$ or $\\theta _k=1.$\nAn algebra {\\bf U}$_{\\theta },$ with $\\theta _i\\leq 1$\nis generated over {\\bf k}$[G]$ by all $x_k$ with $\\theta _k=1.$ \n\\label{hop}\n\\end{proposition}\n\\begin{proof} \nIf $\\theta _k\\leq 1,$ $1\\leq k\\leq n,$ then Definition \\ref{pet1} shows that $R_k=\\{ k\\} $\nprovided that $\\theta _k=1$ and $R_k=\\emptyset $ otherwise.\nHence by Claim 8 the algebra {\\bf U} is generated over {\\bf k}$[G]$ \nby all $x_k$ with $\\theta _k=1.$ In particular {\\bf U} is a Hopf subalgebra of\n$U_q^+(sl_n).$ \n\nConversely, let {\\bf U} be a Hopf subalgebra. According to Claim 8 the algebra {\\bf U} is generated \nover {\\bf k}$[G]$ by the elements $a$ of the form $\\Psi ^{T_k}(k,m)$ \nwith $[k:m]$ being the simple {\\bf U}-roots.\nWe have $\\Delta (a)=\\sum a^{(1)}\\otimes a^{(2)}$ with \n$a^{(1)}, a^{(2)}\\in \\, ${\\bf U}. Since $[k:m]$ $=D(a)$ $=D(a^{(1)})+D(a^{(2)})$ and $[k:m]$ is simple,\nwe have either $D(a^{(1)})=0,$ or $D(a^{(2)})=0.$ Thus $a$ is a skew primitive element;\nthat is, $a=x_k$ is the only option for $a$ (see the second statement of Theorem \\ref{strA} for $q^t\\neq1,$ and comments after that theorem for $q^t=1).$\nIn particular all simple roots are of length 1, while Definition \\ref{tet} implies \n$\\theta _k\\leq 1.$ \\end{proof}\n\nNow we consider three special cases.\n\n\\begin{example} \\rm \nConsider the root sequence with the maximal possible components,\n$r({\\bf U})=(n,n-1,n-2,\\ldots ,2,1).$\n In this case by definition $T_n=R_n=\\{ n\\}.$ For $k2,$ \nthen this is the case for $\\Gamma $-homogeneous\nright coideal subalgebras of $u_q(\\frak{ sl}_{n+1}).$\n\\label{raz2}\n\\end{lemma}\n\\begin{proof} \nDue to the triangular decompositions (\\ref{tr}), (\\ref{tr1})\nthe values of super-letters $[x_kx_{k+1}\\ldots x_m],$ $[x_k^-x_{k+1}^-\\ldots x_m^-]$\nform a set of PBW-generators over {\\bf k}$[H]$ for $U_q(\\frak{ sl}_{n+1}).$ \n\nLet us fix the following order on the skew-primitive generators\n\\begin{equation} \nx_1>x_2>\\ldots >x_n>x_1^->x_2^->\\ldots >x_n^-.\n\\label{orr}\n\\end{equation}\nBy Lemma \\ref{nco1} and Proposition \\ref{pro} (see the arguments above Eq. (\\ref{vad22}))\nthe subalgebra {\\bf U} has PBW-generators of the form\n\\begin{equation}\nc=[u]+\\sum \\alpha _iW_i+\\sum_j\\beta_jV_j \\in \\hbox{\\bf U},\n\\label{vad10}\n\\end{equation}\nwhere $W_i$ are the basis super-words starting with less than $[u]$ super-letters, \n$D(W_i)=D(u),$ and $V_j$ are $G$-super-words of $D$-degree less than $D(u),$ while\nthe leading term $[u]$ equals either $[x_kx_{k+1}\\ldots x_m]$ or \n$[x_k^-x_{k+1}^-\\ldots x_m^-].$ \nCertainly the leading terms here are defined by the degree function into the\nadditive monoid $\\Gamma ^+\\oplus \\Gamma ^-$ generated by $X\\cup X^-$\n(but not into the group $\\Gamma $!).\nIn particular all $W_i$ in (\\ref{vad10}) have the same constitution in \n$X\\cup X^-$ as the leading term $[u]$ does. Thus all $W_i$'s and \nthe leading term $[u]$\nbelong to the same component of the triangular decomposition. Hence\nit remains to show that there are no terms $V_j.$\n\nIf $q$ is not a root of 1 then\nby Corollary \\ref{odn11} the algebra {\\bf U} is $\\Gamma $-homogeneous.\nHence (in both cases) the PBW-generators may be chosen to be $\\Gamma $-homogeneous as well. \nIn this case all terms $V_j$ have the same $\\Gamma $-degree and smaller \n$\\Gamma ^+\\oplus \\Gamma ^-$-degree. However this is impossible.\n\nIndeed, if the leading term is $[x_k^-x_{k+1}^-\\ldots x_m^-]$ \nthen the $\\Gamma ^+\\oplus \\Gamma ^-$-degree\nof $V_j$ should be less than $x_k^-+x_{k+1}^-+\\ldots +x_m^-.$ Hence due to\ndefinitions (\\ref{orr}) and (\\ref{ord}) we have $V_j\\in U_q^-(\\frak{ sl}_{n+1}),$\n(respectively, $V_j\\in u_q^-(\\frak{ sl}_{n+1})$),\nand the $\\Gamma $-degree of $V_j$ coincides with the $\\Gamma ^+\\oplus \\Gamma ^-$-degree.\nA contradiction.\n\nSuppose that the leading term is $[x_k^+x_{k+1}^+\\ldots x_m^+].$\n Let $d=\\sum s_ix_i+\\sum r_ix_i^-$\nbe the $\\Gamma ^+\\oplus \\Gamma ^-$-degree of $V_j.$ Since \n\\begin{equation} \nd2,$ then\n$u_q(\\frak{ sl}_{n+1})$ has\njust a finite number of $\\Gamma $-homogeneous \nright coideal subalgebras containing the coradical.\n\\label{fin2}\n\\end{corollary}\n\\begin{proof}\nThis follows from the above lemma and Corollary \\ref{fin1} applied to $U_q^{\\pm }(\\frak{ sl}_{n+1}),$\n$u_q^{\\pm }(\\frak{ sl}_{n+1}).$\n \\end{proof}\n\nOur next goal is to understand when tensor product (\\ref{tru}) is a subalgebra\nand then to find a way to calculate the total number of ($\\Gamma $-homogeneous)\nright coideal subalgebras.\n\\begin{lemma} The tensor product $(\\ref{tru})$\nis a right coideal subalgebra if and only if \n\\begin{equation}\n [{\\bf U}^+,{\\bf U}^-]\\subseteq \\, {\\bf U}^-\\otimes _{{\\bf k}[F]} {\\bf k}[H]\n\\otimes _{{\\bf k}[G]} {\\bf U}^+.\n\\label{rud}\n\\end{equation}\n\\label{raz1}\n\\end{lemma}\n\\begin{proof} Of course if {\\bf U} is a subalgebra then (\\ref{rud}) holds.\nConversely, it is clear that {\\bf U} is a right coideal. Relation (\\ref{rud})\nimplies $u^+\\cdot v^-$ $=[u^+, v^-]$ $+p(u^+,v^-) v^-\\cdot u^+$ $\\in \\, ${\\bf U}, \nwhere $u^+\\in {\\bf U}^+,$ $v^-\\in {\\bf U}^-.$\nHence $(u^-\\cdot u^+)(v^-\\cdot v^+)$ $=u^-(u^+\\cdot v^-)v^+$\n$\\in \\,${\\bf U}, with arbitrary $v^+\\in {\\bf U}^+,$ $u^-\\in {\\bf U}^-.$\n\nSince ${\\bf U}={\\bf U}^-\\cdot H\\cdot {\\bf U}^+,$\nit remains to check that ${\\bf U}^-\\cdot H$ $=H\\cdot {\\bf U}^-,$ and\n${\\bf U}^+\\cdot H$ $=H\\cdot {\\bf U}^+.$ \nSince ${\\bf U}^+$ contains $G,$ it is homogeneous with respect to the grading \n(\\ref{grad}). If $u\\in ({\\bf U}^+)^{\\chi },$ $f\\in F,$ then $uf=\\chi (f)fu.$\nHence ${\\bf U}^+\\cdot F=F\\cdot {\\bf U}^+.$ Similarly \n${\\bf U}^-\\cdot G=G\\cdot {\\bf U}^-.$ \\end{proof}\n\n\n\\section{Consistency condition}\nIn this section we are going to find sufficient condition for consistency relation\n (\\ref{rud}) to be valid. \nIn what follows we denote by $\\Psi_-^{\\hbox{\\bf s}}(i,j)$ a polynomial that appears from\n$\\Psi ^{\\hbox{\\bf s}}(i,j)$ given in (\\ref{cbr1}) under the substitutions $x_t\\leftarrow x_t^-,$\n $1\\leq t\\leq n$ with skew commutators defined by (\\ref{sqo}) in $U_q^-(\\frak{ sl}_{n+1}).$ By\npr$W^{\\hbox{\\bf s}}(k,m)$ (respectively, pr$W_-^{\\hbox{\\bf s}}(i,j)$)\nwe denote a subspace spanned by proper derivatives \nof $\\Psi ^{\\hbox{\\bf s}}(k,m)$ (respectively, of $\\Psi_-^{\\hbox{\\bf s}}(i,j)$), see Theorem \\ref{26}. \nConsider two elements \n$\\Psi ^{\\hbox{\\bf s}}(k,m)$ and $\\Psi _-^{T}(i,j).$ Let us display them graphically\nas defined in (\\ref{gra}):\n\\begin{equation}\n\\begin{matrix}\n\\stackrel{k-1}{\\circ } \\ & \\cdots \\ & \\stackrel{i-1}{\\bullet } \n\\ & \\stackrel{i}{\\bullet }\\ \\ & \\stackrel{i+1}{\\circ }\\ & \\cdots &\n\\ & \\stackrel{m}{\\bullet } \\ & \\ & \\stackrel{j}{\\cdot } \\cr\n\\ \\ & \\ \\ & \\circ \n\\ & \\circ \\ \\ & \\bullet \\ & \\cdots &\n\\ & \\bullet \\ & \\cdots \\ & \\bullet\n\\end{matrix}\n\\label{gra1}\n\\end{equation}\nWe shall prove that \n\\begin{equation}\n [\\Psi ^{\\hbox{\\bf s}}(k,m),\\Psi _-^{T}(i,j)]\\in \n{\\rm pr}\\, W_-^{T}(i,j)\\cdot {\\rm pr}\\, W^{\\hbox{\\bf s}}(k,m)\n\\label{rdi}\n\\end{equation}\nif one of the following two options fulfills:\n\na) Representation (\\ref{gra1}) has no fragments of the form \\label{use}\n\\begin{equation}\n\\begin{matrix}\n\\stackrel{t}{\\circ } \\ & \\cdots & \\stackrel{l}{\\bullet } \\cr\n\\circ \n\\ & \\cdots & \\bullet \n\\end{matrix}\n\\label{gra2}\n\\end{equation}\n\nb) Representation (\\ref{gra1}) has the form \\label{use1}\n\\begin{equation}\n\\begin{matrix}\n\\stackrel{k-1}{\\circ } \\ & \\cdots & \\circ & \\cdots & \\bullet & \\cdots & \\stackrel{m}{\\bullet } \\cr\n\\circ \n\\ & \\cdots & \\bullet & \\cdots & \\circ & \\cdots & \\bullet \n\\end{matrix}\n\\label{gra3}\n\\end{equation}\n(in particular $i=k,$ $j=m$),\nwhere no one intermediate column has points of the same color.\n\nSuppose that diagram (\\ref{gra1}) satisfies condition a). In this case all black-black\ncolumns are located before all white-white columns. Let us choose the closest\nblack-black and white-white pair of columns. Then (\\ref{gra1}) takes up the form\n\\begin{equation}\n\\underbrace{\n\\begin{matrix}\\ & \\ &\\circ & \\bullet \n& \\bullet & \\cdots \n& \\stackrel{t}{\\bullet } \\cr \n\\cdots & \\circ &\n\\bullet & \\bullet \n& \\circ & \\cdots \n& \\bullet \n\\end{matrix}}_{\\hbox{mainly black}}\\,\n\\underbrace{\n\\begin{matrix} \\stackrel{t+1}{\\circ } & \\bullet & \\cdots \\cr\n\\bullet & \\circ & \\cdots \n\\end{matrix}}_{\\hbox{equality}}\\, \n\\underbrace{\n\\begin{matrix} \\stackrel{l}{\\circ } & \\circ & \\bullet \n& \\circ & \\bullet & \\cdots \\cr\n\\circ & \\circ & \\circ \n& \\bullet & \\\n\\end{matrix}}_{\\hbox{mainly white}}\n\\label{gra4}\n\\end{equation}\nHere in the ``mainly black\" zone there are no white-white columns; in the ``equality\" zone\nwe have just black-white, and white-black columns; while in the ``mainly white\"\nzone there are no black-black columns. Of course the ``mainly black\" zone may be empty.\nIn this case we may omit the ``equality\" zone as well, since all the diagram has no \nblack-black columns at all. In the same way the ``mainly white\" zone may be empty too.\n\nRecall that in Definition \\ref{eski} for a fixed pair $(k,m)$ \nwe define ${\\bf S}_{\\circ }=\\, (${\\bf S}$\\, \\cap [k,m-1])\\, \\cup \\, \\{ k-1 \\} ,$\nwhile ${\\bf S}^{\\bullet }=\\, (${\\bf S}$\\, \\cap [k,m-1])\\, \\cup \\, \\{ m \\} ;$\nrespectively $s_0=k-1\\in \\, {\\bf S}_{\\circ },$ and \n$s_{r+1}=m\\in \\, {\\bf S}^{\\bullet }.$ \n\nAll black-black columns are labeled by numbers from ${\\bf S}^{\\bullet }\\cap T^{\\bullet},$\nwhere the bullets correspond to the pairs $(k,m)$ and $(i,j),$ respectively. \nSimilarly all white-white columns are labeled by numbers from\n $(\\overline{\\bf S})_{\\circ }\\cap (\\overline{T})_{\\circ },$\nwhere $\\overline{\\bf S}, \\overline{T}$ are the complements of ${\\bf S},T$ with respect to $[k, m-1],$\n$[i,j-1],$ respectively.\nThus condition a) is equivalent to the inequality \n\\begin{equation}\n\\sup \\{ {\\bf S}^{\\bullet }\\cap T^{\\bullet}\\} <\\inf \\{ (\\overline{\\bf S})_{\\circ }\\cap (\\overline{T})_{\\circ }\\} .\n\\label{ineq}\n\\end{equation}\nWe are reminded that the supremum and infimum of the empty set equal $-\\infty $\nand $\\infty ,$ respectively.\n\nCondition b), in turn, means that $i=k,$ $j=m,$ $T=\\overline{\\bf S}.$\n\n\\smallskip\nWe go ahead with a number of useful notes. If $u$ is a word in $X,$ then by $u^-$ we denote\na word in $X^-$ that appears from $u$ under the substitution $x_i\\leftarrow x_i^-.$\nWe have $p(v,w^-)=\\chi ^v(f_w)=p(w,v),$ while $p(w^-,v)=(\\chi ^w)^{-1}(g_v)=p(w,v)^{-1}.$\nThus $p(v,w^-)p(w^-,v)=1.$ Therefore the Jacobi and antisymmetry identities take up their\noriginal ``colored\" form (see, (\\ref{jak1})):\n\\begin{equation} \n[[u,v],w^-]=[u,[v,w^-]]+p_{wv}[[u,w^-],v];\n\\label{uno}\n\\end{equation}\n\\begin{equation} \n[u,w^-]=-p_{wu}[w^-,u].\n\\label{dos}\n\\end{equation}\nIn the same way\n\\begin{equation} \n[[u^-,v^-],w]=[u^-,[v^-,w]]+p_{vw}^{-1}[[u^-,w],v^-].\n\\label{tres}\n\\end{equation}\nUsing antisymmetry and (\\ref{tres}) we have also\n\\begin{equation} \n[u,[v^-,w^-]]=[[u,v^-],w^-]+p_{vu}[v^-[u,w^-]].\n\\label{cua}\n\\end{equation}\nIn these relations $u,v,w$ are words in $X.$ To simplify further\ncalculations we may extend the brackets to the set of all \n$H$-words: We put $\\chi ^{hu}=\\chi ^{u},$ $g_{hu}=hg_u,$\n$h\\in H,$ and define the skew-brackets by the same formula (\\ref{sqo}).\nIn this case we have\n\\begin{equation} \n[u, hv]=\\chi ^u(h)\\, h[u,v], \\ \\ \\ h\\in H;\n\\label{cuq1}\n\\end{equation}\n\\begin{equation} \n[hu,v]=h[u,v]+p_{uv}(1-\\chi^v(h))\\, h\\, v\\cdot u, \\ \\ \\ h\\in H.\n\\label{cuq2}\n\\end{equation}\n\\begin{equation} \n[hu,v]=\\chi^v(h)\\, h[u,v]+(1-\\chi^v(h))\\, h\\, u\\cdot v, \\ \\ \\ h\\in H.\n\\label{cuq21}\n\\end{equation}\nTo calculate the coefficients it is convienient to have in mind the following \nconsequences of (\\ref{a1rel}):\n\\begin{equation} \n\\chi ^k(g_{k-1}f_{k-1})=\\chi ^{k-1}(g_{k}f_{k})=q^{-1}, \\ \\ \\chi^k(g_if_i)=1,\\ \\hbox{ if } |i-k|>1.\n\\label{cuq22}\n\\end{equation}\nOf course all basic formulae (\\ref{jak1}), (\\ref{jak3}), (\\ref{br1}) and their consequences \nremain valid. However we must stress that\nonce we apply relations (\\ref{rela3}), or other ``inhomogeneous in $H$\" relations\n(for example the third option of (\\ref{derm1}), see below), \nwe have to fix the curvature of the brackets as soon as\nthe inhomogeneous substitution applies to the right factor in the brackets:\n\\begin{equation}\n[u,[x_i,x_i^-]]=u(1-g_if_i)-\\chi ^u(g_if_i)(1-g_if_i)u=(1-\\chi^u(g_if_i))u,\n\\label{cuq3}\n\\end{equation}\nbut not $[u,[x_i,x_i^-]]=[u,1-g_if_i]=[u,1]-[u,g_if_i]=0.$ At the same time\n\\begin{equation}\n[[x_i,x_i^-],u]=(1-g_if_i)u-u(1-g_if_i)=(\\chi^u(g_if_i)-1)\\, g_if_i\\cdot u,\n\\label{cuq4}\n\\end{equation}\nand $[[x_i,x_i^-],u]=[1-g_if_i, u]$ $=[1,u]-[g_if_i,u]$ is valid since the inhomogeneous\nsubstitution has been applied to the left factor in the brackets. In what follows we shall \ndenote for short $h_i=g_if_i,$ and $\\bar{h}_{ki}=h_kh_{k+1}\\cdots h_{i-1},$\nwhere $kk. \\hfill \\end{matrix} \\right.\n\\label{derm2}\n\\end{equation}\nLet us put $u=\\Psi ^{\\hbox{\\bf s}}(1+s_1,m),$ $v=u[k,s_1],$ $w^-=x_k^-.$\nThe definition (\\ref{cbr1}) shows that $\\Psi ^{\\hbox{\\bf s}}(k,m)=[u,v],$\nwhile (\\ref{uno}) implies $[[u,v],w^-]=[u,[v,w^-]].$\n\n If $s_1=k,$ then by (\\ref{cuq3}) we have $[u,[v,w^-]]$\n$=u-\\chi ^u(h_k)u=(1-q^{-1})u.$\n\nIf $s_1>k,$ then (\\ref{derm1}) yields\n$[v,w^-]\\sim h_k\\cdot u[k+1,s_1].$ Therefore $[u,[v,w^-]]\\sim h_k[u, u[k+1,s_1]],$\nsee (\\ref{cuq1}).\nIt remains to note that $[u, u[k+1,s_1]]=\\Psi ^{\\hbox{\\bf s}}(k+1,m)$ due to (\\ref{cbr1}).\n Thus formula (\\ref{derm2}) is proved.\n\\begin{equation}\n[\\Psi ^{\\hbox{\\bf s}}(k,m), x_m^-]\\sim \n\\left\\{ \\begin{matrix}h_m\\cdot \\Psi ^{\\hbox{\\bf s}}(k,m-1),\\hfill & \\hbox{if }k\\leq s_r=m-1;\\hfill \\cr\n\\Psi ^{\\hbox{\\bf s}}(k,m-1), \\hfill & \\hbox{if } k\\leq s_rm.$ In this case $\\mu =m.$ We use induction on $m-k.$ If $m$ $=k,$\nthe formula follows from dual (\\ref{derm1}). If $m>k,$ we put\n$u=\\Psi ^{\\hbox{\\bf s}}(1+s_1,m),$\n$v=u[k,s_1],$\n$w^-=u[k,j]^-.$\nAccording to (\\ref{cbr1}) we have $[u,v]$ $=\\Psi ^{\\hbox{\\bf s}}(k,m),$\nwhile $[u,w^-]=0$ due to (\\ref{rdi1}) with $T=\\emptyset ,$ $T_{\\circ }=\\{ a-1\\} .$\nBy Jacobi identity (\\ref{uno}) and (\\ref{dm3})\nwe get\n\\begin{equation}\n[\\Psi ^{\\hbox{\\bf s}}(k,m),w^-]=[u,[v,w^-]]\n=\\hbox{\\huge [}u, \\sum _{a=k+1}^{1+s_1}\\alpha _a\\, \\bar{h}_{ka}\\, \nu[a,j]^-\\cdot u[a,s_1]\\hbox{\\huge ]}.\n\\label{rrr}\n\\end{equation}\nRelation (\\ref{rdi1}) with $T=\\emptyset ,$ $T_{\\circ }=\\{ k-1\\} $ implies\n$[u,u[a,j]^-]$ $=0$ unless $a=1+s_1.$ Using ad-identities (\\ref{br1}), (\\ref{cuq1})\nwe may continue\n$$\n=\\alpha \\, \\bar{h}_{k\\, 1+s_1}\\, [u,u[1+s_1,j]^-]\n+\\sum _{a =k+1}^{s_1}\\alpha_a \\, \\bar{h}_{ka}\\, u[a,j]^-\\cdot \\Psi ^{\\hbox{\\bf s}}(a,m),\n$$\nwhere $\\alpha \\neq 0.$\nSince $1+s_1>k,$ we may apply the inductive supposition to the first summand.\n\nLet $jm$ with $m\\leftarrow s_l$ we get\n$$\n[\\Psi ^{\\hbox{\\bf s}}(k,m),w^-]=[u,[v,w^-]]=\n\\hbox{\\large [}u,\\sum _{a=k+1}^{1+s_l}\n\\alpha_a \\, \\bar{h}_{ka}\\, u[a,j]^-\\cdot \\Psi ^{\\hbox{\\bf s}}(a,s_l) \\hbox{\\large ]},\n$$\nwhere $\\alpha _a=0$ if $a-1\\in \\, \\hbox{\\bf S}\\, \\cap \\, [k,s_l-1].$\n Relation (\\ref{dm1}) imply\n$[u,u[a,j]^-]=0$ unless $a=1+s_l.$\nHence by ad-identities (\\ref{br1}), (\\ref{cuq1}) we may continue\n$$\n=\\alpha \\, \\bar{h}_{k\\, 1+s_l}\\, [u,u[1+s_l,j]]+\n\\sum _{a=k+1}^{s_l}\n\\alpha_a \\, \\bar{h}_{ka}\\, u[a,j]^-\\cdot \\Psi ^{\\hbox{\\bf s}}(a,m),\n$$\nwhere $\\alpha \\neq 0,$ $\\alpha _a=0$ if $a-1\\in \\, \\hbox{\\bf S}\\, \\cap \\, [k,s_l-1].$\nIt remains to apply (\\ref{dm3}) to the first summand.\n\nIf, finally, $l\\leq r,$ we put\n$u=\\Psi ^{\\hbox{\\bf s}}(1+s_{l+1},m),$\n$v=\\Psi ^{\\hbox{\\bf s}}(k,s_{l+1}),$\n$w^-=u[k,j]^-.$\nThen still $[u,v]$ $=\\Psi ^{\\hbox{\\bf s}}(k,m),$ see decomposition (\\ref{cbrr}),\nand $[u,w^-]=0.$ By the considered above case with $m\\leftarrow s_{l+1}$ we have\n$$\n[v,w^-]=\n\\sum _{a=k+1}^{j+1}\n\\alpha_a \\, \\bar{h}_{ka}\\, u[a,j]^-\\cdot \\Psi ^{\\hbox{\\bf s}}(a,s_{l+1}),\n$$\nwhere $\\alpha _a=0$ if $a-1\\in \\, {\\bf S}\\, \\cap [k,s_{l+1}-1].$\nIn this case $[u,u[a,j]^-]=0$ since $ji=k.$ If $T=\\emptyset ,$ the formula is already proved, see (\\ref{dm31}). \nSuppose that $T\\neq \\emptyset .$\nLet us denote $u=\\Psi ^{\\hbox{\\bf s}}(k,m),$ $v^-=\\Psi ^{T}_-(1+t_1,j),$\n$w^-=u[k,t_1].$ Then according to definition (\\ref{cbr1}) the left hand side of (\\ref{rdi2})\nequals $[u,[v^-,w^-]].$ By (\\ref{rdi1}) we have $[u,v^-]=0.$ Hence Jacobi identity\n(\\ref{cua}) implies $[u,[v^-,w^-]]\\sim [v^-,[u,w^-]].$ Using (\\ref{dm31}) we get\n\\begin{equation}\n[u, w^-]=\\sum _{a=k+1}^{\\mu +1}\\alpha _a \\,\n\\bar{h}_{ka} \\, u[a,j]^-\\cdot \\Psi ^{\\hbox{\\bf s}}(a,m),\n\\label{dmr}\n\\end{equation}\nwhere $\\mu ={\\rm min }\\{ m, t_1\\},$ while $\\alpha _a=0$ if \n$a-1\\in \\, ${\\bf S}$\\, \\cap \\, [k,m-1],$ and formally \n$\\Psi ^{\\hbox{\\bf s}}(m+1,m)=[j+1,j]^-=1.$ \nOf course, $m\\neq t_1$ since $S^{\\bullet }\\cap T^{\\bullet }=\\emptyset .$\n\nIf $m>t_1,$ then we have\n$$\n[v^-,[u, w^-]=\\alpha _{1+t_1} \\,\n[v^-, \\bar{h}_{k\\, 1+t_1} \\, \\Psi ^{\\hbox{\\bf s}}(1+t_1,m)]+\n\\sum _{a=k+1}^{t_1}\\alpha _a \\,\n[ v^-,\\bar{h}_{ka} \\, u[a,t_1]^-\\cdot \\Psi ^{\\hbox{\\bf s}}(a,m)].\n$$\nBy (\\ref{dos}) and (\\ref{rdi1}) we have $[v^-, \\Psi ^{\\hbox{\\bf s}}(a,m)]=0$ if $kk.$ In this case $k-1\\in T\\cap [i,j-1],$ say $k=1+t_l.$\nLet us put $u=\\Psi ^{\\hbox{\\bf s}}(k,m),$ $v^-=\\Psi ^T_-(k,j),$ $w^-=\\Psi ^T_-(i,k-1).$\nDecomposition (\\ref{cbrr}) implies $\\Psi ^T_-(i,j)=[v^-,w^-].$ Since $[u,w^-]=0,$\nwe have $[u,[v^-,w^-]]=[[u,v^-],w^-].$ To find $[u,v^-]$ we may use already considered case: \n$$\n[u,v^-]=\\sum _{a=k+1}^{\\mu +1}\\alpha _a\\bar{h}_{ka}\\Psi ^T_-(a,j)\\cdot \\Psi ^{\\hbox{\\bf s}}(a,m)\n$$\nwith $\\alpha _a=0$ if $a-1\\in {\\bf S}\\cup T,$ $a\\neq \\mu +1.$ Certainly\n$[\\Psi ^{\\hbox{\\bf s}}(a,m), w^-]$ $=[\\Psi ^T_-(a,j), w^-]=0$ since $a>k.$\nBy means of (\\ref{cuq22}) we have\n$$\n\\chi ^{w^-}(\\bar{h}_{ka})=\\chi ^{w}_-(h_kk_{k+1}\\cdots h_{a-1})=\\chi ^{k-1}_-(h_k)=q\\neq 1.\n$$\nNow formula (\\ref{cuq2}) shows that $[[u,v^-],w^-]\\sim w^-\\cdot [u,v^-],$ which is required.\n\n\nIn perfect analogy, if $ki.$\nBy means of (\\ref{cuq22}) we have\n$$\n\\chi ^{v}(\\bar{h}_{ia})=\\chi ^{v}(h_ih_{i+1}\\cdots h_{a-1})=\\chi ^{i-1}(h_i)=q^{-1}\\neq 1.\n$$\nNow formula (\\ref{cuq21}) shows that $[[u,w^-],v]\\sim [u,w^-]\\cdot v,$ which is required.\n\\end{proof}\n\nWe have mentioned above that our main concepts (Definition \\ref{eski}) are not invariant\nwith respect to the replacement of $x_i,$ $x_i^-,$ $1\\leq i\\leq n$\nby $y_i,$ $y_i^-,$ $1\\leq i\\leq n,$ where by definition $y_i=x_{\\varphi (i)},$\n$y_i^-=x_{\\varphi (i)}^-,$ $\\varphi (i)=n-i+1.$\nHence the application of already proved lemmas to generators $y_i,$ $y_i^{-}$\nprovides an additional information. In this way we are going to prove the following two statements.\n\n\\begin{lemma} \nIf $(\\overline{\\bf S})_{\\circ }\\cap (\\overline{T})_{\\circ }\n=(\\overline{\\bf S})^{\\bullet }\\cap (\\overline{T})^{\\bullet }=\\emptyset $ then\n\\begin{equation}\n [\\Psi ^{\\hbox{\\bf s}}(k,m),\\Psi _-^{T}(i,j)]=0.\n\\label{rdi1b}\n\\end{equation}\n\\label{zerb}\n\\end{lemma}\n\n\\begin{lemma} \nIf \n$(\\overline{\\bf S})_{\\circ }\\cap (\\overline{T})_{\\circ }= \\emptyset ,$ while\n$(\\overline{\\bf S})^{\\bullet }\\cap (\\overline{T})^{\\bullet }\\neq \\emptyset $\n then\n$$\n [\\Psi ^{\\hbox{\\bf s}}(k,m),\\Psi _-^{T}(i,j)]\n$$\n\\begin{equation}\n=\\Psi ^T_-(\\mu +1,j)\\left(\n\\sum _{b=\\nu -1}^{\\mu -1}\\alpha _{b}\\, \n\\bar{h}_{b+1 \\, \\mu +1}\\, \\Psi ^T_-(i,b)\\cdot \\Psi ^{\\hbox{\\bf s}}(k,b)\\right) \\Psi ^{\\hbox{\\bf s}}(\\mu +1,m)\n\\label{rdi2b}\n\\end{equation}\nwhere $\\mu =\\min \\{ m,j\\} ,$ $\\nu =\\max \\{ k,i \\} ,$\nwhile $\\alpha _{b}=0$ provided that $b\\notin {\\bf S} \\cap T$\nwith the only exception, $\\alpha _{\\nu -1}\\neq 0.$\n\\label{zerb1}\n\\end{lemma}\n\\begin{proof}\nTo derive these statements from Lemma \\ref{zer} and Lemma \\ref{zer1}\nwe use the {\\it decoding} lemma, Lemma \\ref{dec}.\nLet us apply (\\ref{decod}) to the left hand side of (\\ref{rdi2b}). In this case we have\n$$\n({\\overline{\\varphi(\\hbox{\\bf S})-1}})_{\\circ }=\\varphi \\{ (\\overline{\\bf S})^{\\bullet }+1\\},\\ \n({\\overline{\\varphi(\\hbox{\\bf S})-1}})^{\\bullet }=\\varphi \\{ (\\overline{\\bf S})_{\\circ }+1\\},\n$$\nwhere in the left hand sides the operators (Definition \\ref{eski}) correspond\nto $(\\varphi (m), \\varphi (k)), $ while in the right hand sides to $(k,m).$ In particular \n$(\\overline{\\bf S})^{\\bullet }\\cap (\\overline{T})^{\\bullet }=\\emptyset $ is equivalent to\n$({\\overline{\\varphi (\\hbox{\\bf S})-1}})_{\\circ }\\cap ({\\overline{\\varphi (T)-1}})_{\\circ } =\\emptyset ,$\nwhile\n$(\\overline{\\bf S})_{\\circ }\\cap (\\overline{T})_{\\circ }=\\emptyset $ is equivalent to \n$({\\overline{\\varphi (\\hbox{\\bf S})-1}})^{\\bullet }\\cap ({\\overline{\\varphi (T)-1}})^{\\bullet } =\\emptyset .$\nHence we may \nuse relations (\\ref{rdi1}), (\\ref{rdi2}). Relation (\\ref{rdi1}) proves Lemma \\ref{zerb}.\nIn the case of Lemma \\ref{zerb1} we \nagain apply decoding formula (\\ref{decod}) in order to get (\\ref{rdi2b}). \\end{proof}\n\n\\begin{proposition} \nCondition $a),$ p.$\\pageref{use}$ implies $(\\ref{rdi}).$\n\\label{coa}\n\\end{proposition}\n\\begin{proof} \nWe have seen that condition a) is equivalent to inequality (\\ref{ineq}).\n\nIf ${\\bf S}^{\\bullet }\\cap {T}^{\\bullet }=\\emptyset $ \nthen we may use Lemma \\ref{zer} and Lemma \\ref{zer1}. Let us show that all factors in\n(\\ref{rdi2}) in terms with nonzero $\\alpha _a$ belong to either pr$\\, W^{T}_-(i,j)$\nor pr$\\, W^{\\hbox{\\bf s}}(k,m).$ \n\nIf $a=\\mu +1,$ say $a-1=m2,$ then this is the case for $\\Gamma $-homogeneous\nright coideal subalgebras of $u_q(\\frak{ sl}_{n+1}).$\n\\label{osn5}\n\\end{theorem}\n\\begin{proof} By Lemma \\ref{raz1} we have to show that \n$[U_{\\theta }^+,U_{\\theta ^{\\prime }}^-]\\subseteq {\\bf U}.$\nThe first condition in the theorem means that $\\Psi ^{T_k}(k,\\tilde{\\theta }_k)$ and\n$\\Psi_-^{T_i^{\\prime }}(k,\\tilde{\\theta }_i^{\\prime })$ satisfy condition a) p.\\pageref{use},\nwhile the second one is equivalent to condition b) p.\\pageref{use1} for these elements.\nBy definition $[k:\\tilde{\\theta }_k]$ is a simple $U_{\\theta }^+$-root, such that any other \nsimple $U_{\\theta }^+$-root of the form $[k:m]$ satisfies $m<\\tilde{\\theta }_k.$\nSimilarly each simple $U_{\\theta ^{\\prime }}^-$-root of the form $[i:j]$\nsatisfies $j\\leq \\tilde{\\theta }_i^{\\prime }.$ Condition a) certainly remain valid\nfor subdiagrams, while proper subdiagrams of (\\ref{gra3}) satisfy condition a).\nTherefore, due to Proposition \\ref{coa} and Proposition \\ref{cob}, for each pair of a simple \n$U_{\\theta }^+$-root, $[k:m],$ and a simple $U_{\\theta ^{\\prime }}^-$-root, $[i:j],$ we have \n\\begin{equation}\n [\\Psi ^{T_k}(k,m),\\Psi _-^{T_i^{\\prime }}(i,j)]\\in {\\bf U}.\n\\label{rdii}\n\\end{equation}\nBy Claim 8 the algebras $U^+_{\\theta },$ and $U^-_{\\theta ^{\\prime }}$\n are generated by $\\Psi ^{T_k}(k,m),$\nand $\\Psi _-^{T_i^{\\prime }}(i,j),$ respectively, where $[k:m]$ and $[i:j]$ run through \nthe sets of simple roots. \nTo show that $[U^+_{\\theta },U^-_{\\theta ^{\\prime }}]\\subseteq {\\bf U},$\nit remains to apply ad-identities (\\ref{br1f}), (\\ref{br1}) and evident induction\non degree (we remark that in (\\ref{rdi}) the degree of factors diminishes).\n\nConversely, suppose that $[U^+_{\\theta },U^-_{\\theta ^{\\prime }}]\\subseteq {\\bf U}.$\nLet us choose any pair $(k,i),$ and denote \n$$\nt=\\sup \\left\\{ a\\, |\\, k\\leq a\\leq \\tilde{\\theta }_k,\\, i\\leq a\\leq \\tilde{\\theta }_i^{\\prime },\n\\, a\\in T_k, \\, a\\in T_i^{\\prime } \\right\\},\n$$\n$$ \nl= \\inf \\left\\{ b\\, |\\, k-1\\leq b< \\tilde{\\theta }_k,\\, i-1\\leq b< \\tilde{\\theta }_i^{\\prime },\n\\, b\\notin T_k, \\, b\\notin T_i^{\\prime } \\right\\}.\n$$\n If one of these sets is empty then \ncondition (\\ref{yo1}) is valid. Suppose that $t 0 \\\\\n\\rm & 0 \t\t& \t\\ \\hat{\\sigma_p} < 0\n\\end{array}\\right.\n\\end{equation}\nwith, \n\\begin{equation}\n\\lambda(0) = \\frac{\\mathscr{L}(\\sigma_p = 0,\\hat{\\hat{R_b}})}{\\mathscr{L}(\\hat{\\sigma_p},\\hat{R_b})}\n\\end{equation}\nHence, a large value of $q_0$ implies a large discrepancy between \nthe two hypothesis which is in favor of a discovery ($H_1$). As $f(q_0 \\mid H_0)$ follows a $\\chi^2_1$ distribution, the discovery significance $Z$ \nis simply defined as $Z = \\sqrt{q^{\\rm obs}}$, in units of $\\sigma$ \\cite{Cowan:2010js}.\\\\ \n\n\n\nThe second approach, first introduced by B. Morgan {\\it et al.} \\cite{morgan2}, is based on a generic test of isotropy following the mean recoil deviation $\\langle \\cos\\theta\n\\rangle$ such as:\n\\begin{equation}\n\\langle \\cos{\\theta} \\rangle = \\frac{1}{N}\\sum^N_{i=1}{\\cos{\\theta_i}}\n\\end{equation}\nwhere $\\theta_i$ is the $i^{th}$ angle between the recoil and the Cygnus direction, and $N$ is the number of\nmeasured recoils. Note that this test, as well as the previous one, is by definition coordinate system dependent as the main recoil direction\n$(\\ell,b)$ (see \\cite{billard.disco}) is not considered here as a fitting parameter.\\\\\nEventually, one can evaluate the significance of an observed anisotropy by computing the distributions of $\\langle\\cos\\theta\\rangle$ for\nboth $H_0$ corresponding to the background (isotropic) and $H_1$ the alternative. It is worth noticing that the use of the \n variable $\\langle\\cos\\theta\\rangle$ is particularly interesting in the case of directional detection of Dark Matter as the expected signal should exhibit a dipole feature\n hence maximizing the deviation between $H_0$ and $H_1$.\n\n\n\n\n\n\\section{Influence of a co-rotating Dark Disk}\n\\label{sec:disco}\n \n\nIn order to investigate the effect of a Dark Disk component on the expected significance of a directional dark matter detection, \nwe allow for a wide range on the Dark Disk parameters, see eq.~\\ref{eq:ddparam}, and we evaluate, for each configuration, \nthe expected significance for a 30 kg.year MIMAC-like detector. We highlight the fact that for a co-rotating Dark Disk to contribute to the data, \nthe energy threshold must be low and\/or the WIMP mass large. For concreteness, we present a case study for a 50 GeV\/c$^2$ WIMP \nmass and a total of 100 WIMP events. Figure~\\ref{fig:dispersion} (left) presents the mean significance E(Z) as a function of $V_{DD}$, the rotation velocity of the Dark Disk at \nSolar radius. The black dashed line corresponds to the no Dark Disk case. The result is then presented for various values \nof the relative density $\\rho_{DD}\/\\rho_{H}$. The general feature is that the mean significance is\ndecreasing when increasing the co-rotating velocity of the Dark Disk at Solar radius as it results in a loss of \ndirectionality. This effect is even stronger when increasing the Dark Disk contribution, {\\it i.e.} for large values of the relative velocity $\\rho_{DD}\/\\rho_{H}$. \nInterestingly, a co-rotating Dark Disk can boost the mean significance of a Directional Dark Matter detection. Indeed, for a\nvelocity dispersion $\\sigma_{DD} = 85$ km\/s and a WIMP mass of 50 GeV\/c$^2$, one can see that for rotation velocity $V_{DD} \\leq 140$ \nkm\/s and for any relative density, the mean significance obtained is greater than the one obtained in the no Dark Disk case.\nThis enhancement of the significance at low rotation velocities can be explained by the fact that the Dark Disk is a structure colder than the host halo, {\\it i.e.} has a smaller \nvelocity dispersion, implying an even more anisotropic recoil angular distribution. Hence, a co-rotating Dark Disk will not\nnecessarily degrade the expected performance of directional detection. Of course, for a perfectly \nco-rotating Dark Disk ($V_{DD} = v_\\odot = 220$ km\/s), whatever the velocity dispersion, the recoil angular distribution induced by the \nDark Disk is necessarily isotropic. Only the contribution of the Dark Disk to the total number of events will change.\\\\\nFigure~\\ref{fig:dispersion} (right) presents the mean significance as a function of $V_{DD}$. For any value of the \nvelocity dispersion $\\sigma_{DD}$, the mean significance is continuously \ndecreasing with the rotation velocity of the Dark Disk. However, the case $\\sigma_{DD} = 35$ km\/s (red solid line) \ntends to the no Dark Disk limit due to the\nfact that the contribution to the total number of WIMP events from the Dark Disk falls quickly to zero for $V_{DD} > 140$ km\/s. \nThis also explains the rapid decrease of the significance enhancement in the range $0 - 100$ km\/s. For larger velocity dispersions, \nthe mean significance does not tend to the no Dark Disk limit as the contribution of the Dark Disk to the total number of events \nremains non negligible. Interestingly, one may note that the range of the values of\n $V_{DD}$ inducing an enhancement of the significance depends strongly on $\\sigma_{DD}$. Indeed, \n for large values of the \n velocity dispersion, the Dark Matter signal gets closer to an isotropic distribution. \n This observation implies that lower is the velocity\n dispersion, larger is the range in values of $V_{DD}$ allowing for a boost of the directional signature. \n As a conclusion, larger is the velocity dispersion of the Dark Disk, weaker is the directional discovery significance, \n except for the case of $\\sigma_{DD} = 35$ km\/s as discussed above. However, note that the value of $\\sigma_{DD} = 141$ km\/s\n is extremely large with respect to the recent results from N-Body simulations. Hence, for a 50 GeV\/c$^2$ WIMP mass, one could expect that a co-rotating Dark Disk could \n have a positive, though small ($\\sim$ 10\\%), effect on the directional detection of Dark Matter.\n \n\n\n\n\n\nFor completeness, we studied the evolution of the modifications of the angular distribution $dR\/d\\Omega_r$ for various Dark Disk parameter values. For this purpose, we\ndefined the relative asymmetry $\\mathscr{A}$ as:\n\\begin{equation}\n\\mathscr{A} = \\frac{\\langle\\cos\\theta\\rangle - \\langle\\cos\\theta\\rangle_{H}}{\\langle\\cos\\theta\\rangle_{H}}\n\\label{eq:a}\n\\end{equation}\nwhere $\\langle\\cos\\theta\\rangle_{H}$ corresponds to the mean recoil deviation obtained in the no Dark Disk case ($\\rho_{DD} = 0$). Note that considering the standard halo\nmodel and a WIMP of 50 GeV\/c$^2$, we found $\\langle\\cos\\theta\\rangle_{H} \\simeq 0.51$. Figure~\\ref{fig:dispersion2} presents the relative asymmetry $\\mathscr{A}$ \nin the plane ($V_{DD}, \\sigma_v^{DD}$) for a relative density $\\rho_{DD}\/\\rho_H = 1\/3$ (left) and $\\rho_{DD}\/\\rho_H = 1$ (right). One may notice that there are three\ndifferent regions : no effect (the 1\\% region), a directional discovery enhancement region (low $V_{DD}$) and a region for which the Dark Disk weakens the directional\nsignature (high $V_{DD}$ together with a high $\\sigma_{DD}$ value). The relative density only affects the amplitude of $\\mathscr{A}$, note that the latter spans the range\n[-18,10] for $\\rho_{DD}\/\\rho_H = 1\/3$ and [-40,23] for $\\rho_{DD}\/\\rho_H = 1$. This also affects the area of the no effect region which \n decreases with increasing value of $\\rho_{DD}\/\\rho_{H}$. Interestingly, one can notice that most of the Dark Disk models suggested by N-Body simulations lie in the no effect\n region. Only extreme, yet unrealistic, Dark Disk models may affect significantly the directional signature. It corresponds to the case when both the co-rotational velocity and the velocity dispersion are\n high.\\\\\n\nThis result is in good agreement with previous work \\cite{Green:2010gw} on the effect of a Dark Disk component on directional detection reach which led to the following conclusion. \nThere is only a small variation, with respect to the no Dark Disk case, in the number of WIMP events required to reject isotropy (at 95\\% confidence in 95\\% of experiments) \nor to reject the median direction being random (at 95\\% confidence in 95\\% of experiments). \nNote that the study has been done for a Sulfure detector (a DRIFT-like one) assuming a 20 keV energy threshold and no background.\\\\\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[scale=0.5,angle=0]{Vdd_Sigdd_Rho0333.eps}\n\\includegraphics[scale=0.5,angle=0]{Vdd_Sigdd_Rho1.eps}\n\\caption{ Relative asymmetry $\\mathscr{A}$ in the plane ($V_{DD}, \\sigma_{DD}$) for a relative density $\\rho_{DD}\/\\rho_H = 1\/3$ (top) \nand $\\rho_{DD}\/\\rho_H = 1$ (bottom).\nThe solid and dashed lines correspond to isocontours of a relative asymmetry equal to $\\pm 1$\\% and $\\pm 5$\\% respectively. \nThese studies have been done for 50 GeV\/c$^2$ WIMP mass.} \n\\label{fig:dispersion2}\n\\end{center}\n\\end{figure*} \n\nSo far, we have focused on an isotropic Maxwellian distribution for the Dark Disk particles. \nHowever, as shown in \\cite{Ling:2009cn}, the dark disk itself may exhibit\nanisotropic features in its velocity distribution. One way to investigate its\nvelocity dispersion tensor using current experimental data, is to look at the stellar thick disk. However, comparison with the observed values of the velocity dispersions of the stellar thick disk may be misleading as the Dark Disk \nanisotropy depends strongly on the merger properties such as infall inclinations. Nevertheless, evidence in favor of a departure from \nisotropy comes from full cosmological hydrodynamics simulations \\cite{Ling:2009cn}.\\\\ \nTo study the effect of an anisotropic Dark Disk, \nwe evaluate the value of the relative asymmetry $\\mathscr{A}$ (eq.~\\ref{eq:a}) as a function of the radial dispersion $\\sigma_r$ and \nthe tangential one, defined as $\\sigma_t^2=\\sigma_y^2+\\sigma_z^2$. Figure \\ref{fig:dispersion3} presents the \nrelative asymmetry $\\mathscr{A}$ in the plane ($\\sigma_{r}, \\sigma_{t}$) for a relative density $\\rho_{DD}\/\\rho_H = 1\/3$ (top) \nand $\\rho_{DD}\/\\rho_H = 1$ (bottom). These studies have been done for a \nWIMP mass of 50 GeV\/c$^2$ and a co-rotational velocity $V_{DD} = 150 \\ km\/s$. \nFor convenience the isovalues of the anisotropy parameter $\\beta = 1 - \\sigma_t^2\/2\\sigma_r^2$ are indicated. \nNote that a positive and a negative value of $\\beta$ refer to a radially and tangentially \nanisotropic velocity distribution respectively.\nFirst, it can be noticed that, for a fixed value \nof $\\sigma_t$, the relative asymmetry decreases with increasing $\\sigma_r$, \n{\\it i.e.} perpendicularly to the detector motion direction, as the WIMP flux is becoming more isotropic \nin the detector frame, without enhancing the Dark Disk contribution to the data (see discussion above). \nFor a fixed value of $\\sigma_r$, due to the Earth rotation along the $(Oy)$ axis, a larger dispersion along this axis \nwill mostly boost the Dark Disk contribution to the number of WIMPs events while keeping a strong anisotropy, if $\\sigma_t$ is \nnot too large. Hence, there is an optimal point above which, increasing $\\sigma_t$ and hence both $\\sigma_y$ and $\\sigma_z$ starts to make the flux sufficiently anisotropic to weaken the directional signal.\nEventually it should be highlighted that for any departure from isotropy, the effect on the relative asymmetry remains small, -15\\% at the very\nmost and in the extreme cas of a relative density of $\\rho_{DD}\/\\rho_H = 1$. On its own, the effect of the Dark Disk anisotropy is small compared to the influence coming from the standard parameters such as the one previously studied ($\\sigma_{DD}$ and $V_{DD}$).\\\\\n\n \n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[scale=0.5,angle=0]{SigR_Sig_t_Rho0333_corrected.eps}\n\\includegraphics[scale=0.5,angle=0]{SigR_Sig_t_Rho1_corrected.eps}\n\\caption{Relative asymmetry $\\mathscr{A}$ in the plane ($\\sigma_{r}, \\sigma_{t}$) for a relative density $\\rho_{DD}\/\\rho_H = 1\/3$ (top) \nand $\\rho_{DD}\/\\rho_H = 1$ (bottom). The solid and dashed lines correspond to isocontours of a relative asymmetry equal to $\\pm 1$\\% and $\\pm 5$\\% respectively. \nThese studies have been done for a WIMP mass of 50 GeV\/c$^2$ and a co-rotational velocity $V_{DD} = 150 \\ km\/s$.} \n\\label{fig:dispersion3}\n\\end{center}\n\\end{figure*} \n\n\n\nWe evaluate the effect of the Dark Disk contribution to the Dark Matter \nreach of upcoming directional detectors. Following \\cite{billard.profile}, we compute the directional reach in the $(m_{\\chi}, \\sigma_p)$ plane, {\\it i.e} the \nlower bound of the 3$\\sigma$ discovery region at 90\\% CL for the two approaches: profile likelihood (red lines) and mean recoil deviation (blue lines).\n Figure \\ref{fig:ProspectsFinal} presents the \ndiscovery limit in the ($m_\\chi,\\log_{10}(\\sigma_p)$) plane corresponding to two Dark Matter models: standard halo model only (solid lines) and \n and with an extreme Dark Disk model contribution \\{$\\rho_{DD}\/\\rho_h = 1$, $V_{DD} = 220$ km\/s, $\\sigma_{DD} = 106$ km\/s\\} (dashed lines). \\\\\n The conclusion of this study is twofold. First, we found that for both statistical tests, the effect of an extreme Dark Disk is only mild. Indeed, the directional reach is\n only degraded by a factor of 3 at high WIMP masses and not affected for light WIMP. Second, we found that the two statistical tests give similar results with a maximal\n deviation of a few percent. However, it is worth emphasizing that their intepretation differ as the profile likelihood method favors\n the background plus signal hypothesis ($H_1$)\n whereas the mean recoil deviation method rejects the isotropy hypothesis.\n \n\n \n \n \\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.5,angle=0]{ProspectsDarkDisk.eps}\n\\caption{Discovery limit in the ($m_\\chi,\\log_{10}(\\sigma_p)$) plane corresponding to two Dark Matter models: standard halo model only (solid lines) and \n and with an extreme Dark Disk model contribution \\{$\\rho_{DD}\/\\rho_h = 1$, $V_{DD} = 220$ km\/s, $\\sigma_{DD} = 106$ km\/s\\} (dashed lines). \n We compute the directional reach, {\\it i.e} the \nlower bound of the 3$\\sigma$ discovery region at 90\\% CL, for the two approaches: profile likelihood (red lines) and mean recoil deviation (blue lines).} \n\\label{fig:ProspectsFinal}\n\\end{center}\n\\end{figure}\n\n \n \n\n\n\n\\section{Conclusion}\nA co-rotating Dark Disk, as predicted by recent N-Body simulations, might contribute \n(10\\%-50\\%) to the local Dark Matter density, with a potentially dramatic effect on directional detection. \nIn this letter, we have evaluated the effect of Dark Disk model on the discovery potential \nof upcoming directional detectors. We conclude that, if a co-rotating Dark Disk is present in our Galaxy and has the properties predicted by N-Body simulations \\cite{nezri}, \n the discovery potential of directional detection would be strictly unchanged. Only an extreme and unrealistic Dark Disk model (high co-rotational velocity and high velocity dispersion) \n might affect significantly the Dark Matter reach of upcoming directional detectors, by increasing the discovery limit by a \n factor of three at high WIMP mass ($m_{\\chi} \\sim 1000$ GeV\/c$^2$). Additionally, we also have shown that anisotropic features in the Dark Matter velocity distribution of the Dark Disk will only have a small effect on the expected directional signal. Hence, according to our results we believe that the possibility of the existence of a co-rotational Dark Disk in our galaxy shouldn't be a threat for upcoming directional detection experiments.\\\\\n Interestingly, note that even if the impact of Dark Disk contribution to the local Dark Matter distribution only mildly affects the discovery \n potential of directional detection, it may\n significantly affect the mass and cross section determination \\cite{billard.ident}. Indeed, as explained in \\cite{Green:2010gw}, \n WIMP events arising from the Dark Disk contribution will induce an\n excess at low recoil energies which can lower the estimation of the WIMP mass when considering a standard halo model. \n As outlined in \\cite{Lee:2012pf}, the presence of a Dark Disk restricts the ability to constrain \n the Dark Matter parameters (both from the halo and particle physics). Of course, a measurement of the parameters of the Dark Disk itself remains challenging with the exposure \n of the next generation of directional detectors (30 kg.year). This highlights the fact that even if a co-rotating Dark Disk is not a threat to the discovery potential of directional detection, \n it has to be characterized in order to consistently constrain the Dark Matter properties. \n \n\n \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzgxuu b/data_all_eng_slimpj/shuffled/split2/finalzzgxuu new file mode 100644 index 0000000000000000000000000000000000000000..900c1cda39267080e6ea547bffcdca57d0dc2d2b --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzgxuu @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction} \\label{sec:introduction}\n\n Deep field surveys carried out with the\n {\\em Hubble Space Telescope} have enabled the detection of galaxies out to the cosmic\ndawn at $z$\\xspace$\\gtrsim\\,$11 and provide stringent constraints on the bright end of rest-frame UV luminosity functions (LF)\nof galaxies during the Epoch of Reionization \\citep[e.g.,][]{Oesch16a, Song16a,Finkelstein16a}.\nWhile measuring distributions of galaxy properties, such as the\nLF, provides important constraints on how galaxies evolve over cosmic time,\nit is also useful to target individual sources at high redshift\\xspace in order to conduct detailed case studies.\nSuch studies allow us to address open questions such as properties of the interstellar medium (ISM), how metal enrichment in galaxies proceded in the early universe,\nhow much star formation is dust-obscured or how much dust is present in these early systems,\nand the physical conditions under which star formation and stellar mass assembly took place.\nFollow-up observations with the Atacama Large (Sub-)Millimeter Array (ALMA) probing the dust continuum and line emission from ions and molecules in the dense ISM of high-redshift galaxies appear extremely promising for characterizing the star-forming gas and dust properties of early galaxies and helping to constrain the physical processes at play (see a review by \\citealt{Hodge20a}).\n\nOwing to the relative brightness of the fine-structure line from singly ionized\ncarbon at rest-frame 157.7\\,$\\mu$m and accessibility of the line with\nALMA at high redshift, there is currently great interest in using [C{\\scriptsize II}]\\xspace\nline emission as a tracer to study galaxies during the Epoch of Reionization (EoR). At the\nmoment, most of these follow-up observations\\xspace target some of the brightest\nobjects samples selected at other wavelengths\n\\citep[e.g.,][]{Capak15a, Smit18a, Marrone18a}. [C{\\scriptsize II}]\\xspace has been\ndetected in a handful of normal star-forming galaxies at the EoR\n\\citep[e.g.,][]{Capak15a, Carniani18b} and even resolved on kpc-scales\nto study the gas kinematics for a handful of sources\n\\citep[e.g.,][]{Jones17a, Matthee17a, Carniani18b,\n Hashimoto19a}. However, due to the small field of view of ALMA, it\nis extremely challenging and expensive to carry out\n\\emph{flux-limited} (untargeted) surveys over significant areas. The largest area untargeted survey to date that probes [C{\\scriptsize II}]\\xspace at $z\\sim 6-8$ is the ALMA Spectroscopic Survey in the HUDF (Large Program; ASPECS), which covers 4.6 arcmin$^2$ \\citep{Walter16a,Aravena16a}.\n\nWhile these detections are exciting, interpretation of the [C{\\scriptsize II}]\\xspace line\nluminosity and its connection with galaxy properties remains complex\nand poorly understood. [C{\\scriptsize II}]\\xspace is the dominant coolant in the ISM in\nnearby star forming galaxies, and its luminosity is expected to be\ncorrelated with the star formation rate (SFR). Indeed, such a\ncorrelation is seen in local galaxies\n\\citep{DeLooze14a,Herrera-Camus15a}, but with hints that the \\mbox{$L_{\\rm [CII]}$}\\xspace\/SFR\nratio depends on other galaxy properties such as metallicity and dust\ntemperature \\citep[e.g.][]{Malhotra01a, Luhman03a, Diaz-Santos14a}.\nThis is expected, as the dust\ncontent affects the degree of shielding against hard ionizing photons,\nas well as the amount of photoelectric heating, which affect\nthe ionization state of the line emitting gas and the collision rate. In addition,\ntheoretical modeling has shown that \\mbox{$L_{\\rm [CII]}$}\\xspace\/SFR is also expected to\ndepend on the pressure of the ISM environment, if molecular cloud\nsizes depend on the ambient pressure, and on the density distribution\non small scales within the ISM \\citep{Popping19a}.\n\nAnother motivation for studying the physics of line emission and its\nconnection to galaxy properties and the physics of galaxy formation is\nthe development of multiple experiments that will carry out line\nintensity mapping (LIM) studies. Different LIM experiments will\nenable the detection of various tracers of dense and more diffuse gas\nin and around galaxies, including Lyman-$\\alpha$, H$\\alpha$, 21-cm,\nCO, and [CII] (see \\citealt{Kovetz17a} for a review). LIM promises to\nmap the statistical signal from emission in galaxies over very large\nvolumes (on the order of Mpc-to-Gpc-scales), with the tradeoff of not\nresolving individual galaxies. As such, LIM experiments also probe\nemission from faint galaxies. While LIM holds great promise for\nstudying galaxy evolution and cosmology (e.g., the cosmic star\nformation history, evolution of the ISM and intergalactic medium\n(IGM), and physical processes during the EoR; \\citealt{Kovetz17a}),\nthe power spectra depend on the line luminosities of different galaxy\npopulations within the volume sampled (or the intrinsic line LF).\nTherefore, physically motivated models that can self-consistently\npredict line luminosities are vital for strategizing LIM experiments\nand interpreting LIM data. As shown by \\citet{Yue19a}, a shallower\n\\mbox{$L_{\\rm [CII]}$}\\xspace\\,--\\,SFR relation would imply that the [C{\\scriptsize II}]\\xspace LF drops quickly at\nthe bright end and most of the IM signal comes from faint galaxies.\nThis in turn determines the detection depth needed for [C{\\scriptsize II}]\\xspace LIM\nexperiments. At the moment, most LIM forecasts for tracers such as CO\nand [C{\\scriptsize II}]\\xspace have been made using a series of empirical scaling relations\n(e.g., \\citealt{Gong12a, Uzgil14a, Keating15a, Chung19a}).\n\nCarrying out detailed and realistic predictions of the [C{\\scriptsize II}]\\xspace emission\nfor a large cosmologically representative sample of galaxies is\nextremely challenging. The [C{\\scriptsize II}]\\xspace line can arise from all phases of the\nISM, by being collisionally excited by either electrons, atoms or\nmolecules. Hence, the line strength depends strongly on the density\nand kinematic temperature of these species. Modeling has repeatedly\nshown that different ISM phases are all important to consider when\nderiving [C{\\scriptsize II}]\\xspace emission for a galaxy (e.g., \\citealt{Olsen17a,\n Pallottini19a, Lupi20a}). However, state-of-the-art cosmological\nsimulations of volumes larger than 100\\,Mpc do not resolve particle\nmasses below $\\sim$\\,10$^{6-7}$\\,\\mbox{$M_{\\odot}$}\\xspace (at $z$\\eq0; e.g.,\n\\citealt{Dave19a, Nelson18a}), which corresponds to hydrogen densities\nbelow $n_H$\\,<\\,100\\,${\\rm cm}^{-3}$\\xspace and temperatures above 10$^4$\\,K. In\nparticular the molecular ISM phase --- with typical densities above a\nfew hundred cm$^3$ and and temperatures below 100\\,K --- is not\ntracked in cosmological simulations but knowledge about it is critical\nin order to reliably simulate [C{\\scriptsize II}]\\xspace line emission. Currently, all large\nvolume cosmological simulations adopt phenomenological ``sub-grid''\nrecipes to treat unresolved processes such as star formation, stellar\nfeedback, and chemical enrichment, and these carry significant\nuncertainties \\citep[see review by][]{Somerville15a}. Additionally, a\nsecond type of ``sub-grid'' modeling is required to describe the\ndetailed structures of molecular gas on cloud scales, which must be\ninput into the radiative transfer (RT) and ionization state modeling tools\ndescribed in more detail below.\n\nPrevious works that have attempted to model [C{\\scriptsize II}]\\xspace for galaxy\npopulations mainly fall into three categories: {\\em (i)} extremely simple,\nempirical mappings between \\mbox{$L_{\\rm [CII]}$}\\xspace\\ and halo mass\n\\citep{Visbal10a,Gong12a}, {\\em (ii)} semi-analytic models to predict galaxy\nproperties coupled with machinery to predict the [C{\\scriptsize II}]\\xspace emission in\npost-processing \\citep{Popping16a,Lagache18a,Popping19a}, or {\\em (iii)} small\nsamples of cosmological zoom-in simulations again post-processed to\ncompute the radiative transfer and line emission \\citep{Olsen15a,\n Narayanan15a, Vallini15a, Olsen17a, Katz17a, Pallottini17a,\n Pallottini17b}. A few recent studies have coupled on-the-fly\nradiative transfer, non-equilibrium chemistry modeling, and line spectral\nsynthesis with ultra high resolution hydrodynamic zoom-in simulations\n\\citep{Katz17a,Katz19a,Pallottini19a}.\n\nA variety of tools are brought to bear to compute the emergent line\nemission in the literature. A stellar population synthesis code such\nas \\ncode{Starburst99} is used to derive the amount of intrinsic\nradiation from stellar emission \\citep{Leitherer14a}. Codes such as\n\\ncode{Skirt} or \\ncode{Powderday} are used to perform dust RT\n\\citep{Camps15a, Narayanan15a}, and tools such as \\ncode{RADMC-3D},\n\\ncode{Despotic}, \\ncode{Lime}, \\ncode{Cloudy}, or \\ncode{MAPPINGS}\nare used to compute line RT in the ISM \\citep{Allen08a, Krumholz14a,\n Ferland17a}. For a detailed summary and comparison of these codes,\nwe refer interested readers to \\citet{Olsen18a}.\n\nSome of the biggest differences among the aforementioned codes are the\ndensity range considered\/permitted, the geometry, and the species\nincluded in the chemical network. These differences also imply\ndifferent demands on computational time and memory, each with\ndifferent benefits and trade-offs. The accuracy needed for a given\ngalaxy simulation and the emission line of interest typically determine the\nmethod used. For instance, a photo-dissociation region (PDR) code such\nas \\ncode{Despotic} can be used to calculate line emission from the\nneutral ISM; however, \\ncode{Despotic} does not simulate line emission\noriginating in the ionized phase of the ISM.\n\nIn addition to different methods being adopted for RT and line\nspectral synthesis, the type of simulation used also sets limits on\nthe galaxy sample size obtainable and the level of realistic physics\nthat can be adopted. SAMs are computationally inexpensive and can\neasily generate galaxy catalogues of statistically significant sample\nsizes (e.g., \\citealt{Popping16a, Lagache18a} and\n\\citealt{Popping19a}), and thus are excellent for testing physical\nrecipes and exploring wide ranges of parameter space, but they do not\nprovide any detailed information on sub-galactic structure. In\ncontrast, the highest resolution zoom-in hydrodynamical simulations can\nnumerically resolve the ISM down to scales of $\\sim$ 10\\,pc at high\nredshifts (e.g., \\citealt{Katz17a, Pallottini17b, Pallottini19a}), but\nare computationally expensive. The samples of\ngalaxies with simulated line emission based on numerical\nhydrodynamical simulations are thus limited in number to 1-30 per\nstudy, and therefore also probe a limited parameter space of galaxy\nproperties. Previous studies commonly targeted the most massive halos\nand\/or a handful of sources with properties resembling some known\nproperties of a given observed $z$\\ssim6 galaxy. Over the years, the\nresolution of large volume cosmological hydrodynamics simulations has\nincreased significantly, reaching $\\gtrsim$100\\,pc resolution even\nbefore carrying out additional refinement using zoom-in\ntechniques. This produces a statistically significant and unbiased\nsample of galaxies while reaching down to a relevant spatial scale to\nsimulate emission emerging from the ISM (though sub-grid models are\nstill required). \\cite{Inoue20} gave a successful demonstration of calculating CO line emission in direct post-processing of the cosmological IllustrisTNG simulation using a simulation box size of 75\\,Mpc.\n\nIn this work, we leverage the new \\ncode{simba}\\xspace suite of cosmological\nhydrodynamic simulations (\\citealt{Dave19a}) to select a galaxy sample\nat $z\\sim 6$ for line emission postprocessing that is unprecedented in\nits size (11,137 galaxies) and dynamic range (halo mass $\\sim\n10^9-10^{12} \\mbox{$M_{\\odot}$}\\xspace$; stellar mass $\\sim 10^7-10^{11} \\mbox{$M_{\\odot}$}\\xspace$). The\nsample is drawn from a set of volumes that vary in resolution, such\nthat the sample from each volume is representative of the population\nthat is well resolved in the simulation. We apply and updated version\nof the \\ncode{s\\'{i}game}\\xspace package \\citep{Olsen17a}, which includes\nsub-resolution modeling of the ISM, radiative transfer and line\nspectral synthesis. We use this calculation to provide predictions of\nhow the [C{\\scriptsize II}]\\xspace luminosity at this redshift is related to other\ngalaxy and DM halo properties, and of the [C{\\scriptsize II}]\\xspace LF, and compare our results with\navailable observations and with other models in the literature.\n\nThe paper is structured as follows. In \\Sec{sim}, we describe the\n\\ncode{simba}\\xspace simulations, and in \\Sec{sigame}, the method used to\nsimulate [C{\\scriptsize II}]\\xspace line emission. We present the results in \\Sec{results}\nand discuss the limitations in \\Sec{caveats}. Finally, conclusions\nand implications of our findings are presented in \\Sec{conclusion}.\nThroughout this paper, we adopt a concordance cosmology, with total\nmatter, vacuum and baryonic densities in units of the critical density\n$\\Omega_{\\Lambda}$\\eq0.693, $\\Omega_m$\\eq0.307, $\\Omega_b$\\eq0.048, a\ndimensionless Hubble parameter $h$\\eq0.678, scalar spectral index of\n$n$\\eq0.96 and power spectrum normalization of $\\sigma_8$\\eq0.823\n\\citep{Planck16a}.\n\n\\section{Simulation}\\label{sec:sim}\n\n\\subsection{Cosmological Hydrodynamics Simulation: \\ncode{simba}} \\label{sec:simba}\nWe use galaxies from the \\ncode{simba}\\xspace cosmological galaxy formation simulations \\citep{Dave19a} for this study.\n\\ncode{simba} is a suite of \\ncode{gizmo}-based simulations using meshless finite mass\nhydrodynamics,\nincorporating state-of-the-art feedback modules that provide very good agreement\nwith a wide range of lower-redshift observables.\nThe suite consists of random cubical volumes of 100, 50, and 25\\,\\ensuremath{\\mathrm{cMpc}\\,\\mathrm{h}^{-1}}\\ on a side, each with\n1024$^3$ dark matter particles and 1024$^3$ gas elements. By combining the results\nof these simulations, we achieve unprecedented dynamic range, with the highest resolution\nequalling e.g. that in a study of far-infrared lines using zoom simulations presented\nin \\citet{Olsen17a}.\nThe smoothing lengths and the initial gas mass resolutions for the 100, 50, and 25\\,\\ensuremath{\\mathrm{cMpc}\\,\\mathrm{h}^{-1}}\\ volumes\nare $\\epsilon_{\\rm min}$\\eq0.5, 0.25, and 0.125\\,$h$\\mbox{$^{-1}$}\\xspace\\,kpc\nand $m_{\\rm gas}$\\eq1.8\\E{7}, 2.3\\E{6}, and 2.9\\E{5}\\,\\mbox{$M_{\\odot}$}\\xspace, respectively\n(see Table 1 of \\citealt{Dave19a}).\nThese runs all use identical input physics, begin at\n$z$\\eq249 and assume a Planck-concordant cosmology.\n\n\\ncode{simba}\\xspace is the successor of the \\ncode{mufasa} simulations (\\citealt{Dave16a}), and details\nof the improvements in \\ncode{simba}\\xspace are provided in \\citet{Dave19a}.\nAmong the various updates, key ones relevant to this work are:\n{\\em (i)} \\ncode{simba}\\xspace uses the \\ncode{grackle-3.1} package\nto model radiative cooling and photoionization heating, updated from \\ncode{mufasa}\nto apply radiative processes via isochoric substep cycling, and also computing\nthe neutral hydrogen content accounting for self-shielding\non-the-fly via the prescription in \\citet{Rahmati13a};\n{\\em (ii)} \\ncode{simba}\\xspace explicitly models the growth and feedback of\nsupermassive black holes (SMBH) residing in galaxies, with\nthe growth of the SMBH set by\ntorque-limited accretion of cold gas~\\citep{Angles17a} and\nBondi accretion of hot gas, while black hole feedback\nis modeled via bipolar kinetic outflows\nand injection of X-ray energy;\n{\\em (iii)} \\ncode{simba}\\xspace includes a sub-grid model to form and destroy dust within the ISM of galaxies during the simulation run,\nwith the dust mainly produced by Type II supernovae, asymptotic giant branch stars\nand condensation from metals, and destroyed\npredominantly via sputtering (including supernova shocks) and consumption by star formation\\xspace\n\\citep{Li19a};\n{\\em (iv)} \\ncode{simba}\\xspace employs ejective star formation feedback like \\ncode{mufasa}, but with scalings updated to reflect particle tracking results from the\nFeedback in Realistic Environment (FIRE) simulations~\\citep{Angles17b},\nwith minor modifications to better reproduce EoR galaxy properties\n(See \\citealt{Wu19a}, under review for details).\n\n\n\\ncode{simba}\\xspace has been compared to a wide range of observations across cosmic time, and is found to\nprovide reasonable agreement, including for the galaxy stellar mass function (GSMF)\nand mass metallicity relation \\citep{Dave19a}, black hole properties \\citep{Thomas19a},\ndust properties \\citep{Li19a}, galaxy sizes and profiles \\citep{Appleby2020},\nand cold gas contents including \\mbox{CO\\,($J$\\,=\\,1\\,\\rarr\\,0) } luminosity functions\nfrom $z=0-2$ \\citep{Dave20a}. Minor disagreements with observations\\xspace include an overproduction\nof the very highest mass galaxies at $z\\la 2$, too-large size for low-mass quenched\ngalaxies at $z$\\xspace$\\la 1$, and an underproduction of the dust mass function at $z$\\xspace$\\sim 2$.\nRelevant to this work where we focus on $z=6$,\nWu et al., (2019, under review) examined the EoR properties of \\ncode{simba}\\xspace galaxies\nand found good agreement at $z=6$ with the UV luminosity function, UV slope measurements when a\n\\citet{Calzetti01a} dust model is assumed, and galaxy sizes down to the faintest limits.\nHence \\ncode{simba}\\xspace provides a realistic platform to examine the far-infrared line properties\nof EoR galaxies, which is the goal of this study.\n\nIn \\Fig{GSMF}, we show the GSMF and SFR function at $z$\\,\\ssim6.\nWe point out the robust numerical convergence in the GSMF and SFR function\n(and galaxy properties, see \\citealt{Dave19a} and \\Sec{sample}) without\nrefining or fine-tuning the parameters in the sub-grid models of \\ncode{simba}\nbetween the various simulation boxes at the redshift studied in this work.\nNote that we do not impose any stellar mass limits in the GSMF plotted.\nThis illustrates the exceptional convergence reached across the boxes.\nThis is crucial as we make predictions of the [C{\\scriptsize II}]\\xspace LF in the luminosity range of 5.5\\,$<\\log$\\,(\\mbox{$L_{\\rm [CII]}$}\\xspace\/\\mbox{$L_{\\odot}$}\\xspace)\\,$<$\\,8.5 by combining galaxies in the $25 \\ensuremath{\\mathrm{cMpc}\\,\\mathrm{h}^{-1}}$, $50 \\ensuremath{\\mathrm{cMpc}\\,\\mathrm{h}^{-1}}$, and $100 \\ensuremath{\\mathrm{cMpc}\\,\\mathrm{h}^{-1}}$\nboxes (hereafter Simba-25, Simba-50, and Simba-100, respectively).\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[trim=0 0 0 28, clip, width=.5\\textwidth]{fig1a} \\\\\n\\includegraphics[trim=0 0 0 0, clip, width=.5\\textwidth]{fig1b}\n\\caption{\nTop: Galaxy stellar mass function for the different simulation boxes at $z$\\eq6 compared to\nthe results based on a rest-frame UV selected observational sample (black markers).\nVertical dashed lines show the mass requirement applied to each of the simulation boxes (color-coded)\nto select only galaxies that are resolved by the simulation (see \\Sec{simba}).\nBottom: Same as the top panel, but for the SFR function.\nVertical dashed lines show the thresholds in SFR for each box, below which the boxes become incomplete.\nThe turnover arises from the finite mass resolution of these simulations\n(before applying the selection criteria; see \\Sec{sample}).\nObservational results (corrected for dust attenuation) are plotted as red symbols.\nThe spread marked by the shaded regions is computed from jackknife resampling eight sub-octants of the simulated volumes.\nObservations are from \\citet{Song16a} (top) and \\citet{Bouwens15a} (bottom). The \\ncode{simba}\\xspace predictions are in very good agreement with the observational estimates.\n\\label{fig:GSMF}}\n\\end{figure}\n\n\n\\subsection{Main Sample: 11,137 Galaxies at $z$\\xspace$\\simeq$\\,6} \\label{sec:sample}\nGalaxies from the simulation suite are identified using a galaxy finder that adopts a\n6-dimensional friends-of-friends algorithm (\\ncode{caesar}).\nFor the purpose of this work, we include only galaxies that have at least 64 stellar and gas particles,\nrespectively, to ensure they are resolved in the simulation.\nAs illustrated in \\Fig{GSMF}, these mass requirements correspond to\n$\\log {(M_{\\rm \\star,min}\/M_\\odot)}$\\eq7.24 for Simba-25,\n$\\log {(M_{\\rm \\star,min}\/M_\\odot)}$\\eq8.15 for Simba-50, and\n$\\log {(M_{\\rm \\star,min}\/M_\\odot)}$\\eq9.05 for Simba-100.\nSimilarly, we impose thresholds on the SFR averaged over 10\\,Myr based on the\nturnover seen in the SFR function indicating incompleteness in SFR.\nThis corresponds to $\\log\\left(\\textrm{SFR}\/\\textrm{M}_\\odot\\,\\textrm{yr}^{-1}\\right)>-$1.9, $-$0.8, and $0.4$\nfor Simba-25, -50, and -100 respectively.\nIn addition, we only include galaxies with a molecular gas mass of at least\n$M_{\\rm mol}$\\,=\\,$f_{\\rm H2}\\,M_{\\rm gas}$\\,$>$10$^5$\\,\\mbox{$M_{\\odot}$}\\xspace, as the\nsub-grid model has to form giant molecular clouds (GMCs) of at least 10$^4$\\,\\mbox{$M_{\\odot}$}\\xspace each by\nsampling the cloud mass from a GMC mass function (see \\Sec{sigame}).\nAfter applying these criteria, we have a sample of $N_{\\rm tot}$\\eq11,137 galaxies for a single snapshot at $z$\\,\\eq6.\nThe ranges of their physical properties are listed in \\Tab{param}.\n\nTo illustrate the range of physical properties sampled by the \\ncode{simba}\\xspace galaxies,\n\\Fig{prop} shows the relations between the specific SFR (sSFR),\nSFR-weighted metallicity ($\\langle Z_{\\rm gas}\\rangle_{\\rm SFR}$), molecular gas-to-stellar mass ratio ($M_{\\rm mol}$\/\\mbox{$M_*$}\\xspace),\nand the stellar mass-weighted age in our $z$\\eq6 sample.\nThe three clumps of points represent galaxies from our three simulation volumes, and generally\nshow reasonable convergence.\nThe weighted quantities are indicated with the $\\langle...\\rangle$ notation (e.g., $\\langle Z_{\\rm gas}\\rangle_{\\rm SFR}$), as\ndefined as follows:\n\\begin{equation}\\label{eqn:defineaverage}\n\\langle x \\rangle \\equiv \\frac{\\sum_{i} \\rho_i x_i }{\\sum_i \\rho_i}\\,,\n\\end{equation}\nwhere $x$ is the variable and $\\rho_i$ is the volume of each fluid element $i$.\nThe $\\Sigma_{\\rm SFR}$ in this paper is defined as:\n\\begin{equation}\n\\Sigma_{\\rm SFR} = \\frac{\\textrm{SFR}}{\\pi R_{\\rm 1\/2}^2}\n\\end{equation}\nwhere $R_{\\rm 1\/2}$ is the half-mass radius of gas of a given galaxy.\nThe scaling relations shown are similar to the mass-metallicity\nand fundamental metallicity relation (a.k.a. MZR and FMR; e.g., \\citealt{Maiolino08a, Mannucci10a}),\nwhich are commonly used to gain insights into the interplay between star formation\\xspace, gas accretion, and\nfeedback during the evolution of a galaxy,\nand are shown to illustrate the range of physical properties sampled by the \\ncode{simba}\\xspace galaxies.\nAs can be seen in the second panel, most of the \\ncode{simba}\\xspace galaxies are rich in molecular gas.\nThe trend of decreasing $M_{\\rm mol}\/$\\mbox{$M_*$}\\xspace with increasing \\mbox{$M_*$}\\xspace\narises owing to stellar feedback which preferentially suppresses the stellar mass in lower\nmass systems, thereby increasing\n$M_{\\rm mol}$\/\\mbox{$M_*$}\\xspace.\nThe middle panel also shows how, for a given stellar mass bin,\nthe SFR increases with the molecular gas mass fraction, as expected.\nThat said, there are certainly jumps in $M_{\\rm mol}\/$\\mbox{$M_*$}\\xspace fractions between the different simulations volumes indicating less than ideal convergence, although this becomes less apparent when plotting $M_{\\rm mol}$ against \\mbox{$M_*$}\\xspace as shown in the left panel of Fig.\\,\\ref{fig:mH2} in the Appendix.\nThe bottom panel displays the \\ncode{simba}\\xspace galaxies on top of the ``star-forming main sequence'' (SFMS; e.g., \\citealt{Speagle14a, Iyer18a}),\nwhich most of our galaxies follow at $\\log M_*\\gtrsim9$\\,\\mbox{$M_{\\odot}$}\\xspace --- which is also\nthe stellar mass limit of the observational data. At lower stellar mass, the sSFR\nof \\ncode{simba}\\xspace galaxies falls below the SFMS extrapolation\\footnote{Such extrapolation\nassumes that the SFMS follows the same power law as that at the high mass end, which is not directly observed.}.\nThe color coding in the three panels shows that, at a given stellar mass,\ngalaxies that are higher metallicity have lower gas contents, galaxies\nthat have higher gas contents have higher sSFR, and galaxies that have higher\nsSFR have lower mean stellar age.\n\nGalaxies of similar stellar masses and SFRs may have different sizes,\nsurface densities, gas contents, metallicities, interstellar radiation field strengths,\nstructural properties, and gas dynamics;\nall of which would produce varying [C{\\scriptsize II}]\\xspace luminosities (see e.g., \\citealt{Kaufman99a, Vallini15a, Olsen17a}).\nAs such, comparing the physical properties of observed (i.e., [C{\\scriptsize II}]\\xspace-detected ones in the context of this work) and\nsimulated galaxies is pertinent to establishing the reliability of model predictions.\nIn other words, comparing these global properties of \\ncode{simba} galaxies with those of observed galaxies\nenables one to place the observed ones, given their [C{\\scriptsize II}]\\xspace luminosities,\nin a theoretical framework.\nIt is beyond the scope of this paper to perform a detailed comparative study since the information\navailable for observed galaxies at these redshifts (see \\Sec{ciifir}) remains limited and inhomogeneous.\nFor comparisons of \\ncode{simba}\\xspace galaxies with observations\\xspace at lower redshifts and discussion on\nredshift evolution in these relations, we refer interested readers to \\citet{Dave19a}.\nAt the EoR, the size-luminosity relation of \\ncode{simba}\\xspace galaxies agrees with observations\\xspace\n(\\citealt{Kawamata18a, Wu19a}).\n\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[trim=0 0 0 0, clip, width=.5\\textwidth]{fig2}\n\\caption{\nScaling relations for the \\ncode{simba}\\xspace galaxy sample (circular dots).\nTop: $\\langle Z_{\\rm gas}\\rangle_{\\rm SFR}$ -- \\mbox{$M_*$}\\xspace relation, color-coded by the molecular gas mass fraction.\nMiddle: Molecular gas-to-stellar mass ratio ($M_{\\rm H2}$\/\\mbox{$M_*$}\\xspace) -- \\mbox{$M_*$}\\xspace relation, color-coded by sSFR.\nBottom: SFR--\\mbox{$M_*$}\\xspace relation, color-coded by the mass-weighted stellar age.\n The SFR of the simulated galaxies are averaged over 100\\,Myr.\nMagenta squares, red crosses, and blue plus symbols\ncorrespond to observations\\xspace of UV-selected star-forming galaxies at $z$\\xspace$\\sim$\\,6 \\citep{Capak15a, Jiang16a},\nwith the red markers indicating older galaxies with a crude estimated age of\n$\\gtrsim$\\,100\\,Myr and\nthe blue ones indicating younger galaxies with age $\\lesssim$\\,30\\,Myr.\nDashed lines correspond to empirical relations for\nthe star-forming main sequence (SFMS) and their 1$\\sigma$\\ and 3$\\sigma$\\\nspreads at this redshift \\citep{Speagle14a, Iyer18a}.\nThe sharp cutoffs seen in the last two plots results from the mass cut imposed on each of the simulation boxes\n(Simba-25, Simba-50, and Simba-100) to only include galaxies that are numerically resolved (see \\Sec{sample}).\n\\label{fig:prop}\n}\n\\end{figure}\n\n\n\n\\input{Table_paramSpace.tex}\n\n\n\\section{Method: Simulating Line Emission} \\label{sec:sigame}\n\nWe use an updated version of \\ncode{s\\'{i}game}\\xspace \\citep{Olsen15a, Olsen17a}\nto post-process the \\ncode{simba}\\xspace simulation outputs.\nFor details of the code, we refer interested readers to \\citet{Olsen17a}.\nHere, we briefly summarize the salient points of \\ncode{s\\'{i}game}\\xspace and updates made to the code as part of this work.\nFor each gas fluid element, \\ncode{s\\'{i}game}\\xspace divides the molecular gas mass (i.e., $f_{\\rm H2, i}\\,m_{\\rm gas, i}$) into\nGMC by sampling the Galactic GMC mass function\nover the mass range of 10$^{4-6}$\\,\\mbox{$M_{\\odot}$}\\xspace\n($dn\/dM\\propto M_{\\rm GMC}^{-1.8}$; see e.g., the review by \\citealt{McKee07a, Blitz07a, Kennicutt12a}).\nThe remaining mass of the parent fluid element is assumed to be in the diffuse gas phase, and\nis subsequently distributed into diffuse ionized and neutral gas phases. This division is determined\nby the boundary at which the inner neutral region transitions to the outer ionized region of the diffuse clouds as computed from RT calculations within \\ncode{Cloudy}.\nThat is, the neutral gas phase corresponds to the region beyond a radius where the neutral fraction $x_{\\rm HI} = n_{\\rm HI} \/ (n_{\\rm HI} + n_{\\rm HII}) > \\,$0.5, such that it is dominated by neutral hydrogen --- here $n$ is the number density.\nAs such, \\ncode{s\\'{i}game}\\xspace accounts for\nline emission from three distinct ISM phases.\nThe smoothing length of the parent fluid element is adopted as the size of the diffuse gas clouds, whereas the size of each GMC\n is derived from a pressure-normalized mass-size relation, following\n\\begin{equation}\n\\frac{R_{\\rm GMC}}{\\rm pc} = \\left(\\frac{P_{\\rm ext}\/k_{\\rm B}}{10^4\\,{\\rm cm}^{-3}\\,{\\rm K}}\\right)^{-1\/4}\\left(\\frac{M_{\\rm GMC}}{290~M_\\odot}\\right)^{1\/2},\n\\end{equation}\nwhere $k_{\\rm B}$ is the Boltzmann constant\nand the external cloud pressure $P_{\\rm ext}$ is defined assuming mid-plane hydrostatic equilibrium within the galaxy.\n\nEach GMC radial density profile is assumed to follow a truncated logotropic profile.\nBoth GMCs and diffuse gas phases inherit the metallicity of their parent fluid element.\nThe FUV luminosity of each star is calculated based on its age and metallicity\nand is determined by interpolating over a grid of \\ncode{starburst99}\nstellar population synthesis models \\citep{Leitherer14a} (with the default \\citealt{Kroupa02a} initial mass function).\nEach GMC is irradiated by a local FUV radiation (6--13.6\\,eV), where the strength\nof the radiation field ($G_0$ in Habing units; 1.6\\E{-3}\\,erg\\,cm$^{-2}$\\,s\\,\\mbox{$^{-1}$}\\xspace)\nis determined by summing up the FUV flux from nearby stellar particles and by assuming that the flux falls\noff as $1\/r^2$.\nIn the diffuse gas phase, the FUV radiation field is determined based on the SFR surface density\nof the galaxy.\nFor our sample, the SFR surface density ranges between $\\simeq$\\,1\\,--\\,6200 times the Milky Way.\n\nWe use the photoionzation code \\ncode{cloudy} version 17.01 \\citep{Ferland17a}\nto simulate the thermo-chemistry in the three distinct ISM phases tracked by \\ncode{s\\'{i}game}\\xspace\nby performing detailed balance calculations of the various species, taking into account physical\nprocesses such as H$_2$ photo processes, dust physics (grain-atom\/ion charge\ntransfer),\nand cosmic ray (CR) ionization.\nThe line luminosities are then derived from the cooling rates for different line transitions.\nFor computational purposes, lookup tables are generated for the\nGMC and the diffuse (neutral and ionized) gas phase, respectively.\nThe FUV radiation field impinging on the gas phases\nis assumed to have the same spectral shape as in the solar neighborhood.\n\nCosmic rays are added, with an ionization rate equal to that of the Milky Way scaled linearly by a\nfactor of $(G_{0, \\rm gas}\/G_{0, \\rm MW})$.\nFor the GMC models, the clouds are in theory completely embedded within diffuse gas,\nand thus H-ionizing radiation is turned off in the \\ncode{Cloudy} models (cf. \\citealt{Olsen17a}).\n\nThe main parameters in the GMC phase of the\n\\ncode{cloudy} models considered are the $G_{\\rm 0, GMC}$ of the radiation source,\nradius of the cloud ($R_{\\rm GMC}$), and cloud density profile as a function of cloud radius ($n_H(R_{\\rm GMC})$).\nTurbulent velocity is added to the GMC models according to the velocity dispersion calculated from the\ncloud radius and pressure, assuming clouds are virialized.\nFor the diffuse gas phase, the main model parameters are\ngas density ($n_H$), gas kinetic temperature ($T_k$), diffuse cloud size,\n($R_{\\rm dif}$), and metallicity ($Z$).\n\nCompared to \\citet{Olsen17a}, the main updates made to \\ncode{s\\'{i}game}\\xspace used in this work are as follows:\n\\begin{itemize}\n\\item Instead of fixing the number and width of shells used by \\ncode{cloudy} to model each GMC,\nwe now allow \\ncode{cloudy} to determine the optimal quantities to ensure convergence.\nThis modification leads to more accurate calculation of the\ngrain photoelectric heating of the gas and increases the importance of gas heating\ndue to this mechanism in the GMC models, which\nis the main excitation mechanism for [C{\\scriptsize II}]\\xspace emission.\nNamely, [C{\\scriptsize II}]\\xspace is collisionally excited such that higher kinetic temperature leads to more molecular motions and collisions\ninside GMCs and photo-dissociation regions (PDRs), the main sites for [C{\\scriptsize II}]\\xspace emission in galaxies.\n\\item To ensure good sampling of the parameter space for both GMCs and diffuse gas clouds, the\nnumber of \\ncode{cloudy} models used to create look-up tables is significantly increased\nfrom 1296 to 4096 models by using 8 grid points in each parameter space dimension rather than 6\nas in \\citet{Olsen17a}. The look-up tables are further described in \\Sec{grids}.\n\\item The dust content of the ISM is a crucial factor in setting the [C{\\scriptsize II}]\\xspace luminosity.\nAs often done, we will assume here that dust scales with metallicity via the dust to metal ratio (DTM), but\ninstead of using a solar DTM value of $\\sim0.46$ (as done in \\cite{Olsen17a}),\nwe take a DTM of 0.25 based on the mean value of our \\ncode{simba}\\xspace galaxies (see \\Sec{grids}).\n\\end{itemize}\n\n\\subsection{GMCs and Diffuse Gas Phase Properties and \\ncode{cloudy} Model Grids} \\label{sec:grids}\n\nAs mentioned in the previous section, the parameters passed to \\ncode{cloudy}\nfor GMCs and diffuse gas clouds are;\n$M_{\\rm GMC}$, $G_{\\rm 0, GMC}$, $Z$, and $P_{\\rm ext}$ for the GMC models\nand in $n_H$, $T_k$, $R_{\\rm dif}$, and $Z$ for diffuse gas models.\n\nFor the GMCs, we generate 4096 models spanning\n$\\log (M_{\\rm GMC}\/M_\\odot)\\in$[4.1, 4.3, 4.6, 4.8, 5.1, 5.3, 5.6, 5.8],\n$\\log (G_{\\rm 0, GMC}\/G_{\\rm 0, MW})\\in[$0.3, 1.0, 1.6, 2.3, 3.0, 3.7, 4.3, 5.0],\n$\\log (Z\/Z_\\odot)\\in[-$3, $-$2.5, $-$2.1, $-$1.6, $-$1.2, $-$0.7, $-$0.3, 0.2], and\n$\\log (P_{\\rm ext}\/k_B)\\in[$4.0, 4.9, 5.7, 6.6, 7.4, 8.3, 9.1, 10.0]\\,cm$^{-3}$\\,K.\nFor the diffuse gas, we first determine the SFR surface density of all the galaxies of the sample.\nWe then define a range of FUV grids over $(G_{\\rm 0, GMC}\/G_{\\rm 0, MW})\\in$[0.8, 7.2, 68, 650, 6200]\nbased on the ranges of SFR surface densities found in the simulated galaxies as a hyperparameter.\nFor each of the FUV grid, we generate 4096 models spanning\n$\\log (n_H\/{\\rm cm}^3)\\in$[$-$5.0, $-$4.32, $-$3.66, $-$2.99, $-$2.31, $-$1.64, $-$0.97, $-$0.3],\n$\\log (T_k\/{\\rm K})\\in$[2.5, 3.0, 3.6, 4.2, 4.8, 5.4, 5.9, 6.5],\n$\\log (R_{\\rm dif}\/{\\rm kpc})\\in$[$-$0.7, $-$0.49, $-$0.27, $-$0.06, 0.16, 0.37, 0.59, 0.8], and\n$\\log (Z\/Z_\\odot)\\in[-$1.0, $-$0.83, $-$0.66, $-$0.49, $-$0.31, $-$0.14, 0.03, 0.2].\n\nThe gas kinetic temperature in the GMC phase is left as a free parameter to be determined by solving the thermal balance equation\nin \\ncode{cloudy},\nwhereas in the diffuse gas phase, the temperature is fixed to the grid points representative of the range seen in\nthe gas fluid elements from the \\ncode{simba}\\xspace simulation.\nThe effect of gas heating due to the photo-excitation by the cosmic microwave background (CMB)\nat the EoR is included. The resulting line intensities\nare corrected to give the net flux above the background continuum\n(i.e., not the contrasting flux that observers would measure; see e.g., \\citealt{daCunha13a}),\nand include the diminution effect where the upper levels are sustained by CMB\n(\\citealt{Ferland17a}).\n\n\n\n\\subsection{Dust and Elemental Abundances} \\label{sec:element}\nWhile \\ncode{simba}\\xspace tracks dust in the simulation, we do not create a different \\ncode{cloudy} lookup table for\neach dust-to-mass (DTM) ratio found in the simulated galaxies since this becomes computationally intractable (i.e., it corresponds to\na hyperparameter where each DTM ratio would have a separate set of 4096 \\ncode{cloudy} models).\nInstead, we adopt a DTM ratio based on the median of \\ncode{simba}\\xspace galaxies at $z\\sim6$, corresponding to $\\xi_{\\rm DTM}$\\eq0.25, which is defined as\n\\begin{equation}\n\\xi_{\\rm DTM} = \\frac{M_{\\rm dust}}{f_Z~M_{\\rm gas} + M_{\\rm dust}},\n\\end{equation}\nwhere $M_{\\rm dust}$ and $M_{\\rm gas}$ are the dust mass and total gas mass in solar mass units, and\n$f_Z$ is the mass fraction of metals (i.e., $f_Z~M_{\\rm gas}$ yields the mass of metals in gas-phase).\nThe dust content of each cloud is then set to scale linearly with its metallicity through this DTM.\nThe default set of lookup tables in \\ncode{Cloudy} assumes a DTM of $\\xi_{\\rm DTM}$\\eq0.46 at solar metallicity.\nA more commonly adopted expression for the DTM is:\n\\begin{equation}\n\\textrm{DTM} \\equiv \\frac{\\textrm{DGR}}{\\textrm Z}.\n\\end{equation}\nIn the Milky Way, Z\\,=\\,$Z_\\odot$ and DGR is $\\sim$0.01, yielding $\\log$~DTM\\,=\\,$-$2.\n\nIn \\ncode{cloudy}, one can supply the total metallicity, and the abundance of each of the elements is scaled correspondingly\nassuming Solar composition (i.e., abundance ratios of the Sun).\nIn order to account for abundance patterns of galaxies that differ from Solar,\nwe use the abundance of each metal tracked in \\ncode{simba}\\xspace (i.e., He, C, N, O, Ne, Mg, Si, S, Ca, and Fe).\nFor elements that are not tracked in the simulation, we use Solar abundance ratios.\nSince the mass fraction of each element tracked varies as a function of metallicity, we\nfit a spline curve to the running means across all gas fluid elements.\nThis provides a function that maps a given metallicity to an abundance pattern, such that\nthe relative elemental abundances in the \\ncode{cloudy} input are scaled according to the\nmetallicity of the cloud. In the Appendix we show the particle distribution and scalings found for our sample of \\ncode{simba}\\xspace galaxies. We note that some elements, like carbon and nitrogen, can be quite far from their solar abundance value, which in \\ncode{simba}\\xspace is a result of including enrichment from Type II supernovae (SNe), Type Ia SNe, and Asymptotic Giant Branch (AGB) stars, with separate yield tables for each class of star as described in \\cite{Oppenheimer2006}.\n\n\\section{Results and Discussion}\\label{sec:results}\n\\subsection{[C{\\scriptsize II}]\\xspace -- SFR Relation at $z$\\xspace$\\simeq$\\,6} \\label{sec:ciifir}\n\n\\begin{figure*}[htbp]\n\\centering\n\\includegraphics[trim=0 0 0 0, clip, width=.9\\textwidth]{fig3}\n\\caption{SFR and \\mbox{$L_{\\rm [CII]}$}\\xspace of \\ncode{simba}\\xspace galaxies at $z$\\xspace\\eq6 (hexbin) compared to existing observations\\xspace and models at the EoR.\nRed lines show the running mean and standard deviations of the binned data for \\ncode{simba}\\xspace galaxies.\nResults from a sample of 30 (zoom-in)\n\\ncode{mufasa}\\xspace galaxies at $z$\\,$\\simeq$\\,6 are shown as green shaded regions \\citep{Olsen17a},\nwhereas those from SAM-based predictions at $z$\\xspace\\eq6 are shown as light red shaded regions \\citep{Popping19a},\nand results from zoom-in AMR simulations from \\citet{Pallottini17a} and \\citet{Pallottini17b} are shown as blue stars.\nFits to observations from $z$\\,\\eq0 are shown as gray and blue\nshaded regions \\citep{DeLooze14a, Herrera-Camus15a}.\nSquare symbols show observations at $z\\simeq$6 compiled from \\citet{Ouchi13a, Kanekar13a,\nOta14a, Gonzalez-Lopez14a, Maiolino15a,\nSchaerer15a, Capak15a, Willott15a,\nBradac17a, Inoue16a, Pentericci16a,\nKnudsen16a, Knudsen17a, Decarli17a, Smit17a, Carniani18a} and Uzgil et al. 2020, in prep.\n\\label{fig:ciisfr}}\n\\end{figure*}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[trim=0 20 0 0, clip, width=.55\\textwidth]{fig4}\n\\caption{Same as \\Fig{ciisfr}, but visualized across three panels for clarity.\nTop: Running mean and standard deviations of $z$\\eq6 \\ncode{simba}\\xspace galaxies (red) overplotted with $z$\\eq0 observations\\xspace. The predicted \\mbox{$L_{\\rm [CII]}$}\\xspace\/SFR for galaxies with SFR $\\gtrsim 1$ are 1-2 dex lower than observed galaxies in the local Universe, and the slope of the \\mbox{$L_{\\rm [CII]}$}\\xspace vs. SFR relation is shallower.\nMiddle: \\ncode{simba}\\xspace galaxies (red line) overplotted with other $z$\\eq6 models in the literature (see legend). Our model predictions are in reasonable agreement with those of other studies, especially at higher SFR, but we predict a shallower \\mbox{$L_{\\rm [CII]}$}\\xspace vs. SFR relation than other studies.\nBottom: \\ncode{simba}\\xspace galaxies (red line and hexbin) overplotted with $z$\\eq6 observations\\xspace. The hexbins are color-coded by the density\nof points, see \\Fig{ciisfr} for colorbar. Our predictions show reasonable overlap with the locus of the heterogeneous observational samples, although we do not produce any galaxies with \\mbox{$L_{\\rm [CII]}$}\\xspace values as high as those of some of the detected galaxies at high SFR $\\gtrsim 10$. See text for a discussion of possible reasons for these discrepancies.\n\\label{fig:ciisfrmultipanel}}\n\\end{figure}\n\n\nIn \\Fig{ciisfr}, we plot the simulated \\mbox{$L_{\\rm [CII]}$}\\xspace and SFR\\footnote{The SFR is computed by dividing the stellar mass formed over the past 100\\,Myr by this timescale.} of the \\ncode{simba}\\xspace galaxies\ntogether with measurements from existing observations at $z\\,\\simeq$\\,6,\nlocal measurements, and other model predictions at $z\\,\\simeq$\\,6.\nThe \\mbox{$L_{\\rm [CII]}$}\\xspace-SFR relation converges across the different simulation volumes as does the \\mbox{$L_{\\rm [CII]}$}\\xspace-$M_{\\rm mol}$ relation as seen in the right panel Fig.\\,\\ref{fig:mH2} in the Appendix.\nFor clarity, information shown in this figure is also plotted across three panels in \\Fig{ciisfrmultipanel}.\nWe fit a linear model to \\mbox{$L_{\\rm [CII]}$}\\xspace and SFR in log-log space to facilitate comparison with literature work, and obtain\n\\begin{equation}\n\\log L_{\\rm [CII]} = (6.82\\pm0.08) + (0.66\\pm0.01)\\times\\log {\\rm SFR},\n\\label{eqn:ciisfr}\n\\end{equation}\nwhere \\mbox{$L_{\\rm [CII]}$}\\xspace is in units of \\mbox{$L_{\\odot}$}\\xspace, and SFR is in units of \\mbox{$M_{\\odot}$}\\xspace\\,yr\\mbox{$^{-1}$}\\xspace.\n\n\n\nThe [C{\\scriptsize II}]\\xspace luminosities of the \\ncode{simba}\\xspace galaxies are consistent with existing upper limits and a handful of detections\n from targeted observations \\citep[e.g.,][]{Ouchi13a, Kanekar13a, Ota14a,\nGonzalez-Lopez14a, Maiolino15a, Schaerer15a, Capak15a, Willott15a, Inoue16a, Pentericci16a,\nKnudsen16a, Inoue16a, Bradac17a, Knudsen17a, Decarli17a, Smit17a, Carniani18a}.\nIn addition, our results are in agreement with the latest upper limits placed at $z\\simeq\\,$6 by the ALMA large program\nASPECS, which is an untargeted survey, placing an upper limit of\n\\mbox{$L_{\\rm [CII]}$}\\xspace$<$\\,2\\E8\\,\\mbox{$L_{\\odot}$}\\xspace for galaxies with UV-derived SFR of $\\sim$\\,0.25--50\\,\\mbox{$M_{\\odot}$}\\xspace\\,yr\\mbox{$^{-1}$}\\xspace (\\citealt{Walter16a}, Uzgil et al., in prep.)\\footnote{ASPECS consists of two bands (Bands 3 and 6). In Band 6, blind spectral scans over an 85 pointing mosaic\nin the {\\it Hubble} Ultra Deep Field (HUDF) with an areal footprint of 4.2 arcmin$^2$ were observed, reaching down to $\\sigma_{\\rm cont}$\\eq9.3\\,$\\mu$Jy\\,beam$^{-1}$\\xspace for the continuum and $\\sigma_{\\rm ch}$\\eq0.3\\,mJy\\,beam$^{-1}$\\xspace per $\\Delta v$\\eq75\\,km\\,s$^{-1}$\\xspace channel\nfor the line cube. At $z$\\eq6, the sensitivity reaches a 5\\,$\\sigma$ limit of\n \\mbox{$L_{\\rm [CII]}$}\\xspace$\\simeq$\\,10$^{8.3}$\\,\\mbox{$L_{\\odot}$}\\xspace at $z$\\xspace\\eq6, assuming a linewidth of $\\Delta v$\\eq200\\,km\\,s$^{-1}$\\xspace.\\label{footnote:aspecs}}.\nIn particular, the ASPECS sources with upper limits on \\mbox{$L_{\\rm [CII]}$}\\xspace shown in \\Fig{ciisfr} with blue squares are\na combination of Lyman-$\\alpha$ emitters (LAEs)\n with spectroscopic redshifts from the MUSE survey \\citep{Inami17a},\n and Lyman break galaxies (LBGs) \\citep{Bouwens15a}.\n The Lyman-$\\alpha$ luminosities of the LAEs are L$_{Ly\\alpha}$\\eq0.7\\,$-$\\,1.5\\E{42}\\,erg\\,s\\mbox{$^{-1}$}\\xspace, with\n UV-based SFR\\,$<$\\,4\\,\\mbox{$M_{\\odot}$}\\xspace\\,yr\\mbox{$^{-1}$}\\xspace, and stellar mass of $\\log(M_*\/M_\\odot)$\\eq8.04\\,--\\,8.75\n (note that only two of the six MUSE LAEs have stellar mass constraints).\n The LBGs have H-band magnitudes of H$_{\\rm 160}$\\eq27.5--30.9\\,mag, corresponding to a UV-based\n SFR of 0.25\\,--\\,48\\,\\mbox{$M_{\\odot}$}\\xspace\\,yr\\mbox{$^{-1}$}\\xspace, and have stellar masses between $\\log(M_*\/M_\\odot)$\\eq7.99\\,--\\,9.37.\nIn general, the running mean in [C{\\scriptsize II}]\\xspace luminosity of the \\ncode{simba}\\xspace sample is lower than the average of existing detections of {\\em targeted} observations\\xspace at $z$\\ssim6.\n\n\nOur results are consistent with those based on the \\ncode{Serra} simulation suite by \\citet{Pallottini17a} and \\citet{Pallottini17b},\nwhich is a suite of cosmological zoom-in AMR simulations that resolve the gas down to\n10\\,pc-scales at $z\\simeq$\\,6.\nIt is also in reasonably good agreement with the sample of 30 \\ncode{mufasa} galaxies\nanalyzed in \\citet{Olsen17a}, thus broadly confirming these results\nusing a larger sample from its successor simulation (\\ncode{simba}),\nwhile reaching comparable resolution over cosmological volumes.\nThat said, our results yield a flatter \\mbox{$L_{\\rm [CII]}$}\\xspace--SFR relation than other models at $z\\,\\simeq$\\,6,\nsuch as those based on SAMs and semi-empirical models \\citep{Vallini15a, Lagache18a, Popping19a}.\nWe discuss the potential causes of the differences seen between our results and other models in the literature in \\Sec{caveats}.\n\n\n\n\n\\subsection{ [C{\\scriptsize II}]\\xspace Luminosity Function at $z$\\xspace$\\simeq$\\,6}\n\\begin{figure}[phtb]\n\\centering\n\\includegraphics[trim=0 0 0 0, clip, width=.5\\textwidth]{fig5}\n\\caption{[C{\\scriptsize II}]\\xspace LF predicted at $z$\\xspace$\\simeq$\\,6 based on the cosmological hydrodynamics simulation \\ncode{simba}\\xspace.\nShaded regions are obtained by jackknife resampling of the simulation sub-volumes.\nThe flattening and turnover at the faintest end is due to incompleteness of\nhaloes with $\\log$ \\mbox{$L_{\\rm [CII]}$}\\xspace$\\lesssim$\\,6\\,\\mbox{$L_{\\odot}$}\\xspace.\nResults from SAM-based models are overplotted as dashed lines \\citep{Popping16a, Popping19a}.\nOur results are fully consistent with the SAM-based model predictions within the error bars and the\nupper limits from ASPECS (blue symbol; Uzgil et al., in prep.).\n\\label{fig:ciilf}}\n\\end{figure}\n\nIn \\Fig{ciilf}, we show predictions for the [C{\\scriptsize II}]\\xspace LF at $z$\\xspace$\\simeq\\,$6\nbased on the simulated \\mbox{$L_{\\rm [CII]}$}\\xspace of galaxies in Simba-25, Simba-50, and Simba-100.\nWe note that previously, LF predictions were only possible in models that made\nmore simplified assumptions to connect \\mbox{$L_{\\rm [CII]}$}\\xspace to dark matter halos.\nNonetheless, our results are in agreement with those based on SAMs by \\citet{Lagache18a} and \\citet{Popping19a},\nand with constraints from the latest limits placed using data from ASPECS (Uzgil et al., in prep.; see footnote~\\ref{footnote:aspecs}).\n\nUsing the \\citet{Capak15a} targeted sample, \\citet{Hemmati17a}\nreport a volume density that is almost an order of magnitude higher\nthan our results at the bright end at $\\log $ \\mbox{$L_{\\rm [CII]}$}\\xspace$\\gtrsim$\\,8.5\\,\\mbox{$L_{\\odot}$}\\xspace, although they\nare consistent within the error bars. Note that the actual uncertainties on the [C{\\scriptsize II}]\\xspace LF constrained by\nthe \\citet{Capak15a} sample are likely to be larger than those reported by \\citet{Hemmati17a}, as\nincompleteness and selection bias are not corrected for.\nAs shown in \\Fig{ciisfr}, most of the \\ncode{simba}\\xspace galaxies have \\mbox{$L_{\\rm [CII]}$}\\xspace\\eq10$^{6-8}$\\,\\mbox{$L_{\\odot}$}\\xspace.\nThus, it is unsurprising to see a discrepancy in the [C{\\scriptsize II}]\\xspace LF between the \\citet{Capak15a} sample and our sample\ndue to the lack of overlap in terms of \\mbox{$L_{\\rm [CII]}$}\\xspace.\n\n\\citet{Miller16a} derive a [C{\\scriptsize II}]\\xspace LF using the Bolshoi-Planck dark matter only simulation catalog from\n\\citet{Behroozi13b}, the abundance matched SFR from \\citet{Hayward13c},\nand the empirical \\mbox{$L_{\\rm [CII]}$}\\xspace--SFR relations established at $z$\\ssim0 by \\citet{DeLooze14a}.\nConsistent with our results, \\citet{Miller16a} report a [C{\\scriptsize II}]\\xspace LF that underpredicts the observational constraints placed\nby \\citet{Hemmati17a} using the \\citet{Capak15a} sample and that placed based on a\nblind search of five deep fields centered on IR-bright galaxies and quasar host galaxies at $z$\\ssim6.\nBy simulating only regions with \\mbox{$L_{\\rm [CII]}$}\\xspace matched to the central galaxies observed in\nthe deep fields (8.7\\,$<\\,\\log$\\mbox{$L_{\\rm [CII]}$}\\xspace\/\\mbox{$L_{\\odot}$}\\xspace$<$\\,9), \\citet{Miller16a} find a good agreement between the [C{\\scriptsize II}]\\xspace LF\nand the observational constraints.\nOn this basis, they argue that the [C{\\scriptsize II}]\\xspace-detected sources in the deep fields are indeed in biased overdense regions.\nThis may partially explain the discrepancy seen between the observed and the \\ncode{simba}\\xspace based [C{\\scriptsize II}]\\xspace LF ---\nsince the largest simulation box of \\ncode{simba}\\xspace is 100\\,cMpc, and it may not contain these rare highly biased regions.\n\n\n\n\\subsection{\\mbox{$L_{\\rm [CII]}$}\\xspace\\,--\\,M$_{\\rm halo}$ Relation} \\label{sec:halo}\nForecasts for upcoming [C{\\scriptsize II}]\\xspace LIM surveys have been obtained using\nscaling relations between [C{\\scriptsize II}]\\xspace luminosity and halo mass, where\nthe latter quantity is obtained from large volume N-body simulations (e.g., \\citealt{Silva15a, Kovetz17a}).\nThe large cosmological volumes provided by N-body simulations are needed to make mock lightcones\nfor LIM \\citep[e.g.][]{yang2020}; however,\nmost previous work in this area has made simple empirical assumptions regarding the relationship between\ndark matter halo properties and [C{\\scriptsize II}]\\xspace line luminosity.\nHere we study the \\mbox{$L_{\\rm [CII]}$}\\xspace\\,--\\,M$_{\\rm halo}$ relation based on\nthe central galaxies in our cosmological hydrodynamic simulation.\n\nIn \\Fig{ciimhalo}, \\ncode{simba}\\xspace galaxies are shown in \\mbox{$L_{\\rm [CII]}$}\\xspace versus M$_{\\rm halo}$ with a fit made following the formalism of \\citet{Silva15a}, where SFR is expressed in terms of M$_{\\rm halo}$:\n \\begin{equation}\n{\\rm SFR} = M_0 \\times \\left(\\frac{M_{\\rm halo}} {M_a} \\right)^a \\left(1 + \\frac{M_{\\rm halo}}{M_b}\\right)^b.\n\\end{equation}\nTogether with the expected linear relation between \\mbox{$L_{\\rm [CII]}$}\\xspace and SFR, the following equation relates M$_{\\rm halo}$ to \\mbox{$L_{\\rm [CII]}$}\\xspace:\n\\begin{equation}\n\\log L_{\\rm [CII]} = M'_0 + a' \\log \\left(\\frac{M_{\\rm halo}} {M'_a} \\right) + b' \\log\\left(1 + \\frac{M_{\\rm halo}}{M'_b}\\right),\n\\label{eqn:lciisfr}\n\\end{equation}\nwhere \\mbox{$L_{\\rm [CII]}$}\\xspace is in units of \\mbox{$L_{\\odot}$}\\xspace, and M$_{\\rm halo}$ is in units of \\mbox{$M_{\\odot}$}\\xspace. With the current set of parameters adopted in the sub-grid model (see \\Sec{sigame}), the fitted parameters are\n$a'$\\eq0.65,\n$b'$\\,=\\,$-$9.85,\n$M'_0$\\eq3.62,\n$M'_a$\\eq1.50\\E{7}, and\n$M'_b$\\eq2.90\\E{13}.\n\nThe \\mbox{$L_{\\rm [CII]}$}\\xspace--\\,M$_{\\rm halo}$ relation of central galaxies in \\ncode{simba}\\xspace has\na scatter of $\\simeq$\\,0.5\\,dex around the fit.\nWe also show the running mean in the same figure which follows the parametric form.\nIn the same figure, we show a comparison with the \\mbox{$L_{\\rm [CII]}$}\\xspace--\\,M$_{\\rm halo}$ relation from\na SAM \\citep{Popping19a,yang2020}, which is steeper than both \\citet{Silva15a} and\nthis work; but is however, consistent within the scatter of \\ncode{simba}\\xspace galaxies.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[trim=0 0 0 0, clip, width=.5\\textwidth]{fig6}\n\\caption{\\mbox{$L_{\\rm [CII]}$}\\xspace$-$\\,M$_{\\rm halo}$ of \\ncode{simba}\\xspace galaxies studied in this work, color-coded by sSFR (dot symbols).\nThe black line shows the best-fit parametric model, whereas the orange dots show\nthe mean\nwhen the \\ncode{simba}\\xspace data is binned in 30 logarithmic intervals in M$_{\\rm halo}$ (i.e., non-parametric).\nThe blue dashed line shows the m1 model of \\citet{Silva15a} at a comparable redshift.\nThe red line shows the model from \\cite{yang2020} based on SAMs by \\citet{Popping19a}. Although there is qualitative agreement between the different models, the remaining discrepancies could have significant implications for predictions for upcoming line intensity mapping surveys.\n\\label{fig:ciimhalo}}\n\\end{figure}\n\n\n\n\\section{Discussion: Discrepancies and Caveats} \\label{sec:caveats}\n\n\\begin{turnpage}\n\\input{Table_models}\n\\end{turnpage}\n\n\nAs discussed in \\Sec{results}, our predicted [C{\\scriptsize II}]\\xspace LF is consistent\nwith that predicted from other models based on SAMs, and the \\ncode{simba}\\xspace\ngalaxies lie in the same region of \\mbox{$L_{\\rm [CII]}$}\\xspace--\\,SFR as the galaxies\nstudied by one of the most detailed cosmological zoom-in AMR\nsimulations at the same redshift \\citep{Pallottini17a, Pallottini17b}.\nHowever, compared to other models, our model underpredicts the [C{\\scriptsize II}]\\xspace\nluminosity at the bright end (and high SFR; \\Fig{ciisfrmultipanel})\nand yields a flatter \\mbox{$L_{\\rm [CII]}$}\\xspace--\\,SFR relation. This discrepancy may\narise from for instance different ranges of properties for galaxies in\nthe samples, different predicted scaling relations between galaxy\nproperties in different galaxy evolution models, a limited number of\nmassive halos in our simulation, and the different sub-grid treatments\nof physical processes in \\ncode{simba}\\xspace and in \\ncode{s\\'{i}game}\\xspace compared to other\napproaches adopted in the literature used for comparison here\n(\\Tab{model}). For instance, \\citet{Vallini15a} simulate the [C{\\scriptsize II}]\\xspace\nline emission by post-processing the UV radiation field of an\nSPH-based simulated galaxy using \\ncode{Licorice}, and calculate the\n[C{\\scriptsize II}]\\xspace emission using a combination of an analytical model and the\nphoto-dissociation region (PDR) code \\ncode{ucl\\_pdr}\n(\\citealt{Bayet09b} and references therein; to account for\ncontributions from PDR). While the sub-grid modeling of\n\\citet{Popping19a} follows a similar approach as \\ncode{s\\'{i}game}\\xspace, the former\nis based on a SAM while the latter is applied to hydrodynamical\nsimulations. As a result, there are relatively subtle differences\nbetween the two, such as the assumption of exponential gas disks for\nall galaxies in the former. In addition, the former approach assumes\nthat all GMCs in each galaxy share the same metallicity based on the\nglobal metallicity, and adopts \\ncode{despotic} instead of\n\\ncode{cloudy} in performing the thermochemistry calculation. As\nmentioned in \\Sec{introduction}, \\ncode{despotic} does not account for\nline emission in the ionized phase while the latter does. In contrast\nto \\citet{Popping19a}, \\citet{Lagache18a} use \\ncode{cloudy} to\npost-process their SAM. While using \\ncode{cloudy} is the same\napproach as \\ncode{s\\'{i}game}\\xspace (including this work) and works by e.g.,\n\\citet{Katz19a} and \\citet{Pallottini19a}, \\citet{Lagache18a} adopt\ndifferent sub-grid approaches and assumptions compared to those made\nfor hydrodynamical simulations. In particular, their model does not\naccount for [C{\\scriptsize II}]\\xspace emission coming from regions outside of PDRs, the\ncomplexity of the multiphase ISM, and the detailed structures of\nmolecular clouds. This illustrates the various differences between\nexisting models which can contribute to the discrepant \\mbox{$L_{\\rm [CII]}$}\\xspace--\\,SFR\nslopes. Using SAMs, \\citet{Popping19a} experiment with different\nassumptions made in the sub-grid approaches and indeed find\ndifferences in the resulting [CII] luminosity.\n\n\\citet{Lagache18a} report that after selecting galaxies with the same range of stellar mass, SFR, and gas-phase metallicities\nas the \\ncode{Mufasa} sample studied by \\citet{Olsen17a} --- i.e.,\nwith $M_*\\in$\\,(0.7--8)\\E{9}\\,\\mbox{$M_{\\odot}$}\\xspace, SFR$\\in[$3--23$]$\\,\\mbox{$M_{\\odot}$}\\xspace\\,yr\\mbox{$^{-1}$}\\xspace, and $Z_{\\rm gas}\\in[$0.15--0.45$]$\\,$Z_{\\odot}$ --- the \\mbox{$L_{\\rm [CII]}$}\\xspace--\\,SFR relation of their SAM galaxies is flatter and\nmore consistent with that of \\citet{Olsen17a}.\nYet, this relation remains flatter than that obtained when applying the same set of criteria to the\nSAM galaxies of \\citet{Popping19a} (9,653 galaxies after selection).\nIn \\Fig{ciisfr_selectedolsen17}, we show the \\mbox{$L_{\\rm [CII]}$}\\xspace --\\,SFR for\na subset of \\ncode{simba}\\xspace galaxies selected based on these criteria (1,136 galaxies)\ncompared to \\citet{Olsen17a} and \\citet{Popping19a}.\nA relation with a flatter slope than \\citet{Popping19a}'s SAM-based models persists for the \\ncode{simba}\\xspace galaxies.\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[trim=0 0 0 0, clip, width=.5\\textwidth]{fig7}\n\\caption{\nRegion of \\mbox{$L_{\\rm [CII]}$}\\xspace--\\,SFR in log-space spanned by \\ncode{simba}\\xspace galaxies (teal line) and\ngalaxies from Popping et al.~(\\citeyear{Popping19a}; magenta line)\nselected with the same range in stellar mass, SFR, and gas-phase metallicities\nas \\citet{Olsen17a} (green shaded). A relation with a flatter slope than other models persists for the \\ncode{simba}\\xspace galaxies (see \\Sec{caveats}).\n\\label{fig:ciisfr_selectedolsen17}}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[trim=0 0 0 0, clip, width=.5\\textwidth]{fig8}\n\\caption{Normalized distributions of stellar mass, SFR, molecular gas mass, and metallicity between the subset of galaxies\nfrom \\citet{Popping19a} (blue) and this work (green) after applying the \\citet{Olsen17a} selection.\nWhile distributions in SFR are comparable, the subset of galaxies from the \\citet{Popping19a} sample have higher\nmolecular gas mass and metallicity compared to the \\ncode{simba}\\xspace subset.\n\\label{fig:p19thiswork_dist_selectedolsen17}}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[trim=10 150 30 100, clip, width=.5\\textwidth]{fig9}\n\\caption{\nDistributions of \\mbox{$L_{\\rm [CII]}$}\\xspace across all \\ncode{simba}\\xspace galaxies, in the first (red) and third (cyan) quartiles in molecular gas mass\nfrom different SFR bins (second through last panels). See text for the SFR covered by each bin.\nThe top panel shows the distributions across all SFR.\nAt a fixed SFR, galaxies with higher $M_{\\rm mol}$ have higher \\mbox{$L_{\\rm [CII]}$}\\xspace.\n\\label{fig:ciisfr_quartile_mol}}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[trim=10 150 30 100, clip, width=.5\\textwidth]{fig10}\n\\caption{\nSame as \\Fig{ciisfr_quartile_mol} but for metallicity.\nAt a fixed SFR, galaxies with higher $\\langle Z_{\\rm gas}\\rangle_{\\rm SFR}$ have higher \\mbox{$L_{\\rm [CII]}$}\\xspace.\n\\label{fig:ciisfr_quartile_Zgas}}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[trim=10 150 30 100, clip, width=.5\\textwidth]{fig11}\n\\caption{\nSame as \\Fig{ciisfr_quartile_mol} but for SFR surface density.\nOverall, galaxies with higher $\\Sigma_{\\rm SFR}$ have higher \\mbox{$L_{\\rm [CII]}$}\\xspace, but this trend is not seen after accounting for the influence caused by different SFR.\n\\label{fig:ciisfr_quartile_sfrsd}}\n\\end{figure}\n\n\n\nAs shown and discussed in the literature (e.g., \\citealt{Vallini15a, Olsen17a, Lagache18a, Pallottini19a, Popping19a}),\na lower metallicity can result in a lower \\mbox{$L_{\\rm [CII]}$}\\xspace at given SFR.\nIn fact, as shown in \\Fig{p19thiswork_dist_selectedolsen17}, while the distributions in SFR\nbetween the subset of galaxies from the \\citet{Popping19a} sample and this work are comparable\nafter applying the \\citet{Olsen17a} selection cut, the former are more metal rich, with higher molecular gas masses\ncompared to the latter.\nThe trend of decreasing molecular gas mass and metallicity with \\mbox{$L_{\\rm [CII]}$}\\xspace is also seen in Figures~\\ref{fig:ciisfr_quartile_mol} and \\ref{fig:ciisfr_quartile_Zgas},\nwhere we show the different \\mbox{$L_{\\rm [CII]}$}\\xspace distributions when the full set of 11,137 \\ncode{simba}\\xspace galaxies are selected\nusing the first and third quartiles in molecular gas mass and metallicity, respectively.\nTo account for the effect of SFR in \\mbox{$L_{\\rm [CII]}$}\\xspace, we bin the galaxies into three SFR bins, $\\in[0.01, 10.3]$, $\\in[0.3, 10.3]$, $\\in[10.3, 329]$\\,\\mbox{$M_{\\odot}$}\\xspace\\,yr\\mbox{$^{-1}$}\\xspace\\footnote{Binning in SFR is performed in log-space to avoid small number statistics in each bin and to ensure that the number of galaxies in each bin is of the same order of magnitude (see e.g., hexbins in \\Fig{ciisfr}).}.\nAt least in the lowest SFR bin, the variation seen in the [C{\\scriptsize II}]\\xspace luminosity is mostly driven by the lower metallicity and molecular gas mass (see also \\citealt{Narayanan17a}).\nAcross all SFR, galaxies with lower \\mbox{$L_{\\rm [CII]}$}\\xspace correspond to those with the least molecular gas mass, metallicity,\nand SFR surface density (Figures~\\ref{fig:ciisfr_quartile_mol}, \\ref{fig:ciisfr_quartile_Zgas}, and \\ref{fig:ciisfr_quartile_sfrsd}).\n\n\n\nAs mentioned in \\Sec{introduction}, models in the literature adopt different approaches and assumptions in post-processing the simulations. This could also yield different simulated line luminosities.\n A steeper slope, more compatible with other models in the literature, namely \\citet{Vallini15a} and\n \\citet{Popping19a},\n would result from adopting the local ISM abundance ratios\\footnote{The C\/H ratio of\n the local ISM is comparable to the Solar value \\citep{Cowie86a, AllendePrieto02a, Asplund09a}.}\n instead of the abundance pattern tracked in \\ncode{simba}\\xspace (see \\Sec{element} and \\citealt{Olsen17a}).\nBoth \\citet{Vallini15a} and \\citet{Popping19a} adopt a Solar abundance pattern (of relevance to this work is the C\/H ratio)\nand scale the C\/H ratio according to the gas-phase metallicity of each galaxy to determine the [C{\\scriptsize II}]\\xspace emissivity.\nThat is, their models do not consider abundance patterns that differ from Solar.\nThe resulting \\mbox{$L_{\\rm [CII]}$}\\xspace could differ significantly owing to the amount of cooling via C$^+$.\nOn the other hand, \\citet{Lagache18a} accounted for abundance ratios that differ from Solar. Specifically, they\nscale the element abundances based on the median of their sample. Unsurprisingly,\nthe slopes of the \\mbox{$L_{\\rm [CII]}$}\\xspace--\\,SFR relation of \\ncode{mufasa} and \\ncode{simba}\\xspace galaxies\nare the most compatible to the \\citet{Lagache18a} model.\nAs mentioned by \\citet{Lagache18a} and \\citet{Katz19a},\ndetails in modeling the interstellar radiation field intensity, self-shielding,\nand the choice and implementation of stellar feedback are all effects that can\ncause differences between existing models (see also \\citealt{Pallottini17a}).\n\n\nIn contrast to \\ncode{mufasa}, where dust is not tracked in the simulation,\ncausing \\citet{Olsen17a} to adopt a constant DTM ratio,\n\\ncode{simba}\\xspace tracks dust in the simulation; however, we do not create a different \\ncode{cloudy} lookup table for\neach DTM ratio since it would become computationally intractable.\nThe mean DTM of \\ncode{simba}\\xspace galaxies is $\\xi_{\\rm DTM}$\\eq0.25, which yields a [C{\\scriptsize II}]\\xspace luminosity approximately\n$\\lesssim$\\,0.5\\,dex higher than for models with a DTM ratio set to the Solar value of\n$\\xi_{\\rm DTM}$\\eq0.46 (see Appendix; cf. \\citealt{Olsen17a} who finds that \\mbox{$L_{\\rm [CII]}$}\\xspace only increased by $\\sim0.15$ \\,dex at a given SFR when decreasing the DTM by a factor 2 from Solar DTM for their \\ncode{Mufasa} SFMS sample\\footnote{This work uses\nthe first release of \\ncode{s\\'{i}game}\\xspace; see main improvements in \\ncode{s\\'{i}game}\\xspace in \\Sec{sigame}.}).\nNote that in reality, the DTM ratio varies from galaxy to galaxy (and within galaxies themselves), and is correlated with\nthe galaxy metallicity \\citep[e.g.,][]{Remy-Ruyer14a, Popping17a, DeVis19a}.\nSince the standard deviation of the DTM ratio of the \\ncode{simba}\\xspace sample studied here is $\\sigma$\\ssim0.15,\nwe do not expect the simulated [C{\\scriptsize II}]\\xspace luminosity to deviate more than 0.5\\,dex as a result of\nvariations in the DTM ratio.\n\nA final important caveat is that the effect of AGN was not included in the present modeling with \\ncode{cloudy}, although \\ncode{cloudy} does have the capability to do so and this has been shown to be relevant at least for CO line emission at high redshift \\citep{Vallini19}, and hence likely also relevant for [CII]. The effect of AGN is an additional feature that we wish to include in the future.\n\n\n\n\n\\section{Summary and Conclusions} \\label{sec:conclusion}\n\nIn this work, we presented the first prediction of the [C{\\scriptsize II}]\\xspace luminosity\nfunction (LF) during the Epoch of Reionization (EoR; $z\\simeq$\\,6) based\non large-volume cosmological hydrodynamical simulations (\\ncode{simba}\\xspace)\ncoupled with radiative transfer and line spectral synthesis\ncalculations. We simulate the [C{\\scriptsize II}]\\xspace line luminosity for a sample of\n11,137 galaxies identified in the combined 25, 50, and\n100\\,\\ensuremath{\\mathrm{cMpc}\\,\\mathrm{h}^{-1}}\\ boxes (with 2\\,$\\times$\\,1024$^3$ particles each) at $z\\simeq$\\,6. The runs for the three simulation\nboxes have identical input physics, and\nproduce converged GSMF and SFR functions without fine-tuning\nthe parameters in the sub-grid models of \\ncode{simba}\\xspace. In addition, both\nGSMF and SFR functions are in good agreement with observations at this redshift. This\nis crucial as we make predictions for the [C{\\scriptsize II}]\\xspace LF in the luminosity\nrange of 5.5\\,$<$\\,$\\log$(\\mbox{$L_{\\rm [CII]}$}\\xspace\/\\mbox{$L_{\\odot}$}\\xspace)$<$\\,8.5 by combining the boxes.\n\n\nWe use an updated version of \\ncode{s\\'{i}game}\\xspace to post-process the \\ncode{simba}\\xspace output. Three major improvements are\nimplemented relative to the previous version of \\ncode{s\\'{i}game}\\xspace presented in \\citet{Olsen17a} to produce the results presented;\n{\\em (i)} We do not fix the number and width of shells for each GMC model, but instead allow \\ncode{cloudy} to determine the optimal quantities to ensure convergence.\nThis modification leads to more accurate calculation of the grain photoelectric heating of the gas and\nincreases the importance of gas heating due to this mechanism in the GMC models --- the main excitation mode for [C{\\scriptsize II}]\\xspace emission;\n{\\em (ii)} The number of \\ncode{cloudy} models used to create look-up tables is significantly increased\nfrom 1296 to 4096 models to ensure better sampling of the physical properties of the ISM,\nand\n{\\em (iii)} A DTM ratio of 0.25 is adopted based on the mean value of the \\ncode{simba}\\xspace galaxy sample\nrather than using a solar DTM value.\nFinally, \\citet{Olsen17a} simulated line emission for a subset of 30 zoom-in \\ncode{Mufasa} galaxies along\nthe SFMS, whereas with \\ncode{simba}\\xspace, we are able to expand the parameter space in M$_{\\rm halo}$, SFR, M$_*$, M$_{\\rm gas}$, SFR surface density, and metallicity at comparable resolution without the need for ``zooming in'' on specific galaxies.\n\nWe summarize the main results of this paper in the following:\n\\begin{itemize}\n\\item The simulated \\mbox{$L_{\\rm [CII]}$}\\xspace is consistent with the range observed in $z$\\xspace\\ssim6 galaxies, with\na spread of $\\simeq$\\,0.3\\,dex at the high SFR end of $>$\\,100\\,\\mbox{$M_{\\odot}$}\\xspace\\,yr\\mbox{$^{-1}$}\\xspace\nwhich increases to $\\simeq$0.6\\,dex at the lower end of the SFR. The predicted \\mbox{$L_{\\rm [CII]}$}\\xspace-SFR is consistent with targeted observed samples within the uncertainties due to selection and incompleteness effects. On the other hand, our model does not produce galaxies with values of \\mbox{$L_{\\rm [CII]}$}\\xspace as high as those for some galaxies observed\nin targeted heterogenous samples reported in the literature, at a given SFR.\n\n\\item The [C{\\scriptsize II}]\\xspace LF is consistent with the upper limits placed by the only existing\nuntargeted flux-limited [C{\\scriptsize II}]\\xspace survey at the EoR (ASPECS) and those predicted by semi-analytic models.\n\n\\item Our model yields a \\mbox{$L_{\\rm [CII]}$}\\xspace--\\,SFR relation similar to \\ncode{mufasa} \\citep{Olsen17a} but\nis flatter compared to some other models in the literature. The flatter slope results from\ndifferent galaxy properties sampled by different simulations, implementation and assumptions of sub-grid recipes between different simulations and the post-processing steps (see \\Tab{model}).\n\n\\item At a fixed SFR, galaxies with higher molecular gas mass, metallicity, and SFR surface density have higher [C{\\scriptsize II}]\\xspace luminosity.\n\n\\item We present the \\mbox{$L_{\\rm [CII]}$}\\xspace--\\,M$_{\\rm halo}$ relation for the central galaxies in\n\\ncode{simba}\\xspace at $z\\sim6$.\nOur relation is steeper than those based on N-body simulations and SAMs; but are consistent within the scatter of $\\simeq$\\,0.5--0.6\\,dex.\n\n\n\\end{itemize}\n\nThe differing results presented in the literature on simulating [C{\\scriptsize II}]\\xspace\nline emission at the EoR highlights the challenges modelers face in\nthis field. As discussed in this paper, SAMs are computationally\nefficient and can simulate the line emission for a statistically\nsignificant sample of galaxies, but SAMs have their limitations. They\ndo not contain information regarding the structure of the ISM of\ngalaxies, or the 3D distribution and morphology of galaxies, to name a\nfew. Cosmological hydrodynamic simulations, on the other hand, can\nprovide more detailed information on the temperature and density of\nthe intergalactic medium and interstellar medium, the 3D structures of\ngalaxies, and the local properties of galaxies (e.g., each gas element\nhas a different metallicity), but are computationally demanding. In\naddition, large volume cosmological simulations still lack the\nresolution needed to resolve the multi-phase ISM in detail. Our convergence tests among different resolution simulations indicate good convergence in stellar masses and star formation rates but less than ideal convergence in the molecular gas fractions, indicating that some sub-grid models may still need refinement to improve convergence properties. Zoom-in\nsimulations have been used to achieve higher resolution, but the\ncomputational cost limits the number of galaxies (and thus, the galaxy\nparameter space studied) that can be ``re-simulated'' with the zoom-in\napproach in reasonable time. Calibrating parameters based on higher\nresolution simulations with resolved ISM properties will be necessary to test the\nsub-grid models and assumptions made, for example adopting the\ndistributions based on pc-scale hydrodynamic simulations (e.g.,\n\\citealt{Tress20a}), and such work is underway by \\ncode{s\\'{i}game}\\xspace group members.\n\nDeep galaxy surveys over the past decade have provided constraints on the cosmic star formation\nhistory and supermassive black hole growth history \\citep[e.g.,][]{Madau14a, Wilkins19a, Hickox18a, Aird19a},\nwhile surveys and intensity mapping experiments planned in the next decade, with facilities such as\nthe {\\it James Webb Space Telescope}, {\\it WFIRST}, {\\it Euclid}, LSST,\nCONCERTO, HERA, SPHEREx, EXCLAIM, and TIM \\citep[see reviews by][]{Kovetz17a, Cooray19a},\nwill expand and sharpen our view of galaxy evolution by discovering galaxies in new ways, obtaining photometric and spectroscopic redshifts, and\nprobing a wider parameter space in galaxy properties and large scale environment.\nDue to the brightness of the fine-structure line [C{\\scriptsize II}]\\xspace and its accessibility at high redshift,\nit is one of the main spectral lines that may be observed at the EoR to study the ISM properties of galaxies and\nto secure their spectroscopic redshifts. LIM experiments such as CONCERTO, TIME, and\nCCAT-p will measure the [C{\\scriptsize II}]\\xspace line power spectrum from galaxies at the EoR; however, the detection limits\nand interpretation of the observed power spectrum depend on the [C{\\scriptsize II}]\\xspace line luminosities of different EoR galaxy populations within\nthe volume sampled. The fainter populations are below the current detection limit of existing\nfacilities, but their signal can be predicted, highlighting the importance of building theoretical frameworks\nto simulate the [C{\\scriptsize II}]\\xspace line at the EoR using cosmological simulations.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\subsection*{Local pressure estimation}\nThe 3-D pressure from molecular interactions is estimated classically by IK method \\cite{13irving1950statistical} through the expression shown in Eq. \\ref{eq1}. Here, the first term represents the kinetic energy contribution and second represents the virial contribution. \n\\begin{equation}\nP(r_p) = P_K(r_p) + P_V(r_p)\n\\label{eq1}\n\\end{equation}\nKinetic contribution is,\n\\begin{equation}\nP_K(r_p) = \\sum^N_{i=1} m_i v_i \\circledast v_i \\delta(r_i-r_p)\n\\label{eq2}\n\\end{equation}\nVirial contribution is\n\\begin{equation}\nP_V(r_p) = \\sum_{i=1}^{N-1} \\sum_{j=i+1}^{N} r_{ij} \\circledast F_{ij} \\delta(r_i-r_j)\\delta(r_i-r_p)\n\\label{eq3}\n\\end{equation}\nHere $P$ is the pressure, $m$ is mass of $i^{th}$ atom, $v$ is velocity, $r_i$ and $r_j$ are the position vectors of $i^{th}$ and $j^{th}$ atoms respectively, $N$ is number of atoms, $r_p$ is the position vector of $p^{th}$ grid point, $r_{ij} = r_i -r_j$, $F_{ij}$ is the force, and $\\delta$ is the Dirac delta function \\cite{dirac1981principles}. Though this expression is theoretically correct, practically it needs infinite sampling, which makes it less appealing for finite computer simulations. Specifically, for molecular dynamics simulations this is computationally expensive due to its convolution nature. To evade this situation, Hardy introduced \\cite{12hardy1982formulas,14hardy2004two} interpolation functions to distribute the kinetic contribution and a bond function to distribute the virial contribution to the local grid points. This resulted in the modified expression for pressure as\n\\begin{equation}\nP(r_p) = \\sum_{i=1}^{N} m_i v_i \\circledast v_i w(r_i - r_p) + \\sum_{i=1}^{N-1} \\sum_{j=i+1}^{N} r_{ij} \\circledast F_{ij} B_{ij}(r_p)\n\\label{eq4}\n\\end{equation}\nHere, $w$ is the weight function (interpolation function) and $B$ is the bond function and defined as\n\\begin{equation}\nB_{ij}(r_p) = \\int_{0}^{1} w(\\lambda r_{ij} + r_i - r_p)~ d\\lambda\n\\label{eq5}\n\\end{equation}\nA weight function has to be normalized and should follow\n\\begin{equation}\n\\int_{R^3} w(r)dr^3 = 1\n\\label{eq6}\n\\end{equation}\n\n\\subsection*{3-D pressure formulation}\nFor a 3-D system, if the distribution is assumed to be spherically symmetric, then \n\\begin{equation}\n\\int_{R^3} w(r)dr^3 = \\int_0^\\infty \\hat{w}(r) 4\\pi r^2 dr = 1\n\\label{eq7}\n\\end{equation}\nIf the spread of the function is limited to a certain spread radius $r_s$ then the equation becomes\n\\begin{equation}\n\\int_{R^3} w(r)dr^3 = \\int_0^{r_s} \\hat{w}(r) 4\\pi r^2 dr = 1\n\\label{eq8}\n\\end{equation}\nHere, $\\hat{w}(r)$ is the weight function and used by researchers \\cite{8yang2012generalized,22admal2010unified} for 3-D grid, is given as:\n\\begin{equation}\n\\hat{w}(r) = C_1 [1 - 3r^2\/r_s^2 + 2r^3\/r_s^3]\n\\label{eq9}\n\\end{equation}\nhere, $C_1$ is the normalization constant.\nFor 3-D systems, \n\\begin{equation}\n\\int_{R^3} w(r)dr^3 = \\int_0^{r_s} 4\\pi r^2 C_1 [1 - 3r^2\/r_s^2 + 2r^3\/r_s^3] dr = 1\n\\label{eq10}\n\\end{equation}\nthe constant of integration takes the form $C_1 = 15 \/ 4 \\pi r_s^3$ and $r(x,y,z)$ is a function in three coordinates.\n\n\\subsection*{2-D pressure formulation}\nIn this section we will explain the formulation of 2-D local pressure method by reformulating the 3-D weight function which will significantly reduce the computational cost without losing any desired details in the results. Typically, a 3-D local pressure method requires $N^2 \\times N_X \\times N_Y \\times N_Z \\times N_B$ operations ($N$ is the number of atoms; $N_X$, $N_Y$ and $N_Z$ are the number of grid points along x, y and z-directions respectively; $N_B$ is the number of discrete points for bond function integration). Here, the first term $N^2$ is the cost of inter-atomic pair potential force determination, which can be reduced to $O(N)$ using cell list algorithms \\cite{26welling2011efficiency}. This will make the 3-D pressure estimation cost as $N \\times N_X \\times N_Y \\times N_Z \\times N_B$ as shown in the Fig. \\ref{figure1}a. \n\n\\begin{figure}[!httbp]\n\\includegraphics[width=1\\linewidth]{figure1.png}\n\\caption{Local pressure estimation in 3-D and 2-D grids. (a) Schematic of pressure estimation in a 3-D grid from a molecular system. (b) Estimation of pressure in a 2-D grid by averaging the 3-D grid data (This is the traditional approach). (c) Direct estimation of pressure in 2-D grids from the MD simulation system (this work).}\n\\label{figure1} \n\\end{figure}\n\nWhile estimating the pressure in a 2-D grid, traditionally, the pressure in the 3-D grid is averaged to obtain it as seen in the Fig. \\ref{figure1}b. This expensive step will become unnecessary if we can directly estimate the pressure in 2-D grids as shown in Fig. \\ref{figure1}c. Though it looks like a trivial case, the results are very promising by reducing the computational effort to $N \\times N_X \\times N_Z \\times N_B$. In this work, we propose that while extending the pressure estimation theory to a 2-D grid, the spherical distribution volume has to be changed to a cylindrical volume as shown in Fig. \\ref{figure2}a. This is the case with most of the 2-D non-homogeneous systems. \n\nThe thermodynamic property variations along the y-axis is considered unchanged over long period of time and hence the $r(x,z)$ depends only on $x$ and $z$. \nThe resulting 2-D weight function will follow:\n\\begin{equation}\n\\int_{R^3} w(r)dr^3 = \\int_0^{r_s} 2\\pi r D C_1 [1 - 3r^2\/r_s^2 + 2r^3\/r_s^3] dr = 1\n\\label{eq11}\n\\end{equation}\n\nHere, $D$ is the depth of the system along $Y$ (direction of homogeneity) as shown in Fig. \\ref{figure2}a, $r_s$ is the \\textit{spread} radius and the constant of integration is $C_1 = 10\/3 \\pi D r_s^2$. \n\nFigure \\ref{figure2}b shows the variation of bond function for a pair of atoms in the case of 2-D system kept at $1.5~ nm$ apart. The isometric view shows the variation of magnitude of bond function for a spread radius of $0.5~ nm$. \n\n\\begin{figure}[!httbp]\n\\includegraphics[width=1\\linewidth]{figure2.png}\n\\caption{Weight and bond function developed for 2-D pressure formulation. (a) Cylindrical volume of influence associated with an atom located at , where is the spread radius, L, D, H are length, depth and height respectively. (b) Visualization of bond function for two atoms separated at a distance of 1.5 nm. The gradient image (upper) shows variation along surface and the contour plot of the same is shown below.}\n\\label{figure2} \n\\end{figure}\n\nThus, based on the new weight function, the density of the system ($\\rho$) is defined as:\n\\begin{equation}\n\\rho(r_p) = \\sum_{i=1}^{N} m_i \\hat{w}(r_i - r_p)\n\\label{eq12}\n\\end{equation}\nlocal number density at a grid point is\n\\begin{equation}\nn(r_p) = \\sum_{i=1}^{N} \\hat{w}(r_i - r_p)\n\\label{eq13}\n\\end{equation}\nand temperature as \n\\begin{equation}\nT(r_p) = \\sum_{i=1}^{N} \\frac{m_i v_i \\circledast v_i}{3 n(r_p) k_B} \\hat{w}(r_i - r_p)\n\\label{eq14}\n\\end{equation}\n\nThe selection of our interpolation function is arbitrary to demonstrate the 2-D formulation and instead any of the popular functions can be used. With that in mind, we have formulated 2-D forms for some selected functions, along with their 3-D functions are shown below.\n\n\\noindent Quadratic:\n\\begin{align}\n&\\hat{w}_{3D}(r) = \\frac{15(1-r^2\/r_s^2)}{8 \\pi r_s^3}\\\\\n&\\hat{w}_{2D}(r) = \\frac{2(1-r^2\/r_s^2)}{D \\pi r_s^2}\n\\label{eq15}\n\\end{align}\n\n\\noindent Exponential:\n\\begin{align}\n&\\hat{w}_{3D}(r) = \\frac{2.2671}{ r_s^3} exp(\\frac{r_s^2}{r^2-r_s^2})\\\\ \n&\\hat{w}_{2D}(r) = \\frac{2.1435}{D r_s^2} exp(\\frac{r_s^2}{r^2-r_s^2})\n\\label{eq16}\n\\end{align}\n\n\\noindent Trignometric:\n\\begin{align}\n&\\hat{w}_{3D}(x,y,z) = \\frac{1}{ 8 r_s^3} (1+cos(\\frac{\\pi x}{r_s}))(1+cos(\\frac{\\pi y}{r_s}))(1+cos(\\frac{\\pi z}{r_s}))\\\\ \n&\\hat{w}_{2D}(x,z) = \\frac{1}{ 4D r_s^2} (1+cos(\\frac{\\pi x}{r_s}))(1+cos(\\frac{\\pi z}{r_s}))\n\\label{eq17}\n\\end{align}\n\nFor grid dependent and finite support weight functions like B-splines, a rectangular prism volume could be used instead of cylindrical volume.\n\n\\subsection*{1-D pressure formulation}\nFor completeness, we have also derived the 1-D variation of pressure and density which is very suitable for 1-D inhomogeneous systems like pressure in thin films, lipid bilayers etc. The $r(z)$ will now depend only on the z-axis and the x and y axis variations are assumed to be negligible over time. \n\\begin{equation}\n\\int_{R^3} w(r)dr^3 = \\int_0^{r_s} 2 L D C_1 [1 - 3r^2\/r_s^2 + 2r^3\/r_s^3] dr = 1\n\\label{eq18}\n\\end{equation}\n\nThis will give the integration constant as $C_1 = 1\/LDr_s$. This is also consistent with the derivation of Hardy stress \\cite{12hardy1982formulas} and will be shown with example results in the next section.\n\n\\subsection*{Results and discussion}\n\nIn order to demonstrate and validate the new 2-D pressure formulation, we apply it to study the pressure, surface tension and density variations of argon liquid films suspended in argon vapor using MD simulations. In our chosen example (argon liquid film suspended in vapor as shown in Fig. \\ref{figure3}a) and also for lipid bilayer \\cite{7vanegas2014importance}, the inhomogeneity is in two dimensions (say, $X$ and $Z$ axes) and there is no bulk density variation along the third dimension ($Y$ axis) over ensemble average. \n\n\\begin{figure}[!ht]\n\\includegraphics[width=1\\linewidth]{figure3.png}\n\\caption{Two-dimensional density and pressure profile in argon multiphase system using the new 2-D formulation. (a) A $10 ~nm$ thick argon film suspended with $7.5 ~nm$ thick vapor on both sides along the z-direction. Two-dimensional (b) density and (c) pressure distribution obtained for the system equilibrated at $90~K$. The saturation density (NIST data) corresponding to liquid (Liq) and vapor (Vap) are marked in the density plot colorbar, while the saturation pressure (Sat) corresponding to the saturated fluid at $90~ K$ (NIST data) is marked in the pressure plot colorbar showing good agreement with the simulation results.}\n\\label{figure3} \n\\end{figure}\n\n\\begin{figure*}[!ht]\n\\includegraphics[width=1\\linewidth]{figure4.png}\n\\caption{Sensitivity study of spread radius $r_s$ on bond function, pressure and density. (a) Pressure variation across the argon film for different values of $r_s$. \"IK 2 A\" is the case study using the established Irving-Kirkwood's modified 1-D implementation \\cite{11weng2000molecular} for comparison. The \"IK 2 A\" and the new 2-D formation based profiles show good agreement when the volume of smearing became comparable. (b) Density variation across the film for different values of which confirms that the overall system bulk properties is not affected by the spread radius. (c-f) Contour plots of bond function with $r_s$ ranging from $1 ~nm$, $0.5~ nm$, $0.3~ nm$ and $0.1 ~nm$ for two atoms kept $1.2~ nm$ apart in a $3 ~nm \\times 3 ~nm$ domain. The plots visually show how the bond function controls the spreading of the pressure and density across the grids for different spread radii.}\n\\label{figure4} \n\\end{figure*}\n\nThe computational domain is shown in Fig. \\ref{figure3}a. The argon liquid film is $10~ nm$ thick with $7.5~ nm$ thick argon vapor on either side along the z-direction. The X-Y cross section size is $5~ nm \\times 5~ nm$. Periodic boundary conditions are applied in all directions. The vapor and liquid domains in this molecular system are first equilibrated separately \\cite{11weng2000molecular} for $1000 ~ps$ in order to get a stable suspended film and are then brought them together. The system is then equilibrated for $1000 ~ps$ followed by production run for another $1000 ~ps$ on which statistical analysis is performed. The modified Stoddard-Ford LJ potential \\cite{24stoddard1973numerical} is used with argon \u2013 argon LJ parameters as $\\sigma_{Ar-Ar} = 0.34 ~nm$ and $\\epsilon_{Ar-Ar} = 1.005841 ~kJ\/mol$. The time step of velocity verlet integration is $5~ fs$ and the thermostat to keep temperature constant is chosen as velocity scaling algorithm. MD simulations for different temperatures, spread radius and cutoff radius were performed. A validated, self-written C++ molecular dynamics code is used for all simulations \\cite{daisy2016molecular}.\n\nIt is found that the thermodynamic properties like pressure of argon is best captured by using a cutoff radius of $5 \\sigma$ or greater \\cite{weng2000molecular}. This corresponds to $1.8 ~nm$ for argon and we have used the same for all the simulations presented in this work. In the literature, it is common to consider the cutoff radius $r_c$ of MD simulations and spread radius $r_s$ of local pressure calculation as the same. However, considering same cutoff and spread radius will lead to increased number of grid point influence, increasing the computational cost and also limits the finer local details. Hence, in this work, the dependency between spread radius and cutoff radius is removed and considered them as separate entities, which enables us to retain the accuracy of the simulation without introducing any artifacts by choosing a higher cutoff radius. Therefore, the spread radius can be adjusted to capture the localized effects as desired.\n\nUsing the developed 2-D formulation, the temporally averaged 2-D contours of density and pressure at $90 ~K$ are estimated and shown in Figs. \\ref{figure3}b and \\ref{figure3}c. The density and pressure results are compared with the saturation properties from NIST thermodynamic properties database \\cite{25lemmon2005thermophysical} and found to be in very good agreement, which highlights the accuracy of the pressure and density calculation in the new formulation. \n\n\\begin{figure*}\n\\includegraphics[width=1\\linewidth]{figure5.png}\n\\caption{Comparison of MD simulation results with the standard thermodynamic physical data from NIST \\cite{25lemmon2005thermophysical}. (a) 1-D density profile and (b) 1-D pressure profile, deduced from the new 2-D formulation method, plotted over the molecular simulation of argon film. The interface locations capture the expected change in pressure and density. Comparison of MD simulation results and thermodynamic data for (c) pressure vs. density, and (d) surface tension vs. temperature showing excellent agreement. Pressure is estimated by temporal and spatial averaging of vapor and liquid regions separately.}\n\\label{figure5} \n\\end{figure*}\n\nThe sensitivity of spread radius on pressure and density results is studied using the system shown in Fig. \\ref{figure3}a by varying the spread radius to $0.2 ~nm$, $1~ nm$ and $1.8 ~nm$ and estimating the 2-D properties of pressure and density. The 2-D values are then averaged along the axis to obtain a 1-D pressure and 1-D density profile varying along the z-direction as shown in Figs. \\ref{figure4}a and \\ref{figure4}b respectively. Alongside, the pressure and density calculation based on the already-established 1-D IK method \\cite{11weng2000molecular} with a slab thickness of 0.2 nm are also plotted. The results in Figs. \\ref{figure4}a and \\ref{figure4}b show that density and pressure smoothen and spreads to a larger area as the spread radius is increased. Also, when the spread radius is small and comparable to the slab thickness of IK method, both density and pressure matches very well. As expected, the bulk region (vapor only and liquid only) properties are found to be not sensitive to the spread radius since it primarily captures the local effects.\n\n\\begin{figure*}\n\\begin{center}\n\n\\noindent \\includegraphics[width=.9\\linewidth]{figure6.png}\n\\caption{Laplace pressure study in a cylindrical liquid argon droplet. (a) Molecular model of cylindrical argon in a 3-D periodic box. (b) Density of the system after ensemble averaging using our 2-D method. (c) Pressure of the system ensemble averaged using our 2-D method. (d, e) Pressure and density variation in the cylindrical Argon system using cylindrical coordinate conversion.}\n\\label{figure6} \n\n\\end{center}\n\\end{figure*}\n\n\n\nFurther, to understand the dependency of the bond function to the spread radius, the bond function for two atoms placed at $1.5 ~nm$ apart are plotted with varying spread radius of $1 ~nm$, $0.5 ~nm$, $0.3 ~nm$ and $0.1 ~nm$ (Figs. \\ref{figure4}c-f). The resulting images show an important result: the spread radius determines the degree of sharpness required to capture the local features as desired. Further, as long as the integral of bond function is unity and conserved, it does not give erroneous values for surface tension, density or pressure. However, care should be taken while selecting the grid cell size for smearing as the results may be less accurate when the spread radius becomes comparable to grid size (although the resulting artifacts can possibly be alleviated using finite support weight functions like B-Splines, which however needs further investigation).\n\n\nNext, we validate the 2-D pressure formulation by performing multiple simulations with varying temperature of the argon system ($90~ K$, $100 ~ K$, $110 ~ K$, $120 ~ K$, $130 ~ K$, and $140 ~ K$) and comparing the simulation results with the experimental thermodynamic properties of argon from NIST database \\cite{25lemmon2005thermophysical}. The spread radius and cutoff radius are chosen as $0.5 ~ nm$ and $1.8 ~ nm$ respectively for these simulations. We would like re-emphasize the fact that spread radius does not alter any continuum level quantities and the choice of $0.5 ~ nm$ as the spread radius is merely arbitrary. Thermodynamic quantities of pressure, density and surface tension are estimated using the developed 2-D methodology. The 2-D results are averaged along the x-direction to obtain a 1-D pressure and 1-D density profile varying along the z-direction. A visualization of pressure and density variation along the height of the domain is shown in Fig. \\ref{figure5}a and \\ref{figure5}b which is consistent with previous argon film studies \\cite{10lee2012pressure,11weng2000molecular}. The comparison of pressure vs. density and surface tension vs. temperature are plotted in Figs. \\ref{figure5}c and \\ref{figure5}d, respectively, and show very good agreement with the experimental data \\cite{25lemmon2005thermophysical}. \n\nSince the above system is inhomogeneous only in one-dimension, we performed another validation on a curvilinear system which is inhomogeneous in two-dimensions. We estimate the pressure difference in a cylindrical droplet as shown in Fig. \\ref{figure6}a, and compare the result with the classical Young-Laplace equation. The droplet is symmetric in the plane of the figure with a depth of $3 ~nm$ and has periodic boundary conditions in all directions with sides of $11 ~nm$ each. The droplet is equilibrated for $1000 ~ps$ and then production runs are done for another $2000 ~ps$. The pressure and density is estimated at every 20 steps and averaged using the method introduced in this work. However, during the course of the simulation, the center of the droplet may vary around the original location. In order to avoid a skewed averaging, center of mass of every data set is found and readjusted to the center of the domain before averaging. The resulting ensemble averaged density and pressure is shown in Figs. \\ref{figure6}b and \\ref{figure6}c respectively. The variation of the pressure and density from the center of the droplet towards outside is shown in Figs. \\ref{figure6}d and \\ref{figure6}e. \n\nThe excess pressure inside the drop is given by the classical Young-Laplace equation:\n\\begin{equation}\nP_{in}-P_{out}=\\frac{2\\gamma}{R}\n\\label{eq19}\n\\end{equation}\n\nwhere $P_{out}$ and $P_{in}$ are the outside and inside pressures of the drop, $\\gamma$ is the surface tension, and $R$ is the radius of the drop. The radius $R$ is estimated by identifying the interface using our interface detection algorithm \\cite{daisy2017robust, yd2015new, yesudasandaisy_2015}. All parameters in Eq. (\\ref{eq19}) are estimated independently from the MD simulations. For the system simulated, we obtain from the density profile, and the surface tension is estimated. In order to estimate the radial variation of the properties like normal pressure, density, tangential pressure and surface tension, we used the 2-D rotation matrix in combination with B-spline interpolation polynomials. The left hand side of Eq. (\\ref{eq19}) results in a value of $3.3~ MPa$, while the right hand side results in $2.5~ MPa$, and thus, is in good agreement with the Young Laplace equation. We expect the agreement to improve further for larger drop sizes (however, with added computational cost). These simulations confirm the validity and accuracy of the new 2-D formulation method developed and presented in this work.\n\n\\subsection*{Conclusions}\n\nIn conclusion, a grid based method for two-dimensional estimation of pressure and density was developed and validated. The methodology was applied to a suspended argon liquid film in argon vapor with varying temperatures, and results were in very good agreement NIST experimental database values. The method was also applied to the classical problem of pressure difference calculation in a cylindrical drop and the results were found to be in good agreement with the Young Laplace equation. Further, the dependency between spread radius and cutoff radius was disconnected which allows for high accuracy of the simulation by choosing a higher cutoff radius without introducing any artifacts. The spread radius can be adjusted to capture the localized effects in the system as desired. The developed method will be significantly faster (computationally) than the existing 3-D grid method, and can be very useful in determining stresses occurring in lipid bilayers and other systems where inhomogeneity exists only in two of the three dimensions. This work also supports the fact that for the conversion of virial stress to a continuum level property, both kinetic component and force component of virial stress should be considered.\n\n\n\\textbf{Nomenclature}\n\n\\begin{tabular}{ l l }\n1-D & One-dimensional\\\\\n2-D& Two-dimensional\\\\\n3-D & Three-dimensional\\\\\n$B_{ij}$& Bond function between $i^{th}$ and $j^{th}$ atoms\\\\\n$C_1$& Constant of integration\\\\\nD & Depth\\\\\n$F_{ij}$ & Force between $i^{th}$ and $j^{th}$ atoms\\\\\nH & Height\\\\\nIK & Irving-Kirkwood\\\\\nL & Length\\\\\nLJ & Lennard Jones\\\\\nMD & Molecular Dynamics\\\\\nN& Number of atoms\\\\\nP & Pressure\\\\\n$P_{in}$ & Pressure inside cylindrical drop\\\\\n$P_K$ & Kinetic component of pressure\\\\\n$P_{out}$ & Pressure outside cylindrical drop\\\\\n$P_V$ & Virial component of pressure\\\\\nR& Radius of cylindrical drop\\\\\nT& Temperature\\\\\n$k_B$& Boltzmann constant\\\\\n$kJ$& kilo Joules\\\\\n$m_i$ & Mass of $i^{th}$ atom\\\\\nn & Number density of atoms in $p^{th}$ grid point\\\\\nnm& nano meter\\\\\nps& pico second\\\\\n$r_i$ & Position coordinate of $i^{th}$ atom\\\\\n$r_p$ & Position coordinate of $p^{th}$ grid point\\\\\n$r_s$ & Spread radius\\\\\n$r_c$ & Cutoff radius\\\\\n$v_i$ & Velocity of $i^{th}$ atom\\\\\n$w$ & Weight function\\\\\n$\\rho$& Density\\\\\n$\\lambda$& Dummy integration variable\\\\\n$\\delta$ & Dirac Delta function\\\\\n$\\epsilon$& Lennard Jones energy well depth\\\\\n$\\sigma$& Lennard Jones zero energy distance\\\\\n\\end{tabular}\n\n\\textbf{Acknowledgment}. We acknowledge Prof. Xiantao Li of Penn State University for sharing computer code snippets and the helpful discussions with Dr. Maroo.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{section-intro}}\nBhabha scattering is the process employed to \nmeasure the luminosity at electron-positron colliders,\nbecause of its clear experimental signature.\nAt machines operating at a $1 - 10$ $\\rm{GeV}$ centre-of-mass energy $\\sqrt{s}$, \nthe relevant kinematic region is that of large-angle Bhabha scattering.\nSmall-angle Bhabha scattering, instead, is an invaluable\nluminosity monitor at high-energy colliders in the TeV region.\n\nIn order to minimize the luminosity error,\na precise theoretical computation of radiative\ncorrections to the Bhabha-scattering cross section is required.\nThe electroweak next-to-leading order (NLO) corrections\nto Bhabha scattering were computed\na long time ago in \\cite{Consoli:1979xw}.\nIn recent years, studies\nhave been focusing on pure quantum electrodynamics (QED) contributions\nbeyond the one-loop level.\nThe two-loop virtual corrections for massless\nelectrons were obtained in \\cite{Bern:2000ie}.\nHowever, this result was not immediately useful since the available Monte Carlo programs\nemploy a non-vanishing electron mass $m$.\n\nThe virtual and real second-order contributions to\nBhabha scattering, enhanced by factors of $\\ln (s \/ m^2)$\nand $\\ln^2 (s \/ m^2)$ were completed in \\cite{Arbuzov:1995vi,Arbuzov:1995vj,Arbuzov:1998du,Glover:2001ev}.\nThis result was recently improved in \\cite{Penin:2005kf,Penin:2005eh,Penin:2005iv},\nwhere the photonic non-logarithmic term\nwas evaluated at leading order in the ratio $m^2 \\slash s$.\nThe diagrams with fermion loops remained uncovered in this approach.\n\n\nAn important breakthrough in the field was the\nuse of the Laporta-Remiddi algorithm (\\cite{Laporta:1996mq,Laporta:2001dd}),\nin order to reduce the Bhabha-scattering\ncross section to a few Master Integrals\n(MIs).\nThe technique of differential equations\nproved useful in evaluating several MIs\n(see i.e. \\cite{Bonciani:2003cj,Czakon:2004wm,Czakon:2005gi}).\nThe results were represented \nin terms of Harmonic Polylogarithms (HPLs) introduced in \\cite{Remiddi:1999ew} or\nof Generalized Harmonic Polylogarithms (GPLs)\n(details in the context of Bhabha scattering can be found in\n\\cite{Czakon:2005jd} and in references therein).\nThese results led eventually to the exact result\nof \\cite{Bonciani:2004gi,Bonciani:2004qt} for the virtual and real next-to-next-to-leading\norder (NNLO) corrections\nto the Bhabha-scattering cross section\ninvolving one electron loop.\nNon-approximated expressions for all NNLO contributions,\nexcept for double box diagrams, can be found in\n\\cite{Bonciani:2005im,Bonciani:2006qu}.\nThe MIs for the loop-by-loop contributions were studied e.g. in \\cite{Fleischer:2006ht}. \n\n\\begin{table*}[htb]\n\\label{tab-boxes}\n\\caption{\nThe $N_f=1$ four-point master integrals entering the six basic two-loop box diagrams.\nNP denotes non-planar topologies, and references with a dagger give divergent parts only.}\n\\vspace{2mm}\n\\begin{tabular}{|l|c|c|c|c|c|c|l|}\n\\hline\nMI & B1 & B2 & B3 & B4 & B5 & B6& \\\\\n\\hline \\hline\n{\\tt B7l4m1} & + & -- & -- & -- & -- & -- & \\cite{Smirnov:2001cm,Czakon:2006pa} \\\\\n{\\tt B7l4m1N} & + & -- & -- & -- & -- & --\n&\\cite{Heinrich:2004iq,Czakon:2006pa}\n\\\\ \n{\\tt B7l4m2} & --\n& + & -- & -- & -- & -- & \n\\cite{Heinrich:2004iq,Czakon:2006pa}\n\\\\ \n{\\tt B7l4m2[d1-d3]} & -- & + & -- & -- & -- & --\n&\\cite{Czakon:2006pa}\n\\\\ \n{\\tt B7l4m3} & -- & -- &\n+ & -- & -- & -- & \nNP \\cite{Heinrich:2004iq}$^{\\dagger}$\n\\\\\n{\\tt B7l4m3[d1-d2]} & -- & -- & + & -- & -- & -- &\nNP\n\\\\\n\n\\hline\n{\\tt B6l3m1} & + & -- & + & -- & -- & -- &\\cite{Czakon:2006pa}\\\\\n{\\tt B6l3m1d} & + & -- & + & -- & -- & -- &\\cite{Czakon:2006pa}\\\\\n{\\tt B6l3m2} & -- & + & -- & + & -- & -- &\\cite{Czakon:2006pa}\\\\\n{\\tt B6l3m2d} & -- & + & -- & + & -- & -- &\\cite{Czakon:2006pa}\\\\\n{\\tt B6l3m3} & -- & -- & + & -- & -- & -- &\nNP\n\\\\\n{\\tt B6l3m3[d1-d5]} & -- & -- & + & -- & -- & -- &\nNP\n\\\\\n\\hline\n{\\tt B5l2m1} & + & -- & + & -- & -- & -- & \\cite{Czakon:2004tg} \\\\\n{\\tt B5l2m2} & -- & + & -- & + & -- & +\n&\\cite{Czakon:2006pa}\\\\ \n{\\tt B5l2m2[d1-d2]} & -- & + & -- & + & -- & +\n&\\cite{Czakon:2006pa}\\\\ \n{\\tt B5l2m3} & + & -- & + & -- & -- & -- &\\cite{Czakon:2006pa}\\\\\n{\\tt B5l2m3[d1-d3]} & + & -- & + & -- & -- &\n--&\\cite{Czakon:2006pa}\\\\ \n{\\tt B5l3m} & -- & + & + & + & -- & -- &\\cite{Czakon:2006pa}\\\\\n{\\tt B5l3m[d1-d3]}& -- & + & + & + & -- & -- &\\cite{Czakon:2006pa}\\\\\n{\\tt B5l4m} & -- & + & + & + & + & -- & \\cite{Bonciani:2003cj} \\\\\n{\\tt B5l4md} & -- & + & + & + & + & -- & \\cite{Czakon:2004tg}\n\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\nThe complete set of the needed master integrals is known from\n\\cite{Czakon:2004wm}.\nTable \\ref{tab-boxes} reproduces all two-loop box master integrals for $N_f = 1$.\nNotations are exactly those of \\cite{Czakon:2004wm}.\nThe {\\tt B7l4m3d2} is, e.g., a box MI with 7 internal lines (7l), four of them being massive (4m), with a higher power of one of the numerators (a line being dotted (d));\nit is one of several such topologies and of several ones with dots, so '3d2'.\nIn order to improve the Bhabha-scattering theoretical prediction,\nwe investigate two classes of NNLO QED corrections.\nIn Section \\ref{section-planar},\nwe briefly discuss a method based on expansion of Mellin-Barnes (MB)\nrepresentations (\\cite{Usyukina:1975yg,Boos:1991rg,Smirnov:1999gc,Tausk:1999vh})\nand review the results of \\cite{Czakon:2006pa},\nwhere all planar two-loop box MIs were obtained.\nThe non-planar MIs are indicated in Table \\ref{tab-boxes}.\nIn Section \\ref{section-nf2}, we apply the same method\nto evaluate the MIs arising from diagrams containing \nheavy fermions, like muons and tau-leptons;\nin the following, we will call them the $N_f > 1$ contributions.\nTheir topologies are shown in Figure \\ref{fig-nf2}.\n\nThe MB-representations are valid for arbitrary kinematics.\nAlthough their actual evaluations are restricted to the high-energy limit (small lepton masses at fixed scattering angle), \nthey are well suited for practical applications.\nWhen dealing with $N_f > 1$ MIs,\na second fermion mass $M$ is involved. \nSince our\npurpose is to evaluate the complete QED Bhabha-scattering cross section at high\nenergies, we assume a hierarchy of all three\nscales, namely $m \\ll M \\ll s,t$, where $t$ is the usual\nMandelstam invariant related to the scattering angle. \nWith the summation techniques described in Section \\ref{section-techn}, divergent parts have been evaluated exactly.\nNote that the\ntreatment of hadronic contributions is a separate problem, which is\nbetter solved by using dispersion relations\n(see \\cite{Kniehl:1988id}). \n\n\\begin{figure}\n\\label{fig-nf2}\n\\includegraphics[height=9cm,width=9.cm,angle=0,scale=0.8]{top.eps}\n\\vspace*{-0.3cm}\n\\caption{The topologies of the eight master integrals\nfor the heavy-fermion corrections.\nBold lines represent heavy fermions.}\n\\end{figure}\n\n\\vfill\n\n\\begin{figure}[htbp]\n\\label{fig-diagnf2}\n\\vspace*{-0.1cm}\n\\hspace*{0.5cm}\n\\includegraphics[height=10.5cm,width=8.cm,angle=0,scale=0.8]{diagsNf2_thick.eps}\n\\vspace*{-0.9cm}\n\\caption{The diagrams for Bhabha scattering with two fermion flavors.\nInternal fermionic loops represent heavy leptons and the other fermion lines are electrons.}\n\\end{figure}\n\n\\section{The planar two-loop boxes for $N_f=1$\\label{section-planar}}\nThere are 24 planar two-loop box MIs to be determined for the $N_f=1$ case, see Table \\ref{tab-boxes}.\nSo far, we could not determine all of them analytically with the exact mass dependence; the status is reviewed in \\cite{Czakon:2006hb}.\nFor this reason, we decided to treat these MIs uniquely in the approximation of small $m$ at fixed scattering angle.\nThe results have been published in the mean time \\cite{Czakon:2006pa}, so we will make here only few introductory remarks and show one example.\nIn the list of MIs, we preferred to include Feynman integrals without any numerators.\nThe pragmatic reason was the independence of their defintion on the internal flow of momenta.\nIt is sufficient to indicate the lines with dots.\nPerforming explicit calculations, one is, of course, faced with the observation that the singularities of dotted MIs and those with numerators are quite different.\nEvaluating the small mass expansions, we preferred in some cases to treat instead of dotted integrals those with numerators.\nDue to unique algebraic relations between all the integrals, there is no principal difference in these two approaches, and further details are discussed in \\cite{Czakon:2006pa}.\nAnother observation concerns the MB-representations.\nIn principle, one may use the representations for the basic 7-line boxes as given e.g. in \\cite{Smirnov:book4} and shrink lines.\nHowever, when calculating MIs with numerators, additional representations have to be determined.\nWe observed further, that it is sometimes not evident how to get an effective represenation for dotted MIs from more general ones.\nFor the MI {\\tt B5l2m2} (a 5-liner) we got, by shrinking of lines in {\\tt B7l4m1} (first planar double box)and after expanding in $\\varepsilon$, 11 MB-integrals with at most 4 integrations.\nFrom our direct derivation, we got 4 integrals, at most 3-dimensional.\nFor the related dotted MI {\\tt B5l2m2d2}, we got from line shrinking 102 integrals, and by direct derivation only one, again 3-dimensional.\n\n\n\\begin{figure}\n \\begin{center}\n \\epsfig{file=FigNum.eps,width=6cm}\n \\end{center}\n \\caption{\\label{5lin}\nThe 5-line topology {\\tt B5l2m3}. The momentum distribution has been chosen to\nmake the derivation of the MB representation easier.\n}\n\\end{figure}\n\n\nAs an example, we reproduce here for the MI B5l2m3($k_2 p_3$), shown in Figure \\ref{5lin}, the basic $d$-dimensional MB-represenation:\n\\begin{eqnarray}\n{\\tt B5l2m3(p_e \\cdot k_2)} \n\\nonumber \\\\ \n=\n\\frac{ (-1)^{a_{12345}}\\;\ne^{2\\varepsilon\\gamma_E}}{ \\prod_{j=1}^5\\Gamma[a_{i}]\n \\Gamma[4 - 2\\varepsilon- a_{123} ](2\\pi i)^4}\n\\nonumber\\\\\n\\int_{-i \\infty}^{+i \\infty} d \\alpha \\int_{-i \\infty}^{+i \\infty}\nd \\beta \\int_{-i \\infty}^{+i \\infty} d \\gamma \\int_{-i \\infty}^{+i \\infty} d \\delta\n\\nonumber \\\\ \n(-s)^{\\gamma} \\;(-t)^{(4 - 2 \\epsilon - a_{12345} - \\beta - \\delta - \\gamma)} \\nonumber\n\\\\\n\\Gamma[-\\beta] \\;\\Gamma[-\\gamma] \\;\\Gamma[-\\delta] \\; \\Gamma[a_3 + \\alpha + 2 \\;\\beta] \\; \\nonumber\\\\\n\\frac{\\Gamma[2 - \\epsilon - a_{45} + \\alpha - \\delta - \\gamma]}\n{\\Gamma[7 - 3 \\epsilon- a_{12345} - \\beta] } \\nonumber \\\\\n\\frac{\\Gamma[2 - \\epsilon - a_{13} - \\beta ]\n\\;\\Gamma[2- \\epsilon - a_{23} - \\alpha - \\beta ]}\n{ \\Gamma[a_5 - \\alpha + 2 \\;\\gamma] \\;\\Gamma[1 + a_5 - \\alpha + 2 \\;\\gamma]} \\nonumber \\\\\n \\Gamma[-4 + 2 \\;\\epsilon + a_{12345} + \\beta + \\delta + \\gamma] \\nonumber \\\\\n\\biggl\\{ \\Gamma[4 - 2 \\;\\epsilon - a_{1235} - \\beta - \\delta - \\gamma] \\nonumber \\\\\n \\biggl[ (p_e \\cdot p_2) \\;\\Gamma[1 + a_5 + \\gamma] \\;\\Gamma[-\\alpha + \\gamma] \n\\nonumber\\\\\n- (p_e \\cdot p_1) \\;\\Gamma[a_5 + \\gamma] \\;\\Gamma[1 - \\alpha + \\gamma] \\biggr]\n\\nonumber \\\\\n\\Gamma[a_5 - \\alpha + 2 \\;\\gamma] \\;\\Gamma[1 + a_5 - \\alpha + 2 \\;\\delta + 2 \\;\\gamma] \\nonumber \\\\\n+\n [(p_e \\cdot p_3) - (p_e \\cdot p_1)] \\;\n\\nonumber \\\\\n\\Gamma[5- 2 \\epsilon - a_{1235} - \\beta - \\delta - \\gamma] \\nonumber \\\\\n\\Gamma[a_5 + \\gamma] \\;\\Gamma[-\\alpha + \\gamma] \\;\\Gamma[1 + a_5 - \\alpha + 2 \\;\\gamma] \\;\n\\nonumber\\\\\n \\Gamma[a_5 - \\alpha + 2 \\;(\\delta + \\gamma)] \\biggr\\}\\nonumber\n\\label{numB5l2m3}\n\\end{eqnarray}\n\nThe small mass expansion of the result is:\n\\begin{eqnarray}\n{\\tt B5l2m3(k_2\\cdot p_2)} \n =\\frac{1}{4}\\left(\\frac{s}{u}\\right)^2 \\biggl\\{\n{\\rm L}^2\\;(6\\;x\\;\\zeta_2 \n\\nonumber \\\\\n+ 2\\;x\\;\\ln(x) + 2\\;x^2\\;\\ln(x)\n+ x\\;\\ln^2(x)) \\nonumber \\\\\n+{\\rm L}\\;(16\\;x\\;\\zeta_2 - 8\\;x^2\\;\\zeta_2\n\\nonumber \\\\\n - 4\\;x\\;\\zeta_3 -\n2\\;\\ln(x) + 2\\;x^2\\;\\ln(x) \\nonumber \\\\\n+4\\;x\\;\\zeta_2\\;\\ln(x) + 2\\;x\\;\\ln^2(x) - 2\\;x^2\\;\\ln^2(x)\n\\nonumber \\\\\n -\n 12\\;x\\;\\zeta_2\\;\\ln(1 + x) \\nonumber \\\\\n-2\\;x\\;\\ln^2(x)\\;\\ln(1 + x)\n- 4\\;x\\;\\ln(x)\\;{{\\rm{Li_2}}}( -x)\n\\nonumber \\\\\n + 4\\;x\\;{{\\rm{Li_3}}}( -x)) \\biggr\\} \\nonumber \\\\\n+ \\frac{1}{120} \\left(\\frac{s}{u}\\right)^2 \\biggl\\{ +120\\;\\zeta_2\n \\nonumber \\\\\n+ 360\\;x\\;\\zeta_2\n\\nonumber \\\\\n- 120\\;x^2\\;\\zeta_2 - 1560\\;x\\;\\zeta_4 - 480\\;x\\;\\zeta_3\n\\nonumber \\\\\n-240\\;x^2\\;\\zeta_3 - 240\\;x\\;\\zeta_2\\;\\ln(x)\n\\nonumber \\\\\n- 480\\;x^2\\;\\zeta_2\\;\\ln(x) - 360\\;x\\;\\zeta_3\\;\\ln(x)\n\\nonumber \\\\\n+30\\;\\ln^2(x) + 60\\;x\\;\\ln^2(x) - 30\\;x^2\\;\\ln^2(x)\n\\nonumber \\\\\n -\n 180\\;x\\;\\zeta_2\\;\\ln^2(x) \n\\nonumber \\\\\n-20\\;x\\;\\ln^3(x) - 20\\;x^2\\;\\ln^3(x) - 5\\;x\\;\\ln^4(x)\n\\nonumber \\\\\n+ 720\\;x^2\\;\\zeta_2\\;\\ln(1 + x)\n\\nonumber \\\\\n +120\\;x\\;\\zeta_3\\;\\ln(1 + x)\n\\nonumber \\\\\n - 120\\;x\\;\\zeta_2\\;\\ln(x)\\;\\ln(1 + x)\n\\nonumber \\\\\n +\n120\\;x^2\\;\\ln^2(x)\\;\\ln(1 + x) \\nonumber \\\\\n+\n 180\\;x\\;\\zeta_2\\;\\ln^2(1 + x) \n\\nonumber \\\\\n+ 30\\;x\\;\\ln^2(x)\\;\\ln^2(1 + x)\n \\nonumber \\\\\n+\n120\\;x\\;\\ln(x)\\;{\\rm{S_{1,2}}} \n(-x) \\nonumber \\\\\n+60\\;x\\;(-8\\;\\zeta_2 - \\ln^2(x) + 2\\;\\ln(x)\\;\n\\nonumber \\\\\n(2\\;x + \\ln(1 + x)))\\;{{\\rm{Li_2}}}( -x)\n\\nonumber \\\\\n- 240\\;x^2\\;{{\\rm{Li_3}}}( -x) +\n 240\\;x\\;\\ln(x)\\;{{\\rm{Li_3}}}( -x)\n\\nonumber \\\\\n - 120\\;x\\;\\ln(1 + x)\\;{{\\rm{Li_3}}}( -x)\n\\nonumber \\\\\n-360\\;x\\;{{\\rm{Li_4}}}( -x) - 120\\;x\\;{{\\rm{S_{2,2}}}}(-x)\n\\biggr\\}\\nonumber\n\\label{B5l2m2N1ms}\n\\end{eqnarray}\nThe expression is more complicated than those for the planar 7-line MIs concerning both the functions appearing as well as the dependence on all three Mandelstam variables $s,t,u$; the latter is typical for non-planar diagrams, where the {\\tt B5l2m3} topology contributes. \n\n\n\\section{The master integrals for the $N_f > 1$ corrections\\label{section-nf2}}\nThe differential Bhabha-scattering cross section with respect\nto the solid angle $\\Omega$ can be written\nby means of an expansion in the fine-structure constant $\\alpha$,\n\\begin{equation}\n\\frac{d \\sigma}{d \\Omega}= \\frac{d \\sigma_0}{d \\Omega} + \n\\left(\\frac{\\alpha}{\\pi}\\right) \\frac{d \\sigma_1}{d \\Omega} +\n\\left(\\frac{\\alpha}{\\pi}\\right)^2 \\frac{d \\sigma_2}{d \\Omega}+\\ldots,\n\\end{equation}\nwhere $\\sigma_0$ is the Born contribution and $\\sigma_i$ ($i=1,2,\\ldots$)\nrepresent the higher-order radiative corrections.\nIf we are interested in the NNLO virtual contributions,\nwe need to evaluate the diagrams of Figure 2,\nwhere the fermion self-energy is required for wave-function\nrenormalization.\nNote that results for the photonic vacuum polarization diagrams\ncan be found in \\cite{Kallen:1955fb}.\n\nAfter interfering the two-loop amplitude with\nthe tree-level one, summing over the spins of the\nfinal state and averaging over those of the initial\nstate, we get a large number of integrals.\nWe use the \\texttt{DiaGen\/IdSolver} \\cite{Czakon:2004uu2}\nimplementation of the Laporta-Remiddi algorithm\n\\cite{Laporta:1996mq} in order to reduce all the needed\nFeynman integrals to a limited set of MIs.\nApart from products of one-loop integrals,\nwe get the eight MIs of Figure \\ref{fig-nf2}, as already pointed out in \\cite{Czakon:2004wm}.\nThe corresponding Feynman integrals with $n$ propagators $D_i$ are defined as follows:\n\\begin{equation}\n\\label{feynmi}\nD(\\{\\nu_i\\}_n) = -\n\\frac{(e^\\gamma)^{2\\varepsilon}}\n{\\pi^d}\\int\\frac{d^dk_1d^dk_2}{\\prod_{i}^{n}D_i^{\\nu_i}},\n\\end{equation}\nwhere $\\gamma$ is the Euler-Mascheroni constant and we introduced\nthe shorthand notations\n\\begin{eqnarray}\n\\{\\nu\\}_n&\\equiv& \\nu_1,\\ldots,\\nu_n,\\nonumber\\\\\n\\nu_{ab\\ldots c}&\\equiv&\\nu_{a}+ \\nu_{b} +\\ldots + \\nu_{c}.\n\\end{eqnarray}\n\nIn contrast to \\cite{Czakon:2006pa}, we do not consider MIs with \nscalar products in the numerators.\nWe have then to allow for higher powers of propagators.\nOf course, there are algebraic relations between\nMIs with scalar products in the numerators\nand MIs with propagators raised to higher powers.\n\nWe construct our MB representations using the standard approach described in\n\\cite{Smirnov:book4}. The Feynman-parameter\nintegrals are derived one after the other for the two subloops. \nIn each step we replace the sum over monomials in the Feynman parameters\nby an appropriate MB representation. Due to the relatively simple structure of the\nconsidered diagrams, it is easier to begin with the propagator-type subloop.\n\nAs far as the electron self-energy is concerned, we only need the sunrise\ntopology with the electron on its mass shell, $p_1^2=m^2$. \nA number of results for this mass configuration of the sunrise diagram can be found in the\nliterature. \nAnalytic expressions for the residues of the poles in dimensional regularization \nand the finite parts were given already\nin \\cite{Berends:1998vk}.\nThe explicit result for the ${\\cal O} (\\epsilon)$ terms, where $\\epsilon \\equiv (4-d)\/2$\nand $d$ are the space-time dimensions,\ncan be found in \\cite{Argeri:2002wz}.\nNote that the inclusion of the ${\\cal O} (\\epsilon)$ terms\nfor the sunrise MIs is mandatory when\nderiving the complete squared amplitude,\nsince the reduction to MIs generates inverse powers of $\\epsilon$.\n\nFor the self energy, we use our MB representation in order to reproduce the \nknown result for the MIs.\nIts general form, for arbitrary powers $\\nu_i$ of\nthe propagators, is given by\n\\begin{eqnarray}\\label{def:SUN}\n&&\\texttt{SE3l2M1m}(\\{\\nu\\}_3) =\n(m^2)^{4-\\nu_{123}-2 \\epsilon}\n\\nonumber \\\\\n&&\n\\times \\frac{(-1)^{\\nu_{123}} e^{2 \\gamma \\epsilon}}{\\prod_{i=1}^3 \\Gamma({\\nu_i})}\n \\int \\frac{dz}{2\\pi i} \\left( \\frac{m^2}{M^2}\\right)^{z+\\nu_{12}-2+\\epsilon}\n\\nonumber\\\\\n&&\\times\\frac{\\prod_{i=1}^6 \\Gamma_i}\n{\\Gamma(2 z + \\nu_{12}) \\Gamma(z-\\nu_3+4-2 \\epsilon)}.\n\\end{eqnarray}\nFurthermore, we defined\n\\begin{xalignat}{2}\n\\Gamma_1 &\\equiv\\Gamma(-z), \\nonumber\\\\\n\\Gamma_2&\\equiv\\Gamma(z+\\nu_1), \\nonumber\\\\\n\\Gamma_3&\\equiv\\Gamma(z+\\nu_2), \\nonumber\\\\ \n\\Gamma_4&\\equiv\\Gamma(-z+\\nu_3-2+\\epsilon), \\nonumber\\\\\n\\Gamma_5&\\equiv\\Gamma(2 z-\\nu_3+4-2\\epsilon), \\nonumber\\\\ \n\\Gamma_6&\\equiv\\Gamma(z+\\nu_{12}-2+\\epsilon).\n\\end{xalignat}\nThe integration contour is a straight line parallel\nto the imaginary axis separating the poles\ngenerated by $\\Gamma_1$ and $\\Gamma_4$ from\nthose coming from $\\Gamma_2$, $\\Gamma_3$, $\\Gamma_5$ and $\\Gamma_6$.\n\nThe two sunrise MIs are defined by the following values\nfor the powers of the propagators,\n\\begin{eqnarray}\n\\texttt{SE3l2M1m}&\\equiv&\\texttt{SE3l2M1m}(1,1,1),\\nonumber\\\\\n\\texttt{SE3l2M1md}&\\equiv&\\texttt{SE3l2M1m}(1,2,1).\n\\end{eqnarray}\n\nHaving a MB representation at hand, one needs\nto perform an analytic continuation in $\\varepsilon$\nfrom a range where the integral is regular to the vicinity\nof the origin, uncovering the singular structure on the way.\nThis is done\nby an automatized procedure implemented in the\nMathematica package {\\tt MB.m} \\cite{Czakon:2005rk}.\nThe resulting MB representations are verified\nnumerically in the Euclidean region against the sector\ndecomposition approach as described in \\cite{Binoth:2003ak}.\n\nFor the sunrise MIs, a straightforward application of the Cauchy theorem\nto the MB representation of Eq.~\\eqref{def:SUN}\nleads to a sum over residua which can be easily\nevaluated.\nTherefore, we reproduced the results of \\cite{Argeri:2002wz}.\nIn general, however, one has to deal with multiple\nMB representations.\nFor the vertex and box MIs, the evaluation of\nthe needed sums is far from being trivial.\n\nAs explained in the introduction, our purpose is\nto calculate the integrals by assuming a hierarchy\nof all scales, namely $m^2 \\ll M^2 \\ll s,t$.\nFirst of all we \nidentify the leading contributions in the electron mass\nfollowing the procedure described in \\cite{Czakon:2006pa}.\nThen, by using the Cauchy theorem to express\nthe integrals through sums over residua,\nwe evaluate these sums with the aid of {\\tt XSUMMER} \\cite{Moch:2005uc}.\n\nFor the sunrise MIs the results depend on one variable,\n$R\\equiv m^2 \\slash M^2$,\nand read as\n\\begin{eqnarray}\n\\texttt{SE3l2M1m}\\ &=& \\ M^2\\ (m^2)^{-2 \\epsilon} \n\\nonumber\\\\\n&&\\times \\Bigl[ \\sum_{k=-2}^{1}\\ S_k\\ \\epsilon^k\\ +\\ {\\cal O} (\\epsilon^2) \\Bigr], \n\\nonumber\\\\\nS_{-2}&=& 1 , \n\\\\\n S_{-1}&=& 3 + 2 \\ln \\left(R\\right) ,\\nonumber\\\\\n S_0 &=&\n7 + \\zeta_2 + 6 \\ln \\left(R\\right) + 2 \\ln^2 \\left(R\\right) , \\nonumber\\\\\nS_1\n&=&\n15 + 3\\zeta_2\n - \\frac{2}{3}\\zeta_3+ \\left(14 + 2\\zeta_2\\right) \n\\nonumber \\\\\\nonumber\n&& \\times \\ln \\left(R\\right)\n+ 6 \\ln^2 \\left(R\\right) +\\frac{4}{3} \\ln^3 \\left(R\\right), \n\\end{eqnarray}\n\\begin{eqnarray}\n\\texttt{SE3l2M1md} &=&\n(m^2)^{-2 \\epsilon}\n\\Bigl[ \n\\sum_{k=-2}^{1} S^d_k \\epsilon^k + {\\cal O} (\\epsilon^2) \n\\Bigr], \n\\nonumber\\\\\n S_{-2}^d &=& \\frac{1}{2}, \\\\\n S^d_{-1}&=& \\frac{1}{2} \\Bigl[1+ 2\\ln \\left(R\\right)\\Bigr], \\nonumber \\\\\n S_0^d &=& \\frac{1}{2}\\left(1 + \\zeta_2\\right) +\\ln \\left(R\\right) + \\ln^2 \\left(R\\right),\\nonumber \\\\\n S_1^d&=&\n\\frac{1}{6} \\left(3 + 3\\zeta_2 - 2\\zeta_3\\right)\n+ \\left(1 + \\zeta_2\\right)\n\\nonumber \\\\ \\nonumber\n&& \\times \\ln \\left(R\\right) \n+\\ln^2 \\left(R\\right) + \\frac{2}{3} \\ln^3 \\left(R\\right).\n\\end{eqnarray}\n\nFor vertices, the external electrons are on their mass shell,\n$p_i^2 = m^2$, $i=1,2$, and we introduce the Mandelstam invariant $s \\equiv (p_1+p_2)^2$\n(see Figure \\ref{fig-nf2}).\nSince each of the two vertices is related to two MIs,\nwe have to consider\n\\begin{eqnarray}\\label{def:vert}\n\\texttt{V4l2M1m} &\\equiv&\\texttt{V4l2M1m}(1,1,1,1),\\nonumber \\\\\n\\texttt{V4l2M1md}&\\equiv&\\texttt{V4l2M1m}(1,1,1,2),\\nonumber\\\\\n\\texttt{V4l2M2m} &\\equiv&\\texttt{V4l2M2m}(1,1,1,1),\\nonumber \\\\\n\\texttt{V4l2M2md}&\\equiv&\\texttt{V4l2M2m}(1,2,1,1).\n\\end{eqnarray}\n\nWe follow the same strategy employed for the sunrise diagrams.\nFirst of all we derive the exact multi-dimensional MB representation,\nand then we perform first a small-mass expansions in $m_s$, and then in $M_s$,\ndefined as the ratios of the fermion masses and the centre-of-mass\nenergy, $m_s \\equiv -m^2 \\slash s$,\n$M_s \\equiv -M^2 \\slash s$.\nThe MB representations read as\n\\begin{multline}\n\\texttt{V4l2M1m} (\\{\\nu\\}_4) =\n(m^2)^{4-\\nu_{1234}-2 \\epsilon} \n\\\\\n\\times \\frac{(-1)^{\\nu_{1234}} e^{2 \\gamma \\epsilon}}{\\prod_{i=1}^{4}\n\\Gamma({\\nu_i})}\n\\int \\frac{dz_1 dz_2}{(2\\pi i)^2}\n\\\\\n\\times m_s^{z_2-4+\\nu_{1234}+2\\epsilon}\\,\n M_s^{-z_1+2-\\nu_{12}-\\epsilon}\n\\\\\n\\times\\frac{\\prod_{i=1}^8 \\Gamma_i}{\\Gamma(2 z_1+\\nu_{12})\n\\Gamma(z_1-\\nu_{34}+4-2\\epsilon)},\\qquad\n\\end{multline}\nwith\n\\begin{xalignat}{2}\n\\Gamma_1&\\equiv\\Gamma(z_1+\\nu_1),\\nonumber\\\\\n\\Gamma_2&\\equiv\\Gamma(z_1+\\nu_2),\\nonumber\\\\\n\\Gamma_3&\\equiv\\Gamma(z_1+\\nu_{12}-2+\\epsilon),\\nonumber\\\\\n\\Gamma_4&\\equiv\\Gamma(-z_2),\\nonumber\\\\\n\\Gamma_5&\\equiv\\Gamma(2 z_2+\\nu_4),\\nonumber\\\\\n\\Gamma_6&\\equiv\\Gamma(-z_2-\\nu_{34}+2-\\epsilon),\\nonumber \\\\\n\\Gamma_7&\\equiv\\Gamma(z_1-z_2-\\nu_4+2-\\epsilon),\\nonumber\\\\\n\\Gamma_8&\\equiv\\Gamma(-z_1+z_2+\\nu_{34}-2+\\epsilon),\n\\end{xalignat}\nand\n\\begin{multline}\n\\texttt{V4l2M2m}(\\{\\nu\\}_4)=\n(m^2)^{4-\\nu_{1234}-2 \\epsilon}\n\\\\\n\\times \\frac{ (-1)^{\\nu_{1234}}e^{2 \\gamma \\epsilon}}\n{\\prod_{i=1}^4 \\Gamma({\\nu_i})} \\int \\frac{dz_1 dz_2}{(2\\pi i)^2}\n\\\\\n\\times m_s^{z_2-4+\\nu_{1234}+2\\epsilon} \\, M_s^{-z_1+2-\\nu_{12}-\\epsilon}\n\\\\\n\\times \\frac{\\prod_{i=1}^9 \\Gamma_i}\n{\\prod_{j=10}^{12} \\Gamma_j},\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\n\\end{multline}\nwith\n\\begin{xalignat}{2}\n\\Gamma_1&\\equiv\\Gamma(-z_1),\\nonumber\\\\\n\\Gamma_2&\\equiv\\Gamma(z_1+\\nu_1),\n\\nonumber\\\\\n\\Gamma_3&\\equiv\\Gamma(z_1+\\nu_2),\\nonumber\\\\\n\\Gamma_4&\\equiv\\Gamma(z_1+\\nu_{12}-2+\\epsilon),\n\\nonumber\\\\\n\\Gamma_5&\\equiv\\Gamma(2 z_1-\\nu_{34}+4-2 \\epsilon),\\nonumber\\\\\n\\Gamma_6&\\equiv\\Gamma(-z_2),\n\\nonumber\\\\\n\\Gamma_7&\\equiv\\Gamma(z_1-z_2-\\nu_3+2-\\epsilon),\\nonumber\\\\\n\\Gamma_8&\\equiv\\Gamma(z_1-z_2-\\nu_4+2-\\epsilon),\n\\nonumber\\\\\n\\Gamma_9&\\equiv\\Gamma(-z_1+z_2+\\nu_{34}-2+\\epsilon),\\nonumber\\\\\n\\Gamma_{10}&\\equiv\\Gamma(2 z_1+\\nu_{12}),\\nonumber\\\\\n\\Gamma_{11}&\\equiv\\Gamma(z_1-\\nu_{34}+4-2\\epsilon),\\nonumber\\\\\n\\Gamma_{12}&\\equiv \\Gamma(2z_1-2z_2-\\nu_{34}+4-2\\epsilon).\n\\end{xalignat}\n\nA careful analysis\nof the powers of $m_s$ and $M_s$ under the MB integrals\nleads to the following results,\n\\begin{eqnarray}\n\\texttt{V4l2M1m} &=&\n(m^2)^{-2 \\epsilon} \n\\Bigl[ \\sum_{k=-2}^0 V^1_{k} \\epsilon^k + {\\cal O}(\\epsilon) \\Bigr], \\nonumber\\\\\nV^{1}_{-2} &=& \\frac{1}{2},\\nonumber\\\\\nV^{1}_{-1} &=& \\frac{5}{2},\\\\\nV^{1}_{0} &=& \\frac{1}{2}\\left[ 19 - 3\\zeta_2 - \\ln^2(m_s) \\right],\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n\\texttt{V4l2M1md} &=& \n \\frac{(m^2)^{-2 \\epsilon}}{m^2}\n\\Bigl[ \\sum_{k=-2}^0 V^{1d}_{k} \\epsilon^k + {\\cal O}(\\epsilon) \\Bigr], \\nonumber\\\\\nV^{1d}_{-2} &=& \\frac{1}{2},\\nonumber\\\\\nV^{1d}_{-1} &=& 1+\\frac{1}{2} \\ln(m_s) ,\\\\\nV^{1d}_{0} &=& 2-\\zeta_2 +\\ln(m_s)+\\frac{1}{4}\\ln^2(m_s),\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n\\texttt{V4l2M2m} &=&\n(m^2)^{-2 \\epsilon} \n\\Bigl[ \\sum_{k=-2}^0 V^2_{k} \\epsilon^k + {\\cal O}(\\epsilon) \\Bigr], \\nonumber\\\\\nV^{2}_{-2} &=& \\frac{1}{2},\\nonumber\\\\\nV^{2}_{-1} &=& \\frac{5}{2} +\\ln(m_s) ,\\nonumber\\\\\nV^{2}_{0} &=& \\frac{1}{2}(19+\\zeta_2) + 5\\ln(m_s)\\nonumber\\\\\n& +& \\ln^2(m_s),\n\\end{eqnarray}\n\\begin{eqnarray}\n\\texttt{V4l2M2md} &=&\n(m^2)^{-2 \\epsilon} \\frac{1}{s} \n\\Bigl[ V^{2d}_{0} + {\\cal O}(\\epsilon) \\Bigr], \\nonumber\\\\\nV^{2d}_{0} &=&\n\\frac{1}{6}\\Bigl[ 12\\zeta_3 - 6\\zeta_2 \\ln(M_s)\\nonumber\\\\\n &-& \\ln^3(M_s) \\Bigr].\n\\end{eqnarray}\n\nFor box diagrams, the external momenta are again on their mass shell, \nand we have additionally $(p_1-p_3)^2 = t$.\nAfter introducing $M_t \\equiv - M^2 \\slash t$, \nthe appropriate MB representation is given by\n\\begin{multline}\n\\texttt{B5l2M2m}(\\{\\nu\\}_5)=\n(m^2)^{4-\\nu_{12345}-2 \\epsilon}\n\\\\\n\\times \\frac{(-1)^{\\nu_{12345}}e^{2 \\gamma \\epsilon}}\n{\\prod_{i=1}^{5} \\Gamma({\\nu_i})}\n \\int \\frac{dz_1 dz_2 dz_3}{(2\\pi i)^3}\\\\\n\\times M_t^{-z_1+2-\\nu_{12}-\\epsilon}\nm_s^{z_3-4+\\nu_{12345}+2 \\epsilon}\\\\\n\\times \\left(\\frac{t}{s}\\right)^{z_2-z_1+2-\\nu_{12}-\\epsilon}\n \\frac{\\prod_{i=1}^{11} \\Gamma_i}{\\prod_{j=12}^{14} \\Gamma_j },\\qquad\n\\end{multline}\nwith\n\\begin{xalignat}{2}\n\\Gamma_1&\\equiv\\Gamma(z_1+\\nu_1),\\\\\n\\Gamma_2&\\equiv\\Gamma(z_1+\\nu_2),\\nonumber\\\\\n\\Gamma_3&\\equiv\\Gamma(z_1+\\nu_{12}-2+\\epsilon),\\nonumber\\\\\n\\Gamma_4&\\equiv\\Gamma(-z_2),\\nonumber\\\\\n\\Gamma_5&\\equiv\\Gamma(z_2+\\nu_4), \\nonumber \\\\\n\\Gamma_6&\\equiv\\Gamma(-z_3),\\nonumber\\\\\n\\Gamma_7&\\equiv\\Gamma(-z_1+z_2),\n\\nonumber \\\\ \\nonumber\n\\Gamma_8&\\equiv\\Gamma(2 z_1\\!-\\!2 z_2-\\nu_{3445}+4-2 \\epsilon),\n\\nonumber\\\\\n\\Gamma_{9}&\\equiv\\Gamma(z_1-z_2-z_3-\\nu_{34}+2-\\epsilon),\\nonumber\\\\\n\\Gamma_{10}&\\equiv\\Gamma(z_1\\!-\\!z_2\\!-\\!z_3-\\nu_{45}+2-\\epsilon),\n\\nonumber\\\\\n\\Gamma_{11}&\\equiv\\Gamma(-z_1 +z_{2}+z_3+\\nu_{345}-2+\\epsilon),\\nonumber\\\\\n\\Gamma_{12} &\\equiv \\Gamma(2 z_1+\\nu_{12}),\n\\nonumber\\\\\n\\Gamma_{13} &\\equiv \\Gamma(z_1-\\nu_{345}+4-2 \\epsilon),\\nonumber\\\\ \\nonumber\n\\Gamma_{14} &\\equiv \\Gamma(2(z_1\\!-\\!z_2\\!-\\!z_3)-\\nu_{3445}+4-2\\epsilon).\n\\end{xalignat}\nWe have to compute two MIs,\n\\begin{eqnarray}\\label{dfB}\n\\texttt{B5l2M2m}&\\equiv&\\texttt{B5l2M2m}(1,1,1,1,1),\\nonumber \\\\\n\\texttt{B5l2M2md}&\\equiv&\\texttt{B5l2M2m}(1,2,1,1,1),\n\\end{eqnarray}\nand an expansion in the high-energy limit of the appropriate\nthree-fold MB representations leads to the following results,\n\\begin{eqnarray}\\label{def:box1}\n\\texttt{B5l2M2m} &=&\n(m^2)^{-2 \\epsilon} \n\\Bigl[ \\sum_{k=-2}^0 B_{k} \\epsilon^k + {\\cal O}(\\epsilon) \\Bigr], \\nonumber\\\\\n B_{-2}&=& \\frac{1}{s}\\ln(m_s), \\nonumber \\\\\n B_{-1}&=&\n\\frac{1}{ s}\n \\Bigl(- \\zeta_2 + 2 \\ln(m_s) + \\frac{1}{2}\\ln^2(m_s) \\nonumber \\\\ &+& \\ln(m_s)\\ln(m_t)\n\\Bigr),\n\\nonumber\n\\end{eqnarray}\n\\begin{eqnarray}\n B_0&=&\n\n\\frac{1}{s}\\Bigl[\n-2 \\zeta_2\n- 2 \\zeta_3\n+ 4 \\ln(m_s)\n+ \\ln^2(m_s)\n\\nonumber\\\\\n&+& \\frac{1}{3} \\ln^3(m_s)\n- 4 \\zeta_2 \\ln(m_t)\\nonumber\\\\\n&+& 2 \\ln(m_s)\\ln(m_t)\n+ \\ln(m_s)\\ln^2(m_t)\n\\nonumber\\\\\n&-&\\frac{1}{6}\\ln^3(m_t)\\nonumber\\\\\n&-& \\Bigl( 3 \\zeta_2 \n+ \\frac{1}{2} \\ln^2(m_s)\n- \\ln(m_s) \\ln(m_t)\n \\nonumber\\\\\n&+& \\frac{1}{2} \\ln^2(m_t) \\Bigr) \\ln\\left( 1+ \\frac{t}{s}\\right)\\nonumber\\\\\n&-& \\Bigl( \\ln(m_s) -\\ln(m_t) \\Bigr) \\text{Li}_2 \\left(-\\frac{t}{s}\\right)\n\\nonumber\\\\\n&+& \\text{Li}_3 \\left(-\\frac{t}{s}\\right)\n\\Bigr],\n\\end{eqnarray}\n\n\\begin{eqnarray}\\label{def:box2}\n\\texttt{B5l2M2md} &=&\n(m^2)^{-2 \\epsilon} \n\\Bigl[ \\sum_{k=-1}^0 B^d_{k} \\epsilon^k + {\\cal O}(\\epsilon) \\Bigr], \n\\nonumber\\end{eqnarray}\n\\begin{eqnarray}\n B^d_{-1}&=&\n -\\frac{1}{st} \\Bigl[\\ln(m_s)\\ln(m_t)-\\ln(m_s)L(R)\\Bigr],\n \\nonumber\\\\\n B^d_0&=&\n \\frac{1}{st}\\Bigl\\{\n-2 \\zeta_3\n+ \\zeta_2 \\ln(m_s)\n+ 4 \\zeta_2 \\ln(m_t)\\nonumber\\\\\n&-& 2 \\ln(m_s)\\ln^2(m_t)\n+ \\frac{1}{6}\\ln^3(m_t)\n \\nonumber \\\\\n&-& 2 \\zeta_2 L(R)\n+ 2 \\ln(m_s) \\ln(m_t)L(R)\n\\nonumber\\\\\n&-& \\frac{1}{6}L^3(R)\n\\\\\n&+& \\Bigl( 3 \\zeta_2 \n+ \\frac{1}{2} \\ln^2(m_s) - \\ln(m_s) \\ln(m_t)\n\\nonumber\\\\\n&+& \\frac{1}{2}\\ln^2(m_t) \\Bigr) \\ln\\left( 1+ \\frac{t}{s}\\right)\\nonumber\\\\\n&+& \\Bigl( \\ln(m_s) -\\ln(m_t) \\Bigr) \\text{Li}_2 \\left(-\\frac{t}{s}\\right)\n\\nonumber\\\\\n&-& \\text{Li}_3 \\left(-\\frac{t}{s}\\right)\n\\Bigr\\}.\\nonumber\n\\end{eqnarray}\n\n\n\n\\section{Summation techniques\\label{section-techn}}\n\nIn any realistic computation we have to check the\nstructure of the ultraviolet (UV) and infrared (IR) divergencies.\nBy combining the Mellin-Barnes method with recently developed\nsummation techniques we are able to evaluate exactly \n(i.e. without a high-energy approximation) the residues of the UV and IR poles\nfor each MI.\n\nA simple example is enough to illustrate our procedure. \nWe consider\nthe following one-fold integral in the complex plane,\nrelated to the single pole of \\texttt{V4l2M1md},\n\\begin{equation}\nI\n\\equiv\n\\frac{1}{2\\pi i} \\int_{c-i\\infty}^{c+i \\infty} d z \\ M_s^{-z}\\\n\\frac{ \\prod_{i=1}^3 \\Gamma_i }{\\Gamma(2 z + 2)},\n\\end{equation}\nwhere we recall that $M_s \\equiv - M^2 \\slash s $, the integration contour is a straight\nline parallel to the imaginary axis, $c= - 1 \\slash 2$\nand we introduced\n\\begin{xalignat}{2}\n\\Gamma_1&\\equiv \\Gamma(-z),\\nonumber\\\\\n\\Gamma_2&\\equiv \\Gamma(z),\\nonumber\\\\\n\\Gamma_3&\\equiv \\Gamma^2(z+1).\n\\end{xalignat}\nAfter closing the integration contour to the right of the complex plane and taking\nresidua, the integral $I$ can be written by means of two inverse binomial sums,\n\\begin{eqnarray}\nI&=&\n\\sum_{n=1}^{\\infty}\n\\left(-1 \\right)^n\\ M_s^{-n}\n\\frac{1}{\\binom{2 n}{n}} \\left( \\frac{1}{n} - \\frac{2}{2 n+1}\\right)\\nonumber\\\\\n&-& 2 - \\ln\\left(M_s \\right).\n\\end{eqnarray}\nInverse binomial sums were recently studied by means of the \nlog-sine approach in \\cite{Davydychev:2003mv}.\nAnother approach was developed in \\cite{Weinzierl:2004bn}\nby generalizing the summation algorithms\nintroduced in \\cite{Moch:2001zr}.\nA straightforward application of these techniques leads to\na compact result,\n\\begin{equation}\nI\n=\n\\frac{1-x_M}{1+x_M} \\ln (y_M) \\left(1+4 M_s \\right)\n-\n\\ln \\left(M_s \\right),\n\\end{equation}\nwhere we introduced the variables $x_M$ and $y_M$,\n\\begin{eqnarray}\nx_M &\\equiv& \\frac{\\sqrt{4 M^2 -s} - \\sqrt{-s}}{\\sqrt{4 M^2 -s} + \\sqrt{-s}},\\nonumber\\\\\ny_M &\\equiv& \\frac{\\sqrt{4 M^2 -t} - \\sqrt{-t}}{\\sqrt{4 M^2 -t} + \\sqrt{-t}}.\n\\end{eqnarray}\n\nAs an example, after additionally introducing the following variables,\n\\begin{eqnarray}\nx_m &\\equiv& \\frac{\\sqrt{4 m^2 -s} - \\sqrt{-s}}{\\sqrt{4 m^2 -s} + \\sqrt{-s}},\\nonumber\\\\\ny_m &\\equiv& \\frac{\\sqrt{4 m^2 -t} - \\sqrt{-t}}{\\sqrt{4 m^2 -t} + \\sqrt{-t}},\n\\end{eqnarray}\nwe get the non-approximated\nexpressions for the residues of the poles of the two box diagrams\ndefined in Eq.~\\eqref{dfB},\n\\begin{eqnarray}\nB_{-2}&=& - \\frac{1}{m^2} \\frac{x_m}{1-x_m^2} H(0;x_m),\\nonumber \\\\\nB_{-1}&=& \n\\frac{1}{2 m^2} \\frac{x_m}{1-x_m^2}\n\\Bigl\\{-\nH^2(0;x_m) \\nonumber\\\\\n&+& 2 \\Bigl[\n\\zeta_2 - 2 H(0,-1;x_m)\n\\Bigr]\n\\nonumber\\\\\n&+& 2 H(0;x_m)\n\\Bigl[\n2 H(-1;x_m)\\nonumber\\\\\n&-&\\frac{1+y_M}{1-y_M} H(0;y_M)\n-\n2 \\nonumber \\\\\n&-& \\ln \\left(\\frac{m^2}{M^2}\n\\right)\n\\Bigr]\n\\Bigr\\},\n\\end{eqnarray}\nand \n\\begin{eqnarray}\n B^d_{-1}&=& - \\frac{1}{m^2 M^2} \\frac{x_m\\,y_M}{(1-x_m^2)(1-y_M^2)}\\nonumber \\\\\n &\\times&H(0;x_m) H(0;y_M),\n\\end{eqnarray}\nwhere we used the HPLs introduced in \\cite{Remiddi:1999ew}.\n\nFor completeness, we add here also the exact expressions for the diveregent parts of the vertex MIs:\n\\begin{eqnarray}\n V^{1d}_{-1} &=& \\frac{1}{2}\n \\left\\{\n \\frac{1+x_M}{1-x_M} H(0;x_M) +2 + \\ln R\n \\right\\}\n\\nonumber \\\\\n V^{2}_{-1} &=& \\frac{5}{2} + \\frac{1+x_m}{1-x_m} H(0;x_m).\n\\end{eqnarray}\n\n\\section{Summary}\nFrom \\cite{Czakon:2004wm} we know the table of MIs for massive\ntwo-loop Bhabha scattering. \nWe were able to express all\nthe Feynman integrals occurring in the amplitude\nthrough these MIs by algebraic relations. \nWe presented at the workshop all the planar two-loop box MIs.\nThe $N_f=1$ MIs has been published in the meantime \\cite{Czakon:2006pa}.\nIn this contribution, we provide the expanded results for all the MIs\nentering the Bhabha-scattering amplitude with two fermion flavors in\nthe limit of small fermion masses at fixed scattering angle.\nThe MIs may be also found at our webpage \\cite{web-masters:2006nn}.\nThese MIs were one of the last missing ingredients for the evaluation of the\nvirtual two-loop contribution to the differential cross-section. \n\nThe computation of the last nine\nnon-planar two-loop box MIs is under way.\n\n\\section*{Acknowledgements}\nWe would like to thank S. Moch for useful discussions.\n\nWork supported in part by Sonderforschungsbereich\/Transregio 9--03 of DFG\n`Computergest{\\\"u}tzte Theo\\-re\\-ti\\-sche Teil\\-chen\\-phy\\-sik', by\nthe Sofja Kovalevskaja Award of the Alexander von Humboldt Foundation\n sponsored by the German Federal Ministry of Education and Research,\nand by the Polish State Committee for Scientific Research (KBN),\nresearch projects in 2004--2005.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{\\bf Introduction}\nLet $G$ be a simply connected Lie group \nand $\\frak{g}$ the Lie algebra of $G$.\nWe consider the space $\\bigwedge \\frak{g}_{\\mathbb{C}}^{\\ast}$ of $\\mathbb{C}$-valued left $G$-invariant differential forms on $G$.\nWe assume that $G$ has a lattice (i. e. cocompact discrete subgroup) $\\Gamma$.\nWe consider the compact homogeneous space $G\/\\Gamma$ and $\\bigwedge \\frak{g}_{\\mathbb{C}}^{\\ast}$ as a subcomplex of the de Rham complex $A^{\\ast}_{\\mathbb{C}}(G\/\\Gamma)$ of $G\/\\Gamma$.\nSuppose $G$ is nilpotent.\nThen we have the unique unipotent algebraic group ${\\bf U}_{\\Gamma}$ called the Malcev completion of $\\Gamma$ such that there is a injection $\\Gamma\\to {\\bf U}_{\\Gamma}$ with the Zariski-dense image.\nWe can represent the coordinate ring of ${\\bf U}_{\\Gamma}$ by using Chen's iterated integrals on $G\/\\Gamma$ (see \\cite{CH}).\nSince the inclusion $\\bigwedge \\frak{g}_{\\mathbb{C}}^{\\ast}\\subset A^{\\ast}_{\\mathbb{C}}(G\/\\Gamma)$ induces a cohomology isomorphism by Nomizu's theorem \\cite{Nom}, $\\bigwedge \\frak{g}_{\\mathbb{C}}^{\\ast}$ is the Sullivan minimal model of $ A^{\\ast}_{\\mathbb{C}}(G\/\\Gamma)$ (see \\cite{H}).\nThis implies $H^{0}(\\bar B (\\bigwedge \\frak{g}_{\\mathbb{C}}^{\\ast}))\\cong H^{0}(\\bar B( A^{\\ast}_{\\mathbb{C}}(G\/\\Gamma)))$ where $\\bar B (\\bigwedge \\frak{g}_{\\mathbb{C}}^{\\ast})$ and $\\bar B( A^{\\ast}_{\\mathbb{C}}(G\/\\Gamma))$ are the reduced bar constructions of $\\bigwedge \\frak{g}_{\\mathbb{C}}^{\\ast}$ and $A^{\\ast}_{\\mathbb{C}}(G\/\\Gamma)$ respectively (see \\cite{CH2}).\nHence we can represent the coordinate ring of ${\\bf U}_{\\Gamma}$ by using Chen's iterated integrals of left-invariant forms.\n\nSuppose $G$ is solvable.\nThen Chen's iterated integrals on $G\/\\Gamma$ does not give sufficient information on the fundamental group of $G\/\\Gamma$.\nFor example, let $G=\\mathbb{R}\\ltimes_{\\phi} \\mathbb{R}^{2}$ such that $\\phi(t)=\\left(\n\\begin{array}{cc}\ne^{ t}&0\\\\\n0&e^{-t}\n\\end{array}\n\\right) $.\nThen $G$ has a lattice $\\Gamma$ and the inclusion $\\bigwedge \\frak{g}^{\\ast}_{\\mathbb{C}}\\subset A_{\\mathbb{C}}^{\\ast}\n(G\/\\Gamma)$ induces a cohomology isomorphism (see \\cite{Hatt}).\nSince we have $H^{1}(G\/\\Gamma,\\mathbb{C})=H^{1}(\\bigwedge \\frak{g}^{\\ast}_{\\mathbb{C}})=\\mathbb{C}$, by Chen's results,\niterated integrals represent the coordinate ring of a additive algebraic group ${\\mathbb G}_{\\rm ad}=\\mathbb{C}$ (see \\cite{M}).\nBut since $\\Gamma$ is solvable and not abelian, we can't embed $\\Gamma$ in ${\\mathbb G}_{\\rm ad}$.\n\n\n \n \n \n In \\cite{Mos2}, as an extension of the Malcev completion, Mostow constructed a Zariski-dense embedding of $\\Gamma$\nin an algebraic group ${\\bf H}_{\\Gamma}$ called algebraic hull of $\\Gamma$.\nIn \\cite{M}, Miller gave extensions of Chen's iterated integrals called exponential iterated integrals.\nIn this paper we represent the coordinate ring of ${\\bf H}_{\\Gamma}$ by using Miller's exponential iterated integrals of left-invariant differential forms on $G\/\\Gamma$.\n\n\n\\section{\\bf Relative completions and algebraic hulls}\n\nLet $G$ be a discrete group (resp. a Lie group).\nWe call a map $\\rho:G\\to GL_{n}(\\mathbb{C})$ a representation, if $\\rho$ is a homomorphism of groups (resp. Lie groups).\nIn this paper we denote by $T_{n}(\\mathbb{C})$ the group of $n\\times n$ upper triangular matrix and denote by $U_{n}(\\mathbb{C})$ the group of $n\\times n$ upper triangular unipotent matrix.\n\n\\subsection{\\bf algebraic groups and pro-algebraic groups}\nIn this paper an algebraic group means an affine algebraic variety $\\bf G$ over $\\mathbb{C}$ with a group structure such that the multiplication and inverse are morphisms of varieties.\nAll algebraic groups arise as Zariski-closed subgroups of $GL_{n}(\\mathbb{C})$.\nA pro-algebraic group is an inverse limit of algebraic groups.\nIf a pro-algebraic group is an inverse limit of unipotent algebraic groups, it is called pro-unipotent.\nLet $\\bf G$ be a pro-algebraic group.\nWe denote by ${\\bf U}({\\bf G})$ the maximal pro-unipotent normal subgroup called the pro-unipotent radical.\nIf ${\\bf U}({\\bf G})=e$, $\\bf G$ is called reductive.\nWe denote by $\\mathbb{C}[\\bf G]$ the coordinate ring of $\\bf G$.\nThe group structure on $\\bf G$ induces a Hopf algebra structure on $\\mathbb{C}[\\bf G]$.\nIt is known that we have the anti-equivalence between algebras and affine varieties induces an anti-equivalence between pro-algebraic groups and reduced Hopf algebras.\n\n\\begin{theorem}{\\rm (\\cite{Mos1},\\cite{HM})} \\label{sppp}\nLet $\\bf G$ be a pro-algebraic group. Then the exact sequence \n\\[\\xymatrix{\n\t1\\ar[r]&{\\bf U}(\\bf G) \\ar[r]& {\\bf G} \\ar[r] & {\\bf G}\/{\\bf U}(\\bf G) \\ar[r] &1 \n\t }\n\\]\nsplits.\n\\end{theorem}\nLet $G$ be a discrete group or Lie group.\nWe denote by $A(G)$ the inverse limit of all representations $\\phi :G\\to \\bf G$ with Zariski-dense images.\nWe call the pro-unipotent radical ${\\bf U}(A(G))$ of $A(G)$ the unipotent hull of $G$ and denote it ${\\bf U}_{G}$.\n\n\n\\subsection{\\bf Relative completion}\nLet $\\rho:G\\to {\\bf S}$ be a representation of $G$ to a diagonal algebraic group ${\\bf S}$ with the Zariski-dense image.\nLet $\\phi: G\\to {\\bf G}$ be a representation of $G$ to an algebraic group $\\bf G$ with the Zariski-dense image.\nWe call $\\phi$ a $\\rho$-relative representation if we have the commutative diagram \n\\[\\xymatrix{\n\t1\\ar[r]&{\\bf U}(\\bf G) \\ar[r]& {\\bf G} \\ar[r] &{\\bf S} \\ar[r] &1 \\\\\n\t& &G\\ar^{\\phi}[u]\t \\ar^{\\rho}[ru]\n }.\n\\]\n\nIf $\\bf S$ is contained in an algebraic torus, for any $\\rho$-relative \nrepresentation $\\phi:G\\to {\\bf G}$ there exists a faithful \nrepresentation ${\\bf G}\\hookrightarrow T_{n}(\\mathbb{C})$ such that ${\\bf G}\\cap \nU_{n}(\\mathbb{C})={\\bf U}({\\bf G})$(see \\cite{M}).\n\n\nWe denote by $\\mathcal G_{\\rho}(G)$ the inverse limit of $\\rho$-relative representations $\\phi_{i}:G\\to {\\bf G}_{i}$.\nWe call $\\mathcal G_{\\rho}(G)$ the $\\rho$-relative completion of $G$.\nIf $\\bf S$ is trivial, $\\mathcal G_{\\rho}(G)$ is the classical Malcev (or unipotent) completion.\n\n\n\\subsection{\\bf Algebraic hulls}\nWe define the algebraic hulls of polycyclic groups (resp. Lie groups) which constructed in {\\cite{Mos2}.\n\n\n \n\nA group $\\Gamma$ is polycyclic if it admits a sequence \n\\[\\Gamma=\\Gamma_{0}\\supset \\Gamma_{1}\\supset \\cdot \\cdot \\cdot \\supset \\Gamma_{k}=\\{ e \\}\\]\nof subgroups such that each $\\Gamma_{i}$ is normal in $\\Gamma_{i-1}$ and $\\Gamma_{i-1}\/\\Gamma_{i}$ is cyclic.\nFor a polycyclic group $\\Gamma$, we denote ${\\rm rank}\\,\\Gamma=\\sum_{i=1}^{i=k} {\\rm rank}\\,\\Gamma_{i-1}\/\\Gamma_{i}$.\nLet $G$ be a simply connected solvable Lie group and $\\Gamma$ be a lattice of $G$.\nThen $\\Gamma$ is torsion-free polycyclic and $ \\dim G={\\rm rank}\\,\\Gamma$.\n\nLet $G$ be a simply connected solvable Lie group or torsion-free polycyclic group.\nConsider the algebraic completion $A(G)$.\nThen it is known that the unipotent hull ${\\bf U}_{G}={\\bf U}(A(G))$ is finite dimensional (see \\cite{Mos2}).\nBy Theorem \\ref{sppp}, we have a splitting $A(G)=(A(G)\/{\\bf U}_{G})\\ltimes_{\\phi} {\\bf U}_{G}$.\nLet $K$ be the kernel of the action $\\phi:(A(G)\/{\\bf U}_{G})\\to {\\rm Aut}({\\bf U}_{G})$.\nThen $K$ is the maximal reductive normal subgroup of $A(G)$ (see \\cite{Mos2}).\nDenote ${\\bf H}_{G}=A(G)\/K$ and call it the algebraic hull of $G$.\n\n\n\\begin{theorem}{\\rm (\\cite{Mos2},\\cite{R})} \\label{AL}\nLet $G$ be a simply connected solvable Lie group (resp. torsion-free \npolycyclic group).\nThen $G \\to {\\bf H}_{G}$ is injective and ${\\bf H}_{G}$ is a finite dimensional algebraic group such that:\\\\\n$(1)$ $\\dim {\\bf U}({\\bf H}_{G})=\\dim G$ (resp. $={\\rm rank}\\, G $).\\\\\n$(2)$The centralizer of ${\\bf U}({\\bf H}_{G})$ in ${\\bf H}_{G}$ is \ncontained in ${\\bf U}({\\bf H}_{G})$.\\\\\nConversely if an algebraic group $\\bf H$ with an injective homomorphism $\\\n\\psi:G\\to {\\bf H}$ with the Zariski-dense image satisfies the properties $(1)$ and $(2)$,\n then ${\\bf H}$ is isomorphic to ${\\bf H}_{G}$.\n\\end{theorem}\n\n\n\n\n\n\n\\subsection{Direct constructions of algebraic hulls}\nThe idea of this subsection is based on classical works of semi-simple splitting (see \\cite{Re}, \\cite{OV} and the references given there).\nLet $\\frak{g}$ be a solvable Lie algebra, and $\\frak{n}=\\{X\\in \\frak{g}\\vert {\\rm ad}_{X}\\, \\, {\\rm is\\,\\, nilpotent}\\}$.\n$\\frak{n}$ is the maximal nilpotent ideal of $\\frak{g}$ and called the nilradical of $\\frak{g}$.\n Then we have $[\\frak{g}, \\frak{g}]\\subset \\frak{n}$.\n Let $D(\\frak{g})$ be the Lie algebra of derivations of $\\frak{g}$.\n By the Jordan decomposition, we have ${\\rm ad}_{X}={\\rm ad}_{sX}+{\\rm ad}_{nX}$ such that ${\\rm ad}_{sX} $ is a semi-simple operator and ${\\rm ad}_{nX}$ is a nilpotent operator.\n Since we have $d_{X}$, $n_{X}$ $\\in D(\\frak{g})$,\n we have the map ${\\rm ad}_{s}:\\frak{g}\\to D(\\frak{g})$.\nSince $\\rm ad$ is trigonalizable (Lie's theorem), this map is homomorphism with the kernel $\\frak{n}$.\n Let $\\bar{\\frak{g}} ={\\rm Im} \\,{\\rm ad}_{s}\\ltimes\\frak{g}$.\nand $\\bar{\\frak{n}}=\\{X-{\\rm ad}_{sX}\\in \\bar{\\frak{g}} \\vert X\\in\\frak{g}\\}$.\n\\begin{proposition}\\label{spli}\n$\\bar{\\frak{n}}$ is a nilpotent ideal of $\\bar{\\frak{g}}$ and we have a decomposition $\\bar{\\frak{g}}={\\rm Im}\\, {\\rm ad}_{s} \\ltimes\\bar{\\frak{n}}$.\n\\end{proposition}\n\\begin{proof}\n By ${\\rm ad}_{X-{\\rm ad}_{sX}}={\\rm ad}_{X}-{\\rm ad}_{sX}$ on $\\frak{g}$, ${\\rm ad}_{X-{\\rm ad}_{sX}}$ is a nilpotent operator and\nhence $\\bar{\\frak{n}}$ consists of nilpotent elements. \n By Lie's theorem, we have a basis \n\\[X_{1},\\dots,X_{l},X_{l+1}\\dots ,X_{n}\\]\n of $\\frak{g} \\otimes \\mathbb{C}$ such that ${\\rm ad}$\n is represented by upper triangular matrices.\nSince the nilradical $\\frak{n}$ is an ideal, $\\frak{n}\\otimes \\mathbb{C}$ is ${\\rm ad}$-invariant subspace of $\\frak{g} \\otimes \\mathbb{C}$.\nWe choose $X_{1},\\dots,X_{l}$ a basis of $\\frak{n}\\otimes \\mathbb{C}$.\nBy $[\\frak{g}, \\frak{g}]\\subset \\frak{n}$, we have ${\\rm ad}_{X}(\\frak{g}\\otimes \\mathbb{C})\\subset \\frak{n}\\otimes \\mathbb{C}=\\langle X_{1},\\dots ,X_{l}\\rangle$, and hence ${\\rm ad}$ represented as\n\\[{\\rm ad}_{X}=\\left(\n\\begin{array}{cccccc}\na_{11}(X)&\\dots&&&\\dots&a_{1l}(X)\\\\\n&\\ddots&&&&\\vdots\\\\\n&&a_{ll}(X)&\\dots &&a_{lm}(X)\\\\\n&&&0&\\dots&0\\\\\n&&&&\\ddots&\\vdots\\\\\n&&&&&0\n\\end{array}\n\\right).\n\\]\nThus we have \n${\\rm ad}_{sX}(X_{i})=a_{11}(X)X_{i} $ for $ 1\\le i \\le l$ and\n$ {\\rm ad}_{sX}(X_{i})=0$ for $l+1\\le i \\le n$.\nBy this we have \n\\[\n[X_{i}+{\\rm ad}_{sY},X_{j}+{\\rm ad}_{sZ}]\\in\\langle X_{1},\\dots ,X_{l}\\rangle=\\frak{n}\\otimes \\mathbb{C}\\]\nfor any $1\\le i,j\\le n$, $Y,Z\\in \\frak{g}$.\nThis implies $[\\bar{\\frak{g}}, \\bar{\\frak{g}}]\\subset\\frak{n}$.\n By $\\frak{n}\\subset \\bar\\frak{n}$, $ \\bar{\\frak{n}}$ is an ideal of $\\bar{\\frak{g}}$ and\nwe have $\\bar{\\frak{g}}=\\{{\\rm ad}_{sX}+Y-{\\rm ad}_{sY}\\vert X, Y\\in \\frak{g}\\}={\\rm Im}\\, {\\rm ad}_{s}\\ltimes\\bar{\\frak{n}}$.\n\\end{proof}\nBy this proposition, we have the inclusion $i:\\frak{g}\\to D(\\bar{\\frak{n}})\\ltimes \\bar{\\frak{n}}$ given by $i(X)={\\rm ad}_{sX}+X-{\\rm ad}_{sX}$ for $X\\in\\frak{g}$.\n\nLet $G$ be a simply connected solvable Lie group and $\\frak{g}$ be the Lie algebra of $G$.\nFor the adjoint representation ${\\rm Ad}:G\\to {\\rm Aut}(\\frak{g})$,\nwe consider the semi-simple part ${\\rm Ad}_{s}: G\\to {\\rm Aut}(\\frak{g})$ as similar to the Lie algebra case.\nDenote by $T$ the universal covering of ${\\rm Ad}_{s}( G)$.\nLet $\\bar{N}$ be the simply connected Lie group which corresponds to $\\bar{\\frak{n}}$.\nThen by Proposition \\ref{spli}, we have $T\\ltimes G=T\\ltimes \\bar N$.\nBy the proof of this proposition, the action $T\\to{\\rm Aut} (\\bar \\frak{n})$ is the extension of the action of ${\\rm Im}\\, {\\rm ad}_{s} $.\nHence we have ${\\rm Ad}_{s}(G)\\ltimes G={\\rm Ad}_{s}(G)\\ltimes \\bar N$.\n\n\n\n\nA simply connected nilpotent Lie group is considered as the real points of a unipotent $\\mathbb{R}$-algebraic group (see \\cite[p. 43]{OV}) by the exponential map.\nWe have the unipotent $\\mathbb{R}$-algebraic group $\\bf \\bar{N}$ with ${\\bf \\bar{N}}(\\mathbb{R})=\\bar{N}$.\nWe identify $\\rm Aut_{a}({\\bf \\bar{N}})$ with $\\rm Aut(\\frak{n}_{\\mathbb{C}})$ and $\\rm Aut_{a}({\\bf \\bar{N}})$ has the $\\mathbb{R}$-algebraic group structure with ${\\rm Aut}_{a}({\\bf \\bar{N}})(\\mathbb{R})= {\\rm Aut} (N)$.\nSo we have the $\\mathbb{R}$-algebraic group $ \\rm Aut_{a} (\\bf\\bar{N})\\ltimes \\bar{N}$.\nThen by ${\\rm Ad}_{s}(G)\\ltimes G={\\rm Ad}_{s}(G)\\ltimes \\bar N\\subset \\rm Aut_{a} (\\bf\\bar{N})\\ltimes \\bar{N}$, we consider the Zariski-closure $\\bf G$ of $G$ in $ \\rm Aut_{a} (\\bf\\bar{N})\\ltimes \\bar{N}$.\nSince ${\\rm Ad}_{s}(G)$ is a group of diagonal automorphisms, we have ${\\bf U}({\\bf G})= \\bar{\\bf N}$.\nBy $\\dim G=\\dim \\bar N$, we can easily check that $\\bf G$ satisfies the properties (1), (2) in Theorem \\ref{AL} and hence it is the algebraic hull ${\\bf H}_{G}$ of $G$.\nHence the inclusion $i:\\frak{g}\\to D(\\bar{\\frak{n}})\\ltimes \\bar{\\frak{n}}$ induces the algebraic hull $I:G\\to {\\bf H}_{G}$ of $G$.\nSince $i:\\frak{g}\\to D(\\bar{\\frak{n}})\\ltimes \\bar{\\frak{n}}$ is given by $i(X)={\\rm ad}_{sX}+X-{\\rm ad}_{sX}\\in D(\\bar{\\frak{n}})\\ltimes \\bar{\\frak{n}}$ , the composition $G\\to {\\bf H}_{G}\\to {\\bf H}_{G}\/{\\bf U}({\\bf H}_{G})$ is induced by the Lie algebra homomorphism ${\\rm ad}_{s}:\\frak{g}\\to D(\\frak{g})$ by ${\\bf U}({\\bf G})= \\bar{\\bf N}$. \nThus we have the following lemma.\n\\begin{lemma}\n The algebraic hull $ G\\to {\\bf H}_{G}$ is an ${\\rm Ad}_{s}$-relative representation.\n\\end{lemma}\n\n\\subsection{\\bf Algebraic hulls and relative completions of solvable groups}\n\n\n\\begin{theorem}\\label{relal}\nLet $G$ be a simply connected Lie group.\nThen the algebraic hull $ {\\bf H}_{G}$ is isomorphic to the ${\\rm Ad}_{s}$-relative completion ${\\mathcal G}_{{\\rm Ad}_{s}}(G)$ of $G$.\n\\end{theorem}\n\\begin{proof}\nConsider a commutative diagram\n\\[\\xymatrix{\n\t{\\bf H}^{\\prime} \\ar[r]^{\\Phi}&{\\bf H}_{G} \\\\\n\tG\\ar[u]\t \\ar[ru]\n }\n\\] \nfor some ${\\rm Ad}_{s}$-relative representation $G\\to {\\bf H}^{\\prime}$.\nSince $G\\to {\\bf H}^{\\prime}$ and $G\\to {\\bf H}_{G}$ have Zariski-dense images, $\\Phi : {\\bf H}^{\\prime} \\to {\\bf H}_{G}$ is surjective and the restriction $\\Phi :{\\bf U}({\\bf H}^{\\prime}) \\to {\\bf U} ({\\bf H}_{G})$ is also surjective.\nBy ${\\bf U} ({\\bf H}_{G})={\\bf U}_{G}$, $\\Phi :{\\bf U}({\\bf H}^{\\prime }) \\to {\\bf U} ({\\bf H}_{G})$ is an isomorphism. \nSince $G\\to {\\bf H}^{\\prime}$ and $G\\to {\\bf H}_{G}$ are ${\\rm Ad}_{s}$-relative representations, $\\Phi$ induces the isomorphism ${\\bf H}^{\\prime}\/ {\\bf U} ({\\bf H}^{\\prime}) \\to {\\bf H}_{G}\/{\\bf U}({\\bf H}_{G})$.\nHence $\\Phi : {\\bf H}^{\\prime}\\to {\\bf H}_{G}$ is an isomorphism.\nBy the definition of ${\\rm Ad}_{s}$-relative completion of $G$, we have the theorem. \n\\end{proof}\n\n\n\\begin{theorem}\\label{alrr}\nLet $G$ be a simply connected solvable Lie group and $\\Gamma$ a lattice of $G$.\nThen the algebraic hull ${\\bf H}_{\\Gamma}$ of $\\Gamma$ is isomorphic to ${\\rm Ad}_{s\\vert_{\\Gamma}}$-relative completion ${\\mathcal G}_{{\\rm Ad}_{s\\vert_{\\Gamma}}}(\\Gamma)$ of $\\Gamma$. \n\\end{theorem}\n\\begin{proof}\nFor the algebraic hull $\\psi :G\\to {\\bf H}_{G}$ of $G$, we consider the Zariski-closure of $\\psi(\\Gamma)$ in ${\\bf H}_{G}$.\nThen by $\\dim G={\\rm rank}\\, \\Gamma$ we can easily check that this algebraic group satisfies (1) and (2) in Theorem \\ref{AL} and hence it is the algebraic hull ${\\bf H}_{\\Gamma}$ of $\\Gamma$.\nBy the above theorem, $\\Gamma\\to {\\bf H}_{\\Gamma}$ is a ${\\rm Ad}_{s\\vert_{\\Gamma}}$-relative representation.\nAs similar to the above proof, we have the theorem.\n\\end{proof}\n\n\n\\section{Exponential iterated integral on solvmanifolds}\nIn this section we consider Miller's exponential iterated integrals.\nLet $M$ be a $C^{\\infty}$-manifold and $\\Omega_{x}M$ be a space of piecewise smooth loops\n$\\lambda :[0,1]\\rightarrow M$ with $\\lambda(0)=x$.\nFor $1$-forms $\\omega_{1},\\dots ,\\omega_{n}\\in A^{\\ast}_{\\mathbb{C}}(M)$, the iterated integral $\\int \\omega_{1} \\omega_{2} \\cdot \\cdot \\cdot \\omega_{n}:\\Omega_{x}M\\to \\mathbb{C}$ is defined by\n\\[\\int_{\\lambda} \\omega_{1} \\omega_{2} \\cdot \\cdot \\cdot \\omega_{n} = \\int_{0\\le t_{1}\\le t_{2} \\le \\cdot \\cdot \\le t_{n} \\le 1}F(t_{1})F(t_{2})\\cdot \\cdot \\cdot F(t_{n}) dt_{1} dt_{2}\\cdot \\cdot \\cdot dt_{n} \\\\\n\\lambda \\in \\Omega_{x}M\n\\]\nwhere $F_{i}(t)dt=\\lambda ^{\\ast} \\omega_{i} \\in A^{1}([0,1])$.\nIn \\cite{M}, for $\\delta_{1}, \\delta_{2}, \\cdot \\cdot \\cdot ,\\delta_{n}, \\omega_{12}, \\omega_{23} ,\\cdot \\cdot \\cdot, \\omega_{n-1n} \\in A^{1}_{\\mathbb{C}}(M)$ Miller defined the exponential iterated integral $\\int e^{\\delta_{1}}\\omega_{12}e^{\\delta_{2}}\\omega_{23}\\cdot\\cdot \\cdot e^{\\delta_{n-1}}\\omega_{n-1n}e^{\\delta_{n}} :\\Omega_{x}M\\to \\mathbb{C}$ as \n\\[\\int_{\\lambda} e^{\\delta_{1}}\\omega_{12}e^{\\delta_{2}}\\omega_{23}\\cdot\\cdot \\cdot e^{\\delta_{n-1}}\\omega_{n-1n}e^{\\delta_{n}}\\]\n\\[=\n\\sum_{m_{1},m_{2},\\cdot \\cdot \\cdot m_{n} \\ge 0}\\int_{\\lambda}\\underbrace{\\delta_{1}\\dots \\delta_{1}}_{m_{1} \\ terms}\\omega_{12}\\underbrace{\\delta_{2}\\dots \\delta_{2}}_{m_{2} \\ terms}\\cdot \\cdot \\cdot \\omega_{n-1n}\\underbrace{\\delta_{n}\\dots \\delta_{n}}_{m_{n} \\ terms}.\n\\]\n Then this infinite sum converges (see \\cite{M}).\nLet $L\\subset A^{1}_{\\mathbb{C}}(M)$ be a finitely generated $\\mathbb{Z}$-module of $1$-forms such that $dL=0$.\nWe denote $E^{L}(M,x)$ the $\\mathbb{C}$-vector space of functions on $\\Omega_{x}M$ generated by \n\\[\\{ \\int e^{\\delta_{1}}\\omega_{12}\\cdot \\cdot \\cdot \\omega_{n-1n}e^{\\delta_{n}}\\vert \\delta_{1},\\cdot \\cdot \\cdot \\delta_{n} \\in L ,\n \\ \\omega_{12}, \\omega_{23} ,\\cdot \\cdot \\cdot, \\omega_{n-1n} \\in A^{\\ast}_{\\mathbb{C}}(M)\\}.\\]\nIf $I \\in E^{L}(M,x)$ is constant on homotopy classes of loops $\\lambda:[0,1]\\to M$ relative to $\\{0, 1\\}$, we call $I$ a closed exponential iterated integral.\nLet $H^{0}(E^{L}(M,x))$ denote the subspace of closed exponential iterated integrals. \nTake a $\\mathbb{Z}$-basis $\\{\\delta_{1}, \\delta_{2}, \\dots ,\\delta_{n}\\}$ of $L$.\nThen we have the diagonal representation $\\rho:\\pi_{1}(M,x)\\to D_{n}(\\mathbb{C})$ such that \n$\n\\rho(\\lambda)={\\rm diag} (\\int_{\\lambda}e^{\\delta_{1}},\\dots ,\\int_{\\lambda}e^{\\delta_{n}})\n$\nfor $\\lambda\\in\\pi_{1}(M,x)$. \nConsider the $\\rho$-relative completion ${\\mathcal G}_{\\rho}(\\pi_{1}(M,x))$ of the fundamental group of $M$.\nMiller showed the following theorem.\n\\begin{theorem}{\\rm (\\cite[Theorem 6.1]{M})}\nThe space $H^{0}(E^{L}(M,x))$ is a Hopf algebra and we have a Hopf algebra isomorphism\n\\[H^{0}(E^{L}(M,x))\\cong \\mathbb{C}[{\\mathcal G}_{\\rho}(\\pi_{1}(M,x))].\n\\]\n\\end{theorem}\n\\begin{remark}\\label{SAII}\nFor any $\\rho$-relative representation $\\phi:\\pi_{1}(M,x)\\to {\\bf G}$, Miller showed that $\\phi$ is the monodromy of a flat connection $\\omega$ on $M\\times \\mathbb{C}^{n}$ whose connection form is an upper triangular matrix.\nThen the monodromy of $\\omega$ is given by $ I+\\sum^{\\infty}_{i=1} \\int \\underbrace{\\omega \\omega \\cdot \\cdot \\cdot \\omega}_{i \\ terms}$ and its matrix entries are exponential iterated integrals.\nIn the proof of Theorem 6.1 of \\cite{M}, Miller showed that these matrix entries generate the coordinate ring $\\mathbb{C}[{\\bf G}]$.\n\\end{remark}\nConsider a simply connected solvable Lie group $G$ with a lattice $\\Gamma$.\nTake a diagonalization of the semi-simple part ${\\rm ad}_{s}$ of the adjoint representation ${\\rm ad}$ on $\\frak{g}$. \nWrite\n${\\rm ad}_{s}={\\rm diag}(\\delta_{1},\\dots ,\\delta_{n})\n$\nwhere $\\delta_{1},\\dots ,\\delta_{n}$ are characters of $\\frak{g}$.\nBy $\\delta_{1},\\dots ,\\delta_{n}\\in {\\rm Hom}(\\frak{g},\\mathbb{C})$,\nwe regard $\\delta_{1},\\dots ,\\delta_{n}$ as left-invariant closed $1$-forms.\nLet $L$ be the $\\mathbb{Z}$-module generated by $\\delta_{1},\\dots ,\\delta_{n}$.\nConsider the algebraic hull ${\\bf H}_{\\Gamma}$ of $\\Gamma$.\nSince we have $\\pi_{1}(G\/\\Gamma,x)\\cong\\Gamma$, by Theorem \\ref{alrr}, we have:\n\\begin{corollary}\\label{ALGH}\nWe have a Hopf algebra isomorphism\n\\[\nH^{0}(E^{L}(G\/\\Gamma,x))\\cong \\mathbb{C}[{\\bf H}_{\\Gamma}].\n\\]\n\\end{corollary}\n\nLet $E^{L}( \\frak{g}^{\\ast}_{\\mathbb{C}})$ denote the subvector space of $E^{L}(G\/\\Gamma,x)$ generated by \n\\[\\{ \\int e^{\\delta_{1}}\\omega_{12}\\cdot \\cdot \\cdot \\omega_{n-1n}e^{\\delta_{n}}\\vert \\delta_{1},\\cdot \\cdot \\cdot \\delta_{n} \\in L \n\\ \\ \\ \\omega_{12}, \\omega_{23} ,\\cdot \\cdot \\cdot, \\omega_{n-1n} \\in \\frak{g}^{\\ast}_{\\mathbb{C}}\\}.\\]\nStudying the proof of \\cite[Lemma 5.1]{M}, we can see that $E^{L}( \\frak{g}^{\\ast}_{\\mathbb{C}})$ is closed under the multiplication.\nWe define the subring \n\\[H^{0}(E^{L}(\\frak{g}^{\\ast}_{\\mathbb{C}}))=E^{L}(\\frak{g}^{\\ast}_{\\mathbb{C}})\\cap H^{0}(E^{L}(G\/\\Gamma,x))\\]\n of $H^{0}(E^{L}(G\/\\Gamma,x))$.\n\n\\begin{theorem}\\label{INVT}\nWe have $H^{0}(E^{L}(\\frak{g}^{\\ast}_{\\mathbb{C}}))= H^{0}(E^{L}(G\/\\Gamma,x))$.\n\\end{theorem}\n\\begin{proof}\nConsider the algebraic hull $\\psi:G\\to {\\bf H}_{G}$ of $G$.\nSince $\\psi:G\\to {\\bf H}_{G}$ is ${\\rm Ad}_{s}$-relative, we can assume ${\\bf H}_{G}\\subset T_{r}(\\mathbb{C})$ and $U_{r}(\\mathbb{C})\\cap {\\bf H}_{G}= {\\bf U}({\\bf H}_{G})$ as in Section 2.2.\nLet $\\psi_{\\ast}:\\frak{g}\\to {\\frak t}_{r}(\\mathbb{C})$ be the derivative of $\\psi$ where $ {\\frak t}_{r}(\\mathbb{C})$ is the Lie algebra of $T_{r}(\\mathbb{C})$.\nWe write\n\\[\\psi_{\\ast}=\\left(\n\\begin{array}{ccccc}\n\\omega_{11}&\\omega_{12}&\\cdots&&\\omega_{1r}\\\\\n&\\ddots&\\ddots&&\\vdots\\\\\n&&\\ddots& &\\\\\n&&&\\ddots&\\omega_{r-1r}\\\\\n&&&&\\omega_{rr}\n\n\\end{array}\n\\right) \n\\]\n as we consider $\\psi_{\\ast}\\in {\\rm Hom}(\\frak{g},\\mathbb{C})\\otimes T_{r}(\\mathbb{C})$.\nThen we have \n\\[(d\\psi_{\\ast}-\\psi_{\\ast}\\wedge\\psi_{\\ast})(X,Y)=\\psi_{\\ast}([X,Y])-[\\psi_{\\ast}(X),\\psi_{\\ast}(Y)]=0\n\\]\nfor $X,Y\\in \\frak{g}$.\nHence we have the flat connection $d-\\psi_{\\ast}$ on the vector bundle $G\\times \\mathbb{C}^{r}$.\nConsider the parallel transport $T=I+\\sum^{\\infty}_{i=1} \\int \\psi_{\\ast} \\cdot \\cdot \\cdot \\psi_{\\ast}$ of this connection.\nLet $P_{e}G$ be the space of the paths $\\gamma:[0,1]\\to G$ with $\\gamma(0)=e$ where $e$ is the identity element of $G$.\nWe consider the spaces $P_{e}G\/\\sim$ of homotopy classes of $\\gamma \\in P_{e}G$ relative to $\\{0,1\\}$.\nSince $G$ is simply connected, we have $P_{e}G\/\\sim=G$.\nIt is easily seen that the parallel transport $T$ on $P_{e}G\/\\sim=G$ is a homomorphism whose derivative is equal to $\\psi_{\\ast}$.\nHence we can identify the parallel transport $T$ on $P_{e}G\/\\sim$ with the representation $\\psi$.\nSince $\\psi$ is ${\\rm Ad}_{s}$-relative and the diagonal entries of $T$ are $\\int e^{\\omega_{11}},\\dots ,\\int e^{\\omega_{rr}}$, we have $\\omega_{11},\\dots ,\\omega_{rr}\\in L$.\n By the proof of Theorem \\ref{alrr}, the injection $\\phi:\\Gamma\\to {\\bf H}_{\\Gamma}$ is the restriction of $\\psi$ on $\\Gamma$.\nThus the representation $\\phi$ is the monodromy $I+\\sum^{\\infty}_{i=1} \\int \\psi_{\\ast} \\cdot \\cdot \\cdot \\psi_{\\ast}$ of the left-invariant flat connection \n$d-\\psi_{\\ast}$ on the vector bundle $G\/\\Gamma\\times \\mathbb{C}^{r}$.\nBy Remark \\ref{SAII}, the ring $\\mathbb{C}[{\\bf H}_{\\Gamma}]$ is generated by matrix entries of $I+\\sum^{\\infty}_{i=1} \\int \\psi_{\\ast} \\cdot \\cdot \\cdot \\psi_{\\ast}$.\nHence the theorem follows from Corollary \\ref{ALGH}.\n\\end{proof}\n\n\\section{\\bf An Example and a remark}\nLet $N$ be a simply connected nilpotent Lie group and $\\frak{n}$ the Lie algebra of $N$.\nWe suppose that $G$ has a lattice $\\Gamma$.\nThen we can represent the coordinate ring of the Malcev completion of $\\Gamma$ by using Chen's iterated integral of left-invariant forms on $N$.\nIn this paper we give another representation of the Malcev completion of the fundamental group of some nilmanifold.\n\n\nConsider the solvable Lie group $G=\\mathbb{R}\\ltimes_{\\mu} \\mathbb{C}^{2}$ such that\n$\\mu (t)=\\left(\n\\begin{array}{cc}\ne^{i\\pi t}&te^{i\\pi t}\\\\\n0&e^{i \\pi t}\n\\end{array}\n\\right)\n$.\nWe have the lattice $\\Gamma\n=2\\mathbb{Z}\\ltimes (\\mathbb{Z}+i\\mathbb{Z})$.\nWe consider the inclusion $\\bigwedge \\frak{g}^{\\ast}_{\\mathbb{C}} \\subset A^{\\ast}(G\/\\Gamma)$.\nThe map $H^{\\ast}( \\bigwedge \\frak{g}^{\\ast}_{\\mathbb{C}})\\to H^{\\ast}(G\/\\Gamma, \\mathbb{C} )$ induced by this inclusion is injective (see \\cite{R}).\nBy $(\\bigwedge \\frak{g}^{\\ast}_{\\mathbb{C}})^{0}=\\mathbb{C}$ and $(\\bigwedge \\frak{g}^{\\ast}_{\\mathbb{C}})^\n{1}\\cap dA^{0}(G\/\\Gamma)=0$, we have an isomorphism\n\\[ H^{0}(B(\\bigwedge \\frak{g}^{\\ast}_{\\mathbb{C}}, x))\\cong H^{0}\n(\\bar{B}(\\bigwedge \\frak{g}^{\\ast}_{\\mathbb{C}}))\\]\n where $H^{0}(B(\\bigwedge \\frak{g}^{\\ast}_{\\mathbb{C}},x) )$ is the space of closed Chen's iterated integrals of the left-invariant forms on the based loop space $\\Omega_{x}G\/\\Gamma$ and $H^{0}(\\bar{B}(\\bigwedge \\frak{g}^{\\ast}_{\\mathbb{C}}))$ is the \nreduced bar construction (see \\cite{CH}).\nSince we have $H^{1}(\\bigwedge \\frak{g}^{\\ast}_{\\mathbb{C}})\\cong \\mathbb{C}$,\n we have an isomorphism $ H^{0}(B(\\bigwedge \\frak{g}^{\\ast}_{\\mathbb{C}},x) )\\cong \\mathbb{C}[{\\mathbb G}_{\\rm ad}]$.\n \nOn the other hand, let $L$ be the sub $\\mathbb{Z}$-module of $\\frak{g}^{\\ast}_{\\mathbb{C}}$ generated \nby \n$\\{ i\\pi dt \\}$.\nThen by Corollary \\ref{ALGH} and Theorem \\ref{INVT}, we have an isomorphism\n\\[H^{0}(E^{L}(\\frak{g}^{\\ast}_{\\mathbb{C}}))\\cong \\mathbb{C}[{\\bf H}_{\\Gamma}].\\]\nSince we have $\\mu (2t)=\\left(\n\\begin{array}{cc}\n1&2t\\\\\n0&1\n\\end{array}\n\\right)$\nfor $t\\in\\mathbb{Z}$, $\\Gamma$ is nilpotent.\nHence ${\\bf H}_{\\Gamma}$ is the Malcev completion of $\\Gamma$.\nSince two compact solvmanifolds having the same fundamental group are \ndiffeomorphic (see \\cite{Mosr} or \\cite{R}),\n $G\/\\Gamma$ is diffeomorphic to a nilmanifold.\nBy these arguments we notice:\n\\begin{remark}\nBy closed Chen's iterated integrals of the $1$-forms $\\frak{g}_{\\mathbb{C}}^{\\ast}$ on $G\/\\Gamma$, we can not represent the coordinate ring of Malcev completion of the fundamental \ngroup of the nilmanifold $G\/\\Gamma$.\nBut the closed $L$-exponential iterated integrals of $\\frak{g}_{\\mathbb{C}}^{\\ast}$ enable \nus to represent it. \n\\end{remark}\n\n\n{\\bf Acknowledgements.} \n\nThe author would like to express his gratitude to Toshitake Kohno for helpful suggestions and stimulating discussions.\nThis research is supported by JSPS Research Fellowships for Young Scientists.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nThe inference problem is one of the most-important and well-studied problems in the field of statistics and machine learning. Inference can be considered as the (1) combination of information from various sources, which could be in the form of probability distributions from a probabilistic graphical model \\cite{graphicalmodels}, belief functions in Dempster-Shafer theory \\cite{dempster1968generalization,josang2012dempster} or tables in a relational database; and (2) subsequent focusing or projection to variables of interest, which corresponds to projection for variables in probabilistic graphical models, or a query in the relational database. Our work is based on the theory of generic inference \\cite{generic-inference} which abstracts and generalises the inference problem across these different areas.\n\nThe utility of generic inference can be understood as an analogue to sorting, which is agnostic to the specific data type, as long as there is a total order. Generic inference generalises\ninference algorithms by abstracting the essential components\nof information in an algebraic structure. In \\cite{LauritzenSpiegelhalter}, an algorithm was defined which solved the inference problem\non Bayesian networks, using a technique called \\emph{local computation}.\nIt was noted in \\cite{shenoy2008axioms} that the same algorithm could be\nused to solve the inference problem on belief functions, and\na sufficient set of axioms were proposed for an algebraic framework that is necessary\nfor the generic inference algorithm. This was extended by Kohlas\ninto a theory of valuation algebras, and a computer implementation of\ninference over valuation algebras along with concrete instantiations was\ndeveloped in \\cite{pouly2010nenok}.\n\nGeneric inference as formulated in \\cite{generic-inference} solves the inference problem in the exact case.\nAs exact inference is an \\#P-hard problem \\cite{valiant1979complexity}, in practice, we need frameworks for approximate inference. Approximation schemes exist for specific instances of valuation\nalgebras (probability potentials \\cite{dechter1998bucket}, belief potentials \\cite{Haenni2002103}); as well as for the generic case \\cite{Haenni20041}, but there is no such generic framework for anytime inference. In this paper, we extend the approximate inference framework in \\cite{Haenni20041} to support anytime inference.\n\nIn anytime algorithms, instead of an\nalgorithm terminating after an unspecified amount of time with a specific\naccuracy, we are able to tune the accuracy via a parameter passed to the\nalgorithm. The algorithm can also be designed to be interruptible,\ngradually improving its accuracy until terminated by the user. Such\nalgorithms are important in online learning where new data is being\nstreamed in \\cite{ueno2006anytime}, in intelligent systems, decision\nmaking under uncertainty \\cite{horsch1998anytime} and robotics \\cite{zilberstein1996using} where\ndue to the limitation of interacting in real-time there may not be\nsufficient time to compute an exact solution. We shall consider \\emph{interruptible} anytime algorithms which can be interrupted at any time and the approximation can be improved by resuming the\nalgorithm. This affords the greatest flexibility from the user's perspective, with applications of such algorithms to real-time systems such as sensor networks and path planning.\n\nTable \\ref{context-work} notes the previous work done in the area of inference algorithms, in both the generic case and for the specific case of probability potentials, and situates our work in context.\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\emph{inference}&generic&probability potentials\\\\\n\\hline\nexact&\\cite{generic-inference}&\\cite{pearl1982reverend}\\\\\n\\hline\napproximate&\\cite{Haenni20041}&loopy belief propagation, \\cite{dechter1998bucket}\\\\\n\\hline\nanytime&[our work]&\\cite{ramos2005anytime}\\\\\n\\hline\n\\end{tabular}\n\\caption{Our work, in relation to various inference algorithms and frameworks}\n\\label{context-work}\n\\end{table}\nWe note that the successive rows in the above table refine upon the previous one, and include it; the approximate inference framework can also\nperform exact inference, and the anytime inference framework presented here gives an\napproximate solution which incrementally improves with time, converging\non the solution obtained from exact inference given sufficient time.\n\nThis article is divided into the following sections. Section 2 reviews the framework of valuation algebras\nand ordered valuation algebras. Section 3 introduces our extension to\nordered valuation algebras to support anytime inference, and proves soundness and completeness\ntheorems for anytime inference. Section 4 describes instances of the framework, including its application to anytime inference in semiring-induced valuation algebras. Section 5 gives a complexity analysis\nof the algorithm. Section 6 shows implementation results of anytime inference on a Bayesian network. Section 7 concludes.\n\n\\section{Valuation Algebras}\n\nValuation algebras are the core algebraic structure in the theory of generic inference. In a valuation algebra, we consider the various pieces of information in an inference problem\n(conditional probability distributions, belief potentials, relational database tables, etc.) as elements\nin an algebraic structure with a set of axioms. We review the axioms of valuation algebra \\cite{generic-inference} below, preceded by some remarks on notation.\n\nAll operations in the valuation\nalgebra are defined on elements denoted by lowercase\nGreek letters: $\\phi, \\psi, \\ldots$. We can think of a valuation as\nthe information contained by the possible values of a set of variables, which\nare denoted by Roman lowercase letters (with possible subscripts): $x,y,\\ldots$ and denote sets of\nvariables by uppercase letters: $S,T,\\ldots$. Each valuation refers\nto the information contained in\na set of variables which we call the \\emph{domain} of a valuation, denoted\nby $d(\\phi)$ for a valuation $\\phi$. For a finite set of variables $D$, $\\Phi_D$\ndenotes the set of valuations $\\phi$ for which $d(\\phi) = D$. Thus, the set\nof all possible valuations for a countable set of variables $V$ is\n\\begin{equation}\n\\Phi = \\bigcup_{D\\subseteq V} \\Phi_D\n\\end{equation}\nIf $\\hat D = \\mathcal{P}_f(V)$ the finite powerset of $V$, and $\\Phi$ the set of valuations\nwith domains in $\\hat D$; we define the following operations on $\\langle \\Phi, \\hat D\\rangle$:\n\\begin{enumerate*}[label=(\\roman*)]\n\\item \\emph{labeling}: $\\Phi \\rightarrow \\hat D; \\phi \\mapsto d(\\phi)$\n\\item \\emph{combination}: $\\Phi \\times \\Phi \\mapsto \\Phi; (\\phi,\\psi) \\mapsto \\phi \\otimes \\psi$\n\\item \\emph{projection}: $\\Phi \\times \\hat D \\rightarrow \\Phi; (\\phi, X) \\mapsto \\phi^{\\downarrow X}\\ \\mathrm{for}\\ X \\subseteq d(\\phi)$\n\\end{enumerate*}\n\nThese are the basic operations of a valuation algebra. Using the view of valuations as pieces of information which refer to questions as valuations, the labelling operation\ntells us which set of variables the valuation refers to; the combination operation\naggregates the information, and the projection operation focuses the information\non a particular question (query) of interest. Projection is also referred to\nas \\emph{focusing} or \\emph{marginalization}. The following axioms are then imposed\non $\\langle \\Phi, \\hat D \\rangle$:\n\n(A1) \\emph{Commutative semigroup}: $\\Phi$ is associative and commutative under $\\otimes$\n\n(A2) \\emph{Labeling}: For $\\phi,\\psi \\in \\Phi$, $d(\\phi \\otimes \\psi) = d(\\phi) \\cup d(\\psi)$.\n\n(A3) \\emph{Projection}: For $\\phi \\in \\Phi, X \\in \\hat D$ and $X \\subseteq d(\\phi)$,\n$d(\\phi^{\\downarrow X}) = X$. Alternatively this is equivalent to the following\n\\emph{elimination} operation, $\\phi^{\\downarrow X} = \\phi^{-(d(\\phi) \\backslash X)}$ where\nall the variables except those in $X$ are eliminated.\n\n(A4) \\emph{Transitivity}: For $\\phi \\in \\Phi$ and $X \\subseteq Y \\subseteq d(\\phi)$,\n$(\\phi^{\\downarrow Y})^{\\downarrow X} = \\phi^{\\downarrow X}$.\n\n(A5) \\emph{Combination}: For $\\phi, \\psi \\in \\Phi$ with $d(\\phi) = X,\\ d(\\psi) = Y$\nand $Z \\in D$ such that $X \\subseteq Z \\subseteq X \\cup Y$,\n$(\\phi \\otimes \\psi)^{\\downarrow Z} = \\phi \\otimes \\psi^{\\downarrow Z \\cap Y}$.\n\n(A6) \\emph{Domain}: For $\\phi \\in \\Phi$ with $d(\\phi) = X$,\n$\\phi^{\\downarrow X} = \\phi$.\n\n\\parskip 0.2cm\nFor the intuitive reading of these axioms, we refer the reader to \\cite{generic-inference,shenoy2008axioms}.\n\nBefore proceeding to approximate inference, we formally define the inference problem:\n\\begin{defn}\nThe \\emph{inference problem} is the task of computing\n\\begin{equation}\n\\phi^{\\downarrow X} = (\\phi_1 \\otimes \\cdots \\otimes \\phi_r)^{\\downarrow X}\n\\end{equation}\nfor a given knowledgebase $\\{\\phi_1,\\ldots,\\phi_r\\} \\subseteq \\Phi$; domain $X$ is the \\emph{query} for the inference problem.\n\\end{defn}\n\nNext we consider \\emph{approximate inference}. Existing approximation schemes, like the mini-bucket scheme \\cite{dechter1998bucket} are either not general enough or do not provide a reliable measure of the approximation and how to improve the approximation in an anytime algorithm. In this article, we have used the ordered valuation algebra framework defined in \\cite{Haenni20041} as a basis for constructing an anytime algorithm. We thus review the extra axioms of the ordered valuation algebra framework, which\nintroduces the notion of a partial order into the valuation algebra, and defines a partial combination operator\n$\\otimes_t$ to construct approximate inference algorithms.\n\nFirstly we define a relation $\\succeq$ which represents an information ordering.\nIf $\\phi,\\phi'$ are\ntwo valuations, then $\\phi \\succeq \\phi'$ means that $\\phi$ is\n\\emph{more complete} than $\\phi'$. Intuitively, the information contained\nin $\\phi$ is more informative and a better approximation than the information\ncontained by $\\phi'$; generally this means $\\phi'$ has a more compact or sparse representation\nthan $\\phi$. Furthermore, we assume that this relation is a \\emph{partial order}. It is also reasonable to assume that approximations are only valid for valuations\nwith equal domains; thus $\\phi \\succeq \\phi'$ implies $d(\\phi) = d(\\phi')$ for all $\\phi,\\phi' \\in \\Phi$.\nThus $\\succeq$ actually defines separate completeness relations $\\succeq_D$ for each sub-semigroup $\\Phi_D$.\n\nWe also impose the condition of each sub-semigroup $\\Phi_D$ having a \\emph{zero}\nelement, denoted by $n_D$, where $\\phi \\otimes n_D = n_D \\otimes \\phi = n_D$. For notational simplicity we\nshall also denote the neutral element by $\\varnothing$ (without a subscript), denoting the appropriate\nneutral element corresponding to a particular domain.\n\nAn ordered valuation algebra is still a valuation algebra, so it retains\nall the axioms (A1)-(A6) introduced previously. The additional axioms are about how $\\succeq$ behaves under\ncombination and marginalization:\n\n(A7) \\emph{Partial order}: There is a partial order $\\succeq$ on $\\Phi$\nsuch that $\\phi \\succeq \\phi'$ implies $d(\\phi) = d(\\phi')$ for all\n$\\phi,\\phi' \\in \\Phi$.\n\n(A8) \\emph{Zero element}: We assume that the zero element for the combination operation, $n_D$ is the least element of the approximation order $\\succeq_D$ for all $D \\subseteq V$. Also, since\nzero elements for a particular domain are unique, $n_{D_1} \\otimes n_{D_2} = n_{D_1 \\cup D_2}$ for\n$D_1,D_2 \\subseteq V$. Also, $n_D^{\\downarrow D'}= n_{D'}$ for all $D' \\subseteq D$.\n\n(A9) \\emph{Combination preserves partial order}: If $\\phi_1,\\phi_1',\\phi_2,\\phi_2' \\in \\Phi$\nare valuations such that $\\phi_1 \\succeq \\phi_1'$ and $\\phi_2 \\succeq \\phi_2'$, then\n$\\phi_1 \\otimes \\phi_2 \\succeq \\phi_1' \\otimes \\phi_2'$\n\n(A10) \\emph{Marginalisation preserves partial order}: If $\\phi,\\phi' \\in \\Phi$ are\nvaluations such that $\\phi \\succeq \\phi'$, then $\\phi^{\\downarrow D} \\succeq \\phi'^{\\downarrow D}$\nfor all $D \\subseteq d(\\phi) = d(\\phi')$.\n\n\\begin{defn}\nThe \\emph{time-bounded combination operator} \\cite{Haenni20041} $\\otimes_t: \\Phi \\times \\Phi \\rightarrow \\Phi$ is used to approximate the exact computation during the propagation phase. $\\otimes_t$ performs a partial combination of two valuations within time $t$ units, where $t \\in \\mathbb{R}^+$. The following properties are satisfied by $\\otimes_t$:\n\n(R1) $\\phi_1 \\otimes \\phi_2 \\succeq \\phi_1 \\otimes_t \\phi_2$.\n\n(R2) $\\phi_1 \\otimes_{t'} \\phi_2 \\succeq \\phi_1 \\otimes_t \\phi_2$ for all $t' > t$.\n\n(R3) $\\phi_1 \\otimes_0 \\phi_2 = n_{d(\\phi_1) \\cup d(\\phi_2)}$.\n\n(R4) $\\phi_1 \\otimes_\\infty \\phi_2 = \\phi_1 \\otimes \\phi_2$.\n\\end{defn}\n\n\\begin{defn}\n \\label{def-ordered-valuation-algebra}\nSuch a system $\\langle \\Phi,V,\\succeq,d,\\otimes,\\downarrow,\\otimes_t \\rangle$ of valuations $\\Phi$, variables $V$, a completeness relation $\\succeq$ and a time-bounded combination operation $\\otimes_t$ is called an \\emph{ordered valuation algebra},\nif the labeling operations $d$, combination $\\otimes$ and marginalization $\\downarrow$\nsatisfy (A1)-(A10).\n\\end{defn}\n\n\\begin{defn}\nA binary join tree (BJT) $N = \\langle V,E\\rangle$ corresponding to a\nknowledgebase $\\{\\phi_1,\\ldots,\\phi_r\\}$ is a covering junction tree for\nthe inference problem, constructed in a manner such that the tree is binary. The valuations in the knowledgebase\nform the leaves of the tree, thus $|V(N)| = 2r - 1, |E(N)| = 2r - 2$, while\nthe query $X \\subseteq d(\\mathit{root}(N))$. Inference takes place by message passing in the BJT (for details of the algorithm, see \\cite{shenoy1997binary, Haenni20041}).\nIn the next section we shall modify this message passing algorithm to cache partial\nvaluations for anytime inference.\n\\end{defn}\n\n\\section{Anytime Ordered Valuation Algebras}\n\nIn this section, we augment ordered valuation algebras in a structure\nwe refer to as \\emph{anytime ordered valuation algebras}. We\nintroduce the extension, and in the following section give examples of\nanytime ordered valuation algebras. The primary purpose of introducing anytime ordered\nvaluation algebras is to develop an anytime inference algorithm\nwithin the framework of generic inference. Such extensions\npreserve the generic structure of valuation algebras, but add restrictions to simplify or add features to\nthe inference algorithm; in another instance, valuation algebras were extended to weighted valuation algebras\nto study communication complexity \\cite{Pouly05minimizingcommunication}.\n\nBefore defining anytime ordered valuation algebras, we define\na couple of prerequisites; the \\emph{composition operation} and a \\emph{truncation function}.\n\n\\begin{defn}\nThe \\emph{composition operator}, $\\oplus:\\Phi \\times \\Phi \\rightarrow \\Phi; (\\phi', \\phi'') \\mapsto \\phi$ combines valuations $\\phi'$ and $\\phi''$ into a valuation $\\phi$ more complete than either ($\\phi \\succeq \\phi', \\phi \\succeq \\phi''$). This is not to be confused with the combination operation $\\otimes$ which generally combines valuations from different domains. The valuations being composed belong to the same\napproximation order $\\succeq_D$, where $D = d(\\phi') = d(\\phi'') = d(\\phi)$. It is natural in this context to consider whether composition should be a supremum operation. However, this cannot be assumed in general.\n\\end{defn}\n\\begin{defn}\nThe \\emph{truncation function} $\\rho: \\Phi \\times \\mathbb{R}^+ \\rightarrow \\Phi$\nperforms a truncation of the information contained in the valuation, according\nto the real valued parameter. Also, $\\rho$ is defined to be \\emph{monotonically increasing}\nwith the real valued parameter, thus $\\rho(\\phi,k)\\succeq \\rho(\\phi,k')\\ \\mathrm{whenever}\\ k \\ge k'$.\n\\end{defn}\nThe time-bounded combination operation $\\otimes_t$ can be recast such that truncation of the original pair of valuations followed by exact combination is equivalent to doing a time-bounded combination:\n\\begin{equation}\n\\phi_1 \\otimes_t \\phi_2 = \\rho(\\phi_1, k_1) \\otimes \\rho(\\phi_2, k_2)\n\\label{otimes-t-k1-k2}\n\\end{equation}\nThe parameters $k_1, k_2$ determining the truncated portions of $\\phi_1, \\phi_2$ will be important\nlater in defining the partial valuations which will be used in the refinement algorithm for\nanytime inference. \nAs $k_1, k_2$ are parameters that depend on the particular valuations $\\phi_1, \\phi_2$ and the time $t$, this assumes a function $K(\\phi_1,\\phi_2,t) = (k_1, k_2)$.\n\nFollowing these two definitions, we extend the system of axioms (A1-A10) for\nordered valuation algebras, with the properties (P1) and (P2):\n\n(P1) The combination operation $\\otimes$ distributes over $\\oplus$:\n\\begin{flalign}\n\\quad& (\\phi_1' \\oplus \\phi_1'')\\otimes(\\phi_2' \\oplus \\phi_2'')=\\nonumber &\\\\\n\\quad& (\\phi_1' \\otimes \\phi_2') \\oplus \\underbrace{(\\phi_1' \\otimes \\phi_2'')\n \\oplus (\\phi_1'' \\otimes \\phi_2') \\oplus (\\phi_1'' \\otimes \\phi_2'')}_{\\Large \\textsc{refine}'(\\phi_1',\\phi_1'',\\phi_2',\\phi_2'')} &\n\\end{flalign}\n\nHere, $\\phi_1' \\otimes \\phi_2' = \\rho(\\phi_1, k_1) \\otimes \\rho(\\phi_2, k_2)$ is a truncated valuation of the exact combined valuation $\\phi_1 \\otimes \\phi_2$; $\\textsc{refine}'$ is the part of the exact valuation that needs to be composed with the truncated valuation $\\phi'_1 \\otimes \\phi'_2$ to complete the valuation. We also use the time-bounded operation $\\textsc{refine}'_t$ for the same operation bounded by a time $t$, with an analogous definition in terms of truncation functions as $\\otimes_t$ in equation \\ref{otimes-t-k1-k2}:\n\\begin{equation}\n\\textsc{refine}^\\prime_t(\\phi_1',\\phi_1'',\\phi_2', \\phi_2'') =\n\\textsc{refine}^\\prime(\\phi_1', \\rho(\\phi_1'', k_1), \\phi_2', \\rho(\\phi_2'', k_2))\n\\end{equation}\nwhere the parameters $k_1, k_2$ are obtained from an assumed function $K'(\\phi_1,\\phi_1',\\phi_2',\\phi_2'', t) = (k_1,k_2)$.\n\n\\vskip 0.2cm\n(P2) The projection operation $\\downarrow$ distributes over $\\oplus$:\n\\begin{equation}\n(\\phi' \\oplus \\phi'')^{\\downarrow D} = \\phi'^{\\downarrow D} \\oplus\n \\phi''^{\\downarrow D}, D \\subseteq d(\\phi).\n\\end{equation}\n\\vskip 0.2cm\nWe can now formally define the anytime ordered valuation algebra.\n\n\\begin{defn}\nAn \\emph{anytime ordered valuation algebra} is an ordered valuation\nalgebra $\\langle V,\\Phi,d,\\otimes,\\downarrow,\\otimes_t,\\succeq\\rangle$ with the additional operations of composition\n$\\oplus$ and the function $\\rho$, making the structure $\\langle V,\n\\Phi,d,\\rho,\\otimes,\\downarrow,\\oplus,\\otimes_t,\\succeq\\rangle$,\nwhich satisfies properties (P1) and (P2).\n\\end{defn}\n\nWe show by construction that\nthe composition operator\\\\ $\\oplus: \\Phi \\times \\Phi \\rightarrow \\Phi$ with (P1, P2) along with the truncation function $\\rho: \\Phi \\times \\mathbb{R}^+\n\\rightarrow \\Phi$ is \\emph{sufficient} to construct a\nrefinement algorithm to improve the accuracy of a valuation.\n\nTo describe a refinement algorithm to improve upon the result provided by\n\\textsc{inward}$(N,t)$, we need to cache the partial valuations at each\nstep so that we can use $\\textsc{refine}'$ to improve upon them. We use\na modified version of the propagation algorithm \\cite{shenoy1997binary,Haenni20041}, where $\\tau$ and\n$\\bar\\tau$ store the partial and complementary partial valuations\nrespectively for a particular BJT node, where the complementary partial valuation $\\bar\\rho(\\phi, k)$ is such that $\\bar\\rho(\\phi, k) \\oplus \\rho(\\phi, k) = \\phi$. In the following procedures, $\\Delta(n) = d(n)\\backslash d(P(n))$ is the set of variables to be eliminated as we propagate messages to the parent node. To get the solution to the inference problem at the final step, we also define $\\Delta(\\mathit{root}(N)) = d(\\mathit{root}(N)) \\backslash X$ where $X$ is the query. There are $r$ valuations in the knowledgebase resulting in $r-1$ combination steps in the BJT. $P(n)$ is the parent of $n$, $\\phi(n)$ is the valuation at node $n$, $\\phi_s(n)$ is the message from $n$ to $P(n)$; $L(n), R(n)$ are the left and right nodes of $n$ respectively and\n\\begin{equation}\n\\mathit{next}(N) = \\{n \\in N: \\phi_s(n) = \\mathit{nil},\n \\phi_s(L(n)) \\ne \\mathit{nil}, \\phi_s(R(n)) \\ne \\mathit{nil} \\}\n\\end{equation}\nBoth $\\textsc{inward}(N,t)$ and $\\textsc{refine}(N,t)$ return valuations which are the (approximate) solution to the inference problem.\n\n\\hrulefill\n\\begin{algorithmic}[1]\n\\Procedure{inward}{$N,t$}\n\\State $s \\leftarrow r-1$;\n\\State initialise timer to $t$ units.\n\\State for all $n \\in \\mathit{leaves}(N)$ do $\\phi_s(n) \\leftarrow \\phi(n)^{-\\Delta(n)}$\n\\While {$\\mathit{next}(N) \\ne \\emptyset$}\n\\State \\textbf{select} $n \\in next(N)$\n\\State $(k_1, k_2) \\gets K(\\phi_s(L(n)), \\phi_s(R(n)), t\/s)$\n\\State $\\phi(n) \\gets \\phi_s(L(n)) \\otimes_{t\/s} \\phi_s(R(n))$\n\\State $\\tau(L(n)) \\gets \\rho(\\phi_s(L(n)), k_1)$\n\\State $\\tau(R(n)) \\gets \\rho(\\phi_s(R(n)), k_2)$\n\\State $\\bar\\tau(L(n)) \\gets \\bar\\rho(\\phi_s(L(n)), k_1)$\n\\State $\\bar\\tau(R(n)) \\gets \\bar\\rho(\\phi_s(R(n)), k_2)$\n\\State $\\phi_s(n) \\leftarrow \\phi(n)^{-\\Delta(n)}$\n\\State $s \\leftarrow s-1$\n\\State $t \\leftarrow \\mathit{timer}()$\n\\EndWhile\n\n\\State \\Return $\\phi_s(\\mathit{root}(N))$\n\\EndProcedure\n\\end{algorithmic}\n\\hrulefill\n\nWe can use the cached partial valuations in $\\tau$ and $\\bar\\tau$ to define\nthe refinement algorithm that follows in a similar manner to the algorithm in \\cite{Haenni2002103}.\n\n\\hrulefill\n\\begin{algorithmic}[1]\n \\Procedure{refine}{$N,t$}\n\\State \\textbf{initialise} timer to $t$ units\n\\State $s\\gets r-1$\n\\While{$next(N)\\not=\\emptyset$}\n\\State \\textbf{select} $n \\in next(N)$\n\\State $(k_1, k_2) \\gets K'(\\tau(L(n)), \\bar\\tau(L(n)), \\tau(R(n)), \\bar\\tau(R(n)), t\/s)$\n\\State $\\nu \\gets \\textsc{refine}'_{t\/s}(\\tau(L(n)), \\bar\\tau(L(n)), \\tau(R(n)), \\bar\\tau(R(n)))$\n\\State $t \\gets timer()$\n\\State $\\tau(L(n)) \\gets \\tau(L(n)) \\oplus \\rho(\\bar\\tau(L(n)), k_1)$\n\\State $\\tau(R(n)) \\gets \\tau(R(n)) \\oplus \\rho(\\bar\\tau(R(n)), k_2)$\n\\State $\\bar\\tau(L(n)) \\gets \\bar\\rho(\\bar\\tau(L(n)), k_1)$\n\\State $\\bar\\tau(R(n)) \\gets \\bar\\rho(\\bar\\tau(R(n)), k_2)$\n\\State $\\phi(n)\\gets \\phi(n) \\oplus \\nu$\n\\State $\\bar\\tau(n)\\gets\\bar\\tau(n) \\oplus \\nu^{-\\Delta(n)}$\n\\State $s\\gets s-1$\n\\EndWhile\\label{euclidendwhile}\n\\State \\textbf{return} $\\phi_s(root(N))$\n\\EndProcedure\n\\end{algorithmic}\n\\hrulefill\t\n\nThis procedure refines the existing valuations in the binary join tree $N$, taking at most time $t$ units. We ensure that the algorithm\nis interruptible in lines 9--12 using appropriate caching of partial valuations. A diagram of the truncation of a valuation is shown below\nto illustrate anytime refinement.\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.7]{valuation.pdf}\n\\end{figure}\n\nHere, and in the following proof, the notation $\\phi^k := \\rho(\\phi,k)$\nand $\\phi_k := \\bar\\rho(\\phi,k)$. We shall also abbreviate the notation $\\tau(L(n))$ as $\\tau_L$ and $\\bar\\tau(L(n))$ as $\\bar\\tau_L$ (accordingly for $R(n)$), and $\\bar\\tau(n)$ as $\\bar\\tau$. The shaded region $\\phi^k$ is the\npart that has already been combined, while $\\phi_k$ represents the\ncached part that has not been combined yet. The dotted\nregion represents the part of $\\phi$ that is yet uncomputed, due to\ntruncated messages from child nodes; line 14 in $\\textsc{refine}(N,t)$ shrinks the uncomputed portion by extending $\\bar\\tau$.\n\n\\begin{thm}[Soundness of anytime inference]\nIf $\\phi_{[t_0,t_1,\\ldots,t_j]}$ is the valuation returned after the following invocations:\n$\\left[\\textsc{inward}(N_0,t_0>0),\\ \\textsc{refine}(N_1, t_1), \\ldots,\\ \\textsc{refine}(N_j, t_j) \\right]$, where $N_{k+1}$ is the modified BJT with the cached valuations after step $k$, then\n$\\phi_{[t_0]} \\preceq \\phi_{[t_0,t_1]} \\preceq \\cdots \\preceq \\phi_{[t_0,t_1,\\ldots,t_j]} \\preceq \\cdots \\preceq \\phi$ where $\\phi$ is the exact valuation. The sequence becomes strictly increasing (upto the exact valuation) if $t_i > t_\\epsilon$ for all $i>0$ where $t_\\epsilon$ is the minimum time required for the refinement to update one valuation.\n\\end{thm}\n\\begin{proof}\nWe split the proof into two parts: (S1) proving that the sequence of valuations\nreturned from successive calls to $\\textsc{refine}$ are partially ordered and (S2) showing the upper bound is the exact valuation, to which the partial valuations\nconverge after a finite time.\n\nProving (S1) is trivial; for each node, $\\phi$ is updated once (line 13), thus $\\phi' = (\\phi \\oplus \\nu) \\succeq \\phi$, where $\\phi'$ is the valuation at node $n$ after a call to $\\textsc{refine}$. Using transitivity of the partial order, we obtain (S1). In the case\nwhen $t_i > t_\\epsilon$, at least one valuation is updated, resulting in $\\nu \\succ \\varnothing$, which gives $\\phi' \\succ \\phi$.\n\nTo prove (2) we shall note the following statements\n\n(T1) $(\\phi_k)^m = (\\phi^{k+m})_k$\n\n(T2) $\\phi^k \\oplus \\phi_k = \\phi$\n\n(T3) $\\phi^k \\oplus (\\phi_k)^m = \\phi^{k+m}$\n\n(T4) $(\\phi_k)_m = \\phi_{k+m}$\n\nFor notational simplicity, only for the following proof, we denote $\\phi\\psi := \\phi\\otimes\\psi$ and $+ := \\oplus$.\n\nSince each node is only updated once, we can consider a particular node; let's denote by $\\phi$ the valuation at node $n$ after $\\textsc{inward}(N,t_0)$. If\n$(k_1, k_2)$ are the parameters obtained from $K'$ in $\\textsc{refine}(N_1,t_1)$ then the updated valuation\n$\\phi' = \\phi + \\tau_L\\bar\\tau_R^{k_2} + \\bar\\tau_L^{k_1}\\tau_R +\n\\bar\\tau_L^{k_1}\\bar\\tau_R^{k_2}$, where $\\phi = \\tau_L\\tau_R$.\n\nHere we note that we can replace $(\\bar\\tau_{L,R})^k$ with their exact counterpart $(\\bar\\tau^{\\infty}_{L,R})^k$, where\nwe use the $\\bar\\tau^\\infty$ to denote the exact valuation. This can be done as the truncation function is invariant\nunder extension of the valuation to incorporate previously uncomputed information. Following this, we shall\ndrop the superscript and use $\\bar\\tau_L$ to denote $\\bar\\tau^\\infty_L$.\n\nThen if we consider a subsequent call, $\\textsc{refine}(N_2,t_2)$,\n$\\phi'' = \\phi' + \\tau_L'\\bar\\tau_R^{\\prime m_2} + \\bar\\tau_L^{\\prime m_1}\\tau_R' +\n\\bar\\tau_L^{\\prime m_1}\\bar\\tau_R^{\\prime m_2}$.\nwhere the additional prime indicates the the value for this iteration, and\n$(m_1,m_2)$ are the parameters obtained from $K'$.\n\nFrom lines 9--12 in \\textsc{refine} we get: $\\tau_L' = \\tau_L + \\bar\\tau_L^{k_1}$,\n$\\tau_R' = \\tau_R + \\bar\\tau_R^{k_2}$,\n$\\bar\\tau'_L = (\\bar\\tau_L)_{k_1}$,\n$\\bar\\tau'_R = (\\bar\\tau_R)_{k_2}$\n\nExpanding $\\phi''$ we get:\n\\begin{eqnarray*}\n\\phi''&=&\\tau_L\\tau_R + \\tau_L\\bar\\tau_R^{k_2} + \\bar\\tau_L^{k_1}\\tau_R +\n\\bar\\tau_L^{k_1}\\bar\\tau_R^{k_2}\\\\\n&&+\\ (\\tau_L + \\bar\\tau_L^{k_1})(\\bar\\tau_{R,k_2})^{m_2} +\n(\\bar\\tau_{L,k_1})^{m_1}(\\tau_R + \\bar\\tau_R^{k_2}) + (\\bar\\tau_{L,k_1})^{m_1}(\\bar\\tau_{R,k_2})^{m_2}\\\\\n&=&\\tau_L\\tau_R + \\tau_L\\bar\\tau_R^{k_2} + \\bar\\tau_L^{k_1}\\tau_R +\n\\bar\\tau_L^{k_1}\\bar\\tau_R^{k_2} + \\tau_L(\\bar\\tau_{R,k_2})^{m_2}\\\\\n&&+ (\\bar\\tau_L^{k_1})(\\bar\\tau_{R,k_2})^{m_2} +\n(\\bar\\tau_{L,k_1})^{m_1}\\tau_R + (\\bar\\tau_{L,k_1})\\bar\\tau_R^{k_2} +\n(\\bar\\tau_{L,k_1})^{m_1}(\\bar\\tau_{R,k_2})^{m_2}\\\\\n&=&\\tau_L\\tau_R + \\tau_L\\bar\\tau_R^{k_2+m_2} +\n\\bar\\tau_L^{k_1+m_1}\\tau_R + \\bar\\tau_L^{k_1+m_1}\\bar\\tau_R^{k_2+m_2}\n\\end{eqnarray*}\nHere we use (T1,T3) to simplify the expression. Note that\nthis is the same form as $\\phi' = \\phi + \\tau_L\\bar\\tau_R^{k_2} + \\bar\\tau_L^{k_1}\\tau_R + \\bar\\tau_L^{k_1}\\bar\\tau_R^{k_2}$, with $k_1 \\rightarrow k_1+m_1,\\ k_2 \\rightarrow k_2 + m_2$. Thus, subsequent calls to $\\textsc{refine}$ will always\nresult in $\\phi$ having the same form by induction. From the definition of the\ntruncation function, $\\phi^k \\succeq \\phi^{k'}$ for $k \\ge k'$, from which (S1)\nfollows as well, by preservation of partial order under combination and composition.\nTo show (S2) we note that for finite valuations, there exists $k$, such that $\\phi^k = \\phi$. As the exponent is monotonically increasing with subsequent calls\nto $\\textsc{refine}$, we shall eventually get $\\phi_{[t_0,t_1,\\ldots,t_j]} = \\tau_L\\tau_R\n+ \\tau_L\\bar\\tau_R + \\bar\\tau_L\\tau_R + \\bar\\tau_L\\bar\\tau_R =\n(\\tau_L+\\bar\\tau_L)(\\tau_R+\\bar\\tau_R)$, the exact valuation at node $n$. Thus, we\nshall eventually get the exact valuation at the root after finite invocations of $\\textsc{refine}$.\n\n\\end{proof}\n\n\n\\begin{thm}[Completeness of anytime inference]\nIf $\\phi_{[t_0,t]}$ is the valuation returned after the following invocations: $\\left[\\textsc{inward}(N,t_0>0),\\ \\textsc{refine}(N', t)\\right]$, where $N'$ is the modified BJT with the cached valuations after the call to $\\textsc{inward}(N,t_0)$, then\nthere exists a $T$ such that for all $t \\ge T$\n$\\phi_{[t_0,t]} = \\phi = (\\bigotimes_{\\psi \\in \\Phi} \\psi)^{\\downarrow X}$, the\nexact solution to the inference problem.\n\\end{thm}\n\\begin{proof}\nWe consider two cases:\n\n\\textbf{Case 1}: $\\textsc{inward}(N,t_0)$ has performed exact inference.\n\nWe shall show that $\\textsc{refine}(N,t)$ is a null operation which\ndoes not change $\\phi,\\tau,\\bar\\tau$; then the statement of the theorem follows if we set $T = t_0$.\n\n$\\phi' = \\phi \\oplus \\nu$ (line 13), so if we show $\\nu = \\varnothing$, we are done.\n\n$\\nu = \\textsc{refine}'_{t\/s}(\\tau_L,\\bar\\tau_L,\\tau_R,\\bar\\tau_R)$, but $\\bar\\tau_L = \\bar\\tau_R = \\varnothing$ as $\\bar\\tau$ represents the partial valuation that has not been combined, which is null for the exact inference case. Thus $\\nu = \\varnothing$.\n\n\\textbf{Case 2}: $\\textsc{inward}(N,t_0)$ gives a partial result.\n\nIn general, $\\nu$ is also a partial valuation due to the time restriction. \nSince we are operating on finite datasets,\nthe combination operation at a particular node in $\\textsc{refine}'$ takes a finite amount of time, say $t_n$. Thus $\\textsc{refine}'_{t_n}$ at a node $n$ is the\nexact refinement, making $\\phi(n)$ exact after line 13, and thus $m(root(N))$ is exact after completion of the propagation. So we set $T = \\sum_{n \\in V} t_n$ to\nget the time bound, such that for all $t \\ge T$ we get the exact result.\n\\end{proof}\n\n\\section{Instances of anytime ordered valuation algebras}\n\nIn the following sections, we describe instances of anytime ordered valuation algebras. Specifically we show that the important class of semiring induced valuation algebras,\n\\cite{kohlas2008semiring}, can be considered as anytime ordered valuation algebras. We also remark on the application of our framework to belief potentials.\n\n\\subsection{Semiring induced valuation algebras}\n\nSemiring induced valuation algebras are a subclass of valuation algebras with several useful instances like probability potentials and disjunctive normal forms. We use the definition of semiring induced valuation algebras from \\cite{kohlas2008semiring} and review the following standard notation. The semiring is denoted by $\\mathcal{A} = \\langle A,+,\\times\\rangle$ with the semiring operations $+,\\times$ on a set $A$, where $+, \\times$ are assumed to be commutative and associative, with $\\times$ distributing over $+$. Lowercase letters like $x$\nare variables, with a corresponding finite set of values for $x$, called the \\emph{frame} of $x$\nand denoted by $\\Omega_x$. Each $\\Omega_x$ also has an associated total order on its elements. If the frame has two elements, then it is the frame of a \\emph{binary\nvariable}. If the binary elements represent true and false, then we call the variable \\emph{propositional}.\nFor a domain $D \\subseteq V$ where $V$ is the set of all variables in the system,\nthe corresponding set of possible values becomes the Cartesian product $\\Omega_D = \\prod \\{\\Omega_x: x \\in D\\}$, whose\nelements $\\mathbf{x} \\in \\Omega_D$ are called \\emph{D-configurations} or \\emph{D-tuples}. For a subset $D' \\subseteq D$,\n$\\mathbf{x}^{\\downarrow D'} \\in \\Omega_{D'}$ is the projection of $\\mathbf{x}$ to $D'$. Where $D$ is empty,\nwe use the convention that the frame is a singleton set: $\\Omega_\\phi = \\{\\diamond\\}$.\nAny set of $D$-configurations can be ordered using a lexicographical order.\n\n\\begin{defn}\nAn $\\mathcal{A}$-valuation $\\phi$ with domain $D$ associates a value in $A$ with each configuration $\\mathbf{x} \\in \\Omega_D$, i.e. $\\phi$ is a function $\\phi: \\Omega_D \\rightarrow A$. \n\\end{defn}\n\nThe set of all such $\\mathcal{A}$-valuations with a domain $D$ is denoted by $\\Phi_D$, and the union of all such sets with $D \\subseteq V$ is the set of all $\\mathcal{A}$-valuations $\\Phi$. The operations $+,\\times$ on $A$ then induce a valuation algebra structure on $\\langle \\Phi, \\mathcal{P}_f(V) \\rangle$ where $\\mathcal{P}_f(V)$ is the finite powerset of the set of variables $V$\n\\cite[Theorem 2]{kohlas2008semiring}, using the following definitions of combination and projection:\n\n\\begin{enumerate}\n\\item \\emph{Combination}: $\\otimes: \\Phi \\times \\Phi \\rightarrow \\Phi$ defined for $\\mathbf{x} \\in \\Omega_{d(\\phi) \\cup d(\\psi)}$ by\n\\begin{equation}\n\\phi \\otimes \\psi (\\mathbf{x}) = \\phi(\\mathbf{x}^{\\downarrow d(\\phi)}) \\times \\psi(\\mathbf{x}^{\\downarrow d(\\psi)})\n\\end{equation}\n\n\\item \\emph{Projection}: $\\downarrow: \\Phi \\times D \\rightarrow \\Phi$ defined for all $\\phi \\in \\Phi$ and $T \\subseteq d(\\phi)$ for $\\mathbf{x} \\in \\Omega_T$ by\n\\begin{equation}\n\\phi^{\\downarrow T}(\\mathbf{x}) = \\sum_{\\mathbf{z} \\in \\Omega_{d(\\phi)}:\\ \\mathbf{z}^{\\downarrow T} = \\mathbf{x}} \\phi(\\mathbf{z})\n\\end{equation}\n\\end{enumerate}\n\n\\begin{thm}\n\\label{semiring-ova}\nSemiring induced valuation algebras, provided the underlying semiring has a zero element, form an ordered valuation algebra.\n\\end{thm}\n\\begin{proof}\nTo show semiring induced valuation algebras are an ordered valuation algebra, we have to show\n(A7-A10):\n\n(A7) The \\emph{preorder} $\\succeq$ is defined by $\\phi \\succeq \\phi'$ iff $\\phi(\\mathbf{x}) \\succeq_A \\phi'(\\mathbf{x})$ for all $\\mathbf{x} \\in \\Omega_{d(\\phi)}$, where $\\succeq_A$ is the preorder on the semiring \\cite[Prop. 1, p1362]{kohlas2008semiring} defined as $b \\succeq_A a$ iff $a = b$ or there exists $c$ such that $a+c=b$, with $d(\\phi) = d(\\phi')$ as it only makes sense to compare valuations on the same domain. However we need a \\emph{partial order} for this axiom, which is possible if the additive monoid is positive, has a zero element and is cancellative:\n\\begin{lem}\nThe preorder $\\preceq$ defined on a positive, cancellative, commutative monoid, $\\langle A, + \\rangle$ with a zero element, is a partial order.\n\\end{lem}\n\\begin{proof}\nA preorder implies $a \\preceq b$ iff $a + c = b$. For a partial order, we need asymmetry: if $a \\preceq b$ and $b \\preceq a$, then $a=b$.\n\n$a \\preceq b$ implies there exists $c$ such that $a + c = b$; similarly there exists $d$ such that $b + d = a$; substituting gives us $b + d + c = b + 0$,\nthe cancellative property implies $d + c = 0$ and the positivity property\nimplies $c = d = 0$, implying $a = b$, and we have a partial order.\n\\end{proof}\n\n(A8) \\emph{Zero element}: Most common\ninstances of semiring induced valuation algebras have a zero element. Specifically semirings with \\emph{zero elements} induce valuation algebras with\nthe zero element $n_D$ such that $n_D(\\mathbf{x}) = 0$ for all $\\mathbf{x} \\in \\Omega_D$.\n\n(A9, A10) \\emph{Combination and marginalisation preserve partial order}. This follows from the fact that $\\times$ and $+$\npreserve partial order in the underlying semiring structure.\n\n\\end{proof}\n\nHaving shown that semiring induced valuation algebras satisfy the ordered valuation algebra axioms (A7--A10) provided the underlying semiring has a zero element and the additive commutative monoid is cancellative and positive, we proceed to define the composition and truncation functions\nfor semiring induced valuation algebras.\n\n\\begin{enumerate}\n\\item We denote the composition operator on semiring induced valuation algebras as\n $(\\phi \\oplus \\phi') (\\mathbf{x}) = \\phi (\\mathbf{x}) + \\phi' (\\mathbf{x}),\\ d(\\phi) = d(\\phi')\n $\n\\item The function $\\rho$ is defined on the semiring induced valuation algebra as\n$\\rho(\\phi, k)=$ the first $k$ (lexicographically ordered on $\\mathbf{x}$) elements of $\\mathrm{graph}(\\phi)$; where\n$\\mathrm{graph}(\\phi) = \\{(\\mathbf{x},\\phi(\\mathbf{x}))\\ \\vert\\ \\mathbf{x} \\in \\Omega_{d(\\phi)} \\}$. For efficient implementation, we only store $(\\mathbf{x},\\phi(\\mathbf{x}))$ where $\\phi(\\mathbf{x}) \\ne 0$.\n\nIn case the semiring has a total order (as in the case of probability potentials), we order\nthe configurations in decreasing weight order: $[ (\\mathbf{x}_i, \\phi(\\mathbf{x}_i)),\n\\ldots ]$ where $\\phi(\\mathbf{x}_i) \\ge \\phi(\\mathbf{x}_j)$ for $i \\le j$.\n\\end{enumerate}\n\nWe also define the time-bounded combination operation $\\phi_1 \\otimes_t \\phi_2$,\nwhere $L_{\\phi_1} = \\left[(\\mathbf{x_1}, \\phi_1(\\mathbf{x_1})), \\ldots \\right]$,\nand $L_{\\phi_2} = \\left[(\\mathbf{y_1}, \\phi_2(\\mathbf{y_1})), \\ldots \\right]$. $\\mathbf{xy}$ denotes the configuration in $\\Omega_{d(\\phi_1) \\cup d(\\phi_2)}$ such that\n$(\\mathbf{xy})^{\\downarrow d(\\phi_1)} = \\mathbf{x}$ and\n$(\\mathbf{xy})^{\\downarrow d(\\phi_2)} = \\mathbf{y}$.\n\nWe define helper functions $\\textsc{insert}$, which inserts a combination into\nthe configuration space provided there is a common support and $\\textsc{combine-extend}$ which incrementally adds combinations into the configuration and updates\nthe state, going from the state $\\rho(\\phi_1, i) \\otimes \\rho(\\phi_2, j)$\nto $\\rho(\\phi_1, i + i') \\otimes \\rho(\\phi_2, j + j')$. Finally\nwe define $\\textsc{combine}$ which performs the combination\noperation within the allocated time constraint.\n\n\\hrulefill\n\n\n\\begin{algorithmic}[1]\n\\Function{insert}{$\\phi_1, \\phi_2, i, j, L$}\n\\State $\\mathbf{x} = L_{\\phi_1}; \\mathbf{y} = L_{\\phi_2}$\n\\If{$\\mathbf{x}_i^{\\downarrow D_1 \\cap D_2} = \\mathbf{y}_r^{\\downarrow D_1 \\cap D_2}$}\n\\State insert $[\\mathbf{x}_i\\mathbf{y}_j, \\phi_1(\\mathbf{x}_i) \\times \\phi_2(\\mathbf{y}_j)]$ into $L$.\n\\EndIf\n\\EndFunction\n\\end{algorithmic}\n\\hrulefill\n\n\\begin{algorithmic}[1]\n\\Function{combine-extend}{$\\phi_1, \\phi_2, \\langle i,j,L \\rangle, i', j'$}\n\\For{$k \\gets 1$ to $i+i'$}\n\\For{$m \\gets j$ to $j+j'$}\n\\State $\\textsc{insert}(\\phi_1, \\phi_2, k,m,L)$\n\\EndFor\n\\EndFor\n\\For{$k \\gets i$ to $i+i'$}\n\\For{$m \\gets 1$ to $j+j'$}\n\\State $\\textsc{insert}(\\phi_1, \\phi_2, k,m,L)$\n\\EndFor\n\\EndFor\n\\State \\Return $\\langle i,j,L \\rangle$\n\\EndFunction\n\\end{algorithmic}\n\n\n\\hrulefill\n\\begin{algorithmic}[1]\n\\Function{combine}{$\\phi_1, \\phi_2,t$}\n\\State $L \\leftarrow \\langle \\rangle; i \\leftarrow 1; j \\leftarrow 1;\nn_1 \\gets |L_{\\phi_1}|; n_2 \\gets |L_{\\phi_2}|$\n\\State initialise timer to $t$ units\n\\While{$\\mathit{timer}() > 0$ and $i \\le n_1$ and $j \\le n_2$}\n\\State $\\langle i, j, L \\rangle \\gets \\textsc{combine-extend}(\\phi_1, \\phi_2, \\langle i,j,L \\rangle,0,1) $\n\\If{not $\\mathit{timer}() > 0$}\n\\State \\textbf{break}\n\\EndIf\n\\State $\\langle i, j, L \\rangle \\gets \\textsc{combine-extend}(\\phi_1, \\phi_2, \\langle i,j,L \\rangle,1,0) $\n\\EndWhile\n\\If{$i > n_1$}\n\\State $m \\gets j+1$\n\\While{$\\mathit{timer}() > 0$ and $m \\le n_2$}\n\\State $\\langle i,j,L \\rangle \\gets \\textsc{combine-extend}(\\phi_1,\\phi_2,\\langle i,j,L \\rangle, 0, 1)$\n\\State $m \\gets m + 1$\n\\EndWhile\n\\Else\n\\State $m \\gets i+1$\n\\While{$\\mathit{timer}() > 0$ and $m \\le n_1$}\n\\State $\\langle i,j,L \\rangle \\gets \\textsc{combine-extend}(\\phi_1,\\phi_2,\\langle i,j,L \\rangle, 1,0)$\n\\State $m \\gets m + 1$\n\\EndWhile\n\n\\EndIf\n\\State \\Return valuation corresponding to $L$\n\\EndFunction\n\\end{algorithmic}\n\\hrulefill\n\n\\begin{thm}\nSemiring induced valuation algebras, provided the underlying semiring has a zero element, along with the composition operator and the truncation function\ndefined above form an anytime ordered valuation algebra.\n\\end{thm}\n\n\\begin{proof}\nSemiring induced valuation algebras form an ordered valuation algebra as shown in Theorem~\\ref{semiring-ova}. To show that they also constitute an anytime ordered valuation algebra, we have to show properties (P1, P2), i.e. combination and projection distribute over $\\oplus$:\n\n(P1) If $p_1 = p_1' \\oplus p_1''$ and $p_2 = p_2' \\oplus p_2''$ then we have to show that: $\np_1 \\otimes p_2 = (p_1' \\otimes p_2') \\oplus (p_1' \\otimes p_2'') \\oplus (p_1'' \\otimes p_2') \\oplus (p_1'' \\otimes p_2'')$.\n\nLHS applied to $\\mathbf{x}$ is $p_1(\\mathbf{x}^{\\downarrow S}) \\times p_2(\\mathbf{x}^{\\downarrow\nT})$, where $d(p_1) = S$ and $d(p_2) = T$.\n\\begin{eqnarray*}\n\\text{RHS is }(p_1'(\\mathbf{x}^{\\downarrow S}) \\times p_2'(\\mathbf{x}^{\\downarrow T})) +\n(p_1'(\\mathbf{x}^{\\downarrow S}) \\times p_2''(\\mathbf{x}^{\\downarrow T})) +\\\\\n(p_1''(\\mathbf{x}^{\\downarrow S}) \\times p_2'(\\mathbf{x}^{\\downarrow T})) +\n(p_1''(\\mathbf{x}^{\\downarrow S}) \\times p_2''(\\mathbf{x}^{\\downarrow T}))\\\\\n= (p_1'(\\mathbf{x}^{\\downarrow S}) + p_1''(\\mathbf{x}^{\\downarrow S})) \\times\n(p_2'(\\mathbf{x}^{\\downarrow S}) + p_2''(\\mathbf{x}^{\\downarrow T}) = \\text{LHS}\\\\\n\\text{using distributivity of} \\times \\text{over}\\ +.\n\\end{eqnarray*}\n\n(P2) We have to show that if $p = p' \\oplus p''$ that $p^{\\downarrow D} =\np'^{\\downarrow D} \\oplus p''^{\\downarrow D}$, where $D \\subseteq d(p)$. The LHS applied to $\\mathbf{x}$ is $p^{\\downarrow D}(\\mathbf{x}) = \\sum_{\\mathbf{z}^{\\downarrow D} = \\mathbf{x}} p(\\mathbf{z}) = \\sum_{\\mathbf{z}^{\\downarrow D} = \\mathbf{x}} (p' \\oplus p'')(\\mathbf{z})$, and the RHS is\n\\begin{eqnarray*}\n(p'^{\\downarrow D} \\oplus p''^{\\downarrow D})(\\mathbf{x})&=&p'^{\\downarrow D}(\\mathbf{x}) + p''^{\\downarrow D}(\\mathbf{x}))\\\\\n=\\sum_{\\mathbf{z}^{\\downarrow D} = \\mathbf{x}} p'(\\mathbf{z}) +\n\\sum_{\\mathbf{z}^{\\downarrow D} = \\mathbf{x}} p''(\\mathbf{z})&=&\n\\sum_{\\mathbf{z}^{\\downarrow D} = \\mathbf{x}} (p' \\oplus p'')(\\mathbf{z})\n\\end{eqnarray*}\nwhere we use the associativity and commutativity of $+$.\n\\end{proof}\n\nAs stated earlier, several common instances of valuation algebra can\nbe considered as semiring induced. We present a couple of important examples below:\n\n\\begin{exmp}\n\\emph{Probability potentials} are semiring induced valuation algebras on $\\mathbb{R}^+$ with the semiring operations being the arithmetic addition and multiplication. Also known as \\emph{arithmetic potentials}, these describe (unnormalised) probability distributions, and thus inference in probabilistic graphical models.\n\\end{exmp}\n\\begin{exmp}\n\\label{dnf-val}\n\\emph{Disjunctive normal forms} (abbreviated as DNF) are of the form $\\alpha_1 \\vee \\alpha_2 \\cdots \\vee \\alpha_n$ where $\\alpha_i$ is of the form $x_1 \\wedge x_2 \\wedge \\cdots \\wedge x_k$ and $x_j$ is a literal; either a logical variable or its negation. All frames are binary reflecting true and false values respectively. DNF potentials are induced by the semiring with + and $\\times$ being defined as\n$a + b = \\max(a, b)$ and $a \\times b = \\min(a, b)$; which are equivalent to\nthe definition of logical-or and logical-and.\n\\end{exmp}\n\nThere are many other examples of semiring induced valuation algebras, a\ndetailed introduction to which can be found in \\cite{kohlas2008semiring}. In certain cases,\nthe valuation algebra induced by the semiring has the idempotent property, i.e. $\\phi \\otimes \\phi = \\phi$; then we may use more efficient architectures for local computation such as the Lauritzen-Spiegelhalter architecture \\cite{kohlas2003information}.\n\nIt is also pertinent to mention that for DNF potentials, one can alternately consider the valuation algebra over the formulae itself instead of the models \\cite{kohlas1999propositional}, which simplifies computation extensively. This alternative representation is also an anytime ordered valuation algebra, but we have omitted the proof for the purposes of brevity.\n\n\\subsection{Belief functions}\n\nBelief potentials are a generalisation of probability potentials to subsets of the\nconfiguration space in Dempster-Shafer's theory of evidence \\cite{shafer1976mathematical}. The advantage of belief potentials over standard probability theory is in their\nability to express partially available information in a manner not possible\nin probability theory. This is the reason for the usage of belief functions in sensor network literature, which involves fusion of information from various sources \\cite{murphy1998dempster,denoeux2000neural,sentz2002combination,yu2005alert}.\n\nFor the instance of belief functions, with the composition operator defined as\n$[\\phi \\oplus \\phi']_m(A) = [\\phi]_m(A) + [\\phi']_m(A)$, where $[\\phi]_m$ is the mass function associated with the belief function $\\phi$, our framework specialises to anytime inference in belief potentials as described in \\cite{Haenni2002103}.\n\n\\begin{thm}\nBelief functions, along with the composition operator defined above, and the truncation operation\n$\\rho(\\phi,k)$ as the potential that contains the $k$ focal sets\nof $\\phi$ with the highest masses, form an anytime ordered valuation algebra.\n\\end{thm}\n\n\\begin{proof}\nBelief functions already form an ordered valuation algebra \\cite{Haenni20041}, as well as permit\nanytime inference \\cite{Haenni2002103}. The anytime inference algorithm in \\cite{Haenni2002103} turns\nout to be a specific case of the generic anytime inference framework presented in this article. In particular\nif we denote $\\oplus := +$ in their notation, and the truncation function $\\rho(\\phi,k) := \\rho_k(\\phi)$ then\n\\cite[Theorem 9,10]{Haenni2002103} shows that belief functions also form an anytime ordered valuation\nalgebra according to the axioms in Section 3.\n\\end{proof}\n\n\\section{Complexity Analysis}\n\nThe anytime inference algorithm presented in Section 3 hides the time complexity of approximate inference by restricting\nthe accuracy of the valuations. While we don't have an explicit control over the accuracy, we can improve it by allocating more time to the refinement algorithm. In this section, we take an alternative approach of focussing on accuracy and estimating the\ntime complexity, which also allows us to use a tuning parameter which scales from zero accuracy (null valuations) to the valuation obtained after exact inference.\n\nSince complexity of exact (and approximate) inference depends upon the complexity of the combination operation (usually\nthe more time-consuming operation among combination and focussing), we consider the specific instance of semiring-induced valuation algebras. As there are $n$ valuations, $\\phi_1, \\ldots, \\phi_n$, the resulting BJT $N$ will have $2n-1$ nodes, $n$ of which are the valuations themselves at the leaves of the tree. We denote the maximum frame size of a variable in the semiring induced valuation algebra as $m := \\max \\{ |\\Omega_x|, x \\in V \\}$. As we are representing semiring induced valuations in memory in terms of a tuple of\nthe configuration and its associated value, the number of words required to represent\nthe configuration is a key component in the time and communication complexity. The upper bound on the size of the configuration space for a valuation is thus $m^{|d(\\phi)|}$.\n\n\\begin{defn}\nThe \\emph{approximation parameter} $k$ is a tunable parameter that goes from 0 to $m^\\omega$, where $\\omega$ is the treewidth of the binary join tree $N$. \n\\end{defn}\n$m^\\omega$ is the maximum\nsize of the configuration space that we have to process during the inward or outward propagation phase of the Shenoy-Shafer algorithm. Now we can define the following.\n\n\\begin{defn}\nThe approximate combination operation $\\otimes^k: \\Phi \\times \\Phi \\rightarrow \\Phi$ is defined as combining\nthe elements of the configuration space of the valuations in a semiring-induced valuation algebra, until we get $k$ resultant elements.\n\\end{defn}\n\\begin{lem}\nThe complexity of the approximate combination operator $\\otimes^k$ is $O(k)$.\n\\end{lem}\n\\begin{proof}\nThe worst-case scenario is when the configuration spaces are independent (no variables in common). Then there is no requirement for common support and we can take\nthe pairwise multiplication of the elements of the configuration space, till we get $k$ elements, giving us $O(k)$ complexity.\n\\end{proof}\n\nThe \\textsc{inward-approx}$(N,k)$ algorithm is defined similarly to the \\textsc{inward} algorithm, with the\ninstances of the time-bound combination operator $\\otimes_t$ replaced by the approximate combination operator\n$\\otimes^k$. In the following, $K(\\phi,\\psi,k)$ returns $(k_1, k_2)$ such that $\\rho(\\phi, k_1) \\otimes \\rho(\\psi, k_2)$\nhas at most $k$ elements.\n\n\\hrulefill\n\\begin{algorithmic}\n\\Function{inward-approx}{$N,k$}\n\\State for all $n \\in \\mathit{leaves}(N)$ do $\\phi_s(n) \\leftarrow \\phi(n)^{-\\Delta(n)}$\n\\While {$\\mathit{next}(N) \\ne \\emptyset$}\n\\State select $n$ from $\\mathit{next}(N)$\n\\State $(k_1, k_2) \\gets K(\\phi_s(L(n)), \\phi_s(R(n)), k)$\n\\State $\\phi(n) \\leftarrow \\phi_s(L(n)) \\otimes^k \\phi_s(R(n))$;\n\\State $\\phi_s(n) \\leftarrow \\phi(n)^{-\\Delta(n)}$\n\\State $\\tau(L(n)) \\gets \\rho(\\phi_s(L(n)), k_1)$\n\\State $\\tau(R(n)) \\gets \\rho(\\phi_s(R(n)), k_2)$\n\\State $\\bar\\tau(L(n)) \\gets \\bar\\rho(\\phi_s(L(n)), k_1)$\n\\State $\\bar\\tau(R(n)) \\gets \\bar\\rho(\\phi_s(R(n)), k_2)$\n\\State $\\phi_s(n) \\leftarrow \\phi(n)^{-\\Delta(n)}$\n\\State $s \\leftarrow s-1$\n\n\\EndWhile\n\\EndFunction\n\\end{algorithmic}\n\\hrulefill\n\n\\begin{thm}\nThe time complexity of \\textsc{inward-approx}$(N,k)$ in the Shenoy-Shafer architecture, with the approximation parameter of $k$, given that there\nare $n$ valuations in the knowledgebase is $O((n-1)k)$. \n\\end{thm}\n\\begin{proof}\nThere are $n-1$ combinations as the number of combinations in the binary join tree is the same as the number of non-leaf nodes. As each combination has\na complexity of $O(k)$, we get a complexity of $O((n-1)k)$. Projection has a complexity of $O(k)$ as there are $k$ elements in the configuration space, so at most $k-1$ summations, which is the case when we are\nmarginalising to the null set (equivalent to eliminating all the variables), thus it does not change the asymptotic complexity.\n\\end{proof}\n\nWe get the same time complexity for an analogous $\\textsc{refine-approx}$ algorithm, with a modification to lines 6--7 of\n$\\textsc{refine}$ to combine at most $k$ elements.\n\n\\textbf{Matching in the exact inference case}. In the exact inference case, the complexity is known to be in the class \\#P-hard. In the discussion on complexity \\cite{generic-inference}, Kohlas and Pouly derive the estimate $O(|V|. f(\\omega))$ where $\\omega$ is the treewidth, with $f(x) = m^x$ for the case of semiring induced valuation algebras with variables having a upper bound frame size of $m$. $|V|$ is the number of vertices in the join tree. Substituting $|V| = n, k = m^\\omega$ in the time complexity $O(n-1)k$ and taking $k = m^\\omega$, we get the same time complexity as the exact inference case; thus the approximate time complexity obtained in terms of the approximation parameter $k$ gives us a transition from $k=0$ (null valuations, obtained when we set the $t=0$ in $\\textsc{inward}(N,t)$) to $k=m^\\omega$, the exact inference case.\n\n\\textbf{Estimation of accuracy from elapsed time}.\nIt can be useful to derive an estimate of the accuracy of a valuation given the elapsed time of the algorithm in specific cases. Here, we shall consider the example of probability potentials. The time-bound\ncombination operator combines the configurations with the largest weight first so that we get diminishing returns; the accuracy also depends on the sparsity of the probability potential. For simplicity we consider uniform distributions, where the weights are uniformly distributed in the configuration space. Then we can state the following:\n\\begin{lem}\nThe fractional error estimate compared to the exact probability potential is\n\\begin{equation}\n\\epsilon(t) = 1 - \\max{\\left(1,\\frac{t}{m^\\omega c(n-1)}\\right)}\n\\end{equation}\nwhere $\\omega$ is the treewidth, $c$ is the constant time required to combine two elements in the configuration space, and $n$ is the number of valuations in the knowledgebase.\n\\end{lem}\n\\begin{proof}\nAs each configuration has an uniform weight, the accuracy of combination at the root node (which is the solution to the inference problem obtained from the inward propagation algorithm) is directly proportional to the allocated time which is on average $t\/(n-1)$ as there are $n-1$ combinations. Considering that each combination takes $c$ units, and in the worst-case each configuration has weight $1\/m^\\omega$ (for a normalised potential; for unnormalised, this introduces a constant factor which is cancelled out by considering a fractional error estimate), we get the fractional error estimate as above.\n\\end{proof}\nAs can be easily seen, $\\epsilon(0) = 1$, and $\\epsilon(O((n-1).m^\\omega)) = 0$ where $O((n-1).m^\\omega)$ is the exact inference time complexity.\n\n\\section{Implementation}\nWe implemented the anytime inference algorithm using the Python programming language, on a Core i5 CPU with 4GB RAM. While we have shown anytime inference in a Bayesian network here, the framework, being generic, can be applied to other valuation algebras which satisfy the necessary axioms.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.6]{inference_child.pdf}\n\\caption{Anytime inference progress in the \\textsc{child} dataset}\n\\end{figure}\n\nThe figure shows progress of anytime inference on the \\textsc{child} dataset, which was used as a\ncase study for exact inference in \\cite{cowell2007probabilistic}. The progress is shown as a function of the fractional error estimate with time units (the actual total time for the series of successive refinements, up to the exact valuation is $<10\\text{s}$):\n\\begin{equation}\n\\epsilon(t) = 1 - \\frac{\\sum L_{\\phi_t}}{\\sum L_{\\phi}}\n\\end{equation}\nHere the sum is over the weights of the configurations $L_\\phi$ of a valuation $\\phi$; $\\phi_t$ is the valuation obtained at the root\nafter time $t$, and $\\phi$ is the exact valuation. As expected, the fractional error estimate\nconverges to zero as we obtain the exact valuation.\n\n\\section{Conclusion}\nIn this work, we have shown that we can construct anytime algorithms for\ngeneric classes of valuation algebras, provided certain conditions are satisfied. We have also shown that the important subclass of semiring induced valuation algebras admit an anytime inference algorithm as they meet the aforementioned conditions. This is useful as semiring induced valuation algebras include\nimportant valuation algebra instances like probability potentials,\nDNF potentials and relational algebras, among others.\n\nFrom a broader perspective, the advantage of operating in the generic framework of valuation algebras has been addressed before \\cite{generic-inference}; we can target a large class of problems using a unified framework; the inference or projection problem can be found in various forms: Fourier transforms, linear programming and constraint satisfaction problems. Enriching the valuation algebra structure through extensions is thus useful. Anytime inference in particular has a wide spectrum of applications. We also plan to study the applicability of our framework across these various domains in future work.\n\nWe are currently working on implementation of other instances of anytime ordered valuation algebras, as well as conducting a complexity analysis of the algorithm in a distributed setting using the Bulk Synchronous Parallel \\cite{bsp} model.\n\\bibliographystyle{ecai}\n\\pagebreak\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzhdcb b/data_all_eng_slimpj/shuffled/split2/finalzzhdcb new file mode 100644 index 0000000000000000000000000000000000000000..84fb10c355aec3843ec4a042aaee05bb670de9cd --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzhdcb @@ -0,0 +1,5 @@ +{"text":"\\section{\\label{}}\n\n\\section{INTRODUCTION}\n\nSupersymmetry (SUSY) is one of the most promising \nextensions to the Standard Model (SM). Since low-energy SUSY is \nbroken there exist numerous free parameters that make it a\nhighly challenging task to reveal the underlying model \nat the Large Hadron Collider (LHC) and at the International Linear Collider\n(ILC). It is planned that the ILC starts with an energy of\n$\\sqrt{s}=500$~GeV, which will be upgraded to about 1~TeV\n\\cite{ITRP}. However, already at the first energy stage, the ILC could\nreach higher energy up to about $\\sqrt{s}=650$~GeV at cost of\nluminosity. In this study we sketch a possible motivation to\napply this higher energy option. Particularly interesting are case\nstudies which apply interplay of search strategies at the LHC and the\nILC~\\cite{lhclc-report}. We extend in this paper the methods\nfor combined LHC\/ILC analyses developed \nin~\\cite{Moortgat-Pick:2005vs}.\n\nAn interesting possibility for the determination of the supersymmetric\nmodel is to study the gaugino\/higgsino particles, which are expected\nto be among the lightest supersymmetric particles. In this paper we\nconsider two basic supersymmetric models: the minimal supersymmetric\nstandard model (MSSM) and the next-to-minimal supersymmetric standard\nmodel (NMSSM). The MSSM contains four neutralinos $\\tilde{\\chi}^0_i$,\nthe mass eigenstates of the photino, zino and neutral higgsinos, and\ntwo charginos $\\tilde{\\chi}^\\pm_i$, being mixtures of wino and charged\nhiggsino. The neutralino\/chargino sector depends at tree level on four\nparameters: the U(1) and SU(2) gaugino masses $M_1$ and $M_2$, the\nhiggsino mass parameter $\\mu$, and the ratio $\\tan\\beta$ of the vacuum\nexpectation values of the Higgs fields. For the determination of\nthese parameters, straightforward strategies \\cite{parameters,choi}\nhave been worked out even if only the light neutralinos and charginos\n$\\tilde{\\chi}^0_1$, $\\tilde{\\chi}^0_2$ and $\\tilde{\\chi}^\\pm_1$ are\nkinematically accessible at the first stage of the ILC~\\cite{ckmz}.\n\nThe NMSSM \\cite{nmssm}\nis the simplest extension of the MSSM\nby an additional Higgs singlet field.\nNew parameters in the neutralino sector are the vacuum expectation\nvalue $x$ of the singlet field and\nthe trilinear couplings $\\lambda$ and $\\kappa$ in the superpotential,\nwhere the product $\\lambda x = \\mu_{\\rm eff}$ replaces the\n$\\mu$-parameter of the MSSM \\cite{Franke,Choi:2004zx}.\nThe additional fifth neutralino may significantly change the phenomenology\nof the neutralino sector.\nIn scenarios where the lightest supersymmetric particle is a nearly\npure singlino, the existence of displaced vertices leads\nto a particularly interesting experimental signature\n\\cite{singlinohugonie,Singlinos}. In case only a\npart of the particle spectrum is kinematically accessible the\ndistinction between the models may become challenging. \n\nIt has already been worked out that there exist\nMSSM and NMSSM scenarios with\nthe same mass spectra of the light neutralinos but\ndifferent neutralino mixing. In this case\nbeam polarization is crucial for distinguishing the two models\n\\cite{Sitges}.\nWe present a scenario where\nall kinematically accessible neutralinos and charginos have\nsimilar masses and almost identical cross sections,\nwithin experimental errors, in MSSM and NMSSM.\nAlthough the second lightest neutralino in the NMSSM has a significant\nsinglino component, the models cannot be distinguished\nby the experimental results at the LHC or at\nthe ILC$_{500}$ with $\\sqrt{s}=500$ GeV alone \nif only measurements of masses, cross sections and\ngaugino branching ratios are considered. \nPrecision measurements of the neutralino branching ratio \ninto the lightest Higgs particle and of \nthe mass difference between the lightest and next-to-lightest \nSUSY particle \\cite{gunion-mrenna}\nmay give first evidence for the SUSY model but are \ndifficult to realize in our case. \nTherefore the identification of the\nunderlying model requires precision measurements of the\nheavier neutralinos by combined analyses of LHC and ILC\nas described in the following section.\n\n\n\\section{CASE STUDY}\n\n\nWe study an NMSSM scenario with the parameters \n\\begin{eqnarray}\n&&M_1=360~\\mbox{\\rm GeV},\\quad M_2=147~\\mbox{\\rm GeV},\\quad \\tan\\beta=10,\\quad\n\\lambda=0.5,\\quad x=915~\\mbox{\\rm GeV},\\quad \\kappa=0.2.\n\\label{eq-para-nm}\n\\end{eqnarray}\nThe hierarchy $M_1 > M_2$ of the U(1) and SU(2) mass parameters\nleads to very similar masses\nof the lightest neutralino $\\tilde{\\chi}_1^0$, which is assumed to be the\nlightest supersymmetric particle (LSP), and of the light chargino\n$\\tilde{\\chi}_1^\\pm$. This mass degeneration \nis also typical for minimal anomaly mediated SUSY breaking\n(mAMSB) scenarios.\nRather small mass differences may be resolved experimentally \nby applying the ISR method \\cite{Hensel} at the linear \ncollider \\cite{gunion-mrenna} as well as at the LHC \\cite{amsb-lhc}.\n\nThe NMSSM parameters lead to the following gaugino\/higgsino masses\nand eigenstates: \n\\begin{eqnarray}\nm_{\\tilde{\\chi}^0_1}=138~\\mbox{\\rm GeV}, \\quad\\quad \n\\tilde{\\chi}^0_1=(-0.02, +0.97, -0.20, +0.09, -0.07), \\label{ez-chi01-nm} \\\\\nm_{\\tilde{\\chi}^0_2}=337~\\mbox{\\rm GeV},\\quad\\quad\n\\tilde{\\chi}^0_2=(+0.62, +0.14, +0.25, -0.31, +0.65), \\label{ez-chi02-nm} \\\\\nm_{\\tilde{\\chi}^0_3}=367~\\mbox{\\rm GeV}, \\quad\\quad\n\\tilde{\\chi}^0_3=(-0.75, +0.04, +0.01, -0.12, +0.65), \\label{ez-chi03-nm} \\\\\nm_{\\tilde{\\chi}^0_4}=468~\\mbox{\\rm GeV}, \\quad\\quad\n\\tilde{\\chi}^0_4=(-0.03, +0.08, +0.70, +0.70, +0.08), \\label{ez-chi04-nm} \\\\\nm_{\\tilde{\\chi}^0_5}=499~\\mbox{\\rm GeV}, \\quad\\quad\n\\tilde{\\chi}^0_5=(+0.21, -0.16, -0.64, +0.62, +0.37), \\label{ez-chi05-nm}\n\\end{eqnarray}\nwhere the neutralino eigenstates are given in the basis\n$(\\tilde{B}^0, \\tilde{W}^0, \\tilde{H}^0_1,\\tilde{H}^0_2, \\tilde{S})$.\nAs can be seen from eqs.~(\\ref{ez-chi02-nm}) and (\\ref{ez-chi03-nm}), \nthe particles $\\tilde{\\chi}^0_2$ and $\\tilde{\\chi}^0_3$ have a rather \nstrong singlino admixture.\n\nThe Higgs sector does not allow the identification of the NMSSM\n\\cite{NMSSMhiggs} if scalar and pseudoscalar Higgs bosons with\ndominant singlet character escape detection. A scan with NMHDECAY\n\\cite{Ellwanger:2004xm} in our scenario over the remaining parameters\nin the Higgs sector, $A_\\lambda$ and $A_\\kappa$, results in parameter\npoints which survive the theoretical and experimental constraints in\nthe region $2740~\\textrm{GeV} < A_\\lambda < 5465$~GeV and\n$-553~\\textrm{GeV} < A_\\kappa < 0$. For $-443~\\textrm{GeV} < A_\\kappa\n< -91$~GeV the second lightest scalar ($S_2$) and the lightest\npseudoscalar ($P_1$) Higgs particle have very pure singlet character\nand are heavier than the mass difference\n$m_{\\tilde{\\chi}^0_3} - m_{\\tilde{\\chi}^0_1}$,\nhence the decays of the neutralinos $\\tilde{\\chi}^0_2$ and\n$\\tilde{\\chi}^0_3$, which will be discussed in the following, are not\naffected by $S_2$ and $P_1$, see figure~\\ref{higgs-sector} (left panel). For\nour specific case study we choose $A_{\\lambda}=4000$~GeV and\n$A_{\\kappa}=-200$~GeV, which leads to $m_{S_2}=311$~GeV,\n$m_{P_1}=335$~GeV and $m_{S_3}$, $m_{P_2}$ and $m_{H^{\\pm}}>4$~TeV.\nFurthermore the lightest scalar Higgs $S_1$ has MSSM-like character in\nthis parameter range with a mass of about 124~GeV. Also the branching\nratio of $\\tilde{\\chi}^0_2$ in the lightest Higgs particle differs\nonly by a factor two in both scenarios. In case that a precise\nmeasurement of this BR is possible first hints for the\ninconsistency of the model could be derived at the ILC.\n\n\\subsection{Strategy for the gaugino\/higgsino sector}\nIn our NMSSM scenario in the gaugino\/higgsino sector only the\nlight chargino $\\tilde{\\chi}^{\\pm}_1$ and the light neutralinos\n$\\tilde{\\chi}_1^0$ and\n$\\tilde{\\chi}_2^0$\nare accessible at the ILC$_{500}$.\nWe calculate the masses of the charginos and neutralinos and\nthe cross sections for the pair production\nof the light chargino $e^+e^- \\rightarrow \\tilde{\\chi}^+_1\n\\tilde{\\chi}^-_1$ and for the associated production of the light neutralinos\n$e^+e^- \\rightarrow \\tilde{\\chi}_1^0 \\tilde{\\chi}_2^0$ with \npolarized and unpolarized beams.\n \nThe masses and cross sections in different beam polarization configurations\nprovide the experimental input for deriving the supersymmetric parameters \nwithin the MSSM using standard methods\n\\cite{choi,ckmz}:\n\\begin{itemize}\n\\item We assume an uncertainty of $\\mathcal{O}(1-2\\%)$ for the masses\n$m_{\\tilde{\\chi}^{\\pm}_1}$,\n$m_{\\tilde{\\chi}^0_1}$, $m_{\\tilde{\\chi}^0_2}$,\n$m_{\\tilde{\\nu}_e}$, $m_{\\tilde{e}_L}$ and $m_{\\tilde{e}_R}$.\nThe errors of the cross sections,\n$e^+e^- \\rightarrow \\tilde{\\chi}^+_1 \\tilde{\\chi}^-_1$ and\n$e^+e^- \\rightarrow \\tilde{\\chi}_1^0 \\tilde{\\chi}_2^0$, are composed\nof the error due to the mass uncertainties, polarization uncertainty and\none standard deviation statistical error based on\n$\\int {\\cal L}=100$~fb$^{-1}$ for each polarization configuration.\nDeviations in the cross sections due to \nthe polarization uncertainty of \n$\\Delta P_{e^{\\pm}}\/P_{e^{\\pm}}=0.5\\%$ are generally small; it \nis expected that the error\ncould even be reduced up to $\\Delta P_{e^{\\pm}}\/P_{e^{\\pm}}=0.2\\%$--$0.1\\%$,\n\\cite{Power}. The assumed uncertainties in total are listed in \ntable~\\ref{lc-input}.\n\\item\nFrom the chargino mass $m_{\\tilde{\\chi}^{\\pm}_1}$ and\nthe cross section\n$e^+e^- \\rightarrow \\tilde{\\chi}^+_1 \\tilde{\\chi}^-_1$\nmeasured at two energies, $\\sqrt{s}=400$~GeV and $500$~GeV,\nwe determine bounds for the\nelements $U_{11}$ and $V_{11}$ of the chargino mixing matrices:\n\\begin{equation}\nU^2_{11}=[0.84,1.0], \\quad V^2_{11}=[0.83,1.0].\n\\end{equation}\nPolarized beams allow the resolution of ambiguities and the improvement\nof the accuracy.\n\\item Using the mixing matrix elements $U_{11}$ and $V_{11}$,\nthe masses $m_{\\tilde{\\chi}^{\\pm}_1}$,\n$m_{\\tilde{\\chi}^0_1}$ and $m_{\\tilde{\\chi}^0_2}$,\nand the cross sections for\n$e^+e^- \\rightarrow \\tilde{\\chi}^0_1 \\tilde{\\chi}^0_2$,\nwe derive constraints for\nthe parameters $M_1$, $M_2$, $\\mu$ and $\\tan\\beta$:\n\\begin{eqnarray} \nM_1 &=& 377 \\pm 42 \\mbox{ GeV}, \\label{MSSMresult1}\\\\\nM_2 &=& 150\\pm 20 \\mbox{ GeV}, \\\\\n\\mu &=& 450 \\pm 100 \\mbox{ GeV}, \\\\\n\\tan\\beta &=& [1,30]. \\label{MSSMresult4}\n\\end{eqnarray}\nNote that, in our scenario with $M_1 > M_2$, the crucial observable to\ndetermine the parameter $M_1$ is $m_{\\tilde{\\chi}^0_2}$ and not\n$m_{\\tilde{\\chi}^0_1}$ as often assumed. Such a hierarchy could be\nnaturally embedded in mAMSB scenarios. For even larger $M_1\\gg M_2$\nthe heavier neutralinos $\\tilde{\\chi}^0_{3,4}$ become crucial for $M_1$\ndetermination see \\cite{Moortgat-Pick:2005vs,m1-paper}. \nSince the heavier \nneutralino and chargino states are not produced, some of the \nparameters ---in our case $\\mu$ and $\\tan\\beta$--- \ncan only be determined with a considerable\nuncertainty.\n\nWithin these limits an explicit MSSM scenario, \n\\begin{eqnarray}\n&&M_1=375~\\mbox{\\rm GeV},\\quad M_2=152~\\mbox{\\rm GeV},\\quad \\tan\\beta=8,\\quad\n\\mu=360~\\mbox{\\rm GeV},\n\\label{eq-para-ms}\n\\end{eqnarray}\nleads to the \nsame (lighter) neutralino\/chargino masses and cross sections:\n\\begin{eqnarray}\nm_{\\tilde{\\chi}^0_1}=138~\\mbox{\\rm GeV}, \\quad\\quad \n\\tilde{\\chi}^0_1=(+0.03, -0.96, +0.26, -0.13), \\label{ez-chi01-ms}\\\\\nm_{\\tilde{\\chi}^0_2}=344~\\mbox{\\rm GeV}, \n\\quad\\quad\n\\tilde{\\chi}^0_2=(+0.72, +0.22, +0.48, -0.46), \\label{ez-chi02-ms} \\\\\nm_{\\tilde{\\chi}^0_3}=366~\\mbox{\\rm GeV}, \\quad\\quad\n\\tilde{\\chi}^0_3= (-0.04, +0.10, -0.70, -0.71), \\label{ez-chi03-ms} \\\\\nm_{\\tilde{\\chi}^0_4}=410~\\mbox{\\rm GeV}, \\quad\\quad\n\\tilde{\\chi}^0_4=(-0.70, +0.18, +0.47, -0.52), \\label{ez-chi04-ms}\n\\end{eqnarray}\nwhere the neutralino mixing states are given in the basis\n$(\\tilde{B}^0, \\tilde{W}^0, \\tilde{H}^0_1,\\tilde{H}^0_2)$.\nComparing eqs.~(\\ref{ez-chi01-ms})--(\\ref{ez-chi03-ms}) with \neqs.~(\\ref{ez-chi01-nm})--(\\ref{ez-chi03-nm}) shows that \nthe three lightest neutralino masses are the same within the \nexperimental uncertainties. We checked that also the accessible\ncross sections at the ILC$_{500}$ \nand the BR's of $\\tilde{\\chi}^0_2$ are consistent.\n\n\\item After the determination of the fundamental MSSM\nparameters we calculate the heavy chargino\nand neutralino masses and expected mixing characters.\nFor the masses we obtain:\n\\begin{equation}\nm_{\\tilde{\\chi}^0_3} = 443\\pm 107 \\mbox{ GeV},\\quad\\quad \nm_{\\tilde{\\chi}^0_4} = 490\\pm 110 \\mbox{ GeV}, \\quad\\quad\nm_{\\tilde{\\chi}^{\\pm}_2} = 475 \\pm 125 \\mbox{ GeV}. \\label{eq_mp3_nm}\n\\end{equation}\n\\end{itemize}\nThe predicted gaugino admixture of $\\tilde{\\chi}^0_3$, $\\tilde{\\chi}^0_4$\nwithin the allowed parameter ranges, eqs.(\\ref{MSSMresult1})--(\\ref{MSSMresult4}),\nare shown in figure~\\ref{higgs-sector} (right panel). Obviously, the heavy neutralino\n$\\tilde{\\chi}^0_3$ should be almost a pure higgsino within the MSSM prediction. \nThe predicted properties of the\nheavier particles can now be compared with \nmass measurements of such SUSY particles via the analysis\nof cascade decays at the LHC~\\cite{lhclc-report}.\n\nWe emphasize that although we started with an NMSSM scenario where\n$\\tilde{\\chi}^0_2$ and $\\tilde{\\chi}^0_3$ have large singlino\nadmixtures, the MSSM parameter strategy does not fail and\nthe experimental results from the ILC$_{500}$ with $\\sqrt{s}=400$~GeV\nand $500$~GeV lead to a consistent parameter determination in the MSSM.\nHence in the considered scenario the analyses at the ILC$_{500}$ or\nLHC alone do not allow a clear discrimination between MSSM and NMSSM.\nAll predictions for the heavier gaugino\/higgsino masses are\nconsistent with both models.\nHowever, the ILC$_{500}$ analysis predicts an almost pure\nhiggsino-like state for $\\tilde{\\chi}^0_3$\nand a mixed gaugino-higgsino-like $\\tilde{\\chi}^0_4$, see\nfigure~\\ref{higgs-sector} (right panel). \nThis allows the identification of the underlying supersymmetric model\nin combined analyses at the LHC and the\nILC$_{650}^{{\\cal L}=1\/3}$.\n\n\\subsection{Interplay between LHC and ILC}\nThe expected large cross sections for squark and gluino\nproduction at the LHC\ngive access to a large spectrum of coloured as\nwell as non-coloured supersymmetric particles via the cascade decays.\nHeavy gaugino-states appear almost only in cascade decays and \nthere exist some true simulations how to measure the heavier gauginos in\nsuch decays at the LHC \\cite{Giacomo}. Particularly helpful for the\nidentification \nof the particles involved in the cascades, e.g.\\ for more\nmodel-independent analyses, \nare mass predictions from the ILC analysis which lead to an increase\nof statistical \nsensitivity for the LHC analysis and open the possibility of identifying even\nmarginal signals in the squark cascades~\\cite{lhclc-report}. \nHowever, since higgsino-like charginos and neutralinos\ndo not couple to squarks, their detection via cascade decays is not possible.\n\nIn our original NMSSM scenario \nthe neutralinos $\\tilde{\\chi}^0_2$ and\n$\\tilde{\\chi}^0_3$ have a large\nbino-admixture and therefore appear\nin the squark decay cascades. The dominant decay mode of\n$\\tilde{\\chi}^0_2$ has a branching ratio $BR(\\tilde{\\chi}^0_2\\to\n\\tilde{\\chi}^{\\pm}_1 W^{\\mp})\\sim 50\\%$, while\nfor the $\\tilde{\\chi}^0_3$ decays $BR(\\tilde{\\chi}^0_3\\to\n\\tilde{\\ell}^{\\pm}_{L,R} \\ell^{\\mp})\\sim 45\\%$ is largest.\nSince the heavier neutralinos, $\\tilde{\\chi}^0_4$, $\\tilde{\\chi}^0_5$,\nare mainly higgsino-like,\nno visible edges from these particles occur in the cascades.\nIt is expected to see the edges for\n$\\tilde{\\chi}^0_2 \\to \\tilde{\\ell}^{\\pm}_R \\ell^{\\mp}$,\n$\\tilde{\\chi}^0_2 \\to \\tilde{\\ell}^{\\pm}_L \\ell^{\\mp}$,\n$\\tilde{\\chi}^0_3 \\to \\tilde{\\ell}^{\\pm}_R \\ell^{\\mp}$\nand for\n$\\tilde{\\chi}^0_3 \\to \\tilde{\\ell}^{\\pm}_L \\ell^{\\mp}$\n\\cite{Giacomo_private}.\n\nWith a precise\nmass measurement of\n$\\tilde{\\chi}^0_1$,$\\tilde{\\chi}^0_2$, $\\tilde{\\ell}_{L,R}$ and\n$\\tilde{\\nu}$ from the ILC$_{500}$\nanalysis, a clear identification and separation of\nthe edges of the two gauginos at the LHC is\npossible without imposing specific model assumptions.\nWe therefore assume a precision of\nabout 2\\% for the measurement of $m_{\\tilde{\\chi}^0_3}$, in analogy to\n\\cite{Giacomo}:\n\\begin{eqnarray} \n\\label{eq:mchi3LHC}\n&& m_{\\tilde{\\chi}^0_3}=\n367\\pm 7 \\mbox{ GeV}.\n\\end{eqnarray}\nThe precise mass measurement of $\\tilde{\\chi}^0_3$ is compatible with\nthe mass predictions of the ILC$_{500}$ but not with the prediction of\nthe mixing character,\nsee e.g.\\ eq.~(\\ref{ez-chi03-ms}). \nHowever, it is not clear that\nthe measured particle at the LHC is indeed the $\\tilde{\\chi}^0_3$. Often\nin the constrained MSSM, as e.g. also in our MSSM comparison scenario,\nthe second heaviest neutralino $\\tilde{\\chi}^0_3$ is nearly a pure\nhiggsino and does not couple in the cascade decays. In those cases,\nthe heaviest neutralino $\\tilde{\\chi}^0_4$ has frequently a sufficiently\nlarge gaugino component and could be measured in cascades, as shown in\n\\cite{lhclc-report,Giacomo}.\n\nTherefore is is inevitable to \ndiscuss the following cases of possible particle {\\it identification}\nof the measured gaugino mass $m_{\\tilde{\\chi}^0_3}$ at the LHC:\n\\begin{itemize}\n\\item interpretation of the measured particle as\n$\\tilde{\\chi}^0_3$ and feeding it back in the ILC analysis leads to\nimproved parameter determination and mass prediction for\n$m_{\\tilde{\\chi}^0_4}$, $m_{\\tilde{\\chi}^{\\pm}_2}$.\nUsing eq.~(\\ref{eq:mchi3LHC}) for the ILC$_{500}$ analysis\nleads in our case, after rechecking with the allowed cross sections of\n$\\tilde{\\chi}^0_1\\tilde{\\chi}^0_2$ and\n$\\tilde{\\chi}^{+}_1\\tilde{\\chi}^-_1$ production, \nto rather precise mass predictions:\n\\begin{equation}\nm_{\\tilde{\\chi}^0_4}=[384, 393]\\mbox{ GeV \\quad and \\quad}\nm_{\\tilde{\\chi}^{\\pm}_2}=[360, 380]\\mbox{ GeV}.\n\\label{eq:heavymass-pred}\n\\end{equation}\n\\item interpretation of the measured particle as $\\tilde{\\chi}^0_4$\nand feeding it back in the parameter determination of the ILC analysis\nleads to inconsistency with the measured cross sections of\n$\\tilde{\\chi}^0_1\\tilde{\\chi}^0_2$ and $\\tilde{\\chi}^{+}_1\\tilde{\\chi}^-_1$ production.\n\\end{itemize}\nThe combined LHC$\\leftrightarrow$ILC$_{500}$ analysis leads therefore\nto a correct interpretation of the measured particles in the cascades.\nHowever, a neutralino $\\tilde{\\chi}^0_3$ with sufficiently large\ngaugino admixture to couple to squarks is incompatible with the\nallowed parameter ranges of\neqs.~(\\ref{MSSMresult1})--(\\ref{MSSMresult4}) in the MSSM, \ncf.\\ figure~\\ref{higgs-sector} (right panel).\n\nWe point out\nthat a measurement of the neutralino masses\n$m_{\\tilde{\\chi}^0_1}$, $m_{\\tilde{\\chi}^0_2}$, $m_{\\tilde{\\chi}^0_3}$\nwhich could take place at the LHC alone is not sufficient to distinguish\nthe SUSY models since rather similar mass spectra could exist, cf. \neqs.~(\\ref{ez-chi01-nm})--(\\ref{ez-chi03-nm}) \nwith eqs.~(\\ref{ez-chi01-ms})--(\\ref{ez-chi03-ms}).\n\nTherefore the cross sections in different beam polarization\nconfigurations\nat the ILC have to be included in the analysis. \nThe combined results from the LHC and the ILC$_{500}$ analyses\nand the rather precise predictions for the missing\nchargino\/neutralino masses, eq.~(\\ref{eq:heavymass-pred}),\nconstitute a serious motivation\nto apply immediately the low-luminosity but higher-energy option\nILC$_{650}^{{\\cal L}=1\/3}$, \nwhich finally leads to the right identification\nof the underlying model. The expected polarized and\nunpolarized cross sections, including the statistical error on the basis of\none third of the luminosity of the ILC$_{500}$, are given in\nTable~\\ref{tab_heavy}. The neutralino\n$\\tilde{\\chi}^0_3$ as well as the higgsino-like heavy neutralino\n$\\tilde{\\chi}^0_4$ and the chargino $\\tilde{\\chi}^{\\pm}_2$ are now\naccessible at the ILC$^{{\\cal L}=1\/3}_{650}$. \nAlready the high rates for $\\tilde{\\chi}^0_1\\tilde{\\chi}^0_3$ production\ngive last true evidence for the obvious contraction with an\ncorresponding MSSM scenario.\nTogether with the \nmass measurements of $m_{\\tilde{\\chi}^0_4}=468$~GeV and \n$m_{\\tilde{\\chi}^{\\pm}_2}=474$~GeV, \nwhich are also in strong disagreement with the mass prediction, \neq.~(\\ref{eq:heavymass-pred}), one has sufficient observables which point\nto the NMSSM. Extensions of existing fit programs for the NMSSM\nmay lead to an exact resolution of the \nunderlying parameters~\\cite{fit-porod}.\n\n\\section{CONCLUSIONS}\nWe have presented a scenario in the next-to-minimal supersymmetric\nstandard model\n(NMSSM) that could not be distinguished from the MSSM at either the LHC or at \nthe first stage of the International Linear Collider with\n$\\sqrt{s}= 500$ GeV. It turns out that the most promising sector for distinction \nis the gaugino\/higgsino sector.\nAlthough a light neutralino has a significant\nsinglet component in the NMSSM, the masses of the accessible\nlight neutralinos and charginos, as well as the production cross sections,\nlead to identical values in the two models within experimental errors.\nThe comparison of the predicted masses and mixing character of the heavier neutralinos and charginos\nwith the measured masses \nin combined analyses with the LHC followed\nby a precise measurement of the cross sections at the ILC at\n$\\sqrt{s}= 650$ GeV leads to a clear identification of the supersymmetric\nmodel.\n \n\nThe exemplary scenario shows that the interplay between the two\nexperiments could be crucial for the determination of the supersymmetric\nmodel. A possible feed-back of ILC$_{500}$\/LHC results \ncould motivate the immediate use of the low-luminosity option of\nthe ILC at $\\sqrt{s}=650$~GeV in order to resolve model ambiguities even at an early stage of\nthe experiment and outline future search strategies at the\nupgraded ILC at 1 TeV.\n\n\n\n\n\n\\begin{table}[t!]\n\\renewcommand{\\arraystretch}{1.3}\n\\centering\n\\begin{tabular}{|c||c|c|c|c|}\n\\hline\n$m_{\\tilde{\\chi}^0_1}$\/GeV$=138\\pm 2.8$ &\n & \\multicolumn{2}{|c|}{$\\sigma(e^+e^-\\to\\tilde{\\chi}^{\\pm}_1\\tilde{\\chi}^{\\mp}_1)$\/fb} & $\\sigma(e^+e^-\\to\\tilde{\\chi}^{0}_1\\tilde{\\chi}^{0}_2)$\/fb \\\\\n$m_{\\tilde{\\chi}^0_2}$\/GeV$=337\\pm 5.1$ & $(P_{e^-},P_{e^+})$\n& $\\sqrt{s}=400$~GeV & $\\sqrt{s}=500$~GeV & $\\sqrt{s}=500$~GeV \\\\\n\\cline{2-5}\n$m_{\\tilde{\\chi}^{\\pm}_1}$\/GeV$=139\\pm 2.8$ &\nUnpolarized & $323.9 \\pm 33.5$ & $287.5 \\pm 16.5$ & $4.0 \\pm 1.2$\\\\\n$m_{\\tilde{e}_L}$\/GeV$=240\\pm 3.6$ & $(-90\\%,+60\\%)$ &\n $984.0 \\pm 101.6$ & $873.9 \\pm 50.1$ & $12.1 \\pm 3.8$ \\\\\n$m_{\\tilde{e}_R}$\/GeV$=220\\pm 3.3$ & $(+90\\%,-60\\%)$ &\n$13.6 \\pm 1.6$ & $11.7 \\pm 1.2$ & $0.2 \\pm 0.1$\\\\\n$m_{\\tilde{\\nu}_e}$\/GeV$=226\\pm 3.4$ & &&&\\\\ \\hline\n\\end{tabular}\n\\caption{Masses with \n1.5\\% ($\\tilde{\\chi}^0_{2,3}$, $\\tilde{e}_{L,R}$, $\\tilde{\\nu}_e$)\nand 2\\% ($\\tilde{\\chi}^0_1$, $\\tilde{\\chi}^{\\pm}_1$)\nuncertainty and cross sections with an error\ncomposed of\nthe error due to the mass uncertainties, polarization uncertainty and\none standard deviation statistical error based on\n$\\int {\\cal L}=100$~fb$^{-1}$,\nfor both unpolarized beams and polarized beams with\n$(P_{e^-},P_{e^+})=(\\mp 90\\%,\\pm 60\\%)$ and $\\Delta P(e^\\pm)\/P(e^{\\pm})=0.5\\%$,\nin analogy to the study in \\cite{lhclc-report}.\n\\label{lc-input}\n}\n\\end{table}\n\n\n\\begin{table}\n\\renewcommand{\\arraystretch}{1.2}\n\\centering\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline &\n \\multicolumn{3}{c|}{%\n$\\sigma(e^+e^- \\to \\tilde{\\chi}^0_1\\tilde{\\chi}^0_j)\/$fb at\n$\\sqrt{s}=650$~GeV} & $\\sigma(e^+e^- \\to \\tilde{\\chi}^{\\pm}_1\\tilde{\\chi}^{\\mp}_2)\/$fb at\n$\\sqrt{s}=650$~GeV \\\\ \\cline{2-4}\n & \\makebox[21mm]{$j=3$} & \\makebox[21mm]{$j=4$} & \\makebox[21mm]{$j=5$} &\\\\\n\\hline \\hline\nUnpolarized beams & $12.2 \\pm 0.6$\n& $5.5 \\pm 0.4$ & $\\le 0.02$ & $2.4 \\pm 0.3$ \\\\ \\hline\n$P(e^-)=-90\\%$, $P(e^+)=+60\\%$ &\n$36.9 \\pm 1.1$ & $14.8 \\pm 0.7$ & $\\le 0.07$ & $5.8 \\pm 0.4$ \\\\ \\hline\n$P(e^-)=+90\\%$, $P(e^+)=-60\\%$ &\n$0.6 \\pm 0.1$ & $2.2 \\pm 0.3$ & $\\le 0.01$ & $1.6 \\pm 0.2$\\\\ \\hline\n\\end{tabular}\n\\caption{Expected cross sections for\nthe associated production of\nthe heavier neutralinos and charginos in the NMSSM scenario\nfor the\nILC$^{{\\cal L}=1\/3}_{650}$ option with one sigma\nstatistical error\nbased on $\\int {\\cal L} = 33$~fb$^{-1}$\nfor both unpolarized and polarized beams.\n\\label{tab_heavy}}\n\\end{table}\n\n\n\n\n\n\\begin{figure*}[t]\n\\setlength{\\unitlength}{1cm}\n\\begin{center}\n\\begin{picture}(10,5)\n\\put(-3.4,4.55){\\small $m_{H}\/$GeV}\n\\put(-.5,4.1){\\color{Blue}\\small $P_1$}\n\\put(-.5,1.8){\\color{Red}\\small $S_2$}\n\\put(-.5,.8){\\color{Green}\\small $S_1$}\n\\put(0,-.5){\\small $A_{\\kappa}$\/GeV}\n\\put(-3,-1){\\includegraphics[width=70mm]{0206-higgsmass.epsy}}\n\\put(8.7,3.5){\\small{\\color{Red} \\phantom{$\\longleftarrow$ }$\\tilde{\\chi}^0_3$\\quad} and\n{\\color{Blue}\\quad $\\tilde{\\chi}^0_4$}}\n\\put(4.9,4.6){\\small $m_{\\tilde{\\chi}^0_i}$\/GeV}\n\\put(4.5,1.2){\\small LHC:}\n\\put(4.5,.9){\\small measurement}\n\\put(4.95,.6){\\small of \\color{green}$m_{\\tilde{\\chi}^0_i}\\rightarrow$}\n\\put(7.5,5){\\small Contradiction within MSSM}\n\\put(8.8,4.5){\\small ILC: prediction of}\n\\put(8.8,4){\\small mixing character of}\n\\put(8.3,-.5){\\small gaugino character}\n\\put(6,-1){\\includegraphics[width=70mm]{0206-chi3chi4}}\n\\end{picture}\n\\end{center}\n\\vspace{-.2cm}\n\\caption{Left: The possible\nmasses of the two light scalar Higgs bosons, $m_{S_1}$, $m_{S_2}$, and of the\nlightest pseudoscalar Higgs boson $m_{P_1}$ as function of the trilinear\nHiggs parameter $A_{\\kappa}$ in the NMSSM.\nIn our chosen scenario, $S_1$ is MSSM-like and\n$S_2$ and $P_1$ are heavy singlet-dominated Higgs particles.\nRight: Predicted masses and gaugino admixture for the heavier neutralinos\n$\\tilde{\\chi}^0_3$ and $\\tilde{\\chi}^0_4$ within the consistent\nparameter ranges derived at the ILC$_{500}$ analysis\nin the MSSM and measured mass $m_{\\tilde{\\chi}^0_i}=367\\pm 7$~GeV\nof a neutralino with sufficiently high gaugino admixture in cascade\ndecays at the LHC. We took a lower bound of sufficient gaugino admixture\nof about 10\\% for\nthe heavy neutralinos, cf.\\ \\cite{Giacomo}. \\label{higgs-sector}}\n\\end{figure*}\n\n\n\n\n\n\\begin{acknowledgments}\nWe are very grateful to G.~Polesello and P.~Richardson for\nconstructive discussions.\nS.H.\\ is supported by the G\\\"oran Gustafsson Foundation.\nThis work is supported by the\nEuropean Community's Human Potential Programme under contract\nHPRN-CT-2000-00149 and by the Deutsche Forschungsgemeinschaft (DFG) under\ncontract No.\\ \\mbox{FR~1064\/5-2}.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Acknowledgement}\nWe would like to thank Gautam Tiwari and Anirudh Raju for valuable discussions on the work of this paper.\n\n\n\n\n\\bibliographystyle{IEEEbib}\n\n\\section{Conclusion} \nIn this paper, we proposed a multi-task semantic RNN-T architecture for streamable end-to-end spoken language understanding. The proposed semantic decoder and semantic beam search empower the model to consider more contextual information, both from the predicted word piece and slot tags in the history, and produce higher quality word-piece and slot tag hypothesis in the next time step. Moreover, the model with the proposed losses, both cross-entropy loss and the aligned RNN-T loss for slot tagging, outperformed the two-stage and one-stage E2E SLU models. \n\n\\section{Related Work}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{figures\/fig_1.png}\n\\caption{A high-level diagram of comparing the proposed E2E SLU model to the previous two-stage~\\cite{s2i,nlu, nlu2, bertnlu,larson2012spoken} (a) and one-stage \\cite{google, radfar2021fans} (b) E2E SLU models. Dotted lines represent the conditioning of output label on its history, as a part of decoders.} \n\\label{fig:high_level}\n\\vspace{-2.5mm}\n\\end{figure*}\n\nA multi-task learning framework for E2E SLU was first introduced in \\cite{google}; the authors investigated several encoder-decoder structures for joint training of ASR and NLU tasks, in which the multi-task structure achieves the best performance. \nThis work was followed by proposed pre-training based approaches to improve the performance \\cite{fluent, pretrain1, pretrain2}; all of these models are designed specifically for intent classification. \nAnother category of work attempts a combined parameter transfer from well trained end-to-end ASR systems and end-to-end NLU models such as pretrained BERT~\\cite{devlin2018bert} through teacher-student learning~\\cite{semantic1, semantic2, semantic3}.\nNote that both of these categories rely on at least a two-stage training process and all operate with non-streamable inference. \n\nLeveraging RNN-T, in \\cite{s2i}, a two-stage E2E SLU structure was proposed where the RNN-T based ASR subsystem interacts with an NLU subsystem through an interface, which is not streamable.\n\nIn the most recent work, \\cite{streaming_e2e} proposed a CTC-based streamable E2E SLU framework which employs a unidirectional RNN to make multiple intent predictions. The NLU output is generated either directly from the audio signal or based on an intermediate ASR output. \nNamely, the advantage of semantic posterior that we employ, was not considered in making word-piece predictions.\nTo the best of our knowledge, there is no existing E2E SLU model, either streamable or non-streamable, which takes both word-piece and slot tag in beam search decoding on joint multi-task sequence prediction for word-piece, slot tag, and intent label.\n\\section{Introduction}\nWith the widespread application of intelligent voice assistants, e.g. Alexa, Siri, and Google Home, SLU systems have generated increased interest in the recent years.\nAn SLU system predicts semantic information implied by an audio signal.\nThis semantic content is commonly represented as intent, slot tags, named entities and\/or part-of-speech taggings.\nToday's SLU technology typically accomplishes this task in two separate stages, which we refer to as ASR-NLU approaches for SLU~\\cite{larson2012spoken}: an ASR system first transcribes the audio signals~\\cite{deepspeech, las}, and then the transcripts are passed to an NLU system to extract corresponding intent and slot tags~\\cite{nlu, nlu2, bertnlu}; an example is presented in Table~\\ref{tab:semantc_labels}.\nGiven the extracted semantic labels, downstream applications of the voice assistant can produce an appropriate response to the user.\n\nRecently, complete E2E-SLU based approaches have attracted attention due to their efficiency and reduced model complexity compared with an ASR-NLU pipeline, making them suitable candidates for deployment on low-resource devices~\\cite{google, lai2021semi, chuang2019speechbert, radfar2021fans}. \nFurthermore, performance improvements in both tasks, driven by joint training, has been observed in several studies \\cite{google, fluent}.\n\nMost of existing E2E-SLU models still adopt a two-stage setup as shown in Fig.~\\ref{fig:high_level}(a), where the NLU subsystem waits for the transcripts of the whole utterance produced by the ASR subsystem to generate semantic labels~\\cite{e2eslu, google, fluent, semantic1}. Meanwhile, the NLU subsystem is typically non-streamable. \nOne-stage approaches to E2E-SLU have been proposed as well \\cite{google, radfar2021fans}; however, again the NLU subsystem remains non-streamable. \nFor example, in \\cite{radfar2021fans}, slot tag prediction only occurs after the intent is extracted at the end of an utterance.\nMoreover, in all the above approaches, the ASR and NLU label generations do not interact with one another during the forward pass of inference (Fig.~\\ref{fig:high_level}(b)). \nAs a result, this design can lead to three main limitations. \nFirst, the NLU posterior or hypothesis does not provide any feedback upon word-piece generation, while its feedback could be helpful to narrow down potential word-piece candidates generated in the next time step.\nSecondly, the decoding of the NLU label predictions is not streamable \\cite{google}, given that the model is an encoder-decoder framework augmented by attention.\nFinally, the inference speed of an SLU system may be affected by the nature of the cascaded setup and non-streamable NLU subsystem, all the while low latency is crucial for a responsive virtual assistant. \n\nTo address these limitations, we propose a streamable E2E-SLU model based on RNN-T \\cite{rnnt, rnnt-asr} with a novel semantic beam search decoder which predicts word-pieces and NLU labels jointly, as illustrated in Fig.~\\ref{fig:high_level}(c).\nSpecifically, we introduce a semantic decoder to aggregate not only the word-pieces but also slot candidates during the beam search, which we call \\emph{semantic beam search}. \nFurthermore, we propose different multi-task loss functions to learn the alignment between word-pieces and slot tags along with the intent prediction.\n\n\\begin{table}[t]\n\\centering\n\\small\n\\tabcolsep=0.1cm\n\\caption{\\small An example of a transcription, slot tags, and intent.}\n\\label{tab:semantc_labels}\n\\begin{tabular}{cc}\n\\toprule\n\\textbf{transcription} & turn on the kitchen light \\\\ \\midrule\n\\textbf{slot tags} & [DeviceLocation]: kitchen \\\\\n& [ApplianceType]: light, [Other]:turn,on,the \\\\ \\midrule \n\\textbf{intent} & TurnOnApplianceIntent \\\\ \\bottomrule\n\\end{tabular}\n\\vskip -10pt\n\\end{table}\n\n\\section{Acknowledgement}\nWe would like to thank Gautam Tiwari and Anirudh Raju for valuable discussions on the work of this paper.\n\n\n\n\n\\bibliographystyle{IEEEbib}\n\n\\section{Methodology}\n\\label{sec:methodology}\n\nThe inputs of the multi-task RNN-T are $D$-dimensional audio features of length $T$, $\\mathcal{X} = (\\mathbf{x}_1, \\mathbf{x}_2, ..., \\mathbf{x}_T)$, $\\mathbf{x}_k \\in \\mathbb{R}^{D}$. The outputs are transcript tokens of length $U$, $\\mathbf{y}^w = (y^w_1, y^w_2, ..., y^w_U)$, $y^w_u \\in \\mathcal{W}$, its corresponding slot tags (also of length $U$), $\\mathbf{y}^s = (y^s_1, y^s_2, ..., y^s_U)$, $y^s_u \\in \\mathcal{S}$, and the intent $y^i \\in \\mathcal{I}$; here $\\mathcal{W}$, $\\mathcal{S}$, and $\\mathcal{I}$ are the predefined set of token labels (or token vocabulary), slot tags, and intents. Both transcript tokens and slot tags are encoded as one-hot vectors. \n\nThe model defines a conditional distribution of $p(\\mathcal{W},\\mathcal{S},\\mathcal{I}|\\mathcal{X})$, and we factorize it as follows (Fig.~\\ref{fig:modelb}),\n\\begin{align}\n & p(\\hat{\\mathbf{y}}^w,\\hat{\\mathbf{y}}^s,y^i|\\mathcal{X}) = \\nonumber \\\\ \n & \\prod_{k=1}^{T+U} p(\\hat{y}^w_k|\\mathcal{X},t_k,y^w_0,...,y^w_{u_{k-1}},y^s_0,...,y^s_{u_{k-1}}) p(y^i|\\hat{y}^w_k) \\nonumber \\\\\n & \\prod_{j=1}^{T+U} p(\\hat{y}^s_j|\\mathcal{X},t_j,y^s_0,...,y^s_{u_{j-1}},y^w_0,...,y^w_{u_{j-1}})\n \\label{eq:cond_dist}\n\\end{align}\nwhere $\\hat{\\mathbf{y}}^{w}=(\\hat{y}^w_1,...,\\hat{y}^w_{T+U}) \\subset \\{\\mathcal{W} \\cup \\langle b^w \\rangle\\}^{T+U}$ , $\\hat{\\mathbf{y}}^{s}=(\\hat{y}^s_1,...,\\hat{y}^s_{T+U}) \\subset \\{\\mathcal{S} \\cup \\langle b^s \\rangle\\}^{T+U}$ correspond to any possible alignment path with $T$ blank symbols and $U$ token\/slot labels such that after removing all blank symbols, $b^w$ and $b^s$, in $\\hat{\\mathbf{y}}^{w}$ and $\\hat{\\mathbf{y}}^{s}$ correspondingly, it yields $\\mathbf{y}^{w}$ and $\\mathbf{y}^{s}$. $y^{w}_0$ and $y^{s}_0$ are the start of sentence and slot symbol respectively.\n\n\\subsection{Multi-task Semantic RNN-T} \nThe multi-task Semantic RNN-T architecture consists of three components, an audio encoder, a semantic decoder and a multiple-output joint network as shown in Fig.~\\ref{fig:modelb}. The audio encoder is a unidirectional RNN \\cite{lstm} that takes the audio features $\\mathcal{X} = (\\mathbf{x}_1, \\mathbf{x}_2, ..., \\mathbf{x}_T)$ as inputs and generates the hidden representations, $\\mathcal{H} = (\\mathbf{h}_1, \\mathbf{h}_2, ...,\\mathbf{h}_T)$ auto-regressively. The semantic decoder takes in the word-pieces along with slot tags and outputs hidden label embeddings, $\\mathcal{G} = (\\mathbf{g}_1, \\mathbf{g}_2, ..., \\mathbf{g}_U)$.\nThe encoder and semantic decoder output, $\\mathbf{h}_t$ and $\\mathbf{g}_u$, respectively, are then fed into a joint network to predict next word-piece, $y^w_{u+1}$, and slot tag, $y^s_{u+1}$.\n\nThe semantic decoder has two separate prediction networks for encoding word-pieces and slot tags, correspondingly, and a fusion layer is employed to aggregate the prediction network outputs.\nEach of the prediction networks is a recurrent neural network consisting of an embedding layer, an output layer, and a recurrent hidden layer.\nThe outputs of the two prediction networks, $\\mathbf{g}_u^w$ and $\\mathbf{g}_u^s$, are then fused together producing the semantic decoder output, $\\mathbf{g}_u$.\nWhile we investigated both the addition, $\\mathbf{g}_u = \\mathbf{g}_u^w + \\mathbf{g}_u^s$, and the concatenation with a projection, $\\mathbf{g}_u = W([\\mathbf{g}_u^w; \\mathbf{g}_u^s]$) , as fusion methods, we did not observe a significant performance difference between them, so we report all results with the addition fusion method. \n\nGiven the audio feature vector $\\mathbf{h}_t$ and semantic decoder output $\\mathbf{g}_u$, the multi-output joint network yields distributions for the word-piece and slot tag at the next time step $u+1$.\nThe joint network is composed of a feed-forward neural network and two separate classification layers to produce joint logits, also called lattice, for transcript tokens and slot tags,\n$Z^{w} \\in \\mathbb{R}^{T \\times U \\times V^{w}}$ and $Z^s \\in \\mathbb{R}^{T\\times U \\times V^s}$, where $V^w$ and $V^s$ stand for the word piece size and slot value size, respectively. Each element of $Z^{w}$ and $Z^{s}$ represents the probability of the next word piece $p(y^w_{u+1}|t, u)$,\nand the next slot tag $p(y^s_{u+1}|t, y^s_u)$, correspondingly.\nThe intent classification layer is appended upon the prediction network for the word-piece.\nThe reason for separating intent prediction from the slot tag hypotheses is to reduce the effect of [Other] slot hypotheses (see Table~\\ref{tab:semantc_labels}) in prior time steps on the intent prediction for the final state.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.42\\textwidth]{figures\/model-b-final.png}\n\\vspace{-8pt}\n\\caption{The proposed Multi-task Semantic RNN-T SLU model.} \n\\label{fig:modelb}\n\\vspace{-10pt}\n\\end{figure}\n\\vspace{-4pt}\n\\subsection{Semantic Beam Search}\nTo jointly decode the word piece and slot tag sequences at inference time, we propose a semantic beam search algorithm based on the search algorithm in \\cite{rnnt} (only applied to the word-pieces) to find the top-n best output pairs of word pieces and slot tags. The motivation is to provide the decoder with the most possible candidate pairs of the fixed beam size in every time step and select the best aligned sequences through the decoding path. The semantic beam search parameters of each decoding are local word beam size ($B_{wp}$), local slot beam size ($B_{slot}$), local candidate pair beam size ($B_{local}$), and global candidate pair beam size ($B_{beam}$). $B_{wp}$ and $B_{slot}$ define the number of top possible word piece and slot candidates selected by the top log probability respectively. Given the $B_{wp} \\times B_{slot}$ candidate pairs (combining the top possible word pieces and slot tags), $B_{local}$ of candidate pairs with the highest addition of log probabilities are then selected. Finally, among $B_{local} \\times B_{beam}$ candidate pairs, we preserve $B_{beam}$ ones for the next decoding step.\n\n\\subsection{Loss Functions}\n\\label{subsec:losses}\n\\subsubsection{Word-Piece Loss}\nThe word-piece prediction is optimized with the RNN-T loss \\cite{rnnt}, denoted as $L_{rnnt}(wp)$, which computes the alignment probability summation, $p(\\hat{\\mathbf{y}}^w|\\mathcal{X})$ with a forward-backward algorithm.\n\n\\subsubsection{Intent Classification Loss}\n\nThe intent classification is optimized by minimizing the cross-entropy loss between the intent logits and the ground truth intent label, summed over a batch of utterances.\n\\begin{equation} \n\\label{eq:intent_loss}\n\\begin{split}\n &L_{ce}(intent) = -\\sum y^i \\times \\log(p(\\hat{y}^i|t,u, y^w))\n\\end{split}\n\\end{equation}\n\n\\subsubsection{Slot Tagging Loss} \nGiven the generated transcript tokens and slot tags from prediction networks (Fig.~\\ref{fig:modelb}), this loss is designed to learn the alignment between the two sequences. In particular, we investigate two losses:\n\\begin{itemize}[leftmargin=12pt]\n\\item \\emph{Cross Entropy Loss}: The cross-entropy loss is computed at each state of the slot lattice $Z^s$ and averaged over $T$ time steps for each decoder state.\n\\begin{equation}\n \\begin{split}\n L_{ce}(slot) = -\\sum_{u=1}^U\\frac{1}{T}\\sum_{t=0}^Ty^s \\log(p(\\hat{y}^s|t,u,y^w)) \\\n \\end{split}\n\\end{equation}\n\\item \\emph{Aligned RNN-T Loss}: This loss consists of two terms as follows,\n\\begin{equation}\n\\label{joint}\n\\begin{split}\n L_{rnnt, align}(slot) = L_{rnnt}(slot) + L_{align}(slot)\n\\end{split}\n\\end{equation}\nSimilar to the word-piece loss, the first term, $L_{rnnt}(slot) $ is used to learn the alignment between the audio inputs and the slot tags with the standard RNN-T loss \\cite{rnnt}. The second term $L_{align}(slot)$, is responsible for learning the alignment between the word pieces and their corresponding slot tags at each state, $L_{align}(slot) = -\\log(p(\\mathbf{y}^w, \\mathbf{y}^s | \\mathcal{X}))$. Based on a conditional independence assumption, we use $p(y^w_{u+1}, y^s_{u+1}|t, y^s_u, y^w_u) = p(y^w_{u+1}|t, y^w_u) \\cdot p(y^s_{u+1}|t, y^s_u)$ at each state $(t, u+1)$ and are able to simply reapply the same transducer forward-backward algorithm on this combined lattice to efficiently compute $L_{align}(slot)$.\n\\end{itemize}\n\n\\section{Experimental Results}\n\n\\begin{table}[t]\n\\centering\n\\tabcolsep=0.1cm\n\\caption{Relative Improvements of ASR and NLU metrics (\\%) for Multi-task Semantic RNN-T over the two-stage SLU~\\cite{s2i}.}\n\\label{tab:mt_sem_rnnt}\n\\begin{tabular}{ccccc}\n\\toprule\nModel & WERR & SemERR & IRERR & ICERR \\\\ \\midrule\nTwo-stage SLU~\\cite{s2i} & 0 & 0 & 0 & 0 \\\\ \nOne-stage version of~\\cite{s2i} & 0.6 & 1.2 & 0.1 & -5.8 \\\\\nMulti-task Semantic RNN-T & \\textbf{1.4} & \\textbf{9.5} & \\textbf{14.4} & \\textbf{5.1} \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-3mm}\n\\end{table}\n\n\\begin{table*}[t]\n\\centering\n\\tabcolsep=0.1cm\n\\caption{Comparisons of different proposed multi-task losses}\n\\label{tab:different_losses}\n\\resizebox{0.8\\linewidth}{!}{%\n\\begin{tabular}{cccccc}\n\\toprule\nModel & Loss Type & WERR & SemERR & IRERR & ICERR \\\\ \\midrule\nTwo-stage SLU~\\cite{s2i} & - & 0 & 0 & 0 & 0 \\\\\n\\multirow{2}{*}{Multi-task Semantic RNN-T} & $ L_{rnnt}(wp) + L_{ce}(slot) + L_{ce}(intent) $ & \\textbf{1.41} & \\textbf{9.49} & \\textbf{14.38} & \\textbf{5.13} \\\\\n& $L_{rnnt}(wp) + L_{rnnt,align}(slot) + L_{ce}(intent) $ & -0.99 & \\textbf{7.43} & \\textbf{12.04} & -1.26 \\\\ \\bottomrule\n\\end{tabular}}%\n\\vspace{-2.5mm}\n\\end{table*}\n\n\\begin{table}[t]\n\\centering\n\\tabcolsep=0.1cm\n\\caption{Comparisons of different slot beam sizes, $B_{slot}$, in semantic beam search configuration, ($B_{wp}, B_{slot}, B_{local}, B_{beam}$), for Multi-task Sem-RNN-T}\n\\vspace{2.0mm}\n\\label{tab:beamsearch_seperate_intent}\n\\resizebox{0.8\\linewidth}{!}{%\n\\begin{tabular}{ccccc}\n\\toprule\nSemantic Beam Search & WERR & SemERR & IRERR & ICERR \\\\ \\midrule\n(1,1,1,1)-Greedy Search & 0 & 0 & 0 & 0 \\\\\n(10,1,10,8) & 8.5 & 0.6 & 0.6 & 1.1 \\\\\n(10,2,10,8) & \\textbf{8.6} & \\textbf{11.9} & \\textbf{9.5} & \\textbf{2.9} \\\\ \n(10,4,10,8) & \\textbf{8.6} & \\textbf{11.9} & \\textbf{9.5} & 2.8 \\\\ \\bottomrule\n\\end{tabular}}%\n\\vspace{-4.5mm}\n\\end{table}\n\n\\begin{table}[h]\n\\centering\n\\tabcolsep=0.1cm\n\\caption{ASR and NLU performances (\\%) on the public Fluent Speech Commands dataset}\n\\label{tab:fsc}\n\\begin{tabular}{c|c|c|c|c}\n\\toprule\nModel & Streaming & SemDec. & WER & IRER \\\\ \\midrule\nTwo-stage SLU~\\cite{s2i} & N & N & 0.61 & 0.85 \\\\\nOne-stage version of~\\cite{s2i} & Y & N & 0.54 & 0.91 \\\\\nMT Semantic RNN-T (Ours) & \\textbf{Y} & \\textbf{Y} & \\textbf{0.55} & \\textbf{0.84} \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-3mm}\n\\end{table}\n\n\\subsection{Dataset} \nTo evaluate the multi-task semantic RNN-T model, we use 1,300 hours of speech utterances from our in-house de-identified far-field SLU dataset, containing not only transcriptions but also slot tags and intents.\nThis is broken into training and test sets of 910 hours and 195 hours, respectively.\nThe device-directed far-field, speech data is captured using a smart speaker across multiple English locales (e.g. en-US, en-IN, etc.).\nThe input audio features fed into the network consist of 64-dimensional LFBE features, which are extracted every 10 ms with a window size of 25 ms from audio samples. The features of each frame are then stacked with the left two frames, followed by a downsampling of factor 3 to achieve a low frame rate, with 192 feature dimensions.\nThe subword tokenizer \\cite{sennrich-etal-2016-neural, kudo2018subword} is used to create tokens from the transcriptions; we use 4000 word-pieces in total. There are 63 intent classes and 183 slot tags annotated in this dataset.\nWe also conducted experiments on the public SLU corpus, Fluent Speech Commands (FSC) dataset \\cite{fluent}, which contains 15 hrs (23k utterances), 11 intents and 3 slots, including the $``Other\"$ class.\n\n\\subsection{Baselines and Model Configurations}\nWe compare the proposed method with the state-of-the-art RNN-T based two-stage SLU model ~\\cite{s2i}. We also compare to another baseline by extending \\cite{s2i} to the one-stage version (as in Fig.~\\ref{fig:high_level}) introduced in \\cite{google}. The proposed model has the following configurations. The audio encoder is a 5-layer LSTM with 736 neurons and output size of 512-dimension. The word piece prediction network is of a 512-dim embedding layer followed by a 2-layer LSTM with 736 neurons and output size of 512-dim. The slot tag prediction network is of a 128-dim embedding layer and a 2-layer LSTM with 256 neurons and output size of 512-dim. The intent decoder consists of two 128-dim dense layers with $ReLU$ as an activation function. The joint network is a fully-connected feed-forward component with one hidden layer followed by a $tanh$ activation function. Overall, the size of both the proposed model and the baselines sum to approximately 40 million parameters. For the FSC data set, given the small amount of transcribed data to train the ASR module of million parameters well, we followed \\cite{s2i} by first pre-training the audio encoder, the word piece prediction network, and the joint network on 910 hrs Alexa data, before finetuning on the FSC dataset. \n\n\\subsection{Metrics}\nWe use four metrics to evaluate the performance of an E2E SLU system.\n(i) \\textbf{Word Error Rate (WER)}: WER is a word-level metric used for evaluating the word-piece recognition performance. It calculates Levenshtein distance or edit distance that is the shortest distance required for transforming word-piece hypothesis to the ground truth by using insertion, deletion and substitution.\n(ii) \\textbf{Semantic Error Rate (SemER)}: The SemER metric jointly evaluates the performance of intent classification and slot filling or say NLU performance.\nBy comparing a word sequence reference and their accompanying slot tags, performance is calculated as: \n\\begin{equation}\n\\label{eq:semer}\n SemER = \\frac{\\#Deletion+\\#Insertion+\\#Substitution}{\\#Correct+\\#Deletion+\\#Substitution},\n\\end{equation}\nwhere $Correct$ is when slot tag and slot value (words) are correctly identified, $Deletion$ is when a slot tag present in the reference is not the hypothesis, $Insertion$ is an extraneous slot tag included by hypothesis, and\n$Substitution$ is when a slot tag from hypothesis is included but with the incorrect slot value. Intent classification errors are counted as substitution errors.\n(iii) \\textbf{Interpretation Error Rate (IRER)}: The IRER metric is an utterance-level metric for evaluating the joint intent classification and slot filling performance without partial credit. Namely, it is the fraction of utterances where either the intent or any of the slots are predicted incorrectly. \n(iv) \\textbf{Intent Classification Error Rate (ICER)}: The ICER metric measures the error rate of intent classification, which is an utterance-level evaluation metric. \nThe results of all experiments are presented as the relative error rate reductions (WERR\/SemERR\/IRERR\/ICERR). For example, given model A's WER ($\\text{WER}_A$) and a baseline B's WER ($\\text{WER}_B$), the WERR of A over B is computed as\n$\\text{WERR} = (\\text{WER}_B - \\text{WER}_A)\/\\text{WER}_B.$\n\n\\vspace{-2pt}\n\\subsection{Results}\nThe results of multi-task semantic RNN-T over the baselines are shown in Table~\\ref{tab:mt_sem_rnnt}. Jointly training ASR and NLU tasks with a semantic decoder shows consistent improvements across tasks and metrics by providing more contextual information. Of note is that the NLU metrics such as SemER and IRER are significantly improved while the improvement of WER is comparably small. We attribute this to the annotation bias of the slot tag distribution of the dataset: Around 60\\% of the utterances have their slot tags mapped to the [Other] label. Therefore, the semantic information provided by slot tags for the word-piece generation may be limited.\n\nIn Table~\\ref{tab:different_losses}, we compare different loss combinations as introduced in Sec~\\ref{subsec:losses}. Using cross-entropy loss for slot tagging has demonstrated the best overall performances, and greatly improves the two-stage SLU model in terms of both NLU and ASR metrics. Imposing aligned RNN-T loss also significantly improves the NLU metrics such as SemER and IRER, but slightly degrades the WER and ICER. We believe this is because the loss is more sensitive to the misalignments between the word-piece and slot tags produced by the separate prediction networks. \n\nFinally, we validate the effectiveness of the semantic beam search algorithm. We fixed the best-performing parameters of $B_{wp}$, $B_{local}$, $B_{beam}$, and varied $B_{slot}$ to 1, 2, 4 and showed the results in Table~\\ref{tab:beamsearch_seperate_intent}. As it can be seen, changing the parameters from a greedy search to the beam search, from (1,1,1,1) to (10,1,10,8), improves all metrics, while mainly improving WER. When further increasing the slot beam size from 1 to 2 or 4, the improvements over NLU metrics become significant, \n$11.9\\%$ and $9.5\\%$ in terms of SemERR and IRERR. Again, the improvement of WER from increasing slot beam search size is limited and may be attributed to the inherent bias of the slot tag annotations.\n\nTable~\\ref{tab:fsc} presents the results on FSC dataset~\\cite{fluent}, where the streaming capability and the use of semantic information during decoding (Yes\/No: Y\/N) are also shown in the table. In our model (MT Semantic RNN-T), we used the additive fusion to obtain the semantic decoder output, with the semantic beam search size set by $B_{wp}$=10, $B_{slot}$=2, $B_{local}$=10, and $B_{beam}$=16. Note that due to the lower complexity and limited number of intents and slots in FSC, all the models in our experiments lead to $<1\\%$ WER and IRER values. The proposed model improved the WER of 2-stage SLU by 9.8\\% while improved IRER of 1-stage SLU by 7.7\\% relatively. \n\\section{Conclusion} \nIn this paper, we proposed a multi-task semantic RNN-T architecture for streamable end-to-end spoken language understanding. The proposed semantic decoder and semantic beam search empower the model to consider more contextual information, both from the predicted word piece and slot tags in the history, and produce higher quality word-piece and slot tag hypothesis in the next time step. Moreover, the model with the proposed losses, both cross-entropy loss and the aligned RNN-T loss for slot tagging, outperformed the two-stage and one-stage E2E SLU models. \n\n\\section{Experimental Results}\n\n\\begin{table}[t]\n\\centering\n\\tabcolsep=0.1cm\n\\caption{Relative Improvements of ASR and NLU metrics (\\%) for Multi-task Semantic RNN-T over the two-stage SLU~\\cite{s2i}.}\n\\label{tab:mt_sem_rnnt}\n\\begin{tabular}{ccccc}\n\\toprule\nModel & WERR & SemERR & IRERR & ICERR \\\\ \\midrule\nTwo-stage SLU~\\cite{s2i} & 0 & 0 & 0 & 0 \\\\ \nOne-stage version of~\\cite{s2i} & 0.6 & 1.2 & 0.1 & -5.8 \\\\\nMulti-task Semantic RNN-T & \\textbf{1.4} & \\textbf{9.5} & \\textbf{14.4} & \\textbf{5.1} \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-3mm}\n\\end{table}\n\n\\begin{table*}[t]\n\\centering\n\\tabcolsep=0.1cm\n\\caption{Comparisons of different proposed multi-task losses}\n\\label{tab:different_losses}\n\\resizebox{0.8\\linewidth}{!}{%\n\\begin{tabular}{cccccc}\n\\toprule\nModel & Loss Type & WERR & SemERR & IRERR & ICERR \\\\ \\midrule\nTwo-stage SLU~\\cite{s2i} & - & 0 & 0 & 0 & 0 \\\\\n\\multirow{2}{*}{Multi-task Semantic RNN-T} & $ L_{rnnt}(wp) + L_{ce}(slot) + L_{ce}(intent) $ & \\textbf{1.41} & \\textbf{9.49} & \\textbf{14.38} & \\textbf{5.13} \\\\\n& $L_{rnnt}(wp) + L_{rnnt,align}(slot) + L_{ce}(intent) $ & -0.99 & \\textbf{7.43} & \\textbf{12.04} & -1.26 \\\\ \\bottomrule\n\\end{tabular}}%\n\\vspace{-2.5mm}\n\\end{table*}\n\n\\begin{table}[t]\n\\centering\n\\tabcolsep=0.1cm\n\\caption{Comparisons of different slot beam sizes, $B_{slot}$, in semantic beam search configuration, ($B_{wp}, B_{slot}, B_{local}, B_{beam}$), for Multi-task Sem-RNN-T}\n\\vspace{2.0mm}\n\\label{tab:beamsearch_seperate_intent}\n\\resizebox{0.8\\linewidth}{!}{%\n\\begin{tabular}{ccccc}\n\\toprule\nSemantic Beam Search & WERR & SemERR & IRERR & ICERR \\\\ \\midrule\n(1,1,1,1)-Greedy Search & 0 & 0 & 0 & 0 \\\\\n(10,1,10,8) & 8.5 & 0.6 & 0.6 & 1.1 \\\\\n(10,2,10,8) & \\textbf{8.6} & \\textbf{11.9} & \\textbf{9.5} & \\textbf{2.9} \\\\ \n(10,4,10,8) & \\textbf{8.6} & \\textbf{11.9} & \\textbf{9.5} & 2.8 \\\\ \\bottomrule\n\\end{tabular}}%\n\\vspace{-4.5mm}\n\\end{table}\n\n\\begin{table}[h]\n\\centering\n\\tabcolsep=0.1cm\n\\caption{ASR and NLU performances (\\%) on the public Fluent Speech Commands dataset}\n\\label{tab:fsc}\n\\begin{tabular}{c|c|c|c|c}\n\\toprule\nModel & Streaming & SemDec. & WER & IRER \\\\ \\midrule\nTwo-stage SLU~\\cite{s2i} & N & N & 0.61 & 0.85 \\\\\nOne-stage version of~\\cite{s2i} & Y & N & 0.54 & 0.91 \\\\\nMT Semantic RNN-T (Ours) & \\textbf{Y} & \\textbf{Y} & \\textbf{0.55} & \\textbf{0.84} \\\\ \\bottomrule\n\\end{tabular}\n\\vspace{-3mm}\n\\end{table}\n\n\\subsection{Dataset} \nTo evaluate the multi-task semantic RNN-T model, we use 1,300 hours of speech utterances from our in-house de-identified far-field SLU dataset, containing not only transcriptions but also slot tags and intents.\nThis is broken into training and test sets of 910 hours and 195 hours, respectively.\nThe device-directed far-field, speech data is captured using a smart speaker across multiple English locales (e.g. en-US, en-IN, etc.).\nThe input audio features fed into the network consist of 64-dimensional LFBE features, which are extracted every 10 ms with a window size of 25 ms from audio samples. The features of each frame are then stacked with the left two frames, followed by a downsampling of factor 3 to achieve a low frame rate, with 192 feature dimensions.\nThe subword tokenizer \\cite{sennrich-etal-2016-neural, kudo2018subword} is used to create tokens from the transcriptions; we use 4000 word-pieces in total. There are 63 intent classes and 183 slot tags annotated in this dataset.\nWe also conducted experiments on the public SLU corpus, Fluent Speech Commands (FSC) dataset \\cite{fluent}, which contains 15 hrs (23k utterances), 11 intents and 3 slots, including the $``Other\"$ class.\n\n\\subsection{Baselines and Model Configurations}\nWe compare the proposed method with the state-of-the-art RNN-T based two-stage SLU model ~\\cite{s2i}. We also compare to another baseline by extending \\cite{s2i} to the one-stage version (as in Fig.~\\ref{fig:high_level}) introduced in \\cite{google}. The proposed model has the following configurations. The audio encoder is a 5-layer LSTM with 736 neurons and output size of 512-dimension. The word piece prediction network is of a 512-dim embedding layer followed by a 2-layer LSTM with 736 neurons and output size of 512-dim. The slot tag prediction network is of a 128-dim embedding layer and a 2-layer LSTM with 256 neurons and output size of 512-dim. The intent decoder consists of two 128-dim dense layers with $ReLU$ as an activation function. The joint network is a fully-connected feed-forward component with one hidden layer followed by a $tanh$ activation function. Overall, the size of both the proposed model and the baselines sum to approximately 40 million parameters. For the FSC data set, given the small amount of transcribed data to train the ASR module of million parameters well, we followed \\cite{s2i} by first pre-training the audio encoder, the word piece prediction network, and the joint network on 910 hrs Alexa data, before finetuning on the FSC dataset. \n\n\\subsection{Metrics}\nWe use four metrics to evaluate the performance of an E2E SLU system.\n(i) \\textbf{Word Error Rate (WER)}: WER is a word-level metric used for evaluating the word-piece recognition performance. It calculates Levenshtein distance or edit distance that is the shortest distance required for transforming word-piece hypothesis to the ground truth by using insertion, deletion and substitution.\n(ii) \\textbf{Semantic Error Rate (SemER)}: The SemER metric jointly evaluates the performance of intent classification and slot filling or say NLU performance.\nBy comparing a word sequence reference and their accompanying slot tags, performance is calculated as: \n\\begin{equation}\n\\label{eq:semer}\n SemER = \\frac{\\#Deletion+\\#Insertion+\\#Substitution}{\\#Correct+\\#Deletion+\\#Substitution},\n\\end{equation}\nwhere $Correct$ is when slot tag and slot value (words) are correctly identified, $Deletion$ is when a slot tag present in the reference is not the hypothesis, $Insertion$ is an extraneous slot tag included by hypothesis, and\n$Substitution$ is when a slot tag from hypothesis is included but with the incorrect slot value. Intent classification errors are counted as substitution errors.\n(iii) \\textbf{Interpretation Error Rate (IRER)}: The IRER metric is an utterance-level metric for evaluating the joint intent classification and slot filling performance without partial credit. Namely, it is the fraction of utterances where either the intent or any of the slots are predicted incorrectly. \n(iv) \\textbf{Intent Classification Error Rate (ICER)}: The ICER metric measures the error rate of intent classification, which is an utterance-level evaluation metric. \nThe results of all experiments are presented as the relative error rate reductions (WERR\/SemERR\/IRERR\/ICERR). For example, given model A's WER ($\\text{WER}_A$) and a baseline B's WER ($\\text{WER}_B$), the WERR of A over B is computed as\n$\\text{WERR} = (\\text{WER}_B - \\text{WER}_A)\/\\text{WER}_B.$\n\n\\vspace{-2pt}\n\\subsection{Results}\nThe results of multi-task semantic RNN-T over the baselines are shown in Table~\\ref{tab:mt_sem_rnnt}. Jointly training ASR and NLU tasks with a semantic decoder shows consistent improvements across tasks and metrics by providing more contextual information. Of note is that the NLU metrics such as SemER and IRER are significantly improved while the improvement of WER is comparably small. We attribute this to the annotation bias of the slot tag distribution of the dataset: Around 60\\% of the utterances have their slot tags mapped to the [Other] label. Therefore, the semantic information provided by slot tags for the word-piece generation may be limited.\n\nIn Table~\\ref{tab:different_losses}, we compare different loss combinations as introduced in Sec~\\ref{subsec:losses}. Using cross-entropy loss for slot tagging has demonstrated the best overall performances, and greatly improves the two-stage SLU model in terms of both NLU and ASR metrics. Imposing aligned RNN-T loss also significantly improves the NLU metrics such as SemER and IRER, but slightly degrades the WER and ICER. We believe this is because the loss is more sensitive to the misalignments between the word-piece and slot tags produced by the separate prediction networks. \n\nFinally, we validate the effectiveness of the semantic beam search algorithm. We fixed the best-performing parameters of $B_{wp}$, $B_{local}$, $B_{beam}$, and varied $B_{slot}$ to 1, 2, 4 and showed the results in Table~\\ref{tab:beamsearch_seperate_intent}. As it can be seen, changing the parameters from a greedy search to the beam search, from (1,1,1,1) to (10,1,10,8), improves all metrics, while mainly improving WER. When further increasing the slot beam size from 1 to 2 or 4, the improvements over NLU metrics become significant, \n$11.9\\%$ and $9.5\\%$ in terms of SemERR and IRERR. Again, the improvement of WER from increasing slot beam search size is limited and may be attributed to the inherent bias of the slot tag annotations.\n\nTable~\\ref{tab:fsc} presents the results on FSC dataset~\\cite{fluent}, where the streaming capability and the use of semantic information during decoding (Yes\/No: Y\/N) are also shown in the table. In our model (MT Semantic RNN-T), we used the additive fusion to obtain the semantic decoder output, with the semantic beam search size set by $B_{wp}$=10, $B_{slot}$=2, $B_{local}$=10, and $B_{beam}$=16. Note that due to the lower complexity and limited number of intents and slots in FSC, all the models in our experiments lead to $<1\\%$ WER and IRER values. The proposed model improved the WER of 2-stage SLU by 9.8\\% while improved IRER of 1-stage SLU by 7.7\\% relatively. \n\\section{Introduction}\nWith the widespread application of intelligent voice assistants, e.g. Alexa, Siri, and Google Home, SLU systems have generated increased interest in the recent years.\nAn SLU system predicts semantic information implied by an audio signal.\nThis semantic content is commonly represented as intent, slot tags, named entities and\/or part-of-speech taggings.\nToday's SLU technology typically accomplishes this task in two separate stages, which we refer to as ASR-NLU approaches for SLU~\\cite{larson2012spoken}: an ASR system first transcribes the audio signals~\\cite{deepspeech, las}, and then the transcripts are passed to an NLU system to extract corresponding intent and slot tags~\\cite{nlu, nlu2, bertnlu}; an example is presented in Table~\\ref{tab:semantc_labels}.\nGiven the extracted semantic labels, downstream applications of the voice assistant can produce an appropriate response to the user.\n\nRecently, complete E2E-SLU based approaches have attracted attention due to their efficiency and reduced model complexity compared with an ASR-NLU pipeline, making them suitable candidates for deployment on low-resource devices~\\cite{google, lai2021semi, chuang2019speechbert, radfar2021fans}. \nFurthermore, performance improvements in both tasks, driven by joint training, has been observed in several studies \\cite{google, fluent}.\n\nMost of existing E2E-SLU models still adopt a two-stage setup as shown in Fig.~\\ref{fig:high_level}(a), where the NLU subsystem waits for the transcripts of the whole utterance produced by the ASR subsystem to generate semantic labels~\\cite{e2eslu, google, fluent, semantic1}. Meanwhile, the NLU subsystem is typically non-streamable. \nOne-stage approaches to E2E-SLU have been proposed as well \\cite{google, radfar2021fans}; however, again the NLU subsystem remains non-streamable. \nFor example, in \\cite{radfar2021fans}, slot tag prediction only occurs after the intent is extracted at the end of an utterance.\nMoreover, in all the above approaches, the ASR and NLU label generations do not interact with one another during the forward pass of inference (Fig.~\\ref{fig:high_level}(b)). \nAs a result, this design can lead to three main limitations. \nFirst, the NLU posterior or hypothesis does not provide any feedback upon word-piece generation, while its feedback could be helpful to narrow down potential word-piece candidates generated in the next time step.\nSecondly, the decoding of the NLU label predictions is not streamable \\cite{google}, given that the model is an encoder-decoder framework augmented by attention.\nFinally, the inference speed of an SLU system may be affected by the nature of the cascaded setup and non-streamable NLU subsystem, all the while low latency is crucial for a responsive virtual assistant. \n\nTo address these limitations, we propose a streamable E2E-SLU model based on RNN-T \\cite{rnnt, rnnt-asr} with a novel semantic beam search decoder which predicts word-pieces and NLU labels jointly, as illustrated in Fig.~\\ref{fig:high_level}(c).\nSpecifically, we introduce a semantic decoder to aggregate not only the word-pieces but also slot candidates during the beam search, which we call \\emph{semantic beam search}. \nFurthermore, we propose different multi-task loss functions to learn the alignment between word-pieces and slot tags along with the intent prediction.\n\n\\begin{table}[t]\n\\centering\n\\small\n\\tabcolsep=0.1cm\n\\caption{\\small An example of a transcription, slot tags, and intent.}\n\\label{tab:semantc_labels}\n\\begin{tabular}{cc}\n\\toprule\n\\textbf{transcription} & turn on the kitchen light \\\\ \\midrule\n\\textbf{slot tags} & [DeviceLocation]: kitchen \\\\\n& [ApplianceType]: light, [Other]:turn,on,the \\\\ \\midrule \n\\textbf{intent} & TurnOnApplianceIntent \\\\ \\bottomrule\n\\end{tabular}\n\\vskip -10pt\n\\end{table}\n\n\\section{Methodology}\n\\label{sec:methodology}\n\nThe inputs of the multi-task RNN-T are $D$-dimensional audio features of length $T$, $\\mathcal{X} = (\\mathbf{x}_1, \\mathbf{x}_2, ..., \\mathbf{x}_T)$, $\\mathbf{x}_k \\in \\mathbb{R}^{D}$. The outputs are transcript tokens of length $U$, $\\mathbf{y}^w = (y^w_1, y^w_2, ..., y^w_U)$, $y^w_u \\in \\mathcal{W}$, its corresponding slot tags (also of length $U$), $\\mathbf{y}^s = (y^s_1, y^s_2, ..., y^s_U)$, $y^s_u \\in \\mathcal{S}$, and the intent $y^i \\in \\mathcal{I}$; here $\\mathcal{W}$, $\\mathcal{S}$, and $\\mathcal{I}$ are the predefined set of token labels (or token vocabulary), slot tags, and intents. Both transcript tokens and slot tags are encoded as one-hot vectors. \n\nThe model defines a conditional distribution of $p(\\mathcal{W},\\mathcal{S},\\mathcal{I}|\\mathcal{X})$, and we factorize it as follows (Fig.~\\ref{fig:modelb}),\n\\begin{align}\n & p(\\hat{\\mathbf{y}}^w,\\hat{\\mathbf{y}}^s,y^i|\\mathcal{X}) = \\nonumber \\\\ \n & \\prod_{k=1}^{T+U} p(\\hat{y}^w_k|\\mathcal{X},t_k,y^w_0,...,y^w_{u_{k-1}},y^s_0,...,y^s_{u_{k-1}}) p(y^i|\\hat{y}^w_k) \\nonumber \\\\\n & \\prod_{j=1}^{T+U} p(\\hat{y}^s_j|\\mathcal{X},t_j,y^s_0,...,y^s_{u_{j-1}},y^w_0,...,y^w_{u_{j-1}})\n \\label{eq:cond_dist}\n\\end{align}\nwhere $\\hat{\\mathbf{y}}^{w}=(\\hat{y}^w_1,...,\\hat{y}^w_{T+U}) \\subset \\{\\mathcal{W} \\cup \\langle b^w \\rangle\\}^{T+U}$ , $\\hat{\\mathbf{y}}^{s}=(\\hat{y}^s_1,...,\\hat{y}^s_{T+U}) \\subset \\{\\mathcal{S} \\cup \\langle b^s \\rangle\\}^{T+U}$ correspond to any possible alignment path with $T$ blank symbols and $U$ token\/slot labels such that after removing all blank symbols, $b^w$ and $b^s$, in $\\hat{\\mathbf{y}}^{w}$ and $\\hat{\\mathbf{y}}^{s}$ correspondingly, it yields $\\mathbf{y}^{w}$ and $\\mathbf{y}^{s}$. $y^{w}_0$ and $y^{s}_0$ are the start of sentence and slot symbol respectively.\n\n\\subsection{Multi-task Semantic RNN-T} \nThe multi-task Semantic RNN-T architecture consists of three components, an audio encoder, a semantic decoder and a multiple-output joint network as shown in Fig.~\\ref{fig:modelb}. The audio encoder is a unidirectional RNN \\cite{lstm} that takes the audio features $\\mathcal{X} = (\\mathbf{x}_1, \\mathbf{x}_2, ..., \\mathbf{x}_T)$ as inputs and generates the hidden representations, $\\mathcal{H} = (\\mathbf{h}_1, \\mathbf{h}_2, ...,\\mathbf{h}_T)$ auto-regressively. The semantic decoder takes in the word-pieces along with slot tags and outputs hidden label embeddings, $\\mathcal{G} = (\\mathbf{g}_1, \\mathbf{g}_2, ..., \\mathbf{g}_U)$.\nThe encoder and semantic decoder output, $\\mathbf{h}_t$ and $\\mathbf{g}_u$, respectively, are then fed into a joint network to predict next word-piece, $y^w_{u+1}$, and slot tag, $y^s_{u+1}$.\n\nThe semantic decoder has two separate prediction networks for encoding word-pieces and slot tags, correspondingly, and a fusion layer is employed to aggregate the prediction network outputs.\nEach of the prediction networks is a recurrent neural network consisting of an embedding layer, an output layer, and a recurrent hidden layer.\nThe outputs of the two prediction networks, $\\mathbf{g}_u^w$ and $\\mathbf{g}_u^s$, are then fused together producing the semantic decoder output, $\\mathbf{g}_u$.\nWhile we investigated both the addition, $\\mathbf{g}_u = \\mathbf{g}_u^w + \\mathbf{g}_u^s$, and the concatenation with a projection, $\\mathbf{g}_u = W([\\mathbf{g}_u^w; \\mathbf{g}_u^s]$) , as fusion methods, we did not observe a significant performance difference between them, so we report all results with the addition fusion method. \n\nGiven the audio feature vector $\\mathbf{h}_t$ and semantic decoder output $\\mathbf{g}_u$, the multi-output joint network yields distributions for the word-piece and slot tag at the next time step $u+1$.\nThe joint network is composed of a feed-forward neural network and two separate classification layers to produce joint logits, also called lattice, for transcript tokens and slot tags,\n$Z^{w} \\in \\mathbb{R}^{T \\times U \\times V^{w}}$ and $Z^s \\in \\mathbb{R}^{T\\times U \\times V^s}$, where $V^w$ and $V^s$ stand for the word piece size and slot value size, respectively. Each element of $Z^{w}$ and $Z^{s}$ represents the probability of the next word piece $p(y^w_{u+1}|t, u)$,\nand the next slot tag $p(y^s_{u+1}|t, y^s_u)$, correspondingly.\nThe intent classification layer is appended upon the prediction network for the word-piece.\nThe reason for separating intent prediction from the slot tag hypotheses is to reduce the effect of [Other] slot hypotheses (see Table~\\ref{tab:semantc_labels}) in prior time steps on the intent prediction for the final state.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.42\\textwidth]{figures\/model-b-final.png}\n\\vspace{-8pt}\n\\caption{The proposed Multi-task Semantic RNN-T SLU model.} \n\\label{fig:modelb}\n\\vspace{-10pt}\n\\end{figure}\n\\vspace{-4pt}\n\\subsection{Semantic Beam Search}\nTo jointly decode the word piece and slot tag sequences at inference time, we propose a semantic beam search algorithm based on the search algorithm in \\cite{rnnt} (only applied to the word-pieces) to find the top-n best output pairs of word pieces and slot tags. The motivation is to provide the decoder with the most possible candidate pairs of the fixed beam size in every time step and select the best aligned sequences through the decoding path. The semantic beam search parameters of each decoding are local word beam size ($B_{wp}$), local slot beam size ($B_{slot}$), local candidate pair beam size ($B_{local}$), and global candidate pair beam size ($B_{beam}$). $B_{wp}$ and $B_{slot}$ define the number of top possible word piece and slot candidates selected by the top log probability respectively. Given the $B_{wp} \\times B_{slot}$ candidate pairs (combining the top possible word pieces and slot tags), $B_{local}$ of candidate pairs with the highest addition of log probabilities are then selected. Finally, among $B_{local} \\times B_{beam}$ candidate pairs, we preserve $B_{beam}$ ones for the next decoding step.\n\n\\subsection{Loss Functions}\n\\label{subsec:losses}\n\\subsubsection{Word-Piece Loss}\nThe word-piece prediction is optimized with the RNN-T loss \\cite{rnnt}, denoted as $L_{rnnt}(wp)$, which computes the alignment probability summation, $p(\\hat{\\mathbf{y}}^w|\\mathcal{X})$ with a forward-backward algorithm.\n\n\\subsubsection{Intent Classification Loss}\n\nThe intent classification is optimized by minimizing the cross-entropy loss between the intent logits and the ground truth intent label, summed over a batch of utterances.\n\\begin{equation} \n\\label{eq:intent_loss}\n\\begin{split}\n &L_{ce}(intent) = -\\sum y^i \\times \\log(p(\\hat{y}^i|t,u, y^w))\n\\end{split}\n\\end{equation}\n\n\\subsubsection{Slot Tagging Loss} \nGiven the generated transcript tokens and slot tags from prediction networks (Fig.~\\ref{fig:modelb}), this loss is designed to learn the alignment between the two sequences. In particular, we investigate two losses:\n\\begin{itemize}[leftmargin=12pt]\n\\item \\emph{Cross Entropy Loss}: The cross-entropy loss is computed at each state of the slot lattice $Z^s$ and averaged over $T$ time steps for each decoder state.\n\\begin{equation}\n \\begin{split}\n L_{ce}(slot) = -\\sum_{u=1}^U\\frac{1}{T}\\sum_{t=0}^Ty^s \\log(p(\\hat{y}^s|t,u,y^w)) \\\n \\end{split}\n\\end{equation}\n\\item \\emph{Aligned RNN-T Loss}: This loss consists of two terms as follows,\n\\begin{equation}\n\\label{joint}\n\\begin{split}\n L_{rnnt, align}(slot) = L_{rnnt}(slot) + L_{align}(slot)\n\\end{split}\n\\end{equation}\nSimilar to the word-piece loss, the first term, $L_{rnnt}(slot) $ is used to learn the alignment between the audio inputs and the slot tags with the standard RNN-T loss \\cite{rnnt}. The second term $L_{align}(slot)$, is responsible for learning the alignment between the word pieces and their corresponding slot tags at each state, $L_{align}(slot) = -\\log(p(\\mathbf{y}^w, \\mathbf{y}^s | \\mathcal{X}))$. Based on a conditional independence assumption, we use $p(y^w_{u+1}, y^s_{u+1}|t, y^s_u, y^w_u) = p(y^w_{u+1}|t, y^w_u) \\cdot p(y^s_{u+1}|t, y^s_u)$ at each state $(t, u+1)$ and are able to simply reapply the same transducer forward-backward algorithm on this combined lattice to efficiently compute $L_{align}(slot)$.\n\\end{itemize}\n\n\\section{Related Work}\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{figures\/fig_1.png}\n\\caption{A high-level diagram of comparing the proposed E2E SLU model to the previous two-stage~\\cite{s2i,nlu, nlu2, bertnlu,larson2012spoken} (a) and one-stage \\cite{google, radfar2021fans} (b) E2E SLU models. Dotted lines represent the conditioning of output label on its history, as a part of decoders.} \n\\label{fig:high_level}\n\\vspace{-2.5mm}\n\\end{figure*}\n\nA multi-task learning framework for E2E SLU was first introduced in \\cite{google}; the authors investigated several encoder-decoder structures for joint training of ASR and NLU tasks, in which the multi-task structure achieves the best performance. \nThis work was followed by proposed pre-training based approaches to improve the performance \\cite{fluent, pretrain1, pretrain2}; all of these models are designed specifically for intent classification. \nAnother category of work attempts a combined parameter transfer from well trained end-to-end ASR systems and end-to-end NLU models such as pretrained BERT~\\cite{devlin2018bert} through teacher-student learning~\\cite{semantic1, semantic2, semantic3}.\nNote that both of these categories rely on at least a two-stage training process and all operate with non-streamable inference. \n\nLeveraging RNN-T, in \\cite{s2i}, a two-stage E2E SLU structure was proposed where the RNN-T based ASR subsystem interacts with an NLU subsystem through an interface, which is not streamable.\n\nIn the most recent work, \\cite{streaming_e2e} proposed a CTC-based streamable E2E SLU framework which employs a unidirectional RNN to make multiple intent predictions. The NLU output is generated either directly from the audio signal or based on an intermediate ASR output. \nNamely, the advantage of semantic posterior that we employ, was not considered in making word-piece predictions.\nTo the best of our knowledge, there is no existing E2E SLU model, either streamable or non-streamable, which takes both word-piece and slot tag in beam search decoding on joint multi-task sequence prediction for word-piece, slot tag, and intent label.","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIn recent years numerous works have been dedicated to the calculus of variations\non time scales and their generalizations ---\nsee \\cite{MyID:140,MyID:175,MyID:180,MyID:170,MyID:141,MyID:183,MyID:187,MyID:171,MyID:174}\nand the references therein.\nMost of them deal with delta or nabla derivatives of first-order\n\\cite{Ric:Del,Atici06,Atici09,Zbig:Del,CD:Bohner:2004,FT:Rem,iso:ts,zeidan,Agn:Del,Agn:Del2},\nonly a few with higher-order derivatives \\cite{FT,MT}.\nDepending on the type of the functional being considered, different time scale Euler-Lagrange\ntype equations are obtained. For variational problems of first-order\nthe Euler-Lagrange equations are valid for an arbitrary\ntime scale $\\mathbb{T}$, while for the problems with higher-order delta (or nabla)\nderivatives they are only valid in a certain class of time scales, more precisely,\nthe ones for which the forward (or backward) jump operator is a polynomial\nof degree one \\cite{FT,MT}. Here we consider variational problems\ninvolving Hilger derivatives of higher order, and prove a necessary optimality condition\nof the Euler-Lagrange type on an arbitrary time scale, \\textrm{i.e.},\nwithout imposing any restriction to the jump operators.\n\n\n\\section{Preliminaries}\n\\label{sec:Prel}\n\nHere we recall some basic results and notation needed in the sequel.\nFor the theory of time scales we refer the reader to\n\\cite{Agarwal,livro,Hilger90,Hilger97}.\n\nA time scale $\\mathbb{T}$ is an arbitrary nonempty closed subset of the real\nnumbers $\\mathbb{R}$. The functions $\\sigma:\\mathbb{T} \\mbox{$\\rightarrow$} \\mathbb{T}$ and $\\rho:\\mathbb{T} \\mbox{$\\rightarrow$}\n\\mathbb{T}$ are, respectively, the forward and backward jump operators:\n$\\sigma(t)=\\inf{\\{s\\in\\mathbb{T}:s>t\\}}$ with\n$\\inf\\emptyset=\\sup\\mathbb{T}$ (\\textrm{i.e.}, $\\sigma(M)=M$ if\n$\\mathbb{T}$ has a maximum $M$);\n$\\rho(t)=\\sup{\\{s\\in\\mathbb{T}:st$, $\\rho(t)=t$, or $\\rho(t)0$ there exists a neighborhood $U$\nof $t$ such that\n$$\n\\left|f(\\sigma(t))-f(s)-f^{\\Delta}(t)(\\sigma(t)-s)\\right|\n\\leq\\varepsilon|\\sigma(t)-s|,\\quad\\mbox{ for all $s\\in U$}.\n$$\nWe call $f^{\\Delta}(t)$ the \\emph{delta-derivative} of $f$ at $t$.\nWe note that if the number $f^\\Delta(t)$ exists then it is unique\nin $\\mathbb{T}^\\kappa$ (see \\cite{Hilger90,Hilger97}).\nIn the special cases $\\mathbb{T}=\\mathbb{R}$ and $\\mathbb{T}=\\mathbb{Z}$,\n$f^\\Delta$ reduces to the standard derivative $f'(t)$\nand the forward difference $\\Delta f(t) = f(t+1)-f(t)$, respectively.\nWhenever $f^\\Delta$ exists, the following formula holds:\n$f^\\sigma(t)=f(t)+\\mu(t)f^\\Delta(t)$,\nwhere we abbreviate $f\\circ\\sigma$ by $f^\\sigma$.\nLet $f^{\\Delta^{0}} = f$. We define the\n$r$th-delta derivative of $f:\\mathbb{T}^{\\kappa^r}\\rightarrow\n\\mathbb{R}$, $r\\in\\mathbb{N}$, to be the function $\\left(f^{\\Delta^{r-1}}\\right)^\\Delta$,\nprovided $f^{\\Delta^{r-1}}$ is delta differentiable on $\\mathbb{T}^{\\kappa^r}$.\n\nA function $f:\\mathbb{T} \\to \\mathbb{R}$ is called rd-continuous if\nit is continuous at the right-dense points in $\\mathbb{T}$ and its\nleft-sided limits exist at all left-dense points in $\\mathbb{T}$. A\nfunction $f:\\mathbb{T} \\to \\mathbb{R}^n$ is rd-continuous if all its\ncomponents are rd-continuous. The set of all rd-continuous functions\nis denoted by $C_{rd}$. Similarly, $C^r_{rd}$ will denote the set of\nfunctions with delta derivatives up to order $r$ belonging to\n$C_{rd}$. A function $f$ is of class $f\\in C_{prd}^r$ if\n$f^{\\Delta^i}$ is continuous for $i = 0,\\ldots,r-1$, and\n$f^{\\Delta^r}$ exists and is rd-continuous for all, except possibly\nat finitely many $t \\in \\mathbb{T}^{\\kappa^r}$.\n\nA piecewise rd-continuous function $f:\\mathbb{T} \\to \\mathbb{R}$\npossess an antiderivative $F^{\\Delta}=f$, and in this case the delta\nintegral is defined by $\\int_{c}^{d}f(t)\\Delta t=F(d)-F(c)$ for all\n$c,d\\in\\mathbb{T}$. It satisfies\n$$\\int_t^{\\sigma(t)}f(\\tau)\\Delta\\tau=\\mu(t)f(t).$$\nIf $\\mathbb{T}=\\mathbb{R}$, then $\\int\\limits_{a}^{b} f(t) \\Delta\nt=\\int\\limits_{a}^{b}f(t)dt$, where the integral on the right hand\nside is the usual Riemann integral; if $\\mathbb{T}=\\mathbb{Z}$ and $a0$ such\nthat $\\mathcal{L}(y(\\cdot))\\leq\\mathcal{L}(\\bar{y}(\\cdot))$ for any\nadmissible $\\bar{y}\\in\\textrm{C}_{prd}^r$ with\n$\\|y-\\bar{y}\\|_{r,\\infty}<\\delta$, where\n$$||y||_{r,\\infty} := \\sum_{i=0}^{r} \\left\\|y^{\\Delta^i}\\right\\|_{\\infty},$$\n$y^{\\Delta^0} = y$ and $||y||_{\\infty}:= \\sup_{t \\in\n[a,\\rho^{r}(b)]_{\\mathbb{T}}} |y(t)|$.\nFor simplicity of notation we introduce the operator $[y]$ defined by\n$[y](t) = \\left(t,y(t),y^\\Delta(t),\\ldots,y^{\\Delta^r}(t)\\right)$.\nThen, functional \\eqref{eq:prb} can be written as\n\\begin{equation*}\n\\mathcal{L}(y(\\cdot)) = \\int_{a}^{\\rho^{r-1}(b)} L[y](t) \\Delta t.\n\\end{equation*}\nWe assume that $(t,u_1,\\ldots,u_{r+1}) \\rightarrow\nL(t,u_1,\\ldots,u_{r+1})$ has continuous partial derivatives\n$\\frac{\\partial L}{\\partial u_{i}}$ for all\n$t\\in[a,\\rho^{r}(b)]_{\\mathbb{T}} $, $i=1,\\ldots,r+1$,\nand $t\\rightarrow L[y](t)$ and $t\\rightarrow\\frac{\\partial L}{\\partial u_{i}}[y](t)$,\n$i=1,\\ldots,r+1$, are piecewise rd-continuous for all admissible functions $y(\\cdot)$.\n\n\n\\subsection{The higher-order Euler-Lagrange equation}\n\\label{subsec:HO}\n\nWe now prove the Euler-Lagrange equation for problem\n\\eqref{eq:prb}--\\eqref{problema }.\n\n\\begin{rem}\n\\label{rem:Pneeds2rp1points}\nIn order for the problem\nto be nontrivial we require the time scale $\\mathbb{T}$\nto have at least $2r+1$ points. Indeed, if the time scale has only $2r$\npoints, then it can be written as\n$\\mathbb{T}=\\{a,\\sigma(a),\\ldots,\\sigma^{2r-1}(a)\\}$ and\n\\begin{multline}\n\\label{snormal}\n\\int_{a}^{\\rho^{r-1}(b)}\nL(t,y(t),y^\\Delta(t),\\ldots,y^{\\Delta^r}(t))\\Delta t \\\\\n=\\int_{a}^{\\sigma^{r}(a)}\nL(t,y(t),y^\\Delta(t),\\ldots,y^{\\Delta^r}(t))\\Delta t\n=\\sum_{i=0}^{r-1}\\int_{\\sigma^i(a)}^{\\sigma^{i+1}(a)}L(t,\ny(t),y^\\Delta(t),\\ldots,y^{\\Delta^r}(t))\\Delta t \\\\\n=\\sum_{i=0}^{r-1}(\\sigma^{i+1}(a)-\\sigma^i(a))L(\\sigma^i(a),y(\\sigma^i(a)),\ny^\\Delta(\\sigma^i(a)),\\ldots,y^{\\Delta^r}(\\sigma^i(a))).\n\\end{multline}\nHaving in mind the boundary conditions and the formula\n$f^\\Delta(t)=\\frac{f(\\sigma(t))-f(t)}{\\mu(t)},$ we can conclude that\nthe sum in \\eqref{snormal} is constant for every admissible function\n$y(\\cdot)$.\n\\end{rem}\n\n\\begin{thm}\nIf $y(\\cdot)$ is a weak local minimizer for the problem\n\\eqref{eq:prb}--\\eqref{problema }, then $y(\\cdot)$ satisfies the\nEuler-Lagrange equation\n\\begin{multline}\n\\label{eq:EL} \\frac{\\partial L}{\\partial y^{\\Delta^r} }[y](t)\n- \\int_a^{\\sigma(t)} \\frac{\\partial L}{\\partial y^{\\Delta^{r-1}}}[y](\\tau_r) \\Delta \\tau_r\\\\\n+ \\sum_{i=0}^{r-3} (-1)^i \\int_a^{\\sigma(t)} \\int_a^{\\sigma(\\tau_r)}\n\\cdots \\int_a^{\\sigma(\\tau_{r-i})} \\frac{\\partial L}{\\partial\ny^{\\Delta^{r-2-i}}}[y](\\tau_{r-1-i})\n\\Delta\\tau_{r-1-i} \\cdots \\Delta\\tau_{r-1}\\Delta\\tau_{r}\\\\\n(-1)^r \\int_a^{\\sigma(t)} \\left\\{ \\int_a^{\\sigma(\\tau_r)} \\left[\n\\cdots \\int_a^{\\sigma(\\tau_2)} \\frac{\\partial L}{\\partial\ny}[y](\\tau_1) \\Delta\\tau_1 + c_1 \\cdots \\right] \\Delta\\tau_{r-1}\n-(-1)^{r-1} c_{r-1}\\right\\} \\Delta\\tau_r - c_r = 0\n\\end{multline}\nfor some constants $c_1, \\ldots, c_r$ and all $t \\in [a,\\rho^{r}(b)]_{\\mathbb{T}}$.\n\\end{thm}\n\n\\begin{proof}\nWe first introduce some notation: $y_0(t)=y(t)$,\n$y_1(t)=y^\\Delta(t)$, \\ldots, $y_{r-1}(t)=y^{\\Delta^{r-1}}(t)$,\n$u(t)=y^{\\Delta^r}(t)$. Then problem \\eqref{eq:prb}--\\eqref{problema\n} takes the following form:\n\\begin{equation*}\n\\begin{gathered}\n\\mathcal{L}[y(\\cdot)]=\\int_{a}^{\\rho^{r-1}(b)}L(t,y_0(t),y_1(t),\n\\ldots,y_{r-1}(t),u(t))\\Delta t\\longrightarrow\\min, \\\\\n \\left\\{ \\begin{array}{l}\ny_i^{\\Delta}(t)=y^{i+1}(t), \\quad i=0,\\ldots,r-2,\\\\\ny_{r-1}^{\\Delta}(t)=u(t),\n\\end{array} \\right.\\\\\ny^j(a)=y_a^j,\\ y^j\\left(\\rho^{r-1}(b)\\right)=y_b^j,\\\nj=0,\\ldots,r-1 \\, .\n\\end{gathered}\n\\end{equation*}\nWith the notation $x=(y_0,y_1,\\ldots,y_{r-1})$, our problem\n\\eqref{eq:prb}--\\eqref{problema } can be written as the optimal\ncontrol problem\n\\begin{equation}\\label{op:1}\n\\begin{gathered}\n\\mathcal{L}[x(\\cdot)]=\\int_{a}^{\\rho^{r-1}(b)}L(t,x(t),u(t))\\Delta t\\longrightarrow\\min, \\\\\nx^\\Delta(t)=Ax(t)+Bu(t) \\, ,\\\\\n\\varphi (x(a),x(\\rho^{r-1}(b))= \\left[ \\begin{array}{l}\nx(a)-x_a\\\\\nx(\\rho^{r-1}(b))-x_b\\end{array} \\right]=0 \\, ,\n\\end{gathered}\n\\end{equation}\nwhere\n\\begin{equation*}\nA = \\left(\\begin{array}{ccccc}\n 0 & 1 & 0 & \\cdots & 0 \\\\\n 0 & 0 & 1 & \\cdots & 0 \\\\\n \\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & 0 & 0 & \\cdots & 1 \\\\\n 0 & 0 & 0 & \\cdots & 0 \\\\\n\\end{array}\\right) \\, ,\n\\quad B =\\left( \\begin{array}{c}\n0 \\\\\n\\vdots \\\\\n1\n\\end{array}\\right).\n\\end{equation*}\nNote that assumption (A1) of \\cite[Theorem~9.4]{zeidan2} holds:\nmatrix $I+\\mu(t)A$ is invertible, and the matrix\n$\\nabla \\varphi(x(a),x(\\rho^{r-1}(b))$ has full rank. Therefore, if\n$(x(\\cdot),u(\\cdot))$ is a weak local minimum for \\eqref{op:1}, then\nthere exists a constant $\\lambda$ and a function\n$p:[a,\\rho^{r-1}(b)]_{\\mathbb{T}}\\rightarrow \\mathbb{R}^r$, $p\\in C_{prd}^1$,\nsuch that $(\\lambda, p(\\cdot))\\neq 0$ and the following\nconditions hold:\n\\begin{equation*}\n-p^{\\Delta}(t)=A^Tp^{\\sigma}(t)+\\lambda\\left[\\frac{\\partial\nL}{\\partial x}(t,x(t),u(t))\\right]^T,\n\\end{equation*}\n\\begin{equation}\\label{op:3}\nB^Tp^{\\sigma}(t)+\\lambda\\frac{\\partial L}{\\partial u}(t,x(t),u(t))=0\n\\end{equation}\nfor all $t\\in[a,\\rho^{r}(b)]_{\\mathbb{T}}$. Consequently, if $y(\\cdot)$ is a\nweak local minimizer for \\eqref{eq:prb}--\\eqref{problema }, then\n\\begin{equation}\\label{op:4}\np_{r-1}^{\\sigma}(t)=-\\lambda\\frac{\\partial L}{\\partial u}[y](t)\n\\end{equation}\nholds for all $t\\in [a,\\rho^{r}(b)]_{\\mathbb{T}}$, where\n$p_{r-1}^{\\sigma}(t)$ is defined recursively by\n\\begin{align}\np_{0}^{\\sigma}(t)&\n=-\\int_a^{\\sigma(t)}\\lambda\\frac{\\partial L}{\\partial y_0}[y](\\tau_1)\\Delta\\tau_1-c_1\n\\, ,\\label{op:5}\\\\\np_{i}^{\\sigma}(t)&=-\\int_a^{\\sigma(t)}\\left[\\lambda\\frac{\\partial\nL}{\\partial y_i}[y](\\tau_{i+1})\n+p_{i-1}^{\\sigma}(\\tau_{i+1})\\right]\\Delta\\tau_{i+1}-c_{i-1}\\label{op:6},\\\ni=1,\\ldots,r-1 \\, ,\n\\end{align}\nwith $c_i$, $i = 0,\\ldots, r- 1$, constants. From\n\\eqref{op:4}--\\eqref{op:6} we obtain that equation\n\\begin{multline}\n\\label{eq:EL:0} \\lambda\\frac{\\partial L}{\\partial u}[y](t)\n- \\int_a^{\\sigma(t)} \\lambda\\frac{\\partial L}{\\partial y_{r-1}}[y](\\tau_r) \\Delta \\tau_r\\\\\n+ \\sum_{i=0}^{r-3} (-1)^i \\int_a^{\\sigma(t)} \\int_a^{\\sigma(\\tau_r)}\n\\cdots \\int_a^{\\sigma(\\tau_{r-i})}\\lambda \\frac{\\partial L}{\\partial\ny_{r-2-i}}[y](\\tau_{r-1-i})\n\\Delta\\tau_{r-1-i} \\cdots \\Delta\\tau_{r-1}\\Delta\\tau_{r}\\\\\n(-1)^r \\int_a^{\\sigma(t)} \\left\\{ \\int_a^{\\sigma(\\tau_r)} \\left[\n\\cdots \\int_a^{\\sigma(\\tau_2)} \\lambda\\frac{\\partial L}{\\partial\ny_0}[y](\\tau_1) \\Delta\\tau_1 + c_1 \\cdots \\right] \\Delta\\tau_{r-1}\n-(-1)^{r-1} c_{r-1}\\right\\} \\Delta\\tau_r - c_r = 0\n\\end{multline}\nholds for all $t \\in [a,\\rho^{r}(b)]_{\\mathbb{T}}$.\nWe show next that $\\lambda\\neq 0$. First observe that if $f\\in\nC_{prd}^1$ and $f^{\\sigma}(t)=0$ for all $t\\in [a,b]_{\\mathbb{T}}^{\\kappa}$,\nthen $f(t)=0$ for all $t\\in [\\sigma(a),b]_{\\mathbb{T}}$. Suppose,\ncontrary to our claim, that $\\lambda=0$ in equation \\eqref{op:3} and\n\\eqref{op:4}. Then, we can write the system of equations\n\\begin{equation}\n\\label{eq:syst:ab} \\left\\{ \\begin{array}{ll}\np_0^\\Delta (t)&=0 \\, , \\\\\np_i^\\Delta (t)&=-p_{i-1}^\\sigma (t), \\quad i=1,\\ldots, r-1 \\, , \\\\\np_{r-1}^\\sigma (t)&=0,\n\\end{array} \\right.\n\\end{equation}\nfor all $t\\in [a,\\rho^{r}(b)]_{\\mathbb{T}}$. From the last equation we have\n$p_{r-1}(t)=0$, $\\forall t\\in[\\sigma (a),\\rho^{r-1}(b)]_{\\mathbb{T}}$. This\nimplies that $p_{r-1}^\\Delta(t)=0$, $\\forall t\\in[\\sigma\n(a),\\rho^{r}(b)]_{\\mathbb{T}}$, and consequently $p_{r-2}^\\sigma(t)=0$,\n$\\forall t\\in[\\sigma (a),\\rho^{r}(b)]_{\\mathbb{T}}$. Therefore,\n$p_{r-2}(t)=0$, $\\forall t\\in[\\sigma^2 (a),\\rho^{r-1}(b)]_{\\mathbb{T}}$.\nRepeating this procedure we have $p_{1}(t)=0$ for all\n$t\\in[\\sigma^{r-1}(a),\\rho^{r-1}(b)]_{\\mathbb{T}}$. Hence,\n$0=p_{1}^\\Delta(t)=-p_0^\\sigma (t)=-p_0^\\Delta\n(t)\\mu(t)-p_0(t)=-p_0(t)$ for all\n$t\\in[\\sigma^{r-1}(a),\\rho^{r}(b)]_{\\mathbb{T}}$. Note that the first\nequation of \\eqref{eq:syst:ab} implies $p_0(t)=c$ for some constant\n$c$ and all $t\\in[a,\\rho^{r-1}(b)]_{\\mathbb{T}}$. Since the time scale has\nat least $2r+1$ points (see Remark~\\ref{rem:Pneeds2rp1points}), the\nset $t\\in[\\sigma^{r-1}(a),\\rho^{r-1}(b)]_{\\mathbb{T}}$ is nonempty and we\nconclude that $p_0(t)=0$ for all $t\\in[a,\\rho^{r-1}(b)]_{\\mathbb{T}}$.\nSubstituting this into the second equation we get $p_1^\\Delta (t)=d$ for\nsome constant $d$ and all $t\\in[a,\\rho^{r-1}(b)]_{\\mathbb{T}}$. Having in\nmind that $p_1(t_0)=0$ for some $t_0 \\in[a,\\rho^{r-1}(b)]_{\\mathbb{T}}$ we\nobtain $p_1(t)=0$ for all $t \\in[a,\\rho^{r-1}(b)]_{\\mathbb{T}}$. Repeating\nthis procedure we conclude that $p_i(t)=0$, $i=1,\\ldots,r-1$, for\nall $t\\in[a,\\rho^{r-1}(b)]_{\\mathbb{T}}$. This contradicts the fact that\n$(\\lambda,p(\\cdot))\\neq 0$. Hence, equation \\eqref{eq:EL:0} can be\ndivided by $\\lambda$ and \\eqref{eq:EL} is proved.\n\\end{proof}\n\n\n\\subsection{Corollaries}\n\\label{subsec:appl}\n\nFor illustrating purposes we consider now the two simplest\nsituations, \\textrm{i.e.}, $r=1$ and $r = 2$.\n\n\\begin{cor}[\\textrm{cf.} \\cite{CD:Bohner:2004,zeidan}]\nIf $y(\\cdot)$ is a weak local minimizer for the problem\n\\begin{equation*}\n\\mathcal{L}(y(\\cdot)) = \\int_{a}^{b}\n L(t,y(t),y^\\Delta(t))\\Delta\nt\\longrightarrow\\min\n\\end{equation*}\nsubject to boundary conditions $y(a)=y_a$ and $y(b)=y_b$,\nthen $y(\\cdot)$ satisfies the Euler-Lagrange equation\n\\begin{equation*}\n\\frac{\\partial L}{\\partial y^\\Delta}\\left(t,y(t),y^\\Delta(t)\\right)\n= \\int_a^{\\sigma(t)} \\frac{\\partial L}{\\partial y}\\left(\\tau,y(\\tau),\ny^\\Delta(\\tau)\\right) \\Delta \\tau + c_1\n\\end{equation*}\nfor some constant $c_1$ and all $t \\in [a,b]_{\\mathbb{T}}^\\kappa$.\n\\end{cor}\n\n\\begin{cor}[\\textrm{cf.} \\cite{FT,MT}]\nIf $y(\\cdot)$ is a weak local minimizer for the problem\n\\begin{equation*}\n\\mathcal{L}(y(\\cdot)) = \\int_{a}^{\\rho(b)}\n L(t,y(t),y^\\Delta(t),y^{\\Delta \\Delta})\\Delta\nt\\longrightarrow\\min\n\\end{equation*}\nsubject to boundary conditions\n$y(a)=y_a^0$, $y(\\rho(b))=y_b$,\n$y^\\Delta(a)=y_a^1$, and $y^\\Delta(\\rho(b))=y_b^1$,\nthen $y(\\cdot)$ satisfies the Euler-Lagrange equation\n\\begin{multline*}\n\\frac{\\partial L}{\\partial\ny^{\\Delta\\Delta}}\\left(t,y(t),y^\\Delta(t),y^{\\Delta\\Delta}(t)\\right)\n- \\int_a^{\\sigma(t)} \\frac{\\partial L}{\\partial\ny^\\Delta}\\left(\\tau_2,y(\\tau_2),\ny^\\Delta(\\tau_2),y^{\\Delta\\Delta}(\\tau_2)\\right)\n\\Delta \\tau_2 \\\\\n+ \\int_a^{\\sigma(t)} \\left[ \\int_a^{\\sigma(\\tau_2)} \\frac{\\partial\nL}{\\partial y}\\left(\\tau_1,y(\\tau_1),\ny^\\Delta(\\tau_1),y^{\\Delta\\Delta}(\\tau_1)\\right)\\Delta\\tau_1 + c_1\n\\right] \\Delta\\tau_2 - c_2 = 0\n\\end{multline*}\nfor some constants $c_1$ and $c_2$ and all $t \\in [a,\\rho(b)]_{\\mathbb{T}}^\\kappa$.\n\\end{cor}\n\n\n\\subsection{An example}\n\\label{subsec:ex}\n\nLet $\\mathbb{T}=[a,b]\\cap h\\mathbb{Z}$, where $h\\mathbb{Z}:=\\{h z | z \\in\n\\mathbb{Z}\\}$, $h>0$. Then for any $f\\in C_{prd}^{r}$ we\nhave\n\\begin{align}\n\\label{eq:exGC} {\\underbrace{\\left[\\int_a^{\\sigma(t)}\n\\left(\\int_a^\\sigma \\cdots \\int_a^\\sigma f \\right) \\Delta\n\\tau\\right]}_{j-i\\text{ integrals}}}^{\\Delta^j} = f^{\\Delta^i\n\\sigma^{j-i}} \\, , \\quad i \\in \\{0,\\ldots,j-1\\} \\, ,\n\\end{align}\nwhere $f^{\\Delta^i \\sigma^{j-i}}(t)$ stands for\n$f^{\\Delta^i}(\\sigma^{j-i}(t))$. We will show this by induction. For\n$j = 1$\n\\begin{equation*}\n\\int_a^{\\sigma(t)}f(\\xi)\\Delta\\xi=\\int_a^{t}f(\\xi)\\Delta\\xi\n+\\int_t^{t+h}f(\\xi)\\Delta\\xi=\\int_a^{t}f(\\xi)\\Delta\\xi+hf(t),\n\\end{equation*}\nand then $\\left[\\int_a^{\\sigma(t)}f(\\xi)\\Delta\\xi\\right]^\\Delta\n=f(t)+hf^\\Delta(t) = f^\\sigma$. Now assume that \\eqref{eq:exGC} is\ntrue for all $j = 1,\\ldots,k$. Then for $j=k+1$\n\\begin{multline*}\n{\\underbrace{\\left[\\int_a^{\\sigma(t)} \\left( \\int_a^\\sigma \\cdots\n\\int_a^\\sigma f \\right)\n\\Delta \\tau\\right]}_{k+1-i\\text{ integrals}}}^{\\Delta^{k+1}}\n= \\left( \\underbrace{\\int_a^{t} \\int_a^\\sigma \\cdots\n\\int_a^\\sigma}_{k+1-i} f \\Delta \\tau +\nh\\underbrace{\\int_a^{\\sigma(t)} \\cdots \\int_a^\\sigma}_{k-i} f \\Delta\n\\tau\n\\right)^{\\Delta^{k+1}} \\\\\n= \\left(\\underbrace{\\int_a^{\\sigma(t)} \\cdots \\int_a^\\sigma}_{k-i}\nf \\Delta \\tau \\right)^{\\Delta^{k}} +\n\\left[h\\left(\\underbrace{\\int_a^{\\sigma(t)} \\cdots\n\\int_a^\\sigma}_{k-i} f \\Delta \\tau\n\\right)^{\\Delta^{k}}\\right]^{\\Delta}\n= f^{\\Delta^i \\sigma^{k-i}} + \\left(hf^{\\Delta^i \\sigma^{k-i}}\\right)^\\Delta\n= f^{\\Delta^i \\sigma^{k+1-i}} \\, .\n\\end{multline*}\nDelta differentiating $r$ times both sides of equation \\eqref{eq:EL}\nand in view of \\eqref{eq:exGC}, we obtain the $h$-Euler-Lagrange\nequation in delta differentiated form:\n\\begin{equation*}\nL_{y^{\\Delta^r}}^{\\Delta^r}(t,y,y^\\Delta,\\ldots,y^{\\Delta^r}) +\n\\sum_{i=0}^{r-1} (-1)^{r-i} L_{y^{\\Delta^{i}}}^{\\Delta^i\n\\sigma^{r-i}}(t,y,y^\\Delta,\\ldots,y^{\\Delta^r}) =0.\n\\end{equation*}\n\n\n\\section*{Acknowledgments}\n\nThis work was partially supported by the \\emph{Portuguese Foundation\nfor Science and Technology} (FCT) through the \\emph{Center for Research\nand Development in Mathematics and Applications} (CIDMA) of University of Aveiro.\nThe first author was also supported by FCT through the PhD fellowship\nSFRH\/BD\/39816\/2007; the second author is currently a researcher\nat the University of Aveiro with the support\nof Bia{\\l}ystok University of Technology, via a project of\nthe Polish Ministry of Science and Higher Education ``Wsparcie\nmiedzynarodowej mobilnosci naukowcow''; the third author\nwas partially supported by the Portugal--Austin (USA)\nproject UTAustin\/MAT\/0057\/2008.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\nA way to identify dense groups of points in $\\mathbb{R}^k$ is to construct connected components of points where direct connections are given for all pairs of points whose Euclidean separation is less than a `linking length' $b$. \nThis task is particularly common when processing cosmological simulations of the $\\Lambda$ cold-dark matter model to find the statistics of halos, \nwhich are virialised objects with a mean density of approximately $200\\times$ the critical density of the universe \\citep{Gunn_1972,Bertschinger_1985,Eke_1996}.\nSimulations of these objects discretise the (primarily dark) matter distribution into $N$ bodies \\citep{Davis_1985}, and at any given time-scale of interest\na catalogue of the connected components (or `friends-of-friends' groups) in $\\mathbb{R}^3$ is constructed (e.g. \\citealp{Jenkins_2001, Reed_2003, Reed_2007, Crocce_2010, Courtin_2011, Angulo_2012},\nalthough other alternatives exist, see \\citealp{Knebe_2011} for an overview).\nThe data sets of these simulations have grown from 32,768 particles \\citep{Davis_1985} to the trillions of particles this decade \\citep{Skillman_2014}, making the production of these group catalogues challenging.\n\nThe ubiquitous algorithm for finding these friends-of-friends (hereafter FOF) groups is to perform a breadth-first search \\citep[e.g.][]{Huchra_1982}.\nIn this algorithm, finding connected components proceeds in the following manner: a stack of boundary points is maintained (initialised with a single point), and at each step a point is removed (marked as linked) and replaced by all its (unlinked) neighbours within the linking length, and this proceeds until the stack is empty, and the component is complete.\nThis fixed-radius neighbour search is performed via organisation of the points into a {$k$-d tree}, a binary space partitioning structure where neighbour searches can be performed in $O(\\log n)$ operations, $n$ being the total number of points.\nExamples of such codes include \\citet{Behroozi_2013a}, the FOF code from the NbodyShop\\footnote{\\url{http:\/\/faculty.washington.edu\/trq\/hpcc\/}, see also \\url{https:\/\/github.com\/N-BodyShop\/fof} for the code}, which is the almost unmodified ancestor of more recent codes such as \\href{https:\/\/github.com\/junkoda\/cola_halo}{Cola} \\citep{Koda_2016,Carter_2018}, {\\sc yt}\\ \\citep{Turk_2011} and probably many others unknown to this author. \nAs far as I am aware {$k$-d trees\\ } are used to perform the neighbour finding step in the non-public codes also, such as \\citet{Kwon_2010, Fu_2010} and {\\sc Arepo}\\ \\citep[][and also the non-public version of its predecessor {\\sc Gadget}-2]{Arepo}.\nSome of these codes have been designed to create the group catalogue in parallel (often on the same cluster as the simulation), to mitigate the analysis problems.\n\nRecently \\citet{Feng_2017} have released an open source ($k$-dimensional) FOF algorithm {\\href{http:\/\/rainwoodman.github.io\/kdcount}{kdcount}}\\footnote{see \\url{http:\/\/rainwoodman.github.io\/kdcount}} that is used in {\\sc nbodykit}\\ \\citep{Nbodykit}. This algorithm uses the dual tree method \\citep[e.g.][]{Moore_2001} which exploits the fact that the searching points are hierarchically organised, allowing neighbour calculations (either inclusions or exclusions) to be calculated (typically excluded) for entire branches of the search tree.\nTheir algorithm is not strictly breadth-first, a consequence of which is the need to merge components using a (customised) disjoint-set algorithm \\citep{Tarjan_1975}. \n\nAn alternative method for neighbour searches is the mapping of points on a fixed-grid, for example in the `chaining mesh' method of \\citet[sec 8.4.1]{Hockney_1988} for a short-range component of the Coulomb force,\nand in the correlation function code Corrfunc (ascl:1703.003).\nBy choosing a cell width greater than the search radius, one guarantees that all neighbours are within the 26 adjacent cells (in 3-d). \nSince the extent of the short range force is generally a multiple of the interparticle separation, this mesh is coarse w.r.t. the particles, corresponding to a modest memory footprint. Unfortunately, in the application FOF one is generally interested in linking lengths of $0.2\\times$ the interparticle spacing \\citep[e.g.][]{Davis_1985}, implying meshes of (at least) $125\\times$ the particle count, and correspondingly a prohibitively large memory footprint. \n\nA method to avoid such large data structures is to store only the filled cells, mapping them into a 1-d hash-table \\citep{Yuval_1975, Bentley_1979} such that neighbouring cells can be (speculatively) searched at the map (hash) of their location.\nSuch a method has been employed for fixed-radius neighbour searches \\citep[e.g.][sometimes referred to as locality sensitive hashing]{Teschner_2003, Hastings_2005}. \nThis is $O(1)$ for look-ups, though limitations include the expense of the hash function, the cost of resolving collisions (cells mapped to the same index) and the decreased coherence of memory accesses.\n\nSpatial hashing has been successfully implemented by \\citet{Wu_2007} and \\citet{Vijayalaksmi_2012} for the related clustering algorithm DBSCAN, which is a generalisation of FOF to connecting components only about a subset of `core' points \\citep{Ester_96}. \n\nIn practice these codes are not applied to FOF calculations, possibly because they have not been optimised for this specialised use-case.\nSpatial hashing appears to be less common in computational physics, with exceptions such as the parallelisation scheme of \\citet{Warren_1993} and in the level set tracking methods of \\citet{Brun_2012}.\n\nThis paper describes a novel algorithm for performing FOF in 3-d by grouping points into fine mesh whose cells are sufficiently compact to guarantee their points will be connected. These filled cells are grouped into $4^3$ blocks which are stored via spatial hashing, the use of blocks decreasing the average number of hash-look-ups per filled cell. \nThe merging of cells happens `on the fly' as the blocks are inserted in a raster order, i.e. neighbours queries are only performed over blocks previously inserted in the table, and then the components are connected via the disjoint-sets algorithm, in a manner similar to \\citet{Feng_2017}. \nAn example implementation is provided at \\url{https:\/\/github.com\/pec27\/hfof}.\n\nThis paper is organised as follows. Section~\\ref{sec:method} describes the spatial hashing and linking algorithm, optimisations, and the method applied for periodic domains. Section~\\ref{sec:comp} describes the comparison codes and test sets. Section~\\ref{sec:results} analyses the performance and compares with other codes and Section~\\ref{sec:conc} concludes.\n\n\\section{Spatial hashing for fixed-distance neighbour linking}\\label{sec:method}\n\nIn this section a methodology for FOF group finding via spatial hashing is described. \nWhilst this algorithm is not limited to cosmological simulations, these are the motivation, and some consideration of their features for this purpose as described in Sec.~\\ref{sec:cosmo}.\nSec.~\\ref{sec:cells} describes the arrangement of points into cells compact enough to guarantee connectivity, and their aggregation into blocks to reduce the number of lookups. Sec.~\\ref{sec:hash_func} describes the hash function and \\ref{sec:periodicity} describes the adjustments to account for periodic domains.\n\n\n\\subsection{Matter distribution in cosmological simulations}\\label{sec:cosmo}\n\nIn the cosmological context, the clustering of matter produces halos which at the low-mass regime have a mass function approximating a power law\n\\begin{equation}\\label{eq:mass_func}\n\\frac{{\\rm d}N}{{\\rm d} M} \\propto M^{-\\alpha}\n\\end{equation}\nwith $\\alpha \\approx 1.9$ (e.g. \\citealp{Reed_2007}), and notably $\\alpha>1$ implies a divergent low-mass tail,\n i.e. there should be an infinite number density of low-mass clusters, our discrimination of them limited only by our finite mass resolution (this is not strictly speaking true of the real universe, where diffusion damping terms will limit very low mass halos, but these are rarely resolved in cosmological simulations). A corollary of this is that the groups found are likely to be dominated (by number) by single particle groups\\footnote{in the analysis of cosmological simulations groups with small (e.g. $<20$) particles are generally ignored, but at the stage of constructing FOF groups these have yet to be filtered}, and also that the number of groups is a significant fraction of the total number of particles (typically around one-third for cosmological simulations). \nAs such a FOF algorithm needs to be efficient in the cases where the neighbourhood within a linking length is empty.\n\nAt the other extreme is that of high mass groups. Given the previous paragraph it may be tempting to think that most points are in small groups, however this is not the case. \nThis can also be seen from Eqn.~(\\ref{eq:mass_func}), since $\\int M {\\rm d}M \/ \\int {\\rm d}M$ (i.e. the mass-weighted average halo mass) would have a divergent high-mass contribution, i.e. the average particle is in a group of $\\gg 1$ particles, the exact number depending upon the mass function to higher masses (which in discrete simulations often depends upon artificial limitations such as the box size). \nAs such the linking component of a FOF algorithm needs to scale well, in order to handle the connection of points to large groups.\n\nWhilst both of these extremes need to be handled by group finding algorithms, I find in general the former seems more demanding, in that a significant fraction of the particles have zero neighbours within the linking length, and the majority of the computational time is spent confirming that these particles are truly isolated (see for example the 2nd panel of Fig.~\\ref{fig:adj_idx}). It is helpful to keep this in mind during the following section.\n\n\\subsection{Cell and block organisation}\\label{sec:cells}\nAt the finest level, each particle is assigned to a cell according to its position in a lattice with cell-width\n\\begin{equation}\nc = \\frac{b}{\\sqrt{3}}\n\\end{equation}\nwhere $b$ is the linking length. Since the maximum distance between vertices in a unit hypercube in $\\mathbb{R}^k$ is $\\sqrt{k}$, this guarantees that any points in the same cell must belong to the same FOF group, which essentially reduces the problem of linking points to one of linking cells, and hereafter I will almost exclusively talk in terms of cells. The filled cells are sorted in raster order (i.e. sorted by $z$ then $y$ then $x$), which immediately places a bound on the the complexity of the algorithm to be at least $O(n\\log n)$, similar to that of the {$k$-d tree}\\ construction.\n\n\\begin{figure*}\n\\includegraphics[width=2\\columnwidth]{figures\/Fig1.png}\n\\caption{2 and 3-dimensional representation of the method, from left to right. \\emph{Far-left}, a filled cell (\\emph{yellow square}) and the neighbouring 2-dimensional `stencil' (\\emph{grey squares}) of 20 neighbouring cells which could contain points within $\\sqrt{2}$ cell widths (the linking length $b$ denoted with the \\emph{red arrow}), where the exact locus for the \\emph{yellow square} is given by the \\emph{red shaded region} and some example points are given by \\emph{filled red dots}. \\emph{Centre-left}, points from a (tiny slice of a) real simulation (\\emph{black dots}) which are grouped into filled cells (\\emph{dark yellow squares}) which are themselves grouped into $4^2$ blocks (\\emph{light yellow squares}). These blocks are then mapped (\\emph{light yellow arrows}) to the hash-table in the \\emph{centre-right panel} at entries (\\emph{light yellow segments}) whose positions that are hashes modulo $2^3$ (values in \\emph{black}) of their spatial indices (in \\emph{blue}), with collisions demoted to the next available position (\\emph{magenta}). \\emph{Far-right} the 3-d stencil of the (116) neighbouring cells that can be with $\\sqrt{3}$ cell widths of the innermost cell.}\n\\label{fig:adj_idx}\n\\end{figure*}\n\nThis relationship of cell size to linking length is illustrated in Fig.~\\ref{fig:adj_idx} (first panel), where the locus of potential neighbours for positions in the central cell is highlighted. \nThis lattice size guarantees that any neighbouring particle within a distance $ \\frac{L}\\Delta + 1$ and $P_{\\rm x}> \\frac{L P_{\\rm y}}\\Delta +1$ (i.e. ordering by $\\Phi$ is still ordering by $i$ then $j$ then $k$, the additional $+1$s being required to `buffer' the rightmost values from the next row), and the \nprimes are found from the Miller-Rabin test \\citep{Rabin_1980} as the smallest values satisfying these inequalities. \nA 64-bit integer is used to store $\\Phi$, since this safely covers $\\frac{L}\\Delta$ up to approximately 2,600,000, or $N<10^{18}$ particles (for $b=0.2$), which is still well out of reach of all current cosmological simulations. \nThe choice of hash function is given by\n\\begin{equation}\\label{eq:hash}\nH(i,j,k) = \\Phi(i,j,k) \\times Q \\mod 2^n\n\\end{equation}\nwhere $2^n$ is the size of the hash table and $Q$ is a prime number of similar magnitude to $2^n$. \n\nThis choice of Eqns.~(\\ref{eq:block_index}) \\& (\\ref{eq:hash}) has the convenient property that the relative hashes of adjacent blocks are \nstraightforward to compute as\n\\begin{eqnarray}\nH(i+1,j,k) - H(i,j,k) =& QP_{\\rm x} & \\mod 2^n , \\nonumber \\\\\nH(i,j+1,k) - H(i,j,k) =& QP_{\\rm y} &\\mod 2^n , \\nonumber \\\\\nH(i,j,k+1) - H(i,j,k) =& Q & \\mod 2^n , \\nonumber \\\\\n\\dots && \\nonumber\n\\end{eqnarray}\nwith all adjacent blocks following from substitution of indices into Eqns.~(\\ref{eq:block_index})~\\&~(\\ref{eq:hash}).\n\nWhilst the use of a multiplicative hash in Eqn.~(\\ref{eq:hash}) is easily understandable in terms of simplicity, the use of prime multipliers in Eqn.~(\\ref{eq:block_index}) for conversion from a 3-d index to 1-d may not be immediately clear. \nA simplification of Eqn.~(\\ref{eq:block_index}) would be to omit the requirement that the $P_{\\rm x}$ and $P_{\\rm y}$ multipliers be prime, however this causes problems in the cases when one these values is a multiple of a power of 2. In particular if it is a multiple of $2^m$ (for some nonzero $m$) then it is no-longer co-prime with $2^n$ and the hash function outputs identical values for any increment that is a multiple of $2^{n-m}$ along the corresponding axis (or if one is thinking in terms of bit-wise representations the increments do not affect the lowest $m$ bits of the hash). This use of prime multipliers is similar to \\citet{Teschner_2003} in which indices were combined as $P_1 i \\xor P_2 j \\xor P_3 k$, however the use of the $\\xor$-function in that work complicates the evaluation of hashes for adjacent indices (i.e. the relative hashes are no-longer independent of $i$,$j$,$k$) because $\\xor$ is not distributive over addition. \n\nNotably building the hash table incrementally (as one adds blocks ordered by $\\Phi$) turns out to have the additional benefit that the speculative neighbour searches \nare performed when the hash table is only partially full. Taking the continuous approximation for random hashing that the expected rate of collisions (i.e. false elements found) when inserting\/searching a single element into a table of fill $\\lambda$ is $F=\\frac{\\lambda}{1-\\lambda}$ (i.e. the geometric sum of $\\lambda^n$), the \\emph{average} collision rate (as the table is incrementally built) is given by the convex function\n\\begin{equation}\n\\bar{F}(\\lambda) = \\frac{1}{\\lambda}\\int_0^\\lambda \\frac{s \\, {\\rm d}s}{1-s}= -1 - \\frac{\\log\\, 1-\\lambda}{\\lambda}\n\\end{equation}\nwhere $\\log$ refers to the natural logarithm. Assuming $\\lambda=60\\%$ this gives $F=1.5$ and $\\bar{F} \\approx 0.53$, an improvement of almost a factor 3. \nThis is super-linear in the load (the mean load at look-up being reduced by a factor 2), due to the convexity of $\\bar{F}$.\n\nFor some comparison of the actual performance of collision rates I have included in Table~\\ref{tab:simtable} (Section~\\ref{sec:results}) the theoretical and actual collision rate when filling the hash-tables for the four different data sets, and as expected for three of the data sets there are collisions in excess of random, though they are sub-dominant. Unusually, the baryonic simulation has an actual collision rate below that expected for random ($23.7\\%$ vs $49.7\\%$) - this is likely a symptom of this data set having many contiguous blocks, since the multiplicative hash in Eqns.~(\\ref{eq:block_index})~\\&~(\\ref{eq:hash}) is actually \\emph{guaranteed} to give a distinct hash when only a single index is incremented by 1.\n\nOne might reasonably wonder if going through the filled blocks in raster order is in fact the optimal approach. Other approaches such as ordering by the index on Peano-Hilbert curve (such as performed in \\citealp{Springel_2005}), better preserves spatial locality, which in this context correspond to \nneighbour searches that are more coherent in memory. \nSuch schemes introduce the additional complication that the relationship between the values of the 1-d index no longer depends only on the relative positions of the block\\footnote{For example a block which is `above' the current (e.g. $i-1$) may have a smaller Hilbert key at one point on the curve, but this will not be true at all points.} and thus a na\\\"ive implementation requires double the number of neighbour block searches. Examining the various choices of space-filling curve (e.g. Morton\/Hilbert) and implementation choices for neighbour searches may produce an interesting direction for future research.\n\nAnother avenue for extension is the use of this method on dimensions other than 3. In principle this is straightforward since all of the above methods can be transferred to lower (i.e. 2) or higher dimensions (e.g. the 6-D phase-space FOF of \\citealp{Diemand_2006}), although one might need some care since various choices about block-size, hash-function etc. have been calibrated for the 3-D case.\n\n\\begin{algorithm}\n\\caption{Linking of friends of friends groups}\n\\label{alg:link}\n\\begin{algorithmic}[1]\n\\State \\emph{\\% Precompute indices of blocks reachable from positions 1-64}\n\\State {\\sc Adj}$(p)\\gets $ blocks reachable from position $p$\n\\State \\emph{\\% Precompute cells reachable within these block pairs $a \\in${\\sc Adj}$(p)$}\n\\State {\\sc CellReachStencil}$(p_{\\rm adj}, p, a) \\gets 1$ or 0 (if reachable)\n\\Procedure{FOF}{$Q, P_{\\rm x}, P_{\\rm y}, \\lambda=0.6$ (desired load)}\n\\For{all particles ${\\bf x}_i$} \n \\State Assign a block $\\Phi_i$ \\Comment See Eqn.~(\\ref{eq:block_index})\n \\State Assign position $p_i$ (1-64) of cell within block. \n\\EndFor\n\\State Sort (by blocks and then cell)\n\\State $B \\gets$ hash-table of size $2^n$, with $n$ s.t. $\\lambda 2^n>$count(blocks)\n\\State Create array $C$ to hold cell data\n\\For{Each block $\\Phi_i$ and cell at position $p$ in block $\\Phi_i$}\n \\State Append cell $c(\\Phi_i, p)$ to $C$\n \\State Assign parent$(c) \\gets c$ (i.e. in own new set)\n \\State{\\emph{\\% Loop over cells already in my block}}\n \\For{cell $c_{\\rm adj}$ at position $p_{\\rm adj}$ already in block $\\Phi_i$}\n \\State \\Call{CompareAndLink}{$c$, $c_{\\rm adj}$, $p$, $p_{\\rm adj},0$}\n \\EndFor\n\n \\State \\emph{\\% Loop over cells in adjacent blocks} \n \\For{$a$ in {\\sc Adj}$(p)$ where $\\Phi_i+a \\in B$}\n \\State $\\Phi_{\\rm adj} \\gets \\Phi_i+a$\n \\For{cell $c_{\\rm adj}$ at position $p_{\\rm adj}$ in block $\\Phi_{\\rm adj}$}\n \\State \\Call{CompareAndLink}{$c$, $c_{\\rm adj}$, $p$, $p_{\\rm adj},a$}\n \\EndFor\n \\EndFor\n \\State Append cell $c$ to block $\\Phi$\n \\State At the last cell, insert $\\Phi_i$ into $B$.\n\\EndFor\n\\State $\\forall c \\in C$, assign root($c$) as FOF label for $c$\n\\State Return .\n\\EndProcedure\n\\Procedure{CompareAndLink}{$c_{\\rm me}$, $c_{\\rm adj}$, $p$, $p_{\\rm adj},a$}\n \\State \\emph{\\% If the cells within $b$ and roots are distinct then}\n \\State \\emph{\\% compare points pairwise}\n \\If {{\\sc CellReachStencil}($p_{\\rm adj},p,a$)}\n\n \\If {${\\rm root}(c_{\\rm adj}) \\neq {\\rm root}(c_{\\rm me})$}\n \\If {$\\exists \\;{\\bf x} \\in c_{\\rm me}, {\\bf y} \\in c_{\\rm adj} : \\left| {\\bf x} - {\\bf y} \\right|0\\}}\n\\frac{1}{-\\phi'(E)} g^2\\, dv\\, dx \n-\\frac{1}{8\\pi}\\int |\\nabla_{x}U_{g}|^2 dx. \n\\end{equation}\nIt is natural to expect that positive definiteness of this quadratic \nform should \nimply stability for $f_{0}$. Ever since the seminal work of {\\sc Antonov} \\cite{An}\nthere have been vigorous efforts in the astrophysics community to establish\nthis positive definiteness and to derive stability results in this way.\n\nAn important step in this direction was to show that the above\nquadratic form is positive definite on linearized, dynamically accessible\nperturbations. To make this precise\nwe define the Lie-Poisson bracket of two functions $f_1, f_2$ \nof $x$ and $v$ as \n\\begin{equation}\n\\{f_{1},f_{2}\\} := \\nabla _{x}f_{1}\\cdot \\nabla _{v}f_{2}-\\nabla\n_{v}f_{1}\\cdot \\nabla _{x}f_{2}. \\label{lie}\n\\end{equation}\nThen the following holds:\n\\begin{lemma} \\label{ks}\nLet $\\phi^{\\prime }<0$ and let $h\\in\nC_{c}^{\\infty }(\\R^6)$ be spherically symmetric with support in the set \n$\\{f_{0}>0\\}$ and such that $h(x,-v)=-h(x,v)$. Then \n\\[\nD^2{{\\cal H}_C} (f_0)[\\{f_{0},h\\}]\n\\geq \n-\\frac{1}{2} \\int_{f_{0}>0} \\phi^{\\prime }(E)\n\\left[ |x\\cdot v|^2\n\\left\\vert \\left\\{ E,\\frac{h}{x\\cdot v}\\right\\} \\right\\vert^2\n+ \\frac{1}{r}U_{0}^{\\prime}\\, h^2\n\\right] \\,dv\\,dx.\n\\]\n\\end{lemma}\nHere $U_0'$ denotes the radial derivative of the steady state potential.\nSince $U_0$ is radially increasing, the right hand side in the estimate above\nis indeed positive for $h\\neq 0$. We refer to \\cite{KS,SDLP} for astrophysical\ninvestigations where this result is used to analyze\nlinearized stability. We do not go into the reasons why perturbations of\nthe form $\\{f_0,h\\}$ are called dynamically accessible for the linearized system,\nbut for the sake of completeness we provide a proof of this elegant result in\nthe Appendix. Despite its significance the result is still quite a distance \naway from a true, nonlinear stability result. \nThere are at least two serious mathematical\ndifficulties. Firstly, it is very challenging to use the positivity of \n$D^2 {{\\cal H}_C} (f_{0})[g]$ to control the higher order remainder in the expansion\nof the energy-Casimir functional \\cite{Wa}. This is due to the\nnon-smooth nature of $f_{0}=\\phi(E)$ in all important examples. Secondly,\neven if one succeeds in controlling the higher order terms, the positivity \nof $D^2 {{\\cal H}_C} (f_{0})[g]$ in the lemma is only valid for certain\nperturbation of the form $g=\\{f_{0},h\\}$. This class of perturbations is\ninvariant under solutions of the linearized Vlasov-Poisson system,\nbut it is not invariant under solutions of the nonlinear system.\n\nTo overcome these difficulties a variational approach was\ninitiated by {\\sc Wolansky} \\cite{Wo1} and then developed \nsystematically by {\\sc Guo}\nand {\\sc Rein} \\cite{G1,G2,GR1,GR2,GR3,GR4,R1,R2,RG}. Their \nmethod entirely avoids the delicate\nanalysis of the second order term $D^2 {{\\cal H}_C} (f_{0})$ in \n(\\ref{h2}), and it has led to the first rigorous nonlinear stability \nproofs for a large class of steady states. More precisely, a large class \nof steady states is obtained as minimizers of energy-Casimir functionals\nunder a mass constraint $\\int f = M$,\nand their minimizing property then entails their stability.\nIn particular, all polytropes $f_{0}(x,v)=(E_{0}-E)_{+}^{k}$ \nwith $07\/2$ the corresponding steady state has infinite mass\nand is therefore unphysical. In addition, many new stable galaxy models were\nestablished. The variational method has also been investigated in \n\\cite{DSS,H,LMR,SS,Wo2}.\n\nDespite its considerable success, the variational approach has drawbacks\nand limitations, the main one being that by its very nature it can not\naccess the stability of steady states which are only local, but not global\nminimizers of the energy-Casimir functional.\nSince the existence of the steady state as a (global) minimizer\nis aimed for, certain growth conditions on the Casimir function $\\Phi$ are needed,\nwhich are not satisfied for all steady states with $\\phi' <0$. \nMost notably, the King model obtained by\n\\[\nf_0(x,v) = (e^{E_0 - E}-1)_+\n\\]\nis the single most important model which is currently out of the reach. \nIt describes isothermal galaxies and is widely used in astrophysics.\nThe corresponding Casimir function (\\ref{q}) \nhas very slow growth for $f \\to \\infty$, and as a result the\nvariational method fails. \n\nThe aim of the present paper is to develop a new approach\nto nonlinear stability\nresults for steady states which need not be global minimizers of the \ncorresponding energy-Casimir functional by exploiting Lemma~\\ref{ks}.\nAlthough we are aiming for a general approach,\nwe focus here on the King model and as a first step \nestablish its nonlinear stability against\nspherically symmetric, dynamically accessible perturbations.\n\nThe paper proceeds as follows. In the next section we formulate our results.\nThe nonlinear stability of the King model is an easy corollary of\nthe following main theorem: In a certain neighborhood of the King model\nthe potential energy distance of a perturbation can be controlled\nin terms of the energy-Casimir distance.\nIn particular, within a certain class of perturbations, which\nis invariant under solutions of the nonlinear Vlasov-Poisson system,\nthe King model is a local minimizer of the corresponding energy-Casimir\nfunctional. \nThe resulting stability estimate is more explicit that the\nones obtained by the variational approach. The main part of the work\nis then done in Section~3 where the local minimizing property\nof the King model is established. In an appendix we give a proof of \nLemma~\\ref{ks}.\n\n\\section{Main results}\n\\setcounter{equation}{0} \n\nWe start with a steady state $f_0$ with induced potential\n$U_0$ and spatial density $\\rho_0$, satisfying the relation\n\\begin{equation}\nf_{0}(x,v)=\\phi_0(E) := \\left( e^{E_{0}-E}-1\\right) _{+},\\ E:=\\frac{1}{2}|v|^2+U_{0}(x). \\label{king}\n\\end{equation}\nThe cut-off energy $E_{0}<0$ is a given negative constant and \n$(\\cdot)_+$ denotes the positive part. \nThe corresponding Casimir function in the sense of the introduction is\n\\begin{equation}\n\\Phi_{0}(f) := (1+f)\\ln (1+f)-f. \\label{q}\n\\end{equation}\nThe existence of such King models,\ni.e., of suitable solutions of the resulting semilinear Poisson equation\n(\\ref{semilinpoisson}), is established in \\cite{RR}. Such a model has\ncompact support\n\\[\n\\supp f_0 = \\{(v,v) \\in \\R^6 \\mid E(x,v) \\leq E_0\\} =: \\{E \\leq E_0\\},\n\\]\nand it is spherically symmetric. A state $f$ is called \n{\\em spherically symmetric}\nif for any rotation $A\\in \\mathrm{SO}(3)$, \n\\[\nf(x,v) = f(A x,A v),\\ x,v \\in \\R^3. \n\\] \nIt is well known that non-negative, smooth, and compactly supported\ninitial data $f(0) \\in C^1_c(\\R^6)$ launch unique global smooth solutions\n$t\\mapsto f(t)$\nof the Vlasov-Poisson system \\cite{Pf,LP,Sch}. \nIf the initial datum is spherically symmetric then this symmetry is preserved,\nand the modulus of the particle angular momentum squared,\n\\[\nL:= |x \\times v|^2 = |x|^2 |v|^2 - (x\\cdot v)^2 ,\n\\]\nis conserved along characteristics of the Vlasov equation.\nHence for any smooth function $\\Phi$ such that $\\Phi(0,L)=0,\\ L\\geq 0$,\nthe functional \n\\[\n\\int\\!\\!\\!\\!\\int \\Phi(f,L)\\,dv\\,dx\n\\]\nis conserved along spherically symmetric solutions of the \nVlasov-Poisson system;\nunless explicitly stated otherwise integrals $\\int$ always extend over $\\R^3$.\nWe consider the following class of perturbations:\n\\begin{eqnarray*}\n\\mathcal{S}_{f_{0}} := \\Bigl\\{f\\in L^1(\\R^6) \n&\\mid& \nf\\geq 0 \\ \\mbox{spherically symmetric}, \\\\\n&& \\int\\!\\!\\!\\!\\int\\Phi(f,L)=\\int\\!\\!\\!\\!\\int \\Phi(f_{0},L) \n\\text{ for all }\\Phi\\in C^2([0,\\infty[^2) \\text{ with } \\\\\n&&\n\\Phi(0,L)=\\partial_f \\Phi(0,L)= 0,\\ L\\geq 0,\\ \\mbox{and}\\ \n\\partial_f^2 \\Phi \\ \\mbox{bounded}\\Bigr\\}.\n\\end{eqnarray*}\nAs noted above, the class $\\mathcal{S}_{f_{0}} \\cap C^1_c(\\R^6)$ is\ninvariant under solutions of the Vlasov-Poisson system.\nMoreover, functions in $\\mathcal{S}_{f_{0}}$ are equi-measurable to\n$f_0$, i.e., for every $\\tau>0$ the sets $\\{ f>\\tau\\}$ and $\\{ f_0>\\tau\\}$ have \nthe same measure,\nin particular, $||f||_p = ||f_0||_p$ for any $L^p$-norm, $p\\in [1,\\infty]$. \n\nFor the Casimir function $\\Phi_0$ defined in (\\ref{q})\nwe define the energy-Casimir functional\nas in the introduction.\nThen for $f\\in \\mathcal{S}_{f_{0}}$ we have\n\\begin{eqnarray*}\n{{\\cal H}_C}(f)-{{\\cal H}_C}(f_{0}) \n&=&\n\\int\\!\\!\\!\\!\\int [\\Phi_{0}(f)-\\Phi_{0}(f_{0})+(E-E_{0})(f-f_{0})]\\,dv\\,dx \\\\\n&&-\\frac{1}{8\\pi }\\int |\\nabla U_{f}-\\nabla U_0|^2 dx;\n\\end{eqnarray*}\nnotice that $\\int f = \\int f_0$ which allows us to bring \nin the term $E_0(f-f_0)$. Now\n\\[\n(E-E_0) (f-f_0) \\geq -\\Phi_0'(f_0) (f-f_0)\n\\]\nwith equality on the support of $f_0$, and hence for \n$f\\in \\mathcal{S}_{f_{0}}$,\n\\begin{eqnarray} \\label{convest}\n\\Phi_{0}(f)-\\Phi_{0}(f_{0})+(E-E_{0})(f-f_{0}) \n&\\geq&\n\\frac{1}{2} \\inf_{0\\leq \\tau \\leq ||f_0||_\\infty} \\Phi_0''(\\tau)\\; (f-f_0)^2 \\nonumber \\\\\n&\\geq& \nC_0 (f-f_0)^2\n\\end{eqnarray}\nwhere $C_0:= 1\/(2+ 2 ||f_0||_\\infty)$; notice again that \n$||f||_\\infty = ||f_0||_\\infty$ for any \n$f\\in \\mathcal{S}_{f_{0}}$. \nThe deviation from the steady state is going to be measured\nby the quantity\n\\begin{eqnarray} \\label{d}\nd(f,f_{0})\n&:=&\n\\int\\!\\!\\!\\!\\int \n[\\Phi_{0}(f)-\\Phi_{0}(f_{0})+(E-E_{0})(f-f_{0})]\\,dv\\,dx \\nonumber \\\\\n&&\n{}+\\frac{1}{8\\pi }\n\\int |\\nabla U_{f}-\\nabla U_0|^2 dx,\n\\end{eqnarray}\nwhich, as we have seen, controls both $||f-f_0||_2$ and \n$||\\nabla U_{f}-\\nabla U_0||_2$,\nand satisfies the following relation to the energy-Casimir functional:\n\\begin{equation} \\label{decrel}\nd(f,f_{0}) = {{\\cal H}_C}(f)-{{\\cal H}_C}(f_{0}) + \n\\frac{1}{4\\pi }\\int |\\nabla U_{f}-\\nabla U_0|^2 dx.\n\\end{equation}\nOur stability result is the following:\n\\begin{theorem} \\label{main}\nThere exist constants $\\delta >0$ and $C>0$ such that \nfor any solution $t\\mapsto f(t)$ of the Vlasov-Poisson system\nwith $f(0) \\in \\mathcal{S}_{f_{0}}\\cap C_{c}^{1}(\\R^6)$ and \n\\[\nd(f(0),f_{0}) \\leq \\delta \n\\]\nthe estimate\n\\[\nd(f(t),f_{0}) \\leq C \\; d(f(0),f_{0})\n\\]\nholds for all time $t>0$.\n\\end{theorem}\n\\noindent\n{\\bf Remark:}\nIn order to better understand the perturbation class $\\mathcal{S}_{f_{0}}$\nwe show that it contains \n{\\em spherically symmetric, dynamically accessible\nperturbations} \nby which we mean the following:\nLet $H=H(x,v) \\in C^2(\\R^6)$ be spherically symmetric,\nand let $g=g(s,x,v)$ denote the solution of the linear problem\n\\[\n\\partial_s g + \\nabla_v H\\cdot \\nabla_x g - \\nabla_x H\\cdot \\nabla_v g = 0,\n\\quad \\mbox{i.e.},\\quad\n\\partial_s g(s) = \\{H,g(s)\\},\n\\]\nwith initial datum $g(0) = f_0$; we assume that $H$ is such that \nthis solution exists on some interval $I$ about $s=0$. Then for any $s \\in I$,\n$g(s)$ is spherically symmetric and equi-measurable with $f_0$.\nMoreover, for any function $\\Phi$ as considered in the definition\nof $\\mathcal{S}_{f_{0}}$,\n\\[\n\\partial_s \\Phi(g(s),L) = \\partial_f \\Phi(g(s),L) \\{ H,g(s)\\},\n\\]\nand by a simple computation, $\\{H,L\\} = 0$.\nHence\n\\begin{eqnarray*}\n\\frac{d}{ds} \\int\\!\\!\\!\\!\\int \\Phi(g(s),L)\\,dv\\,dx\n&=& \n\\int\\!\\!\\!\\!\\int \n\\left( \\partial_f \\Phi \\{H,g(s)\\} + \\partial_L \\Phi \\{H,L\\}\\right)\\,dv\\,dx \\\\\n&=&\n\\int\\!\\!\\!\\!\\int \\{H,\\Phi(g(s),L)\\}\\,dv\\,dx = 0\n\\end{eqnarray*}\nafter an integration by parts.\nThis means that $g(s) \\in \\mathcal{S}_{f_{0}}$ for any such \ngenerating function $H$ and any $s$. The only undesirable restriction \nin the class $\\mathcal{S}_{f_{0}}$\nor in the generating functions $H$ respectively is the spherical symmetry\nwhich hopefully can be removed in the future.\n\n\\smallskip\n\nThe stability result Theorem~\\ref{main} is easily deduced from the\nfollowing theorem:\n\\begin{theorem} \\label{lower}\nThere exist constants $\\delta _{0}>0,$ and $C_0>0$ such that\nfor all $f\\in \\mathcal{S}_{f_{0}}$ with $d(f,f_{0})\\leq \\delta _{0}$ \nthe following estimate holds:\n\\[\n{{\\cal H}_C}(f)-{{\\cal H}_C}(f_{0})\\geq C_0 ||\\nabla U_{f}-\\nabla U_0||_2^2. \n\\]\n\\end{theorem}\nBefore going into the proof of this theorem, which will occupy the rest of this\npaper, we conclude this section by deducing our stability result from it.\n\n\\noindent\n{\\bf Proof of Theorem~\\ref{main}}. \nLet $\\delta:= \\delta_0 (1+1\/(4 \\pi C_0))^{-1}$\nwith $\\delta_0$ and $C_0$ from Theorem~\\ref{lower}. Consider a solution\n$t\\mapsto f(t)$ of the Vlasov-Poisson system with \n$f(0) \\in \\mathcal{S}_{f_{0}}\\cap C_{c}^{1}(\\R^6)$ and \n\\[\nd(f(0),f_{0}) \\leq \\delta < \\delta_0. \n\\]\nThen by continuity we can choose some maximal $t^\\ast \\in ]0,\\infty]$ such that\n\\[\nd(f(t),f_{0}) < \\delta_0,\\ t\\in [0,t^\\ast[. \n\\]\nNow $f(t) \\in \\mathcal{S}_{f_{0}}$ for all $t$, and hence\nTheorem~\\ref{lower}, the relation (\\ref{decrel}) of $d$ to the energy-Casimir \nfunctional, and the fact that the latter is a conserved quantity\nyield the following chain of estimates for $t\\in[0,t^\\ast[$:\n\\begin{eqnarray*}\nd(f(t),f_0)\n&=&\n{{\\cal H}_C}(f(t)) - {{\\cal H}_C}(f_0) + \\frac{1}{4\\pi} ||\\nabla U_{f(t)}-\\nabla U_0||_2^2\\\\\n&\\leq&\n{{\\cal H}_C}(f(t)) - {{\\cal H}_C}(f_0) + \\frac{1}{4 \\pi C_0} \\left({{\\cal H}_C}(f(t)) - {{\\cal H}_C}(f_0)\\right)\\\\\n&=&\n\\left(1+\\frac{1}{4 \\pi C_0}\\right) \\, \\left({{\\cal H}_C}(f(0)) - {{\\cal H}_C}(f_0)\\right)\\\\\n&\\leq&\n\\left(1+\\frac{1}{4 \\pi C_0}\\right)\\,d(f(0),f_0) < \\delta_0.\n\\end{eqnarray*}\nThis implies that $t^\\ast = \\infty$, and Theorem~\\ref{main}\nis established.\n\\prfe\n\n\\section{Proof of Theorem~\\ref{lower}}\n\\setcounter{equation}{0} \n\nTheorem~\\ref{lower} is proven by contradiction. \nThere are two main ingredients: The first part (Subsection~\\ref{redsec}) is a\ngeneral argument to establish that if the estimate in the theorem fails, then \nthere exists a non-zero function $g$ such that $D^2 {{\\cal H}_C} (f_{0})[g]\\leq 0$. The\nsecond (Subsection~\\ref{hsec}) is to use the measure-preserving property \nincorporated in our perturbation class $\\mathcal{S}_{f_{0}}$ to conclude that \n$g=\\{f_{0},h\\}$ for some function $h$. This leads to a\ncontradiction to Lemma~\\ref{ks} (Subsection~\\ref{contrasec}).\n\n\\subsection{Existence of $g\\neq 0$ with $D^2 {{\\cal H}_C} (f_{0})[g]\\leq 0$} \n\\label{redsec}\n\nThe aim of this subsection is to prove the following result:\n\n\\begin{lemma} \\label{reduction}\nAssume that Theorem \\ref{lower} were false. Then there is a\nfunction $g\\in L^2(\\R^6)$ which is spherically symmetric,\nsupported in the set $\\{E \\leq E_0\\}$, even in $v$, i.e., $g(x,-v)=g(x,v)$, and \nsuch that\n\\begin{equation} \\label{normal}\n\\frac{1}{8\\pi }||\\nabla U_{g}||_2^2 =1,\n\\end{equation}\n\\begin{equation} \\label{hle}\nD^2 {{\\cal H}_C} (f_{0})[g] = \\frac{1}{2}\\int\\!\\!\\!\\!\\int_{\\{f_{0}>0\\}}\n\\Phi_{0}^{\\prime \\prime }(f_{0})\\,g^2\\,dv\\,dx - 1 \\leq 0,\n\\end{equation}\n\\begin{equation} \\label{preserve}\n\\int\\!\\!\\!\\!\\int \\partial_f\\Phi(f_{0},L)\\,g\\,dv\\,dx = 0\n\\end{equation}\nfor all functions $\\Phi$ as specified in the definition of \n$\\mathcal{S}_{f_{0}}$.\n\\end{lemma}\n\n\\noindent{\\bf Proof.}\\ \nIf Theorem \\ref{lower} were false, then for any $n\\in \\mathbb N$ there exists \n$f_{n}\\in \\mathcal{S}_{f_{0}}$ such that \n\\[\nd(f_{n},f_{0})<\\frac{1}{n}\n\\]\nbut\n\\begin{equation} \\label{hsmall}\n{{\\cal H}_C}(f_{n})-{{\\cal H}_C}(f_{0}) < \\frac{1}{8\\pi\\, n}\n||\\nabla U_{f_{n}}-\\nabla U_0||_2^2, \n\\end{equation}\nin particular, $f_n \\neq f_0$.\nWe define \n\\begin{equation} \\label{sigma}\n\\sigma_{n}:=\\frac{1}{\\sqrt{8\\pi}} ||\\nabla U_{f_{n}}-\\nabla U_0||_2 ,\\\ng_{n} := \\frac{1}{\\sigma _{n}} \\left(f_{n}-f_{0}\\right)\n\\end{equation}\nso that \n\\begin{equation} \\label{normaln}\n\\frac{1}{8\\pi }||\\nabla U_{g_{n}}||_2^2 =1 \n\\end{equation}\nand $f_n = f_0 + \\sigma_n g_n$.\nBy (\\ref{decrel}) and (\\ref{sigma}),\n\\begin{equation}\n\\sigma _{n}^2\\leq d(f_{n},f_{0}) < \\frac{1}{n}. \\label{sigmazero}\n\\end{equation}\n\n\\noindent\n{\\em Bounds on $(g_n)$ and weak limit.}\\\\\nAs a first step in the\nproof of the lemma we need to establish bounds on the\nsequence $(g_n)$ which allow us extract a subsequence that\nconverges to some $g$ which will be our candidate for the\nfunction asserted in the lemma.\nBy (\\ref{sigma}), (\\ref{d}), (\\ref{decrel}), and (\\ref{hsmall}) we find that\n\\begin{eqnarray} \\label{q''}\n&&\n\\frac{1}{\\sigma _{n}^2}\\int\\!\\!\\!\\!\\int\n[\\Phi_{0}(f_{0}+\\sigma_{n}g_{n})-\\Phi_{0}(f_{0})+(E-E_{0})\\sigma _{n}g_{n}]\n\\,dv\\,dx - 1 \\nonumber \\\\\n&&\n\\qquad\\qquad = \\frac{1}{\\sigma _{n}^2}\\left(d(f_n,f_0)\n-\\frac{1}{4\\pi } \\int |\\nabla U_{f_{n}}-\\nabla U_0|^2 dx \\right) \\nonumber\\\\ \n&&\n\\qquad\\qquad = \\frac{1}{\\sigma _{n}^2} \\left({{\\cal H}_C}(f_{n})-{{\\cal H}_C}(f_{0})\\right)\n< \\frac{1}{\\sigma _{n}^2} \n\\frac{1}{8\\pi\\, n } ||\\nabla U_{f_{n}}-\\nabla U_0||_2^2 \n= \\frac{1}{n}.\\quad \n\\end{eqnarray}\nRecalling the estimate (\\ref{convest}) which is applicable since\n$f_n \\in \\mathcal{S}_{f_{0}}$ this implies that\n\\[\n1+\\frac{1}{n} > \\frac{1}{\\sigma _{n}^2}\\int\\!\\!\\!\\!\\int [\\ldots]\\,dv\\,dx\n\\geq C_0 \\frac{1}{\\sigma _{n}^2} \\int\\!\\!\\!\\!\\int |f_n-f_0|^2\\,dv\\,dx \n= C_0\\, \\int\\!\\!\\!\\!\\int |g_n|^2\\,dv\\,dx,\n\\]\nwhich means that the sequence $(g_n)$ is bounded in $L^2(\\R^6)$.\nMoreover, since the integrand $[\\ldots]$ in (\\ref{q''}) is non-negative \nwe find that\n\\[\n1+\\frac{1}{n} > \\frac{1}{\\sigma _{n}^2}\n\\int\\!\\!\\!\\!\\int_{\\{E\\geq E_0\\}} [\\ldots]\\,dv\\,dx\n\\geq \\frac{1}{\\sigma _{n}}\n\\int\\!\\!\\!\\!\\int_{\\{E\\geq E_0\\}} (E-E_0)\\, g_n\\,dv\\,dx,\n\\]\nwhere we used the fact that on the set $\\{E\\geq E_0\\}$ the steady\nstate distribution $f_0$ and hence also $\\Phi_0(f_0)$ vanish\nwhile $\\Phi_0(f_n)\\geq 0$.\nBy (\\ref{sigmazero}),\n\\begin{equation} \\label{zerooutside1}\n\\int\\!\\!\\!\\!\\int_{\\{E\\geq E_0\\}} (E-E_0)\\, g_n\\,dv\\,dx \\leq 2 \\sigma_n \\to 0,\\\nn\\to \\infty.\n\\end{equation}\nNow fix any $E_0< E_1 < 0$. Since $\\lim_{|x|\\to \\infty} U_0 (x) = 0$\nit follows that $E=E(x,v) > E_1$ for $x$ or $v$ large so that the set\n$\\{E\\leq E_1\\}\\subset \\R^6$\nis compact. Hence $(g_n)$ is bounded in $L^1(\\{E\\leq E_1\\})$. \nIn addition\n\\begin{equation} \\label{zerooutside2}\n\\int\\!\\!\\!\\!\\int_{\\{E\\geq E_1\\}} g_n\\,dv\\,dx \n\\leq \n\\int\\!\\!\\!\\!\\int_{\\{E\\geq E_0\\}} \\frac{E-E_0}{E_1-E_0}\\, g_n\\,dv\\,dx \\to 0;\n\\end{equation}\nnotice that $g_n \\geq 0$ outside the support of $f_0$, i.e.,\non the set $\\{E > E_0\\}$. \nThus we have shown that $(g_n)$ is bounded in $L^1\\cap L^2(\\R^6)$.\nWe extract a subsequence, denoted again by $(g_n)$ such that\n\\[\ng_n \\rightharpoonup g\\ \\mbox{weakly in}\\ L^2(\\R^6).\n\\]\nSince $g_n\\geq 0$ on $\\{E>E_0\\}$ and since (\\ref{zerooutside2}) holds\nfor any $E_0< E_1 < 0$ we conclude that $g$ vanishes a.~e.\\ outside\nthe set $\\{E\\leq E_0\\}$ as desired. Since the functions $g_n$\nare spherically symmetric so is $g$.\n\n\\smallskip\n\n\\noindent\n{\\em Proof of (\\ref{normal}).}\\\\\nIn order to pass the week convergence into (\\ref{normaln}) \nwe need better bounds for the sequence $(g_n)$.\nIndeed, we can bound its kinetic energy:\n\\[\n\\int\\!\\!\\!\\!\\int_{\\{E\\geq E_0\\}}\n\\left(\\frac{1}{2} |v|^2 +U_0(x) -E_0\\right)\\,g_n\\,dv\\,dx\n= \\int\\!\\!\\!\\!\\int_{\\{E\\geq E_0\\}}\n\\left(E-E_0\\right)\\,g_n\\,dv\\,dx \\to 0\n\\]\nby (\\ref{zerooutside1}) so that \n\\[\n\\int\\!\\!\\!\\!\\int_{\\{E\\geq E_0\\}}\n\\frac{1}{2} |v|^2\\,g_n\\,dv\\,dx\n\\]\nis bounded, while\n\\[\n\\int\\!\\!\\!\\!\\int_{\\{E\\leq E_0\\}} \\frac{1}{2} |v|^2\\,g_n\\,dv\\,dx \n\\leq (E_0 - U_0(0)) \\int\\!\\!\\!\\!\\int_{\\{E\\leq E_0\\}}\ng_n\\,dv\\,dx\n\\]\nis bounded as well; recall that $U_0$ is spherically symmetric \nand radially increasing. Now well known interpolation arguments \nimply that the sequence of induced spatial densities $(\\rho_{g_n})$ \nis bounded in $L^1\\cap L^{7\/5} (\\R^3)$, \ncf.\\ \\cite[Ch.~1, Lemma~5.1]{R3}, so without loss of generality,\n\\[\n\\rho_{g_n} \\rightharpoonup \\rho_g\\ \\mbox{weakly in}\\ L^{7\/5}(\\R^3).\n\\]\nLet again $E_00$\nsuch that $U_0(R_1)=E_1$ which implies that $E(x,v)\\geq E_1$\nfor $|x|\\geq R_1$. Then by (\\ref{zerooutside2}),\n\\[\n\\int_{\\{|x|\\geq R_1\\}} |\\rho_{g_n}|\\, dx\n\\leq\n\\int\\!\\!\\!\\!\\int_{\\{E\\geq E_1\\}} g_n\\,dv\\,dx\n\\to 0.\n\\]\nThe fact that the sequence $(\\rho_{g_n})$ remains concentrated in this\nway gives the desired compactness:\n\\[\n\\nabla U_{g_n} \\to \\nabla U_g \\ \\mbox{strongly in}\\ L^2(\\R^3),\n\\]\ncf.\\ \\cite[Ch.~2, Lemma~3.2]{R3}. \nHence we can pass to the limit in (\\ref{normaln}) and find that\n$g$ satisfies the condition (\\ref{normal}) in the lemma.\n\n\\smallskip\n\n\\noindent\n{\\em Proof of (\\ref{hle}).}\\\\ \nSince $||g_{n}||_2$ is bounded it follows from (\\ref{sigmazero}) that \n\\[\n||\\sigma _{n}g_{n}||_2 \\leq C\\sigma _{n} \\to 0.\n\\]\nTherefore, after extracting again a subsequence,\n$\\sigma _{n}g_{n} \\to 0$ almost\neverywhere.\nBy Egorov's Theorem there exists for every\n$m\\in \\mathbb N$ a measurable \nsubset $K_{m} \\subset \\{E \\leq E_0\\}$ with the property that \n\\[\n\\vol \\left(\\{E \\leq E_0\\}\\setminus K_m\\right) <\\frac{1}{m}\n\\ \\mbox{and}\\\n\\lim_{n \\to \\infty}\\sigma_{n}g_{n}=0 \\ \n\\mbox{uniformly on}\\ K_{m};\n\\]\nnote that the set $\\{E \\leq E_0\\}$ has finite measure. In addition we can \nassume that $K_m \\subset K_{m+1},\\ m\\in \\mathbb N$.\nOn the set $\\{E \\leq E_0\\}$,\n\\[\n[\\Phi_0(f_n) - \\Phi_0(f_0) + (E-E_0) \\sigma_n g_n] \n = \\frac{1}{2} \\Phi_0''(f_0)(\\sigma_n g_n)^2\n+ \\frac{1}{6} \\Phi_0'''(f_0+\\tau \\sigma_n g_n)(\\sigma_n g_n)^3\n\\]\nfor some $\\tau \\in [0,1]$. Since both $f_0$ and\n$f_0+\\sigma_n g_n = f_n$ are non-negative the same is true\nfor $f_0+\\tau \\sigma_n g_n$, and we can use the estimate\n\\[\n|\\Phi_0''' (f)| = \\frac{1}{(1+f)^2} \\leq 1,\\ f\\geq 0\n\\]\nthe estimate (\\ref{q''}), and the fact that the integrand\n$[\\ldots]$ in (\\ref{q''}) is\nnon-negative by (\\ref{convest}) to conclude that\n\\begin{eqnarray*}\n\\int\\!\\!\\!\\!\\int_{K_m} \\frac{1}{2} \\Phi_0''(f_0)\\, |g_n|^2\\,dv\\,dx\n&=&\n\\frac{1}{\\sigma_n^2} \\int\\!\\!\\!\\!\\int_{K_m} [\\ldots]\\,dv\\,dx\\\\\n&&\n{}- \\frac{1}{\\sigma_n^2} \\int\\!\\!\\!\\!\\int_{K_m}\n\\frac{1}{6} \\Phi_0'''(f_0+\\tau \\sigma_n g_n)\\,(\\sigma_n g_n)^3\\,dv\\,dx\\\\\n&<&\n1+\\frac{1}{n} + \\sup_{K_m}|\\sigma_n g_n| \\int\\!\\!\\!\\!\\int |g_n|^2\\,dv\\,dx.\n\\end{eqnarray*}\nTaking the limit $n\\to \\infty $ in this estimate we find that\n\\[\n\\int\\!\\!\\!\\!\\int_{K_m} \\frac{1}{2} \\Phi_0''(f_0)\\, g^2\\,dv\\,dx \\leq 1.\n\\] \nIf we observe the choice of the sets $K_m$, let $m\\to \\infty$, \nand recall the fact that $g=0$ outside the set $\\{E\\leq E_0\\}$ \nthe proof of (\\ref{hle}) is complete.\n\n\\smallskip\n\n\\noindent\n{\\em Proof of (\\ref{preserve}).}\\\\ \nTo prove (\\ref{preserve}), the measure-preserving property of the set \n$\\mathcal{S}_{f_{0}}$ plays the crucial role. \nLet $\\Phi=\\Phi(f,L)$ be a function as specified in\nthe definition of that set respectively in the lemma.\nBy Taylor expansion with respect to the first argument,\n\\[\n\\Phi(f_n,L)-\\Phi(f_{0},L)\n=\\partial _{f}\\Phi(f_{0},L)\\,\\sigma _{n}g_{n}+\n\\frac{1}{2}\\partial_{f}^2\\Phi(f_{0}+\\tau\\sigma_{n}g_{n},L)\\,\n(\\sigma _{n}g_{n})^2\n\\]\nfor some $\\tau\\in[0,1]$. If we integrate this identity\nand observe that, since $f_n \\in \\mathcal{S}_{f_{0}}$,\n\\[\n\\int\\!\\!\\!\\!\\int \\Phi(f_n,L)\\,dv\\,dx=\\int\\!\\!\\!\\!\\int \\Phi(f_{0},L)\\,dv\\,dx,\n\\]\nit follows that\n\\[\n\\int\\!\\!\\!\\!\\int\n\\partial_{f}\\Phi(f_{0},L)\\,g_{n}\\,dv\\,dx \n=\n-\\frac{1}{2}\\sigma_{n}\n\\int\\!\\!\\!\\!\\int\\partial_f^2 \\Phi(f_{0}+\\tau \\sigma_{n}g_{n},L)\\,g_n^2\\,dv\\,dx\n\\to 0.\n\\]\nAs to the latter limit we note that $\\partial_f^2 \\Phi$ is bounded,\n$(g_n)$ is bounded in $L^2(\\R^6)$, and $\\sigma _{n}\\to 0$. \nOn the other hand $\\partial_{f}\\Phi(f_{0},L)$ is supported\non the compact set $\\{E\\leq E_0\\}$ and hence bounded.\nSince $g_{n}\\to g$ in $L^2(\\R^6)$, the identity (\\ref{preserve}) \nfollows as $n \\to \\infty$.\n\n\\smallskip\n\\noindent\n{\\em Conclusion of the proof of Lemma~\\ref{reduction}.}\\\\\nThe function $g$ constructed above has all the properties\nrequired in the lemma, except that it need not be even in $v$. Hence\nwe decompose it into its even and odd parts with respect to $v$: $g=g_\\mathrm{even}+g_\\mathrm{odd}$. \nWe claim that (\\ref{normal}), (\\ref{hle}), and (\\ref{preserve}) \nremain valid for the even part. Since \n\\[\n\\rho_{g}=\\int g\\, dv=\\int g_\\mathrm{even}dv=\\rho_{g_\\mathrm{even}}\n\\] \nwe have $\\nabla U_{g_\\mathrm{even}}=\\nabla U_{g}$, and (\\ref{normal})\nremains valid. \nSince $\\Phi_0^{\\prime \\prime }(f_0)$ is even in $v$, \n\\begin{eqnarray*}\n1\n&\\geq&\n\\int\\!\\!\\!\\!\\int \\frac{1}{2} \n\\Phi_0''(f_0)\\, (g_\\mathrm{even}+g_\\mathrm{odd})^2\\,dv\\,dx\n=\n\\int\\!\\!\\!\\!\\int \\frac{1}{2} \n\\Phi_0''(f_0)\\, \\left((g_\\mathrm{even})^2+(g_\\mathrm{odd})^2\\right)\\,dv\\,dx \\\\\n&\\geq&\n\\int\\!\\!\\!\\!\\int \\frac{1}{2} \n\\Phi_0''(f_0)\\, (g_\\mathrm{even})^2\\,dv\\,dx,\n\\end{eqnarray*}\ni.e., (\\ref{hle}) remains valid. Finally, let $\\Phi$ be as in\nthe definition of the set $\\mathcal{S}_{f_{0}}$. Then\n$\\partial_f \\Phi(f_0,L)$ is even in $v$ so that in (\\ref{preserve})\nthe odd part of $g$ drops out, and the assertions of Lemma~\\ref{reduction}\nhold for $g_\\mathrm{even}$.\n\\prfe\n\n\\subsection{Characteristics and $g=\\{f_{0},h\\}$}\\label{hsec}\n\nIn this subsection we construct a spherically symmetric\nfunction $h$ such that $g=\\{f_{0},h\\}$.\nTo this end we need to introduce variables which are adapted to the\nspherical symmetry:\n\\[\nr:=|x|,\\ w:= \\frac{x\\cdot v}{r},\\ L:=|x\\times v|^2;\n\\]\n$w$ is the radial velocity, and $L$ has already been used above.\nAny spherically symmetric function of $x$ and $v$ such as\nthe desired $h$ can be written\nin terms of these variables, so $h=h(r,w,L)$. Then\n\\[\n\\{f_{0},h\\} \n=\n\\phi_0'(E) \\{E,h\\}\n= - \\phi_0'(E)\\,\\left[w\\, \\partial_r +\n\\left(\\frac{L}{r^3}-U_0'(r)\\right)\\,\\partial_w \\right]\\, h,\n\\]\nand the equation we wish to solve for $h$ reads\n\\begin{equation} \\label{heqn}\n\\left[w\\, \\partial_r +\n\\left(\\frac{L}{r^3}-U_0'(r)\\right)\\,\\partial_w \\right]\\, h= - \\frac{1}{\\phi_0'(E)}\\,g.\n\\end{equation}\nIn order to analyze its characteristic system\n\\[\n\\dot r = w,\\ \\dot w = \\frac{L}{r^3}-U_0'(r)\n\\]\nwe define for fixed $L>0$ the effective potential\n\\[\n\\Psi_L(r):= U_0(r) + \\frac{L}{2 r^2}\n\\]\nand observe that in terms of the variables $r,w,L$ the \nconserved particle energy takes the form\n\\[\nE=E(x,v)=E(r,w,L) = \\frac{1}{2}w^2 +\\Psi_L(r);\n\\]\n$L$ only plays the role of a parameter here since $\\dot L = 0$.\nWe need to analyze the effective potential $\\Psi_L$.\nThe boundary condition for $U_0$ at infinity implies that \n$\\lim_{r\\to \\infty} \\Psi_L(r)=0$, and clearly,\n$\\lim_{r\\to 0} \\Psi_L(r)=\\infty$. Moreover,\n\\[\nU_0'(r)=\\frac{4\\pi }{r^2}\\int_{0}^{r}\\rho_{0}(s)\\,s^2 ds =:\n\\frac{m_0(r)}{r^2}>0,\\ r>0,\n\\]\nsince $U_0$ is increasing, thus $U_0(0) < E_0$, and by (\\ref{king}),\n$\\rho_{0}(0)>0$. Now\n\\[\n\\Psi_L'(r) = 0 \\ \\Leftrightarrow\\ \\frac{m_0(r)}{r^2}- \\frac{L}{r^3} =0\n\\ \\Leftrightarrow\\ m_0(r)-\\frac{L}{r} = 0,\n\\]\nand since the left hand side of the latter equation\nis strictly increasing for $L>0$ with \n$\\lim_{r\\to \\infty}(m_0(r)-L\/r) = M >0$ and \n$\\lim_{r\\to 0}(m_0(r)-L\/r) = -\\infty$ there exists a unique $r_L >0$\nsuch that\n\\[\n\\Psi_L'(r_L)=0,\\ \\Psi_L'(r)<0\\ \\mbox{for}\\ r0\\ \\mbox{for}\\ r>r_L.\n\\]\nMoreover, since \n\\[\n\\frac{d}{dr} \\left(m_0(r)-\\frac{L}{r}\\right) \n= 4 \\pi r^2 \\rho_0(r) + \\frac{L}{r^2} >0\n\\]\nthe implicit function theorem implies that the mapping\n$]0,\\infty[ \\ni L \\mapsto r_L$ is continuously differentiable.\nFor $r=r_L$,\n\\begin{equation} \\label{psi''>0}\n\\Psi''_L (r_L) = - 2\\frac{m_0(r)}{r^3} + 3\\frac{L}{r^4} \n+ 4 \\pi \\rho_0(r) = 4 \\pi \\rho_0(r) + \\frac{L}{r^4} > 0.\n\\end{equation}\nThe behavior of $\\Psi_L$ implies that for any $L>0$ and\n$\\Psi_L(r_L)0$ and\nrunning in the set $\\{E\\leq E_0\\}$, so that\n$\\Psi_L(r_L)0$\nwe define $h$ as follows:\nWe let $E:= \\frac{1}{2} w^2 + \\Psi_L(r)$ so that \n$r_{-}(E,L)\\leq r \\leq r_{+}(E,L)$, and \n\\begin{equation}\nh(r,w,L) := - \\mathrm{sign}\\, w \\frac{1}{\\phi_0'(E)}\\int_{r_{-}(E,L)}^{r}\n\\frac{g(s,\\sqrt{2E-2\\Psi_L(s)},L)}{\\sqrt{2E-2 \\Psi_L(s)}} ds. \\label{hg}\n\\end{equation}\nOutside the set $\\{E\\leq E_0\\}$ we let $h=0$. \nWe need to make sure that we can consistently set\n$h(r,0,L)=0$, i.e., the integral in the above definition\nmust vanish for $r=r_+(E,L)$. To this end let\n$\\Phi$ be as specified in the definition of the class\n$\\mathcal{S}_{f_{0}}$. We want to apply the change of variables\n$(x,v) \\mapsto (r,w,L) \\mapsto (r,E,L)$ to the integral in (\\ref{preserve}).\nNow\n\\begin{equation} \\label{xvtorel}\ndx\\,dv = 8\\pi^2 dr\\,dw\\,dL = 8\\pi^2 \n\\frac{dr\\,dE\\,dL}{\\sqrt{2E-2\\Psi_L(r)}},\n\\end{equation}\nwhere we note that $g$ is even in $v$ and hence in $w$ \nso that we can restrict the integral to $w>0$. We obtain\nthe identity\n\\begin{eqnarray*}\n0\n&=&\n\\int\\!\\!\\!\\!\\int \\partial_f\\Phi(f_{0},L)\\,g\\,dv\\,dx\n=\n8\\pi^2 \\int_0^\\infty \\int_0^\\infty \\int_0^\\infty\n\\partial_f\\Phi(f_{0},L)\\,g\\,dr\\,dw\\,dL\\\\\n&=&\n8\\pi^2\n\\int\\!\\!\\!\\!\\int_M\n\\int_{r_{-}(E,L)}^{r_+(E,L)}\n\\frac{g(r,\\sqrt{2E-2\\Psi_L(r)},L)}{\\sqrt{2E-2 \\Psi_L(r)}} dr\\,\n\\partial_f\\Phi(\\phi_0(E),L)\\,dE\\,dL,\n\\end{eqnarray*}\nwhere $M:=\\{(E,L)(x,v)|f_0(x,v)>0\\}$. \nThe class of test functions $\\partial_f\\Phi(\\phi_0(E),L)$ is sufficiently \nlarge to conclude that for almost all $E$ and $L$,\n\\[\n\\int_{r_{-}(E,L)}^{r_{+}(E,L)}\\frac{g(r,\\sqrt{2E-2\\Psi_L(r)},L)}\n{\\sqrt{2E-2\\Psi_L(r)}} dr =0\n\\]\nas desired. \n\n\\subsection{Contradiction to Lemma~\\ref{ks}}\n\\label{contrasec}\n\n\nAs defined above, $h$ need not be smooth or even integrable,\nso in order to derive a contradiction to Lemma~\\ref{ks}\nwe need to regularize it. In order to do so it should first\nbe noted that $h$ as defined in (\\ref{hg}) is measurable,\nwhich follows by Fubini's Theorem and the fact that by the change\nof variables formula the function\n\\[\n(s,r,E,L) \\mapsto \\frac{g(s,\\sqrt{2 E -2\\Psi_L(s)},L)}\n{\\sqrt{2 E -2\\Psi_L(s)}} \n\\mathbf{1}_{[\\Psi_L(r_L),E_0]}(E)\\mathbf{1}_{[r_-(E,L),r_+(E,L)]}(s)\n\\mathbf{1}_{[0,r]}(s)\n\\]\nis integrable; $r\\leq \\max\\{|x| \\mid (x,v) \\in \\supp f_0\\}$.\n\n\\smallskip\n\n\\noindent{\\em The cut-off $h$.}\\\\\nAs a first step in regularizing $h$ we define for $m$ large the set\n\\[\n\\Omega_{m}:= \\left\\{(x,v)\\in \\R^6 \\mid\nE \\leq E_{0}-\\frac{1}{m},\\ L\\geq \\frac{1}{m}\\right\\}.\n\\]\nWe want to approximate $h$ by $h\\mathbf{1}_{\\Omega _{m}}$. \nIn order to analyze this approximation the following lemma will be useful:\n\\begin{lemma} \\label{line}\nThere exists a constant $C_{m}>0$ such that for\n$L\\geq \\frac{1}{m}$ and $\\Psi_L(r_L) < E \\leq E_0$, \n\\[\n\\int_{r_{-}(E,L)}^{r_{+}(E,L)}\\frac{dr}{\\sqrt{2 E-2 \\Psi_L(r)}} 0$ such that\nfor all $L\\geq 1\/m$, $\\Psi_L(r_L)< E\\leq E_0$,\nand $r\\in[r_-(E,L),r_+(E,L)]$,\n\\begin{equation} \\label{claim}\n \\frac{|\\Psi'_L(r)|}{\\sqrt{\\Psi_L(r)-\\Psi_L(r_L)}}\n\\geq \\eta _{m}. \n\\end{equation}\nTo see this let $m\\in \\mathbb N$ and $L,E,r$ be as specified.\nThen\n\\[\nE_0 \\geq E \\geq \\Psi_L(r) = U_0(r) + \\frac{L}{2 r^2} \n\\geq U_0(0) + \\frac{1}{2 m r^2}\n\\]\nand hence $r\\geq (2m(E_0 - U_0(0))^{-1\/2}$. \nLet $R:=\\max\\{|x| \\mid (x,v)\\in \\supp f_0\\}$, i.e., $U_0(R)=E_0$.\nThen\n\\[\nL \\leq 2 r^2 \\left(E_0 - U_0(r)\\right) \n\\leq 2 R^2 \\left(E_0 - U_0(0)\\right).\n\\]\nHence if we assume that (\\ref{claim}) were false, \nwe can find a sequence $(r_{n},L_{n})\\to (\\bar{r},\\bar{L})$ \nin the set \n$[(2m(E_0 - U_0(0))^{-1\/2},R]\\times[1\/m, 2 R^2 \\left(E_0 - U_0(0)\\right)]$\nsuch that \n\\[\n\\lim_{n\\to \\infty }\\frac{\\Psi'_{L_n}(r_n)}\n{\\sqrt{\\Psi_{L_{n}}(r_{n}))- \\Psi_{L_{n}}(r_{L_{n}})}}=0.\n\\]\nIf $\\bar{r}\\neq r_{\\bar{L}}$ it follows that\n$\\Psi_{\\bar L} (\\bar r) > \\Psi_{\\bar L} (r_{\\bar L})$ and\n$\\Psi'_{\\bar L}(\\bar r) = 0$ which is a contradiction\nto the uniqueness of the minimizer $r_{\\bar L}$ of \n$\\Psi_{\\bar L}$. So assume that $\\bar{r} = r_{\\bar{L}}$.\nNow recall that $\\Psi'_{L_n}(r_{L_n})=0$.\nBy Taylor expansion at $r=r_{L_n}$ we find intermediate values\n$\\theta_n$ and $\\tau_n$ between $r_n$ and $r_{L_n}$ such that\n\\[\n\\frac{|\\Psi'_{L_{n}}(r_{n})|}\n{\\sqrt{\\Psi_{L_{n}}(r_{n}) - \\Psi_{L_{n}}(r_{L_n})}}\n=\n\\frac{|\\Psi''_{L_{n}}(\\theta_{n})\\,(r_{n}-r_{L_{n}})|}\n{\\sqrt{\\frac{1}{2} \\Psi''_{L_{n}}(\\tau_{n})\\,(r_{n}-r_{L_{n}})^2}}\n\\to\n\\sqrt{2 |\\Psi''_{\\bar L}(r_{\\bar L})|} \\neq 0,\\ n\\to \\infty,\n\\]\nwhere we recall (\\ref{psi''>0}).\nThis contradiction completes the proof of (\\ref{claim}).\n\nTo complete the proof of the lemma we split the integral as\n\\[\n\\int_{r_{-}(E,L)}^{r_{+}(E,L)}\\frac{dr}{\\sqrt{2E-2\\Psi_L(r)}}\n=\\int_{r_{-}(E,L)}^{r_L}\\ldots + \\int_{r_L}^{r_{+}(E,L)}\\ldots \n=: I_1 + I_2.\n\\]\nIn the first term, we make a change of variables \n$u=\\sqrt{\\Psi_L(r)-\\Psi_L(r_L)}$ so that\n$\\frac{du}{dr}=\\frac{1}{2 u}\\Psi_L'(r) <0$ on $[r_{-}(E,L),r_L[$.\nBy (\\ref{claim}), \n\\begin{eqnarray*}\nI_1\n&=&\n\\int_{\\sqrt{E-\\Psi_L(r_L)}}^{0}\n\\frac{1}{\\sqrt{2(E-\\Psi_L(r_L)-u^2)}}\\frac{dr}{du}du \\\\\n&\\leq&\n\\frac{\\sqrt{2}}{\\eta _{m}}\n\\int_{0}^{\\sqrt{E-\\Psi_L(r_L)}}\n\\frac{du}{\\sqrt{E-\\Psi_L(r_L)-u^2}}\n=\n\\frac{\\sqrt{2}}{\\eta _{m}}\n\\int_{0}^{1}\\frac{ds}{\\sqrt{1-s^2}}<\\infty \n\\end{eqnarray*}\nby a further change of variables $u=\\sqrt{E-\\Psi_L(r_L)} s$. \nThe same type of estimate holds for the second part\n$I_2$ of the integral under investigation, and the proof of the\nlemma is complete.\n\\prfe\n\nWe now show that the cut-off function $h\\mathbf{1}_{\\Omega _{m}}$\nis square integrable and solves the equation\n$\\{f_{0},h\\mathbf{1}_{\\Omega _{m}}\\} =g\\mathbf{1}_{\\Omega _{m}}$\nin the sense of distributions, more precisely:\n\\begin{lemma} \\label{dis}\nFor any $m\\in \\mathbb N$ large, \n$h\\mathbf{1}_{\\Omega _{m}}\\in L^2(\\R^6)$, and \nfor any spherically symmetric test function \n$\\psi=\\psi(r,w,L) \\in C^1([0,\\infty[\\times \\R \\times [0,\\infty[)$,\n\\[\n\\int_{E\\leq\nE_{0}}\\{f_{0},\\psi \\}h\\mathbf{1}_{\\Omega _{m}}=-\n\\int_{E\\leq E_{0}}g\\mathbf{1}_{\\Omega _{m}}\\psi.\n\\]\n\\end{lemma}\n\\noindent{\\bf Proof.}\\ \nWe first prove that $h\\mathbf{1}_{\\Omega _{m}}\\in L^2(\\R^6)$. \nSince the integrand is even in $v$ we can apply\nthe change of variables (\\ref{xvtorel}):\n\\begin{eqnarray*}\n&&\n\\int\\!\\!\\!\\!\\int \\mathbf{1}_{\\Omega _{m}} h^2 dv\\,dx\\\\\n&&\n\\qquad\n= 8 \\pi^2 \\int\\!\\!\\!\\!\\int_{S_m} \n\\int_{r_{-}(E,L)}^{r_{+}(E,L)}h^2(r,\\sqrt{2E-2\\Psi_L(r)},L)\n\\frac{dr}{\\sqrt{2E-2\\Psi_L(r)}}\\,dE\\,dL,\n\\end{eqnarray*}\nwhere \n\\begin{equation} \\label{sm}\nS_m:= \n\\left\\{(E,L)=(E,L)(x,v)\n\\mid (x,v)\\in \\supp f_0,\\ E\\leq E_0-\\frac{1}{m},\\ L\\geq \\frac{1}{m}\\right\\}.\n\\end{equation}\nLet $(E,L) \\in S_m$. In the estimates below we write\n$w(r)=\\sqrt{2E-2\\Psi_L(r)}$ and $r_\\pm(E,L) = r_\\pm$\nfor brevity. Then by the definition\n(\\ref{hg}) of $h$ and the fact that $1\/|\\phi_0'(E)| \\leq 1$ for $E\\leq E_0$,\n\\begin{eqnarray*}\n\\int_{r_{-}}^{r_{+}}h^2(r,w(r),L) \\frac{dr}{w(r)}\n&=&\n\\int_{r_{-}}^{r_{+}}\n\\left[\\frac{1}{\\phi_0'(E)} \\int_{r_{-}}^{r}\ng(s,w(s),L) \\frac{ds}{w(s)}\\right]^2\n\\frac{dr}{w(r)} \\\\\n&\\leq&\n\\int_{r_{-}}^{r_{+}}\n\\left[\\int_{r_{-}}^{r_{+}}\ng(s,w(s),L) \\frac{ds}{w(s)}\\right]^2\n\\frac{dr}{w(r)} \\\\\n&\\leq&\n\\left(\\int_{r_{-}}^{r_{+}}\\frac{dr}{w(r)}\\right)^2\n\\int_{r_{-}}^{r_{+}}\ng^2(r,w(r),L) \\frac{dr}{w(r)}\\\\\n&\\leq&\nC_m^2\n\\int_{r_{-}}^{r_{+}}\ng^2(r,w(r),L) \\frac{dr}{w(r)};\n\\end{eqnarray*}%\nin the last two estimates\nwe used the Cauchy-Schwarz inequality and Lemma~\\ref{line}.\nA further integration with respect to $E$ and $L$ and the change\nof variables $(r,E,L)\\mapsto (x,v)$ shows that \n$||h\\mathbf{1}_{\\Omega _{m}}||_2$ is bounded in terms of $C_m$ and\n$||g||_2$.\n\nNow let $\\psi=\\psi (r,w,L)$ be a test function \nas specified in the lemma. Along characteristic curves\nof (\\ref{heqn}), which as before we parameterize by $r$\ndistinguishing between $w>0$ and $w<0$,\na simple computation shows that\n\\[\n\\{E,\\psi\\}= -\\mathrm{sign}\\, w \\sqrt{2 E- 2 \\Psi_L(r)}\n \\frac{d}{dr}[\\psi (r,\\mathrm{sign}\\, w \\sqrt{2 E- 2 \\Psi_L(r)},L)].\n\\]\nBy the change of variables used repeatedly above, using the\nabbreviation $w(r):=\\sqrt{2 E- 2 \\Psi_L(r)}$, and recalling\nthe definition (\\ref{sm}) we find that \n\\begin{eqnarray*}\n&&\n\\int\\!\\!\\!\\!\\int \\{f_{0},\\psi \\}h\\mathbf{1}_{\\Omega _{m}}dv\\,dx\n=\\int_{\\{w>0\\}} \\ldots + \\int_{\\{w<0\\}} \\ldots\\\\\n&&\n= -\\int\\!\\!\\!\\!\\int_{S_m} \\phi_0'(E)\n\\int_{r_{-}}^{r_{+}} w(r)\n\\frac{d}{dr}[\\psi (r,w(r),L)]\\,\nh(r,w(r),L) \\frac{dr}{w(r)} dE\\,dL\\\\\n&& {}-\\int\\!\\!\\!\\!\\int_{S_m} \\phi_0'(E)\n\\int_{r_{-}}^{r_{+}} (-w(r))\n\\frac{d}{dr}[\\psi (r,-w(r),L)]\\,\nh(r,-w(r),L) \\frac{dr}{w(r)} dE\\,dL\\\\\n&&\n= \\int\\!\\!\\!\\!\\int_{S_m} \\phi_0'(E)\n\\int_{r_{-}}^{r_{+}} \\psi (r,w(r),L)\\, \n\\left(-\\frac{g(r,w(r),L)}{\\phi_0'(E)\\, w(r)}\\right)\\,dr\\, dE\\,dL\\\\\n&&\n{}- \\int\\!\\!\\!\\!\\int_{S_m} \\phi_0'(E)\n\\int_{r_{-}}^{r_{+}} \\psi (r,-w(r),L)\\, \n\\frac{g(r,-w(r),L)}{\\phi_0'(E)\\, w(r)}dr\\, dE\\,dL\\\\\n&&\n= - \\int\\!\\!\\!\\!\\int \\psi g \\mathbf{1}_{\\Omega _{m}}dv\\,dx;\n\\end{eqnarray*}\nnotice that $h(r,\\pm w(r),L)=0$ for $r=r_\\pm(E,L)$, which \ntogether with the definition (\\ref{hg}) of $h$ along the\ncharacteristics was essential in the integration by parts\nabove, and also that $g$ is even in $v$ respectively $w$.\nThe proof of Lemma~\\ref{dis} is now complete.\n\\prfe\n\n\\smallskip\n\n\\noindent\n{\\em Regularization of $h\\mathbf{1}_{\\Omega _{m}}$}.\\\\\nThe function $h\\mathbf{1}_{\\Omega _{m}}$ is not smooth,\nhence Lemma~\\ref{ks} cannot be applied to it, and therefore\nwe smooth it. For fixed $m\\in \\mathbb N$ the function $h\\mathbf{1}_{\\Omega _{m}}$\nis, as a function of $r,w,L$, supported in a set of the form\n\\begin{equation} \\label{supphm}\nQ_m := [R_0,R_1]\\times [-W_0,W_0]\\times [L_0,L_1]\n\\end{equation} \nwith $00$, and $00$ we have for all \n$m\\geq m_{0}$, \n\\[\nD^2 {{\\cal H}_C} (f_{0})[g\\mathbf{1}_{\\Omega _{m}}]\n\\geq -\\frac{1}{2}\\int_{\\{E0.\n\\]\nHence $m \\to \\infty$ leads to a contradiction\nto Lemma~\\ref{reduction}, and the proof of Theorem~\\ref{lower}\nis complete.\n\n\\section{Appendix: Proof of Lemma~\\ref{ks}}\n\\setcounter{equation}{0}\n\nLet\n\\[\nU_{h}(x) := \\int\\!\\!\\!\\!\\int \\{f_{0},h\\} dv\\frac{dy}{|x-y|}\n\\]\nbe the potential induced by $-\\{f_{0},h\\}$, which is spherically symmetric.\nA short computation using the definition (\\ref{lie}) of the Lie-Poisson bracket\nshows that\n\\[\n\\int \\{f_{0},h\\} dv = \\nabla _{x}\\cdot \\int v\\, \\phi_0'(E)\\, h(x,v)\\,dv.\n\\]\nBy the spherical symmetry of $U_h$ and $h$, \n\\begin{eqnarray*}\nU_{h}^{\\prime }(r)\n&=&\n\\frac{1}{r^2}\\int_{|x|\\leq r} \\nabla _{x}\\cdot \\int v\\, \\phi_0'(E)\\, h(x,v)\\,dv\\\\\n&=&\n\\frac{1}{r^2}\\int_{|x|= r} \\int \\phi_0'(E)\\, h(x,v) v\\cdot\\frac{x}{r} dv\\, d\\omega(x)\n= 4 \\pi \\int w \\phi_0'(E)\\, h(x,v)\\, dv.\n\\end{eqnarray*}\nTherefore, by the Cauchy-Schwarz inequality, \n\\[\n\\frac{1}{8\\pi }\\int |\\nabla _{x}U_{h}|^2dx\n\\leq\n2\\pi \\int \\left[ -\\int w^2 \\phi_0'(E)\\,dv\\right]\\,\n\\left[ -\\int \\phi_0'(E)\\, h^2 dv\\right]\\, dx.\n\\]\nSince \n\\[\nw^2 \\phi_0'(E) = w \\frac{d}{dw}\\phi_0\\left(\\frac{1}{2} w^2 +\\frac{L}{2 r^2}+U_0(r)\\right)\n\\]\nan integration by parts with respect to $w$ yields \n\\[\n-\\int w^2 \\phi_0'(E)\\,dv = \\frac{\\pi}{r^2}\\int_0^\\infty\\int_{-\\infty}^\\infty\n\\phi_0\\left(\\frac{1}{2} w^2 +\\frac{L}{2 r^2}+U_0(r)\\right)\\, dw\\, dL = \\rho_0(r).\n\\]\nHence \n\\[\nD^2 {{\\cal H}_C} (f_{0})[\\{f_{0},h\\}] \n\\geq \n-\\frac{1}{2} \\int\\!\\!\\!\\!\\int\n\\phi_0' (E)\\,\\left(|\\{E,h\\}|^2 - 4\\pi \\rho_{0} h^2\\right)\\,dv\\,dx.\n\\]\nSince $h$ is odd in $v$ respectively $w$ the function\n\\[\n\\mu(r,w,L)\n:= \\frac{1}{r w} h(r,w,L)\n\\]\nis smooth away from $r=0$; in passing we notice that the functions $h_n$\nto which we applied Lemma~\\ref{ks} in the proof of Theorem~\\ref{lower}\nhave support bounded away from $r=0$, but this is not necessary for Lemma~\\ref{ks}.\nSince $h=r w \\mu$, \n\\[\n\\{E,h\\} = rw \\{E,\\mu \\} + \\mu \\{E,rw\\},\n\\]\nand hence\n\\begin{eqnarray*}\n|\\{E,h\\}|^2 &=&(rw)^2|\\{E,\\mu \\}|^2+rw\\{E,rw\\}\\{E,\\mu\n^2\\}+\\mu ^2|\\{E,rw\\}|^2 \\\\\n&=&\n(rw)^2|\\{E,\\mu \\}|^2 + \\{E,\\mu ^2rw\\{E,rw\\}\\}-\\mu^2rw\\{E,\\{E,rw\\}\\}.\n\\end{eqnarray*}\nThe first term is as claimed in Lemma~\\ref{ks}. \nThe second term leads to \n$\\{f_{0},\\mu ^2rw\\{E,rw\\}\\}$ whose integral with respect to $x$ and $v$\nvanishes; if we cut a small ball of radius $\\epsilon$\nabout $x=0$ from the $x$-integral\nthen the surface integral appearing after the integration by parts\nwith respect to $x$ vanishes for $\\epsilon \\to 0$ since $r \\mu^2 \\leq C\/r$. \nBy the Poisson equation, \n\\[\n\\{E,\\{E,rw\\}\\}=-2 w U_0' - w (r U_0')'=- r w\\, \\left(4\\pi \\rho_{0} + \\frac{1}{r}U_0'\\right).\n\\]\nHence the third term above becomes\n$ 4\\pi \\rho _{0}h^2 + h^2 \\frac{1}{r}U_0'$, and Lemma~\\ref{ks} is proven.\n\\prfe\n\n\\noindent\n{\\bf Acknowledgment.}\nThe research of the first author \nis supported in part by an NSF grant. This article is dedicated\nto the memory of Xudong Liu.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzhgpv b/data_all_eng_slimpj/shuffled/split2/finalzzhgpv new file mode 100644 index 0000000000000000000000000000000000000000..611c961f732e1a0b2f2be629c417810b5e15025e --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzzhgpv @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\\label{Sec:Intro}\n{ASIPs} (\\emph{Application Specific Instruction-set Processors}) play a central role in contemporary embedded systems-on-a-chip (SoCs) replacing hardwired solutions which offer no programmability for enabling reuse or encompassing late specification changes. ASIPs are tuned for cost-effective execution of targeted application sets. An ASIP design flow involves profiling, architecture exploration, generation\/selection of instruction-set extensions (ISEs) and synthesis of the corresponding hardware while enabling the user taking certain decisions. \n\nCustom processors either adhere to the configurable\/extensible processor paradigm \\cite{ARC,Gonzalez00} or can be ASIPs completely designed from scratch. Configurability lies in tuning architectural parameters (e.g. cache sizes) and enabling\/disabling features \\cite{Yiannacouras06} while extensibility of a processor comes in modifying the instruction set architecture by adding forms of custom functionality. Designing a custom processor from scratch is a more aggressive approach requiring a significant investment of effort in developing all the necessary software development tools (compiler, binary utilities, debugger\/simulator) and possibly a real-time OS, while in the configurable processor case, the RTOS is usually targeted to the base ISA and the software toolchain can be incrementally updated. There exist two basic themes for architecture extension: tight integration of custom functional units and storage \\cite{NiosII} or loose coupling of hardware accelerators \nto the processor through a bus interface \\cite{MicroBlaze}. Recent works \\cite{Sirowy07} \nadvocate in favor of both approaches, proving that both techniques can be considered \nsimultaneously by formalizing the problem as a form of two-level partitioning. \n\nIt is often in ASIP\/custom processor design that certain practical issues arising from seemingly invariant elements of the design flow are not addressed:\n\\begin{enumerate}\n\\item[a)] {Assumptions of the intermediate representation (IR) to which the application code is mapped, directly affect solution quality as in the case of ISE synthesis.} \n\\item[b)] {The exploration infrastructure tied up to the conventions of software development tools.} \n\\item[c)] {Adaptability to different compilers\/simulators.}\n\\item[d)] {Support for low-level entry for application migration within a processor family and reverse engineering.}\n\\end{enumerate}\n\nIn this paper, all these issues are successfully addressed by integrating custom instruction (CI) generation and selection techniques with a flexible IR infrastructure that can reflect certain designer decisions that is cumbersome to apply otherwise. Our approach is substantiated in the form of the YARDstick prototype tool \\cite{Kavvadias07}. For example using an IR with intrinsic support for bit-level operations may yield significantly different ISEs to the case of an unaugmented IR. Also, in YARDstick it is possible to directly measure the effect of certain machine-dependent compiler transformations, such as register allocation, to the quality and impact of the generated ISEs, an issue recognized but never quantified in other works \\cite{ClarkN03,Castro04}. Further, YARDstick provides profiling facilities for determining static and dynamic application metrics such as data types, memory hierarchy statistics, and execution frequencies. Application entry can be either high-level (e.g. ANSI C) or low-level (assembly code for a target architecture or virtual machine). A number of recent custom functionality identification and selection techniques have been implemented while hardware estimators (speedup, area) and bindings to third-party tools for hardware synthesis from CDFGs are provided. \n\nIt is important to note that the interpretation of custom functionalities depends on the context; they can represent instruction-set extensions (ISEs) to a baseline ISA requiring to be accounted in the control path of the processor (decoding logic, extending the interrupt services), custom instructions of an ASIP enabled by a programmable controller or hardwired functions meant to be used as non-programmable hardware accelerators, loosely connected to the processor (i.e. accessible through the local bus).\n\n\\section{Related work}\n\\label{Sec:RelatedWork}\nLast years, a number of research efforts have regarded the automated application-specific \nextension of embedded processors \\cite{Alippi99,Yu04a,ClarkN05,Goodwin03,Pozzi06,Biswas07,Pothineni07}. \nA few open instruction generation frameworks exist \\cite{Pattlib}; an \nadvantage of their work being delivering a format for storing, manipulating and \nexchanging instruction patterns. In order to use their pattern library (Pattlib), the \npotential user should adapt his compiler for generating and manipulating patterns in \nthe cumbersome GCC RTL (Register Transfer Language) \\cite{GCC} intermediate representation. Some issues with the Pattlib approach regard the significant efforts for adapting the GCC compiler to emit information in ``pattlib'' format, and that the IR for their selected backend (SPARC V8) is not architecture-neutral and cannot be easily altered. \n \nApplication-specific instructions have been generated for the Xtensa configurable \nprocessor \\cite{Goodwin03} that may comprise of VLIW (Very Long Instruction Word), SIMD \n(Single-Instruction Multiple-Data) or fused (chained) RTL operations. However, as induced by the architecture template of Xtensa, control-transfer instructions ({\\it cti}) are not considered to be included in the resulting complex instructions. A sophisticated framework for the design of tightly-coupled custom coprocessing datapaths and their integration to existing processors has been presented in \\cite{ClarkN05}. While providing a complete solution to programmable acceleration, their work still has some drawbacks: the possibility \nof direct communication to fast local data memory is excluded and for this reason, \nbeneficial addressing modes cannot be identified. In \\cite{Atasu03,Pozzi06} a multi-output \ninstruction generation algorithm is presented which selects maximal-speedup convex \nsubgraphs for each basic block data-dependence graph (DDG), with worst case exponential \ncomplexity, while \\cite{Yu04a} added path profiling to extend beyond basic block scope.\nAn important conclusion was that useful instruction identification scope does not extend \nfurther than 2 or 3 consecutive basic blocks. Still, memory operations are not regarded \nin the formation of custom instructions, while pattern identification can only take \nplace post register allocation.\n\n\\section{YARDstick}\n\\label{Sec:YARDstick}\nThe main role of YARDstick is to facilitate design space exploration (DSE) in heterogeneous flows for ASIP design where the development tools (compiler, binary utilities, simulator\/ debugger) in many cases, lack DSE capabilities and\/or have been designed with different interfaces in mind. Thus, it is often that significant development effort is required in adding features as afterthoughts and dealing with interoperability issues, especially at the {\\it compiler} and {\\it simulator} boundaries.\n\n\\subsection{The YARDstick kernel}\n\\label{Sec:YARDstickKernel}\nThe current YARDstick infrastructure, as illustrated in Fig.~\\ref{Fig:1}, comprises of three kernel components ({\\it libByoX, libPatCUTE, libmachine}), the target architecture specification tools (the BXIR frontend) and a set of backends for exporting control-flow graphs, basic blocks and custom instructions for visualization, simulation and RTL synthesis purposes. {\\it libByoX} and {\\it libPatCUTE} are target-independent, and only {\\it libmachine} has to be retargeted for different IR specifications.\n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=8.0cm]{fig1-c.eps}\n \\caption{The YARDstick infrastructure.}\n \\label{Fig:1}\n \\vspace{-0.25cm}\n\\end{figure}\n\n\\subsubsection{{\\it libByoX}}\n\\label{Sec:libbyox}\n{\\it libByoX} implements the core YARDstick API and provides frontends\/manipulators for internal data structures. \nThe ByoX (Bring Your Own Compiler and Simulator) library provides:\n\\begin{itemize}\n\\item {The ISeqinfo parser for ISeq (historical name for ``Instruction Sequence'') entries. ISeq is a flat CDFG (with\/without SSA) format of application IR that is used for recording the data-dependence graphs for the application basic blocks.}\n\\item {The CFGinfo parser for control-flow graph (CFG) files that attribute the corresponding ISeq files with typed control-flow edges.}\n\\item {Simple file interface for the ISeq and CFG formats as well as for results of compiler analyses, e.g. control\/data flow analyses evaluating register liveness and natural loops, that can be passed to ByoX as defined by their corresponding BNFs.}\n\\item {An IR manipulation API for writing external analyses and optimizations.}\n\\item {Parameterization for a template machine context without inherent restrictions to its ISA.}\n\\end{itemize}\n\nIn ISeq, the following application information is recorded:\n\\begin{itemize}\n\\item\t{The global symbol table.}\n\\item {The procedure list, consisting of data dependence entries, the local symbol table and a statement list per procedure. It is possible to generate different facets of the local symbol table, e.g. single reference per direction (input or output) for each variable versus allowing multiple definition points for the same variable.}\n\\end{itemize}\n\n\\subsubsection{{\\it libPatCUTE}}\n\\label{Sec:libpatcute}\nFurther, a number of custom instruction generation\/selection methods have been implemented as part of the PatCUTE (Pattern-based Custom UniT Exploration) library. CI generation involves the identification of MIMO (Multiple-Input Multiple-Output) or MISO (Multiple-Input Single-Output) ISeq patterns under user-defined constraints. The CI generation methods available in {\\it libPatCUTE} are:\n\\begin{itemize}\n\\item {MAXMISO \\cite{Alippi99} for identifying maximal subgraphs with a single-output node using a linear complexity algorithm.}\n\\item {MISO exploration under constraints for the maximum number of input\/output operands, and for two types of operation node-related constraints \\cite{Kavvadias05}.} \n\\item {MIMO CI generation. In our case, we do not search for maximal MIMO patterns \\cite{Pothineni07}, however, we employ a fast heuristic by assuming similarly to \\cite{Pothineni07} that the performance gain provided by a pattern $P$ is higher than any pattern that is a subgraph of $P$. The user could disable the heuristic and apply an exponential complexity algorithm as well.}\n\\end{itemize}\n\nWhen CI generation is invoked, a CI list is constructed from the resulting ISeq patterns, which can be filtered via graph or graph-subgraph isomorphism tests \\cite{VFLib2} during the process of removing redundant cases. A subset of the library can be selected by using either a configurable greedy selector (supporting cycle-gain and cycle-gain per area priority metrics) or a 0-1 knapsack-based one. An important YARDstick characteristic is that CIs can be expressed in ISeq in the same way to either application CFGs or subregions thereof, thus existing data structures and analyses can be reused for further manipulation of the generated CIs. For example, pattern libraries can be imported to YARDstick.\n\n\\subsubsection{{\\it libmachine}}\n\\label{Sec:libmachine}\nThe {\\it libmachine} library is the only core YARDstick component that needs retargeting for a user-defined target architecture. Target architectures are specified in the BXIR (ByoX IR) \nformat which supports semantics for defining global-scope (data types, operation grouping) and operation-level information (operands, interpretation semantics for each IR operator, area\/latency cost for corresponding hardware implementations and cycle timings). \n\n\\subsubsection{Backend engines}\n\\label{Sec:backends}\nApplication CFGs, (basic blocks) BBs and patterns can be processed by a number of backends for exporting to:\n\\begin{itemize}\n\\item {ANSI C subset code for incorporation to user tools (simulators, validators etc).}\n\\item {GDL (VCG) \\cite{VCG} and dot (Graphviz) \\cite{Graphviz} files for visualization.}\n\\item {An extended CDFG \\cite{CDFGtool} format for scheduling and translation to synthesizable VHDL (applicable to BBs and CI patterns).}\n\\item {GGX XML \\cite{AGG} files for algebraic graph transformation.}\n\\end{itemize}\n\n\\subsection{Structure of a YARDstick environment}\n\\label{Sec:YARDstickStructure}\nThe YARDstick kernel can be utilized as an infrastructure for application analysis and exploration of custom functionality extensions. Fig.~\\ref{Fig:2} shows our YARDstick framework which reuses third-party compilation and simulation tools. The compiler frontend ({\\it gcc} is such an example) accepts input in C\/C++ or other high-level languages of interest. The application program is compiled to a low-level representation that can be represented by a form of ``assembly'' code after frontend processing, conversion to its internal IR, application of machine-independent optimizations and a set of compiler backend processes with only code selection being obligatory. The assembly-level code can then be macro-expanded, instrumented for profiling and converted to ISeq by an appropriate SALTO pass \\cite{SALTO}. This flow assumes that a working SALTO backend library has been ported for the target architecture. Assembly code can be assembled and linked by the target machine binary utilities ({\\it binutils} or equivalent tools) and the resulting ELF executables can be evaluated on an instruction- or cycle-accurate simulator. Alternatively, ISeq files can be generated as compiler IR dumps directly from the compiler for the target machine. This is the case for a modified version of Machine-SUIF \\cite{MachSUIF} for which the basic block profile is automatically obtained by converting the IR to a C subset and executing the low-level C code on a native machine.\n\nAt the simulation boundary, YARDstick expects information on the dynamic profile of the application (basic block execution frequencies, program trace, cache memory access statistics) on a target machine. From within YARDstick, static and dynamic application metrics can be evaluated and visualized. An application analyzer ({\\it iseqtool}) and CI generator ({\\it igensel}) linked to {\\it libByoX} and {\\it libPatCUTE} are used to obtain the application profile and custom instructions, respectively. \n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=8.0cm]{fig2-c.eps}\n \\caption{A high-level look to a YARDstick-based framework.}\n \\label{Fig:2}\n \\vspace{-0.25cm}\n\\end{figure}\n\n\\subsection{Usage of the YARDstick API}\n\\label{Sec:APIUsage}\nThe YARDstick API provides methods for manipulation of ISeq entities and extraction of useful information to internal data structures such as local operand lists, operation-level \nanalysis (e.g. finding zero-predecessor\/-successor instruction nodes) and application of backend processing. Fig.~\\ref{Fig:3} shows an example of API usage for updating the necessary data structures for basic-block-based CI generation. \n\nIn more detail, a basic block ISeq cluster is denoted by `bb'. First, `init\\_opnd\\_library' initializes an empty operand list container, named {\\it Lopnd} which is updated by calls to the `find\\_{\\it type}\\_opnds' functions, where {\\it type} denotes operand type and can be one of \\{input,output,cnst\\}. When the unique register operands and constants option is enabled, operands are treated as in SSA form and have a single representation per input and output sublist by applying `collapse\\_to\\_unique\\_opnds'. After clearing temporary storage for the best cut to be identified in the specific iteration of the CI generation algorithm, either a MIMO or a MISO-based method can be selected for performing the actual process.\n\n\\begin{figure}[tb]\n\\centering\n{\n{\\footnotesize\n\\begin{verbatim}\nvoid evaluate_bb_ci(ISeq bb)\n{\n ... \n \/\/ Setup operand list\n UIOCList Lopnd = init_opnd_library();\n\n \/\/ Find unique i\/o registers and constants\n find_input_opnds(bb, Lopnd, input_opnds);\n find_output_opnds(bb, Lopnd, output_opnds, \n instr_has_successor);\n find_cnst_opnds(bb, Lopnd, cnst_opnds);\n\n if (unique i\/o instances for operands\/constants)\n collapse_to_unique_opnds();\n \n clear_best_cut();\n \n \/\/ CI generation for the BB\n if (MIMO method)\n MIMO_identification(bb);\n else if (MaxMISO or constrained MISO method)\n MAXMISO_identification(bb);\n}\n\\end{verbatim}\n}\n}\n\\vspace{-0.125cm}\n\\caption{Updating internal data structures for BB-level CI generation.}\n\\label{Fig:3}\n\\end{figure}\n\n\\section{Case studies of design space exploration with YARDstick}\n\\label{Sec:DSE}\nFor proof-of-concept, we have evaluated YARDstick under various scenarios that reflect realistic problems in evaluating and exploring the design space when developing new ASIPs or enhancing customizable architectures. For the case studies we have used three different target architectures: \n\\begin{enumerate}\n\\item [1)] {The SUIFvm IR \\cite{MachSUIF} augmented by a set of incremental extensions to it, called {\\it SUIFvmenh}.}\n\\item [2)] {The {\\it SUIFrmenh} architecture (SUIF `real machine' enhanced) supported by an in-house backend written for Machine-SUIF, that introduces a finite register set of configurable size to {\\it SUIFvmenh}. {\\it SUIFrmenh} also resolves type casting (conversion) operations mapping them from the {\\it cvt} SUIFvm instructions to the proper instructions accepted by the {\\it SUIFrmenh} backend: zero- and sign-extend, truncation and {\\it mov} explicitly denoting the source and destination data types.}\n\\item [3)] {The DLX integer subset ({\\it iDLX}) for which the formatted assembly dumps are viewed as a kind of human-readable machine-level IR.}\n\\end{enumerate}\n\nThe target IR architectures are summarized in Table~\\ref{Tab:1}. In the experiments of the following subsections, all control transfer instructions ({\\it beqz, bnez, j, jr, jal, jalr}) were forbidden from CI pattern formation for the {\\it iDLX} IR, while for the SUIFvm-based IRs, branch operations were permitted. The two different forbidden instruction constraint sets were chosen in order to highlight distinct potential requirements and were not meant to be directly contrasted. The DLX-based IR would be a choice when the objective is to optimize pre-existing DLX legacy assembly (or binaries). Instead, {\\it SUIFvmenh} implements a representative RISC-like IR not restricting the processor template, which is more suitable for developing ASIPs from scratch. \n\n\\begin{table}\n \\renewcommand{\\arraystretch}{0.925}\n \\vspace{-0.25cm}\n \\caption{Different IR settings for CI generation.}\n \\centering\n {\\footnotesize\n \\begin{tabular}{|l|l|}\n \\hline\n \\multicolumn{1}{|m{1.5cm}|}{\\centering IR}\n &\\multicolumn{1}{m{5.0cm}|}{\\centering Operations}\\\\\n \\hline\n SUIFvmenh & SUIFvm plus: type conversion (sxt, zxt),\\\\\n & partial predication (select),\\\\ \n & bit manipulation (bitinsert, bitextract, concat) \\\\\n \\hline\n SUIFrmenh & SUIFvmenh with finite register set \\\\\n & (12, 16, 32 or 64 registers); here 32 is used \\\\\n \\hline\n iDLX & The DLX integer instruction set \\\\\n \\hline\n \\end{tabular}\n }\n \\label{Tab:1}\n\\end{table}\n\nFor the experiments we used applications from a set of embedded benchmarks consisting of 5 cryptographic ({\\it crc32, deraiden, enraiden, idea, sha}) and 5 media-oriented applications ({\\it adpcm\\_dec, adpcm\\_enc, fir, fsme, mc}) which are shown in Table~\\ref{Tab:2}. \n\n\\begin{table}\n \\centering\n \\renewcommand{\\arraystretch}{0.925}\n \\caption{Summary of examined benchmarks.} \n {\\footnotesize\n \\begin{tabular}{|l|l|}\n \\hline\n \\multicolumn{1}{|m{2.0cm}|}{\\centering Benchmark}\n &\\multicolumn{1}{m{5.0cm}|}{\\centering Description}\\\\\n \\hline\n {\\it crc32} & Cyclic redundancy check \\\\ \\hline\n {\\it deraiden} \\cite{Raiden} & Decoding raiden cipher \\\\ \\hline\n {\\it enraiden} \\cite{Raiden} & Encoding raiden cipher \\\\ \\hline \n {\\it idea} & IDEA cryptographic kernel \\\\ \\hline\n {\\it sha} & Secure Hash Algorithm producing an 160-bit \\\\ \n & message digest for a given input \\\\ \\hline\n {\\it adpcmdec} & Adaptive Differential Pulse Code Modulation \\\\ \n & (ADPCM) decoder \\\\ \\hline\n {\\it adpcmenc} & Adaptive Differential Pulse Code Modulation \\\\ \n & (ADPCM) encoder \\\\ \\hline\n {\\it fir} & FIR filter \\\\ \\hline\n {\\it fsme} & Full-search motion estimation \\\\ \\hline\n {\\it mc} & Motion compensation \\\\ \\hline\n \\end{tabular}\n }\n \\label{Tab:2}\n\\end{table}\n\n\\subsection{Effect of compilation specifics: Case study of media processing kernels}\n\\label{Sec:CaseStudy}\nIt has been argued recently \\cite{Bonzini06} that traditional compiler transformations and the trivial solution of applying CI identification at the end of the optimization phase pipeline do not necessarily yield the best performance when targeting a custom processor. On the contrary, source code and IR-level transformations have to be especially tuned for exposing beneficial application-specific hardware extensions. \n\nIn this subsection, the effect of the choice in compilation specifics is highlighted for popular case study applications: the ADPCM codec, an FIR filter, and typical implementations of motion estimation\/compensation. We investigate specific effects that the compiler imposes when used for exploring the potential for custom instructions:\n\\begin {enumerate}\n\\item[a)] {The effect of register allocation on the quality of the generated CIs. For this purpose, we have targeted the {\\it SUIFrmenh} backend. A 32-entry register file was assumed while the procedure calling convention for {\\it SUIFrmenh} was the same to an in-house GCC-based DLX backend.}\n\\item[b)]\t{The suitability of using a highly-optimizing (but aimed to general-purpose processors) compiler such as {\\it gcc} targeted to DLX which is our case, against a well-known research compiler ({\\it MachSUIF} targeted to SUIFvm) which has been extensively used for exploring the transformation space for new CIs.}\n\\end{enumerate}\n\nFor accounting only the true data dependencies amongst operations, it is necessary to remove all false dependencies. This can be achieved by a simple IR-level transformation pass (for example, such pass was implemented in the Machine-SUIF compiler for the {\\it SUIFvmenh} target) which involves the use of the pseudocode of Fig.~\\ref{Fig:4}. The algorithm in Fig.~\\ref{Fig:4} can be used for an in-order instruction schedule, i.e. no backward data dependence edges exist within a basic block. For a given set of dependence edges $\\bigcup\\{(i \\rightarrow j)_k\\}$ between instructions $mi(i),mi(j)$ of instruction IDs $i,j$ respectively, the range $[i,j]$ is considered. The destination operands of machine instructions in range are iterated and compared to the operand ($opnd$) for which we want to remove all false dependencies with it as the data dependency. If $opnd$ is written at least once, a false dependency is identified (marked as $TRUE$) and the corresponding data dependence edge is annulled.\n\n\\begin{figure}[tb]\n\\centering\n{\n{\\footnotesize\n\\begin{verbatim}\nboolean is_false_dependency(BB* bb, InstrID mi_lpos, \n mi_hpos, LOpnd opnd)\n{\n boolean false_dependency_f = FALSE;\n ...\n \/\/ Iterate through the [mi_lpos..mi_hpos] range\n foreach machine instruction (mi) in range do \n if the current mi is within the specified range\n get destination operand dstop of mi \n if dstop is ((a base register or address symbol) \n and writes memory)\n if dstop is equal to opnd\n \/\/ a false dependency has been found\n false_dependency_f |= TRUE;\n fi\n fi\n fi\n od\n \n return false_dependency_f;\n}\n\\end{verbatim}\n}\n}\n\\vspace{-0.125cm}\n\\caption{Removing false data dependence edges from basic blocks.}\n\\label{Fig:4}\n\\end{figure}\n\nApplication speedups obtained prior and post register allocation (the latter indicated by a `ra' suffix to the benchmark name) are shown in Fig.~\\ref{Fig:5}. In contrast to common belief \\cite{ClarkN03,Castro04}, the introduction of a finite register set and the mapping of the instruction selection temporaries to this set, does not always have a negative impact on the evaluated speedups. While it is clear that there is a measurable effect (an overhead of 22.15\\%) due to register allocation for a single output ($N_{o}=1$), the extent of this overhead is reduced for larger number of register outputs. Thus, for $N_{o}=\\{2,4,\\infty\\}$ the corresponding overheads have been calculated as 17\\%, 2.5\\% and -21.3\\%, the latter meaning that the register allocated IR enables higher speedups compared to obtaining the IR prior register allocation for the constraint of unlimited number of register outputs. This important outcome infers that the overhead of spills and fills occuring due to register pressure, can be efficiently hidden when multi-output (MIMO) instructions are used for the estimations. In addition to that, CIs have the side-effect of eliminating the need for certain temporary variables within a CI pattern, given that they need not be alive outside the pattern. \n\n\\begin{figure}[tb]\n \\SetFigLayout{2}{1}\n \\centering\n \\subfigure[$N_{i}$ = 4]{\n \\includegraphics[width=8.0cm]{fig5a.eps}\n \\label{Fig:5:a}}\n \\subfigure[$N_{i}$ = 8]{\n \\includegraphics[width=8.0cm]{fig5b.eps}\n \\label{Fig:5:b}}\n \\caption{Effect of register allocation on application speedup for different number of input\/output register operands.}\n \\label{Fig:5}\n \\vspace{-0.5cm}\n\\end{figure}\n\nFig.~\\ref{Fig:6} shows the results on relative application speedups for different number of input\/output register operands for the two selected compiler targets. An unlimited number of inputs was also set but the corresponding results where within 0.4\\% of the $N_{i}=8$ case.\nThe difference in the average speedup achieved for the given numbers of inputs for the same application is about 44\\% (ranging from 20\\% to 61\\%). This is partially due to the fact that stack argument allocation applied for {\\it iDLX} only, adds memory access operations for saving and restoring function arguments that are not usually included in new CIs. Even when MIMO instructions are identified incorporating the callee saved sequence (a series of {\\it sw} instructions), the obtained speedups are severely limited by the data memory bandwidth assumed in the estimations which is 1R\/1W port for all target architectures.\n\n\\begin{figure}[tb]\n \\SetFigLayout{2}{1}\n \\centering\n \\subfigure[$N_{i}$ = 4]{\n \\includegraphics[width=8.0cm]{fig6a.eps}\n \\label{Fig:6:a}}\n \\subfigure[$N_{i}$ = 8]{\n \\includegraphics[width=8.0cm]{fig6b.eps}\n \\label{Fig:6:b}}\n \\caption{Application speedup for different number of input\/output register operands on {\\it SUIFvmenh} and {\\it iDLX}.}\n \\label{Fig:6}\n \\vspace{-0.125cm}\n\\end{figure}\n\n\\subsection{Transformation to more suitable IRs}\n\\label{Sec:IRTransformations}\nAlthough not explicitly stated in related works, the effect of IR selection significantly affects the quality of the CI generation results. In YARDstick, GGX XML graph representations of ISeq patterns can be automatically generated and then transformed by hand-written AGG rules to use different IR operators for implementing equivalent functionality.\n\nMost compilers (one exception is the commercial CoSy \\cite{ACE}) do not account for bit-level manipulations that are desirable in application domains such as network processing and genetic algorithms (GAs). To highlight this issue we have defined three custom IR operators, namely {\\it bitinsert, bitextract} and {\\it concat} with the semantics of Table~\\ref{Tab:3}. As motivational examples, we have used the well-known single- ({\\it crcsp}) and double-point ({\\it crcdp}) crossover operators, which are encountered in typical GAs such as the SGA \\cite{Goldberg89}. It should be noted that the ANSI C implementations of the crossover operators where hand-tuned, with optimizations including the conversion of all function calls inside the {\\it crcsp} and {\\it crcdp} functions to macro-inclusions. Fig.~\\ref{Fig:7} shows the result of applying a rule-based transformation in AGG \\cite{AGG} for replacing a {\\it SUIFvmenh} IR segment (Fig.~\\ref{Fig:7:a}) with a use of the {\\it bitextract} IR operator as seen in the resulting graph (Fig.~\\ref{Fig:7:b}. To highlight the importance of the right choice of compiler IR, Fig.~\\ref{Fig:8} visualizes the VCG representations of the custom instruction generated for the {\\it crcsp} genetic operator, without (Fig.~\\ref{Fig:8:a}) and with the use of the bit-level IR operators (Fig.~\\ref{Fig:8:b}).\n\n\\begin{table}\n \\renewcommand{\\arraystretch}{0.95}\n \\caption{Custom IR operators improving bit-level compiler support. $r_{d}$, $r_{s}$ are register operands, {\\it hpos,lpos} denote a bit range, and {\\it n} is the number of arguments for a variadic operator.} \n \\centering\n {\\footnotesize\n \\begin{tabular}{|l|c|}\n \\hline\n \\multicolumn{1}{|m{2.0cm}|}{\\centering Operator}\n &\\multicolumn{1}{m{5.0cm}|}{\\centering Semantics}\\\\\n \\hline\n {\\it bitinsert} & $r_{d}[lpos..hpos] \\Leftarrow r_{s}$ \\\\ \n {\\it bitextract} & $r_{d} \\Leftarrow r_{s}[lpos..hpos]$ \\\\ \n {\\it concat} & $r_{d} \\Leftarrow r_{s(0)} \\& r_{s(1)} \\& \\ldots \\& r_{s(n-1)}$ \\\\\n \\hline\n \\end{tabular}\n }\n \\label{Tab:3}\n \\vspace{-0.125cm}\n\\end{table}\n\n\\begin{figure}[tb]\n \\SetFigLayout{2}{1}\n \\centering\n \\subfigure[Visualization of an example host {\\it SUIFvmenh} IR graph.]{\n \\includegraphics[width=8.0cm]{fig7a.eps}\n \\label{Fig:7:a}}\n \\subfigure[The resulting graph after the application of a transformation rule for `bitextract'.]{\n \\includegraphics[width=8.0cm]{fig7b.eps}\n \\label{Fig:7:b}}\n \\caption{An example of IR graph rewriting via AGG transformation rules.}\n \\label{Fig:7}\n \\vspace{-0.125cm}\n\\end{figure}\n\n\\begin{figure}[tb]\n \\SetFigLayout{2}{1}\n \\centering\n \\subfigure[The {\\it crcsp}-induced CI without the use of bit-level operators.]{\n \\includegraphics[width=8.0cm]{fig8a-c.eps}\n \\label{Fig:8:a}}\n \\subfigure[The {\\it crcsp}-induced CI using bit-level operators.]{\n \\includegraphics[width=8.0cm]{fig8b-c.eps}\n \\label{Fig:8:b}}\n \\caption{Visualization of the {\\it crcsp} genetic operator CI for different compiler IRs.}\n \\label{Fig:8}\n \\vspace{-0.125cm}\n\\end{figure}\n\nThe performance gains for the generated hardware depend heavily on the target IR used for mapping the application code as can be clearly seen by the results of Table~\\ref{Tab:4}. In Table~\\ref{Tab:4}, the first three columns are self-explanatory. Column `Cycles...' shows the cycles required for a sequential schedule of the corresponding GA operator assuming the usage of the generated CIs. The last two columns indicate the number of cycles and area of the CI. The area requirement is calculated relatively to the area (multiplier area unit or MAU) of a 32-bit single-cycle multiplier producing a 64-bit result.\n\nFor computing schedules with unlimited resources, the generated ISeq files of the custom instructions were automatically converted with YARDstick to CDFGs compatible with an extended version of the CDFG toolset \\cite{CDFGtool} and processed by an ASAP scheduler. If the bit-level operators are not used, the minimum number of cycles required for the {\\it crcsp} operator are 76 for a sequential schedule and 12 for scheduling with unlimited resources, while for the {\\it crcdp} these limits are 111 and 14, respectively. When the bit-level operators are used, the sequential schedules prior to the inclusion of custom instructions require 13 and 18 cycles for {\\it crcsp} and {\\it crcdp} respectively with an ASAP schedule of 5 cycles for both. In the latter case, a single-cycle MIMO custom instruction is identified for each genetic operator when a $N_{i}\/N_{o}=\\{8\/2\\}$ constraint is used with impressive area benefits as well. \n\n\\begin{table}\n \\renewcommand{\\arraystretch}{0.975}\n \\caption{CI characteristics for hand-optimized ANSI C implementations of {\\it crcsp} and {\\it crcdp}.} \n \\centering\n {\\footnotesize\n \\begin{tabular}{|l|l|r|r|r|r|}\n \\hline\n \\multicolumn{1}{|m{1.0cm}|}{\\centering \\footnotesize GA operator}\n &\\multicolumn{1}{m{1.0cm}|}{\\centering \\footnotesize Bit-level operations}\n &\\multicolumn{1}{m{1.25cm}|}{\\centering \\footnotesize CI gen. constraints \\\\ $N_{i}\/N_{o}$}\n &\\multicolumn{1}{m{1.25cm}|}{\\centering \\footnotesize Cycles \\\\ (seq. schedule)}\n &\\multicolumn{1}{m{0.75cm}|}{\\centering \\footnotesize CI cycles}\n &\\multicolumn{1}{m{1.0cm}|}{\\centering \\footnotesize CI area (MAU)}\\\\\n \\hline\n \n {\\it crcsp} & No & 4\/1 & 76 & -- & -- \\\\ \n {\\it crcsp} & No & 8\/1 & 41 & 3 & 0.977 \\\\ \n {\\it crcsp} & No & 8\/2 & 5 & 3 & 1.867 \\\\ \n \\hline\n \n {\\it crcsp} & Yes & 4\/1 & 13 & -- & -- \\\\ \n {\\it crcsp} & Yes & 8\/1 & 6 & 1 & 0.142 \\\\ \n {\\it crcsp} & Yes & 8\/2 & 1 & 1 & 0.153 \\\\ \n \\hline\n \n {\\it crcdp} & No & 4\/1 & 111 & -- & -- \\\\ \n {\\it crcdp} & No & 8\/1 & 58 & 3 & 1.466 \\\\ \n {\\it crcdp} & No & 8\/2 & 5 & 3 & 2.800 \\\\ \n \\hline\n \n {\\it crcdp} & Yes & 4\/1 & 18 & -- & -- \\\\ \n {\\it crcdp} & Yes & 8\/1 & 8 & 1 & 0.147 \\\\ \n {\\it crcdp} & Yes & 8\/2 & 1 & 1 & 0.164 \\\\ \n \\hline\n \\end{tabular}\n }\n \\label{Tab:4}\n \\vspace{-0.25cm}\n\\end{table}\n\n\\subsection{Effect of data memory access model}\n\\label{Sec:MemoryAccessModel}\nThe extent and scope of using custom instructions is constrained by the data bandwidth to the data memory unit and local storage (register file) as defined by the number of input\/output ports and the resolution of dependencies for load\/store operations. In certain approaches \\cite{ClarkN05,Leupers06} which deal with predefined architectures such as the MIPS CorExtend and ARM OptimoDE systems, this limitation imposes a definitive factor. However, for exploration purposes when developing an ASIP from scratch, it is useful to consider different storage consistency models. Following the notation introduced in \\cite{Biswas07} for state consistency between application-specific functional units (AFUs) with local storage and data memory, it is possible to consider two such models in YARDstick:\n\\begin{itemize}\n\\item {{\\it Consistent data memory}, where the AFU directly accesses data in the on-chip data memory and there is no need for local AFU storage. We also make the conservative assumption that loads and stores ought always be serialized.}\n\\item {{\\it Ideal consistent AFU memory}, where each load\/store to main memory is transformed to an access to local AFU memory. We assume that data memory status is updated by DMA accesses occuring in parallel to processor instructions.}\n\\end{itemize} \n\nTo investigate the effect of memory model choice on application speedup due to CIs we first generated CIs without allowing memory inclusion (``noMEM''), then allowed local memory and performed estimations for the consistent data memory model (``CDM'') and subsequently we assumed an ideal consistent AFU memory (``idealCAM''). The corresponding results are illustrated in Fig.~\\ref{Fig:9} for indicative $N_{i}\/N_{o}$ combinations and for a single-issue processor. \n\n\\begin{figure}[tb]\n \\SetFigLayout{2}{1}\n \\centering\n \\subfigure[$N_{i}\/N_{o}$ = 4\/2]{\n \\includegraphics[width=8.0cm]{fig9a.eps}\n \\label{Fig:9:a}}\n \\subfigure[$N_{i}\/N_{o}$ = 8\/4]{\n \\includegraphics[width=8.0cm]{fig9b.eps}\n \\label{Fig:9:b}}\n \\caption{Effect of data memory accesses to the speedup induced by CIs. Accesses to data memory are assumed to require a single clock cycle overhead.}\n \\label{Fig:9}\n \\vspace{-0.25cm}\n\\end{figure}\n\nAs can be seen by the collected results, the inclusion of data memory access operations in CIs has a significant positive impact in the attained speedups: from 15.5\\% to 33.3\\% for the given input\/output constraints. Especially for the {\\it SUIFvmenh} target, the speedup improvements were up to 43.4\\%. Another important observation is that the consistent AFU memory model has a limited effect with improvements of up to 6.3\\% in average and 8.9\\% for the {\\it SUIFvmenh} applications alone. However, for a larger cycle overhead to accessing data storage, the speedup improvements are more considerable. For another exploration example, we have estimated that 2- and 5-cycle load\/store accesses to a data memory module through the local bus (an address cycle followed by either one word data access or consecutive byte data access cycles) result in higher speedups. More specifally, the ``CDM'' case performs better to ``noMEM'' by 34.7\\% and 49.9\\%, respectively for the 2- and 5-cycle overheads. When comparing the two different models that allow memory accesses to be part of CIs, the corresponding values are 7\\% and 20.9\\% in favor of ``idealCAM'' for the given cycle overheads. \n\n\\subsection{Greedy CI selection under priority metrics}\n\\label{Sec:GreedySelection}\nFor implementing a greedy CI selector, the key idea is to assign priorities to the CI patterns and the more proficient instances are chosen by starting with the highest prioritized one. We have used the following two priority functions: \n\n\\begin{equation}\n\\label{Eq:1}\n\\text{Cycle gain}: Priority(\\sum_{j} C_{i,j}) = \\sum_{j} \\{P_{i,j} \\times f_{i,j}\\}\n\\end{equation}\n\nthat forces for best performance regardless AFU area requirements and: \n\n\\begin{equation}\n\\label{Eq:2}\n\\text{Cycle gain\/Area}: Priority(\\sum_{j} C_{i,j}) = \\sum_{j} \\{(P_{i,j} \\times f_{i,j})\\}\/A_{i}\n\\end{equation}\n\nwhere $C_{i,j}$ denotes the $i$-th candidate instruction with $j$ different instances in the entire program, $f_{i,j}$ the basic block execution frequency metric associated with the specific instance, and $A_{i}$ the area cost for the candidate. These priority functions force different objectives: equation~\\ref{Eq:1} maximizes performance gain for each isomorphic candidate CI over the entire program when area is not an issue while equation~\\ref{Eq:2} quantifies the available area budget as well.\n\nA summary of the measurements for the application set is given in Table~\\ref{Tab:5}. \nTaking {\\it sha.dlx} for example, although tens of candidate instructions are identified, \nonly a few (7 for achieving 95\\% of the maximum speedup compared to 20 for achieving totality) contribute significantly to the execution time for either priority function. The \nnumber of required extensions for reaching the 95\\% speedup levels ranges from 2 ({\\it fir.dlx}) to 40 ({\\it idea.dlx}), while the area requirement is less than 3.4 multiples of the area of a 32-bit single-cycle multiplier for all applications with the exception of {\\it idea.dlx} which demands up to 10.23 MAU. \n\nFinally, Fig.~\\ref{Fig:10} compares the pros and cons for the priority functions \nused in the custom instruction selection process for the {\\it sha.dlx} application example. For the {\\it sha} application, CI selection under the `Cycle gain' priority function reaches the 95\\% of the maximum speedup for one instruction less and a slight area increase (0.04 MAU) compared to `Cycle gain\/Area'.\n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width=8.0cm]{fig10.eps}\n \\caption{Custom instruction selection under priority metrics for {\\it sha}\n ($N_{i}\/N_{o}=\\infty\/\\infty$).}\n \\label{Fig:10}\n \\vspace{-0.175cm}\n\\end{figure}\n\n\\begin{table}\n \\renewcommand{\\arraystretch}{1.0}\n \\caption{Speedup-AFU area for `Cycle gain'\/`Cycle gain\/Area' \n for the input\/output constraint $N_{i}\/N_{o}=\\{8\/4\\}$.} \n \\centering\n {\\footnotesize\n \\begin{tabular}{|l|r|r|r|r|}\n \\hline\n \\multicolumn{1}{|m{1.2cm}|}{\\centering Benchmark}\n &\\multicolumn{1}{m{1.0cm}|}{\\centering 0.95$\\times$ max. speedup}\n &\\multicolumn{1}{m{1.2cm}|}{\\centering Area (MAU)}\n &\\multicolumn{1}{m{1.0cm}|}{\\centering At max. speedup}\n &\\multicolumn{1}{m{1.2cm}|}{\\centering Area (MAU)}\\\\\n \\hline\n \n {\\it adpcmdec} & 4\/4 & 0.895\/0.895 & 6 & 0.983 \\\\ \n \\hline\n \n {\\it adpcmdec.dlx} & 11\/11 & 1.123\/1.123 & 17 & 1.721 \\\\ \n \\hline\n \n {\\it adpcmenc} & 4\/4 & 0.998\/0.998 & 6 & 1.086 \\\\\n \\hline\n \n {\\it adpcmenc.dlx} & 16\/16 & 1.475\/1.375 & 22 & 2.074 \\\\\n \\hline\n \n {\\it crc32.dlx} & 3\/3 & 0.12\/0.12 & 3 & 0.12 \\\\\n \\hline\n \n {\\it deraiden} & 4\/4 & 2.657\/2.657 & 4 & 2.657 \\\\\n \\hline\n \n {\\it enraiden} & 3\/3 & 1.949\/1.949 & 3 & 1.949 \\\\\n \\hline\n \n {\\it fir} & 4\/4 & 1.398\/1.398 & 5 & 1.398 \\\\\n \\hline\n \n {\\it fir.dlx} & 2\/2 & 0.155\/0.155 & 2 & 0.155 \\\\\n \\hline\n \n {\\it fsme.dlx} & 9\/9 & 1.143\/1.143 & 11 & 1.546 \\\\\n \\hline\n \n {\\it fsme.dlx} & 6\/6 & 1.03\/1.03 & 10 & 1.65 \\\\\n \\hline\n \n {\\it idea.dlx} & 40\/50 & 10.23\/9.325 & 69 & 13.002 \\\\\n \\hline\n \n {\\it mc} & 5\/5 & 1.824\/1.824 & 7 & 2.53 \\\\\n \\hline\n \n {\\it mc.dlx} & 7\/7 & 1.489\/1.489 & 12 & 2.516 \\\\\n \\hline\n \n {\\it sha.dlx} & 7\/7 & 1.671\/1.671 & 20 & 3.378 \\\\\n \\hline\n \\end{tabular}\n }\n \\label{Tab:5}\n \\vspace{-0.25cm}\n\\end{table}\n\n\\section{Usage environment}\n\\label{Sec:Usage}\nYARDstick has been used along with the SUIF\/Machine-SUIF \\cite{MachSUIF}, GCC \\cite{GCC}, and COINS \\cite{COINS} compilers and the ArchC \\cite{ArchC} simulation framework. Functional and cycle-accurate simulators generated by version 1.5.1 of ArchC can be used with YARDstick without any modifications. Most of the YARDstick functionality is also accessible through a cross-platform GUI \\cite{Kavvadias07} compatible to recent Tcl\/Tk versions (8.5.a5 and newer). \n\nSupported platforms include GNU\/Linux (RedHat 9.0), Cygwin and Win32 (Windows\/XP SP2) on x86-compatible processors.\n\n\\section{Conclusions}\n\\label{Sec:Conclusions}\nYARDstick is a retargetable application analysis and custom instruction generation\/selection environment providing a compiler-\/simulator-agnostic infrastructure. YARDstick aims in separating design space exploration from compiler\/simulator idiosyncrasies. Different compilers\/simulators can be plugged-in via file-based interfaces; further, both high- (e.g. ANSI C) and low-level (assembly for an architecture or a virtual machine) input can be analyzed by the infrastructure.\n\nIn order to prove the applicability and usefulness of YARDstick in ASIP development, we have evaluated a variety of exploration scenarios on a benchmark set consisting of well-known embedded applications and kernels. In this context, we have investigated effects of the compilation process, such as the selection of the target IR and the impact of register allocation, on the characteristics of the identified hardware extensions. Also, different memory models involving local storage for application-specific functional units were examined and quantified, and for the entire set of applications, custom instructions were generated under different input\/output constraints.\n\n\n\n\n\\nocite{*}\n\n\\bibliographystyle{compj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{PSO Introduction}\n\nParticle Swarm Optimisation (PSO,~\\cite{kennedy1995particle}) is a metaheuristic algorithm \nwhich is widely used to solve search and optimisation tasks. \nIt employs a number of particles as a swarm of potential solutions.\nEach particles shares knowledge about the current overall best solution and also retains a memory of \nthe best solution it has encountered itself previously. Otherwise the particles, after random initialisation,\nobey a linear dynamics of the following form\n\\begin{eqnarray}\n\\label{eq:PSOVelocityUpdate}\n\\mathbf{v}_{i,t+1}&=&\\omega\\mathbf{v}_{i,t}+\\alpha_{2} \\mathbf{R}_{1}(\\mathbf{p}_{i}-\\mathbf{x}_{i,t})+\\alpha_{2} \\mathbf{R}_{2}(\\mathbf{g}-\\mathbf{x}_{i,t})\n\\cr\n\\mathbf{x}_{i,t+1}&=&\\mathbf{x}_{i,t}+\\mathbf{v}_{i,t+1}\n\\label{eq:PSOPositionUpdate}\n\\end{eqnarray}\nHere $\\mathbf{x}_{i,t}$ and $\\mathbf{v}_{i,t}$, $i=1,\\dots,N$, $t=0,1,2,\\dots$,\nrepresent, respectively, the $d$-dimensional position in the search space and the velocity \nvector of the $i$-th particle in the swarm at time $t$. \nThe velocity update\ncontains an inertial term parameterised by $\\omega$ and includes attractive forces \ntowards the personal best location $\\mathbf{p}_{i}$ and towards \nthe globally\nbest location $\\mathbf{g}$, which are parameterised by $\\alpha_{1}$ and\nand $\\alpha_{2}$, respectively. \nThe symbols $\\mathbf{R}_{1}$ and $\\mathbf{R}_{2}$ denote diagonal matrices whose non-zero entries \nare uniformly distributed in the unit interval. The number of particles $N$ is quite low in \nmost applications, usually amounting to a few dozens.\n\nIn order to function as an optimiser, the algorithm uses a nonnegative cost function \n$F:\\mathbb{R}^d \\to \\mathbb{R}$, where without loss of generality $F(\\mathbf{x}^*)=0$ \nis assumed at an optimal solution $\\mathbf{x}^*$. In many problems, where PSO is applied, there are also states with near-zero costs can be considered as good solutions.\nThe cost function is evaluated for the state of each particle at each time step. \nIf $F(\\mathbf{x}_{i,t})$ is better than $F(\\mathbf{p}_{i})$, then the personal\nbest $\\mathbf{p}_{i}$ is replaced by $\\mathbf{x}_{i,t}$. \nSimilarly, if one of the particles arrives at a state\nwith a cost less than $F(\\mathbf{g})$, then $\\mathbf{g}$ is replaced in all particles by the \nposition of the particle that has discovered the new solution. \nIf its velocity is non-zero, a particle will depart from the current best location,\nbut it may still have a chance to return guided by the force terms in the dynamics.\n\nNumerous modifications and variants have been \nproposed since the algorithm's inception \\cite{kennedy1995particle} and it continues to enjoy \nwidespread usage. Ref.~\\cite{poli2008analysis} groups around 700 PSO papers into 26 discernible\napplication areas. Google Scholar reveals over 150,000 results for ``Particle Swarm Optimisation''\nin total and 24,000 for the year 2014.\n\nIn the next section we will report observations from a simulation of a particle swarm and move on to\na standard matrix formulation of the swarm dynamics in order to describe some of the existing analytical \nwork on PSO. In Sect.~\\ref{Critical} we will argue for a formulation of PSO as a random dynamical system\nwhich will enable us to derive a novel exact characterisation of the dynamics of one-particle system, \nwhich will then be generalised towards the more realistic case of a multi-particle swarm.\nIn Sect.~\\ref{Simulations} we will compare the theoretical predictions with simulations on\na representative set of benchmark functions. Finally, in Sect.~\\ref{discussion} we will discuss the\nassumption we have made in the theoretical solution in Sect.~\\ref{Critical} and address the applicability\nof our results to other metaheuristic algorithms and to practical optimisation problems.\n\n\\section{Swarm dynamics}\n\n\\subsection{Empirical properties}\n\nThe success of the algorithm in locating good solutions depends on the dynamics of the particles \nin the state space of the problem. In contrast to many evolution strategies, it is not straight \nforward to interpret the particle swarm as following a landscape defined by the cost function. \nUnless the current best\npositions $\\mathbf{p}$ or $\\mathbf{g}$ change, the particles do not interact with each other and \nfollow an intrinsic dynamics that does not even indirectly obtain any gradient information.\n\nThe particle dynamics depends on the parameterisation of the Eq.~\\ref{eq:PSOVelocityUpdate}.\nTo obtain the best result one needs to select parameter settings that \nachieve a balance between the particles exploiting the knowledge of good known locations and exploring regions of the problem space that have not been visited before. Parameter values often need to be experimentally determined, and poor selection may result in premature convergence of the swarm to poor local minima or in a divergence of the particles towards regions that are irrelevant for the problem.\n\nEmpirically we can execute PSO against a variety of problem functions with a range of $\\omega$ and \n$\\alpha_{1,2}$ values. Typically the algorithm shows performance of the form depicted \nin Fig.~\\ref{fig:PSOPerformance}. The best solutions found show a curved relationship \nbetween $\\omega$ and $\\alpha=\\alpha_1+\\alpha_2$, with $\\omega\\approx 1$ at small $\\alpha$, and\n$\\alpha\\gtrapprox 4$ at small $\\omega$. \nLarge values of both $\\alpha$ and $\\omega$ are found \nto cause the particles to diverge leading to results far from optimality, while at small values for both\nparameters the particles converge to a nearby solution which sometimes is acceptable. \nFor other cost functions similar relationships are observed in numerical tests (see Sect.~\\ref{Simulations})\nunless no good solutions found due to problem complexity or run time limits, see Sect.~\\ref{runtime}.\nFor simple cost functions, such as a single well potential, there are also parameter combinations with \nsmall $\\omega$ and small $\\alpha$ will usually lead to good results.\nThe choice of $\\alpha_1$ and $\\alpha_2$ at constant $\\alpha$ may\nhave an effect for some cost functions, but does not seem to have a big effect in most cases.\n\n\\begin{figure}[th]\n\\noindent \\begin{centering}\n\\includegraphics[width=0.75\\linewidth]{fn13_2000_valley_F}\n\\par\\end{centering}\n\\vspace{-5mm}\n\\caption{Typical PSO performance as a function of its $\\omega$ and $\\alpha$ parameters. \nHere a 25 particle swarm was run for pairs of $\\omega$ and $\\alpha$ values ($\\alpha_1=\\alpha_2=\\alpha\/2$). \nCost function here was the $d=10$ non-continuous rotated Rastrigin function~\\cite{CEC2013}. \nEach parameter pair was repeated 25 times and the minimal costs after 2000 iterations were \naveraged. \\label{fig:PSOPerformance}}\n\\end{figure}\n\n\\subsection{Matrix formulation}\n\nIn order to analyse the behaviour of the algorithm it is convenient to use a matrix formulation \nby inserting the velocity explicitly in the second equation (\\ref{eq:PSOPositionUpdate}).\n\\begin{equation}\n\\mathbf{z}_{t+1}=M \\mathbf{z}_t + \\alpha_1 \\mathbf{R}_1 (\\mathbf{p},\\mathbf{p})^{\\top}+\n\\alpha_2 \\mathbf{R}_2 (\\mathbf{g},\\mathbf{g})^{\\top} \\label{eq:matrix_from}\n\\end{equation}\nwith $\\mathbf{z}=(\\mathbf{v},\\mathbf{x})^{\\top}$ and \n\\begin{equation}\nM=\\left(\\begin{array}{cc} \n\t\t\t\\omega \\mathbf{I}_d & -\\alpha_1 \\mathbf{R}_1- \\alpha_2 \\mathbf{R}_2 \\\\\n\t\t\t\\omega \\mathbf{I}_d & \\mathbf{I}_d-\\alpha_1 \\mathbf{R}_1- \\alpha_2 \\mathbf{R}_2 \n\t\\end{array}\\right),\n\\label{matrix}\n\\end{equation}\nwhere $\\mathbf{I}_d$ is the unit matrix in $d$ dimensions. Note that the two occurrence of $\\mathbf{R}_1$ in \nEq.~\\ref{matrix} refer to the same realisation of the random variable. Similarly, the two $\\mathbf{R}_2$'s are \nthe same realisation, but different from $\\mathbf{R}_1$.\nSince the second and third term on the right in Eq.~\\ref{eq:matrix_from} are \nconstant most of the time, the analysis of the algorithm can focus on the properties of the matrix $M$.\nIn spite of its wide applicability, PSO has not been subject to deeper theoretical study, which may\nbe due to the multiplicative noise in the simple quasi-linear, quasi-decoupled dynamics. In previous\nstudies the effect of the noise has largely been ignored.\n\n\\subsection{Analytical results}\n\nAn early exploration of the PSO dynamics~\\cite{kennedy1998behavior} considered a single particle in \na one-dimension space where the personal and global best locations were taken to be the same. \nThe random components were replaced by their averages such that apart from random initialisation\nthe algorithm was deterministic. \nVarying the parameters was shown to result \nin a range of periodic motions and divergent behaviour for the case of $\\alpha_1+\\alpha_2\\ge4$. \nThe addition of the random vectors was seen as beneficial as it adds noise to the deterministic search.\n\nControl of velocity, not requiring the enforcement of an arbitrary maximum value as in \nRef.~\\cite{kennedy1998behavior}, is derived in an analytical manner by~\\cite{clerc2002particle}. \nHere eigenvalues derived from the dynamic matrix of a simplified version of the PSO algorithm \nare used to imply various search behaviours. Thus, again the $\\alpha_1+\\alpha_2\\ge4$ case is\nexpected to diverge. For $\\alpha_1+\\alpha_2<4$ various cyclic and quasi-cyclic motions are shown \nto exist for a non-random version of the algorithm.\n\nIn Ref.~\\cite{trelea2003particle} again a single particle was considered in a one dimensional \nproblem space, using a deterministic version of PSO, setting $\\mathbf{R}_{1}=\\mathbf{R}_{2}=0.5$. \nThe eigenvalues of the system were determined as functions of $\\omega$ and a combined $\\alpha$, which\nleads to three conditions: The particle is shown to converge when $\\omega<1$, $\\alpha>0$ and \n$2\\omega-\\alpha+2>0$. Harmonic oscillations occur for \n$\\omega^2+\\alpha^2-2\\omega\\alpha-2\\omega-2\\alpha+1<0$ \nand a zigzag motion is expected if \n$\\omega<0$ and $\\omega-\\alpha+1<0$. \nAs with the preceding papers the discussion of the random numbers in the algorithm views \nthem purely as enhancing the search capabilities by adding a \\emph{drunken walk} \nto the particle motions. Their replacement by expectation values was thus believed to \nsimplify the analysis with no loss of generality. \n\nWe show in this contribution that the \niterated use of these random factors $\\mathbf{R}_{1}$ and $\\mathbf{R}_{2}$\nin fact adds a further level of complexity to the \ndynamics of the swarm which affects the behaviour of the algorithm in a non-trivial way. In Ref.~\\cite{jiang2007stagnation} these factors were given some consideration. Regions of convergence and divergence separated by a curved line were predicted. This line separating these regions (an equation for which is given in Ref.~\\cite{cleghorn2014generalized}) fails to include some parameter settings that lead to convergent swarms. Our analytical solution of the stability problem for the swarm dynamics explains\nwhy parameter settings derived from the deterministic approaches are not in line with\nexperiences from practical tests. For this purpose we will now formulate the PSO algorithm as\na random dynamical system and present an analytical solution for the swarm dynamics in a\nsimplified but representative case.\n\n\\section{Critical swarm conditions for a single particle\\label{Critical}}\n\n\\subsection{PSO as a random dynamical system}\n\nAs in Refs.~\\cite{kennedy1998behavior,trelea2003particle} the dynamics of the particle swarm will \nbe studied here as well in the single-particle case. This can be justified\nbecause the particles interact only\nvia the global best position such that, while $\\mathbf{g}$ (\\ref{eq:PSOVelocityUpdate}) is unchanged,\nsingle particles exhibit qualitatively the same dynamics as in the swarm. \nFor the one-particle case we have necessarily $\\mathbf{p}=\\mathbf{g}$,\nsuch that shift invariance allows us to set both to zero, which leads us to the following\nis given by the stochastic-map formulation of the PSO dynamics (\\ref{eq:matrix_from}).\n\\begin{equation}\n\\mathbf{z}_{t+1}=M \\mathbf{z}_t \n\t\\label{eq:pso_expl-1}\n\\end{equation}\nExtending earlier approaches we will explicitly consider the randomness of the dynamics,\ni.e.~instead of averages over $\\mathbf{R}_1$ and $\\mathbf{R}_2$ we consider a random\ndynamical system with dynamical matrices $M$ chosen from the set\n\\begin{equation}\n\t{\\cal M}_{\\alpha,\\omega}=\\left\\{\n\\left(\\begin{array}{cc} \n\t\t\t\\omega \\mathbf{I}_d & -\\alpha \\mathbf{R} \\\\\n\t\t\t\\omega \\mathbf{I}_d & \\mathbf{I}_d-\\alpha \\mathbf{R} \n\t\\end{array}\\right)\\!,\\,\\,\n\t\\mathbf{R}_{ij}=0 \\mbox{ for } i\\ne j \\mbox{ and } \\mathbf{R}_{ii}\\in\\left[0,1\\right], \n\\right\\} \n\\label{eq:matrix_set}\n\\end{equation}\nwith $\\mathbf{R}$ being in both rows the same realisation of a random diagonal matrix \nthat combines the effects of $\\mathbf{R}_1$ and $\\mathbf{R}_2$ (\\ref{eq:PSOPositionUpdate}).\nThe parameter $\\alpha$ is the sum $\\alpha_1+\\alpha_2$ with $\\alpha_1,\\alpha_2\\ge0$ and $\\alpha >0$.\nAs the diagonal elements of $\\mathbf{R}_1$ and $\\mathbf{R}_2$ are \nuniformly distributed in $\\left[0,1\\right]$, the distribution of the random variable \n$\\mathbf{R}_{ii} = \\frac{\\alpha_1}{\\alpha} \\mathbf{R}_{1,ii} + \\frac{\\alpha_2}{\\alpha} \\mathbf{R}_{2,ii}$ \nin Eq.~\\ref{eq:pso_expl-1} is given by a convolution\nof two uniform random variables, namely\n\\begin{equation}\nP_{\\alpha_1,\\alpha_2}(r)=\\begin{cases}\n\t\\frac{\\alpha r}{\\max\\{\\alpha_1,\\alpha_2\\}}&\\mbox{if } 0\\le r\\le \\min\\{\\frac{\\alpha_1}{\\alpha},\\frac{\\alpha_2}{\\alpha}\\}\\\\\n\t\\frac{\\alpha}{\\max\\{\\alpha_1,\\alpha_2\\}} & \\mbox{if } \\min\\{\\frac{\\alpha_1}{\\alpha},\\frac{\\alpha_2}{\\alpha}\\}0$. The finite-time approximation of the\nLyapunov exponent (see Eq.~\\ref{eq:lyapunov})\n\\begin{equation}\n\t\\lambda(t)=\\frac{1}{t} \\log \\left\\langle\\left\\| \\left(\\mathbf{x}_t,\\mathbf{v}_t\\right)\\right\\| \\right\\rangle\n\\end{equation}\nwill be changed by an amount of $\\frac{1}{t} \\log \\kappa $ by the scaling.\nAlthough this has no effect on the asymptotic behaviour, we will have to expect \nan effect on the stability of the swarm for finite times which may be relevant for\npractical applications. For the same parameters, the swarm will be more stable\nif $\\kappa<1$ and less stable for $\\kappa>1$, \nprovided that the initial conditions are scaled in the same way. \nLikewise, if $\\|\\mathbf{p}\\|$ is increased,\nthen the critical contour will move inwards, see Fig.~\\ref{fig:PSOAverageValley_p}.\nNote that in this figure, the low number of iterations lead\nto a few erroneous trials at parameter pairs outside the outer contour which have been omitted here.\nWe also do not consider the behaviour near $\\alpha=0$ which is complex but irrelevant for PSO.\nThe contour (\\ref{eq:implicite_a_w}) can be seen as the limit $\\kappa \\to 0$ such that\nonly an increase of $\\|\\mathbf{p}\\|$ is relevant for comparison with the theoretical\nstability result. When comparing the stability results with numerical simulations for\nreal optimisation problems, we will need to take into account the effects caused by differences between $\\mathbf{p}$ and $\\mathbf{g}$ in a multi-particle swarm with finite runtimes.\n\n\\begin{figure}[th]\n\\begin{center}\n\\includegraphics[width=0.65\\linewidth]{th2_and_emp_200_2000_20000_a.eps}\n\\end{center}\n\\vspace{-5mm}\n\\caption{\nBest parameter regions for 200 (blue), 2000 (green), and 20000 (magenta) iterations: For more\niterations the region shifts towards the critical line.\nCost averaged over 100 runs and 28 CEC benchmark functions. \nThe red (outer) curve represents the zero Lyapunov exponent for $N=1$, $d=1$, $\\alpha_{1}=\\alpha_{2}$.\n\\label{fig:PSOAverageValley}}\n\\end{figure}\n\n\\begin{figure}[th]\n\\begin{center}\n\\includegraphics[width=0.65\\linewidth,trim=0cm 3.5mm 0cm 0cm]{pso_p100.eps}\n\\end{center}\n\\vspace{-5mm}\n\\caption{\nFor $\\mathbf{p}\\ne\\mathbf{g}$ we define neutral stability as the equilibrium between\ndivergence and convergence. Convergence means here that the particle approaches the line\nconnecting $\\mathbf{p}$ and $\\mathbf{g}$.\nCurves are for a one-dimensional problem with $\\mathbf{p}=0.1$ and $\\mathbf{g}=0$ scaled (see \nSect.~\\ref{Personalvsglobal}) by $\\kappa=1$ (outer curve) $\\kappa=0.1$ and $\\kappa=0.04$ (inner curve).\nResults are for 200 iterations and averaged over 100000 repetitions. \n\\label{fig:PSOAverageValley_p}}\n\\end{figure}\n\n\n\\section{Optimisation of benchmark functions\\label{Simulations}}\n\nMetaheuristic algorithms are often tested in competition against benchmark functions designed to \npresent different problem space characteristics. \nThe 28 functions~\\cite{CEC2013} contain a mix of unimodal, basic multimodal and composite functions. \nThe domain of the functions in this test set are all defined to be $[-100, 100]^d$ \nwhere $d$ is the dimensionality of the problem. Particles were initialised within the same domain.\nWe use 10-dimensional problems throughout. \nOur implementation of PSO performed no spatial or velocity clamping. In all trials a swarm of 25 particles was used. \nWe repeated the algorithm 100 times, on each occasion allowing 200, 2000, 20000 iterations to pass before \nrecording the best solution found by the swarm. For the competition 50000 fitness evaluation were allowed which\ncorresponds to 2000 iterations with 25 particles. Other iteration numbers were included for comparison.\nThis protocol was carried out for pairs of $\\omega\\in[-1.1,1.1]$ and $\\alpha\\in[0,5]$ \nThis was repeated for all 28 functions. \nThe averaged solution costs as a function of the two parameters \nshowed curved valleys similar to that in Fig.~\\ref{fig:PSOPerformance} for all problems. \nFor each function we obtain different best values along (or near) the theoretical curve \n(\\ref{eq:implicite_a_w}). There appears to be no\npreferable location within the valley. \nSome individual functions yield best performance \nnear $\\omega=1$. This is not the case near $\\omega=0$, although the global average performance \nover all test functions is better in the valley near $\\omega=0$ than near $\\omega=1$, see\nFig~\\ref{fig:PSOAverageValley}.\n\nAt medium values of $\\omega$ \nthe difference between the analytical solutions for the cases \n$\\alpha_1=\\alpha_2$ and $\\alpha_1=0$ is strongest, see Fig.~\\ref{fig:PSOAverageValley}. \nIn simulations this shows to a lesser extent, thus revealing a shortcoming of the one-particle\napproximation. Because in the multi-particle case, $\\mathbf{p}$ and $\\mathbf{g}$ are often\ndifferent, the resulting vector will have a smaller norm than in the one-particle case, where \n$\\mathbf{p}=\\mathbf{g}$. The case $\\mathbf{p}\\ne\\mathbf{g}$ violates a the assumption of\nthe theory the dynamics can be described based unit vectors. While a particle far away from\nboth $\\mathbf{p}$ and $\\mathbf{g}$ will behave as predicted from the one-particle case,\nat length scales smaller than $\\Vert \\mathbf{p}-\\mathbf{g}\\Vert$ the retractive forces will\ntend to be reduced such that the inertia becomes more effective and the particle is locally less\nstable which shows numerically in optimal parameters that are smaller than predicted.\n\n\n\\section{Discussion\\label{discussion}}\n\n\\subsection{Relevance of criticality}\n\nOur analytical approach predicts a locus of $\\alpha$ and $\\omega$ pairings that maintain the critical\nbehaviour of the PSO swarm.\nOutside this line the swarm will diverge unless steps are taken to constrain it. Inside, the swarm \nwill eventually converge to a single solution.\nIn order to locate a solution within the search space, the swarm needs to converge at some point, so the line represents an upper bound on the exploration-exploitation mix that a swarm manifests. \nFor parameters on the critical line, fluctuations are still arbitrary large. Therefore, subcritical \nparameter values can be preferable if the settling time is of the same order as the scheduled \nruntime of the algorithm. If, in addition, a typical length scale of the problem is known, then the\nfinite standard deviation of the particles in the stable parameter region \ncan be used to decide about the distance of the \nparameter values from the critical curve. These dynamical quantities can be approximately set, \nbased on the theory presented here, such that a precise control of the behaviour of the algorithm is in\nprinciple possible. \n\nThe observation of the distribution of empirically optimal parameter values along the critical curve,\nconfirms the expectation that critical or near-critical behaviour is the main reason for success\nof the algorithm. Critical fluctuations are a plausible tool in search problem if apart from\ncertain smoothness assumption nothing is known about the cost landscape: The majority of excursions\nwill exploit the smoothness of the cost function by local search, whereas the fat tails of the \ndistribution allow the particles to escape from local minima.\n\n\n\n\\subsection{Switching dynamics at discovery of better solutions\\label{sub:The-role-of}}\n\nEq.~\\ref{eq:matrix_from} shows that the discovery of a better solution\naffects only the constant terms of the linear dynamics of a particle, whereas its\ndynamical properties are governed by the linear coefficient matrices. \nHowever, in the time step after a particle has found a new solution the corresponding force term \nin the dynamics is zero (see Eq.~\\ref{eq:PSOPositionUpdate}) such that the particle dynamics \nslows down compared to the theoretical solution which assumes a finite\ndistance from the best position at all (finite) times. As this affects usually only one particle\nat a time and because new discoveries tend to become rarer over time, this effect will be small\nin the asymptotic dynamics, although it could justify the empirical optimality of parameters \nin the unstable region for some test cases.\n\nThe question is nevertheless, how often these changes occur. A weakly\nconverging swarm can still produce good results if it often discovers\nbetter solutions by means of the fluctuations it performs before settling\ninto the current best position. For cost functions that are not `deceptive',\ni.e.~where local optima tend to be near better optima, parameter\nvalues far inside the critical contour (see Fig.~\\ref{fig:Theory_and_Numerical})\nmay give good results, while in other cases more exploration is needed.\n\n\\subsection{The role of personal best and global best\\label{current_best}\\label{runtime}}\n\nA numerical scan of the $(\\alpha_1,\\alpha_2)$ plane shows \na valley of good fitness values, which, at small fixed positive $\\omega$, is roughly linear and described \nby the relation $\\alpha_1+\\alpha_2= \\mbox{const}$, i.e.~only the joint parameter\n$\\alpha=\\alpha_1+\\alpha_2$ matters.\nFor large $\\omega$, and accordingly small predicted optimal $\\alpha$ values, \nthe valley is less straight. This may be \nbecause the effect of the known solutions is \nrelatively weak, so the interaction of the two components becomes more important.\nIn other words if the movement of the particles is mainly due to inertia, \nthen the relation between the global and local best is non-trivial, \nwhile at low inertia the particles can adjust their $\\mathbf{p}$ vectors\nquickly towards the $\\mathbf{g}$ vector such that both terms become interchangeable.\n\nFinally, we should mention that more particles, longer runtime as well as lower search \nspace dimension increase the potential for \nexploration. They all lead to the empirically determined optimal parameters being closer \nto the critical curve.\n\n\n\\section{Conclusion}\n\nPSO is a widely used optimisation scheme which is theoretically not well understood. Existing theory \nconcentrates on a deterministic version of the algorithm which does not possess useful exploration\ncapabilities. We have studied the algorithm by means of a product of random matrices which allows us to\npredict useful parameter ranges and may allow for more precise settings \nif a typical length scale of the problem is known.\nA weakness of the current approach is that it focuses on the standard PSO~\\cite{kennedy1995particle} which\nis known to include biases~\\cite{clerc2006confinments,spears2010biases}, \nthat are not necessarily justifiable,\nand to be outperformed on benchmark set and in practical applications by many of the existing PSO variants.\nSimilar analyses are certainly possible and are expected to be carried out for some of the variants, \neven though the field of metaheuristic search is often portrayed as largely inert to theoretical advances.\nIf the dynamics of particle swarms is better understood, the algorithms may become useful as\nefficient particle filters which have many applications beyond heuristic optimisation.\n\n\\subsection*{Acknowledgments}\n\nThis work was supported by the Engineering and Physical Sciences Research Council (EPSRC), grant number EP\/K503034\/1.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nLow-density parity-check (LDPC) convolutional codes \\cite{JimenezLDPCCC}, also known as spatially coupled LDPC (SC-LDPC) codes \\cite{Kudekar_ThresholdSaturation}, can be obtained from a sequence of individual LDPC block codes by distributing the edges of their Tanner graphs over several adjacent blocks \\cite{LentmaierTransITOct2010}.\nThe resulting spatially coupled codes exhibit a \\emph{threshold saturation} phenomenon, which has attracted a lot of interest in the past few years: The threshold of an iterative belief propagation (BP) decoder, obtained by density evolution (DE), can be improved to that of the optimal maximum-a-posteriori (MAP) decoder, for properly chosen parameters.\nIt follows from threshold saturation that it is possible to achieve capacity by spatial coupling of simple regular LDPC codes, which show a significant gap between BP and MAP threshold in the uncoupled case.\nA first analytical proof of threshold saturation was given in \\cite{Kudekar_ThresholdSaturation} for the binary erasure channel (BEC), considering a specific ensemble with uniform random coupling. An alternative proof based on potential functions was then presented in \\cite{Yedla2012,Yedla2014,KudekarWaveLike}, which was extended from scalar recursions to vector recursions in \\cite{Yedla2012vector}.\nBy means of vector recursions, the proof of threshold saturation can be extended to spatially coupled ensembles with structure, such as SC-LDPC codes based on protographs \\cite{Mitchell_ProtographSCLDPC}.\n\nThe concept of spatial coupling is not limited to LDPC codes.\nAlso codes on graphs with stronger component codes can be considered.\nIn this case the structure of the component codes has to be taken into account in a DE analysis.\nInstead of a simple check node update, a constraint node update within BP decoding of a generalized LDPC code involves an a-posteriori probability (APP) decoder applied to the associated component encoder.\nIn general, the input\/output transfer functions of the APP decoder are multi-dimensional because the output bits of the component encoder have different protection.\nFor the BEC, however, it is possible to analytically derive explicit transfer functions \\cite{AshikhminEXIT} by means of a Markov chain analysis of the decoder metric values in a trellis representation of the considered code \\cite{LentDE_PG-LDPC}.\nThis technique was applied in \\cite{LentDE_BBC, Lent_DE_SCGLDPC} to perform a DE analysis of braided block codes (BBCs) \\cite{JimenezBBC} and other spatially coupled generalized LDPC codes.\nThreshold saturation could be observed numerically in all the considered cases.\nBBCs can be seen as a spatially coupled version of product codes, and are closely related to staircase codes \\cite{Smith12}, which have been proposed for high-speed optical communications.\nIt was demonstrated in \\cite{PfisterISIT12,PfisterGC13} that BBCs show excellent performance even with the iterative hard decision decoding that is proposed for such scenarios.\nThe recently presented spatially coupled split-component codes \\cite{Truhachev_SplitComponent} demonstrate the connections between BBCs and staircase codes.\n\nIn this paper, we study codes on graphs whose constraint nodes represent convolutional codes \\cite{Wiberg95, WibergPhD96,Loeliger}.\nWe denote such codes as turbo-like codes (TCs).\nWe consider three particular concatenated convolutional coding schemes: Parallel concatenated codes (PCCs) \\cite{BerrouTC}, serially concatenated codes (SCCs) \\cite{Benedetto98Serial}, and braided convolutional codes (BCCs) \\cite{ZhangBCC}.\nOur aim is to investigate the impact of spatial coupling on the BP threshold of these TCs.\nFor this purpose we introduce some special block-wise spatially coupled ensembles of PCCs (SC-PCCs) and SCCs (SC-SCCs) \\cite{Moloudi_SCTurbo}.\nIn the case of BCCs, which are inherently spatially coupled, we consider the original block-wise ensemble from \\cite{ZhangBCC,Moloudi_DEBCC} and generalize it to larger coupling memories.\nFurthermore, we introduce a novel BCC ensemble in which not only the parity bits but also the information bits are coupled over several time instants \\cite{Moloudi_SPCOM14}.\n\n\nFor these spatially coupled turbo-like codes (SC-TCs), we perform a threshold analysis for the BEC analogously to \\cite{LentmaierTransITOct2010,LentDE_BBC, Lent_DE_SCGLDPC}.\nWe derive their exact DE equations from the transfer functions of the convolutional component decoders \\cite{Kurkoski, tenBrinkEXITConv}, whose computation is similar to that for generalized LDPC codes in \\cite{LentDE_PG-LDPC}.\nIn order to evaluate and compare the ensembles at different rates, we also derive DE equations for the punctured ensembles.\nUsing these equations, we compute BP thresholds for both coupled and uncoupled TCs \\cite{Moloudi_ISTW14} and compare them with the corresponding MAP thresholds \\cite{Measson2009,Measson_Turbo}.\nOur numerical results indicate that threshold saturation occurs if the coupling memory is chosen sufficiently large.\nThe improvement of the BP threshold is specially significant for SCCs and BCCs, whose uncoupled ensembles suffer from a poor BP threshold.\nWe then consider the construction of families of rate-compatible SC-TCs which achieve close-to-capacity performance for a wide range of code rates.\n\nMotivated by the numerical results, we prove threshold saturation analytically.\nWe show that, by few assumptions in the ensembles of uncoupled TCs, in particular considering identical component encoders, it is possible to rewrite their DE recursions in a form that corresponds to the recursion of a scalar admissible system.\nThis representation allows us to apply the proof technique based on potential functions for scalar admissible systems proposed in \\cite{Yedla2012,Yedla2014}, which simplifies the analysis.\nFor the general case, the analysis is significantly more complicated and requires the coupled vector recursion framework of \\cite{Yedla2012vector}.\nFinally, for the example of PCCs, we generalize the proof to non-symmetric ensembles with different component encoders by using the framework in \\cite{Yedla2012vector}.\n\nThe remainder of the paper is organized as follows. \nIn Section~\\ref{sec:CompactGraphCC}, we introduce a compact graph representation for the trellis of a convolutional code that is amenable for a DE analysis.\nFurthermore, we derive explicit input\/output transfer functions of the BCJR decoder for transmission over the BEC.\nThen, in Section~\\ref{sec:CompactGraphTCs}, we describe uncoupled ensembles of PCCs, SCCs and BCCs by means of the compact graph representation. SC-TCs, their spatially coupled counterparts, are introduced in Section~\\ref{sec:SCTCs}.\nIn Section~\\ref{sec:DE}, we derive exact DE equations for uncoupled and coupled ensembles of TCs.\nIn Section~\\ref{sec:RandomP}, we consider random puncturing and derive the corresponding DE equations and analyze SC-TCs as a family of rate compatible codes.\nNumerical results are presented and discussed in Section~\\ref{Sec6}.\nThreshold saturation, which is observed numerically in the results section, is proved analytically in Section \\ref{Sec7}. \nFinally, the paper is concluded in Section~\\ref{Sec8}.\n\n\\section{Compact Graph Representation and Transfer Functions of Convolutional Codes}\n\\label{sec:CompactGraphCC}\nIn this section, we introduce a graphical representation of a convolutional code, which can be seen as a compact form of its corresponding factor graph \\cite{Loeliger}. \nThis compact graph representation makes the illustration of SC-TCs simpler and is convenient for the DE analysis. \nWe also generalize the method in \\cite{Kurkoski, tenBrinkEXITConv} to derive explicit input\/output transfer functions of the BCJR decoder of rate-$k\/n$ convolutional codes on the BEC, which will be used in Section~\\ref{sec:DE} to derive the exact DE for SC-TCs.\n\n\n\\subsection{Compact Graph Representation}\nConsider a rate-$k\/n$ systematic convolutional encoder of code length $nN$ bits, i.e., its corresponding trellis has $N$ trellis sections. At each time instant $\\tau=1,\\ldots,N$, corresponding to a trellis section, the encoder encodes $k$ input bits and generates $n-k$ parity bits.\nLet $\\bs{u}^{(i)}=(u_{1}^{(i)}, u_{2}^{(i)}, \\dots, u_{N}^{(i)})$, $i=1,\\dots,k$, and $\\bs{v}_{\\text{p}}^{(i)}=(v_{\\text{p},1}^{(i)}, v_{\\text{p},2}^{(i)}, \\dots, v_{\\text{p},N}^{(i)})$, $i=1,\\dots,n-k$, denote the $k$ input sequences and the $n-k$ parity sequences, respectively. We also denote by $\\bs{v}^{(i)}=(v_{1}^{(i)}, v_{2}^{(i)}, \\dots, v_{N}^{(i)})$, $i=1,\\dots,n$, the $i$th code sequence, with $\\bs{v} ^{(i)}=\\bs{u}^{(i)}$ for $i=1,\\dots,k$ and $\\bs{v} ^{(i)}=\\bs{v}_{\\text{p}}^{(i-k)}$ for $i=k+1,\\dots,n$. The conventional factor graph of a convolutional encoder is shown in Fig.~\\ref{factorgraph}(a), where black circles represent code bits, each black square corresponds to the code constraints (allowed combinations of input state, input bits, output bits, and output state) of one trellis section, and the double circles are (hidden) state variable nodes. \n\nFor convenience, we will represent a convolutional encoder with the more compact graph representation depicted in Fig.~\\ref{factorgraph}(b). \nIn this compact graph representation, each input sequence $\\bs{u}^{(i)}$ and each parity sequence $\\bs{v}^{(i)}_{\\text{p}}$ is represented by a single black circle, referred to as variable node, i.e., each circle represents $N$ bits. Furthermore, the code trellis is represented by a single empty square, referred to as factor node. The factor node is labeled by the length $N$ of the trellis. \n\\begin{figure}[!t]\n \\centering\n \\includegraphics[width=0.8\\linewidth]{Fig1.pdf}\n\\caption{(a) Factor graph representation of a rate-$k\/n$ systematic convolutional code. (b) Compact graph representation of the same code.}\n\\label{factorgraph}\n\\end{figure} \nEach node in the compact graph represents a sequence of nodes belonging to the same type, similar to the nodes in a protograph of an LDPC code.\nVariable nodes in the original factor graph may represent different bit values, even if they belong to the same type in the compact graph.\nHowever, assuming a tailbiting trellis, the probability distribution of these values after decoding will be equal for all variables that correspond to the same node type.\nAs a consequence, a DE analysis can be performed in the compact graph, independently of the trellis length $N$, which plays a similar role as the lifting factor of a protograph ensemble.\nIf a terminated convolutional encoder, which starts and ends in the zero state, is used instead, the bits that are close to the start and end of the trellis will have a slightly stronger protection. Since this effect will not have a significant impact on the performance, we will neglect this throughout this paper and assume equal output distributions for all bits of the trellis, even when termination is used.\n\n\\subsection{Transfer Function of the BCJR Decoder of a Convolutional Code} \\label{TransferFunction}\n\nConsider the BCJR decoder of a memory $\\nu$, rate$-k\/n$ convolutional encoder and transmission over the BEC. \nWithout loss of generality, we restrict ourselves within this paper to encoders with $k=1$ or $n-k=1$, which can be implemented with $2^\\nu$ states in controller canonical form or observer canonical form, respectively. \nWe would like to characterize the transfer function between the input erasure probabilities (i.e., prior to decoding) and output erasure probabilities (i.e., after decoding) on both the input bits and the output bits of the convolutional encoder. Note that the erasure probabilities at the input of the decoder depend on both the channel erasure probability and the a-priori erasure probabilities on systematic and parity bits (provided, for example, by another decoder). Thus, in the more general case, we consider non-equal erasure probabilities at the input of the decoder.\n\nConsider the extrinsic erasure probability of the $l$th code bit, $l=1,2, \\dots, n$, which is the erasure probability of the $l$th code bit when it is estimated based on the other code bits\\footnote{Without loss of generality we assume that the first $k$ bits are the systematic bits.}. This extrinsic erasure probability, at the output of the decoder, is denoted by $p_{l}^{\\text {ext}}$. The probabilities $p_{l}^{\\text {ext}}$ depend on the erasure probabilities of all code bits (systematic and parity) at the input of the decoder, \n\\begin{align}\n\\label{eq:Transfer1}\np_ {l}^{\\text {ext}}=f_l(p_1,p_2, \\dots,p_n),\n\\end{align}\nwhere $p_l$ is the erasure probability of the $l$th code bit at the input of the decoder and $f_l(p_1,p_2, \\dots,p_n)$ is the transfer function of the BCJR decoder for the $l$th code bit. For notational simplicity, we will often omit the argument of $f_l(p_1,p_2, \\dots,p_n)$ and write simply $f_l$.\n\n\nLet $\\bs{r}^{(i)}=(r_{1}^{(i)}, r_{2}^{(i)}, \\dots, r_{N}^{(i)})$,\n$i=1,\\dots,n$, be the vectors of received symbols at the output of the channel, with $r_{j}^{(i)}\\in\\{0,1,?\\}$, where $?$ denotes an erasure. The branch metric of the trellis edge departing from state $\\sigma'$ at time $\\tau-1$ and ending to state $\\sigma$ at time $\\tau$, $\\tau=1,\\dots,N$, is\n\n\\begin{align}\n\\gamma_{\\tau}(\\sigma',\\sigma)=\\prod_{l=1}^{n} p\\left(r_{\\tau}^{(l)}\\; \\big| \\; v_{\\tau}^{(l)}\\right)\\cdot p\\left(v_{\\tau}^{(l)}\\right),\n\\end{align}\nwhere $p\\big(v_{\\tau}^{(l)}\\big)$ is the a-priori probability on symbol $v_{\\tau}^{(l)}$.\n\nThe forward and backward metrics of the BCJR decoder are\\begin{align}\n&\\alpha_{\\tau}(\\sigma)=\\sum_{\\sigma'}\n\\gamma_{\\tau}(\\sigma',\\sigma) \\cdot \\alpha_{\\tau-1}(\\sigma')\\\\\n&\\beta_{\\tau-1}(\\sigma')=\\sum _{\\sigma}\\gamma_{\\tau}(\\sigma',\\sigma)\n\\cdot \\beta_{\\tau}(\\sigma').\n\\end{align}\n\nFinally, the extrinsic output likelihood ratio is given by \n\\begin{align*}\n&L^{(l)}_{\\text{out},\\tau}=\\\\\n&\\frac{\\sum\\limits_{(\\sigma',\\sigma):v_{\\tau}^{(l)}=0}\\alpha_{\\tau-1}(\\sigma')\\cdot\n\\gamma_{\\tau}(\\sigma',\\sigma)\\cdot\n\\beta_{\\tau}(\\sigma)}{\\sum\\limits_{(\\sigma',\\sigma):v_{\\tau}^{(l)}=1}\n\\big(\\alpha_{\\tau-1}(\\sigma')\\cdot \\gamma_{\\tau}(\\sigma',\\sigma)\\cdot \\beta_{\\tau}(\\sigma)}\\cdot\\frac{p\\left(v_{\\tau}^{(l)}=1\\right)}{p\\left(v_{\\tau}^{(l)}=0\\right)}.\n\\end{align*}\n\n\n\n\nLet the $2^\\nu$ trellis states be $s_1,s_2,\\ldots,s_{2^\\nu}$. Then, we define the forward and backward metric vectors as $\\boldsymbol{\\alpha}_{\\tau}=(\\alpha_{\\tau}(s_1),\\ldots,\\alpha_{\\tau}(s_{2^\\nu}))$ and $\\boldsymbol{\\beta}_{\\tau}=(\\beta_{\\tau}(s_1),\\ldots,\\beta_{\\tau}(s_{2^\\nu}))$, respectively. \nFor transmission on the BEC, the nonzero entries of vectors $\\boldsymbol{\\alpha}_{\\tau}$ and $\\boldsymbol{\\beta}_{\\tau}$ are all equal. Thus, we can normalize them to $1$.\n\n\n\nWe consider transmission of the all-zero codeword. The sets of values that vectors $\\boldsymbol{\\alpha}_{\\tau}$ and\n $\\boldsymbol{\\beta}_{\\tau}$ can take on are denoted by $\\mathcal{M}_{\\alpha}=\\{\\boldsymbol{m}_{\\alpha}^{(1)},\\ldots,\\boldsymbol{m}_{\\alpha}^{(|\\mathcal{M}_{\\alpha}|)}\\}$ and $\\mathcal{M}_{\\beta}=\\{\\boldsymbol{m}_{\\beta}^{(1)},\\ldots,\\boldsymbol{m}_{\\beta}^{(|\\mathcal{M}_{\\beta}|)}\\}$, respectively. It is important to remark that these sets are finite. Furthermore, the sequence $\\dots,\\boldsymbol{\\alpha}_{\\tau-1},\\boldsymbol{\\alpha}_{\\tau},\\boldsymbol{\\alpha}_{\\tau+1},\\dots$\nforms a Markov chain, which can be properly described by a probability transition matrix, denoted by $\\boldsymbol{M}_{\\alpha}$.\nThe $(i,j)$ entry of $\\boldsymbol{M}_{\\alpha}$ is the probability of transition from state\n$\\boldsymbol{m}_{\\alpha}^{(i)}$ to state $\\boldsymbol{m}_{\\alpha}^{(j)}$. Denote the steady\nstate distribution vector of the Markov chain by $\\boldsymbol{\\pi}_{\\alpha}$, which can be computed as the solution to\n\\begin{equation}\n\\boldsymbol{\\pi}_{\\alpha}=\\boldsymbol{M}_{\\alpha}\\cdot \\boldsymbol{\\pi}_{\\alpha}. \n\\end{equation}\nSimilarly, we can define the transition matrix for the sequence of backward metrics $\\dots,\\boldsymbol{\\beta}_{\\tau+1},\\boldsymbol{\\beta}_{\\tau},$ $\\boldsymbol{\\beta}_{\\tau-1},\\dots$, denoted by $\\boldsymbol{M}_{\\beta}$, and compute the steady state distribution vector $\\bs{\\pi}_{\\beta}$.\n\n\\begin{example}\nConsider the rate-$2\/3$, $4$-state convolutional encoder with generator\nmatrix \n\\begin{equation*\n\\boldsymbol{G} (D)=\\left( \\begin{array}{ccc}1&0&\\frac{1}{1+D+D^2}\\\\0&1&\\frac{1+D^2}{1+D+D^2}\\end{array}\\right).\n\\end{equation*}\n$\\mathcal{M}_{\\alpha}$\nand $\\mathcal{M}_{\\beta}$ are equal and have cardinality $5$,\n\\begin{align*}\n&\\mathcal{M}_{\\alpha}=\\mathcal{M}_{\\beta}=\\\\\n&\\{(1,0,0,0),(1,1,0,0),(1,0,0,1),(1,0,1,0),(1,1,1,1)\\}.\n\\end{align*}\n\nConsider equal erasure probability for all code bits at the input of the decoder, i.e., $p_1=p_2=p_3 = p$. Then,\n\\begin{align*}\n\\resizebox{\\hsize}{!}{$\n\\boldsymbol{M}_{\\alpha}=\\left[\\begin{array}{ccccc}\n(1-p)^2(2p+1)&(1-p)^2&(1-p)^3&0&0\\\\\np^2(1-p)&0&p(1-p)^2&p^3-2p+1&(1-p)^2\\\\\np^2(1-p)&p(1-p)&p(1-p)^2&0&0\\\\\np^2(1-p)&p(1-p)&p(1-p)^2&0&0\\\\\np^3&p^2&p^2(3-2p)&p^2(2-p)&p(2-p)\n\\end{array}\\right] \\; .\n$ \n}\n\\end{align*} \\hfill $\\triangle$\n\\end{example}\n\\vspace{3mm}\n\nIn order to compute the erasure probability of the $l$th bit at the output of the decoder, we have to\ncompute the probability of $L^{(l)}_{\\text{out},\\tau}=1$. Define the \nmatrices $\\boldsymbol{T}_{l}$, $l=1,2,\\dots,n$, where the $(i,j)$ entry of $\\boldsymbol{T}_{l}$ is computed as\n\\[\nT_{l}(i,j)=p\\left(L^{(l)}_{\\text{out},\\tau}=1\\; | \\;\\boldsymbol{\\alpha}_{\\tau}=\\boldsymbol{m}_{\\alpha}^{(i)},\\boldsymbol{\\beta}_{\\tau+1}=\\boldsymbol{m}_{\\beta}^{(j)}\\right).\n\\]\nThenhresho, the extrinsic erasure probability of the $l$th output, $p_ {l}^{\\text{ext}}$, introduced in~\\eqref{eq:Transfer1}, is obtained as\n\\begin{align}\np_ {l}^{\\text {ext}}&=f_l(p_1,p_2,\\dots,p_n)=p\\left(L^{(l)}_{\\text{out},\\tau}=1\\right) \\nonumber\\\\\n&=\\sum^{|\\mathcal{M}_{\\alpha}|}_{i=1}\\sum^{|\\mathcal{M}_{\\beta}|}_{j=1}p\\left(\nL^{(l)}_{\\text{out},\\tau}=1\\; | \\; \\boldsymbol{\\alpha}_{\\tau}=\\boldsymbol{m}_{\\alpha}^{(i)},\\boldsymbol{\\beta}_{\\tau+1}=\\boldsymbol{m}_{\\beta}^{(j)}\\right)\\nonumber\\\\ \n&\\quad\\quad\\quad\\quad\\quad \\cdot p\\left(\\boldsymbol{\\alpha}_{\\tau}=\\boldsymbol{m}_{\\alpha}^{(i)}\\right)\\cdot p\\left(\\boldsymbol{\\beta}_{\\tau+1}=\\boldsymbol{m}_{\\beta}^{(j)}\\right)\\nonumber\\\\\n& =\\boldsymbol{\\pi}_{\\alpha} \\cdot\\boldsymbol{T}_{l}\\cdot \\boldsymbol{\\pi}_{\\beta} . \n\\end{align}\n\n\\begin{example}\nConsider the rate$-2\/3$ convolutional encoder with generator matrix\n\\begin{equation*\n\\boldsymbol{G} (D)=\\left( \\begin{array}{ccc}1&0&\\frac{1}{1+D}\\\\0&1&\\frac{D}{1+D}\\end{array}\\right).\n\\end{equation*}\n\nAssuming $p_1=p_2=p_3\\triangleq p$, the transfer functions for the corresponding decoder are\n\\[\nf_1=f_2=\\frac{p(p^5-4p^4+6p^3-5p^2+2p+1)}{p^6-4p^5+6p^4-6p^3+5p^2-2p+1},\n\\]\n\\[\nf_3=\\frac{p^2(p^2-4p+4)}{p^6-4p^5+6p^4-6p^3+5p^2-2p+1}.\n\\] \\hfill $\\triangle$\n\\end{example}\n\n\\begin{lemma}\n\\label{Lemma1}\nConsider a terminated convolutional encoder where all distinct input\nsequences have distinct encoded sequences. \nFor such a system, the transfer function $f(p_1,p_2,\\dots,p_n)$ of a BCJR decoder with input erasure probabilities $p_1,p_2,\\dots,p_n$, or any convex combination of such transfer functions, is increasing in all its arguments.\n\\end{lemma}\n\\begin{IEEEproof}\nWe prove the statement by contradiction. Recall that the BCJR decoder\nis an optimal APP decoder. \nNow, consider the transmission of the same\ncodeword over two channels, called channel 1 and 2. The erasure probabilities of the $i$th bit\nat the input of the decoder are denoted by $p_i^{(1)}$ and $p_i^{(2)}$ for\ntransmission over channel 1 and 2, respectively. These erasure probabilities are equal for all\n$i=1,\\ldots,n$ except for the $j$th bit, for which\n$p_j^{(1)}f(p_1^{(1)},\\dots,p_j^{(1)},\\ldots,p_n^{(1)}),\n\\end{align}\nSince after puncturing $p_i^{(1)}$ and $p_i^{(2)}$ are equal for all $i$, then \n$f(p_1^{(1)},\\dots,p_{j,\\text{punc}}^{(1)},\\ldots,p_n^{(1)}) =f(p_1^{(2)},\\dots,p_j^{(2)},\\ldots,p_n^{(2)})$. Then, we can rewrite the inequality \\eqref{eq:Contr2} as\n\\begin{align}\n\\label{eq:Contr3}\nf(p_1^{(2)},\\ldots,p_j^{(2)},\\ldots,p_n^{(2)})&>f(p_1^{(1)},\\dots,p_j^{(1)},\\ldots,p_n^{(1)}).\n\\end{align}\nHowever, the inequality \\eqref{eq:Contr3} is in contradiction with \\eqref{contradiction}.\n\\end{IEEEproof}\n\n\\section{Compact Graph Representation of\\\\ Uncoupled Turbo-like Codes}\n\\label{sec:CompactGraphTCs}\n\n\nIn this section, we describe PCCs, SCCs and BCCs using the compact graph representation introduced in the previous section. In Section~\\ref{sec:SCTCs} we then introduce the corresponding spatially coupled ensembles.\n\n\\subsection{Parallel Concatenated Codes}\n\nWe consider a rate $R=1\/3$ PCC built from two rate-$1\/2$ recursive systematic convolutional encoders, referred to as the upper and lower component encoder. \nIts conventional factor graph is shown in Fig.~\\ref{Uncoupled}(a), where $\\Pi$ denotes the permutation.\nThe trellises corresponding to the upper and lower encoders are denoted by $\\text{T}^{\\text{U}}$ and $\\text{T}^{\\text{L}}$, respectively. The information sequence $\\bs{u}$, of length $N$ bits, and a reordered copy are encoded by the upper and lower encoder, respectively, to produce the parity sequences $\\bs{v}^{\\text{U}}$ and $\\bs{v}^{\\text{L}}$. The code sequence is denoted by $\\bs{v}=(\\bs{u},\\bs{v}^{\\text{U}},\\bs{v}^{\\text{L}})$. The compact graph representation of the PCC is shown in Fig.~\\ref{Uncoupled}(b), where\neach of the sequences $\\bs{u}$, $\\bs{v}^{\\text{U}}$ and $\\bs{v}^{\\text{L}}$ is represented by a single variable node and the trellises are replaced by factor nodes $\\text{T}^{\\text{U}}$ and $\\text{T}^{\\text{L}}$ (cf. Fig.~\\ref{factorgraph}).\nIn order to emphasize that a reordered copy of the input sequence is used in $\\text{T}^{\\text{L}}$, the permutation is depicted by a line that crosses the edge which connects $\\bs{u}$ to $\\text{T}^{\\text{L}}$.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.75\\linewidth]{Fig2.pdf}\n\\caption{(a) Conventional factor graph of a PCC. Compact graph representation of a (b) PCC, (c) SCC, (d) BCC.}\n\\label{Uncoupled}\n\\end{figure}\n\n\\subsection{Serially Concatenated Codes}\n\nWe consider a rate $R=1\/4$ SCC built from the serial concatenation of two rate-$1\/2$ recursive systematic component encoders, referred to as the outer and inner component encoder. Its compact graph representation is shown in Fig.~\\ref{Uncoupled}(c), where $\\text{T}^{\\text{O}}$ and $\\text{T}^{\\text{I}}$ are the factor nodes corresponding to the outer and inner encoder, respectively, and the rectangle illustrates a multiplexer\/demultiplexer. The information sequence $\\bs{u}$, of length $N$, is encoded by the outer encoder to produce the parity sequence $\\bs{v}^{\\text{O}}$. Then, the sequences $\\bs{u}$ and $\\bs{v}^{\\text{O}}$ are multiplexed and reordered to create the intermediate sequence $\\tilde{\\bs{v}}^{\\text{O}}$, of length $2N$ (not shown in the graph). Finally, $\\tilde{\\bs{v}}^{\\text{O}}$ is encoded by the inner encoder to produce the parity sequence $\\bs{v}^{\\text{I}}$. The transmitted sequence is $\\bs{v}=(\\bs{u},\\bs{v}^{\\text{O}},\\bs{v}^{\\text{I}})$. \n\n \\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{Fig3.pdf}\n\t\\caption{Block diagram of the encoder of a SC-PCC ensemble with $m=1$. }\n\t\\label{CoupledPCC}\n\\end{figure*}\n\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[width=1\\linewidth]{Fig4.pdf}\n\t\\caption{Compact graph representation of (a) PCC, (b) SC-PCC at time instant $t$, (c) SC-PCC.}\n\t\\label{SC-PCC}\n\\end{figure*}\n\n\\subsection{Braided Convolutional Codes}\n\nWe consider a rate $R=1\/3$ BCC built from two rate-$2\/3$ recursive systematic convolutional encoders, referred to as upper and lower encoders. The corresponding trellises are denoted by $\\text{T}^{\\text{U}}$ and $\\text{T}^{\\text{L}}$. The compact graph representation of this code is shown in Fig.~\\ref{Uncoupled}(d). The parity sequences of the upper and lower encoder are denoted by $\\bs{v}^{\\text{U}}$ and $\\bs{v}^{\\text{L}}$, respectively.\nTo produce the parity sequence $\\bs{v}^{\\text{U}}$, the information sequence $\\bs{u}$ and a reordered copy of $\\bs{v}^{\\text{L}}$ are encoded by $\\text{T}^{\\text{U}}$.\nLikewise, a reordered copy of $\\bs{u}$ and a reordered copy of $\\bs{v}^{\\text{U}}$ are encoded by $\\text{T}^{\\text{L}}$ in order to produce the parity sequence $\\bs{v}^{\\text{L}}$.\nSimilarly to PCCs, the transmitted sequence is $\\bs{v}=(\\bs{u},\\bs{v}^{\\text{U}},\\bs{v}^{\\text{L}})$.\n\n\\section{Spatially Coupled Turbo-like Codes} \n\\label{sec:SCTCs}\nIn this section, we introduce SC-TCs. We first describe the spatial coupling for both PCCs and SCCs. Then, we generalize the original block-wise BCC ensemble \\cite{ZhangBCC} in order to obtain ensembles with larger coupling memories. \n\n\\subsection{Spatially Coupled Parallel Concatenated Codes}\n\nWe consider the spatial coupling of rate-$1\/3$ PCCs, described in the previous section.\nFor simplicity, we first describe the SC-PCC ensemble with coupling memory $m=1$.\nThen we show the coupling for higher coupling memories.\nThe block diagram of the encoder for the SC-PCC ensemble is shown in Fig.~\\ref{CoupledPCC}.\nIn addition, its compact graph representation and the coupling are illustrated in Fig.~\\ref{SC-PCC}.\n\n\nAs it is shown in Fig.~\\ref{CoupledPCC} and Fig.~\\ref{SC-PCC}(a) we denote by $\\bs{u}_t$ the information sequence, and by $\\bs{v}_t^{\\text{U}}$ and $\\bs{v}_t^{\\text{L}}$ the parity sequence of the upper and lower encoder, respectively, at time $t$.\nThe code sequence of the PCC at time $t$ is given by the triple $\\bs{v}_t = \n(\\bs{u}_t,\\bs{v}^{\\text{U}}_t ,\\bs{v}^{\\text{L}}_t )$. With reference to Fig.~\\ref{CoupledPCC} and Fig.~\\ref{SC-PCC}(b),\nin order to obtain the coupled sequence, the information sequence, $\\bs{u}_t$, is divided into two sequences of equal size, $\\bs{u}_{t,0}$ and $\\bs{u}_{t,1}$ by a multiplexer. \nThen, the sequence $\\bs{u}_{t,0}$ is used as a part of the input to the upper encoder at time $t$ and $\\bs{u}_{t,1}$ is used as a part of the input to the upper encoder at time $t+1$.\nLikewise, a reordered copy of the information sequence, $\\tilde{\\bs{u}}_t$, is divided into two sequences $\\tilde{\\bs{u}}_{t,0}$ and $\\tilde{\\bs{u}}_{t,1}$.\n\nTherefore, the input to the upper encoder at time $t$ is a reordered copy of $(\\bs{u}_{t,0},\\bs{u}_{t-1,1})$, and likewise the input to the lower encoder at time $t$ is a reordered copy of $(\\tilde{\\bs{u}}_{t,0},\\tilde{\\bs{u}}_{t-1,1})$.\nIn this ensemble, the coupling memory is $m=1$ as $\\bs{u}_t$ is used only at the time instants $t$ and $t+1$.\n\n\nFinally, an SC-PCC with $m=1$ is obtained by considering a collection of $L$ PCCs at time instants $t=1,\\ldots,L$, where $L$ is referred to as the coupling length, and coupling them as described above, see Fig.~\\ref{SC-PCC}(c).\n\n \nAn SC-PCC ensemble with coupling memory $m$ is obtained by dividing each of the sequences $\\bs{u}_t$ and $\\tilde{\\bs{u}}_t$ into $m+1$ sequences of equal size and spread these sequences respectively to the input of the upper and the lower encoder at time slots $t$ to $t+m$. The compact graph representation of the SC-PCC with coupling memory $m$ is shown in Fig.~\\ref{Coupled}(a) for a given time instant $t$.\n\nThe coupling is performed as follows. Divide the information sequence $\\u_t$ into $m+1$ sequences of equal size $N\/(m+1)$, denoted by $\\bs{u}_{t,j}$, $j=0,\\dots,m$. Likewise, divide $\\tilde{\\bs{u}}_t$, the information sequence $\\bs{u}_t$ reordered by a permutation, into $m+1$ sequences of equal size, denoted by $\\tilde{\\bs{u}}_{t,j}$, $j=0,\\dots,m$. At time $t$, the information sequence at the input of the upper encoder is $(\\u_{t,0},\\u_{t-1,1},\\ldots,\\u_{t-m,m})$, properly reordered by a permutation. Likewise, the information sequence at the input of the lower encoder is $(\\tilde{\\u}_{t,0},\\tilde{\\u}_{t-1,1},\\ldots,\\tilde{\\u}_{t-m,m})$, reordered by a permutation. \nUsing the procedure described above, a coupled chain (a convolutional\nstructure over time) of $L$ PCCs with coupling memory\n$m$ is obtained. \n\nIn order to terminate the encoder of the SC-PCC to the zero state, the information sequences at the end of the chain are chosen in such a way that the code sequences become\n$\\bs{v}_{t}=\\bs{0}$ at time $t=L+1,\\dots,L+m$, and $\\u_t$ is set to $\\bs{0}$ for $t>L$. Analogously to conventional convolutional codes, this results in a rate loss that becomes smaller as $L$ increases. \n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.75\\linewidth]{Fig5.pdf}\n\\caption{Compact graph representation of (a) SC-PCCs, and (b) SC-SCCs of coupling memory $m$ for time instant $t$.}\n\\label{Coupled}\n\\end{figure}\n\n\n\\subsection{Spatially Coupled Serially Concatenated Codes}\n\nAn SC-SCC is constructed similarly to SC-PCCs. Consider a collection of $L$ SCCs at time instants $t=1,\\ldots,L$,\nand let $\\bs{u}_t$ be the information sequence at time $t$. Also, denote by $\\bs{v}_t^{\\mathrm{O}}$ and $\\bs{v}_t^{\\mathrm{I}}$ the parity sequence at the output of the outer and inner encoder, respectively. The information sequence $\\bs{u}_t$ and the parity sequence $\\bs{v}_t^{\\mathrm{O}}$ are multiplexed and reordered into the sequence $\\tilde{\\bs{v}}_t^{\\mathrm{O}}$. The sequence $\\tilde{\\bs{v}}_t^{\\mathrm{O}}$ is divided into $m+1$ sequences of equal length, denoted by $\\tilde{\\bs{v}}_{t,j}^{\\text{O}}$, $j=0,\\dots,m$. Then, at time instant $t$, the\nsequence at the input of the inner encoder is\n$(\\tilde{\\bs{v}}_{t-j,0}^{\\text{O}},\\tilde{\\bs{v}}_{t-1,1}^{\\text{O}}\\ldots,\\tilde{\\bs{v}}_{t-m,m}^{\\text{O}})$, properly reordered by a permutation. This sequence is encoded by the inner encoder into $\\bs{v}_t^{\\mathrm{I}}$. Finally, the code sequence at time $t$ is $\\bs{v}=(\\bs{u}_t,\\bs{v}_t^{\\text{O}},\\bs{v}_t^{\\text{I}})$. Using this construction method, a coupled chain of $L$ SCCs with coupling memory $m$ is obtained. The compact graph representation of SC-SCCs with coupling memory $m$ is shown in Fig.~\\ref{Coupled}(b) for time instant $t$. \n\nIn order to terminate the encoder of the SC-SCC, the information sequences at the end of the chain are chosen in such a way that the code sequences become $\\bs{v}_{t}=\\bs{0}$ at time $t=L+1,\\dots,L+m$. A simple and practical way to terminate SC-SCCs is to set $\\u_t=\\boldsymbol{0}$ for $t=L-m+1,\\dots,L$. This enforces $\\bs{v}_{t}=\\bs{0}$ for $t=L+1,\\dots,L+m$, since we can assume that $\\u_t=\\boldsymbol{0}$ for $t>L$. Using this termination technique, only the parity sequence $\\bs{v}_t^{\\text{I}}$ needs to be transmitted at time instants $t=L-m+1,\\dots,L$.\n\n\n\\subsection{Braided Convolutional Codes}\n\nThe compact graph representation of the original BCCs is depicted in Fig~\\ref{BCCSC}.\nAs for SC-PCCs, let $\\bs{u}_t$, $\\bs{v}_t^{\\text{U}}$ and $\\bs{v}_t^{\\text{L}}$ denote the information sequence, the parity sequence at the output of the upper encoder, and the parity sequence at the output of the lower encoder, respectively, at time $t$. \nAt time $t$, the information sequence \n $\\bs{u}_t$ and a reordered copy of $\\bs{v}_{t-1}^{\\text{L}}$ are encoded by the upper encoder to generate the parity sequence $\\bs{v}_t^{\\text{U}}$.\nLikewise, a reordered copy of the information sequence, denoted by $\\tilde{\\bs{u}}_t$, and a reordered copy of $\\bs{v}_{t-1}^{\\text{L}}$ are encoded by the lower encoder to produce the parity sequence $\\bs{v}_t^{\\text{L}}$. The code sequence at time $t$ is $\\bs{v}=(\\bs{u}_t,\\bs{v}_t^{\\text{U}},\\bs{v}_t^{\\text{L}})$.\n\nAs it can be seen from Fig~\\ref{BCCSC}, the original BCCs are inherently spatially coupled codes\\footnote{The uncoupled ensemble, discussed in the previous section, can be defined by tailbiting a coupled chain of length $L=1$.} with coupling memory one. \nIn the following, we introduce two extensions of BCCs, referred to as Type-I and Type-II, with increased coupling memory, $m>1$.\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{Fig6.pdf}\n\\caption{Compact graph representation of the original BCCs.}\n\\label{BCCSC}\n\\end{figure}\n\nThe compact graph of Type-I BCCs is shown in\nFig.~\\ref{BCCI-II}(a) for time instant $t$.\nThe parity sequence $\\boldsymbol{v}_t^{\\text{U}}$ is randomly divided into $m$ sequences $\\boldsymbol{v}_{t,j}^{\\text{U}}$, $j=1,\\dots,m$, of the same length.\nLikewise, the parity sequence $\\boldsymbol{v}_t^{\\text{L}}$ is randomly divided into $m$ sequences $\\boldsymbol{v}_{t,j}^{\\text{L}}$, $j=1,\\dots,m$. At time $t$, the information sequence $\\bs{u}_t$ and the sequence $(\\boldsymbol{v}_{t-1,1}^{\\text{L}},\\boldsymbol{v}_{t-2,2}^{\\text{L}},\\ldots,\\boldsymbol{v}_{t-m,m}^{\\text{L}})$, properly reordered, are used as input sequences to the upper encoder to produce the parity sequence $\\boldsymbol{v}_t^{\\text{U}}$. Likewise, a reordered copy of the information sequence $\\bs{u}_t$ and the sequence $(\\boldsymbol{v}_{t-1,1}^{\\text{U}},\\boldsymbol{v}_{t-2,2}^{\\text{U}},\\ldots,\\boldsymbol{v}_{t-m,m}^{\\text{U}})$, properly reordered, are encoded by the lower encoder to produce the parity sequence $\\boldsymbol{v}_t^{\\text{L}}$. \n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.75\\linewidth]{Fig7.pdf}\n\\caption{Compact graph representation of (a) Type-I BCCs, and (b) Type-II BCCs of coupling memory $m$ at time instant $t$.}\n\\label{BCCI-II}\n\\end{figure}\n\nThe compact graph of Type-II BCCs is shown in\nFig.~\\ref{BCCI-II}(b) for time instant $t$. Contrary to Type-I BCCs, in addition to the coupling of parity bits, for Type-II BCCs information bits are also coupled. At time $t$, divide the information sequence $\\u_t $ into $m+1$ sequences $\\boldsymbol{u}_{t,j}$, $j=0,\\dots,m$ of equal length. Furthermore, divide the reordered copy of the information sequence, $\\tilde{\\bs{u}}_t$, into $m+1$ sequences $\\tilde{\\boldsymbol{u}}_{t,j}$, $j=0,\\dots,m$. The first input of the upper and lower encoders are now the sequences $(\\bs{u}_{t-0,0},\\bs{u}_{t-1,1},\\ldots,\\bs{u}_{t-m,m})$ and $(\\tilde{\\bs{u}}_{t-0,0},\\tilde{\\bs{u}}_{t-1,1},\\ldots,\\tilde{\\bs{u}}_{t-m,m})$, respectively, properly reordered.\n\n\n\\section{Density Evolution Analysis for SC-TCs over the Binary Erasure Channel}\\label{sec:DE}\n\nIn this section we derive the exact DE for SC-TCs. For the three considered code ensembles, we first derive the DE equations for the uncoupled ensembles and then extend them to the coupled ones.\n\\subsection{Density Evolution Equations and Decoding Thresholds} \nFor transmission over the BEC, it is possible to analyze the asymptotic behavior of TCs and SC-TCs by tracking the evolution of the erasure probability with the number of decoding iterations.\nThis evolution can be formalized in a compact way as a set of equations called DE equations. For the BEC, it is possible to derive a exact DE equations for TCs and SC-TCs.\nBy use of these equations, the BP decoding threshold can be computed.\nThe BP threshold is the largest channel erasure probability $\\varepsilon$ for which the erasure probability at the output of the BP decoder converges to zero as the block length and number of iterations grow to infinity.\n\nIt is also possible to compute the threshold of the MAP decoder, $\\varepsilon_{\\text{MAP}}$, by the use of the area theorem \\cite{Measson_Turbo}.\nAccording to the area theorem, the MAP threshold\\footnote{The threshold given by the area theorem is actually an upper bound on the MAP threshold. However, the numerical results show that the thresholds of the coupled ensembles converge to this upper bound. This indicates that the upper bound on the MAP threshold is a tight bound.} \ncan be obtained from the following equation,\n\\[\n\\int_{\\varepsilon_{\\text{MAP}}}^1\\bar{p}_{\\text{extr}}(\\varepsilon)d\\varepsilon=R \\ ,\n\\]\nwhere $R$ is the rate of the code and $\\bar{p}_{\\text{extr}}(\\varepsilon)$ is the average extrinsic erasure probability for all transmitted bits.\n\n\n\\subsection{Parallel Concatenated Codes}\n\\subsubsection{Uncoupled}\nConsider the compact graph of a PCC in Fig.~\\ref{Uncoupled}(b). \nLet $p_{\\text{U},\\text{s}}^{(i)}$ and $p_{\\text{U},\\text{p}}^{(i)}$ denote \nthe average extrinsic erasure probability from factor node $T^{\\text{U}}$ to $\\bs{u}$ and $\\bs{v}^{\\text{U}}$, respectively, in the $i$th iteration.\\footnote{With some abuse of language, we sometimes refer to a variable node representing a sequence (e.g., $\\u$) as the sequence itself ($\\u$ in this case).} Likewise, denote by $p_{\\text{L},\\text{s}}^{(i)}$ and $p_{\\text{L},\\text{p}}^{(i)}$ the extrinsic erasure probabilities from $T^{\\text{L}}$ to $\\bs{u}$ and $\\bs{v}^{\\text{L}}$, respectively. It is easy to see that the erasure probability from $\\bs{u}_t$ and $\\bs{v}_t^{\\text{U}}$ to $T^{\\text{U}}$ is $\\varepsilon \\cdot p_{\\text{L},\\text{s}}^{(i-1)}$ and $\\varepsilon$, respectively. Therefore, the DE updates for $T^{\\text{U}}$ can be written as\n\\begin{align}\n\\label{DEPCC1}\np_{\\text{U},\\text{s}}^{(i)}=f_{\\text{U,s}}\\left(\nq_{\\text{L}}^{(i)},\\varepsilon\\right),\\\\\n\\label{DEPCC2}\np_{\\text{U},\\text{p}}^{(i)}=f_{\\text{U,p}}\\left(\nq_{\\text{L}}^{(i)},\\varepsilon\\right),\n\\end{align}\nwhere\n\\begin{equation}\n\\label{DEPCC3}\nq_{\\text{L}}^{(i)}=\\varepsilon \\cdot p_{\\text{L},\\text{s}}^{(i-1)},\n\\end{equation}\nand $f_{\\text{U,s}}$ and $f_{\\text{U,p}}$ denote the transfer function of $T^{\\text{U}}$ for the systematic and parity bits, respectively. \n\nSimilarly, the DE update for $T^{\\text{L}}$ can be written as\n\\begin{align}\n\\label{DEPCC4}\np_{\\text{L},\\text{s}}^{(i)}=f_{\\text{L,s}}\\left(\nq_{\\text{U}}^{(i)},\\varepsilon\\right),\\\\ \\label{DEPCC5}\np_{\\text{L},\\text{p}}^{(i)}=f_{\\text{L,p}}\\left(\nq_{\\text{U}}^{(i)},\\varepsilon\\right),\n\\end{align}\nwhere\n\\begin{equation}\n\\label{DEPCC6}\nq_{\\text{U}}^{(i)}=\\varepsilon \\cdot p_{\\text{U},\\text{s}}^{(i-1)},\n\\end{equation}\nand $f_{\\text{L,s}}$ and $f_{\\text{L,p}}$ are the transfer functions of $T^{\\text{L}}$ for the systematic and parity bits, respectively.\n\\subsubsection{Coupled}\nConsider the compact graph of a SC-PCC ensemble in Fig.~\\ref{Coupled}(a).\nThe variable node $\\bs{u}_t$ is connected to factor nodes $T^{\\text{U}}_{t'}$ and $T^{\\text{L}}_{t'}$, at time instants $t'=t,\\ldots,t+m$. We denote by $p_{\\text{U},\\text{s}}^{(i,t')}$ and $p_{\\text{U},\\text{p}}^{(i,t')}$ \nthe average extrinsic erasure probability from factor node $T^{\\text{U}}_{t'}$ at time instant $t'$ to $\\bs{u}$ and $\\bs{v}^{\\text{U}}$, respectively, computed in the $i$th iteration. We also denote by $\\bar{q}^{(i-1,t)}_{\\text{U}}$ the input erasure probability to variable node $\\bs{u}_t$ in the $i$th iteration, received from its neighbors $T^{\\text{U}}_{t'}$. It can be written as\n\\begin{equation}\\label{DESCPCC1}\n\\bar{q}_{\\text{U}}^{(i-1,t)}=\\frac{1}{m+1} \\sum_{j=0}^{m} p_{\\text{U},\\text{s}}^{(i-1,t+j)}.\n\\end{equation}\n\nSimilarly, the average erasure probability from factor nodes $T^{\\text{L}}_{t'}$, $t'=t,\\ldots,t+m$, to $\\bs{u}_t$, denoted by $\\bar{q}_{\\text{L}}^{(i-1,t)}$, can be written as\n\\begin{equation}\\label{DESCPCC2}\n\\bar{q}_{\\text{L}}^{(i-1,t)}=\\frac{1}{m+1} \\sum_{j=0}^{m} p_{\\text{L},\\text{s}}^{(i-1,t+j)}.\n\\end{equation} \n\nThe erasure probabilities from variable node $\\bs{u}_t$ to its neighbors $T^{\\text{U}}_{t'}$ and $T^{\\text{L}}_{t'}$ are\n $\\varepsilon \\cdot \\bar{q}^{(i-1,t)}_{\\text{L}}$ and $\\varepsilon \\cdot \\bar{q}^{(i-1,t)}_{\\text{U}}$, respectively.\n\nOn the other hand, $T^{\\text{U}}_t$ at time $t$ is connected to the set of $\\bs{u}_{t'}$s for $t'=t-m, \\dots, t$. \nThe erasure probability to $T^{\\text{U}}_t$ from this set, denoted by $q_{\\text{L}}^{(i,t)}$, is given by\n\\begin{align}\n\\label{DESCPCC3}\nq_{\\text{L}}^{(i,t)}&=\\varepsilon \\cdot \\frac{1}{m+1} \\sum_{k=0}^{m}\n\\bar{q}_{\\text{L}}^{(i-1,t-k)}\\nonumber\\\\\n&=\\varepsilon \\cdot \\frac{1}{(m+1)^2} \\sum_{k=0}^{m}\\sum_{j=0}^{m} p_{\\text{L},\\text{s}}^{(i-1,t+j-k)}.\n\\end{align}\n\nThus, the DE updates of $T^{\\text{U}}_t$ are\n\\begin{align}\n\\label{DESCPCC4}\np_{\\text{U},\\text{s}}^{(i,t)}=f_{\\text{U,s}}\\left(\nq_{\\text{L}}^{(i,t)},\\varepsilon\\right),\\\\\n\\label{DESCPCC5}\np_{\\text{U},\\text{p}}^{(i,t)}=f_{\\text{U,p}}\\left(\nq_{\\text{L}}^{(i,t)},\\varepsilon\\right).\n\\end{align}\n\nSimilarly, the input erasure probability to $T^{\\text{L}}_t$ from the set of connected $\\bs{u}_{t'}$s at time instants $t'=t-m, \\dots, t$, is\n\\begin{align}\n\\label{DESCPCC6}\n&q_{\\text{U}}^{(i,t)}=\\varepsilon \\cdot \\frac{1}{m+1} \\sum_{k=0}^{m}\n\\bar{q}_{\\text{U}}^{(i-1,t-k)} \\nonumber\\\\\n&=\\varepsilon \\cdot \\frac{1}{(m+1)^2} \\sum_{k=0}^{m}\\sum_{j=0}^{m} p_{\\text{U},\\text{s}}^{(i-1,t+j-k)},\n\\end{align} \nand the DE updates of $T^{\\text{L}}_t$ are\n\\begin{align}\n\\label{DESCPCC7}\np_{\\text{L},\\text{s}}^{(i,t)}=f_{\\text{L,s}}\\left(\nq_{\\text{U}}^{(i,t)},\\varepsilon\\right),\\\\\n\\label{DESCPCC8}\np_{\\text{L},\\text{p}}^{(i,t)}=f_{\\text{L,p}}\\left(\nq_{\\text{U}}^{(i,t)},\\varepsilon\\right).\n\\end{align}\n\nFinally the a-posteriori erasure probability on $\\bs{u}_t$ at time $t$ and iteration $i$ is\n\\begin{equation}\np_a^{(i,t)}=\\varepsilon \\cdot \\bar{q}_{\\text{U}}^{(i,t)} \\cdot \\bar{q}_{\\text{L}}^{(i,t)}.\n\\end{equation}\n DE is performed by tracking the evolution of the a-posteriori erasure probability\n with the number of iterations.\n\\subsection{Serially Concatenated Codes}\n\\subsubsection{Uncoupled}\nConsider the compact graph of the SCC ensemble in Fig.~\\ref{Uncoupled}(c).\nLet $p_{\\text{O},\\text{s}}^{(i)}$ and $p_{\\text{O},\\text{p}}^{(i)}$ denote the erasure probability from $T^{\\text{O}}$ to $\\bs{u}$ and $\\bs{v}^{\\text{O}}$, respectively, computed in the $i$th iteration.\nLikewise, $p_{\\text{I},\\text{s}}^{(i)}$ and $p_{\\text{I},\\text{p}}^{(i)}$ denote the extrinsic erasure probability from $T^{\\text{I}}$ to $\\tilde{\\bs{v}}^{\\text{O}}=(\\bs{u},\\bs{v}^{\\text{O}})$ and $\\bs{v}^{\\text{I}}$. \n\nBoth $\\bs{u}$ and $\\bs{v}^{\\text{O}}$ receive the same erasure probability, $p_{\\text{I},\\text{s}}^{(i-1)}$, from $T^{\\text{I}}$. Therefore, the erasure probabilities that $T^{\\text{O}}$ receives from these two variable nodes are equal and given by\n\\begin{align}\nq_{\\text{I}}^{(i)}=\\varepsilon \\cdot p_{\\text{I},\\text{s}}^{(i-1)}.\\label{DESCC1}\n\\end{align}\nThe DE equations for $T^{\\text{O}}$ can then be written as\n\\begin{align}\n\\label{DESCC2}\np_{\\text{O},\\text{s}}^{(i)}=f_{\\text{O,s}}\\left(\nq_{\\text{I}}^{(i)},q_{\\text{I}}^{(i)}\\right),\\\\\n\\label{DESCC3}\np_{\\text{O},\\text{p}}^{(i)}=f_{\\text{O,p}}\\left(\nq_{\\text{I}}^{(i)},q_{\\text{I}}^{(i)}\\right),\n\\end{align}\nwhere $f_{\\text{O,s}}$ and $f_{\\text{O,p}} $ are the transfer functions of $T^{\\text{O}}$ for the systematic and parity bits, respectively. \n\n The erasure probability that $T^{\\text{I}}$ receives from $\\tilde{\\bs{v}}^{\\text{O}}=(\\bs{u},\\bs{v}^{\\text{O}})$ is the average of the erasure probabilities from $\\bs{u}$ and $\\bs{v}^{\\text{O}}$,\n\\begin{equation}\n\\label{DESCC4}\nq_{\\text{O}}^{(i)}=\\varepsilon \\cdot\\frac{p_{\\text{O},\\text{s}}^{(i)}+p_{\\text{O},\\text{p}}^{(i)}}{2}.\n\\end{equation}\nOn the other hand, the erasure probability to $T^{\\text{I}}$ from $\\bs{v}^{\\text{I}}$ is $\\varepsilon$. Therefore, the DE equations for $T^{\\text{I}}$ can be written as\n\\begin{align}\n\\label{DESCC5}\np_{\\text{I},\\text{s}}^{(i)}=f_{\\text{I,s}}\\left(\nq_{\\text{O}}^{(i)},\\varepsilon\\right),\n\\\\\n\\label{DESCC6}\np_{\\text{I},\\text{p}}^{(i)}=f_{\\text{I,p}}\\left(\nq_{\\text{O}}^{(i)},\\varepsilon\\right).\n\\end{align}\n\\subsubsection{Coupled}\nConsider the compact graph representation of SC-SCCs in Fig.~\\ref{Coupled}(b). \nVariable nodes $\\bs{u}_t$ and $\\bs{v}_t^{\\text{O}}$ are connected to factor nodes $T^{\\text{I}}_{t'}$ at time instants $t'=t,\\ldots,t+m$. \nThe input erasure probability to variable nodes $\\bs{u}_t$ and $\\bs{v}_t^{\\text{O}}$ from these factor nodes, denoted by $\\bar{q}_{\\text{I}}^{(i-1,t)}$, is the same for both $\\bs{u}_t$ and $\\bs{v}_t^{\\text{O}}$ and is obtained as the average of the erasure probabilities from each of the factor nodes $T^{\\text{I}}_{t'}$,\n\\begin{equation}\n\\label{DESCSCC1}\n\\bar{q}_{\\text{I}}^{(i-1,t)}=\\frac{1}{m+1}\\sum_{j=0}^{m}p_{\\text{I},\\text{s}}^{(i-1,t+j)} \\ .\n\\end{equation}\nThe erasure probability to $T^{\\text{O}}_{t}$ from $\\bs{u}_t$ and $\\bs{v}_t^{\\text{O}}$ is\n\\begin{align}\n\\label{DESCSCC2}\nq_{\\text{I}}^{(i,t)}=\\varepsilon \\cdot \\bar{q}_{\\text{I}}^{(i-1,t)}= \\frac{\\varepsilon}{m+1}\\sum_{j=0}^{m}p_{\\text{I},\\text{s}}^{(i-1,t+j)} \\ .\n\\end{align}\nThus, the DE updates of $T^{\\text{O}}_t$ are\n\\begin{align}\n\\label{DESCSCC3}\np_{\\text{O},\\text{s}}^{(i,t)}=f_{\\text{O,s}}\\left(\nq_{\\text{I}}^{(i,t)},q_{\\text{I}}^{(i,t)}\\right) \\ ,\\\\\n\\label{DESCSCC4}\np_{\\text{O},\\text{p}}^{(i,t)}=f_{\\text{O,p}}\\left(\nq_{\\text{I}}^{(i,t)},q_{\\text{I}}^{(i,t)}\\right) \\ . \n\\end{align}\nAt time $t$, $T^{\\text{I}}_t$ is connected to a set of $\\tilde{\\bs{v}}_{t'}^{\\text{O}}$s at time instants $t'=t-m,\\ldots,t$. \nThe erasure probability that $T^{\\text{I}}_t$ receives from this set is the average of the erasure probabilities of all $\\bs{u}_{t'}$s and $\\bs{v}_{t'}^{\\text{O}}$s at times $t'=t-m\\ldots,t$. This erasure probability can be written as\n \\begin{align}\n\\label{DESCSCC5}\nq_{\\text{O}}^{(i,t)}=\\frac{\\varepsilon}{m+1}\\sum_{k=0}^{m}\\frac{p_{\\text{O},\\text{s}}^{(i,t-k)}+p_{\\text{O},\\text{p}}^{(i,t-k)}}{2} \\ .\n\\end{align}\nHence, the DE updates for the inner encoder are given by\n\\begin{align}\n\\label{DESCSCC6}\np_{\\text{I},\\text{s}}^{(i,t)}=f_{\\text{I,s}}\\left(\nq_{\\text{O}}^{(i,t)},\\varepsilon\\right) \\ ,\n\\\\\n\\label{DESCSCC7}\np_{\\text{I},\\text{p}}^{(i,t)}=f_{\\text{I,p}}\\left(\nq_{\\text{O}}^{(i,t)},\\varepsilon\\right) \\ .\n\\end{align}\nFinally, the a-posteriori erasure probability on information bits at time $t$ and iteration $i$ is\n\\begin{equation}\np_{a}^{(i,t)}=\\varepsilon \\cdot p_{\\text{O},\\text{s}}^{(i,t)}\\cdot \\bar{q}_{\\text{I}}^{(i,t)} \\ .\n\\end{equation}\n\\subsection{Braided Convolutional Codes}\n\\subsubsection{Uncoupled}\nConsider the compact graph of uncoupled BCCs in Fig.~\\ref{Uncoupled}(c). These can be obtained by tailbiting BCCs, as shown in Fig.~\\ref{BCCSC}, with coupling length $L=1$.\nLet $p_{\\text{U},k}^{(i)}$ and $p_{\\text{L},k}^{(i)}$ denote the erasure probabilities of messages from $T^{\\text{U}}$ and $T^{\\text{L}}$ through their $k$th connected edge, $k=1,2,3$, respectively.\nThe erasure probability of messages that $T^{\\text{U}}$ receives through its edges are\n\\begin{align}\nq_{\\text{L},1}^{(i)}=\\varepsilon \\cdot p_{\\text{L},1}^{(i-1)} \\ ,\\label{DEBCC1}\\\\\nq_{\\text{L},2}^{(i)}=\\varepsilon \\cdot p_{\\text{L},3}^{(i-1)} \\ ,\\label{DEBCC2}\\\\\nq_{\\text{L},3}^{(i)}=\\varepsilon \\cdot\np_{\\text{L},2}^{(i-1)} \\ .\\label{DEBCC3}\n\\end{align}\nThe exact DE equations of $T^{\\text{U}}$ can be written as\n\\begin{align}\np_{\\text{U},1}^{(i)}=&f_{\\text{U},1}\\left(q_{\\text{L},1}^{(i)} ,q_{\\text{L},2}^{(i)},q_{\\text{L},3}^{(i)}\\right) \\label{DEBCC4}\\ ,\\\\\np_{\\text{U},2}^{(i)}=&f_{\\text{U},2}\\left(q_{\\text{L},1}^{(i)} ,q_{\\text{L},2}^{(i)},q_{\\text{L},3}^{(i)}\\right) \\label{DEBCC5}\\ ,\\\\\np_{\\text{U},3}^{(i)}=&f_{\\text{U},3}\\left(q_{\\text{L},1}^{(i)} ,q_{\\text{L},2}^{(i)},q_{\\text{L},3}^{(i)}\\right) \\label{DEBCC6}\\ , \n\\end{align}\nwhere $f_{\\text{U},k}$ denotes the transfer function of $T^{\\text{U}}$ for its $k$th connected edge.\nSimilarly, the DE equations for $T^{\\text{L}}$ can be written by swapping indexes $\\text{U}$ and $\\text{L}$ in \\eqref{DEBCC1}--\\eqref{DEBCC6}.\n\\subsubsection{Coupled}\nConsider the compact graph representation of Type-I BCCs in Fig.~\\ref{BCCI-II}(a).\nAs in the uncoupled case, the DE updates of factor nodes $T^{\\text{U}}_t$ and $T^{\\text{L}}_t$ are similar due to the symmetric structure of the coupled construction. Therefore, for simplicity, we only describe the DE equations of $T^{\\text{U}}_t$ and the equations for $T^{\\text{L}}_t$ are obtained by swapping indexes $\\text{U}$ and $\\text{L}$ in the equations.\n\nThe first edge of $T^{\\text{U}}_t$ is connected to $\\bs{u}_t$. Thus, the erasure probability that $T^{\\text{U}}_t$ receives through this edge is\n\\begin{equation}\n\\label{eq:T1BCCDE1}\nq_{\\text{L},1}^{(i,t)}=\\varepsilon \\cdot p_{\\text{L},1}^{(i-1,t)}.\n\\end{equation}\nThe second edge of $T^{\\text{U}}_t$ is connected to variable nodes $\\bs{v}_{t'}^{\\text{L}}$ at time instants $t'=t-m,\\ldots,t-1$.\n The erasure probability that $T^{\\text{U}}_t$ receives through its second edge is therefore the average of the erasure probabilities from the variable nodes $\\bs{v}_{t'}^{\\text{L}}$ that are connected to this edge. \nThis erasure probability can be written as\n\\begin{equation}\n\\label{eq:T1BCCDE2}\nq_{\\text{L},2}^{(i,t)}=\\frac{\\varepsilon}{m}\\sum_{j=1}^{m} p_{\\text{L},3}^{(i-1,t-j)}.\n\\end{equation}\nThe third edge of $T^{\\text{U}}_t$ is connected to $\\bs{v}_t^{\\text{U}}$, which is in turn connected to the second edges of factor nodes $T^{\\text{L}}_{t'}$ at time instants $t'=t+1,\\ldots,t+m$. \nThe erasure probability that $\\bs{v}_t^{\\text{U}}$ receives from the set of connected nodes $T^{\\text{L}}_{t'}$ is the average of erasure probabilities from these nodes through their second edges.\nThe erasure probability from $\\bs{v}_t^{\\text{U}}$ to $T^{\\text{U}}_t$ is\n\\begin{equation}\n\\label{eq:T1BCCDE3}\nq_{\\text{L},3}^{(i,t)}=\\frac{\\varepsilon}{m}\\sum_{j=1}^{m} p_{\\text{L},2}^{(i-1,t+j)}.\n\\end{equation}\nThe DE equations of $T^{\\text{U}}_t$ can then be written as\\footnote{The DE equations of the original BCCs are obtained by setting $m=1$ in the DE equations of Type-I BCCs.}\n\\begin{align}\n\\label{eq:T1BCCDE4}\np_{\\text{U},1}^{(i,t)}=&f_{\\text{U},1}\\left(q_{\\text{L},1}^{(i,t)},q_{\\text{L},2}^{(i,t)},q_{\\text{L},3}^{(i,t)}\\right),\\\\ \\label{eq:T1BCCDE5}\np_{\\text{U},2}^{(i,t)}=&f_{\\text{U},2}\\left(q_{\\text{L},1}^{(i,t)},q_{\\text{L},2}^{(i,t)},q_{\\text{L},3}^{(i,t)}\\right),\\\\ \\label{eq:T1BCCDE6}\np_{\\text{U},3}^{(i,t)}=&f_{\\text{U},3}\\left(q_{\\text{L},1}^{(i,t)},q_{\\text{L},2}^{(i,t)},q_{\\text{L},3}^{(i,t)}\\right).\n\\end{align} \nThe a-posteriori erasure probability on $\\bs{u}_t$ at time $t$ and iteration $i$ for Type-I BCCs is\n\\begin{equation}\np_a^{(i,t)}=\\varepsilon \\cdot p_{\\text{U},1}^{(i,t)} \\cdot p_{\\text{L},1}^{(i,t)}.\n\\end{equation}\n\n\nAs we discussed in the previous section, the difference between Type-I and Type-II BCCs is that $\\bs{u}_t$ is also coupled in the latter. Variable node\n$\\bs{u}_t$ in Type-II BCCs is connected to a set of factor nodes $T^{\\text{U}}_{t'}$ and $T^{\\text{L}}_{t'}$ at time instants $t'=t,\\ldots,t+m$.\nThe DE equations of Type-II BCCs are identical to those of Type-I BCCs except for equation \\eqref{eq:T1BCCDE1}. Denote by $\\bar{q}_{\\text{L},1}^{(i-1,t)}$ the input erasure probability to $\\bs{u}_t$ from the connected factor nodes $T^{\\text{L}}_{t'}$ in the $i$th iteration.\nAccording to Fig.~\\ref{BCCI-II}(b), $\\bar{q}_{\\text{L},1}^{(i-1,t)}$ is the average of erasure probabilities\nfrom $T^{\\text{L}}_{t'}$ at time instants $t'=t,\\ldots,t+m$, \n\\begin{equation}\n\\bar{q}_{\\text{L},1}^{(i-1,t)}=\\frac{1}{m+1} \\sum_{j=0}^{m} p_{\\text{L},1}^{(i-1,t+j)}.\n\\end{equation}\nFactor node $T^{\\text{U}}_{t}$ is connected to variable nodes $\\bs{u}_{t'}$ at time instants $t'=t-m,\\ldots,t$. \nThe incoming erasure probability to $T^{\\text{U}}_t$ through its first edge, denoted by $q_{\\text{L},1}^{(i,t)}$, is therefore the average of the erasure probabilities from $\\bs{u}_{t'}$ at times $t'=t-m,\\ldots,t$, \n \\begin{align}\n&q_{\\text{L},1}^{(i,t)}=\\varepsilon \\cdot \\frac{1}{m+1} \\sum_{k=0}^{m}\n\\bar{q}_{\\text{L},1}^{(i-1,t-k)}\\\\\n&=\\varepsilon \\cdot \\frac{1}{(m+1)^2} \\sum_{k=0}^{m}\\sum_{j=0}^{m} p_{\\text{L},1}^{(i-1,t+j-k)}.\\nonumber\n\\end{align}\nFinally, the a-posteriori erasure probability on $\\bs{u}_t$ at time $t$ and iteration $i$ for Type-II BCCs is\n\\begin{equation}\np_a^{(i,t)}=\\varepsilon \\cdot \\bar{q}_{\\text{U}}^{(i,t)} \\cdot \\bar{q}_{\\text{L}}^{(i,t)}.\n\\end{equation}\n\n\n\\section{Rate-compatible SC-TCs via Random Puncturing}\\label{sec:RandomP}\n\n\nHigher rate codes can be obtained by applying puncturing. For analysis purposes, we consider random puncturing. Random puncturing has been considered, e.g., for LDPC codes in \\cite{PishroNik, LDPCPuncture} and for turbo-like codes in \\cite{AGiAa,Kol12}. In \\cite{LDPCPuncture}, the authors introduced a parameter called $\\theta$ which allows comparing the strengths of the codes with different rates. In this section, we consider the construction of rate-compatible SC-TCs by means of random puncturing.\n\n\nWe denote by $\\rho\\in[0,1]$ the fraction of surviving bits after puncturing, referred to as the permeability rate. Consider that a code sequence $\\boldsymbol{v}$ is randomly punctured with permeability rate $\\rho$ and transmitted over a BEC with erasure probability $\\varepsilon$, BEC$(\\varepsilon)$. For the BEC, applying puncturing is equivalent to transmitting $\\bs{v}$ over a BEC with erasure probability $\\varepsilon_{\\rho}=1-(1-\\varepsilon)\\rho$, resulting from the concatenation of two BECs, BEC$(\\varepsilon)$ and BEC$(\\varepsilon_\\rho)$. The DE equations of SC-TCs in the previous section can then be easily modified to account for random puncturing.\n\nFor SC-PCCs, we consider puncturing of parity bits only, i.e., the overall code is systematic. The rate of the punctured code (without considering termination of the coupled chain) is\n$R=\\frac{1}{1+2\\rho}$. The DE equations of punctured SC-PCCs are obtained\nby substituting $\\varepsilon\\leftarrow \\varepsilon_{\\rho}$ in \\eqref{DESCPCC4}, \\eqref{DESCPCC5}, \\eqref{DESCPCC7} and \\eqref{DESCPCC8}. \n\nFor punctured SC-SCCs, we consider the coupling of the punctured SCCs proposed in \\cite{AGiAa,AGiAb}\\footnote{ In contrast to standard SCCs, characterized by a rate-$1$ inner code and for which to achieve higher rates the outer code is heavily punctured, the SCCs proposed in \\cite{AGiAa,AGiAb} achieve higher rates by moving the puncturing of the outer code to the inner code, which is punctured beyond the unitary rate. This allows to preserve the interleaving gain for high rates and yields a larger minimum distance, which results in codes that significantly outperform standard SCCs, especially for high rates. Furthermore, the SCCs in \\cite{AGiAa,AGiAb} yield better MAP thresholds than standard SCCs.}, where $\\rho_0$ and $\\rho_1$ are the permeability rates of the systematic and\nparity bits, respectively, of the outer code (see\n\\cite[Fig.~1]{AGiAb}), and $\\rho_2$ is the permeability rate of the\nparity bits of the inner code. The code rate of the \npunctured SC-SCC is\n$R=\\frac{1}{\\rho_0+\\rho_1+2\\rho_2}$ (neglecting the rate loss due to termination). The DE for punctured SC-SCCs is\nobtained by substituting $\\varepsilon\\leftarrow\\varepsilon_{\\rho_2}$\nin \\eqref{DESCSCC6} and \\eqref{DESCSCC7}, and modifying \\eqref{DESCSCC5} to\n\\begin{align*}\nq_{\\text{O}}^{(i,t)}=\\frac{1}{m+1}\\sum_{k=0}^{m}\\frac{\\varepsilon\n \\cdot p_{\\text{O},\\text{s}}^{(i,t-k)}\n+\\varepsilon_{\\rho_1} \\cdot p_{\\text{O},\\text{p}}^{(i,t-k)}}{2}\n\\end{align*}\nand \\eqref{DESCSCC3}, \\eqref{DESCSCC4} to\n\\begin{align}\np_{\\text{O},\\text{s}}^{(i,t)}=f_{\\text{O,s}}\\left(\nq_{\\text{I}}^{(i,t)},\\tilde{q}_{\\text{I}}^{(i,t)}\\right),\\\\\np_{\\text{O},\\text{p}}^{(i)}=f_{\\text{O,p}}\\left(\nq_{\\text{I}}^{(i,t)},\\tilde{q}_{\\text{I}}^{(i,t)}\\right),\n\\end{align}\nwhere $q_{\\text{I}}^{(i,t)}$ is given in \\eqref{DESCSCC2} and\n\\begin{equation}\n\\tilde{q}_{\\text{I}}^{(i,t)}=\\frac{\\varepsilon_{\\rho_1}}{m+1}\\sum_{j=0}^{m}p_{\\text{I},\\text{s}}^{(i-1,t+j)}.\n\\end{equation}\n\nFor both Type-I and Type-II BCCs, similarly to SC-PCCs, we consider only puncturing of parity bits with permeability rate $\\rho$.\nThe DE equations of punctured SC-BCCs are obtained by substituting $\\varepsilon\\leftarrow \\varepsilon_{\\rho}$ in \\eqref{eq:T1BCCDE2} and \\eqref{eq:T1BCCDE3} and the corresponding equations for $q_{\\text{U},2}^{(i,t)}$ and $q_{\\text{U},3}^{(i,t)}$.\n\n\\section{Numerical Results}\\label{Sec6}\n\nIn Table~\\ref{Tab:BPThresholds}, we give DE results for the SC-TC ensembles, and their uncoupled ensembles for rate $R=1\/2$.\nIn particular, we consider SC-PCC and SC-SCC ensembles with identical 4-state and 8-state component encoders with generator matrix ${\\bf{G}}=(1,5\/7)$ and ${\\bf{G}}=(1,11\/13)$, respectively, in octal notation.\nFor the BCC ensemble, we consider two identical 4-state component encoders and generator matrix\n\\begin{equation}\\label{eqG}\n\\bs{G}_1 (D)= \\left( \\begin{array}{ccc}1&0&1\/7\\\\0&1&5\/7\\end{array}\\right) \\ .\n\\end{equation}\nThe BP thresholds ($\\varepsilon_{\\text{BP}}$) and MAP thresholds ($\\varepsilon_{\\text{MAP}}$) for the uncoupled ensembles are reported in Table~\\ref{Tab:BPThresholds}.\nThe MAP threshold is obtained using the area theorem \\cite{AshikhminEXIT,Measson2009}.\nWe also give the BP thresholds of SC-TCs for coupling memory $m=1$, denoted by $\\varepsilon_{\\text{SC}}^{1}$.\n\\begin{table}[t]\n\n\t\\caption{Thresholds for rate-$1\/2$ TCs, and SC-TCs}\n\n\t\\begin{center}\n\t\t\\begin{tabular}{lcccc}\n\t\t\t\\hline\n\t\t\tEnsemble&states&$\\varepsilon_{\\text{BP}}$ & $\\varepsilon_{\\text{MAP}}$ &$\\varepsilon^1_{\\mathrm{SC}}$ \\\\\n\t\t\t\\hline\n\t\t\n\t\t\n\t\t\n\t\t\t$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$&4&0.4606&0.4689&0.4689\\\\[0.5mm]\n\t\t\t$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$&4&0.3594&0.4981& 0.4708\\\\[0.5mm]\n\t\t\t$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$&8&0.4651&0.4863&0.4862\\\\[0.5mm]\n\t\t\t$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$&8&0.3120&0.4993& 0.4507\\\\\n\t\t\n\t\t\n\t\t\tType-I $\\mathcal{C}_{\\mathrm{BCC}}$&4&0.3013 &0.4993&0.4932\\\\[0.5mm]\n\t\t\tType-II $\\mathcal{C}_{\\mathrm{BCC}}$&4& 0.3013&0.4993& 0.4988\\\\[0.5mm]\n\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\\end{tabular} \n\t\\end{center}\n\t\\label{Tab:BPThresholds}\n\n\\end{table}\n\nAs expected, PCC ensembles yield better BP thresholds than SCC ensembles. However, SCCs have better MAP threshold. The BP decoder works poorly for uncoupled BCCs and the BP thresholds are worse than those of PCCs and SCCs. On the other hand, the MAP thresholds of BCCs are better than those of both PCCs and SCCs. By applying coupling, the BP threshold improves and for $m=1$, the Type-II BCC ensemble has the best coupling threshold.\n\nTable \\ref{Tab:BPThresholdsSCC} shows the thresholds of TCs and SC-TCs for several rates. In the table, for the ensembles $\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$, $\\rho_2$ is the permeability rate of the parity bits of the upper encoder and the lower encoder. For example, $\\rho_2=0.5$ means that half of the bits of $\\bs{v}^{\\text{U}}$ and $\\bs{v}^{\\text{L}}$ are punctured (thus, the resulting code rate is $R=1\/2$). Note that $\\rho_2$ corresponds to permeability $\\rho$ defined in Section~\\ref{sec:RandomP}. Here, we use $\\rho_2$ instead to unify notation with that of SCCs. For the ensembles $\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ (based on the SCCs introduced in \\cite{AGiAa,AGiAb}), for a given code rate $R$ the puncturing rates $\\rho_0$, $\\rho_1$ and $\\rho_2$ (see Section~\\ref{sec:RandomP}) may be optimized. In this paper, we consider $\\rho_0=1$, i.e., the overall code is systematic, and we optimize $\\rho_1$ and $\\rho_2$ such that the MAP threshold of the (uncoupled) SCC is maximized.\\footnote{We remark that nonsystematic codes, i.e., $\\rho_0<1$, lead to better MAP thresholds. In this case, the optimum is to puncture last the parity bits of the inner encoder, i.e., for $R<1\/2$ $\\rho_2=1$ and for $R\\ge 1\/2$ $\\rho_0=0$, $\\rho_1=0$ and $\\rho_2=1\/2R$.} Note that, if $\\rho_0=1$, for a given $R$ the optimization simplifies to the optimization of a single parameter, say $\\rho_2$, since $\\rho_1$ and $\\rho_2$ are related by $\\rho_1=\\frac{1}{R}-1-2\\rho_2$.\\footnote{Alternatively, one may optimize $\\rho_1$ and $\\rho_2$ such that the BP threshold of the SC-SCC is optimized for a given coupling memory $m$.} Rate-compatibility can be guaranteed by choosing $\\rho_1$ and $\\rho_2$ to be decreasing functions of $R$. In the table, we report the coupling thresholds for coupling memory $m=1,2,3$, denoted by $\\varepsilon_{\\text{SC}}^{1}$, $\\varepsilon_{\\text{SC}}^{2}$, and $\\varepsilon_{\\text{SC}}^{3}$, respectively. The gap to the Shannon limit is shown by $\\delta_{\\text{SH}}=(1-R)-\\varepsilon_{\\text{MAP}}$.\n\n\\begin{table*}[t]\n\\caption{Thresholds for punctured spatially coupled turbo codes}\n\\begin{center}\n\\begin{tabular}{ccccccccccc}\n\\hline\nEnsemble& Rate &states& $\\rho_2$ & $\\varepsilon_{\\text{BP}}$ & $\\varepsilon_{\\text{MAP}}$ &$\\varepsilon^1_{\\mathrm{SC}}$ & $\\varepsilon^3_{\\mathrm{SC}}$ & $\\varepsilon^5_{\\mathrm{SC}}$ &$m_{\\text{min}}$ &$\\delta_{\\text{SH}}$ \\\\\n\\hline\n$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$ & $1\/3$ &4& 1.0 & 0.6428 & 0.6553 & 0.6553 & 0.6553 & 0.6553 &1 &0.0113\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ & $1\/3$ &4&1.0 & 0.5405 & 0.6654 & 0.6437 & 0.6650 & 0.6654 &4 &0.0012\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$ & $1\/3$ &8& 1.0 &0.6368& 0.6621 & 0.6617 & 0.6621 & 0.6621&2&0.0045\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ & $1\/3$ &8&1.0 &0.5026&0.6663& 0.6313&0.6647&0.6662&6&0.0003\\\\[0.5mm]\n\\hline\n$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$ & $1\/2$ & 4&0.5 & 0.4606 & 0.4689 & 0.4689 & 0.4689 & 0.4689 &1& 0.0311\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ & $1\/2$ & 4&0.5 & 0.3594 & 0.4981 & 0.4708 & 0.4975 & 0.4981 & 5&0.0019\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$ & $1\/2$ & 8&0.5 &0.4651 & 0.4863&0.4862&0.4863&0.4863&2&0.0137\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ & $1\/2$ & 8&0.5 &0.3120 & 0.4993&0.4507 &0.4970&0.4992&7&0.0007\\\\[0.5mm]\n\\hline\n$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$ & $2\/3$ & 4&0.25 & 0.2732 & 0.2772 & 0.2772 & 0.2772 & 0.2772 &1&0.0561\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ & $2\/3$ & 4&0.25 & 0.2038 & 0.3316 & 0.3303 & 0.3305 & 0.3315 & 6&0.0018\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$ & $2\/3$ & 8&0.25 &0.2945& 0.3080&0.3080& 0.3080& 0.3080&1 &0.0253\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ & $2\/3$ & 8&0.25 &0.1507&0.3326&0.2710&0.3278 &0.3323&7&0.0007\\\\[0.5mm]\n\\hline\n$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$ & $3\/4$ & 4&0.166 & 0.1854 & 0.1876 & 0.1876 & 0.1876 & 0.1876 & 1&0.0624\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ & $3\/4$ & 4&0.166 & 0.1337 & 0.2486 & 0.2155 & 0.2471 & 0.2486 & 5&0.0014\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$ & $3\/4$ & 8&0.166 &0.2103&0.2196&0.2196&0.2196&0.2196&1&0.0304\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ & $3\/4$ & 8&0.166 &0.0865 &0.2495&0.1827&0.2416&0.2488&8&0.0005\\\\[0.5mm]\n\\hline\n$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$ & $4\/5$ & 4&0.125 & 0.1376 & 0.1391 & 0.1391 & 0.1391 & 0.1391 & 1&0.0609\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ & $4\/5$ & 4&0.125 & 0.0942 & 0.1990 & 0.1644 & 0.1968 & 0.1989 & 7&0.0011\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$ & $4\/5$ & 8&0.125 & 0.1628& 0.1698& 0.1698 &0.1698 &0.1698&1 &0.0302\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ & $4\/5$ & 8&0.125 &0.0517&0.1996&0.1302 &0.1885&0.1982&8&0.0004\\\\[0.5mm]\n\\hline\n$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$ & $9\/10$ & 4&0.055 & 0.0578 & 0.0582 & 0.0582 & 0.0582 & 0.0582 & 1&0.0418\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ & $9\/10$ & 4&0.055 & 0.0269 & 0.0996 & 0.0624 & 0.0930 & 0.0988 & 8&0.0012\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$ & $9\/10$ & 8&0.055 &0.0732&0.0761&0.0761& 0.0761&0.0761&1&0.0239\\\\[0.5mm]\n$\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$ & $9\/10$ & 8&0.055 &0.0128& 0.0999 & 0.0384& 0.0765 &0.0931 &16&0.0001\\\\[0.5mm]\n\\hline\n\n\\end{tabular} \n\\end{center}\n\\label{Tab:BPThresholdsSCC}\n\\end{table*}\n\n\n\\begin{table*}[t]\n\n\t\\caption{Thresholds for punctured Braided Convolutional Codes}\n\n\t\\begin{center}\n\t\t\\begin{tabular}{cccccccccc}\n\t\t\t\\hline\n\t\t\tEnsemble& Rate &states& $\\rho_2$ & $\\varepsilon_{\\text{BP}}$ & $\\varepsilon_{\\text{MAP}}$ &$\\varepsilon^1_{\\mathrm{SC}}$ & $\\varepsilon^3_{\\mathrm{SC}}$ & $\\varepsilon^5_{\\mathrm{SC}}$ & $\\delta_{\\text{SH}}$ \\\\\n\t\t\t\\hline\n\t\t\tType-I& $1\/3$ &4& 1.0 & 0.5541 & 0.6653 & 0.6609&0.6644 & 0.6650& 0.0013\\\\[0.5mm]\n\t\t\tType-II & $1\/3$ &4&1.0 &0.5541 & 0.6653& 0.6651 &0.6653 &0.6653 &0.0013\\\\[0.5mm]\n\t\t\n\t\t\t\\hline\n\t\t\tType-I & $1\/2$ & 4&0.5 &0.3013&0.4993&0.4932&0.4980& 0.4988&0.0007 \\\\[0.5mm]\n\t\t\tType-II & $1\/2$ & 4&0.5 &0.3013&0.4993& 0.4988&0.4993&0.4993&0.0007 \\\\[0.5mm]\n\t\t\n\t\t\t\\hline\n\t\t\tType-I & $2\/3$ & 4&0.25 & -- &0.3331&0.3257&0.3315&0.3325&0.0002\\\\[0.5mm]\n\t\t\tType-II & $2\/3$ & 4&0.25 & -- &0.3331& 0.3323&0.3331 &0.3331&0.0002\\\\[0.5mm]\n\t\t\n\t\t\t\\hline\n\t\t\tType-I & $3\/4$ & 4&0.166& -- &0.2491&0.2411&0.2473&0.2484&0.0009\\\\[0.5mm]\n\t\t\tType-II & $3\/4$ & 4&0.166 & -- &0.2491&0.2481&0.2491 &0.2491 &0.0009\\\\[0.5mm]\n\t\t\n\t\t\t\\hline\n\t\t\tType-I & $4\/5$ & 4&0.125 & -- &0.1999&0.1915 &0.1979&0.1991 &0.0001\\\\[0.5mm]\n\t\t\tType-II & $4\/5$ & 4&0.125 & -- &0.1999&0.1986&0.1999&0.1999&0.0001\\\\[0.5mm]\n\t\t\n\t\t\t\\hline\n\t\t\tType-I & $9\/10$ & 4&0.055 & -- &0.0990&0.0893&0.0966&0.0980&0.0010 \\\\[0.5mm]\n\t\t\tType-II & $9\/10$ & 4&0.055& -- &0.0990&0.0954& 0.0990&0.0990&0.0010\\\\[0.5mm]\n\t\t\n\t\t\t\n\t\t\t\\hline\n\t\t\t\n\t\t\\end{tabular} \n\t\\end{center}\n\t\\label{BPThresholdsBCC}\n\n\\end{table*}\n\nFor large enough coupling memory, we observe threshold saturation for both SC-PCCs and SC-SCCs. The value $m_{\\text{min}}$ in Table~\\ref{Tab:BPThresholdsSCC} denotes the smallest coupling memory for which threshold saturation is observed numerically. Interestingly, thanks to the threshold saturation phenomenon, for large enough coupling memory SC-SCCs achieve better BP threshold than SC-PCCs. We remark that SCCs yield better minimum Hamming distance than PCCs \\cite{Benedetto98Serial}.\n\nComparing ensembles with 8-state component encoders and ensembles with 4-state component encoders, we observe that the MAP threshold improves for all the considered cases, since the overall codes become stronger. For PCCs, the BP threshold also improves for 8-state component encoders, but only with puncturing, i.e., for $R>1\/3$. For SCCs, on the other hand, the BP threshold gets worse if higher memory component encoders are used. Due to this fact, a higher coupling memory $m_{\\text{min}}$ is needed for SC-SCCs with 8-state component encoders until threshold saturation is observed, and this effect becomes more pronounced for larger rates. However, the achievable BP thresholds of SC-SCCs are better than those of SC-PCCs for all rates.\n\n\n\nIn Table~\\ref{BPThresholdsBCC}, we give BP thresholds for Type-I and Type-II SC-BCCs with different coupling memories and several rates.\\footnote{The BP threshold of the Type-I BCC with $m=1$ corresponds to the BP threshold of the original BCC.} As for PCCs, $\\rho_2$ is the permeability rate of the parity bits of the upper encoder and the lower encoder. We also report the BP threshold and MAP threshold of the uncoupled ensembles.\nAlmost in all rates, the BP decoder works poorly for uncoupled BCCs and the BP thresholds are worse than those of PCCs and SCCs (an exception are SCCs with $R=1\/3$). This is specially significant for rates $R\\ge 2\/3$, for which the BP thresholds of uncoupled BCCs are very close to zero.\nOn the other hand, the MAP thresholds of BCCs are better than those of both PCCs and SCCs for all rates. As for SC-PCCs and SC-SCCs, the BP thresholds improve if coupling is applied.\nType-II BCCs yield better thresholds than Type-I BCCs and achieve threshold saturation for small coupling memories.\nIn contrast, for the coupling memories considered, threshold saturation is not observed for Type-I BCCs.\n\n\n\nFor comparison purposes, in Table~\\ref{BPThresholdsLDPC} we report the $\\varepsilon_{\\text{BP}}$, $\\varepsilon_{\\text{MAP}}$, and $\\varepsilon_{\\text{SC}}^{1}$ for three rate-$1\/2$ LDPC code ensembles.\nAs it is well known, by increasing the variable node degree, the MAP threshold improves, but the BP threshold decreases.\nSimilarly to TCs, applying the coupling improves the BP threshold.\nAmong all the ensembles shown in Table \\ref{BPThresholdsLDPC}, the $(5,10)$ LDPC ensemble has the best MAP threshold.\nHowever, for this ensemble the gap between the BP and MAP thresholds is larger than that of the other LDPC code ensembles and the coupling (with $m=1$) is not able to completely close this gap, therefore $\\varepsilon_{\\text{SC}}^{1}$ is worse than that of other two SC-LPDC code ensembles. Among all codes in Table~\\ref{BPThresholdsLDPC}, the best $\\varepsilon_{\\text{BP}}$ is achieved by the Type II BCC ensemble. Similar to the $(5,10)$ LDPC code ensemble, the gap between the BP and the MAP threshold is relatively large for BCCs. However, for BCCs the BP threshold increases significantly after applying coupling with $m=1$.\nIn addition, the only way to increase the MAP threshold of the LDPC codes is to increase their variable node degree, but in TCs the BP threshold can be improved by several different methods, e.g., increasing the component code memory, selecting a good ensemble, or increasing the variable node degree.\n\n\n\\begin{table}[t]\n\n\t\\caption{Thresholds for rate-$1\/2$ TCs, SC-TCs, LDPC and SC-LDPC codes}\n\n\t\\begin{center}\n\t\t\\begin{tabular}{lcccc}\n\t\t\t\\hline\n\t\t\tEnsemble&states&$\\varepsilon_{\\text{BP}}$ & $\\varepsilon_{\\text{MAP}}$ &$\\varepsilon^1_{\\mathrm{SC}}$ \\\\\n\t\t\t\\hline\n\t\t \n\t\t\tLDPC $(3,6)$&-& 0.4294& 0.4881& 0.4880\\\\[0.5mm]\n\t\t\tLDPC $(4,8)$&-&0.3834& 0.4977& 0.4944\\\\[0.5mm]\n\t\t\tLDPC $(5,10)$&-&0.3415&0.4994& 0.4826\\\\[0.5mm]\n\t\t\n\t\t\t\\hline\n\t\t\t$\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$&4&0.4606&0.4689&0.4689\\\\[0.5mm]\n $\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$&4&0.3594&0.4981& 0.4708\\\\[0.5mm]\n $\\mathcal{C}_{\\mathrm{PCC}}$\/$\\mathcal{C}_{\\mathrm{SC-PCC}}$&8&0.4651&0.4863&0.4862\\\\[0.5mm]\n $\\mathcal{C}_{\\mathrm{SCC}}$\/$\\mathcal{C}_{\\mathrm{SC-SCC}}$&8&0.3120&0.4993& 0.4507\\\\\n Type-I $\\mathcal{C}_{\\mathrm{BCC}}$&4&0.3013 &0.4993&0.4932\\\\[0.5mm]\n Type-II $\\mathcal{C}_{\\mathrm{BCC}}$&4& 0.3013&0.4993& 0.4988\\\\[0.5mm]\n\t\t\t\\hline\n\t\t\t\n\t\t\\end{tabular} \n\t\\end{center}\n\t\\label{BPThresholdsLDPC}\n\n\\end{table}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.9\\linewidth]{Fig8.pdf}\n\t\\caption{BER results for SC-SCCs with $L=100$ and $m=1$ on the binary erasure channel.}\n\t\\label{SimSCC}\n\\end{figure}\n\nFig.~\\ref{SimSCC} shows the bit error rate (BER) for SC-SCCs with $L=100$ and $m=1$ on the binary erasure channel for two different rates, $R=1\/4$ (solid blue line) and $R=1\/3$ (solid red line). \nHere, we consider the coupling of SCCs with block length $K=1024$, hence the information block length of the SC-SCC ensemble is $K=101376$.\nIn addition, we plot in the figure the BER curves for the uncoupled ensemble (dotted lines) with $K=3072$.\nFor comparison, we also plot the BER using a sliding window decoder with window size $W=3$ and $K=1024$ (dashed lines) which has a decoding latency equal to that of the uncoupled ensemble.\nFor both rates, the BER improves significantly applying coupling and the use of the window decoder entails only a slight performance degradation with respect to full decoder\n\\footnote{In this work, we are focusing on the BER of TC and SC-TC ensembles in the waterfall region.\n\t\t\t However, spatial coupling does also preserve, or even improve, the error floor performance.\n\t\t\t For example, the minimum distance of each SC-TC ensemble is lower bounded by the minimum distance of the corresponding uncoupled TC ensemble. This can be shown by extending the results for BCCs derived in \\cite{MoloudiISITA}.}. \nWe remark that the comparison between SC-TCs and other types of codes is a new and ongoing field of research.\nIn \\cite{zhu2015window} the authors have compared BCC and SC-LDPC codes for rate $1\/2$ and under the assumption of similar latency for both.\nThe results in \\cite{zhu2015window} show that the considered BCC ensemble outperforms the SC-LDPC code ensemble.\n\n\\section{Threshold Saturation}\\label{Sec7}\n\nThe numerical results in the previous section suggest that threshold\nsaturation occurs for SC-TCs. In this section, for some relevant\nensembles, we prove that, indeed, threshold saturation occurs.\nTo prove threshold saturation we use the proof technique based on potential functions introduced in \\cite{Yedla2012,Yedla2012vector}.\nIn the general case, the DE equations of TCs form a vector recursion.\nHowever, we show that, for some relevant TC ensembles, it is possible to rewrite the DE vector recursion in a form which corresponds to the recursion of a scalar admissible system.\nWe can then prove threshold saturation using the framework in \\cite{Yedla2012} for scalar recursions.\nSince the proof for scalar recursions is easier to describe, we first address this case, and we then highlight the proof for the general case of TCs with a vector recursion based on the framework in \\cite{Yedla2012vector}.\n\n\\begin{definition}[\\cite{Yedla2012,Yedla2014}] \\label{def1} A scalar admissible system $(f,g)$, is defined by the recursion\n\\begin{equation}\n\\label{recursion}\nx^{(i)}=f\\Big( g(x^{(i-1)});\\varepsilon\\Big),\n\\end{equation}\nwhere $f : [0,1] \\times [0,1] \\rightarrow [0,1]$ and $g : [0,1] \\rightarrow [0,1]$ satisfy the following conditions.\n\\begin{enumerate}\n\\item $f$ is increasing in both arguments $x,\\varepsilon \\in (0,1]$; \n\\item $g$ is increasing in $x \\in (0,1]$; \n\\item $f(0;\\varepsilon)=f(x;0)=g(0)=0$;\n\\item $f$ and $g$ have continuous second derivatives.\n\\end{enumerate}\n\\end{definition}\n\n\nIn the following we show that the DE equations for some relevant TCs form a scalar admissible system.\n\n\\subsection{Turbo-like codes as Scalar Admissible Systems}\n\\subsubsection{PCC}\nThe DE equations \\eqref{DEPCC1}--\\eqref{DEPCC6} form a vector recursion. However, if the code is built from identical component encoders, i.e., $f_{\\text{U},\\text{s}}=f_{\\text{L},\\text{s}}\\triangleq f_{\\text{s}}$, it follows\n\\[\np_{\\text{U},\\text{s}}^{(i)}=p_{\\text{L},\\text{s}}^{(i)}\\triangleq x^{(i)}.\n\\]\nUsing this and substituting \\eqref{DEPCC3} into \\eqref{DEPCC1} and \\eqref{DEPCC6} into \\eqref{DEPCC4}, the DE can then be written as\n\\begin{equation}\n\\label{recursionPCC}\nx^{(i)}=f_{\\text{s}}(\\varepsilon x^{(i-1)},\\varepsilon ),\n\\end{equation}\nwith initialization $x^{(0)}=1$.\n\\begin{lemma}\n\\label{LemmaPCC}\nThe DE recursion of a PCC with identical component encoders, given in \\eqref{recursionPCC}, forms a scalar admissible system with $f(x;\\varepsilon)=f_s(\\varepsilon\\cdot x,x)$ and $g(x)=x$.\n\\end{lemma}\n\\begin{IEEEproof}\n It is easy to show that all conditions in Definition~\\ref{def1} are satisfied for $g(x)=x$.\nWe now prove that $f(x;\\varepsilon)$ satisfies Conditions 1, 3 and 4. Note that $f(x;\\varepsilon)$ is the transfer function of a rate-$1\/2$\nconvolutional encoder. According to equation \\eqref{eq:Transfer1},\nthis function can be written as $f(p_1,p_2)$, where $p_1=\\varepsilon\n\\cdot x$ and $p_2=\\varepsilon$. Using Lemma~\\ref{Lemma1}, $f(p_1,p_2)$ is increasing with $p_1$ and $p_2$, therefore $f(x;\\varepsilon)$ is increasing with $x$ and $\\varepsilon$ and Condition 1 is satisfied.\n\nTo show that Condition 3 holds, it is enough to realize that for $\\varepsilon=0$ the input sequence can be recovered perfectly\nfrom the received sequence, i.e., $f(x;0)=0$, as there is a one-to-one mapping between input sequences and coded\nsequences. Furthermore, when $x=0$, the input sequence is fully known by a-priori information and the erasure probability at the output of the decoder is zero, i.e., $f(x;0)=0$.\n\nFinally, $f(x;\\varepsilon)$ is a rational function and its poles are outside the interval $x,\\varepsilon \\in [0,1]$ (otherwise we may get infinite output erasure probability for a finite input erasure probability), hence it has continuous first and second derivatives inside this interval.\n\\end{IEEEproof} \n\n\\subsubsection{SCC}\nConsider the DE equations of the SCC ensemble in \\eqref{DESCC1}--\\eqref{DESCC6}, which form a vector recursion. For identical component encoders, $f_{\\text{I},\\text{s}}=f_{\\text{O},\\text{s}}\\triangleq f_{\\text{s}}$ and $f_{\\text{I},\\text{p}}=f_{\\text{O},\\text{p}}\\triangleq f_{\\text{p}}$. \nUsing this and $q_{\\text{I}}^{(i)}\\triangleq x^{(i)}$, by substituting \\eqref{DESCC2}--\\eqref{DESCC6} into \\eqref{DESCC1}, the DE recursion can be rewritten as \n\\begin{equation}\n\\label{eq:SCCrec}\nx^{(i)}=\\varepsilon \\cdot f_{\\text{s}}\\Big(\\varepsilon g(x^{(i-1)}),\\varepsilon\\Big),\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eq:gSCC}\ng(x^{(i)})=\\frac{f_{\\text{s}}\\Big(x^{(i)},x^{(i)}\\Big)+f_{\\text{p}}\\Big(x^{(i)},x^{(i)}\\Big)}{2},\n\\end{equation}\nand the initial condition is $x^{(0)}=1$.\n\n\\begin{lemma}\n\\label{LemmaSCC}\nThe DE recursion of a SCC with identical component encoders, given in \\eqref{eq:SCCrec} and \\eqref{eq:gSCC}, form a scalar admissible system with $f(x;\\varepsilon)=\\varepsilon \\cdot f_{\\text{s}}(\\varepsilon \\cdot x, \\varepsilon)$ and\n\\begin{align*}\ng(x)=\\frac{f_{\\text{s}}(x,x)+f_{\\text{p}}(x,x)}{2}.\n\\end{align*}\n\\end{lemma}\n\\begin{IEEEproof}\nThe proof follows the same arguments as the proof of Lemma~\\ref{LemmaPCC}.\n\\end{IEEEproof}\n\n\n\\subsubsection{BCC}\nSimilarly to PCCs and SCCs, the DE equations of BCCs (see \\eqref{DEBCC4}--\\eqref{DEBCC6}) form a vector recursion. With identical component encoders, due to the symmetric structure of the code, $f_{\\text{U},k}=f_{\\text{L},k}\\triangleq f_k$ and $p_{\\text{U},k}^{(i)}=p_{\\text{U},k}^{(i)}\\triangleq x_k^{(i)}$ for $k=1,2,3$.\nUsing this, \\eqref{DEBCC4}--\\eqref{DEBCC6} can be rewritten as\n\\begin{align}\n\\label{BCC1}\nx_1^{(i)}&=f_1\\Big(\\varepsilon \\cdot x_1^{(i-1)},\\varepsilon \\cdot x_3^{(i-1)},\\varepsilon\\cdot x_2^{(i-1)}\\Big)\\\\\n\\label{BCC2}\nx_2^{(i)}&=f_2\\Big(\\varepsilon \\cdot x_1^{(i-1)},\\varepsilon \\cdot x_3^{(i-1)},\\varepsilon \\cdot x_2^{(i-1)}\\Big)\\\\\n\\label{BCC3}\nx_3^{(i)}&=f_3\\Big(\\varepsilon \\cdot x_1^{(i-1)},\\varepsilon \\cdot x_3^{(i-1)},\\varepsilon \\cdot x_2^{(i-1)}\\Big).\n\\end{align}\n\nThe above DE equations are still a vector recursion.\nTo write the recursion in scalar form, it is necessary to have identical transfer functions for all the edges which are connected to factor nodes $T^{\\text{U}}$ and $T^{\\text{L}}$. This is needed because all variable nodes in a BCC receive a-priori information.\nIn order to achieve this property, we can apply some averaging over the different types of code symbols. In particular, we can randomly permute the order of the encoder outputs $v_\\tau^{(l)}$, $l=1,\\dots,n$. For each trellis section $\\tau$ the order of these $n$ symbols is chosen indepently according to a uniform distribution.\nEquivalently, instead of performing this permutation on the encoder outputs we can define a corresponding component encoder with a time-varying trellis in which the branch labels are permuted accordingly.\nThen, it results $x_1^{(i)}=x_2^{(i)}=x_3^{(i)}\\triangleq x^{(i)}$\nand all transfer functions are equal to the average of the transfer functions $f_1,f_2,f_3$,\n\\[\nf_{\\text{ave}}=\\frac{f_1+f_2+f_3}{3}.\n\\] \nUsing this, the DE equations can be simplified as\n\\begin{equation}\n\\label{eq:BCCScalar}\nx^{(i)}=f_{\\text{ave}}(\\varepsilon \\cdot x^{(i-1)},\\varepsilon \\cdot x^{(i-1)},\\varepsilon \\cdot x^{(i-1)}).\n\\end{equation}\n\\begin{lemma}\n\\label{LemmaBCC}\nThe DE recursion of a BCC with identical component encoders and time varying trellises, given in \\eqref{eq:BCCScalar}, form a scalar admissible system with $f(x;\\varepsilon)=f_{\\text{ave}}(\\varepsilon \\cdot x,\\varepsilon \\cdot x,\\varepsilon \\cdot x)$ and $g(x)=x$.\n\\end{lemma}\n\\begin{IEEEproof}\nThe proof follows the same arguments as the proof of Lemma~\\ref{LemmaPCC}.\n\\end{IEEEproof}\n\n\\subsection{Single System Potential}\n\\begin{definition}[\\cite{Yedla2012,Yedla2014}] \\label{def2}\nFor a scalar admissible system, defined in Definition~\\ref{def1}, the potential function $U(x;\\varepsilon)$ is\n\\begin{align}\n\\label{Potential}\nU(x;\\varepsilon)&=\\int_{0}^{x}\\big{(}z-f(g(x);\\varepsilon)\\big{)}g'(z)dz \\\\\n&=xg(x)-G(x)-F(g(x);\\varepsilon),\\nonumber\n\\end{align}\nwhere $F(x;\\varepsilon)=\\int_{0}^{x}f(z;\\varepsilon) dz$ and $G(x)=\\int_{0}^{x}g(z) dz$.\n\\end{definition}\n\\begin{proposition}[\\cite{Yedla2012,Yedla2014}]\nThe potential function has the following properties.\n\\begin{enumerate}\n\\item $U(x;\\varepsilon)$ is strictly decreasing in $\\varepsilon \\in (0,1]$;\n\\item An $x\\in [0,1]$ is a fixed point of the recursion\n(\\ref{recursion}) if and only if it is a stationary point of the corresponding potential function.\n\\end{enumerate}\n\\end{proposition}\n\\begin{definition}[\\cite{Yedla2012,Yedla2014}] \\label{defBP} If the DE recursion is the recursion of a BP decoder, the BP threshold is \\cite{Yedla2012}\n\\[\n\\varepsilon^{\\text{BP}}=\\sup\\Big\\{\\varepsilon\n \\in[0,1] : U'(x;\\varepsilon)>0,\\; \\forall x\\in (0,1]\\Big\\} \\ .\n\\] \n\\end{definition}\n According to Definition~\\ref{defBP}, for $\\varepsilon < \\varepsilon^{\\text{BP}}$, the derivative of the potential function is always larger than zero for $x\\in (0,1]$, i.e., the potential function has no stationary point in $x\\in (0,1]$. \n\\begin{definition}[\\cite{Yedla2012,Yedla2014}]\n\\label{defPotth}\nFor $\\varepsilon >\\varepsilon^{\\text{BP}} $, the minimum unstable fixed point is $u(\\varepsilon)=\\sup\\big\\{\\tilde{x} \\in [0,1] : f(g(x);\\varepsilon)0, \\min_{x \\in [u(x),1]} U(x;\\varepsilon)> 0 \\Big\\} \\ .\n\\end{align*}\n\\end{definition}\nThe potential threshold depends on the functions $f(x;\\varepsilon)$ and $g(x)$.\n\n\\begin{example}\nConsider rate-$1\/3$ PCCs with identical 2-state component encoders with generator matrix ${\\bs{G}}=(1,1\/3)$.\nFor this code ensemble,\n\\[\nf_{\\text{s}}(\\varepsilon\\cdot x,\\varepsilon)=\\frac{x\\varepsilon^2(2-2\\varepsilon+x\\varepsilon^2)}{(1-\\varepsilon+x\\varepsilon^2)^2} \\ .\n\\]\nTherefore,\n\\[\nF_{\\text{s}}(x;\\varepsilon)=\\frac{x\\varepsilon^2}{1-\\varepsilon+x\\varepsilon^2} \\ ,\n\\]\nand\n\\[\nU(x;\\epsilon)=\\frac{x\\varepsilon^3+(1-\\varepsilon-2\\varepsilon^2)x^2}{2(1-\\varepsilon+x\\varepsilon^2)} \\ . \n\\] \\hfill $\\triangle$\n\\end{example}\n\\begin{example}\nConsider the PCC ensemble in Fig.~\\ref{Uncoupled}(b) with identical component encoders with generator matrix $\\bold{G}=(1,5\/7)$.\nThe DE recursion of this ensemble is given in \\eqref{recursionPCC}, where $f_s$ is the transfer function of the $(1,5\/7)$ component encoder. The corresponding potential function is\n\\begin{equation}\nU(x;\\epsilon)=x^2-G(x)-F_{\\text{s}}(x;\\epsilon)=\\frac{x^2}{2}-F_{\\text{s}}(x;\\epsilon) \\ ,\n\\end{equation} \nwhere $F_{\\text{s}}(x;\\varepsilon)=\\int_{0}^{x}f_{\\text{s}}(\\varepsilon\\cdot z,\\varepsilon) dz$ and $G(x)=\\int_{0}^{x}g(z)dz=\\frac{x^2}{2}$. The potential function is shown in Fig.~\\ref{PotPCC} for several values of $\\varepsilon$.\nAs it is illustrated, for $\\varepsilon<0.6428$ the potential function has no stationary point. The BP threshold and the potential threshold are $\\varepsilon=0.6428$ and $\\varepsilon=0.6553$, respectively (see Definitions~\\ref{defBP} and \\ref{defPotth}).\nThese results match with the DE results in Table~\\ref{Tab:BPThresholdsSCC}. \\hfill $\\triangle$\n\\end{example}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\linewidth]{Fig9V3.pdf}\n\\caption{Potential function of a PCC ensemble.}\n\\label{PotPCC}\n\\end{figure}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\linewidth]{Fig10V3.pdf}\n\\caption{Potential function of a SCC ensemble.}\n\\label{PotSCC}\n\\end{figure}\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\linewidth]{Fig11V3.pdf}\n\\caption{Potential function of a BCC ensemble.}\n\\label{PotBCC}\n\\end{figure}\n\\begin{example} The potential function of the SCC ensemble in Fig.~\\ref{Uncoupled}(c) with identical component encoders with generator matrix $\\bold{G}=(1,5\/7)$ is shown in Fig.~\\ref{PotSCC}. The BP threshold and the potential threshold are \n$\\varepsilon=0.689$ and $\\varepsilon=0.748$, respectively, which match with the DE results in Table~\\ref{Tab:BPThresholdsSCC}. \\hfill $\\triangle$\n\\end{example}\n\\begin{example} Consider the BCC ensemble in\n Fig.~\\ref{Uncoupled}(d) with identical component encoders with generator matrix given in \\eqref{eqG} and time-varying trellises. The potential function of this code is depicted in Fig.~\\ref{PotBCC}. The BP threshold and the potential threshold are $\\varepsilon=0.5522$ and $\\varepsilon=0.6654$, respectively. Note that these values are slightly different from the values in\nTable~\\ref{BPThresholdsBCC}. This is due to the fact that we considered an ensemble with time-varying trellises, which can be modeled by means of a scalar recursion. The ensemble considered in Table~\\ref{BPThresholdsBCC} needs to be analyzed by means of a vector recursion. \\hfill $\\triangle$\n\\end{example}\n\n\\subsection{Coupled System and Threshold Saturation}\n\\begin{theorem}\\label{thm1}\nConsider a spatially coupled system defined by the following recursion at time $t$,\n\\begin{align}\n\\label{SCrecursion}\nx_t^{(i)}=\\frac{1}{1+m}\\sum_{j=0}^{m}f_{t+j}\\Big(\\frac{1}{1+m}\\sum_{k=0}^{m}g(x_{t+j-k}^{(i-1)});\\varepsilon\\Big).\n\\end{align}\nIf $f(x;\\varepsilon)$ and $g(x)$ form a scalar admissible system, for large enough coupling memory and $\\varepsilon < \\varepsilon^*$, the only fixed point of the recursion is\n$x=0$.\n\\end{theorem}\n\\begin{IEEEproof}\nThe proof follows from \\cite{Yedla2012}.\n\\end{IEEEproof}\n\nIn the following we show that the DE recursions of SC-TCs (with identical component encoders) can be written in the form \\eqref{SCrecursion}. As a result, threshold saturation occurs for these ensembles.\n\n\\subsubsection{PCCs}\nConsider the SC-PCC ensemble in Fig.~\\ref{Coupled}(a) with identical component encoders.\nDue to the symmetric coupling structure, it follows that (cf. \\eqref{DESCPCC1} and \\eqref{DESCPCC2})\n\\[\n\\bar{q}_{\\text{U}}^{(i,t)}=\\bar{q}_{\\text{L}}^{(i,t)}\\triangleq x_{t}^{(i)}.\n\\]\nNow, using $x_t^{(i)}$ in \\eqref{DESCPCC3} and \\eqref{DESCPCC6}, we can write\n\\begin{align}\n\\label{eq:QLi}\nq_{\\text{L}}^{(i,t)}=q_{\\text{U}}^{(i,t)}=\\varepsilon \\cdot \\frac{1}{m+1}\\sum_{k=0}^{m}x_{t-k}^{(i-1)}.\n\\end{align}\nFinally, by substituting \\eqref{eq:QLi} into \\eqref{DESCPCC4} and \\eqref{DESCPCC5} and the results into \\eqref{DESCPCC1} and \\eqref{DESCPCC2}, the recursion of SC-PCCs can be rewritten as\n\\begin{equation}\n\\label{SCPCCrecursion}\nx_t^{(i)}=\\frac{1}{1+m}\\sum_{j=0}^{m}f_{\\text{s},t+j}\\Big(\\frac{\\varepsilon}{m+1}\\cdot\\sum_{k=0}^{m}x_{t+j-k}^{(i-1)},\\varepsilon\\Big).\n\\end{equation}\nNote that the recursion in \\eqref{SCPCCrecursion} is identical to the recursion in \\eqref{SCrecursion}.\n\n\\subsubsection{SCCs}\n\nConsider the SC-SCC ensemble in Fig.~\\ref{Coupled}(b) with identical component encoders. Define $x_t^{(i)} \\triangleq q_{\\text{I}}^{(i,t)}$ (see \\eqref{DESCSCC2})\nNow, use it in \\eqref{DESCSCC3}--\\eqref{DESCSCC6}.\nFinally, by substituting the result in \\eqref{DESCSCC2}, the recursion of a SC-SCC an be rewritten as\n\\begin{equation}\n\\label{SCSCCrecursion}\nx_t^{(i)}=\\frac{1}{1+m}\\sum_{j=0}^{m}\\varepsilon \\cdot f_{\\text{s},t+j}\\Big(\\frac{\\varepsilon}{m+1}\\cdot\\sum_{k=0}^{m}g(x_{t+j-k}^{(i-1)}),\\varepsilon\\Big),\n\\end{equation}\nwhere $g(x)$ is shown in equation \\eqref{eq:gSCC}. The recursion in \\eqref{SCSCCrecursion} is identical to the recursion in Theorem \\ref{thm1}. \n\n\\subsubsection{BCCs}\n\nConsider a coupling for BCCs slightly different from the one for Type-II BCCs.\nAt time $t$, each of the parity sequences $\\bs{v}^{\\text{U}}_t$ and $\\bs{v}^{\\text{L}}_t$ is divided into $m+1$ sequences, $\\boldsymbol{v}_{t,j}^{\\text{U}}$, $j=0,\\dots,m$, and $\\boldsymbol{v}_{t,j}^{\\text{L}}$, $j=0,\\dots,m$, respectively (in Type-II BCCs they are divided into $m$ sequences).\nThe sequences $\\boldsymbol{v}_{t-j,j}^{\\text{U}}$ and $\\boldsymbol{v}_{t-j,j}^{\\text{L}}$ are multiplexed and reordered, and are used as the second input of the lower and upper encoder, respectively. Note that in this way of coupling, part of the parity bits at time $t$ are used as input at the same time instant $t$. Now, similarly to uncoupled BCCs, consider identical time-varying trellises. Let $x^{(i)}_t$ denote the extrinsic erasure probability from $T^{\\text{U}}_t$ through all its edges in the $i$th iteration. The erasure probabilities to $\\text{T}^{\\text{U}}_t$ through all its incoming edges are equal and are given by the average of the erasure probabilities from variable nodes $\\bs{v}_{t'}$, $t'= t-m,\\dots,t$,\n\\[\nq_t^{(i)}=\\frac{\\varepsilon}{1+m}\\sum_{k=0}^{m}x_{t-k}^{(i-1)}.\n\\]\nThus, the erasure probabilities from $T^{\\text{U}}_t$ and $T^{\\text{L}}_t$ are identical and equal to $f_{\\text{ave},t}(q_t^{(i)},q_t^{(i)},q_t^{(i)})$. Finally, the recursion at time slot $t$ is\n\\begin{equation}\n\\label{eq:SCBCC}\nx_t^{(i)}=\\frac{1}{1+m}\\sum_{j=0}^{m}f_{\\text{ave},t+j}(q_{t+j}^{(i)},q_{t+j}^{(i)},q_{t+j}^{(i)}).\n\\end{equation}\nThe recursion in (\\ref{eq:SCBCC}) is identical to (\\ref{SCrecursion}).\n\n\\subsection{Random Puncturing and Scalar Admissible System}\n\nIn the following, we show that the DE recursion of punctured TC ensembles can also be rewritten as a scalar admissible system for some particular cases. Then, threshold saturation follows from the discussion in the previous subsection.\n\\subsubsection{PCC} Consider the PCC ensemble with\nidentical component encoders and random puncturing of the parity bits with\npermeability rate $\\rho$. The DE recursion can be rewritten\nas,\n\\[\nx^{(i)}=f_{\\text{s}}(\\varepsilon x^{(i-1)},1-(1-\\varepsilon)\\rho).\n\\]\nThe above equation is a recursion of a scalar admissible system and satisfies the\nconditions in Definition \\ref{def1}, where $g(x)=x$ and\n$f(x;\\varepsilon)=f_s(\\varepsilon\\cdot x, 1-(1-\\varepsilon)\\rho)$.\n\n\\subsubsection{SCC} \n\nConsider random puncturing of the SCC ensemble\nwith identical component encoders. Assuming $\\rho_0=\\rho_1$ (i.e., we puncture also systematic bits of the outer code),\nwe can rewrite the DE recursion as\n\\[\nx^{(i)}=\\varepsilon_{\\rho_1} \\cdot f_{\\text{s}}(\\varepsilon_{\\rho_1} x^{(i-1)},\\varepsilon_{\\rho_2}),\n\\]\nwhere $\\varepsilon_{\\rho_1}=1-(1-\\varepsilon)\\rho_1$ and\n$\\varepsilon_{\\rho_2}=1-(1-\\varepsilon)\\rho_2$. The above equation is\nthe recursion of a scalar admissible system, where\n$f(x;\\varepsilon)=\\varepsilon_{\\rho_1} f_s(\\varepsilon_{\\rho_1}\\cdot\nx,\\varepsilon_{\\rho_2} )$ and $g(x)$ is obtained by equation \\eqref{eq:gSCC}.\n\n\\subsubsection{BCC} \n\nConsider random puncturing of the BCC ensemble with identical\ntime-varying trellises. Assume that the systematic bits and the parity bits of the upper and lower encoders are punctured with the same permeability rate $\\rho$. Then, the DE recursion can be\nrewritten as \\eqref{eq:BCCScalar}, where $\\varepsilon$ should\nbe replaced by $\\varepsilon_{\\rho}=1-(1-\\varepsilon)\\rho$.\n\n\\subsection{Turbo-like Codes as Vector Admissible Systems}\nIn general, the DE recursions of TCs are vector recursions. In this case, it is\npossible to prove threshold saturation using the technique proposed in \\cite{Yedla2012vector} for vector recursions. The proof is similar to that of scalar recursions, albeit more involved. In the following, we show how to rewrite the recursion of punctured PCCs as a vector admissible system recursion. Then, following \\cite{Yedla2012vector}, we can prove threshold saturation. \nUsing the same technique, it is possible to prove threshold saturation for SCCs and BCCs as well.\n\n\nConsider the DE equations of the PCC ensemble in \\eqref{DEPCC1}--\\eqref{DEPCC6}. To reduce the number of the equations, substitute \n\\eqref{DEPCC3} and \\eqref{DEPCC6} into \\eqref{DEPCC1} and \\eqref{DEPCC4},\nrespectively.\nConsider random puncturing of information bits, upper encoder parity bits and lower encoder parity bits with permeability rates\n$\\rho_0$, $\\rho_1$ and $\\rho_2$, respectively. By considering\n$x_1^{(i)}\\triangleq p_{\\text{U,s}} $ and $x_2^{(i)}\\triangleq\np_{\\text{L,s}} $, the DE recursion can be simplified to \n\\[\nx_1^{(i)}=f_{\\text{U,s}}(\\varepsilon_{\\rho_0}\\cdot x_2^{(i-1)},\\varepsilon_{\\rho_1})\n\\]\n\\[\nx_2^{(i)}=f_{\\text{L,s}}(\\varepsilon_{\\rho_0}\\cdot x_1^{(i-1)},\\varepsilon_{\\rho_2}).\n\\]\nThe above equations can be written in vector format as\n\\begin{equation}\n\\label{PCCvector}\n\\bs{x}^{(i)}=\\bs{f}(\\bs{g}(\\bs{x}^{(i-1)});\\varepsilon),\n\\end{equation}\nwhere, $\\bs{x}=[x_1, x_2]$, $\\bs{f}(\\bs{x};\\varepsilon)=[f_{U,\\text{s}}(\\varepsilon_{\\rho_0} \\cdot x_1,\\varepsilon_{\\rho_1}),f _{L,\\text{s}}(\\varepsilon_{\\rho_0} \\cdot x_2,\n\\varepsilon_{\\rho_2})]$ and $\\bs{g}(\\bs{x})=[x_2,x_1]$.\nIs it easy to verify that the recursion in \\eqref{PCCvector} satisfies the\nconditions in \\cite[Def.~1]{Yedla2012vector}, hence \\eqref{PCCvector} is the recursion of a vector\nadmissible system. \nFor this vector admissible system,\nthe line integral is path independent in \\cite[Eq.~(2)]{Yedla2012vector} and the potential function is well defined.\nSo, we can define (see \\cite{Yedla2012vector}) $\\bs{D}=I_{2\\times 2}$, $G=x_1\\cdot x_2$ and\n\\[\nF=\\int_0^{x_1} f_{\\text{U,s}}(\\varepsilon_{\\rho_0} \\cdot\nz,\\varepsilon_{\\rho_1}) \\; dz+\\int_0^{x_2} f_{\\text{L,s}}(\\varepsilon_{\\rho_0} \\cdot z,\\varepsilon_{\\rho_2}) \\; dz.\n\\]\nIt is possible to show that the DE recursion of SC-PCCs can be\nrewritten in the same form as \\cite[Eq.~(5)]{Yedla2012vector} and by\nusing \\cite[Th.~1]{Yedla2012vector}, threshold saturation can be proven.\n\\section{Conclusion}\\label{Sec8}\nIn this paper we investigated the impact of spatial coupling on the BP decoding threshold of turbo-like codes. We introduced the concept of spatial coupling for PCCs and SCCs, and generalized the concept of coupling for BCCs.\nConsidering transmission over the BEC, we derived the exact DE equations for uncoupled and coupled ensembles. \nFor all spatially coupled ensembles, the BP threshold improves and our numerical results suggest that threshold saturation occurs if the coupling memory is chosen sufficiently large. We therefore constructed rate-compatible families of SC-TCs that achieve close-to-capacity performance for a wide range of code rates.\n\nWe showed that the DE equations of SC-TC ensembles with identical component encoders can be properly rewritten as a scalar recursion. \nFor SC-PCCs, SC-SCCs and BCCs we then proved threshold saturation analytically, using the proof technique based on potential functions proposed in \\cite{Yedla2012,Yedla2014}. Finally, we demonstrated how vector recursions can be used to extend the proof to more general ensembles.\n\nA generalization of our results to general binary-input memoryless channels is challenging, because the transfer functions of the component decoders can no longer be obtained in closed form. Even a numerical computation of the exact thresholds is difficult, but Monte Carlo methods and Gaussian approximation techniques could be helpful tools for finding approximate thresholds. EXIT charts, for example, have been widely used for analyzing uncoupled TCs and may be useful for estimating the thresholds of SC-TCs. A connection between EXIT functions and potential functions of spatially coupled systems is given in \\cite{KudekarWaveLike}. An investigation of SC-TC ensembles along this line may be an interesting direction for future work. The simulation results for SC-BCCs over the AWGN channel in \\cite{ZhangBCC} and \\cite{zhu2015window} clearly show that spatial coupling significantly improves the performance, suggesting that threshold saturation also occurs for this channel.\n\nThe invention of turbo codes and the rediscovery of LDPC codes, allowed to approach capacity with practical codes.\nToday, both turbo-like codes and LDPC codes are ubiquitous in communication standards.\nIn the academic arena, however, the interest on turbo-like codes has been declining in the last years in favor of the (considered) more mathematically-appealing LDPC codes.\nThe invention of spatially coupled LDPC codes has exacerbated this situation.\nWithout spatial coupling, it is well known that PCCs yield good BP thresholds but poor error floors, while SCCs and BCCs show low error floors but poor BP thresholds.\nOur SC-TCs, however, demonstrate that turbo-like codes can also greatly benefit from spatial coupling.\nThe concept of spatial coupling opens some new degrees of freedom in the design of codes on graphs: designing a concatenated coding scheme for achieving the best BP threshold in the uncoupled case may not necessarily lead to the best overall performance.\nInstead of optimizing the component encoder characteristics for BP decoding, it is possible to optimize the MAP decoding threshold and rely on the threshold saturation effect of spatial coupling.\nPowerful code ensembles with strong distance properties such as SCCs and BCCs can then perform close to capacity with low-complexity iterative decoding.\nWe hope that our work on spatially coupled turbo-like codes will trigger some new interest in turbo-like coding structures.\n\n\n\n\n\n\n\n\n\\input{SCTCsITTransaction.bbl}\n\n\\balance\n\\begin{IEEEbiographynophoto}{Saeedeh Moloudi}\nreceived the Master Degree in Wireless Communications from Shiraz University, Iran in 2012. Since September 2013, she has been a PhD candidate at the Department of Electrical and Information Technology, Lund University. Her main research interests include design and analysis of coding systems and graph based iterative algorithms.\n\\end{IEEEbiographynophoto}\n\n\\begin{IEEEbiographynophoto}{Michael Lentmaier}\n received the Dipl. Ing. degree in electrical engineering from University of Ulm, Germany in 1998, and the Ph.D. degree in telecommunication theory from Lund University, Sweden in 2003. He then worked as a Post-Doctoral Research Associate at University of Notre Dame, Indiana and at University of Ulm. From 2005 to 2007 he was with the Institute of Communications and Navigation of the German Aerospace Center (DLR) in Oberpfaffenhofen, where he worked on signal processing techniques in satellite navigation receivers. From 2008 to 2012 he was a senior researcher and lecturer at the Vodafone Chair Mobile Communications Systems at TU Dresden, where he was heading the Algorithms and Coding research group. Since January 2013 he is an Associate Professor at the Department of Electrical and Information Technology at Lund University. His research interests include design and analysis of coding systems, graph based iterative algorithms and Bayesian methods applied to decoding, detection and estimation in communication systems. He is a senior member of the IEEE and served as an editor for IEEE Communications Letters (2010-2013), IEEE Transactions on Communications (2014-2017), and IEEE Transactions on Information Theory (since April 2017). He was awarded the Communications Society and Information Theory Society Joint Paper Award (2012) for his paper \"Iterative Decoding Threshold Analysis for LDPC Convolutional Codes\".\n\\end{IEEEbiographynophoto}\n\n\\begin{IEEEbiographynophoto}{Alexandre Graell i Amat}\nreceived the MSc degree in Telecommunications Engineering from the Universitat Polit\u00e8cnica de Catalunya, Barcelona, Catalonia, Spain, in 2001, and the MSc and the PhD degrees in Electrical Engineering from the Politecnico di Torino, Turin, Italy, in 2000 and 2004, respectively. From September 2001 to May 2002, he was a Visiting Scholar at the University of California San Diego, La Jolla, CA. From September 2002 to May 2003, he held a Visiting Appointment at the Universitat Pompeu Fabra and at the Telecommunications Technological Center of Catalonia, both in Barcelona. From 2001 to 2004, he also held a part-time appointment at STMicroelectronics Data Storage Division, Milan, Italy, as a Consultant on coding for magnetic recording channels. From 2004 to 2005, he was a Visiting Professor at the Universitat Pompeu Fabra, Barcelona, Spain. From 2006 to 2010, he was with the Department of Electronics, Telecom Bretagne (former ENST Bretagne), Brest, France. In 2011, he joined the Department of Electrical Engineering, Chalmers University of Technology, Gothenburg, Sweden, where he is currently a Professor. His research interests include the areas of modern coding theory, distributed storage, and optical communications.\nProf. Graell i Amat is currently Editor-at-Large of the IEEE TRANSACTIONS ON COMMUNICATIONS. He was an Associate Editor of the IEEE TRANSACTIONS ON COMMUNICATIONS from 2011 to 2015, and the IEEE COMMUNICATIONS LETTERS from 2011 to 2013. He was the General Co-Chair of the 7th International Symposium on Turbo Codes and Iterative Information Processing, Gothenburg, Sweden, 2012. He received the postdoctoral Juan de la Cierva Fellowship from the Spanish Ministry of Education and Science and the Marie Curie Intra-European Fellowship from the European Commission. He received the IEEE Communications Society 2010 Europe, Middle East, and Africa Region Outstanding Young Researcher Award.\n\\end{IEEEbiographynophoto}\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \\label{Int}\n\nGeneral Theory of Relativity (GR) is the cornerstone of\nastrophysics and cosmology, giving predictions with unprecedented success. \nAt astrophysical scales GR has been tested in, for example, the solar system, stellar dynamics, black hole formation and evolution, among others (see for instance\\cite{FischbachTalmadge1999,Will1993,Kamionkowski:2007wv,*Matts,*Peebles:2013hla}). \nHowever, GR is being currently tested with various phenomena that can be \nsignificant challenges to the GR theory, generating important changes never seen before. Ones of the major challenges of modern cosmology are undoubtedly dark matter (DM) and dark energy. They comprise approximately $27\\%$ for DM and $68\\%$ for dark energy, of our \nuniverse\\cite{PlanckCollaboration2013} allowing the formation of large scale structures\\cite{FW,Diaferio:2008jy}.\nDark matter has been invoked as the mechanism to stabilize spiral galaxies and to provide with a matter distribution component to explain the observed rotation curves.\nNowadays the best model of the universe we have is the \n\\emph{concordance} or $\\Lambda$CDM model that has been successful in explaining the very large-scale structure formation, the statistics of the distribution of galaxy clusters, the temperature anisotropies of the cosmic microwave background radiation (CMB) and many other astronomical observations. \nIn spite of all successes we have mentioned, \nthis model has several problems, for example: predicts too much power on small scale\\cite{Rodriguez-Meza:2012}, then it over predicts the number of observed satellite galaxies\\cite{Klypin1999,Moore1999,Papastergis2011} and predicts halo profiles that are denser and \ncuspier than those inferred observationally\\cite{Navarro1997,Subramanian2000,Salucci}, and also predicts a population of massive and concentrated subclass that are inconsistent \nwith observations of the kinematics of Milky Way satellites\\cite{BoylanKolchin2012}.\n\nOne of the first astronomical observations that brought attention on DM was the observation of rotation curves of spiral galaxies by Rubin and coworkers\\cite{Rubin2001}, \nthese observations turned out to be the main tool to investigate the role of DM at galactic scales: its role in determining the structure, how the mass is distributed, and the dynamics, evolution, and formation of spiral galaxies.\nRemarkably, the corresponding rotation velocities of galaxies, can be explained with the density profiles of different Newtonian DM models like Pseudo Isothermal profile (PISO)\\cite{piso}, Navarro-Frenk-White profile (NFW)\\cite{Navarro1997} \nor Burkert profile\\cite{Burkert}, among \nothers\\cite{Einasto}; except by the fact that until now it is unsolved the problem of cusp and core in the densities profiles. In this way, none of them have the last word because the main questions, about the density distribution and of course the \\emph{nature} of DM, has not been resolved.\n\nAlternative theories of gravity have been used to model DM. For instance a scalar field has been proposed to model DM\\cite{Dick1996,Cho\/Keum:1998}, \nand has been used to study rotation curves of spiral galaxies\\cite{Guzman\/Matos:2000}. This scalar field is coupled minimally to the metric, however, scalar fields coupled non minimally to the metric have also been used to study DM\\cite{Rodriguez-Meza:2012,RodriguezMeza\/Others:2001,RodriguezMeza\/CervantesCota:2004,Rodriguez-Meza:2012b}. Equivalently $F(R)$ models exists in the literature that analyzes rotation curves\\cite{Martins\/Salucci:2007}.\n\nOn the other hand, one of the best candidates to extend GR is the brane theory, whose main characteristic is to add another dimension having a five dimensional bulk where it is embedded a four dimensional manifold called the brane\\cite{Randall-I,*Randall-II}. This model is characterized by the fact that the standard model of particles is confined in the brane and only the gravitational interaction can travel in the bulk\\cite{Randall-I,*Randall-II}. The assumption that the five dimensional Einstein's equations are valid, generates corrections in the four dimensional Einstein's equations confined in the brane bringing information from the extra dimension\\cite{sms}. \nThese extra corrections in the Einstein's equations can help us to elucidate and solve the problems that afflicts the modern cosmology and astrophysics\\cite{m2000,*yo2,*Casadio2012251,*jf1,*gm,*Garcia-Aspeitia:2013jea,*langlois2001large,*Garcia-Aspeitia:2014pna,*jf2,*PerezLorenzana:2005iv,*Ovalle:2014uwa,*Garcia-Aspeitia:2014jda,*Linares:2015fsa,*Casadio:2004nz}.\n\nBefore we start, let us mention here some experimental constraints on braneworld models, most of them about the so-called brane tension, $\\lambda$, which appears explicitly as a free parameter in the corrections of the gravitational equations mentioned above. As a first example we have the measurements on the deviations from Newton's law of the gravitational interaction at small distances. It is reported that no deviation is observed for distances $l \\gtrsim 0.1 \\, {\\rm mm}$, which then implies a lower limit on the brane tension in the Randall-Sundrum II model (RSII): $\\lambda> 1 \\, {\\rm TeV}^{4}$\\cite{Kapner:2006si,*Alexeyev:2015gja}; it is important to mention that these limits do not apply to the two-branes case of the Randall-Sundrum I model (RSI) (see \\cite{mk} for details). \nAstrophysical studies, related to gravitational waves and stellar stability, constrain\nthe brane tension to be $\\lambda > 5\\times10^{8} \\, {\\rm MeV}^{4}$\\cite{gm,Sakstein:2014nfa}, whereas the existence of black hole X-ray binaries suggests that $l\\lesssim 10^{-2} {\\rm mm}$\\cite{mk,Kudoh:2003xz,*Cavaglia:2002si}. Finally, from cosmological observations, the requirement of successful nucleosynthesis provides the lower limit $\\lambda> 1\\, {\\rm MeV}^{4}$, which is a much weaker limit as compared to other experiments (another cosmological tests can be seen in: Ref. \\cite{Holanda:2013doa,*Barrow:2001pi,*Brax:2003fv}).\n \nIn fact, this paper is devoted to study the main observable of brane theory which is the brane tension, whose existence delimits between the four dimensional GR and its high energy corrections. We are given to the task of perform a Newtonian approximation of the modified Tolman-Oppenheimer-Volkoff (TOV) equation, maintaining the effective terms provided by branes which cause subtle differences in the traditional dynamics. \nIn this way we test the theory at galactic scale using high resolution measurements of rotation curves of a sample of low surface brightness (LSB) galaxies with no photometry\\cite{deBlok\/etal:2001}\nand a synthetic rotation curve built from 40 rotation curves of spirals of magnitude around $M_I=-18.5$ where was found that the baryonic components has a very small contribution\\cite{Salucci1},\nassuming PISO, NFW and Burkert DM profiles respectively and with that, we constraint the preferred value of brane tension with observables. \nThat the sample has no photometry means that the galaxies are DM dominated and then we have only two parameters related to the distribution of DM, a density scale and a length scale, adding the brane tension we have three parameters in total to fit. \nThe brane tension fitted values are compared among the traditional DM density profile models of spiral galaxies \n(PISO, NFW and Burkert) and against the same models without the presence of branes and confronted with other values of the tension parameter coming from cosmological and astrophysical observational data.\n\nThis paper is organized as follows: In Sec.\\ \\ref{EM} we show the equations of motion (modified TOV equations) for a spherical symmetry and the appropriate initial conditions. In Sec.\\ \\ref{TOV MOD} we explore the Newtonian limit and we show the mathematical expression of rotation velocity with brane modifications; particularly we show the modifications to velocity rotation expressions of PISO, NFW and Burkert DM profiles and they are compared with models without branes. \nIn Sec.\\ \\ref{Results} we test the DM models plus brane with observations: we use a sample of high resolution measurements of rotation curves of LSB galaxies and a synthetic rotation curve representative of 40 rotation curves of spirals where the baryonic component has a very small contribution.\nFinally in Sec.\\ \\ref{Disc}, we discuss the results obtained in the paper and make some conclusions.\n\nIn what follows, we work in units in which $c=\\hbar=1$, unless explicitly written.\n\n\\section{Review of equations of motion for branes} \\label{EM}\n\nLet us start by writing the equations of motion for galactic stability in a brane embedded in a five-dimensional bulk according to the RSII model\\cite{Randall-II}. Following an appropriate computation (for details see\\cite{mk,sms}), it is possible to demonstrate that the modified four-dimensional Einstein's equations can be written as \n\\begin{equation}\n G_{\\mu\\nu} + \\xi_{\\mu\\nu} + \\Lambda_{(4)}g_{\\mu\\nu} = \\kappa^{2}_{(4)} T_{\\mu\\nu} + \\kappa^{4}_{(5)} \\Pi_{\\mu\\nu} +\n \\kappa^{2}_{(5)} F_{\\mu\\nu} , \\label{Eins}\n\\end{equation}\nwhere $\\kappa_{(4)}$ and $\\kappa_{(5)}$ are respectively the four and five- dimensional coupling constants, which are related in the form: $\\kappa^{2}_{(4)}=8\\pi G_{N}=\\kappa^{4}_{(5)} \\lambda\/6$, where $\\lambda$ is defined as the brane tension, and $G_{N}$ is the Newton constant. For purposes of simplicity, we will not consider bulk matter, which translates into $F_{\\mu\\nu}=0$, and discard the presence of the four-dimensional cosmological constant, $\\Lambda_{(4)}=0$, \nas we do not expect it to have any important effect at galactic scales (for a recent discussion about it see\\cite{Pavlidou:2013zha}). Additionally, we will neglect any nonlocal energy flux, which is allowed by the static spherically symmetric solutions we will study below\\cite{gm}.\n\nThe energy-momentum tensor, the quadratic energy-momentum tensor, and the Weyl (traceless) contribution, have the explicit forms\n\\begin{subequations}\n\\label{eq:4}\n\\begin{eqnarray}\n\\label{Tmunu}\nT_{\\mu\\nu} &=& \\rho u_{\\mu}u_{\\nu} + p h_{\\mu\\nu} \\, , \\\\\n\\label{Pimunu}\n\\Pi_{\\mu\\nu} &=& \\frac{1}{12} \\rho \\left[ \\rho u_{\\mu}u_{\\nu} + (\\rho+2p) h_{\\mu\\nu} \\right] \\, , \\\\\n\\label{ximunu}\n\\xi_{\\mu\\nu} &=& - \\frac{\\kappa^4_{(5)}}{\\kappa^4_{(4)}} \\left[ \\mathcal{U} u_{\\mu}u_{\\nu} + \\mathcal{P}r_{\\mu}r_{\\nu}+ \\frac{ h_{\\mu\\nu} }{3} (\\mathcal{U}-\\mathcal{P} ) \\right] \\, .\n\\end{eqnarray}\n\\end{subequations}\nHere, $p$ and $\\rho$ are, respectively, the pressure and energy density of the stellar matter of interest, $\\mathcal{U}$ is the nonlocal energy density, and $\\mathcal{P}$ is the nonlocal anisotropic stress. Also, $u_{\\alpha}$ is the four-velocity (that also satisfies the condition $g_{\\mu\\nu}u^{\\mu}u^{\\nu}=-1$), $r_{\\mu}$ is a unit radial vector, and $h_{\\mu\\nu} = g_{\\mu\\nu} + u_{\\mu} u_{\\nu}$ is the projection operator orthogonal to $u_{\\mu}$.\n\nSpherical symmetry indicates that the metric can be written as:\n\\begin{equation}\n{ds}^{2}= - B(r){dt}^{2} + A(r){dr}^{2} + r^{2} (d\\theta^{2} + \\sin^{2} \\theta d\\varphi^{2}) \\, .\\label{metric}\n\\end{equation}\nIf we define the reduced Weyl functions $\\mathcal{V} = 6 \\mathcal{U}\/\\kappa^4_{(4)}$, and $\\mathcal{N} = 4 \\mathcal{P}\/\\kappa^4_{(4)}$. First, we define the effective mass as:\n\\begin{equation}\n\\mathcal{M}^\\prime_{eff} = 4\\pi{r}^{2}\\rho_{eff}. \\label{eq:7a}\n\\end{equation}\nThen, from Eqs. \\eqref{Eins} and \\eqref{eq:4} and after straightforward calculations we have the following equations of motion:\n\\begin{subequations}\n \\label{eq:7}\n\\begin{eqnarray}\n p^\\prime &=& -\\frac{G_N}{r^{2}} \\frac{4 \\pi \\, p_{eff} \\, r^3 + \\mathcal{M}_{eff}}{1 - 2G_N \\mathcal{M}_{eff}\/r} ( p + \\rho ) \\, , \\label{eq:7b} \\\\\n \\mathcal{V}^{\\prime} + 3 \\mathcal{N}^{\\prime} &=& - \\frac{2G_N}{r^{2}} \\frac{4 \\pi \\, p_{eff} \\, r^3 + \\mathcal{M}_{eff}}{1 - 2G_N \\mathcal{M}_{eff}\/r} \\left( 2 \\mathcal{V} + 3 \\mathcal{N} \\right)\\nonumber\\\\ \n && - \\frac{9}{r} \\mathcal{N} - 3 (\\rho+p) \\rho^{\\prime} \\, , \\label{eq:7c}\n\\end{eqnarray}\n\\end{subequations}\nwhere a prime indicates derivative with respect to $r$, $A(r) = [1 - 2G_N \\mathcal{M}(r)_{eff}\/r]^{-1}$, and the effective energy density and pressure, respectively, are given as:\n\\begin{subequations}\n\\label{eq:3}\n\\begin{eqnarray}\n\\rho_{eff} &=& \\rho \\left( 1 + \\frac{\\rho}{2\\lambda} \\right) + \\frac{\\mathcal{V}}{\\lambda} \\, , \\label{eq:3a} \\\\\np_{eff} &=& p \\left(1 + \\frac{\\rho}{\\lambda} \\right) + \\frac{\\rho^{2}}{2\\lambda} + \\frac{\\mathcal{V}}{3\\lambda} + \\frac{\\mathcal{N}}{\\lambda} \\, . \\label{eq:3b}\n\\end{eqnarray}\n\\end{subequations}\nEven though we will not consider exterior galaxy solutions, we must anyway take into account the information provided by the Israel-Darmois (ID) matching condition, which for the case under study can be written as\\cite{gm}:\n\\begin{equation}\n \\label{eq:28}\n (3\/2) \\rho^2(R) + \\mathcal{V}^-(R) + 3 \\mathcal{N}^-(R) = 0 \\, .\n\\end{equation}\nIn this case, the superscript ($-$) indicates the interior value of the quantity at the halo surface\\footnote{We denote the surface of the galaxy as a region where does not exist DM or baryons, \\emph{i.e.}, the intergalactic space.} of the galaxy, assuming that $\\rho(r>R)=0$ where $R$ denotes the maximum size of the galaxy. Also, the previous equation takes in consideration the fact that the exterior must be Schwarzschild which in general the following condition must be fulfilled $\\mathcal{V}(r \\geq R) = 0 =\\mathcal{N}(r\\geq R)$ (see\\cite{Garcia-Aspeitia:2014pna} for details).\n\nFor completeness, we just note that the exterior solutions of the metric functions are given by the well known expressions $B(r) = A^{-1}(r) = 1 - 2G_N M_{eff}\/r$.\n\nFinally, we impose $\\mathcal{N}=0$ (see\\cite{Garcia-Aspeitia:2014pna}). Implying that Eq. \\eqref{eq:28} is restricted as:\n\\begin{equation}\n \\label{eq:29}\n -(3\/2) \\rho^2(R) = \\mathcal{V}^-(R) \\, ,\n\\end{equation}\nwith the aim of maintain a galaxy Schwarzschild exterior.\n\n\\section{Low energy limit and rotation curves} \\label{TOV MOD}\n\nTo begin with, we observe, from Eq.\\ \\eqref{eq:7b} in the low energy (Newtonian) limit, that we have: $r^{2}p^{\\prime}=-G_{N}\\mathcal{M}_{eff}\\rho$. Differentiating we found\n\\begin{equation}\n\\frac{d}{dr}\\left(\\frac{r^{2}}{\\rho}\\frac{dp}{dr}\\right)=-4\\pi r^{2}G_{N}\\rho_{\\rm eff}. \\label{eqdiff9}\n\\end{equation}\nFrom here it is possible to note that $d\\Phi\/dr=-\\rho^{-1}(dp\/dr)$ resulting in\n\\begin{equation}\n\\nabla^{2}\\Phi_{\\rm eff}=\\frac{1}{r^{2}}\\frac{d}{dr}\\left(r^{2}\\frac{d\\Phi_{\\rm eff}}{dr}\\right)=4\\pi G_{N}\\rho_{\\rm eff}, \\label{Poisson}\n\\end{equation}\nbeing necessary to define the energy density of DM together with the nonlocal energy density. Notice that the nonlocal energy density can be obtained easily from Eq.\\ \\eqref{eq:7c} in the galaxy interior and also the fluid behaves like dust, implying the condition $p=0$, always fulfilling the low energy condition $4\\pi r^3p_{eff}\\ll\\mathcal{M}_{eff}$ and $2G_{N}\\mathcal{M}_{eff}\/r\\ll1$, between effective quantities and in consequence $4G_{N}\\mathcal{M}_{eff}\\mathcal{V}\/r^2\\sim0$, is negligible.\n\nIn addition, the rotation curve is obtained from the contribution of the effective potential, this expression can be written as:\n\\begin{eqnarray}\nV^2(r) &=& r\\left\\vert\\frac{d\\Phi_{\\rm eff}}{dr}\\right\\vert=\\frac{G_N \\mathcal{M}_{eff}(r)}{r} \n\\nonumber \\\\\n&=& \n\\frac{G_N }{r} \n\\left[\n\\mathcal{M}_{DM}(r) + \\mathcal{M}_{Brane}(r)\n\\right]\n, \\label{rotvel}\n\\end{eqnarray}\nwhere $\\mathcal{M}_{DM}(r)$ is the contribution to the mass from DM, $\\mathcal{M}_{Brane}(r)$ gives the modification to the DM mass that comes from the brane; and $\\mathcal{M}_{eff}(r)$ must be greater than zero. From here, it is possible to study the rotation velocities of the DM, assuming a variety of density profiles.\n\nBefore we start let us define the following dimensionless variables: $\\bar{r}\\equiv r\/r_{\\rm s}$, $v_{0}^{2}\\equiv4\\pi G_{N}r_{\\rm s}^{2}\\rho_{\\rm s}$ and $\\bar{\\rho}\\equiv\\rho_{\\rm s}\/2\\lambda$ where $\\rho_{\\rm s}$, is the central density of the halo and $r_{s}$ is associated with the central radius of the halo. \n\n\\subsection{Pseudo isothermal profile for dark matter}\n\nHere we consider that DM density profile is given by PISO\\cite{piso} written as:\n\\begin{equation}\n\\rho_{\\rm PISO}(r)=\\frac{\\rho_{\\rm s}}{1+\\bar{r}^{2}}. \\label{PIP}\n\\end{equation}\nFrom Eq. \\eqref{rotvel}, together with Eq. \\eqref{PIP}, it is possible to obtain:\n\\begin{eqnarray}\nV_{\\rm PISO}^{2}(\\bar{r}) &=& v_{0}^{2}\n\\left\\lbrace\n\\left(\n1-\\frac{1}{\\bar{r}}\\arctan\\bar{r}\n\\right) \n\\right.\n\\nonumber \\\\\n&& \n\\left. + \\bar{\\rho}\n\\left(\n\\frac{1}{1+\\bar{r}^2}- \\frac{1}{\\bar{r}}\\arctan\\bar{r}\n\\right)\n\\right\\rbrace.\n\\label{RCPISO}\n\\end{eqnarray}\nIn the limit $\\bar{\\rho}\\to0$, we recover the classical rotation velocity associated with PISO for DM.\nThe effective density must be positive defined, then $\\lambda > \\rho_s$ must be fulfilled. The first right-hand term in parenthesis in Eq.\\ \\eqref{RCPISO} is PISO dark matter contribution and the second \nis brane's contribution.\n\n\\subsection{Navarro-Frenk-White profile for dark matter}\nAnother interesting case (motivated by cosmological $N$-body simulations) is the NFW density profile, which is given by\\cite{NFW}:\n\\begin{equation}\n\\rho_{\\rm NFW}(r)=\\frac{\\rho_{\\rm s}}{\\bar{r}(1+\\bar{r})^{2}}. \\label{NFW}\n\\end{equation}\nThis is a density profile that diverges as $r \\rightarrow 0$ \nand \nit is not possible to say\nthat $\\rho_s$ is related with the central density of the DM distribution.\nAlso density goes as $1\/\\bar{r}^3$ when $\\bar{r} \\gg 1$.\nHowever, in this particular case, we will still be calling them the \\emph{central} density and radius of the NFW matter distribution.\nFrom Eq.\\ \\eqref{rotvel}, together with Eq.\\ \\eqref{NFW} we obtain the following rotation curve:\n\\begin{eqnarray}\nV_{\\rm NFW}^{2}(\\bar{r}) &=& v_{0}^{2}\\left\\lbrace\\left(\\frac{(1+\\bar{r})\\ln(1+\\bar{r})-\\bar{r}}{\\bar{r}(1+\\bar{r})}\\right)\\right.\\nonumber\\\\&&\n\\left.+\\frac{2\\bar{\\rho}}{3\\bar{r}}\\left(\\frac{1}{(1+\\bar{r})^{3}}-1\\right) \\right\\rbrace.\n\\label{RCNFW}\n\\end{eqnarray}\nThe first right-hand term in parenthesis in Eq.\\ \\eqref{RCNFW} is NFW dark matter contribution and the second one is the brane's contribution. Notice that we recover also the classical limit when $\\bar{\\rho}\\to0$.\n\nIn addition, it is important to remark that the effective density must be positive defined, then $\\lambda > \\rho_s r_s \/r$. Also, if $\\mathcal{M}(r)$ must be greater than zero, then $r > r_{min}$ where $r_{min}$ is given by solving the following equation:\n\\begin{equation}\n\\frac{2}{3}\\bar{\\rho}=\\frac{(\\alpha+1)^2[(\\alpha+1)\\ln(\\alpha+1)-\\alpha]}{(\\alpha+1)^3-1},\\label{comp}\n\\end{equation}\nwhere we define $\\alpha\\equiv r_{min}\/r_s$ as a dimensionless quantity.\n\n\\subsection{Burkert density profile for dark matter}\n\nAnother density profile was proposed by Burkert\\cite{Burkert}, which it has the form:\n\\begin{equation}\n\\rho_{\\rm Burk}=\\frac{\\rho_{\\rm s}}{(1+\\bar{r})(1+\\bar{r}^{2})}. \\label{Burk}\n\\end{equation}\nAgain, from Eq.\\ \\eqref{rotvel}, together with Eq.\\ \\eqref{Burk} we obtain the following rotation curve:\n\\begin{eqnarray}\nV_{\\rm Burk}^{2}(\\bar{r}) &=&\\frac{v_{0}^{2}}{4\\bar{r}} \\left\\lbrace \\left( \\ln[(1+\\bar{r}^{2})(1+\\bar{r})^{2}]-2\\arctan(\\bar{r}) \\right)\\right.\n\\label{RCBurkert}\n\\\\&&\n\\left.+ \\frac{1}{2}\\bar{\\rho}\\left( \\frac{1}{1+\\bar{r}}+\\frac{1}{1+\\bar{r}^{2}}+\\arctan(\\bar{r})-2 \\right)\\right\\rbrace.\n\\nonumber\n\\end{eqnarray}\nIn the limit $\\bar{\\rho}\\to0$, we recover the classical rotation velocity associated with Burkert density profile\\cite{Burkert}.\nThe effective density must be positive defined, then $\\lambda > \\rho_s$. \nAgain the first right-hand term in parenthesis in Eq.\\ \\eqref{RCBurkert} is\nBurkert DM contribution and the second one comes from the\n brane's contribution.\n\n\\section{Constrictions with galaxies without photometry} \\label{Results}\n\nTo start with the analysis, we $\\chi^{2}$ best fit the observational rotation curves of the sample with:\n\\begin{equation}\n\\chi^{2}=\\sum_{i=1}^{N}\\left(\\frac{V_{theo}-V_{exp \\; i}}{\\delta V_{exp\\; i}}\\right)^{2},\n\\label{chi2Eq}\n\\end{equation}\nwhere $i$ runs from one up to the number of points in the data, $N$; $V_{theo}$, is computed according to the velocity profile under consideration \nand $\\delta V_{exp\\; i}$, is the error in the measurement of the rotational velocity. Notice that \nthe free parameters are only for DM-Branes: $r_{s}$, $\\rho_{s}$ and $\\lambda$. \nIn the tables below we show $\\chi_{red}^{2} \\equiv \\chi^{2}\/(N - n_p -1)$ where $n_p$ is the number of parameters to fit, being in our case, $n_p=3$.\n\nThe analyzed sample of galaxies are twelve high resolution rotation curves of LSB galaxies with no photometry (visible components, such as gas and stars, are negligible) as given in Ref.\\cite{deBlok\/etal:2001}. This sample was used to study DM equation of state (EoS) in Ref.\\cite{Barranco\/etal:2015}. We remark that in this part we use units such that $4 \\pi G_{N}=1$, velocities are in km\/s, and distances are given in kpc.\n\n\\subsection{Results: PISO profile + Branes}\n\nWe have estimated the parameters of the PISO+branes model \nand were compared with PISO model without brane contribution, minimizing the appropriate $\\chi^2$ for the sample of observed rotational curves, using Eq.\\ (\\ref{chi2Eq}) with Eq.\\ (\\ref{RCPISO}) and taking into account that $\\lambda > \\rho_s$ must be fulfilled. \n\nIn Fig.\\ \\ref{PISO1} we show, for each one of the galaxies in the sample,\nthe plots of the PISO theoretical rotation curve (solid line), that best fit of the corresponding observational data (orange symbols); also shown are the errors of the estimation (brown symbols). \nFor each galaxy we have plotted the contribution to the rotation velocity due only to the brane (red long-dashed curve) and only to the dark matter PISO density profile (blue short-dashed curve), see Eq.\\ (\\ref{RCPISO}).\nBrane effects are very clear in galaxies: \nESO 2060140,\nESO 3020120,\nU 11616,\nU 11648,\nU 11748,\nU 11819.\nIn Table \\ref{TablePiso} it is shown the central density, central radius and the brane tension which is the free parameter of the brane theory (only in PISO+branes). As a comparison, it is also shown the central density and radius without brane contribution.\nThe worst fitted galaxies were (high $\\chi_{red}^2$ value): \nU 11648,\nU 11748.\nThe fitted brane tension values presents great dispersion, from the lower value: \n$0.167\\; M_{\\odot}\/\\rm pc^3$, ESO 3020120 to the higher value:\n$108.096\\; M_{\\odot}\/\\rm pc^3$, ESO 4880049.\nIt is useful to have $\\lambda$ in eV, where the conversion from solar masses to eV is: $1 M_{\\odot}\/\\rm pc^3 \\sim 2.915\\times10^{-4}eV^4$. \nThe brane tension parameter has an average value of $\\langle\\lambda\\rangle_{\\rm PISO} = 33.178 \\; M_{\\odot}\/\\rm pc^3$ with a standard deviation $\\sigma_{\\rm PISO} = 40.935 \\; M_{\\odot}\/\\rm pc^3$. Notice that we can't see a clear tendency to a $\\lambda$ value or range of values.\n\n\\begin{figure}\n\\includegraphics[scale=0.33]{vceso30500900PISO+Branes}\n\\includegraphics[scale=0.33]{vceso0140040PISO+Branes} \\\\\n\\includegraphics[scale=0.33]{vceso2060140PISO+Branes} \n\\includegraphics[scale=0.33]{vcESO3020120PISO+Branes} \\\\ \n\\includegraphics[scale=0.33]{vceso4250180PISO+Branes} \n\\includegraphics[scale=0.33]{vceso4880049PISO+Branes} \\\\\n\\includegraphics[scale=0.33]{vcf570_v1PISO+Branes} \n\\includegraphics[scale=0.33]{vcu11454PISO+Branes} \\\\\n\\includegraphics[scale=0.33]{vcu11616PISO+Branes} \n\\includegraphics[scale=0.33]{vcu11648PISO+Branes} \\\\\n\\includegraphics[scale=0.33]{vcu11748PISO+Branes} \n\\includegraphics[scale=0.33]{vcu11819PISO+Branes}\n\\caption{Group of analyzed galaxies using modified rotation velocity for PISO profile: ESO 3050090,\nESO 0140040, ESO 2060140, ESO 3020120, ESO 4250180, ESO 4880049, 570\\_V1, U11454, U11616, U11648, U11748, U11819. We show in the plots: Total rotation curve (solid black line), only PISO curve (short dashed blue curve) and the rotation curve associated with the mass lost by the effect of the brane (red dashed curve).} \n\\label{PISO1}\n\\end{figure}\n\n\n\n\n\\subsection{Results: NFW profile + Branes}\nFor the NFW density profile case we have the following results:\nWe have estimated parameters with and without brane contribution by\nminimizing the corresponding $\\chi^2$, Eq.\\ (\\ref{chi2Eq}) with Eq.\\ (\\ref{RCNFW}), for the sample of observed rotation curves \nand taking into account that\n$\\lambda > \\rho_s r_s \/r$ in order to have an effective density positive defined, always fulfilling Eq.\\ (\\ref{comp}).\n\nIn Fig.\\ \\ref{NFW1}, it is shown, for each galaxy in the sample of the LSB galaxies, the theoretical fitted curve to a preferred brane tension value (solid line), \nthe NFW curve and the rotation curve associated with the mass lost by the effects of branes, see Eq.\\ (\\ref{RCNFW}). \nIn Table \\ref{TableNFW} it is shown, for the sample,\nthe central density, central radius and $\\chi_{red}^2$ values without branes; and\nthe central density, central radius, brane tension\nand $\\chi_{red}^2$ values with branes contribution.\nGalaxy U 11748 is the worst fitted case with $\\chi_{red}^2 = 2.163$.\nFor galaxies: \nESO 4250180,\nESO 4880049,\nand U 11648,\nthere are not clear brane effects. \nGalaxy U 11648 is an \\emph{outlier} with a brane tension value of $4323.28\\; M_{\\odot}\/\\rm pc^3$ that is out of the range of preferred values of the other galaxies in the sample.\nNotice that we have found a preferred range of tension values, from $0.487$ to $9.232$ $M_{\\odot}\/\\rm pc^3$. Without the outlier, the brane tension parameter has an average value of \n$\\langle\\lambda\\rangle_{\\rm NFW}\\simeq 2.51 \\; M_{\\odot}\/\\rm pc^3$ \nwith a standard deviation $\\sigma_{\\rm NFW}\\simeq 3.015 \\; M_{\\odot}\/\\rm pc^3$.\n\n\\begin{figure}\n\\includegraphics[scale=0.33]{vceso30500900NFW+Branes} \n\\includegraphics[scale=0.33]{vceso0140040NFW+Branes} \\\\\n\\includegraphics[scale=0.33]{vceso2060140NFW+Branes} \n\\includegraphics[scale=0.33]{vcESO3020120NFW+Branes} \\\\ \n\\includegraphics[scale=0.33]{vceso4250180NFW+Branes} \n\\includegraphics[scale=0.33]{vceso4880049NFW+Branes} \\\\\n\\includegraphics[scale=0.33]{vcf570_v1NFW+Branes} \n\\includegraphics[scale=0.33]{vcu11454NFW+Branes} \\\\\n\\includegraphics[scale=0.33]{vcu11616NFW+Branes} \n\\includegraphics[scale=0.33]{vcu11648NFW+Branes} \\\\\n\\includegraphics[scale=0.33]{vcu11748NFW+Branes} \n\\includegraphics[scale=0.33]{vcu11819NFW+Branes} \n\\caption{Group of analyzed galaxies using modified rotation velocity for NFW profile: ESO 3050090,\nESO 0140040, ESO 2060140, ESO 3020120, ESO 4250180, ESO 4880049, 570\\_V1, U11454, U11616, U11648, U11748, U11819. We show in the plots: Total rotation curve (solid black line), only NFW curve (short dashed blue curve) and the rotation curve associate with the mass lost by the effect of the brane (red dashed curve).} \n\\label{NFW1}\n\\end{figure}\n\n\n\n\\subsection{Results: Burkert+Branes profile}\n\nIn the case of Burkert DM density profile, we have also estimated the parameters of the Burkert+branes model \nand were compared with Burkert model without branes, minimizing the appropriate $\\chi^2_{red}$, Eq.\\ (\\ref{chi2Eq}) with Eq.\\ (\\ref{RCBurkert}), for the sample of observed rotation curves. We have considered that $\\lambda > \\rho_s$ must be fulfilled.\n\nThe results are shown in Fig.\\ \\ref{Burkert1}, where it is plotted the fit to a preferred brane tension value, remarking the total rotation curve (solid line), the Burkert DM density contribution curve (blue short-dashed line) and the rotation curve associated with the mass lost by the effects of branes (red dashed line), see Eq.\\ (\\ref{RCBurkert}). \nIn Table \\ref{TableBurkert} it is shown the fitted values for the central density, central radius and the corresponding value of the $\\chi_{red}^2$ without brane contribution; and the fitted values for\nthe central density, central radius, brane tension, and theirs $\\chi_{red}^2$ values with brane contribution.\nThe worst fitted (high values of $\\chi_{red}^2$) galaxies are: U 11648 and U 11748.\nGalaxies ESO 3020120, U 11748, and\nU 11819 show a clear brane effects and also are outliers. The main tendency is that $\\lambda$ has values of the order of $10^3 \\;M_{\\odot}\/\\rm pc^3$ or above, approximately.\nThe brane tension parameter, without the outliers, for the DM Burkert profile case has an average value of \n$\\langle\\lambda\\rangle_{\\rm Burk}\\simeq 3192.02 \\;M_{\\odot}\/\\rm pc^3$, \nand a standard deviation of \n$\\sigma_{\\rm Burk}\\simeq 2174.97 \\; M_{\\odot}\/\\rm pc^3$.\n\n\\begin{figure}\n\\includegraphics[scale=0.33]{vceso30500900Burkert+Branes} \n\\includegraphics[scale=0.33]{vceso0140040Burkert+Branes} \\\\\n\\includegraphics[scale=0.33]{vceso2060140Burkert+Branes} \n\\includegraphics[scale=0.33]{vcESO3020120Burkert+Branes} \\\\ \n\\includegraphics[scale=0.33]{vceso4250180Burkert+Branes} \n\\includegraphics[scale=0.33]{vceso4880049Burkert+Branes} \\\\\n\\includegraphics[scale=0.33]{vcf570_v1Burkert+Branes} \n\\includegraphics[scale=0.33]{vcu11454Burkert+Branes} \\\\\n\\includegraphics[scale=0.33]{vcu11616Burkert+Branes} \n\\includegraphics[scale=0.33]{vcu11648Burkert+Branes} \\\\\n\\includegraphics[scale=0.33]{vcu11748Burkert+Branes} \n\\includegraphics[scale=0.33]{vcu11819Burkert+Branes}\n\\caption{Group of analyzed galaxies using modified rotation velocity for Burkert profile: ESO 3050090,\nESO 0140040, ESO 2060140, ESO 3020120, ESO 4250180, ESO 4880049, 570\\_V1, U11454, U11616, U11648, U11748, U11819. We show in the plots: Total rotation curve (solid black line), only Burkert curve (short dashed blue curve) and the rotation curve associate with the mass lost by the effect of the brane (red dashed curve).} \n\\label{Burkert1}\n\\end{figure}\n\n\n\n\\subsection{Results: a synthetic rotation curve}\n\nFinally, we show the fitting results of the DM models plus brane's contribution to a synthetic rotation curve. This synthetic rotation curve was made of 40 rotation curves of galaxies with magnitudes around $M_I = -18.5$\\cite{Salucci1}.\nThese 40 rotation curves came out of 1100 galaxies that gave the universal rotation curve for spirals. For this sample of low luminosity galaxies, of $M_I = -18.5$, it was shown that the baryonic disk has a very small contribution (for details see reference\\cite{Salucci1}).\n\nIn this subsection we are now using units such $G=R_{opt}=V(R_{opt})=1$, where $R_{opt}$ and $V(R_{opt})$ are the optical radius and the velocity at the optical radius, respectively. $R_{opt}$ is the radius encompassing 83 per cent of the total integrated light. For an exponential disk with a surface brightness given by: $I(r) \\propto \\exp(-r\/R_D)$, we have that $R_{opt}=3.2 R_D$\\cite{Salucci1}.\n\nIn figure \\ref{SM185} we show the synthetic rotation curve and the fitting results using PISO, NFW and Burkert profiles with and without brane's contribution.\n\\begin{figure}\n\\includegraphics[scale=0.33]{vcM185PISO} \n\\includegraphics[scale=0.33]{vcM185PISO+Branes} \n\\includegraphics[scale=0.33]{vcM185NFW} \n\\includegraphics[scale=0.33]{vcM185NFW+Branes} \n\\includegraphics[scale=0.33]{vcM185Burkert} \n\\includegraphics[scale=0.33]{vcM185Burkert+Branes} \n\\caption{Synthetic rotation curve of galaxies with magnitud $M_I=-18.5$.\nLeft panels: rotation curves fitted without branes. Right panels: rotation curves fitted with branes.\nFirst row is for PISO model; second row is for NFW model and third row is for Burkert model. \nWe show in the plots: Total rotation curve (solid black line), only DM model curve (short dashed blue curve) and the rotation curve associate with the mass lost by the effect of the brane (red dashed curve).} \n\\label{SM185}\n\\end{figure}\nAs we can see in Table \\ref{TableSynthetic} the same trend is observed in the brane's tension values as compared with the results for the LSB catalog analyzed above using PISO, NFW and Burkert as a DM profiles: lower value is obtained for NFW model and higher values is obtained for Burkert density profile.\n\nGiven that this synthetic rotation curve is built from 40 rotation curves of real spirals, the values of the brane's tension in table \\ref{TableSynthetic} is representative of all these rotation curves. \nThen, for PISO model $\\lambda=60.692$ $M_{\\odot}\/\\rm pc^3$, a value that is greater than the average value of the tension shown in Table \\ref{TablePiso} but inside the interval marked by the standard deviation. \nFor NFW model $\\lambda=226.054$ $M_{\\odot}\/\\rm pc^3$, this value is lower than the average value reported in Table \\ref{TableNFW} and inside the range marked by the standard deviation. \nFor Burkert model $\\lambda=1.58\\times 10^5$ $M_{\\odot}\/\\rm pc^3$ this value is well above than the average value shown in Table \\ref{TableBurkert}; a value outside the range marked by the standard deviation. \n\n\n\\section{Discussion and conclusions} \\label{Disc}\n\nWe have presented in this paper, the effects coming from the presence of branes in galaxy rotation curves for \nthree density profiles used to study the behavior of DM at galactic scales. \nWith this in mind, we were given to the task of study a sample of \nhigh resolution measurements of rotation curves of galaxies without photometry\\cite{deBlok\/etal:2001} \nand a synthetic rotation curve built from 40 rotation curves of galaxies of magnitude around $M_I=-18.5$\nfitting\n the values of $\\rho_{s}$, $r_{s}$ and $\\lambda$ through minimizing the $\\chi^{2}_{red}$ value and we have compared with the standard results of $\\rho_{s}$, $r_{s}$ for each DM density profile without branes. \n The results for every observable in the three different profiles were summarized and compared in Tables \\ref{TablePiso}-\\ref{TableSynthetic}.\n\nFrom here, it is possible to observe how the results show a weaker limit for the value of brane tension \n($\\sim10^{-3}\\; \\rm eV^4-46$ eV$^4$) for the three models, in comparison with other astrophysical and cosmological studies\\cite{Kapner:2006si,Alexeyev:2015gja,mk,gm,Sakstein:2014nfa,Kudoh:2003xz,Cavaglia:2002si,Holanda:2013doa,Barrow:2001pi,Brax:2003fv}; for example, Linares \\emph{et al.}\\cite{Linares:2015fsa} show that weaker values than $\\lambda \\simeq 10^{4}$ MeV$^{4}$, present anomalous behavior in the compactness of a dwarf star composed by a polytropic EoS, concluding that a wide region of their bound \nwill show a non compactness stellar configuration, if it is applied to the study shown in\\cite{Linares:2015fsa}.\n \nIt is important to notice that chosen a value of brane tension that not fulfill our bounds imposed through the paper, generate an anomalous behavior in the center of the galaxy which is characteristic of the model. Remarkable, for higher values of this bound, the modified rotation curves are in good agreement with \nthe observed rotation curves of the sample that we use,\npresenting only the distinctive features of each density profile: For example,\nNFW dark matter density profile prefers lower values of the brane tension (on the average $\\lambda \\sim 0.73\\times 10^{-3}$ eV$^4$), implying clear effects of the brane;\nPISO dark matter\n case has an average value of $\\lambda \\sim 0.96\\times 10^{-2}$ eV$^4$ and show relatively the maximum dispersion on the fitted values of the brane tension;\nwhereas Burkert DM density profile shows negligible brane effects, on the average $\\lambda \\sim 0.93$ eV$^4$ -- $46$ eV$^4$.\n\n In addition, it is important to discuss briefly the changes caused by the presence of branes in the problem of cusp\/core. Notice that in this case the part that play a role is the effective density which is written in terms of brane corrections as: $\\rho_{eff}=\\rho(1-\\rho\/\\lambda)$; it is notorious how the small perturbations alleviate the cusp problem which afflicts NFW, albeit the excessive presence of these terms could generates a negative effective density profile; also PISO and Burkert show modifications when $r\\to0$ but does not pledge its core behavior while the brane tension only takes small values. In this way, the possibility of having a core behavior, help us to constraint the value of brane tension and still keep in the game the NFW profile.\n\nSummarizing, it is really challenging to establish bounds in a dynamical systems like rotation curves in galaxies due to the\nlow densities found in the galactic medium, giving only a weaker limits in comparison with other studies in a most energetic systems. \nOur most important conclusion, is that despite the efforts, we think that it is not straightforward to do that the results fit with other astrophysical and cosmological studies, being impractical and not feasible to\nfind evidence of extra dimensions in galactic dynamics through the determination of the brane tension value, \neven more, we think that exist too much dispersion in the fitted values of the brane tension using this method for some DM density profile models. \nAlso it is important to note that the value of the brane tension is strongly dependent of the characteristic of the galaxy studied, suggesting an average for the preferred value of the brane tension in each case. \nIn addition, we notice that the effects of extra dimensions are stronger in the galactic core, suggesting that the NFW model is not appropriate in the search of constraints in brane theory due to the divergence in the center of the galaxy (see Eq.\\ \\eqref{comp}); PISO and Burkert could be good candidates to explore the galactic core in this framework; however it is necessary a more extensive study before we obtain a definitive conclusion. \n\nAs a final note, we know that it is necessary to recollect more observational data to constraint \nthe models or even give the final conclusion about extra dimensions (for or against), supporting the brane constraints shown through this paper with a more profound study of galactic dynamic or other tests like cosmological evidences presented in CMB anisotropies. \nHowever, this work is in progress and will be reported elsewhere.\n\n\n\\begin{acknowledgements}\nMAG-A acknowledge support from SNI-M\\'exico and CONACyT research fellow. Instituto Avanzado de Cosmolog\\'ia (IAC) collaborations.\n\\end{acknowledgements}\n\n\n\\section{Introduction} \\label{Int}\n\nGeneral Theory of Relativity (GR) is the cornerstone of\nastrophysics and cosmology, giving predictions with unprecedented success. \nAt astrophysical scales GR has been tested in, for example, the solar system, stellar dynamics, black hole formation and evolution, among others (see for instance\\cite{FischbachTalmadge1999,Will1993,Kamionkowski:2007wv,*Matts,*Peebles:2013hla}). \nHowever, GR is being currently tested with various phenomena that can be \nsignificant challenges to the GR theory, generating important changes never seen before. Ones of the major challenges of modern cosmology are undoubtedly dark matter (DM) and dark energy. They comprise approximately $27\\%$ for DM and $68\\%$ for dark energy, of our \nuniverse\\cite{PlanckCollaboration2013} allowing the formation of large scale structures\\cite{FW,Diaferio:2008jy}.\nDark matter has been invoked as the mechanism to stabilize spiral galaxies and to provide with a matter distribution component to explain the observed rotation curves.\nNowadays the best model of the universe we have is the \n\\emph{concordance} or $\\Lambda$CDM model that has been successful in explaining the very large-scale structure formation, the statistics of the distribution of galaxy clusters, the temperature anisotropies of the cosmic microwave background radiation (CMB) and many other astronomical observations. \nIn spite of all successes we have mentioned, \nthis model has several problems, for example: predicts too much power on small scale\\cite{Rodriguez-Meza:2012}, then it over predicts the number of observed satellite galaxies\\cite{Klypin1999,Moore1999,Papastergis2011} and predicts halo profiles that are denser and \ncuspier than those inferred observationally\\cite{Navarro1997,Subramanian2000,Salucci}, and also predicts a population of massive and concentrated subclass that are inconsistent \nwith observations of the kinematics of Milky Way satellites\\cite{BoylanKolchin2012}.\n\nOne of the first astronomical observations that brought attention on DM was the observation of rotation curves of spiral galaxies by Rubin and coworkers\\cite{Rubin2001}, \nthese observations turned out to be the main tool to investigate the role of DM at galactic scales: its role in determining the structure, how the mass is distributed, and the dynamics, evolution, and formation of spiral galaxies.\nRemarkably, the corresponding rotation velocities of galaxies, can be explained with the density profiles of different Newtonian DM models like Pseudo Isothermal profile (PISO)\\cite{piso}, Navarro-Frenk-White profile (NFW)\\cite{Navarro1997} \nor Burkert profile\\cite{Burkert}, among \nothers\\cite{Einasto}; except by the fact that until now it is unsolved the problem of cusp and core in the densities profiles. In this way, none of them have the last word because the main questions, about the density distribution and of course the \\emph{nature} of DM, has not been resolved.\n\nAlternative theories of gravity have been used to model DM. For instance a scalar field has been proposed to model DM\\cite{Dick1996,Cho\/Keum:1998}, \nand has been used to study rotation curves of spiral galaxies\\cite{Guzman\/Matos:2000}. This scalar field is coupled minimally to the metric, however, scalar fields coupled non minimally to the metric have also been used to study DM\\cite{Rodriguez-Meza:2012,RodriguezMeza\/Others:2001,RodriguezMeza\/CervantesCota:2004,Rodriguez-Meza:2012b}. Equivalently $F(R)$ models exists in the literature that analyzes rotation curves\\cite{Martins\/Salucci:2007}.\n\nOn the other hand, one of the best candidates to extend GR is the brane theory, whose main characteristic is to add another dimension having a five dimensional bulk where it is embedded a four dimensional manifold called the brane\\cite{Randall-I,*Randall-II}. This model is characterized by the fact that the standard model of particles is confined in the brane and only the gravitational interaction can travel in the bulk\\cite{Randall-I,*Randall-II}. The assumption that the five dimensional Einstein's equations are valid, generates corrections in the four dimensional Einstein's equations confined in the brane bringing information from the extra dimension\\cite{sms}. \nThese extra corrections in the Einstein's equations can help us to elucidate and solve the problems that afflicts the modern cosmology and astrophysics\\cite{m2000,*yo2,*Casadio2012251,*jf1,*gm,*Garcia-Aspeitia:2013jea,*langlois2001large,*Garcia-Aspeitia:2014pna,*jf2,*PerezLorenzana:2005iv,*Ovalle:2014uwa,*Garcia-Aspeitia:2014jda,*Linares:2015fsa,*Casadio:2004nz}.\n\nBefore we start, let us mention here some experimental constraints on braneworld models, most of them about the so-called brane tension, $\\lambda$, which appears explicitly as a free parameter in the corrections of the gravitational equations mentioned above. As a first example we have the measurements on the deviations from Newton's law of the gravitational interaction at small distances. It is reported that no deviation is observed for distances $l \\gtrsim 0.1 \\, {\\rm mm}$, which then implies a lower limit on the brane tension in the Randall-Sundrum II model (RSII): $\\lambda> 1 \\, {\\rm TeV}^{4}$\\cite{Kapner:2006si,*Alexeyev:2015gja}; it is important to mention that these limits do not apply to the two-branes case of the Randall-Sundrum I model (RSI) (see \\cite{mk} for details). \nAstrophysical studies, related to gravitational waves and stellar stability, constrain\nthe brane tension to be $\\lambda > 5\\times10^{8} \\, {\\rm MeV}^{4}$\\cite{gm,Sakstein:2014nfa}, whereas the existence of black hole X-ray binaries suggests that $l\\lesssim 10^{-2} {\\rm mm}$\\cite{mk,Kudoh:2003xz,*Cavaglia:2002si}. Finally, from cosmological observations, the requirement of successful nucleosynthesis provides the lower limit $\\lambda> 1\\, {\\rm MeV}^{4}$, which is a much weaker limit as compared to other experiments (another cosmological tests can be seen in: Ref. \\cite{Holanda:2013doa,*Barrow:2001pi,*Brax:2003fv}).\n \nIn fact, this paper is devoted to study the main observable of brane theory which is the brane tension, whose existence delimits between the four dimensional GR and its high energy corrections. We are given to the task of perform a Newtonian approximation of the modified Tolman-Oppenheimer-Volkoff (TOV) equation, maintaining the effective terms provided by branes which cause subtle differences in the traditional dynamics. \nIn this way we test the theory at galactic scale using high resolution measurements of rotation curves of a sample of low surface brightness (LSB) galaxies with no photometry\\cite{deBlok\/etal:2001}\nand a synthetic rotation curve built from 40 rotation curves of spirals of magnitude around $M_I=-18.5$ where was found that the baryonic components has a very small contribution\\cite{Salucci1},\nassuming PISO, NFW and Burkert DM profiles respectively and with that, we constraint the preferred value of brane tension with observables. \nThat the sample has no photometry means that the galaxies are DM dominated and then we have only two parameters related to the distribution of DM, a density scale and a length scale, adding the brane tension we have three parameters in total to fit. \nThe brane tension fitted values are compared among the traditional DM density profile models of spiral galaxies \n(PISO, NFW and Burkert) and against the same models without the presence of branes and confronted with other values of the tension parameter coming from cosmological and astrophysical observational data.\n\nThis paper is organized as follows: In Sec.\\ \\ref{EM} we show the equations of motion (modified TOV equations) for a spherical symmetry and the appropriate initial conditions. In Sec.\\ \\ref{TOV MOD} we explore the Newtonian limit and we show the mathematical expression of rotation velocity with brane modifications; particularly we show the modifications to velocity rotation expressions of PISO, NFW and Burkert DM profiles and they are compared with models without branes. \nIn Sec.\\ \\ref{Results} we test the DM models plus brane with observations: we use a sample of high resolution measurements of rotation curves of LSB galaxies and a synthetic rotation curve representative of 40 rotation curves of spirals where the baryonic component has a very small contribution.\nFinally in Sec.\\ \\ref{Disc}, we discuss the results obtained in the paper and make some conclusions.\n\nIn what follows, we work in units in which $c=\\hbar=1$, unless explicitly written.\n\n\\section{Review of equations of motion for branes} \\label{EM}\n\nLet us start by writing the equations of motion for galactic stability in a brane embedded in a five-dimensional bulk according to the RSII model\\cite{Randall-II}. Following an appropriate computation (for details see\\cite{mk,sms}), it is possible to demonstrate that the modified four-dimensional Einstein's equations can be written as \n\\begin{equation}\n G_{\\mu\\nu} + \\xi_{\\mu\\nu} + \\Lambda_{(4)}g_{\\mu\\nu} = \\kappa^{2}_{(4)} T_{\\mu\\nu} + \\kappa^{4}_{(5)} \\Pi_{\\mu\\nu} +\n \\kappa^{2}_{(5)} F_{\\mu\\nu} , \\label{Eins}\n\\end{equation}\nwhere $\\kappa_{(4)}$ and $\\kappa_{(5)}$ are respectively the four and five- dimensional coupling constants, which are related in the form: $\\kappa^{2}_{(4)}=8\\pi G_{N}=\\kappa^{4}_{(5)} \\lambda\/6$, where $\\lambda$ is defined as the brane tension, and $G_{N}$ is the Newton constant. For purposes of simplicity, we will not consider bulk matter, which translates into $F_{\\mu\\nu}=0$, and discard the presence of the four-dimensional cosmological constant, $\\Lambda_{(4)}=0$, \nas we do not expect it to have any important effect at galactic scales (for a recent discussion about it see\\cite{Pavlidou:2013zha}). Additionally, we will neglect any nonlocal energy flux, which is allowed by the static spherically symmetric solutions we will study below\\cite{gm}.\n\nThe energy-momentum tensor, the quadratic energy-momentum tensor, and the Weyl (traceless) contribution, have the explicit forms\n\\begin{subequations}\n\\label{eq:4}\n\\begin{eqnarray}\n\\label{Tmunu}\nT_{\\mu\\nu} &=& \\rho u_{\\mu}u_{\\nu} + p h_{\\mu\\nu} \\, , \\\\\n\\label{Pimunu}\n\\Pi_{\\mu\\nu} &=& \\frac{1}{12} \\rho \\left[ \\rho u_{\\mu}u_{\\nu} + (\\rho+2p) h_{\\mu\\nu} \\right] \\, , \\\\\n\\label{ximunu}\n\\xi_{\\mu\\nu} &=& - \\frac{\\kappa^4_{(5)}}{\\kappa^4_{(4)}} \\left[ \\mathcal{U} u_{\\mu}u_{\\nu} + \\mathcal{P}r_{\\mu}r_{\\nu}+ \\frac{ h_{\\mu\\nu} }{3} (\\mathcal{U}-\\mathcal{P} ) \\right] \\, .\n\\end{eqnarray}\n\\end{subequations}\nHere, $p$ and $\\rho$ are, respectively, the pressure and energy density of the stellar matter of interest, $\\mathcal{U}$ is the nonlocal energy density, and $\\mathcal{P}$ is the nonlocal anisotropic stress. Also, $u_{\\alpha}$ is the four-velocity (that also satisfies the condition $g_{\\mu\\nu}u^{\\mu}u^{\\nu}=-1$), $r_{\\mu}$ is a unit radial vector, and $h_{\\mu\\nu} = g_{\\mu\\nu} + u_{\\mu} u_{\\nu}$ is the projection operator orthogonal to $u_{\\mu}$.\n\nSpherical symmetry indicates that the metric can be written as:\n\\begin{equation}\n{ds}^{2}= - B(r){dt}^{2} + A(r){dr}^{2} + r^{2} (d\\theta^{2} + \\sin^{2} \\theta d\\varphi^{2}) \\, .\\label{metric}\n\\end{equation}\nIf we define the reduced Weyl functions $\\mathcal{V} = 6 \\mathcal{U}\/\\kappa^4_{(4)}$, and $\\mathcal{N} = 4 \\mathcal{P}\/\\kappa^4_{(4)}$. First, we define the effective mass as:\n\\begin{equation}\n\\mathcal{M}^\\prime_{eff} = 4\\pi{r}^{2}\\rho_{eff}. \\label{eq:7a}\n\\end{equation}\nThen, from Eqs. \\eqref{Eins} and \\eqref{eq:4} and after straightforward calculations we have the following equations of motion:\n\\begin{subequations}\n \\label{eq:7}\n\\begin{eqnarray}\n p^\\prime &=& -\\frac{G_N}{r^{2}} \\frac{4 \\pi \\, p_{eff} \\, r^3 + \\mathcal{M}_{eff}}{1 - 2G_N \\mathcal{M}_{eff}\/r} ( p + \\rho ) \\, , \\label{eq:7b} \\\\\n \\mathcal{V}^{\\prime} + 3 \\mathcal{N}^{\\prime} &=& - \\frac{2G_N}{r^{2}} \\frac{4 \\pi \\, p_{eff} \\, r^3 + \\mathcal{M}_{eff}}{1 - 2G_N \\mathcal{M}_{eff}\/r} \\left( 2 \\mathcal{V} + 3 \\mathcal{N} \\right)\\nonumber\\\\ \n && - \\frac{9}{r} \\mathcal{N} - 3 (\\rho+p) \\rho^{\\prime} \\, , \\label{eq:7c}\n\\end{eqnarray}\n\\end{subequations}\nwhere a prime indicates derivative with respect to $r$, $A(r) = [1 - 2G_N \\mathcal{M}(r)_{eff}\/r]^{-1}$, and the effective energy density and pressure, respectively, are given as:\n\\begin{subequations}\n\\label{eq:3}\n\\begin{eqnarray}\n\\rho_{eff} &=& \\rho \\left( 1 + \\frac{\\rho}{2\\lambda} \\right) + \\frac{\\mathcal{V}}{\\lambda} \\, , \\label{eq:3a} \\\\\np_{eff} &=& p \\left(1 + \\frac{\\rho}{\\lambda} \\right) + \\frac{\\rho^{2}}{2\\lambda} + \\frac{\\mathcal{V}}{3\\lambda} + \\frac{\\mathcal{N}}{\\lambda} \\, . \\label{eq:3b}\n\\end{eqnarray}\n\\end{subequations}\nEven though we will not consider exterior galaxy solutions, we must anyway take into account the information provided by the Israel-Darmois (ID) matching condition, which for the case under study can be written as\\cite{gm}:\n\\begin{equation}\n \\label{eq:28}\n (3\/2) \\rho^2(R) + \\mathcal{V}^-(R) + 3 \\mathcal{N}^-(R) = 0 \\, .\n\\end{equation}\nIn this case, the superscript ($-$) indicates the interior value of the quantity at the halo surface\\footnote{We denote the surface of the galaxy as a region where does not exist DM or baryons, \\emph{i.e.}, the intergalactic space.} of the galaxy, assuming that $\\rho(r>R)=0$ where $R$ denotes the maximum size of the galaxy. Also, the previous equation takes in consideration the fact that the exterior must be Schwarzschild which in general the following condition must be fulfilled $\\mathcal{V}(r \\geq R) = 0 =\\mathcal{N}(r\\geq R)$ (see\\cite{Garcia-Aspeitia:2014pna} for details).\n\nFor completeness, we just note that the exterior solutions of the metric functions are given by the well known expressions $B(r) = A^{-1}(r) = 1 - 2G_N M_{eff}\/r$.\n\nFinally, we impose $\\mathcal{N}=0$ (see\\cite{Garcia-Aspeitia:2014pna}). Implying that Eq. \\eqref{eq:28} is restricted as:\n\\begin{equation}\n \\label{eq:29}\n -(3\/2) \\rho^2(R) = \\mathcal{V}^-(R) \\, ,\n\\end{equation}\nwith the aim of maintain a galaxy Schwarzschild exterior.\n\n\\section{Low energy limit and rotation curves} \\label{TOV MOD}\n\nTo begin with, we observe, from Eq.\\ \\eqref{eq:7b} in the low energy (Newtonian) limit, that we have: $r^{2}p^{\\prime}=-G_{N}\\mathcal{M}_{eff}\\rho$. Differentiating we found\n\\begin{equation}\n\\frac{d}{dr}\\left(\\frac{r^{2}}{\\rho}\\frac{dp}{dr}\\right)=-4\\pi r^{2}G_{N}\\rho_{\\rm eff}. \\label{eqdiff9}\n\\end{equation}\nFrom here it is possible to note that $d\\Phi\/dr=-\\rho^{-1}(dp\/dr)$ resulting in\n\\begin{equation}\n\\nabla^{2}\\Phi_{\\rm eff}=\\frac{1}{r^{2}}\\frac{d}{dr}\\left(r^{2}\\frac{d\\Phi_{\\rm eff}}{dr}\\right)=4\\pi G_{N}\\rho_{\\rm eff}, \\label{Poisson}\n\\end{equation}\nbeing necessary to define the energy density of DM together with the nonlocal energy density. Notice that the nonlocal energy density can be obtained easily from Eq.\\ \\eqref{eq:7c} in the galaxy interior and also the fluid behaves like dust, implying the condition $p=0$, always fulfilling the low energy condition $4\\pi r^3p_{eff}\\ll\\mathcal{M}_{eff}$ and $2G_{N}\\mathcal{M}_{eff}\/r\\ll1$, between effective quantities and in consequence $4G_{N}\\mathcal{M}_{eff}\\mathcal{V}\/r^2\\sim0$, is negligible.\n\nIn addition, the rotation curve is obtained from the contribution of the effective potential, this expression can be written as:\n\\begin{eqnarray}\nV^2(r) &=& r\\left\\vert\\frac{d\\Phi_{\\rm eff}}{dr}\\right\\vert=\\frac{G_N \\mathcal{M}_{eff}(r)}{r} \n\\nonumber \\\\\n&=& \n\\frac{G_N }{r} \n\\left[\n\\mathcal{M}_{DM}(r) + \\mathcal{M}_{Brane}(r)\n\\right]\n, \\label{rotvel}\n\\end{eqnarray}\nwhere $\\mathcal{M}_{DM}(r)$ is the contribution to the mass from DM, $\\mathcal{M}_{Brane}(r)$ gives the modification to the DM mass that comes from the brane; and $\\mathcal{M}_{eff}(r)$ must be greater than zero. From here, it is possible to study the rotation velocities of the DM, assuming a variety of density profiles.\n\nBefore we start let us define the following dimensionless variables: $\\bar{r}\\equiv r\/r_{\\rm s}$, $v_{0}^{2}\\equiv4\\pi G_{N}r_{\\rm s}^{2}\\rho_{\\rm s}$ and $\\bar{\\rho}\\equiv\\rho_{\\rm s}\/2\\lambda$ where $\\rho_{\\rm s}$, is the central density of the halo and $r_{s}$ is associated with the central radius of the halo. \n\n\\subsection{Pseudo isothermal profile for dark matter}\n\nHere we consider that DM density profile is given by PISO\\cite{piso} written as:\n\\begin{equation}\n\\rho_{\\rm PISO}(r)=\\frac{\\rho_{\\rm s}}{1+\\bar{r}^{2}}. \\label{PIP}\n\\end{equation}\nFrom Eq. \\eqref{rotvel}, together with Eq. \\eqref{PIP}, it is possible to obtain:\n\\begin{eqnarray}\nV_{\\rm PISO}^{2}(\\bar{r}) &=& v_{0}^{2}\n\\left\\lbrace\n\\left(\n1-\\frac{1}{\\bar{r}}\\arctan\\bar{r}\n\\right) \n\\right.\n\\nonumber \\\\\n&& \n\\left. + \\bar{\\rho}\n\\left(\n\\frac{1}{1+\\bar{r}^2}- \\frac{1}{\\bar{r}}\\arctan\\bar{r}\n\\right)\n\\right\\rbrace.\n\\label{RCPISO}\n\\end{eqnarray}\nIn the limit $\\bar{\\rho}\\to0$, we recover the classical rotation velocity associated with PISO for DM.\nThe effective density must be positive defined, then $\\lambda > \\rho_s$ must be fulfilled. The first right-hand term in parenthesis in Eq.\\ \\eqref{RCPISO} is PISO dark matter contribution and the second \nis brane's contribution.\n\n\\subsection{Navarro-Frenk-White profile for dark matter}\nAnother interesting case (motivated by cosmological $N$-body simulations) is the NFW density profile, which is given by\\cite{NFW}:\n\\begin{equation}\n\\rho_{\\rm NFW}(r)=\\frac{\\rho_{\\rm s}}{\\bar{r}(1+\\bar{r})^{2}}. \\label{NFW}\n\\end{equation}\nThis is a density profile that diverges as $r \\rightarrow 0$ \nand \nit is not possible to say\nthat $\\rho_s$ is related with the central density of the DM distribution.\nAlso density goes as $1\/\\bar{r}^3$ when $\\bar{r} \\gg 1$.\nHowever, in this particular case, we will still be calling them the \\emph{central} density and radius of the NFW matter distribution.\nFrom Eq.\\ \\eqref{rotvel}, together with Eq.\\ \\eqref{NFW} we obtain the following rotation curve:\n\\begin{eqnarray}\nV_{\\rm NFW}^{2}(\\bar{r}) &=& v_{0}^{2}\\left\\lbrace\\left(\\frac{(1+\\bar{r})\\ln(1+\\bar{r})-\\bar{r}}{\\bar{r}(1+\\bar{r})}\\right)\\right.\\nonumber\\\\&&\n\\left.+\\frac{2\\bar{\\rho}}{3\\bar{r}}\\left(\\frac{1}{(1+\\bar{r})^{3}}-1\\right) \\right\\rbrace.\n\\label{RCNFW}\n\\end{eqnarray}\nThe first right-hand term in parenthesis in Eq.\\ \\eqref{RCNFW} is NFW dark matter contribution and the second one is the brane's contribution. Notice that we recover also the classical limit when $\\bar{\\rho}\\to0$.\n\nIn addition, it is important to remark that the effective density must be positive defined, then $\\lambda > \\rho_s r_s \/r$. Also, if $\\mathcal{M}(r)$ must be greater than zero, then $r > r_{min}$ where $r_{min}$ is given by solving the following equation:\n\\begin{equation}\n\\frac{2}{3}\\bar{\\rho}=\\frac{(\\alpha+1)^2[(\\alpha+1)\\ln(\\alpha+1)-\\alpha]}{(\\alpha+1)^3-1},\\label{comp}\n\\end{equation}\nwhere we define $\\alpha\\equiv r_{min}\/r_s$ as a dimensionless quantity.\n\n\\subsection{Burkert density profile for dark matter}\n\nAnother density profile was proposed by Burkert\\cite{Burkert}, which it has the form:\n\\begin{equation}\n\\rho_{\\rm Burk}=\\frac{\\rho_{\\rm s}}{(1+\\bar{r})(1+\\bar{r}^{2})}. \\label{Burk}\n\\end{equation}\nAgain, from Eq.\\ \\eqref{rotvel}, together with Eq.\\ \\eqref{Burk} we obtain the following rotation curve:\n\\begin{eqnarray}\nV_{\\rm Burk}^{2}(\\bar{r}) &=&\\frac{v_{0}^{2}}{4\\bar{r}} \\left\\lbrace \\left( \\ln[(1+\\bar{r}^{2})(1+\\bar{r})^{2}]-2\\arctan(\\bar{r}) \\right)\\right.\n\\label{RCBurkert}\n\\\\&&\n\\left.+ \\frac{1}{2}\\bar{\\rho}\\left( \\frac{1}{1+\\bar{r}}+\\frac{1}{1+\\bar{r}^{2}}+\\arctan(\\bar{r})-2 \\right)\\right\\rbrace.\n\\nonumber\n\\end{eqnarray}\nIn the limit $\\bar{\\rho}\\to0$, we recover the classical rotation velocity associated with Burkert density profile\\cite{Burkert}.\nThe effective density must be positive defined, then $\\lambda > \\rho_s$. \nAgain the first right-hand term in parenthesis in Eq.\\ \\eqref{RCBurkert} is\nBurkert DM contribution and the second one comes from the\n brane's contribution.\n\n\\section{Constrictions with galaxies without photometry} \\label{Results}\n\nTo start with the analysis, we $\\chi^{2}$ best fit the observational rotation curves of the sample with:\n\\begin{equation}\n\\chi^{2}=\\sum_{i=1}^{N}\\left(\\frac{V_{theo}-V_{exp \\; i}}{\\delta V_{exp\\; i}}\\right)^{2},\n\\label{chi2Eq}\n\\end{equation}\nwhere $i$ runs from one up to the number of points in the data, $N$; $V_{theo}$, is computed according to the velocity profile under consideration \nand $\\delta V_{exp\\; i}$, is the error in the measurement of the rotational velocity. Notice that \nthe free parameters are only for DM-Branes: $r_{s}$, $\\rho_{s}$ and $\\lambda$. \nIn the tables below we show $\\chi_{red}^{2} \\equiv \\chi^{2}\/(N - n_p -1)$ where $n_p$ is the number of parameters to fit, being in our case, $n_p=3$.\n\nThe analyzed sample of galaxies are twelve high resolution rotation curves of LSB galaxies with no photometry (visible components, such as gas and stars, are negligible) as given in Ref.\\cite{deBlok\/etal:2001}. This sample was used to study DM equation of state (EoS) in Ref.\\cite{Barranco\/etal:2015}. We remark that in this part we use units such that $4 \\pi G_{N}=1$, velocities are in km\/s, and distances are given in kpc.\n\n\\subsection{Results: PISO profile + Branes}\n\nWe have estimated the parameters of the PISO+branes model \nand were compared with PISO model without brane contribution, minimizing the appropriate $\\chi^2$ for the sample of observed rotational curves, using Eq.\\ (\\ref{chi2Eq}) with Eq.\\ (\\ref{RCPISO}) and taking into account that $\\lambda > \\rho_s$ must be fulfilled. \n\nIn Fig.\\ \\ref{PISO1} we show, for each one of the galaxies in the sample,\nthe plots of the PISO theoretical rotation curve (solid line), that best fit of the corresponding observational data (orange symbols); also shown are the errors of the estimation (brown symbols). \nFor each galaxy we have plotted the contribution to the rotation velocity due only to the brane (red long-dashed curve) and only to the dark matter PISO density profile (blue short-dashed curve), see Eq.\\ (\\ref{RCPISO}).\nBrane effects are very clear in galaxies: \nESO 2060140,\nESO 3020120,\nU 11616,\nU 11648,\nU 11748,\nU 11819.\nIn Table \\ref{TablePiso} it is shown the central density, central radius and the brane tension which is the free parameter of the brane theory (only in PISO+branes). As a comparison, it is also shown the central density and radius without brane contribution.\nThe worst fitted galaxies were (high $\\chi_{red}^2$ value): \nU 11648,\nU 11748.\nThe fitted brane tension values presents great dispersion, from the lower value: \n$0.167\\; M_{\\odot}\/\\rm pc^3$, ESO 3020120 to the higher value:\n$108.096\\; M_{\\odot}\/\\rm pc^3$, ESO 4880049.\nIt is useful to have $\\lambda$ in eV, where the conversion from solar masses to eV is: $1 M_{\\odot}\/\\rm pc^3 \\sim 2.915\\times10^{-4}eV^4$. \nThe brane tension parameter has an average value of $\\langle\\lambda\\rangle_{\\rm PISO} = 33.178 \\; M_{\\odot}\/\\rm pc^3$ with a standard deviation $\\sigma_{\\rm PISO} = 40.935 \\; M_{\\odot}\/\\rm pc^3$. Notice that we can't see a clear tendency to a $\\lambda$ value or range of values.\n\n\\begin{figure}\n\\includegraphics[scale=0.33]{vceso30500900PISO+Branes}\n\\includegraphics[scale=0.33]{vceso0140040PISO+Branes} \\\\\n\\includegraphics[scale=0.33]{vceso2060140PISO+Branes} \n\\includegraphics[scale=0.33]{vcESO3020120PISO+Branes} \\\\ \n\\includegraphics[scale=0.33]{vceso4250180PISO+Branes} \n\\includegraphics[scale=0.33]{vceso4880049PISO+Branes} \\\\\n\\includegraphics[scale=0.33]{vcf570_v1PISO+Branes} \n\\includegraphics[scale=0.33]{vcu11454PISO+Branes} \\\\\n\\includegraphics[scale=0.33]{vcu11616PISO+Branes} \n\\includegraphics[scale=0.33]{vcu11648PISO+Branes} \\\\\n\\includegraphics[scale=0.33]{vcu11748PISO+Branes} \n\\includegraphics[scale=0.33]{vcu11819PISO+Branes}\n\\caption{Group of analyzed galaxies using modified rotation velocity for PISO profile: ESO 3050090,\nESO 0140040, ESO 2060140, ESO 3020120, ESO 4250180, ESO 4880049, 570\\_V1, U11454, U11616, U11648, U11748, U11819. We show in the plots: Total rotation curve (solid black line), only PISO curve (short dashed blue curve) and the rotation curve associated with the mass lost by the effect of the brane (red dashed curve).} \n\\label{PISO1}\n\\end{figure}\n\n\n\n\n\\subsection{Results: NFW profile + Branes}\nFor the NFW density profile case we have the following results:\nWe have estimated parameters with and without brane contribution by\nminimizing the corresponding $\\chi^2$, Eq.\\ (\\ref{chi2Eq}) with Eq.\\ (\\ref{RCNFW}), for the sample of observed rotation curves \nand taking into account that\n$\\lambda > \\rho_s r_s \/r$ in order to have an effective density positive defined, always fulfilling Eq.\\ (\\ref{comp}).\n\nIn Fig.\\ \\ref{NFW1}, it is shown, for each galaxy in the sample of the LSB galaxies, the theoretical fitted curve to a preferred brane tension value (solid line), \nthe NFW curve and the rotation curve associated with the mass lost by the effects of branes, see Eq.\\ (\\ref{RCNFW}). \nIn Table \\ref{TableNFW} it is shown, for the sample,\nthe central density, central radius and $\\chi_{red}^2$ values without branes; and\nthe central density, central radius, brane tension\nand $\\chi_{red}^2$ values with branes contribution.\nGalaxy U 11748 is the worst fitted case with $\\chi_{red}^2 = 2.163$.\nFor galaxies: \nESO 4250180,\nESO 4880049,\nand U 11648,\nthere are not clear brane effects. \nGalaxy U 11648 is an \\emph{outlier} with a brane tension value of $4323.28\\; M_{\\odot}\/\\rm pc^3$ that is out of the range of preferred values of the other galaxies in the sample.\nNotice that we have found a preferred range of tension values, from $0.487$ to $9.232$ $M_{\\odot}\/\\rm pc^3$. Without the outlier, the brane tension parameter has an average value of \n$\\langle\\lambda\\rangle_{\\rm NFW}\\simeq 2.51 \\; M_{\\odot}\/\\rm pc^3$ \nwith a standard deviation $\\sigma_{\\rm NFW}\\simeq 3.015 \\; M_{\\odot}\/\\rm pc^3$.\n\n\\begin{figure}\n\\includegraphics[scale=0.33]{vceso30500900NFW+Branes} \n\\includegraphics[scale=0.33]{vceso0140040NFW+Branes} \\\\\n\\includegraphics[scale=0.33]{vceso2060140NFW+Branes} \n\\includegraphics[scale=0.33]{vcESO3020120NFW+Branes} \\\\ \n\\includegraphics[scale=0.33]{vceso4250180NFW+Branes} \n\\includegraphics[scale=0.33]{vceso4880049NFW+Branes} \\\\\n\\includegraphics[scale=0.33]{vcf570_v1NFW+Branes} \n\\includegraphics[scale=0.33]{vcu11454NFW+Branes} \\\\\n\\includegraphics[scale=0.33]{vcu11616NFW+Branes} \n\\includegraphics[scale=0.33]{vcu11648NFW+Branes} \\\\\n\\includegraphics[scale=0.33]{vcu11748NFW+Branes} \n\\includegraphics[scale=0.33]{vcu11819NFW+Branes} \n\\caption{Group of analyzed galaxies using modified rotation velocity for NFW profile: ESO 3050090,\nESO 0140040, ESO 2060140, ESO 3020120, ESO 4250180, ESO 4880049, 570\\_V1, U11454, U11616, U11648, U11748, U11819. We show in the plots: Total rotation curve (solid black line), only NFW curve (short dashed blue curve) and the rotation curve associate with the mass lost by the effect of the brane (red dashed curve).} \n\\label{NFW1}\n\\end{figure}\n\n\n\n\\subsection{Results: Burkert+Branes profile}\n\nIn the case of Burkert DM density profile, we have also estimated the parameters of the Burkert+branes model \nand were compared with Burkert model without branes, minimizing the appropriate $\\chi^2_{red}$, Eq.\\ (\\ref{chi2Eq}) with Eq.\\ (\\ref{RCBurkert}), for the sample of observed rotation curves. We have considered that $\\lambda > \\rho_s$ must be fulfilled.\n\nThe results are shown in Fig.\\ \\ref{Burkert1}, where it is plotted the fit to a preferred brane tension value, remarking the total rotation curve (solid line), the Burkert DM density contribution curve (blue short-dashed line) and the rotation curve associated with the mass lost by the effects of branes (red dashed line), see Eq.\\ (\\ref{RCBurkert}). \nIn Table \\ref{TableBurkert} it is shown the fitted values for the central density, central radius and the corresponding value of the $\\chi_{red}^2$ without brane contribution; and the fitted values for\nthe central density, central radius, brane tension, and theirs $\\chi_{red}^2$ values with brane contribution.\nThe worst fitted (high values of $\\chi_{red}^2$) galaxies are: U 11648 and U 11748.\nGalaxies ESO 3020120, U 11748, and\nU 11819 show a clear brane effects and also are outliers. The main tendency is that $\\lambda$ has values of the order of $10^3 \\;M_{\\odot}\/\\rm pc^3$ or above, approximately.\nThe brane tension parameter, without the outliers, for the DM Burkert profile case has an average value of \n$\\langle\\lambda\\rangle_{\\rm Burk}\\simeq 3192.02 \\;M_{\\odot}\/\\rm pc^3$, \nand a standard deviation of \n$\\sigma_{\\rm Burk}\\simeq 2174.97 \\; M_{\\odot}\/\\rm pc^3$.\n\n\\begin{figure}\n\\includegraphics[scale=0.33]{vceso30500900Burkert+Branes} \n\\includegraphics[scale=0.33]{vceso0140040Burkert+Branes} \\\\\n\\includegraphics[scale=0.33]{vceso2060140Burkert+Branes} \n\\includegraphics[scale=0.33]{vcESO3020120Burkert+Branes} \\\\ \n\\includegraphics[scale=0.33]{vceso4250180Burkert+Branes} \n\\includegraphics[scale=0.33]{vceso4880049Burkert+Branes} \\\\\n\\includegraphics[scale=0.33]{vcf570_v1Burkert+Branes} \n\\includegraphics[scale=0.33]{vcu11454Burkert+Branes} \\\\\n\\includegraphics[scale=0.33]{vcu11616Burkert+Branes} \n\\includegraphics[scale=0.33]{vcu11648Burkert+Branes} \\\\\n\\includegraphics[scale=0.33]{vcu11748Burkert+Branes} \n\\includegraphics[scale=0.33]{vcu11819Burkert+Branes}\n\\caption{Group of analyzed galaxies using modified rotation velocity for Burkert profile: ESO 3050090,\nESO 0140040, ESO 2060140, ESO 3020120, ESO 4250180, ESO 4880049, 570\\_V1, U11454, U11616, U11648, U11748, U11819. We show in the plots: Total rotation curve (solid black line), only Burkert curve (short dashed blue curve) and the rotation curve associate with the mass lost by the effect of the brane (red dashed curve).} \n\\label{Burkert1}\n\\end{figure}\n\n\n\n\\subsection{Results: a synthetic rotation curve}\n\nFinally, we show the fitting results of the DM models plus brane's contribution to a synthetic rotation curve. This synthetic rotation curve was made of 40 rotation curves of galaxies with magnitudes around $M_I = -18.5$\\cite{Salucci1}.\nThese 40 rotation curves came out of 1100 galaxies that gave the universal rotation curve for spirals. For this sample of low luminosity galaxies, of $M_I = -18.5$, it was shown that the baryonic disk has a very small contribution (for details see reference\\cite{Salucci1}).\n\nIn this subsection we are now using units such $G=R_{opt}=V(R_{opt})=1$, where $R_{opt}$ and $V(R_{opt})$ are the optical radius and the velocity at the optical radius, respectively. $R_{opt}$ is the radius encompassing 83 per cent of the total integrated light. For an exponential disk with a surface brightness given by: $I(r) \\propto \\exp(-r\/R_D)$, we have that $R_{opt}=3.2 R_D$\\cite{Salucci1}.\n\nIn figure \\ref{SM185} we show the synthetic rotation curve and the fitting results using PISO, NFW and Burkert profiles with and without brane's contribution.\n\\begin{figure}\n\\includegraphics[scale=0.33]{vcM185PISO} \n\\includegraphics[scale=0.33]{vcM185PISO+Branes} \n\\includegraphics[scale=0.33]{vcM185NFW} \n\\includegraphics[scale=0.33]{vcM185NFW+Branes} \n\\includegraphics[scale=0.33]{vcM185Burkert} \n\\includegraphics[scale=0.33]{vcM185Burkert+Branes} \n\\caption{Synthetic rotation curve of galaxies with magnitud $M_I=-18.5$.\nLeft panels: rotation curves fitted without branes. Right panels: rotation curves fitted with branes.\nFirst row is for PISO model; second row is for NFW model and third row is for Burkert model. \nWe show in the plots: Total rotation curve (solid black line), only DM model curve (short dashed blue curve) and the rotation curve associate with the mass lost by the effect of the brane (red dashed curve).} \n\\label{SM185}\n\\end{figure}\nAs we can see in Table \\ref{TableSynthetic} the same trend is observed in the brane's tension values as compared with the results for the LSB catalog analyzed above using PISO, NFW and Burkert as a DM profiles: lower value is obtained for NFW model and higher values is obtained for Burkert density profile.\n\nGiven that this synthetic rotation curve is built from 40 rotation curves of real spirals, the values of the brane's tension in table \\ref{TableSynthetic} is representative of all these rotation curves. \nThen, for PISO model $\\lambda=60.692$ $M_{\\odot}\/\\rm pc^3$, a value that is greater than the average value of the tension shown in Table \\ref{TablePiso} but inside the interval marked by the standard deviation. \nFor NFW model $\\lambda=226.054$ $M_{\\odot}\/\\rm pc^3$, this value is lower than the average value reported in Table \\ref{TableNFW} and inside the range marked by the standard deviation. \nFor Burkert model $\\lambda=1.58\\times 10^5$ $M_{\\odot}\/\\rm pc^3$ this value is well above than the average value shown in Table \\ref{TableBurkert}; a value outside the range marked by the standard deviation. \n\n\n\\section{Discussion and conclusions} \\label{Disc}\n\nWe have presented in this paper, the effects coming from the presence of branes in galaxy rotation curves for \nthree density profiles used to study the behavior of DM at galactic scales. \nWith this in mind, we were given to the task of study a sample of \nhigh resolution measurements of rotation curves of galaxies without photometry\\cite{deBlok\/etal:2001} \nand a synthetic rotation curve built from 40 rotation curves of galaxies of magnitude around $M_I=-18.5$\nfitting\n the values of $\\rho_{s}$, $r_{s}$ and $\\lambda$ through minimizing the $\\chi^{2}_{red}$ value and we have compared with the standard results of $\\rho_{s}$, $r_{s}$ for each DM density profile without branes. \n The results for every observable in the three different profiles were summarized and compared in Tables \\ref{TablePiso}-\\ref{TableSynthetic}.\n\nFrom here, it is possible to observe how the results show a weaker limit for the value of brane tension \n($\\sim10^{-3}\\; \\rm eV^4-46$ eV$^4$) for the three models, in comparison with other astrophysical and cosmological studies\\cite{Kapner:2006si,Alexeyev:2015gja,mk,gm,Sakstein:2014nfa,Kudoh:2003xz,Cavaglia:2002si,Holanda:2013doa,Barrow:2001pi,Brax:2003fv}; for example, Linares \\emph{et al.}\\cite{Linares:2015fsa} show that weaker values than $\\lambda \\simeq 10^{4}$ MeV$^{4}$, present anomalous behavior in the compactness of a dwarf star composed by a polytropic EoS, concluding that a wide region of their bound \nwill show a non compactness stellar configuration, if it is applied to the study shown in\\cite{Linares:2015fsa}.\n \nIt is important to notice that chosen a value of brane tension that not fulfill our bounds imposed through the paper, generate an anomalous behavior in the center of the galaxy which is characteristic of the model. Remarkable, for higher values of this bound, the modified rotation curves are in good agreement with \nthe observed rotation curves of the sample that we use,\npresenting only the distinctive features of each density profile: For example,\nNFW dark matter density profile prefers lower values of the brane tension (on the average $\\lambda \\sim 0.73\\times 10^{-3}$ eV$^4$), implying clear effects of the brane;\nPISO dark matter\n case has an average value of $\\lambda \\sim 0.96\\times 10^{-2}$ eV$^4$ and show relatively the maximum dispersion on the fitted values of the brane tension;\nwhereas Burkert DM density profile shows negligible brane effects, on the average $\\lambda \\sim 0.93$ eV$^4$ -- $46$ eV$^4$.\n\n In addition, it is important to discuss briefly the changes caused by the presence of branes in the problem of cusp\/core. Notice that in this case the part that play a role is the effective density which is written in terms of brane corrections as: $\\rho_{eff}=\\rho(1-\\rho\/\\lambda)$; it is notorious how the small perturbations alleviate the cusp problem which afflicts NFW, albeit the excessive presence of these terms could generates a negative effective density profile; also PISO and Burkert show modifications when $r\\to0$ but does not pledge its core behavior while the brane tension only takes small values. In this way, the possibility of having a core behavior, help us to constraint the value of brane tension and still keep in the game the NFW profile.\n\nSummarizing, it is really challenging to establish bounds in a dynamical systems like rotation curves in galaxies due to the\nlow densities found in the galactic medium, giving only a weaker limits in comparison with other studies in a most energetic systems. \nOur most important conclusion, is that despite the efforts, we think that it is not straightforward to do that the results fit with other astrophysical and cosmological studies, being impractical and not feasible to\nfind evidence of extra dimensions in galactic dynamics through the determination of the brane tension value, \neven more, we think that exist too much dispersion in the fitted values of the brane tension using this method for some DM density profile models. \nAlso it is important to note that the value of the brane tension is strongly dependent of the characteristic of the galaxy studied, suggesting an average for the preferred value of the brane tension in each case. \nIn addition, we notice that the effects of extra dimensions are stronger in the galactic core, suggesting that the NFW model is not appropriate in the search of constraints in brane theory due to the divergence in the center of the galaxy (see Eq.\\ \\eqref{comp}); PISO and Burkert could be good candidates to explore the galactic core in this framework; however it is necessary a more extensive study before we obtain a definitive conclusion. \n\nAs a final note, we know that it is necessary to recollect more observational data to constraint \nthe models or even give the final conclusion about extra dimensions (for or against), supporting the brane constraints shown through this paper with a more profound study of galactic dynamic or other tests like cosmological evidences presented in CMB anisotropies. \nHowever, this work is in progress and will be reported elsewhere.\n\n\n\\begin{acknowledgements}\nMAG-A acknowledge support from SNI-M\\'exico and CONACyT research fellow. Instituto Avanzado de Cosmolog\\'ia (IAC) collaborations.\n\\end{acknowledgements}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe construction of the free exact category over a category with finite limits was introduced in ~\\cite{FECLEO}. It was later improved to the construction of the free exact category over a category with finite weak limits (\\emph{weakly lex}) in \\cite{REC}. This followed from the fact that the uniqueness of the finite limits of the original category is never used in the construction; only the existence. In ~\\cite{REC}, the authors also considered the free regular category over a weakly lex one.\n\nAn important property of the free exact (or regular) construction is that such categories always have enough (regular) projectives. In fact, an exact category $\\mathbb{A}$ may be seen as the exact completion of a weakly lex category if and only if it has enough projectives. If so, then $\\mathbb{A}$ is the exact completion of any of its \\emph{projective covers}. Such a phenomenon is captured by varieties of universal algebras: they are the exact completions of their full subcategory of free algebras.\n\nHaving this link in mind, our main interest in studying this subject is to characterize projective covers of certain algebraic categories through simpler properties involving projectives and to relate those properties to the known varietal characterizations in terms of the existence of operations of their varietal theories (when it is the case). Such kind of studies have been done for the projective covers of categories which are: Mal'tsev~\\cite{ECRAC}, protomodular and semi-abelian~\\cite{G}, (strongly) unital and subtractive~\\cite{compl}.\n\nThe aim of this work is to obtain characterizations of the weakly lex categories whose regular completion is a Goursat (=$3$-permutable) category (Propositions~\\ref{cover} and ~\\ref{weaksquare}). We then relate them to the existence of the quaternary operations which characterize the varieties of universal algebras which are $3$-permutable (Remark~\\ref{vars}).\n\n\n\\section{Preliminaries}\\label{pre}\n\nIn this section, we briefly recall some elementary categorical notions needed in the following.\n\nA category with finite limits is \\textbf{regular} if regular epimorphisms are stable under pullbacks, and kernel pairs have coequalizers. Equivalently, any arrow $f: A\\longrightarrow B$ has a unique factorisation $f=i r $ (up to isomorphism), where $r$ is a regular epimorphism and $i$ is a monomorphism and this factorisation is pullback stable.\n\nA \\textbf{relation} $R$ from $X$ to $Y$ is a subobject $\\langle r_1,r_2 \\rangle : R \\rightarrowtail X \\times Y $. The opposite relation of $R$, denoted $R^o$, is the relation from $Y$ to $X$ given by the subobject $\\langle r_2,r_1 \\rangle : R \\rightarrowtail Y \\times X $. A relation $R$ from $X$ to $X$ is called a relation on $X$. We shall identify a morphism $f: X \\longrightarrow Y$ with the relation $\\langle 1_X,f \\rangle: X \\rightarrowtail X \\times Y$ and write $f^o$ for its opposite relation. Given two relations $R \\rightarrowtail X \\times Y $ and $S \\rightarrowtail Y \\times Z $ in a regular category, we write $SR \\rightarrowtail X \\times Z $ for their relational composite. With the above notations, any relation $\\langle r_1, r_2 \\rangle: R \\rightarrowtail X \\times Y$ can be seen as the relational composite $r_2r_1^o$.\nThe following properties are well known and easy to prove (see \\cite{ckp} for instance); we collect them in the following lemma:\n\\begin{lemma}\n\\label{lem}\nLet $f: X \\longrightarrow Y$ be an arrow in a regular category $\\mathbb{C}$, and let $ f=i r $ be its (regular epimorphism, monomorphism) factorisation. Then:\n\\begin{enumerate}\n \\item $f^of$ is the kernel pair of $f$, thus $1_X \\leqslant f^of $; moreover, $1_X = f^of$ if and only if $f$ is a monomorphism;\n \\item $ff^o$ is $(i,i)$, thus $ff^o \\leqslant 1_Y$; moreover, $ff^o = 1_Y$ if and only if $f$ is a regular epimorphism;\n \\item $ff^of = f $ and $f^off^o = f^o$.\n\\end{enumerate}\n\\end{lemma}\n\nA relation $R$ on $X$ is \\textbf{reflexive} if $1_X \\leqslant R$, \\textbf{symmetric} if $R^o \\leqslant R$, and \\textbf{transitive} if $RR \\leqslant R$. As usual, a relation $R$ on $X$ is an \\textbf{equivalence relation} when it is reflexive, symmetric and transitive. In particular, a kernel pair $\\langle f_1,f_2 \\rangle: \\Eq(f)\\rightarrowtail X \\times X$ of a morphism $f: X \\longrightarrow Y$ is an equivalence relation.\n\nBy dropping the assumption of uniqueness of the factorization in the definition of a limit, one obtains the definition of a weak limit. We call \\textbf{weakly lex} a category with weak finite limits.\n\nAn object $P$ in a category is (regular) \\textbf{projective} if, for any arrow $f: P \\longrightarrow X$ and for any regular epimorphism $g: Y \\twoheadrightarrow X$ there exists an arrow $h: P \\longrightarrow Y$ such that $g h = f $. We say that a full subcategory $\\mathbb{C}$ of $\\mathbb{A}$ is a \\textbf{projective cover} of $\\mathbb{A}$ if two conditions are satisfied:\n\\begin{itemize}\n\\item any object of $\\mathbb{C}$ is regular projective in $\\mathbb{A}$;\n\\item for any object $X$ in $\\mathbb{A}$, there exists a ($\\mathbb{C}$-)cover of $X$, that is an object $C$ in $\\mathbb{C}$ and a regular epimorphism $C\\twoheadrightarrow X$.\n\\end{itemize}\n\nWhen $\\mathbb{A}$ admits a projective cover, one says that $\\mathbb{A}$ has \\textit{enough projectives}.\n\\begin{remark}\\label{weaklims}\nIf $\\mathbb{C}$ is a projective cover of a weakly lex category $\\mathbb{A}$, then $\\mathbb{C}$ is also weakly lex~\\cite{REC}. For example, let $X$ and $Y$ be objects in $\\mathbb{C}$ and $\\xymatrix@C=15pt{ X & W \\ar[l] \\ar[r] & Y}$ a weak product of $X$ and $Y$ in $\\mathbb{A}$. Then, for any cover $\\bar{W}\\twoheadrightarrow W$ of $W$, $\\xymatrix@C=15pt{ X & \\bar{W} \\ar[l] \\ar[r] & Y}$ is a weak product of $X$ and $Y$ in $\\mathbb{C}$. Furthermore, if $\\mathbb{A}$ is a regular category, then the induced morphism $W\\twoheadrightarrow X\\times Y$ is a regular epimorphism.\nSimilar remarks apply to all weak finite limits.\n\\end{remark}\n\n\n\n\\section{Goursat categories}\nIn this section we review the notion of Goursat category and the characterizations of Goursat categories through regular images of equivalence relations and through Goursat pushouts.\n\n\\begin{definition} \\emph{\\cite{CLP,ckp}}\nA regular category $\\mathbb{C}$ is called a \\textbf{Goursat category} when the equivalence relations in $\\mathbb{C}$ are $3$-permutable, i.e. $RSR = SRS$ for any pair of equivalence relations $R$ and $S$ on the same object.\n\\end{definition}\n\nWhen $\\mathbb{C}$ is a regular category, $(R,r_1,r_2)$ is an equivalence relation on $X$ and $f: X \\twoheadrightarrow Y$ is a regular epimorphism, we define the \\textbf{regular image of $R$ along $f$} to be the relation $f(R)$ on $Y$ induced by the (regular epimorphism, monomorphism) factorization $\\langle s_1, s_2 \\rangle \\psi$ of the composite $(f\\times f) \\langle r_1,r_2\\rangle$:\n\\[\n \\xymatrix{\nR \\ar@{.>>}[r]^{\\psi} \\ar@{ >->}[d]_{\\langle r_1,r_2\\rangle} & f(R) \\ar@{ >.>}[d]^{\\langle s_1,s_2\\rangle}\\\\\nX \\times X \\ar@{>>}[r]_{f \\times f} & Y \\times Y.\n }\n\\]\nNote that the regular image $f(R)$ can be obtained as the relational composite $f(R)=fRf^o=fr_2r_1^of^o$. When $R$ is an equivalence relation, $f(R)$ is also reflexive and symmetric. In a general regular category $f(R)$ is not necessarily an equivalence relation.\nThis is the case in a \\emph{Goursat category} according to the following theorem.\n\n\\begin{theorem}\\label{CKP} \\emph{\\cite{ckp}} A regular category $\\mathbb{C}$ is a Goursat category if and only if for any regular epimorphism $f: X \\twoheadrightarrow Y$ and any equivalence relation $R$ on $X$, the regular image $f(R)= fRf^o$ of $R$ along $f$ is an equivalence relation.\n\\end{theorem}\n\nGoursat categories are well known in Universal Algebra. In fact, by a classical theorem in \\cite{hm}, a variety of universal algebras is a Goursat category precisely when its theory has two quaternary operations $p$ and $q$ such that the identities $p(x,y,y,z)= x$, $q(x,y,y,z)= z$ and $p(x,x,y,y)= q(x,x,y,y)$ hold. Accordingly, the varieties of groups, Heyting algebras and implication algebras are Goursat categories. The category of topological group, Hausdorff groups, right complemented semi-group are also Goursat categories.\n\nThere are many known characterizations of Goursat categories (see \\cite{ckp,gr,grod,grt} for instance). In particular the following characterization, through Goursat pushouts, will be useful:\n\\begin{theorem}\\label{Goursatpushout}\\emph{\\cite{gr}}\nLet $\\mathbb{C}$ be a regular category. The following conditions are equivalent:\n\\begin{enumerate}\n \\item[(i)] $\\mathbb{C}$ is a Goursat category;\n \\item[(ii)] any commutative diagram of type \\emph{($\\mathrm{I}$)} in $\\mathbb{C}$, where $\\alpha$ and $\\beta$ are regular epimorphisms and $f$ and $g$ are split epimorphisms\n \\[\n \\xymatrix@C=2cm{\n X \\ar@{}[dr]|{(\\mathrm{I})} \\ar@{->>}[r]^{\\alpha} \\ar@<3pt>[d]^{f} & U \\ar@<3pt>[d]^{g} \\ar@<40pt>@{}[d]^(.3){g\\alpha=\\beta f} \\ar@<40pt>@{}[d]^(.7){\\alpha s=t\\beta} \\\\ Y \\ar@{->>}[r]_{\\beta} \\ar@<3pt>[u]^{s}& W, \\ar@<3pt>[u]^t\n }\n\\]\n(which is necessarily a pushout) is a \\textbf{Goursat pushout}: the morphism $ \\lambda : \\Eq(f) \\longrightarrow \\Eq(g)$, induced by the universal property of kernel pair $\\Eq(g)$ of $g$, is a regular epimorphism.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\n\\label{variant} Diagram ($\\mathrm{I}$) is a Goursat pushout precisely when the regular image of $\\Eq(f)$ along $\\alpha$ is (isomorphic to) $\\Eq(g)$. From Theorem~\\ref{Goursatpushout}, it then follows that a regular category $\\mathbb{C}$ is a Goursat category if and only if for any commutative diagram of type ($\\mathrm{I}$) one has $\\alpha(\\Eq(f))= \\Eq(g)$.\n\\end{remark}\n\nNote that Theorem~\\ref{CKP} characterizes Goursat categories through the property that regular images of equivalence relations are equivalence relations, while Theorem~\\ref{Goursatpushout} characterizes them through the property that regular images of certain kernel pairs are kernel pairs.\n\n\n\n\\section{Projective covers of Goursat categories}\nIn this section, we characterize the categories with weak finite limits whose regular completion are Goursat categories.\n\\begin{definition} Let $\\mathbb{C}$ be a weakly lex category:\n\\begin{enumerate}\n\\item a \\textbf{pseudo-relation} on an object X of $\\mathbb{C}$ is a pair of parallel arrows $\\xymatrix {R \\ar@<2pt>[r]^{r_1} \\ar@<-2pt>[r]_{r_2} & X;}$ a pseudo-relation is a relation if $r_1$ and $r_2$ are jointly monomorphic;\n\\item a pseudo-relation $ \\xymatrix {R \\ar@<2pt>[r]^{r_1} \\ar@<-2pt>[r]_{r_2} & X}$ on $X$ is said to be:\n\\begin{itemize}\n \\item \\textbf{reflexive} when there is an arrow $ r: X \\longrightarrow R $ such that $ r_1 r = 1_X = r_2 r $;\n \\item \\textbf{symmetric} when there is an arrow $ \\sigma : R \\longrightarrow R $ such that $ r_2 = r_1 \\sigma $ and $r_1 = r_2 \\sigma $;\n \\item \\textbf{transitive} if by considering a weak pullback\n $$\n \\xymatrix{\n W \\ar[r]^-{p_2} \\ar[d]_{p_1} & R \\ar[d]^{r_1} \\\\ R \\ar[r]_{r_2}& X,\n }\n$$\nthere is an arrow $ t : W \\longrightarrow R$ such that $ r_1 t = r_1 p_1 $ and $r_2 t = r_2 p_2$.\n \\item a \\textbf{pseudo-equivalence relation} if it is reflexive, symmetric and transitive.\n\\end{itemize}\n\\end{enumerate}\n\\end{definition}\n\nRemark that the transitivity of a pseudo-relation $\\xymatrix {R \\ar@<2pt>[r]^{r_1} \\ar@<-2pt>[r]_{r_2} & X}$ does not depend on the choice of the weak pullback of $r_1$ and $r_2$; in fact, if\n$$\n \\xymatrix{\n \\bar{W} \\ar[r]^-{\\bar{p_2}} \\ar[d]_{\\bar{p_1}} & R \\ar[d]^{r_1} \\\\ R \\ar[r]_{r_2}& X,\n }\n$$\nis another weak pullback, the factorization $\\bar{W} \\longrightarrow W$ composed with the transitivity $t: W \\longrightarrow R$ ensures that the pseudo-relation is transitive also with respect to the second weak pullback.\n\\vspace{0.5cm}\n\nThe following property from \\cite{Enrico} (Proposition 1.1.9) will be useful in the sequel:\n\n\\begin{proposition}\\emph{\\cite{Enrico}}\\label{EV}\nLet $\\mathbb{C}$ be a projective cover of a regular category $\\mathbb{A}$. Let $ \\xymatrix {R \\ar@<2pt>[r]^{r_1} \\ar@<-2pt>[r]_{r_2} & X}$ be a pseudo-relation in $\\mathbb{C}$ and consider its (regular epimorphism, monomorphism) factorization in $\\mathbb{A}$\n$$\n \\xymatrix{\n R \\ar[rr]^{(r_1, r_2)} \\ar@{->>}[rd]_p & {} & X \\times X. \\\\ {} & E \\ar@{ >->}[ru]_{(e_1, e_2)} & {}\n }\n$$\nThen, $R$ is a pseudo-equivalence relation in $\\mathbb{C}$ if and only if $S$ is an equivalence relation in $\\mathbb{A}$.\n\\end{proposition}\n\n\n\\begin{definition}\nLet $\\mathbb{C}$ be a weakly lex category. We call $\\mathbb{C}$ a \\textbf{weak Goursat category} if, for any pseudo-equivalence relation $ \\xymatrix { R \\ar@<2pt>[r]^{r_1} \\ar@<-2pt>[r]_{r_2} & X}$ and any regular epimorphism $f: X \\twoheadrightarrow Y$, the composite $\\xymatrix { R \\ar@<2pt>[r]^{fr_1} \\ar@<-2pt>[r]_{fr_2} & X}$ is also a pseudo-equivalence relation.\n\\end{definition}\n\n\nWe use Remark~\\ref{weaklims} repeatedly in the next results.\n\n\\begin{proposition}\\label{cover}\nLet $\\mathbb{C}$ be a projective cover of a regular category $\\mathbb{A}$. Then $\\mathbb{A}$ is a Goursat category if and only if $\\mathbb{C}$ is a weak Goursat category.\n\\end{proposition}\n\n\\begin{proof}\nSince $\\mathbb{C}$ is a projective cover of a regular category $\\mathbb{A}$, then $\\mathbb{C}$ is weakly lex.\n\nSuppose that $\\mathbb{A}$ is a Goursat category. Let $ \\xymatrix { R \\ar@<2pt>[r]^{r_1} \\ar@<-2pt>[r]_{r_2} & X}$ be a pseudo-equiva\\-lence relation in $\\mathbb{C}$ and let $f: X \\twoheadrightarrow Y$ be a regular epimorphism in $\\mathbb{C}$. For the (regular epimorphism, monomorphism) factorizations of $\\langle r_1, r_2\\rangle$ and $\\langle fr_1, fr_2\\rangle$ we get the following diagram\n\\begin{equation}\n\\label{proj}\n \\vcenter{\\xymatrix {\n R \\ar[rr]^(0.4){\\langle r_1, r_2\\rangle} \\ar@{=}[ddd] \\ar@{->>}[rd]_{p} & {} & X\\times X \\ar@{->>}[ddd]^{f \\times f} \\\\\n {} & E \\ar@{ >->}[ru]_(0.4){\\langle e_1, e_2\\rangle} \\ar@{.>}[d]_{w} & {} \\\\\n {} & S \\ar@{ >->}[rd]^(0.4){\\langle s_1, s_2\\rangle} & {} \\\\\n R \\ar@{->>}[ru]^{q} \\ar[rr]_(0.4){\\langle fr_1, fr_2\\rangle} & {} & Y \\times Y,\n}}\n\\end{equation}\n\nwhere $w:E\\longrightarrow S$ is induced by the strong epimorphism $p$\n\\[\n \\xymatrix@C=1cm {\n R \\ar@{->>}[r]^{p} \\ar@{->>}[d]_{q} & E \\ar[d]^{(f \\times f) \\langle e_1, e_2\\rangle} \\ar@{.>}[dl]_{w} \\\\ S \\ar@{ >->}[r]_{\\langle s_1,s_2\\rangle} & Y \\times Y.\n}\n\\]\nThen $w$ is a regular epimorphism and by the commutativity of the right side of $\\eqref{proj}$, one has $S = f(E)$.\nBy Proposition \\ref{EV}, we know that $E$ is an equivalence relation in $\\mathbb{A}$. Since $\\mathbb{A}$ is a Goursat category, then $S = f(E)$ is also an equivalence relation in $\\mathbb{A}$ and by Proposition \\ref{EV}, we can conclude that $ \\xymatrix { R \\ar@<2pt>[r]^{fr_1} \\ar@<-2pt>[r]_{fr_2} & X}$ is a pseudo-equivalence relation in $\\mathbb{C}$.\n\nConversely, suppose that $\\mathbb{C}$ is a weak Goursat category. Let $ \\xymatrix { R \\ar@<2pt>[r]^{r_1} \\ar@<-2pt>[r]_{r_2} & X}$ be an equivalence relation in $\\mathbb{A}$ and $f :X \\twoheadrightarrow Y$ a regular epimorphism. We are going to show that $f(R) = S$\n$$ \\xymatrix {\n R \\ar@{->>}[r]^-{h} \\ar@<.5ex>[d]^{r_2} \\ar@<-.5ex>[d]_{r_1} & f(R)=S \\ar@<.5ex>[d]^{s_2} \\ar@<-.5ex>[d]_{s_1} \\\\ X \\ar@{->>}[r]_{f} & Y\n}\n$$\nis an equivalence relation; it is obviously reflexive and symmetric. In order to conclude that $\\mathbb{A}$ is a Goursat category, we must prove that $S$ is transitive, i.e that $S$ is an equivalence relation.\n\nWe begin by covering the regular epimorphism $f$ in $\\mathbb{A}$ with a regular epimorphism $\\bar{f}$ in $\\mathbb{C}$. For that we take the cover $y : \\bar{Y} \\twoheadrightarrow Y$, consider the pullback of $y$ and $f$ in $\\mathbb{A}$ and take its cover $\\alpha: \\bar{X} \\twoheadrightarrow X \\times_Y \\bar{Y}$\n\\[\n \\xymatrix@C=1cm{\n \\bar{X} \\ar@{->>}[rd]^{\\alpha} \\ar@\/^2pc\/@{->>}[rrd]^{\\bar{f}} \\ar@\/_2pc\/@{->>}[ddr]_{x} & {} & {}\\\\ {} & X \\times_Y \\bar{Y} \\ar@{->>}[r]^{f'} \\ar@{->>}[d]_{y'}& \\bar{Y} \\ar@{->>}[d]^{y}\\\\ {} & X \\ar@{->>}[r]_{f} & Y.\n }\n\\]\nNote that the above outer diagram is a \\emph{regular pushout}, so that\n\\begin{equation}\\label{push}f^o y = x \\bar{f}^o\\;\\;\\mathrm{and}\\;\\; y^of=\\bar{f}x^o \\end{equation}\n(Proposition 2.1 in~\\cite{ckp}).\n\nNext, we take the inverse image $x^{-1}(R)$ in $\\mathbb{A}$, which in an equivalence relation since $R$ is, and cover it to obtain a pseudo-equivalence $W \\rightrightarrows \\bar{X}$ in $\\mathbb{C}$. By assumption $ \\xymatrix { W \\ar@<2pt>[r] \\ar@<-2pt>[r] & \\bar{X} \\ar@{->>}[r]^{\\bar{f}} & \\bar{Y}}$ is a pseudo-equivalence relation in $\\mathbb{C}$ so it factors through an equivalence relation, say $ \\xymatrix { V \\ar@<2pt>[r]^{v_1} \\ar@<-2pt>[r]_{v_2} & \\bar{Y},}$ in $\\mathbb{A}$. We have\n\n$$\n\\xymatrix@C=25pt@R=15pt{\nW \\ar@{=}[rrr] \\ar@{->>}[d]_{w} &{} &{} & W \\ar[ddd] \\ar@{->>}[rd]^v &{} & {} \\\\\nx^{-1}(R)\\ar@{ >->}[dd]_{\\langle\\rho_1, \\rho_2\\rangle} \\ar@{->>}[dr]_{\\pi_R} \\ar@{.>}[rrrr]^-{\\gamma} & {} & {}& {} & V \\ar@{ >->}[ddl]_(0.3){\\langle v_1, v_2\\rangle} \\ar@{.>>}[dr]^{\\lambda} & {} \\\\\n{} & R \\ar@{ >->}[dd]_(.3){\\langle r_1,r_2\\rangle} \\ar@{->>}[rrrr]^(.2){h} & {} & {} & {} & S \\ar@{ >->}[dd]^-{\\langle s_1,s_2\\rangle} \\\\\n\\bar{X} \\times \\bar{X} \\ar@{->>}[dr]_-{x \\times x} \\ar@{-->>}[rrr]^(.7){\\bar{f} \\times \\bar{f}} & {} & {} & \\bar{Y} \\times \\bar{Y} \\ar@{->>}[rrd]^{y \\times y} & {} & {} \\\\\n{} & X \\times X \\ar@{->>}[rrrr]_{f \\times f} & {} & {} &{} & Y \\times Y, }\n$$\nwhere $\\gamma$ and $\\lambda$ are induced by the strong epimorphisms $w$ and $v$, respectively\\\\\n\\begin{center}\n$\n \\xymatrix@C=2cm {\n W \\ar@{->>}[r]^w \\ar@{->>}[dd]_v & x^{-1}(R) \\ar@{ >->}[d]^{\\langle\\rho_1, \\rho_2\\rangle} \\ar@{.>}[ddl]_{\\gamma} \\\\ {} & \\bar{X} \\times \\bar{X} \\ar@{->>}[d]^{\\bar{f} \\times \\bar{f}} \\\\ V \\ar@{ >->}[r]_{\\langle v_1,v_2\\rangle} & \\bar{Y} \\times \\bar{Y}\n}\n$\nand\n$\n \\xymatrix@C=2cm {\n W \\ar@{->>}[r]^v \\ar@{->>}[dd]_{h\\pi_R w} & V \\ar@{ >->}[d]^{\\langle v_1, v_2\\rangle} \\ar@{.>}[ddl]_{\\lambda} \\\\ {} & \\bar{Y} \\times \\bar{Y} \\ar@{->>}[d]^{y \\times y} \\\\ S \\ar@{ >->}[r]_{\\langle s_1,s_2\\rangle} & Y \\times Y.\n}\n$\n\\end{center}\n\nSince $\\gamma$ is a regular epimorphism, we have $V = \\bar{f}(x^{-1}(R))$.\nSince $\\lambda$ is a regular epimorphism, we have $S = y(V)$. One also has $V = y^{-1}(S)$ because\n$$\n\\begin{matrix}\n y^{-1}(S) &=& y^o S y & \\\\\n {} &=& y^o f(R) y &\\\\\n{} &=& y^o f R f^o y &\\\\\n{} &=& \\bar{f} x^o R x \\bar{f}^o & \\text{(by \\eqref{push})}\\\\\n{} &=& \\bar{f}(x^{-1}(R))&\\\\\n{} &=& V.&\n\\end{matrix}\n$$\n\nFinally, $S$ is transitive since\n$$\n\\begin{matrix}\n SS &=& yy^o S yy^o S yy^o &\\text{(Lemma~\\ref{lem}(2))} \\\\\n {} &=& yy^{-1}(S) y^{-1}(S) y^o &\\\\\n{} &=& y VV y^o &\\\\\n{} &=& y V y^o & \\text{(since V is an equivalence relation)}\\\\\n{} &=& y(V)&\\\\\n{} &=& S.&\n\\end{matrix}\n$$\n\n\\end{proof}\n\nWe may also consider weak Goursat categories through a property which is more similar to the one mentioned in Theorem~\\ref{CKP}:\n\n\\begin{lemma} Let $\\mathbb{C}$ be a projective cover of a regular category $\\mathbb{A}$. Then $\\mathbb{C}$ is a weak Goursat category if and only if for any commutative diagram in $\\mathbb{C}$\n\\begin{equation}\n\\label{equivdef}\n \\vcenter{ \\xymatrix {\n R \\ar@{->>}[r]^{\\varphi} \\ar@<.5ex>[d]^{r_2} \\ar@<-.5ex>[d]_{r_1} & S \\ar@<.5ex>[d]^{s_2} \\ar@<-.5ex>[d]_{s_1} \\\\ X \\ar@{->>}[r]_{f} & Y\n}}\n\\end{equation}\nsuch that $f$ and $\\varphi$ are regular epimorphism and $R$ is a pseudo-equivalence relation, then $S$ is a pseudo-equivalence relation.\n\\end{lemma}\n\n\\begin{proof} $(i) \\Rightarrow (ii)$ Since $\\xymatrix { R \\ar@<2pt>[r]^{r_1} \\ar@<-2pt>[r]_{r_2} & X}$ is a pseudo-equivalence relation, by assumption $\\xymatrix { R \\ar@<2pt>[r]^{fr_1} \\ar@<-2pt>[r]_{fr_2} & X}$ is also a pseudo-equivalence relation and then its (regular epimorphism, monomorphism) factorization gives an equivalence relation $\\xymatrix { E \\ar@<2pt>[r]^{e_1} \\ar@<-2pt>[r]_{e_2} & Y}$ in $\\mathbb{A}$ (Proposition~\\ref{EV}). We have the following commutative diagram\n$$ \\xymatrix {\n R \\ar@{=}[r] \\ar@<.5ex>[dd]^{r_2} \\ar@<-.5ex>[dd]_{r_1} & R \\ar@<.5ex>[dd]^(.4){f r_2} \\ar@<-.5ex>[dd]_(.4){f r_1} \\ar@{->>}[rr]^{\\varphi} \\ar@{->>}[rd]^{\\rho} & {} & S \\ar@{.>}[dl]^{\\sigma} \\ar@<.5ex>@\/^4pc\/[ddll]^{s_2} \\ar@<-.5ex>@\/^4pc\/[ddll]_{s_1} \\\\\n {} & {} & E \\ar@<.5ex>[dl]^{e_2} \\ar@<-.5ex>[dl]_{e_1} & {} \\\\\n X \\ar@{->>}[r]_{f} & Y & {} & {}\n}\n$$\nwhere $\\sigma: S \\longrightarrow E$ is induced by the strong epimorphism $\\varphi$\n\\[\n \\xymatrix@C=1cm {\n R \\ar@{->>}[r]^{\\varphi} \\ar@{->>}[d]_{\\rho} & S \\ar[d]^{\\langle s_1, s_2\\rangle} \\ar@{.>}[dl]_{\\sigma} \\\\ E \\ar@{ >->}[r]_{\\langle e_1,e_2\\rangle} & Y \\times Y.\n}\n\\]\nThen $\\sigma$ is a regular epimorphism and $\\xymatrix { S \\ar@<2pt>[r]^{s_1} \\ar@<-2pt>[r]_{s_2} & Y}$ is a pseudo-equivalence relation (Proposition~\\ref{EV}).\n\n $(ii) \\Rightarrow (i)$ Let $ \\xymatrix { R \\ar@<2pt>[r]^{r_1} \\ar@<-2pt>[r]_{r_2} & X}$ be a pseudo-equivalence relation in $\\mathbb{C}$ and $f: X \\twoheadrightarrow Y$ a regular epimorphism. The following diagram is of the type \\eqref{equivdef}\n$$\n\\xymatrix {\n R \\ar@{=}[r] \\ar@<.5ex>[d]^{r_2} \\ar@<-.5ex>[d]_{r_1} & R \\ar@<.5ex>[d]^{fr_2} \\ar@<-.5ex>[d]_{fr_1} \\\\ X \\ar@{->>}[r]_{f} & Y.\n}\n$$\nSince $\\xymatrix { R \\ar@<2pt>[r]^{r_1} \\ar@<-2pt>[r]_{r_2} & X}$ is a pseudo-equivalence relation, then by assumption\\\\ $\\xymatrix { R \\ar@<2pt>[r]^{f r_1} \\ar@<-2pt>[r]_{f r_2} & Y}$ is also a pseudo-equivalence relation.\n\\end{proof}\n\nAlternatively, weak Goursat categories may be characterized through a property more similar to the one mentioned in Remark~\\ref{variant}:\n\n\\begin{proposition}\\label{weaksquare} Let $\\mathbb{C}$ be a projective cover of a regular category $\\mathbb{A}$. The following conditions are equivalent:\n\\begin{enumerate}\n \\item[(i)] $\\mathbb{A}$ is a Goursat category;\n \\item[(ii)] $\\mathbb{C}$ is a weak Goursat category;\n \\item[(iii)] For any commutative diagram of type \\emph{($\\mathrm{I}$)} in $\\mathbb{C}$ where\n$$\\xymatrix@C=2cm{\n F \\ar@<.5ex>[d]^{\\beta_2} \\ar@<-.5ex>[d]_{\\beta_1} \\ar@{->>}[r]^{\\lambda}& G \\ar@<.5ex>[d]^{\\rho_2} \\ar@<-.5ex>[d]_{\\rho_1} \\\\\n X \\ar@{}[dr]|{\\mathrm{(I)}} \\ar@{->>}[r]^{\\alpha} \\ar@<3pt>[d]^{f} & U \\ar@<3pt>[d]^{g} \\\\\n Y \\ar@{->>}[r]_{\\beta} \\ar@<3pt>[u]^{s} & W \\ar@<3pt>[u]^t\n }\n$$\n$F$ is a weak kernel pair of $f$ and $\\lambda$ is a regular epimorphism (in $\\mathbb{C}$), then $G$ is a weak kernel pair of $g$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n$(i) \\Leftrightarrow (ii)$ By Proposition \\ref{cover}.\n\n$(i) \\Rightarrow (iii)$ If we take the kernel pairs of $f$ and $g$, then the induced morphism $\\bar{\\alpha}:\\Eq(f)\\longrightarrow \\Eq(g)$ is a regular epimorphism by Theorem~\\ref{Goursatpushout}. Moreover, the induced morphism $\\varphi:F\\longrightarrow \\Eq(f)$ is also a regular epimorphism. We get\n\\\n \\vcenter{\\xymatrix{\n F \\ar@{->>}[r]^{\\lambda} \\ar@{->>}[d]^{\\varphi} & G \\ar@{.>}[d]^{\\omega} \\ar@<.5ex>@\/^4pc\/[dd]^{\\rho_2} \\ar@<-.5ex>@\/^4pc\/[dd]_{\\rho_1} \\\\\n \\Eq(f) \\ar@<.5ex>[d]^{f_2} \\ar@<-.5ex>[d]_{f_1} \\ar@{->>}[r]^{\\bar{\\alpha}} & \\Eq(g) \\ar@<.5ex>[d]^{g_2} \\ar@<-.5ex>[d]_{g_1} \\\\\n X \\ar@{->>}[r]^{\\alpha} \\ar@<3pt>[d]^{f} & U \\ar@<3pt>[d]^{g}\n \\\\ Y \\ar@{->>}[r]_{\\beta} \\ar@<3pt>[u]^{s} & W, \\ar@<3pt>[u]^t\n }}\n\\]\nwhere $w:G\\longrightarrow \\Eq(g)$ is induced by the strong epimorphism $\\lambda$\n\\[\n \\xymatrix@C=1cm {\n F \\ar@{->>}[r]^{\\lambda} \\ar@{->>}[d]_{\\bar{\\alpha}.\\varphi} & G \\ar[d]^{\\langle\\rho_1, \\rho_2\\rangle} \\ar@{.>}[dl]_-{w} \\\\ \\Eq(g) \\ar@{ >->}[r]_{\\langle g_1,g_2\\rangle} & U.\n}\n\\]\nThis implies that $\\omega$ is a regular epimorphism ($\\omega \\lambda = \\bar{\\alpha} \\varphi$) and then $ \\xymatrix@C=1cm {G \\ar@<2pt>[r]^{\\rho_1} \\ar@<-2pt>[r]_{\\rho_2} & U}$ is a weak kernel pair of $g$.\n\n$(iii) \\Rightarrow (ii)$\nConsider the diagram \\eqref{equivdef} in $\\mathbb{C}$ where $ \\xymatrix { R \\ar@<2pt>[r]^{r_1} \\ar@<-2pt>[r]_{r_2} & X}$ is a pseudo-equivalence relation. We want to prove that $ \\xymatrix { S \\ar@<2pt>[r]^{s_1} \\ar@<-2pt>[r]_{s_2} & Y}$ is also a pseudo-equivalence. Take the (regular epimorphism, monomorphism) factorization of $R$ and $S$ in $\\mathbb{A}$ and the induced morphism $\\mu$ making the following diagram commutative\n$$\n\\xymatrix {\n R \\ar@{->>}[rr]^{\\varphi} \\ar@<.5ex>[dd]^(.4){r_2} \\ar@<-.5ex>[dd]_(.4){r_1} \\ar@{->>}[rd]^{\\rho} & {} & S \\ar@{->>}[rd]^{\\sigma} \\ar@<.5ex>[dd]^(.3){s_2} \\ar@<-.5ex>[dd]_(.3){s_1} & {} \\\\\n {} & U \\ar@{.>}[rr]_(0.3){\\mu} \\ar@<.5ex>[dl]^{u_2} \\ar@<-.5ex>[dl]_{u_1} & {} & V \\ar@<.5ex>[dl]^{v_2} \\ar@<-.5ex>[dl]_{v_1} \\\\\n X \\ar@{->>}[rr]_{f}& {} & Y. & {}\n}\n$$\nSince $\\mu$ is a regular epimorphism, then $V = f(U)$ and consequently, $V$ is reflexive and symmetric.\n\nSince $S$ is a pseudo-relation associated to $V$, then $S$ is also a reflexive and symmetric pseudo-relation. We just need to prove that $V$ is transitive. To do so, we apply our assumption to the diagram\n\n$$\n\\xymatrix@C=25pt@R=20pt{\nF \\ar@{->>}[rr]^{\\lambda} \\ar@{->>}[dd]_{\\delta} \\ar@{->>}[rd] & {} & G \\ar@{->>}[dd]^{\\alpha} \\\\\n{} & \\Eq(r_1) \\times_{\\varphi(\\Eq(r_1))} G \\ar@{->>}[ru] \\ar@{->>}[dl] & {} \\\\\n \\Eq(r_1) \\ar@{->>}[rr]_{\\chi} \\ar@<-.5ex>[d] \\ar@<.5ex>[d] & {} & \\varphi(\\Eq(r_1)) \\ar@<.5ex>[d] \\ar@<-.5ex>[d] \\\\\n R \\ar@{}[drr]|{\\mathrm{(I)}} \\ar@{->>}[rr]^{\\varphi} \\ar@<3pt>[d]^{r_1}& {} & S \\ar@<3pt>[d]^{s_1} \\\\\n X \\ar@{->>}[rr]_{f} \\ar@<3pt>[u]^{e_R} & {} & Y \\ar@<3pt>[u]^{e_S}\n }\n$$\nwhere $G$ is a cover of the regular image $\\varphi(\\Eq(r_1))$ and $F$ is a cover of the pullback $\\Eq(r_1) \\times_{\\varphi(Eq(r_1))} G$. Since $\\delta$ is a regular epimorphism, then $ \\xymatrix { F \\ar@<2pt>[r] \\ar@<-2pt>[r] & R}$ is a weak kernel pair of $r_1$. By assumption $ \\xymatrix { G \\ar@<2pt>[r] \\ar@<-2pt>[r] & S}$ is a weak kernel pair of $s_1$, thus $\\varphi(\\Eq(r_1)) = \\Eq(s_1)$. We then have\n\\[\n\\begin{matrix}\nVV &=& v_2 v_1^o v_1 v_2^o & \\text{(since $V$ is symmetric}) \\\\\n{} &=& v_2 \\sigma \\sigma^o v_1^o v_1 \\sigma \\sigma^o v_2^o & \\text{(Lemma~\\ref{lem}(2))} \\\\\n{} &=& s_2 s_1^o s_1 s_2^o & \\text{($v_i\\sigma=s_i$)}\\\\\n{} &=& s_2 \\varphi r_1^o r_1 \\varphi^o s_2^o &\\text{($\\varphi(\\Eq(r_1)) = \\Eq(s_1)$)}\\\\\n{} &=& f r_2 r_1^o r_1 r_2^o f^o & \\text{($s_i \\varphi = f r_i$ )}\\\\\n{} &=& f u_2 \\rho \\rho^o u_1^o u_1 \\rho \\rho^o u_2^o f^o & \\text{($u_i\\rho=r_i$)} \\\\\n{} &=& f u_2 u_1^o u_1 u_2^o f^o & \\text{(Lemma~\\ref{lem}(2)) } \\\\\n{} &=& f UU f^o & \\text{(since $U$ is an equivalence relation in $\\mathbb{A}$)} \\\\\n{} &=& f U f^o & \\\\\n{} &=& V. &\n\\end{matrix}\n\\]\n\\end{proof}\n\\begin{remark}\\label{vars}\nWhen $\\mathbb{A}$ is a $3$-permutable variety and $\\mathbb{C}$ its subcategory of free algebras, then the property stated in Proposition \\ref{weaksquare} (iii) is precisely what is needed to obtain the existence of the quaternary operations $p$ and $q$ which characterize $3$-permutable varieties. Let $X$ denote the free algebra on one element. Diagram $\\mathrm{(I)}$ below belongs to $\\mathbb{C}$\n$$\n\\xymatrix@C=2cm{\n F \\ar@{=}[r] \\ar@{->>}[d]^{\\mu} & F \\ar@{->>}[d]^{\\lambda \\mu} \\\\\n \\Eq(\\nabla_2 + \\nabla_2) \\ar@<.5ex>[d]^{\\pi_2} \\ar@<-.5ex>[d]_{\\pi_1} \\ar[r]^-{\\lambda} & \\Eq(\\nabla_3) \\ar@<.5ex>[d] \\ar@<-.5ex>[d] \\\\\n 4X \\ar@{}[dr]|{\\mathrm{(I)}} \\ar@{->>}[r]^{1+\\nabla_2+1} \\ar@<3pt>[d]^{\\nabla_2 +\\nabla_2} & 3X \\ar@<3pt>[d]^{\\nabla_3} \\\\\n 2X \\ar@{->>}[r] \\ar@<3pt>[u]^{\\iota_2 + \\iota_1} & X. \\ar@<3pt>[u]^{\\iota_2}\n }\n$$\nIf $F$ is a cover of $\\Eq(\\nabla_2+\\nabla_2))$, then $ \\xymatrix { F \\ar@<2pt>[r] \\ar@<-2pt>[r] & 4X}$ is a weak kernel pair of $\\nabla_2+\\nabla_2$.\nBy assumption $ \\xymatrix { F \\ar@<2pt>[r] \\ar@<-2pt>[r] & 3X}$ is a weak kernel pair of $\\nabla_3$, so that $\\lambda \\mu$ is surjective. We then conclude that $\\lambda$ is surjective and the existence of the quaternary operations $p$ and $q$ follows from Theorem $3$ in \\cite{gr}.\n\\end{remark}\n\n\n\\section*{Acknowledgements}\nThe first author acknowledges partial financial assistance by Centro de Mate--m\\'{a}tica da\nUniversidade de Coimbra---UID\/MAT\/00324\/2013, funded by the\nPortuguese Government through FCT\/MCTES and co-funded by the\nEuropean Regional Development Fund through the Partnership\nAgreement\\\\ PT2020.\\\\\n The second author acknowledges financial assistance by Fonds de la Recher--che Scientifique-FNRS Cr\\'edit Bref S\\'ejour \\`a l'\\'etranger 2018\/V 3\/5\/033 - IB\/JN - 11440, which supported his stay at the University of Algarve, where this paper was partially written.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} diff --git a/data_all_eng_slimpj/shuffled/split2/finalzziebz b/data_all_eng_slimpj/shuffled/split2/finalzziebz new file mode 100644 index 0000000000000000000000000000000000000000..4958b2b5f7f4f7bb39745e3362263c917bbd76e1 --- /dev/null +++ b/data_all_eng_slimpj/shuffled/split2/finalzziebz @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe mathematical scattering theory for potential perturbations of the self-adjoint Laplacian in $L^{2}(\\mathbb{R}^{3})$ is a well developed subject. In particular, the stationary approach leads to a representation formula for the scattering matrix (see, e.g., \\cite{Y-LNM}). \nIn the case of singular perturbations, in particular, in the case of self-adjoint realizations of Laplacians with boundary conditions on hypersurfaces, similar results have been obtained in recent years, either using properties of the so-called Weyl function appearing in the resolvent formula (see \\cite{BMN}) or using a Limiting Absorption Principle (see \\cite{JST} and \\cite{JMPA}). The advantage of the approach by LAP over the former is that no trace class condition is required and so the results are not restricted to the two dimensional situation. \nThe target of the present paper is to put these separate results together and so to provide a representation formula for the scattering matrix in the case where the Laplacian is perturbed both by a regular short-range potential and by a singular one describing self-adjoint boundary conditions. In particular, we extend the abstract results in \\cite{JMPA} to a wider framework which then allows applications to such a setting. In order to develop such a strategy, at first we provide an abstract Kre\\u\\i n's type resolvent formula which, when applied to the concrete case of a potential perturbation of the Laplacian, allows for not compactly supported potentials. In particular, whenever the singular part of the perturbations is absent, this extends from compactly supported potentials in one dimension to short range potentials in three dimensions the kind of results provided in \\cite[Section 5]{BN}. \\par\nHere, in more details, the contents of the paper. In Section 2, following the scheme proposed in \\cite{JFA}, we provide an abstract resolvent formula for a perturbations $A_{\\mathsf B}$ of the self-adjoint $A$ by a linear combination of the adjoint of two bounded trace-like maps $\\tau_{1}:\\text{\\rm dom}(A)\\to\\mathfrak h_{1}$ and $\\tau_{2}:\\text{\\rm dom}(A)\\to\\mathfrak h_{2}$; while the kernel of $\\tau_{2}$ is required to be dense, so $\\tau_{2}^{*}$ plays the role of a singular perturbations, no further hypothesis is required for $\\tau_{1}$ and in applications that allows $\\tau_{1}^{*}$ to represent a regular perturbations by a short-range potential. In Subsection 2.3, by block operator matrices and the Schur complement, we re-write the obtained resolvent formula in terms of the resolvent of the operator corresponding to the non singular part of the perturbations; that plays an important role in the subsequent part regarding LAP and the scattering theory. In Section 3, following the scheme proposed in \\cite{JST} and further generalized in \\cite{JMPA}, at first we provide, under suitable hypothesis, a Limiting Absorption Principle for $A_{\\mathsf B}$ (see Theorem \\ref{LAP}) and then, by a combination of such a LAP with stationary scattering theory in the Birman-Yafaev scheme and the invariance principle, we obtain a representation formula for the scattering matrix of the couple $(A_{\\mathsf B},A)$ (see Theorem \\ref{S-matrix}). Whenever $A$ is the free Laplacian in $L^{2}(\\mathbb{R}^{3})$, such a formula contains, as subcases, both the usual formula for the perturbation given by a short-range potential as given, e.g., in \\cite{Y-LNM} and the formula for the case of a singular perturbation describing self-adjoint boundary conditions on a hypersurface as given in \\cite{JMPA}. Successively, in Section 4,\nin order to apply our abstract results to the case in which $A$ is the free Laplacian and the regular part represents a perturbation by a potential, we give various regularity results for the boundary layer operators associated to $\\Delta+\\mathsf v$, where $\\mathsf v$ is a potential of Kato-Rellich type. In the last Section we present various examples, where the free Laplacian is perturbed both by a singular term, describing either separating boundary conditions (as Dirichlet and Neumann ones) or semi-transparent (as $\\delta$ and $\\delta'$ type ones), and a regular one given by a short range potential $\\mathsf v$ decaying as $|x|^{-\\kappa(1+\\epsilon)}$, $\\epsilon>0$. In order to satisfy all our hypotheses, we need $\\kappa=2$. However, all our hypotheses but a single one (see Lemma \\ref{5.4}) hold with $\\kappa=1$; we conjecture that the requirement $\\kappa=2$ is merely of technical nature and that our results are true for a short range potential decaying as $|x|^{-(1+\\epsilon)}$. Finally, let us remark that whenever one is only interested in the construction of the operators and not in the scattering theory, then it is sufficient to assume that $\\mathsf v$ is a Kato-Rellich potential (see Remark \\ref{suff}). As a byproduct of our abstract construction (see Theorem \\ref{Th-alt-res}), the operator $A_{\\mathsf B}$ corresponds, whenever $A$ is the free Laplacian, to a singular perturbation of a Schr\\\"odinger operator. However, our concern here is the scattering theory with respect to the free Laplacian, thus we regard the regular and the singular parts of the perturbation as a single object; this constitutes the main novelty of our approach. Schr\\\"odinger operators with a Kato-Rellich potential plus a $\\delta$-like perturbation with a $p$-summable strength ($p>2$) have been already considered in \\cite{JDE18}, while for a different construction with a bounded potential and a $\\delta$- or a $\\delta'$-like perturbation with bounded strength we refer to \\cite{BLL}. None of such references considered the scattering matrix (however, \\cite{JDE18} provided a limiting absorption principle).\n\n\\subsection{Some notation and definition.}\n{\\ }\\par\n\\vskip5pt \\noindent $\\bullet$ $\\|\\cdot\\|_{X}$ denotes the norm on the complex Banach space $X$; in case $X$ is a Hilbert space, $\\langle\\cdot,\\cdot\\rangle_{X}$ denotes the (conjugate-linear w.r.t. the first argument) scalar product.\n\\vskip5pt\\noindent $\\bullet$ $\\langle\\cdot,\\cdot\\rangle_{X^{*},X}$ denotes the duality (assumed to be conjugate-linear w.r.t. the first argument) between the dual couple $(X^{*},X)$.\n\\vskip5pt\\noindent $\\bullet$ $L^{*}:\\text{\\rm dom}(L^{*})\\subseteq Y^{*}\\to X^{*}$ denotes the dual of the densely defined linear operator $L:\\text{\\rm dom}(L)\\subseteq X\\to Y$; in a Hilbert spaces setting $L^{*}$ denotes the adjoint operator.\n\\vskip5pt\\noindent $\\bullet$ $\\varrho(A)$ and $\\sigma(A)$ denote the resolvent set and the spectrum of the self-adjoint operator $A$; $\\sigma_{p}(A)$, $\\sigma_{pp}(A)$, $\\sigma_{ac}(A)$, $\\sigma_{sc}(A)$, denote the point, pure point, absolutely continuous and singular continuous spectra.\n\\vskip5pt\\noindent $\\bullet$ $\\mathscr B(X,Y)$, $\\mathscr B(X)\\equiv \\mathscr B(X,X)$, denote the Banach space of bounded linear operator on the Banach space $X$ to the Banach space $Y$; ${\\|}\\cdot {\\|}_{X,Y}$ denotes the corresponding norm.\n\\vskip5pt\\noindent $\\bullet$ ${\\mathfrak S}_{\\infty}(X,Y)$ denotes the space of compact operators on $X$ to $Y$.\n\\vskip5pt\\noindent $\\bullet$ $X\\hookrightarrow Y$ means that $X$ is continuously embedded into $Y$. \n\\vskip5pt\\noindent $\\bullet$ $\\Omega\\equiv\\Omega_{\\rm in}\\subset\\mathbb{R}^{3}$ denotes an open and bounded subset with a Lipschitz boundary $\\Gamma$; $\\Omega_{\\rm ex}:=\\mathbb{R}^{3}\\backslash\\overline\\Omega$.\n\\vskip5pt\\noindent $\\bullet$ $H^{s}(\\Omega)$ and $H^{s}(\\Omega_{\\rm ex})$ denote the scales of Sobolev spaces. \n\\vskip5pt\\noindent $\\bullet$ $H^{s}(\\mathbb{R}^{3}\\backslash\\Gamma):=H^{s}(\\Omega)\\oplus H^{s}(\\Omega_{\\rm ex})$.\n\\vskip5pt\\noindent $\\bullet$ $|x|$ denotes the norm of $x\\in\\mathbb{R}^{n}$.\n $\\langle x\\rangle$ denotes the function $x\\mapsto (1+|x|^{2})^{1\/2}$.\n\\vskip5pt\\noindent $\\bullet$ $L_{w}^{2}(\\mathbb{R}^{3})$, $w\\in\\mathbb{R}$, denotes the set of complex-valued functions $f$ such that $\\langle x\\rangle^{w}f\\in L^{2}(\\mathbb{R}^{3})$. \n\\vskip5pt\\noindent $\\bullet$ $H_{w}^{s}(\\mathbb{R}^{3}\\backslash\\Gamma):=H^{s}(\\Omega)\\oplus H_{w}^{s}(\\Omega_{\\rm ex})$, where $H_{w}^{s}(\\Omega_{\\rm ex})$ denotes the weighted Sobolev space relative to the weight $\\langle x\\rangle^{w}$.\n\\vskip5pt\\noindent $\\bullet$ $\\gamma_{0}^{\\rm in\/\\rm ex}$ and $\\gamma_{1}^{\\rm in\/\\rm ex}$ denote the interior\/exterior Dirichlet and Neumann traces on the boundary $\\Gamma$. \n\\vskip5pt\\noindent $\\bullet$ $\\gamma_{0}:=\\frac12(\\gamma_{0}^{\\rm in}+\\gamma_{0}^{\\rm ex})$, \n $\\gamma_{1}:=\\frac12(\\gamma_{1}^{\\rm in}+\\gamma_{1}^{\\rm ex})$.\n\\vskip5pt\\noindent $\\bullet$ $[\\gamma_{0}]:=\\gamma_{0}^{\\rm in}-\\gamma_{0}^{\\rm ex}$, \n $[\\gamma_{1}]:=\\gamma_{1}^{\\rm in}-\\gamma_{1}^{\\rm ex}$. \n\\vskip5pt\\noindent $\\bullet$ $S\\!L_{z}$ and $D\\!L_{z}$ denote the single- and double-layer operators. \n\\vskip5pt\\noindent $\\bullet$ $S_{z}:=\\gamma_{0}S\\!L_{z}$, $D_{z}:=\\gamma_{1}D\\!L_{z}$.\n\\vskip5pt\\noindent $\\bullet$ $D\\subset\\mathbb{R}$ is said to be discrete in the open set $E\\supset D $ whenever the (possibly empty) set of its accumulations point is contained in $\\mathbb{R}\\backslash E$; $D$ is said to be discrete whenever $E=\\mathbb{R}$.\n\\vskip5pt\\noindent $\\bullet$ Given $x\\ge 0$ and $y\\ge 0$, $x\\lesssim y$ means that there exists $c\\ge 0$ such that $x\\le c\\,y$.\n\n\\section{An abstract Kre\\u\\i n-type resolvent formula}\\label{Sec_Krein} \n\\subsection{The resolvent formula} Let $A:\\text{\\rm dom}(A)\\subseteq \\mathsf H\\to \\mathsf H$ be a self-adjoint operator in the Hilbert space $\\mathsf H$. We denote by $R_{z}:=(-A+z)^{-1}$, $z\\in \\varrho(A)$, its resolvent; one has $R_{z}\\in\\mathscr B(\\mathsf H,\\mathsf H_{A})$, where $\\mathsf H_{A}$ is the Hilbert space given by $\\text{\\rm dom} (A)$ equipped with the scalar product $$\\langle u,u\\rangle_{\\mathsf H_{A}}:=\\langle (A^{2}+1)^{1\/2} u,(A^{2}+1)^{1\/2}v\\rangle_{\\mathsf H}\\,.\n$$ \nLet\n$$\n\\mathfrak h_{k}\\hookrightarrow\\mathfrak h_{k}^{\\circ}\\hookrightarrow\\mathfrak h_{k}^{\\ast}\\,,\\qquad k=1,2\\,,\n$$\nbe auxiliary Hilbert spaces with dense continuous embedding; we do not identify $\\mathfrak h_{k}$ with its dual $\\mathfrak h^{*}_{k}$ (however, we use $\\mathfrak h_{k}\\equiv\\mathfrak h_{k}^{**}$) and we work with the $\\mathfrak h_{k}^{\\ast}$-$\\mathfrak h_{k}$ duality \n$\\langle\\cdot,\\cdot\\rangle_{\\mathfrak h_{k}^{\\ast},\\mathfrak h_{k}}$ defined in terms of the scalar\nproduct of the intermediate Hilbert space $\\mathfrak h_{k}^{\\circ}$. The scalar product and hence the duality are supposed to be conjugate linear with respect to the first variable; notice that $\\langle\\varphi,\\phi\\rangle_{\\mathfrak h_{k},\\mathfrak h^{\\ast}_{k}}=\\langle\\phi,\\varphi\\rangle^{*}_{\\mathfrak h_{k}^{\\ast},\\mathfrak h_{k}}$.\\par\nGiven the bounded linear maps\n$$\\tau_{k}:\\mathsf H_{A}\\rightarrow\\mathfrak h_{k}\\,,\\qquad k=1,2\\,,\n$$\nsuch that \n\\begin{equation}\\label{tau2}\n\\text{$\\text{\\rm ker}(\\tau_{2})$ is dense in $\\mathsf H$ and $\\text{\\rm ran}(\\tau_{2})$ is dense in $\\mathfrak h_{2}$,} \n\\end{equation}\nwe introduce the bounded operators \n$$\n\\tau:\\mathsf H_{A}\\to \\mathfrak h_{1}\\oplus\\mathfrak h_{2}\\,,\\qquad \\tau u:=\\tau_{1}u\\oplus\\tau_{2} u\\,,\n$$\nand\n$$\nG_{z}:\\mathfrak h_{1}^{\\ast}\\oplus \\mathfrak h_{2}^{*}\\to\\mathsf H\\,,\\qquad \nG_{z}:=(\\tau R_{\\bar{z}})^{\\ast}\\,,\\qquad z\\in\\varrho(A)\\,.\n$$\nWe further suppose that there exist reflexive Banach spaces $\\mathfrak b_{k}$, $k=1,2$, with dense continuous embeddings $\\mathfrak h_{k}\\hookrightarrow\\mathfrak b_{k}$ (hence $\\mathfrak b_{k}^{*}\\hookrightarrow\\mathfrak h_{k}^{*}$),\nsuch that\n$\\text{\\rm ran}(G_{z}|\\mathfrak b_{1}^{*}\\oplus\\mathfrak b_{2}^{*})$ is contained in the domain of definition of some (supposed to exist) $(\\mathfrak b_{1}\\oplus\\mathfrak b_{2})$-valued extension of $\\tau$ (which we denote by the same symbol) in such a way that \n\\begin{equation}\n\\tau G_{z}|\\mathfrak b_{1}^{*}\\oplus\\mathfrak b_{2}^{*}\\in{\\mathscr B}(\\mathfrak b_{1}^{*}\\oplus\\mathfrak b_{2}^{*},\\mathfrak b_{1}\\oplus\\mathfrak b_{2})\\,. \\label{tauG}%\n\\end{equation}\nGiven these hypotheses, \nwe set $\\mathsf B=(B_{0},B_{1},B_{2})$, with \n\\begin{equation}\\label{B012}\nB_{0}\\in{\\mathscr B}(\\mathfrak b_{2}^{*},\\mathfrak b_{2,2}^{*})\\,,\\quad B_{1}\\in\\mathscr B(\\mathfrak b_{1},\\mathfrak b_{1}^{*})\\,,\\quad\nB_{2}\\in\\mathscr B(\\mathfrak b_{2},\\mathfrak b^{*}_{2,2})\\,, \\quad\\text{$\\mathfrak b_{2,2}$ a reflexive Banach space,}\n\\end{equation}\n\\begin{equation}\\label{Bse}\n\\quad B_{1}=B_{1}^{*}\\,,\\qquad B_{0}B^{*}_{2}=B_{2}B^{*}_{0}\\,,\n\\end{equation}\nand introduce the map\n\\begin{equation}\\label{Lambda}\nZ_{\\mathsf B}\\ni z\\mapsto\\Lambda_{z}^{\\!\\mathsf B}\\in{\\mathscr B}(\\mathfrak b_{1}\\oplus\\mathfrak b_{2},\\mathfrak b_{1}^{*}\\oplus\\mathfrak b_{2}^{*})\n\\,,\\qquad \n\\Lambda_{z}^{\\!\\mathsf B}:=(M_{z}^{\\mathsf B})^{-1}( B_{1}\\oplus B_{2}) \\,,\n\\end{equation}\nwhere\n\\begin{equation}\\label{ZB}\nZ_{\\mathsf B}:=\\big\\{z\\in{\\mathbb{C}}\\backslash(-\\infty,0]: (M_{w}^{\\mathsf B})^{-1}\n\\in{\\mathscr B}(\\mathfrak b^{*}_{1}\\oplus\\mathfrak b^{*}_{2,2},\\mathfrak b^{*}_{1}\\oplus\\mathfrak b^{*}_{2})\\,,\\ w=z,\\bar{z}\\big\\}\n\\end{equation}\n$$\nM_{z}^{\\mathsf B}:=( 1\\oplus B_{0}) -( B_{1}\\oplus B_{2}) \\tau G_{z} \\in {\\mathscr B}(\\mathfrak b^{*}_{1}\\oplus\\mathfrak b^{*}_{2},\\mathfrak b^{*}_{1}\\oplus\\mathfrak b^{*}_{2,2})\\,.\n$$\n\\begin{theorem}\\label{Th_Krein} Suppose hypotheses \\eqref{tau2}, \\eqref{tauG}, \\eqref{B012} and \\eqref{Bse} hold and that $Z_{\\mathsf B}$ defined in \\eqref{ZB} is not empty. Then, defined $\\Lambda_{z}^{\\!\\mathsf B}$ as in \\eqref{Lambda},\n\\begin{equation}\nR_{z}^{\\mathsf B}:=R_{z}+G_{z}\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar{z}}^{\\ast}\\,,\\quad z\\in\nZ_{\\mathsf B}\\,, \\label{resolvent}%\n\\end{equation}\nis the resolvent of a self-adjoint operator $A_{\\mathsf B}$ and $Z_{\\mathsf B}%\n=\\varrho(A_{\\mathsf B})\\cap\\varrho(A)$.\n\\end{theorem}\n\\begin{proof} By \\eqref{Bse}, one gets\n\\begin{align*}\n\\big( ( 1\\oplus B_{0}) -( B_{1}\\oplus B_{2})\\tau\nG_{\\bar{z}}\\big)( B_{1}\\oplus B^{*}_{2}) =&( B_{1}\\oplus B_{2})\n\\big( ( 1\\oplus B_{0}^{*}) -\\tau G_{\\bar{z}}( B_{1}\\oplus B_{2}^{*})\\big)\\\\\n=&( B_{1}\\oplus B_{2})\n\\big( ( 1\\oplus B_{0}) -( B_{1}\\oplus B_{2})\\tau G_{{z}}\\big)^{*} \\,.\n\\end{align*}\nThis entails, by the definitions \\eqref{Lambda} and \\eqref{ZB},\n\\begin{equation}\\label{PS1}\n(\\Lambda^{\\!\\mathsf B}_{z})^{*}=\\Lambda^{\\!\\mathsf B}_{\\bar z}\\,.\n\\end{equation}\nBy the resolvent identity, there follows\n\\begin{align*}\n& (( 1\\oplus B_{0}) -( B_{1}\\oplus B_{2})\n\\tau G_{z}) -(( 1\\oplus B_{0}) -( B_{1}\\oplus B_{2}) \\tau G_{w}) \\\\ \n&=( B_{1}\\oplus B_{2}) \\tau(G_{w}-G_{z})\n=(z-w)( B_{1}\\oplus B_{2}) \\tau R_{w}G_{z}\\\\\n&=(z-w)(B_{1}\\oplus B_{2}) G_{\\bar{w}}^{\\ast}G_{z}\\,,\n\\end{align*}\nwhich entails \n\\begin{align*}\n& ( ( 1\\oplus B_{0}) -( B_{1}\\oplus B_{2})\n\\tau G_{w}) ^{-1}-(( 1\\oplus B_{0}) -\n(B_{1}\\oplus B_{2}) \\tau G_{z}) ^{-1} \\\\\n=&(z-w)(( 1\\oplus B_{0}) -( B_{1}\\oplus B_{2}) \\tau G_{w}) ^{-1}\n( B_{1}\\oplus B_{2})G_{\\bar{w}}^{\\ast}G_{z}(( 1\\oplus B_{0})\n -(B_{1}\\oplus B_{2}) \\tau G_{z}) ^{-1}\\,,\n\\end{align*}\nand hence\n\\begin{equation}\\label{PS2}\n\\Lambda_{w}^{\\!\\mathsf B}-\\Lambda_{z}^{\\!\\mathsf B}=(z-w)\\Lambda_{w}^{\\!\\mathsf B}G_{\\bar{w}}^{\\ast\n}G_{z}\\Lambda_{z}^{\\!\\mathsf B}\\,.\n\\end{equation}\nBy \\eqref{PS1} and $\\eqref{PS2}$, \n$$(R_{z}^{\\mathsf B})^{*}=R_{\\bar z}^{\\mathsf B}\\,,\\qquad R_{z}^{\\mathsf B}=R_{w}^{\\mathsf B}+(w-z)R_{z}^{\\mathsf B}R_{w}^{\\mathsf B}\\,.\n$$\n(see \\cite[page 113]{JFA}). Hence, $R_{z}^{\\mathsf B}$ is the resolvent of a self-adjoint operator whenever it is injective (see, e.g., \\cite[Theorems 4.10 and 4.19]{Stone}). By \\eqref{resolvent},\n\\begin{align*}\n&(B_{1}\\oplus B_{2})\\tau R_{z}^{\\mathsf B}=(B_{1}\\oplus B_{2})\\big(1+\\tau G_{z}\\Lambda_{z}^{\\!\\mathsf B}\\big)G_{\\bar z}^{*}=\\big((B_{1}\\oplus B_{2})+(B_{1}\\oplus B_{2})\\tau G_{z}\\Lambda_{z}^{\\!\\mathsf B}\\big)G_{\\bar z}^{*}\\\\\n=&\\big((B_{1}\\oplus B_{2})+\\big((1\\oplus B_{0})-\\big((1\\oplus B_{0})-(B_{1}\\oplus B_{2})\\tau G_{z}\\big)\\big)\\Lambda_{z}^{\\!\\mathsf B}\\big)G_{\\bar z}^{*}=(1\\oplus B_{0})\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar z}^{*}\\,.\n\\end{align*}\nThus, if $R_{z}^{\\mathsf B}u=0$ then \n$$\n0\\oplus 0=(1\\oplus B_{0})\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar z}^{*}u=\\big(\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar z}^{*}u\\big)_{1}\\oplus B_{0}\\big(\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar z}^{*}u\\big)_{2}\n$$\nBy\n$$\nG_{z}(\\phi_{1}\\oplus\\phi_{2})=G^{1}_{z}\\phi_{1}+G^{2}_{z}\\phi_{2}\\,, \\qquad G^{k}_{z}:=(\\tau_{k}R_{\\bar{z}})^{\\ast}\\,,\n$$\nthere follows \n\\begin{equation}\n0=R_{z}^{\\mathsf B}u=R_{z}u+G^{1}_{z}\\big(\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar{z}}^{\\ast}\nu\\big)_{1}+G^{2}_{z}\\big(\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar{z}}^{\\ast}u\\big)_{2}=R_{z}u+G^{2}_{z}\n\\big(\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar{z}}^{\\ast}u\\big)_{2}\\,.\\label{z2}%\n\\end{equation}\nSince the denseness of $\\text{\\rm ker}(\\tau_{2})$ is equivalent to $\\text{\\rm ker}(G^{2}_{z})=\\{0\\}$\nand the denseness of $\\text{\\textrm{ran}}(\\tau_{2})$ implies\n$\\text{\\textrm{ran}}(G^{2}_{z})\\cap \\text{\\rm dom}(A)=\\{0\\}$ \n(see \\cite[Remark 2.9]{JFA}), the relation \\eqref{z2} gives $\\big(\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar{z}}^{\\ast\n}u\\big)_{2}=0$. Thus $R_{z}^{\\mathsf B}u=0$ compels $R_{z}u=0$ and hence $u=0$.\\par \nFinally, the equality $Z_{\\mathsf B}=\\varrho(A_{\\mathsf B})\\cap\\varrho(A)$ is consequence of \\cite[Theorem 2.19 and\nRemark 2.20]{CFP}.\n\\end{proof}\n\\begin{remark} By \\eqref{resolvent}, if $u\\in\\text{\\rm dom}(A_{\\mathsf B})$, then $u=u_{0}+G_{z}(\\phi_{1}\\oplus\\phi_{2})$ for some $u_{0}\\in\\mathsf H_{A}$ and $\\phi_{1}\\oplus\\phi_{2}\\in \\mathfrak b_{1}^{*}\\oplus\\mathfrak b_{2}^{*}$; hence, by \\eqref{tauG}, \n$$\n\\tau:\\text{\\rm dom}(A_{\\mathsf B})\\to \\mathfrak b_{1}\\oplus\\mathfrak b_{2}\\,.\n$$ \n\\end{remark}\n\\subsection{An additive representation}\nAt first, let us introduce the Hilbert space $\\mathsf H_{A}^{*}$ defined as the completion of $\\mathsf H$ endowed with the scalar product $$\\langle u,v\\rangle_{\\mathsf H_{A}^{*}}:=\\langle(A^{2}+1)^{-1\/2}u,(A^{2}+1)^{-1\/2}v\\rangle_{\\mathsf H}\\,.\n$$\nNotice that that $R_{z}$ extends to a bounded bijective map (which we denote by the same symbol) on $\\mathsf H_{A}^{*}$ onto $\\mathsf H$. The linear operator $A$, being a densely defined bounded operator on $\\mathsf H$ to $\\mathsf H_{A}^{*}$, extends to a bounded operator $\\overline {\\!A}:\\mathsf H\\to\\mathsf H_{A}^{*}$ given by its closure. Moreover, denoting by $\\langle\\cdot,\\cdot\\rangle_{\\mathsf H_{A}^{*},\\mathsf H_{A}}$ the pairing obtained by extending the scalar product in $\\mathsf H$, since $A$ is self-adjoint and since $\\text{\\rm dom}(A)$ is dense in $\\mathsf H$,\n$$\n\\langle u,Av\\rangle_{\\mathsf H}=\\langle\\, \\overline{\\!A}u,v\\rangle_{\\mathsf H_{A}^{*},\\mathsf H_{A}}\\,,\\qquad u\\in\\mathsf H\\,,\\ v\\in\\mathsf H_{A}, \\,.\n$$\nFurther, we define $\\tau^{*}:\\mathfrak h_{1}^{*}\\oplus\\mathfrak h^{*}_{2}\\to\\mathsf H_{A}^{*}$ by \n\\begin{equation}\\label{tau*}\n\\langle \\tau^{*}\\phi,u\\rangle_{\\mathsf H_{A}^{*},\\mathsf H_{A}}=\\langle\\phi,\\tau u\\rangle_{\\mathfrak h_{1}^{*}\\oplus\\mathfrak h^{*}_{2},\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}\\,,\\qquad u\\in\\mathsf H_{A}\\,,\\ \\phi\\in\\\\mathfrak h_{1}^{*}\\oplus\\mathfrak h^{*}_{2}\\,.\n\\end{equation}\nObviously, $\\tau^{*}(\\phi_{1}\\oplus\\phi_{2})=\\tau_{1}^{*}\\phi_{1}+\\tau^{*}_{2}\\phi_{2}$, where \n$\\tau_{k}^{*}:\\mathfrak h_{k}\\to\\mathsf H_{A}^{*}$, $k=1,2$, are defined in the same way as $\\tau^{*}$.\n\\par\nLet us notice that $R_{z}:\\mathsf H_{A}^{*}\\to \\mathsf H$ is the adjoint, with respect the pairing $\\langle\\cdot,\\cdot\\rangle_{\\mathsf H_{A}^{*},\\mathsf H_{A}}$, of $R_{\\bar z}:\\mathsf H_{A}\\to \\mathsf H$ and it is the inverse of $(-\\overline{\\!A} +z):\\mathsf H\\to\\mathsf H_{A}^{*}$; therefore \n\\begin{equation}\\label{blw}\nG_{z}=R_{z}\\tau^{*}\\,.\n\\end{equation}\n\\begin{lemma}\n\\label{le-green}Let $A_{\\mathsf B}:\\text{\\rm dom}(A_{\\mathsf B})\\subseteq\\mathsf H\\to\\mathsf H$ be the self-adjoint\noperator provided in Theorem \\ref{Th_Krein} and define\n\\begin{equation}\n\\rho_{\\mathsf B}:\\text{\\rm dom}(A_{\\mathsf B})\\rightarrow\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*}\\,,\\quad\n\\rho_{\\mathsf B}(R_{z}^{\\mathsf B}u):=(\\pi_{1}^{\\ast}\\oplus 1)\\Lambda_{z}^{\\!\\mathsf B}G_{\\bar{z}}^{\\ast\n}u\\,,\\qquad u\\in \\mathsf H\\,, \\quad z\\in\\varrho(A_{\\mathsf B})\\cap\\varrho(A)\\,,\\label{rho_def}%\n\\end{equation}\nwhere $\\pi_{1}$ denotes the orthogonal projection onto the subspace\n$\\overline{\\text{\\rm ran}(\\tau_{1})}$. Then, the definition of $\\rho_{\\mathsf B}$ is\nwell-posed, i.e.,\n\\[\nR_{z_{1}}^{\\mathsf B}u_{1}=R_{z_{2}}^{\\mathsf B}u_{2}\\quad\\implies\\quad (\\pi_{1}^{\\ast}\\oplus 1)\\Lambda_{z_{1}}^{\\!\\mathsf B}G_{\\bar{z}_{1}}^{\\ast}u_{1}=(\\pi_{1}^{\\ast}\\oplus 1)\n\\Lambda_{z_{2}}^{\\!\\mathsf B}G_{\\bar{z}_{2}}^{\\ast}u_{2}%\n\\]\nand \n\\begin{equation}\\label{GF}\n\\langle u,\\Delta_{\\mathsf B}v\\rangle_{\\mathsf H}=\\langle A\nu,v\\rangle_{\\mathsf H}+\n\\langle\\tau u,\\rho_{\\mathsf B} v\\rangle_{\\mathfrak h_{1}\\oplus\\mathfrak h_{2},\\mathfrak h^{*}_{1}\\oplus\\mathfrak h_{2}^{*}}\\,,\\quad u\\in \\text{\\rm dom}(A),\\,v\\in\n\\text{\\rm dom}(A_{\\mathsf B})\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nLet $v=R_{z}^{\\mathsf B}u=v_{z}+G_{z}\\Lambda_{z}^{\\!\\mathsf B}\\tau v_{z}$, where $v_{z}%\n:=R_{z}u$ (hence $\\tau v_{z}=G_{\\bar{z}}^{\\ast}u$). Then\n\\begin{align*}\n& \\langle u,A_{\\mathsf B}v\\rangle_{\\mathsf H}-\\langle A\nu,v\\rangle_{\\mathsf H}\\\\\n= & -\\langle u,(-A_{\\mathsf B}+z)v\\rangle_{\\mathsf H}%\n+\\langle(-A+\\bar{z})u,v\\rangle_{\\mathsf H}\\\\\n= & -\\langle u,(-A+z)v_{z}\\rangle_{\\mathsf H}%\n+\\langle(-A+\\bar{z})u,v_{z}+G_{z}\\Lambda_{z}^{\\!\\mathsf B}\\tau v_{z}%\n\\rangle_{\\mathsf H}\\\\\n= & \\langle(-A+\\bar{z})u,G_{z}\\Lambda_{z}^{\\!\\mathsf B}\\tau v_{z}\\rangle\n_{\\mathsf H}=\\langle\\tau u,\\Lambda_{z}^{\\!\\mathsf B}\\tau v_{z}%\n\\rangle_{\\mathfrak h_{1}\\oplus\\mathfrak h_{2},\\mathfrak h^{*}_{1}\\oplus\\mathfrak h_{2}^{*}}\\\\\n= & \\langle (\\pi_{1}\\oplus 1)\\tau u,\\Lambda_{z}^{\\!\\mathsf B}\\tau v_{z}\\rangle\n_{\\mathfrak h_{1}\\oplus\\mathfrak h_{2},\\mathfrak h^{*}_{1}\\oplus\\mathfrak h_{2}^{*}}=\\langle\\tau u,(\\pi_{1}^{\\ast}\\oplus 1)\\Lambda\n_{z}^{\\!\\mathsf B}\\tau v_{z}\\rangle_{\\mathfrak h_{1}\\oplus\\mathfrak h_{2},\\mathfrak h^{*}_{1}\\oplus\\mathfrak h_{2}^{*}}\\,.\n\\end{align*}\nSuppose now that $R_{z_{1}}^{\\mathsf B}u_{1}=R_{z_{2}}^{\\mathsf B}u_{2}$. Then, by the above\nidentities, one gets, for any $u\\in \\text{\\rm dom}(A)$,\n\\[\n\\langle \\tau^{\\ast}(\\pi_{1}^{\\ast}\\oplus 1)(\\Lambda_{z_{1}}^{\\!\\mathsf B}G_{\\bar{z}_{1}%\n}^{\\ast}u_{1}-\\Lambda_{z_{2}}^{\\!\\mathsf B}G_{\\bar{z}_{2}}^{\\ast}u_{2}),u\\rangle\n_{\\mathsf H_{A}^{*},\\mathsf H_{A}}=0\\,.\n\\]\nHence $\\tau^{\\ast}((\\pi_{1}^{\\ast}\\oplus 1)\\Lambda_{z_{1}}^{\\!\\mathsf B}G_{\\bar{z}_{1}}%\n^{\\ast}u_{1}-(\\pi_{1}^{\\ast}\\oplus 1)\\Lambda_{z_{2}}^{\\!\\mathsf B}G_{\\bar{z}_{2}}^{\\ast}%\nu_{2})=0$. However, $\\text{\\rm ker}(\\tau^{\\ast})\\cap\\text{\\rm ran}((\\pi_{1}^{\\ast}\\oplus 1))=\\{0\\}$ since $\\pi_{1}^{\\ast}\\oplus 1$ is the projector onto the subspace\northogonal to $\\text{\\rm ker}(\\tau^{\\ast})$.\n\\end{proof}\nThe next Lemma provides a sort of abstract boundary conditions holding for the elements in $\\text{\\rm dom}(A_{\\mathsf B})$:\n\\begin{lemma}\\label{abc} Let $A_{\\mathsf B}$ be the self-adjoint operator in Theorem \\ref{Th_Krein}. Then, for any $z\\in\\varrho(A_{\\mathsf B})\\cap\\varrho(A)$, one has the representation\n$$\n\\text{\\rm dom}(A_{\\mathsf B})=\\{u\\in\\mathsf H:u_{z}:=u-G_{z}\\rho_{\\mathsf B}u\\in \\text{\\rm dom}(A)\\}\\,,\n$$\n$$\n(-A_{\\mathsf B}+z)u=(-A+z)u_{z}\\,.\n$$\nMoreover, \n$$\nu\\in \\text{\\rm dom}(A_{\\mathsf B})\\quad\\Longrightarrow\\quad (\\pi_{1}^{*}B_{1}\\oplus B_{2})\\tau u=(1\\oplus B_{0})\\rho_{\\mathsf B} u\\,.\n$$\n\\end{lemma}\n\\begin{proof} Since $G_{z}=R_{z}\\tau^{*}$ (see \\eqref{blw} below) and $\\pi_{1}^{*}\\oplus 1$ is the projection onto the orthogonal to $\\text{\\rm ker}(\\tau^{*})$, one has $G_{z}=G_{z}(\\pi_{1}^{*}\\oplus 1) $. \nHence, $u\\in\\text{\\rm dom}(A_{\\mathsf B})$ if and only if $u=R_{z}v+G_{z}(\\pi_{1}^{*}\\oplus 1) \\Lambda_{z}^{\\mathsf B}G^{*}_{\\bar z}v=R_{z}v+G_{z}\\rho_{\\mathsf B}u$. Therefore, \n$$\n\\text{\\rm dom}(A_{\\mathsf B})=\\{u\\in\\mathsf H:u=u_{z}+G_{z}\\rho_{\\mathsf B}u\\,, \\ u_{z}\\in\\text{\\rm dom}(A)\\}\\,.\n$$\nMoreover, given any $u\\in\\text{\\rm dom}(A)$, $u=R^{\\mathsf B}_{z}v$, one has\n$$\n(-A+z)u_{z}=(-A+z)R_{z}v=(-A_{\\mathsf B}+z)R^{\\mathsf B}_{z}v=(-A_{\\mathsf B}+z)u\\,.\n$$\nFinally, given $u=R^{\\mathsf B}_{z}v\\in\\text{\\rm dom} (A_{\\mathsf B})$, one has\n\\begin{align*}\n&(\\pi_{1}^{*}B_{1}\\oplus B_{2})\\tau u=(\\pi_{1}^{*}\\oplus 1)(B_{1}\\oplus B_{2})\\tau R^{\\mathsf B}_{z}v\\\\\n=&(\\pi_{1}^{*}\\oplus 1)\\big((B_{1}\\oplus B_{2})G_{\\bar z}v+ \n(B_{1}\\oplus B_{2})\\tau G_{z}\\big((1\\oplus B_{0})-(B_{1}\\oplus B_{2})\\tau G_{z}\\big)^{-1}(B_{1}\\oplus B_{2})G_{\\bar z}v\\big)\\\\\n=&(\\pi_{1}^{*}\\oplus 1)(1\\oplus B_{0})\\Lambda^{\\mathsf B}_{z}G_{\\bar z}v=(1\\oplus B_{0})(\\pi_{1}^{*}\\oplus 1)\\Lambda^{\\mathsf B}_{z}G_{\\bar z}v=(1\\oplus B_{0})\\rho_{\\mathsf B} u\\,.\n\\end{align*}\n\\end{proof} \nNow, we provide an additive representation of the self-adjoint $A_{\\mathsf B}$ in Theorem \\ref{Th_Krein}. \n\\begin{theorem}\\label{Th-add} Let $A_{\\mathsf B}:\\text{\\rm dom}(A_{\\mathsf B})\\subseteq \\mathsf H\\to\\mathsf H$ be the self-adjoint operator appearing in Theorem \\ref{Th_Krein}. Then\n$$\nA_{\\mathsf B}=\\overline {\\!A}+\\tau^{*}\\!\\rho_{\\mathsf B}\\,,\n$$\nwhere $\\rho_{\\mathsf B}$ is defined in \\eqref{rho_def}. In particular, if $B_{0}^{-1}\\in\\mathscr B(\\mathfrak b_{2,2}^{*},\\mathfrak b_{2}^{*})$, then\n$$\nA_{\\mathsf B}=\\overline {\\!A}+\\tau_{1}^{*}B_{1}\\tau_{1}+\\tau_{2}^{*}B_{0}^{-1}\\!B_{2}\\tau_{2}\\,.\n$$ \n\\end{theorem} \n\\begin{proof} By \\eqref{GF}, for any $u\\in\\text{\\rm dom}(A_{\\mathsf B})$ and $v\\in\\mathsf H_{A}$,\n\\begin{align*}\n\\langle \\Delta_{\\mathsf B}u,v\\rangle_{\\mathsf H_{A}^{*},\\mathsf H_{A}}\\equiv&\\langle \\Delta_{\\mathsf B}u,v\\rangle_{\\mathsf H}=\n\\langle u,Av\\rangle_{\\mathsf H}+\\langle\\rho_{\\mathsf B} u,\\tau v\\rangle_{\\mathfrak h^{*}_{1}\\oplus\\mathfrak h^{*}_{2},\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}\\\\\n=&\\langle\\, \\overline{\\!A}u+\\tau^{*}\\!\\rho_{\\mathsf B} u,v\\rangle_{\\mathsf H_{A}^{*},\\mathsf H_{A}}\\,.\n\\end{align*}\nBy Lemma \\ref{abc} and by $\\tau_{1}^{*}\\pi_{1}^{*}=(\\pi_{1}\\tau_{1})^{*}=\\tau_{1}^{*}$,\n$$\n\\tau^{*}\\!\\rho_{\\mathsf B} =\\tau^{*}(\\pi^{*}_{1}B_{1}\\tau_{1}\\oplus B_{0}^{-1}B_{1}\\tau_{2})=\n\\tau_{1}^{*}B_{1}\\tau_{1}+\\tau_{2}^{*}B_{0}^{-1}\\!B_{2}\\tau_{2}\\,.\n$$\n\\end{proof}\n\n\n\\subsection{An alternative resolvent formula.} At first, let us notice that hypothesis \\eqref{tauG}, can be re-written as\n$$\n\\tau_{j}G^{k}_{z}|\\mathfrak b_{k}\\in\\mathscr B(\\mathfrak b_{k}^{*},\\mathfrak b_{j})\\,,\\quad j,k=1,2\\,,\\qquad G^{k}_{z}:=(\\tau_{k}R_{\\bar z})^{*}\\,.\n$$\nMoreover, \n$$\n{M}_{z}^{\\mathsf B}=(1\\oplus B_{0})+(B_{1}\\oplus B_{2})\\tau G_{z}=\n\\begin{bmatrix}M_{z}^{B_{1}}&B_{1}\\tau_{1}G^{2}_{z}\\\\\nB_{2}\\tau_{2}G^{1}_{z}&M_{z}^{B_{0},B_{2}}\n\\end{bmatrix}\n$$\nwhere\n$$\nM_{z}^{B_{1}}:=1-B_{1}\\tau_{1}G^{1}_{z}\\,,\\qquad \nM_{z}^{B_{0},B_{2}}:=B_{0}-B_{2}\\tau_{2}G^{2}_{z}\\,.\n$$\nThen, supposing all the inverse operators appearing in the next formula exist, by the inversion formula for block operator matrices, one gets\n\\begin{align}\\label{MB-1}\n&({M}_{z}^{\\mathsf B})^{-1}=\n\\begin{bmatrix}(M^{B_{1}}_{z})^{-1}+(M^{B_{1}}_{z})^{-1}B_{1}\\tau_{1}G^{2}_{z}({C}_{z}^{\\mathsf B})^{-1}B_{2}\\tau_{2}G^{1}_{z}(M^{B_{1}}_{z})^{-1}&\n(M^{B_{1}}_{z})^{-1}B_{1}\\tau_{1}G^{2}_{z}({C}_{z}^{\\mathsf B})^{-1}\\\\\n({C}_{z}^{\\mathsf B})^{-1}B_{2}\\tau_{2}G^{1}_{z}(M^{B_{1}}_{z})^{-1}&({C}_{z}^{\\mathsf B})^{-1}\n\\end{bmatrix}\n\\end{align}\nwhere ${C}^{\\mathsf B}_{z}$ denotes the second Schur complement, i.e.,\n\\begin{align*}\n{C}^{\\mathsf B}_{z}:=&M_{z}^{B_{0},B_{2}}-B_{2}\\tau_{2}G^{1}_{z}\n(M_{z}^{B_{1}})^{-1}B_{1}\\tau_{1}G^{2}_{z}\\\\\n=&M_{z}^{B_{0},B_{2}}\\left(1-(M_{z}^{B_{0},B_{2}})^{-1}B_{2}\\tau_{2}G^{1}_{z}\n(M_{z}^{B_{1}})^{-1}B_{1}\\tau_{1}G^{2}_{z}\\right)\\\\\n=&M_{z}^{B_{0},B_{2}}\\left(1-\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{z}\n\\Lambda_{z}^{\\!B_{1}}\\tau_{1}G^{2}_{z}\\right)\\,,\n\\end{align*}\n\\begin{equation}\\label{LB1}\n\\Lambda_{z}^{\\!B_{1}}:=(1-B_{1}\\tau_{1}G^{1}_{z})^{-1}B_{1}\\,,\n\\end{equation}\n\\begin{equation}\\label{LB02}\n\\Lambda_{z}^{\\!B_{0},B_{2}}:=(B_{0}-B_{2}\\tau_{2}G^{2}_{z})^{-1}B_{2}\\,.\n\\end{equation}\nRegarding the well-posedness of \\eqref{MB-1}, taking into account the definition of ${C}^{\\mathsf B}_{z}$, one has \n$$\nZ_{\\mathsf B}=\\big\\{z\\in\\varrho(A): (M^{\\mathsf B}_{z})^{-1}\\in{\\mathscr B}(\\mathfrak b^{*}_{1}\\oplus\\mathfrak b^{*}_{2,2},\\mathfrak b^{*}_{1}\\oplus\\mathfrak b^{*}_{2})\\,,\\ w=z,\\bar{z}\\big\\}\\supseteq \\widehat Z_{\\mathsf B}\\,,\n$$\nwhere\n\\begin{equation}\\label{wZB}\n\\widehat Z_{\\mathsf B}:=\\big\\{z\\in Z_{B_{1}}\\cap Z_{B_{0},B_{2}}:\\left(1-\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{w}\n\\Lambda_{z}^{\\!B_{1}}\\tau_{1}G^{2}_{w}\\right)^{-1}\\in\\mathscr B(\\mathfrak b_{2}^{*}),\\quad w=z,\\bar z\\big\\}\n\\end{equation}\n\\begin{equation}\\label{ZB1}\nZ_{B_{1}}:=\\big\\{z\\in\\varrho(A): (1-B_{1}\\tau_{1}G^{1}_{w})^{-1}\\in{\\mathscr B}(\\mathfrak b^{*}_{1})\\,,\\ w=z,\\bar{z}\\big\\}\\,,\n\\end{equation}\n\\begin{equation}\\label{ZB02}\nZ_{B_{0},B_{2}}:=\\big\\{z\\in\\varrho(A): (B_{0}-B_{2}\\tau_{2}G^{2}_{w})^{-1}\\in{\\mathscr B}(\\mathfrak b^{*}_{2,2},\\mathfrak b^{*}_{2})\\,,\\ w=z,\\bar{z}\\big\\}\\,,\n\\end{equation}\nTherefore, supposing that $\\widehat Z_{\\mathsf B}$ is not empty, for any $z\\in\\widehat Z_{\\mathsf B}$, by \\eqref{resolvent} and by\n$$\n({C}_{z}^{\\mathsf B})^{-1}B_{2}=\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\,,\\qquad \\Sigma^{\\mathsf B}_{z}:=\\left(1-\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{z}\n\\Lambda_{z}^{\\!B_{1}}\\tau_{1}G^{2}_{z}\\right)^{-1}\\,,\n$$\n one has\n\\begin{align*}\n\\Lambda_{z}^{\\!\\mathsf B}=(M^{\\mathsf B}_{z})^{-1}\n\\begin{bmatrix}B_{1}&0\\\\\n0&B_{2}\\end{bmatrix}\n=&\n\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}+\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}&\n\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\\\\n\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}&\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\n\\end{bmatrix}\n\\,.\n\\end{align*}\nTherefore\n\\begin{equation}\\label{res}\nR_{z}^{\\mathsf B}=R_{z}+\\begin{bmatrix}G^{1}_{z}&G^{2}_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}+\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}&\n\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\\\\n\\Sigma^{\\mathsf B}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}&\\Sigma^{\\mathsf B}_{z}\n\\Lambda_{z}^{\\!B_{0},B_{2}}\\end{bmatrix}\n\\begin{bmatrix}{G^{1*}_{\\bar z}}\\\\{G^{2*}_{\\bar z}}\\end{bmatrix}\\,.\n\\end{equation}\nIn particular, taking $\\mathsf B=(1,B_{1},0)$, one gets, for any $z\\in Z_{B_{1}}$,\n\\begin{align}\\label{res1}\nR_{z}^{B_{1}}:=R_{z}^{(1,B_{1},0)}\n=&R_{z}+\\begin{bmatrix}G^{1}_{z}&G^{2}_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda_{z}^{\\!B_{1}}&0\\\\\n0&0\n\\end{bmatrix}\n\\begin{bmatrix}{G^{1*}_{\\bar z}}\\\\{G^{2*}_{\\bar z}}\\end{bmatrix}\n=R_{z}+G^{1}_{z}\\Lambda_{z}^{\\!B_{1}}{G^{1*}_{\\bar z}}\n\\end{align}\nwhile, taking $\\mathsf B=(B_{0},0,B_{2})$, one gets, for any $z\\in Z_{B_{0},B_{2}}$, \n\\begin{align}\\label{res2}\nR_{z}^{B_{0},B_{2}}:=R_{z}^{(B_{0},0,B_{2})}\n=&R_{z}+\\begin{bmatrix}G^{1}_{z}&G^{2}_{z}\\end{bmatrix}\n\\begin{bmatrix}0&\n0\\\\\n0&\\Lambda_{z}^{\\!B_{0},B_{2}}\n\\end{bmatrix}\n\\begin{bmatrix}{G^{1*}_{\\bar z}}\\\\{G^{2*}_{\\bar z}}\\end{bmatrix}\n=R_{z}+G^{2}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}{G^{2*}_{\\bar z}}\\,.\n\\end{align}\nTherefore, by Theorem \\ref{Th_Krein} with $\\mathsf B=(B_{1},0,0)$, one gets\n\\begin{corollary}\\label{cor1} Let $\\tau_{1}\\in\\mathscr B(\\mathsf H_{A},\\mathfrak h_{1})$ such that $\\tau_{1}G^{1}_{z}|\\mathfrak b^{*}_{1}\\in\\mathscr B(\\mathfrak b_{1}^{*},\\mathfrak b_{1})$ and let $B_{1}\\in\\mathscr B(\\mathfrak b_{1},\\mathfrak b^{*}_{1})$ self-adjoint; suppose that \n$Z_{B_{1}}$ defined in \\eqref{ZB1} is not empty. Then \n\\begin{equation}\\label{res1.1}\nR_{z}^{B_{1}}=R_{z}+G^{1}_{z}\\Lambda_{z}^{\\!B_{1}}{G^{1*}_{\\bar z}}\\,,\\qquad z\\in Z_{B_{1}}\\,,\n\\end{equation}\nwhere $\\Lambda_{z}^{\\!B_{1}}$ is defined in \\eqref{LB1}, is the resolvent of a self-adjoint operator $A_{B_{1}}$ and $Z_{B_{1}}=\\varrho(A_{B_{1}})\\cap\\varrho(A)$.\n\\end{corollary}\nBy Theorem \\ref{Th_Krein} with $\\mathsf B=(B_{0},0,B_{2})$, one gets\n\\begin{corollary}\\label{cor02} Let $\\tau_{2}\\in\\mathscr B(\\mathsf H_{A},\\mathfrak h_{2})$ satisfy \\eqref{tau2} be such that $\\tau_{1}G^{1}_{z}|\\mathfrak b^{*}_{2}\\in\\mathscr B(\\mathfrak b_{2}^{*},\\mathfrak b_{2})$ and let $B_{0}\\in\\mathscr B(\\mathfrak b_{2}^{*},\\mathfrak b_{2,2}^{*})$, $B_{2}\\in \\mathscr B(\\mathfrak b_{2},\\mathfrak b_{2,2}^{*})$ be such that $B_{0}B_{2}^{*}=B_{2}B_{0}^{*}$; suppose that\n$Z_{B_{0},B_{2}}$ defined in \\eqref{ZB02} is not empty. Then \n\\begin{equation}\\label{res2.1}\nR_{z}^{B_{0},B_{2}}=R_{z}+G^{2}_{z}\\Lambda_{z}^{\\!B_{0},B_{2}}{G^{2*}_{\\bar z}}\\,,\\qquad z\\in Z_{B_{0},B_{2}}\n\\end{equation}\nwhere $\\Lambda_{z}^{\\!B_{0},B_{2}}$ is defined in \\eqref{LB02}, is the resolvent of a self-adjoint operator $A_{B_{0},B_{2}}$ and $Z_{B_{0},B_{2}}=\\varrho(A_{B_{0},B_{2}})\\cap\\varrho(A)$.\n\\end{corollary}\nSupposing $\\widehat Z_{\\mathsf B}\\not=\\varnothing $, by \\eqref{res}, by \\eqref{res1} and by the relations\n\\begin{align}\\label{GB1}\nG^{B_{1}}_{z}:=(\\tau_{2}R_{\\bar z}^{B_{1}})^{*}=&\n(\\tau_{2}R_{\\bar z}+\\tau_{2} G^{1}_{\\bar z}\\Lambda^{\\!B_{1}}_{\\bar z}{G^{1*}_{z}})^{*}\\\\\n=&G_{z}^{2}+G_{z}^{1}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G_{z}^{2}\\nonumber\n\\end{align}\n\\begin{align*}\n{G^{B_{1}*}_{\\bar z}}=\\tau_{2}R_{z}^{B_{1}}=&\n\\tau_{2}R_{z}+\\tau_{2} G^{1}_{ z}\\Lambda^{\\!B_{1}}_{z}{G^{1*}_{\\bar z}}\\\\=&G_{\\bar z}^{2*}+\\tau_{2}G_{z}^{1}\\Lambda^{\\!B_{1}}_{z}G_{\\bar z}^{1*}\n\\end{align*}\n\\begin{align*}\n\\widehat M_{z}^{\\mathsf B}=B_{0}-B_{2}\\tau_{2}G^{B_{1}}_{z}=&B_{0}-B_{2}\\tau_{2}G_{z}^{2}+\\tau_{2}G_{z}^{1}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G_{z}^{2}\\\\\n=&M_{z}^{B_{0},B_{2}}+B_{2}\\tau_{2}G_{z}^{1}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G_{z}^{2}\\\\\n=&M_{z}^{B_{0},B_{2}}\\big(1+\\Lambda^{\\!B_{0},B_{1}}_{z}\\tau_{2}G_{z}^{1}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G_{z}^{2}\\big)\n\\end{align*}\n\\begin{equation}\\label{wLB1}\n\\widehat \\Lambda^{\\mathsf B}_{z}:=(\\widehat M_{z}^{\\mathsf B})^{-1}B_{2}=(B_{0}-B_{2}\\tau_{2}G^{B_{1}}_{z})^{-1}B_{2}=\\Sigma_{z}^{\\mathsf B}\\Lambda^{\\!B_{0},B_{2}}_{z}\n\\end{equation}\n one gets \n\\begin{align}\n\\Lambda_{z}^{\\mathsf B}=&\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}+\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}&\n\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\\\\n\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda_{z}^{\\!B_{1}}&\n\\widehat \\Lambda^{\\mathsf B}_{z}\\,;\n\\end{bmatrix}\\label{LB-new}\\\\\n=&\\left(1+\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}&0\\\\0&\\widehat \\Lambda^{\\mathsf B}_{z}\n\\end{bmatrix}\\begin{bmatrix}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}&\n\\tau_{1}G^{2}_{z}\\\\\n\\tau_{2}G^{1}_{z}&0\n\\end{bmatrix}\\,\\right)\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}&0\\\\0&\\widehat \\Lambda^{\\mathsf B}_{z}\n\\end{bmatrix}\\,.\\label{LB-new2}\n\\end{align}\nTherefore \n\\begin{align}\\label{res-new}\nR_{z}^{\\mathsf B}=&R_{z}+\\begin{bmatrix}G^{1}_{z}&G^{2}_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}+\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}&\n\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\\\\n\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda_{z}^{\\!B_{1}}&\n\\widehat \\Lambda^{\\mathsf B}_{z}\n\\end{bmatrix}\n\\begin{bmatrix}\n{G^{1*}_{\\bar z}}\\\\ \n{G^{2*}_{\\bar z}}\n\\end{bmatrix}\\\\\n=&R_{z}+\\begin{bmatrix}G^{1}_{z}&G^{2}_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda^{\\!B_{1}}_{z}G^{1*}_{\\bar z}+\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}G^{1*}_{\\bar z}+\n\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}G^{2*}_{\\bar z}\\\\\n\\widehat \\Lambda^{\\!\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda_{z}^{\\!B_{1}}G^{1*}_{\\bar z}+\n\\widehat \\Lambda^{\\mathsf B}_{z}G^{2*}_{\\bar z}\n\\end{bmatrix}\\nonumber\\\\\n=&R_{z}+G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}G^{1*}_{\\bar z}+G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda^{\\!B_{1}}_{z}G^{1*}_{\\bar z}+\nG^{1}_{z}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}G^{2*}_{\\bar z}\\nonumber\\\\\n&+G^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}\\tau_{2}G^{1}_{z}\\Lambda_{z}^{\\!B_{1}}G^{1*}_{\\bar z}+\nG^{2}_{z}\\widehat \\Lambda^{\\mathsf B}_{z}G^{2*}_{\\bar z}\\nonumber\\\\\n=&R^{B_{1}}_{z}+G_{z}^{B_{1}}\\widehat \\Lambda^{\\mathsf B}_{z}G_{\\bar z}^{B_{1}*}\\nonumber\\,.\n\\end{align}\nThis also entails, by \\cite[Theorem 2.19 and Remark 2.20]{CFP}, that if $\\widehat Z_{\\mathsf B}\\not=\\varnothing $, then $\\widehat Z_{\\mathsf B}=Z_{\\mathsf B}=\\varrho(A_{\\mathsf B})\\cap\\varrho(A_{B_{1}})$. Summing up, one has the following \n\\begin{theorem}\\label{Th-alt-res} Assume that hypotheses \\eqref{tauG}, \\eqref{B012} and \\eqref{Bse} hold and that $\\widehat Z_{\\mathsf B}$ defined in \\eqref{wZB} is not empty. Then, for any $z\\in \\varrho(A_{\\mathsf B})\\cap\\varrho(A_{B_{1}})$, the resolvent $R_{z}^{\\mathsf B}$ in \\eqref{resolvent} has the representation \\eqref{res-new} and \n\\begin{equation}\\label{alt-res}\nR^{\\mathsf B}_{z}=R_{z}^{B_{1}}+G_{z}^{B_{1}}\\widehat \\Lambda^{\\mathsf B}_{z}G_{\\bar z}^{B_{1}*}\\,,\\qquad\nz\\in \\varrho(A_{\\mathsf B})\\cap\\varrho(A_{B_{1}})\\,,\n\\end{equation}\nwhere $R_{z}^{B_{1}}$, $G_{z}^{B_{1}}$ and $\\widehat \\Lambda^{\\mathsf B}_{z}$ are defined in \\eqref{res1.1}, \\eqref{GB1} and \\eqref{LB1}.\n\\end{theorem}\n\\begin{remark}Let us notice that the resolvent formula \\eqref{alt-res} is of the same kind of the one in \\eqref{res2.1}, whenever one replaces $A$ with $A_{B_{1}}$. \n\\end{remark}\nBy using the same kind of arguments as in the proof of Lemma \\ref{abc} and defining \n$$\n\\widehat \\rho_{\\mathsf B}:\\text{\\rm dom}(A_{\\mathsf B})\\to\\mathfrak h_{2}^{*}\\,,\\qquad \\widehat \\rho_{\\mathsf B}(R^{\\mathsf B}_{z}u):=G_{z}^{B_{1}}\\widehat\\Lambda_{z}^{\\mathsf B}G^{B_{1}*}_{\\bar z}u\\,,\n$$\none gets the following\n\\begin{lemma}\\label{alt-abc} Let $A_{\\mathsf B}$ be the self-adjoint operator in Theorem \\ref{Th-alt-res}. Then, for any $z\\in\\varrho(A_{\\mathsf B})\\cap\\varrho(A_{B_{1}})$, one has the representation\n$$\n\\text{\\rm dom}(A_{\\mathsf B})=\\{u\\in\\mathsf H:u_{z}:=u-G^{B_{1}}_{z}\\widehat\\rho_{\\mathsf B}u\\in \\text{\\rm dom}(A_{B_{1}})\\}\\,,\n$$\n$$\n(-A_{\\mathsf B}+z)u=(-A_{B_{1}}+z)u_{z}\\,.\n$$\nMoreover, \n$$\nu\\in \\text{\\rm dom}(A_{\\mathsf B})\\quad\\Longrightarrow\\quad B_{2}\\tau_{2} u=B_{0}\\widehat\\rho_{\\mathsf B} u\\,.\n$$\n\\end{lemma}\n\n\\section{The Limiting Absorption Principle and the Scattering Matrix}\\label{Sec-LAP}\nNow we suppose that $\\mathsf H=L^{2}(M,{\\mathcal B},m)\\equiv L^{2}(M)$. \nGiven a measurable $\\varphi:M\\to [1,+\\infty)$, we define the weighted $L^{2}$-space \n\\begin{equation}\\label{Lphi}\nL_{\\varphi}^{2}(M,{\\mathcal B},m)\\equiv L_{\\varphi}^{2}(M):=\\{\\text{$u:M\\to\\mathbb{C}$ measurable} : \\varphi u\\in L^{2}(M)\\}\\,.\n\\end{equation}\nBy $\\varphi\\ge 1$, \n$$\nL^{2}_{\\varphi}(M)\\hookrightarrow L^{2}(M)\\hookrightarrow L_{\\varphi^{-1}}^{2}(M)\\simeq L_{\\varphi}^{2}(M)^{*}\\,.\n$$\nFrom now on $\\langle\\cdot,\\cdot\\rangle $ and $\\|\\cdot\\|$ denote the scalar product and the corresponding norm on $L^{2}(M)$; $\\langle\\cdot,\\cdot\\rangle_{\\varphi} $ and $\\|\\cdot\\|_{\\varphi}$ denote the scalar product and the corresponding norm on $L_{\\varphi}^{2}(M)$. \\par\nThen we introduce the following hypotheses: \\vskip5pt\\par\\noindent\n(H1) $A_{B_{1}}$ is bounded from above and there exists a positive $\\lambda_{1}\\ge\\sup\\sigma(A_{B_{1}})$, such that \n$R^{B_{1}}_{z}\\in \\mathscr B(L^{2}_{\\varphi}(M))$ for any $z\\in \\varrho(A_{B_{1}})$ such that $\\text{Re}(z)> \\lambda_{1}$; \n\\vskip5pt\\par\\noindent\n(H2) $A_{B_{1}}$ satisfies a Limiting Absorption Principle (LAP for short), i.e. there exists a (eventually empty) closed set with zero Lebesgue measure ${e}(A_{B_{1}})\\subset \\mathbb{R}$ such that, for all $\\lambda\\in \\mathbb{R}\\backslash{e}(A_{B_{1}})$, the limits\n\\begin{equation}\\label{LAP1}\nR^{B_{1},\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}R^{B_{1}}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(L_{\\varphi}^{2}(M),L_{\\varphi^{-1}}^{2}(M))$ and the maps $z\\mapsto R^{B_{1},\\pm}_{z}$, where $R^{B_{1},\\pm}_{z}\\equiv R^{B_{1}}_{z}$ whenever $z\\in\\varrho(A_{B_{1}})$, are continuous on $(\\mathbb{R}\\backslash{e}(A_{B_{1}}))\\cup\\mathbb{C}_{\\pm}$ to $\\mathscr B(L_{\\varphi}^{2}(M),L_{\\varphi^{-1}}^{2}(M))$;\n\\vskip5pt\\par\\noindent\n(H3) for any compact set $K\\subset \\mathbb{R}\\backslash{e}(A_{B_{1}})$ there exists $c_{K}>0$ such that for any $\\lambda\\in K$ and for any $u\\in L_{\\varphi^{2}}^{2}(M)\\cap\\text{\\rm ker}(R^{B_{1},+}_{\\lambda}-R^{B_{1},-}_{\\lambda})$ one has\n\\begin{equation}\\label{(H3)}\n\\|R^{B_{1},\\pm}_{\\lambda}u\\|\\le c_{K}\\|u\\|_{\\varphi^{2}}\\,;\n\\end{equation} \nWe split next hypothesis (H4) in two separate points:\\vskip5pt\\par\\noindent\n(H4.1) $A_{\\mathsf B}$ is bounded from above;\n\\vskip5pt\\par\\noindent\n(H4.2) the embedding $\\mathfrak h_{2}\\hookrightarrow\\mathfrak b_{2}$ is compact and there exists a positive $\\lambda_{2}>\\sup\\sigma(A_{B_{1}})$, such that $G_{z}^{B_{1}}\\in \\mathscr B(\\mathfrak h^{*}_{2},L_{\\varphi^{2+\\gamma}}^{2}(M))$ for some $\\gamma>0$ and for any $z\\in\\varrho(A_{B_{1}})$ such that $\\text{Re}(z)>\\lambda_{2}$.\n\\vskip5pt\\par\\noindent\nThen, $A_{\\mathsf B}$ satisfies a Limiting Absorption Principle as well:\n\\begin{theorem}\\label{LAP} Suppose hypotheses (H1)-(H4) hold. Then the limits\n\\begin{equation}\\label{LAP2}\nR^{\\mathsf B,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}R^{\\mathsf B}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(L_{\\varphi}^{2}(M),L_{\\varphi^{-1}}^{2}(M))$ for all $\\lambda\\in\\mathbb{R}\\backslash{e}(A_{\\mathsf B})$, where ${e}(A_{\\mathsf B}):={e}(A_{B_{1}})\\cup\\sigma_{p}(A_{\\mathsf B})$, and \n${e}(A_{\\mathsf B})\\backslash{e}(A_{B_{1}})$ is a (possibly empty) discrete set in $\\mathbb{R}\\backslash {e}(A_{B_{1}})$; the maps $z\\mapsto R^{\\mathsf B,\\pm}_{z}$, where $R^{\\mathsf B,\\pm}_{z}\\equiv R^{\\mathsf B}_{z}$ whenever $z\\in\\varrho(A_{\\mathsf B})$, are continuous on $(\\mathbb{R}\\backslash {e}(A_{\\mathsf B}))\\cup\\mathbb{C}_{\\pm}$ to $\\mathscr B(L_{\\varphi}^{2}(M),L_{\\varphi^{-1}}^{2}(M))$\n\\end{theorem}\n\\begin{proof} We use \\cite[Theorem 3.1]{JMPA} (which builds on \\cite{Reng}). By (H1), \\eqref{alt-res} and (H4.2), $R^{B_{1}}_{z}$ and $R^{\\mathsf B}_{z}$ are in $\\mathscr B(L^{2}_{\\varphi}(M))$ and $z\\mapsto R^{B_{1}}_{z}$ and $z\\mapsto R^{\\mathsf B}_{z}$ are continuous since pseudo-resolvents in $\\mathscr B(L^{2}_{\\varphi}(M))$; $A_{\\mathsf B}$ is bounded from above by (H4.1). Therefore hypothesis (H1) in \\cite{JMPA} holds true. Our hypotheses (H2) and (H3) coincides with the same ones in \\cite{JMPA}. By (H4.2), the embedding $\\mathfrak b^{*}_{2}\\hookrightarrow\\mathfrak h^{*}_{2}$ is compact. From $\\widehat\\Lambda^{\\mathsf B}_{z}\\in\\mathscr B(\\mathfrak b_{2},\\mathfrak b_{2}^{*})$ and \\eqref{alt-res}, follows that \n$R^{\\mathsf B}_{z}-R^{B_{1}}_{z}\\in{\\mathfrak S}_{\\infty}(L^{2}(M),L_{\\varphi^{2+\\gamma}}^{2}(M))$. Therefore hypothesis (H4) in \\cite{JMPA} holds and the statement is a consequence of \\cite[Theorem 3.1]{JMPA}.\n\\end{proof}\nLet us now assume that \\vskip8pt\\noindent\n(H5) the limits\n\\begin{equation}\\label{limG}\nG_{\\lambda}^{B_{1},\\pm}:=\\lim_{\\epsilon\\searrow 0}G^{B_{1}}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(\\mathfrak h_{2} ^{*},L^{2}_{\\varphi^{-1}}(M))$ for any $\\lambda\\in\\mathbb{R}\\backslash{e}(A_{B_{1}})$ and the maps $z\\mapsto G_{z}^{B_{1},\\pm}$, where $G_{z}^{B_{1},\\pm}\\equiv G^{B_{1}}_{z}$ whenever $z\\in\\varrho(A_{B_{1}})$, are continuous on $(\\mathbb{R}\\backslash{e}(A_{B_{1}}))\\cup\\mathbb{C}_{\\pm}$ to $\\mathscr B(\\mathfrak h_{2} ^{*},L^{2}_{\\varphi^{-1}}(M))$; moreover, the linear operators $G_{z}^{B_{1},\\pm}$ are injective. \n\\vskip8pt\\noindent\nThen, by \\cite[Lemma 3.6]{JMPA}, one gets the following:\n\\begin{lemma}\\label{bound}\nAssume that (H1)-(H5) hold. Then, for any open and bounded $I$ s.t. $\\overline{I}\\subset \\mathbb{R}\\backslash{e}(A_{\\mathsf B})$, one has \n\\begin{equation}\\label{supLambda}\n\\sup_{(\\lambda,\\epsilon)\\in I\\times (0,1)}{\\|}\\widehat\\Lambda^{\\mathsf B}_{\\lambda\\pm i\\epsilon}{\\|}_{\\mathfrak h_{2} ,\\mathfrak h_{2}^{*}}<+\\infty\\,.\n\\end{equation}\nMoreover, for any $\\lambda\\in \\mathbb{R}\\backslash{e}(A_{\\mathsf B})$, the limits\n\\begin{equation}\\label{limLambda}\n\\widehat\\Lambda^{\\mathsf B,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\widehat\\Lambda^{\\mathsf B}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(\\mathfrak h_{2} ,\\mathfrak h_{2} ^{*})$ and\n\\begin{equation}\\label{limRes}\nR^{\\mathsf B,\\pm}_{\\lambda}=R^{B_{1},\\pm}_{\\lambda}+G^{B_{1},\\pm}_{\\lambda}\\widehat\\Lambda^{\\mathsf B,\\pm}_{\\lambda}(G^{B_{1},\\mp}_{\\lambda})^{*}\\,.\n\\end{equation}\n\\end{lemma}\nBy the same reasoning as at the end of \\cite[proof of Theorem 5.1]{JMPA}, one can improve the result regarding \\eqref{limLambda}: \n\\begin{corollary}\\label{limbb} Suppose hypotheses (H1)-(H5) hold. Then the limits \\eqref{limLambda} exist in $\\mathscr B(\\mathfrak b_{2},\\mathfrak b_{2}^{*})$.\n\\end{corollary}\nBefore stating the next results, we recall the following: \n\\begin{definition} Given two self-adjoint operators $A_{1}$ and $A_{2}$ in the Hilbert space $\\mathsf H$, we say that completeness holds for the scattering couple $(A_{1},A_{2})$ whenever the strong limits \n$$\nW_{\\pm}(A_ {1},A_{2})\n:=\\text{s-}\\lim_{t\\to\\pm\\infty}e^{itA_{1}}e^{-itA_{2}}P_{2}^{ac}\n\\,,\\qquad\nW_{\\pm}(A_{2},A_{1})\n:=\\text{s-}\\lim_{t\\to\\pm\\infty}e^{itA_{2}}e^{-itA_{1}}P_{1}^{ac}\\,,\n$$ \nexist everywhere in $\\mathsf H$ and \n$$\\text{\\rm ran}(W_{\\pm}(A_ {1},A_{2}))=\\mathsf H_{1}^{ac }\\,,\\qquad\\text{\\rm ran}(W_{\\pm}(A_{2},A_{1}))=\\mathsf H_{2}^{ac }\\,,\n$$\n$$ W_{\\pm}(A_ {1},A_{1})^{*}=W_{\\pm}(A_{2},A_ {1})\\,,\n$$ \nwhere $P_{k}^{ac}$ denotes the orthogonal projector onto the absolutely continuous subspace $\\mathsf H_{k}^{ac}$ of $A_{k}$. Furthermore, we say the asymptotic completeness holds for the scattering couple $(A_{1},A_{2})$ whenever, beside completeness, one has \n$$\n\\mathsf H_{1}^{ac }=(\\mathsf H_{1}^{pp})^{\\perp}\\,,\\qquad \\mathsf H_{2}^{ac }=(\\mathsf H_{2}^{pp})^{\\perp}\\,,\n$$ \nwhere $\\mathsf H_{k}^{pp}$ denotes the pure point subspace of $A_{k}$; equivalently, whenever $\n\\sigma_{sc}(A_{1})=\\sigma_{sc}(A_{2})=\\varnothing $.\n\\end{definition} \nOur next hypothesis is \\vskip8pt\\noindent\n(H6) completeness hold for the scattering couple $(A_{B_{1}},A)$. \n\\vskip8pt\\noindent\n\\begin{theorem}\\label{AC} Suppose that (H1)-(H6) hold. Then completeness holds for the couple $(A_{\\mathsf B},A)$. If furthermore ${e}(A_{B_{1}})$ is a discrete set and $\\sigma_{sc}(A)=\\varnothing $, then asymptotic completeness holds for the couple $(A_{\\mathsf B},A)$.\n\\end{theorem}\n\\begin{proof} By \\eqref{alt-res} and by the same proof as in Lemma \\ref{le-green}, one gets \n\\begin{equation}\\label{alt-gf}\n\\langle u,\\Delta_{\\mathsf B}v\\rangle_{L^{2}(M)}-\\langle A_{B_{1}}\nu,v\\rangle_{L^{2}(M)}=\n\\langle\\tau_{2} u,\\widehat \\rho_{\\mathsf B} v\\rangle_{\\mathfrak h_{2},\\mathfrak h_{2}^{*}}\\,,\\quad u\\in \\text{\\rm dom}(A_{B_{1}}),\\,v\\in\n\\text{\\rm dom}(A_{\\mathsf B})\\,,\n\\end{equation}\nwhere\n\\begin{equation*}\n\\widehat\\rho_{\\mathsf B}:\\text{\\rm dom}(A_{\\mathsf B})\\rightarrow\\mathfrak h_{2}^{*}\\,,\\quad\n\\widehat\\rho_{\\mathsf B}(R_{z}^{\\mathsf B}u):=\\widehat\\Lambda_{z}^{\\mathsf B}G^{B_{1}*}_{\\bar{z}}u\\,,\\qquad u\\in \\mathsf H\\,, \\quad z\\in\\varrho(A_{\\mathsf B})\\cap\\varrho(A_{B_{1}})\\,.\n\\end{equation*}\nThen, by hypotheses (H1)-(H5) and by \\cite[Theorems 2.8 and 3.8]{JMPA} (compare \\eqref{alt-gf} and Lemma \\ref{bound} here with (2.19) and Lemma 3.6 there and notice that hypothesis (H6) there is included in our hypothesis (H4)) one gets the completeness for the couple $(A_{\\mathsf B},A_{B_{1}})$. By (H6) and the chain rule for the wave operators (see \\cite[Theorem 3.4, Chapter X]{Kato}), one then gets completeness for the scattering couple $(A_{\\mathsf B},A)$. To conclude the proof it remains to show that \n$\\sigma_{sc}(A_{\\mathsf B})=\\varnothing$. Let $\\mathsf H_{\\mathsf B}^{pp}$ denote the pure point subspace of $A_{\\mathsf B}$; given $u\\in (\\mathsf H_{\\mathsf B}^{pp})^{\\perp}$, let $\\mu_{u}^{\\mathsf B}$ be the corresponding spectral measure. By Theorem \\ref{LAP} and by standard arguments (see, e.g., the proof of \\cite[Thm. 6.1]{Agmon}), the support of the singular continuous component of $\\mu_{u}^{\\mathsf B}$ is contained in $e(A_{B_{1}})$. Since $e(A_{B_{1}})$ is discrete, such a support is empty and so $u$ has a null projection onto $\\mathsf H_{\\mathsf B}^{sc}$, the singular continuous subspace of $A_{\\mathsf B}$. This gives $(\\mathsf H_{\\mathsf B}^{pp})^{\\perp}=\\mathsf H_{\\mathsf B}^{ac}$, where $\\mathsf H_{\\mathsf B}^{ac}$ denote the absolutely continuous subspace of $A_{\\mathsf B}$.\n\\end{proof}\n\n\\subsection{A representation formula for the scattering matrix}\nAccording to Theorem \\ref{AC}, under the assumptions there stated, the scattering operator \n$$\nS_{\\mathsf B}:=W_{+}(A_{\\mathsf B},A)^{*}W_{-}(A_{\\mathsf B},A)\n$$\nis a well defined unitary map. Let \n\\begin{equation}\\label{F0}\nF:L^{2}(M)_{ac}\\to\\int^{\\oplus}_{\\sigma_{ac}(A)}(L^{2}(M)_{ac})_{\\lambda}\\,d\\eta(\\lambda)\n\\end{equation}\nbe a unitary map which diagonalizes the absolutely continuous component of $A$, i.e., a direct integral representation of $L^{2}(M)_{ac}$, the absolutely continuous subspace relative to $A$, with respect to the spectral measure of the absolutely continuous component of $A$ (see e.g. \\cite[Section 4.5.1]{BW}). We define the scattering matrix $${\\mathcal S}^{\\mathsf B}_{\\lambda}:(L^{2}(M)_{ac})_{\\lambda}\\to (L^{2}(M)_{ac})_{\\lambda}$$ by the relation (see e.g. \\cite[Section 9.6.2]{BW})\n$$\nFS_{\\mathsf B}F^{*}u_{\\lambda}={\\mathcal S}^{\\mathsf B}_{\\lambda}u_{\\lambda}\n\\,.\n$$\nNow, following the same scheme as in \\cite{JMPA}, which uses the Birman-Kato invariance principle and the Birman-Yafaev general scheme in stationary scattering theory, we provide an explicit relation between ${\\mathcal S}^{\\mathsf B}_{\\lambda}$ and $\\Lambda^{\\!\\mathsf B,+}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\Lambda^{\\!\\mathsf B}_{\\lambda+i\\epsilon}$. \\par \nGiven $\\mu\\in \\varrho(A)\\cap\\varrho(A_{\\mathsf B})$, we consider the scattering couple $(R^{\\mathsf B}_{\\mu}, R_{\\mu})$ and the strong limits \n$$\nW_{\\pm}(R^{\\mathsf B}_{\\mu},R_{\\mu})\n:=\\text{s-}\\lim_{t\\to\\pm\\infty}e^{itR^{\\mathsf B}_{\\mu}}e^{-itR_{\\mu}}P^{\\mu}_{ac}\\,,\n$$ \nwhere $P^{\\mu}_{ac}$ is the orthogonal projector onto the absolutely continuous subspace of $R_{\\mu}$; we prove below that such limits exist everywhere in $L^{2}(M)$. Let $S^{\\mu}_{\\Lambda}$ the corresponding scattering operator \n$$\nS_{\\mathsf B}^{\\mu}:=W_{+}(R^{\\mathsf B}_{\\mu},R_{\\mu})^{*}W_{-}(R^{\\mathsf B}_{\\mu},R_{\\mu})\\,.\n$$\nUsing the unitary operator $F_{\\mu}$ which diagonalizes the absolutely continuous component of $R_{\\mu}$, i.e. $(F_{\\mu}u)_{\\lambda}:=\\frac1\\lambda(Fu)_{\\mu-\\frac1\\lambda}$, $\\lambda\\not=0$ such that $\\mu-\\frac1\\lambda\\in \\sigma_{ac}(A)$, one defines the scattering matrix $${\\mathcal S}^{\\mathsf B,\\mu}_{\\lambda}:(L^{2}(M)_{ac})_{\\mu-\\frac1\\lambda}\\to (L^{2}(M)_{ac})_{\\mu-\\frac1\\lambda}$$ corresponding to the scattering operator $S^{\\mu}_{\\mathsf B}$ by the relation\n$$\nF_{\\mu}S^{\\mu}_{\\mathsf B}F_{\\mu}^{*}u^{\\mu}_{\\lambda}={\\mathcal S}^{\\mathsf B,\\mu}_{\\lambda}u^{\\mu}_{\\lambda}\n\\,.\n$$ \nWe introduce a further hypothesis (H7), which we split in four separate points:\\vskip5pt\\par\\noindent\n(H7.1) $A$ is bounded from above and satisfies a Limiting Absorption Principle: there exists a (eventually empty) closed set ${e}(A)\\subset\\mathbb{R}$ of zero Lebesgue measure such that for all $\\lambda\\in \\mathbb{R}\\backslash{e}(A)$ the limits\n\\begin{equation}\\label{LAP0}\nR^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}R_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(L_{\\varphi}^{2}(M),L_{\\varphi^{-1}}^{2}(M))$; \n\\vskip5pt\\par\\noindent\n(H7.2) $G^{1}_{z}\\in\\mathscr B(\\mathfrak h_{1}^{*},L^{2}_{\\varphi}(M))$ for any $z\\in\\varrho(A)$ and the limits\n\\begin{equation}\\label{limG1}\nG_{\\lambda}^{{1},\\pm}:=\\lim_{\\epsilon\\searrow 0}G^{{1}}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(\\mathfrak h_{1} ^{*},L^{2}_{\\varphi^{-1}}(M))$ for any $\\lambda\\in\\mathbb{R}\\backslash{e}(A)$;\n\\vskip5pt\\par\\noindent\n(H7.3) the limits\n\\begin{equation}\\label{limLB1}\n\\Lambda^{\\!B_{1},\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\Lambda^{\\!B_{1},\\pm}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(\\mathfrak h_{1},\\mathfrak h_{1}^{*})$ for any $\\lambda\\in\\mathbb{R}\\backslash{e}(A_{B_{1}})$;\n\\vskip5pt\\par\\noindent\n(H7.4) the limits\n\\begin{equation}\\label{limtG1}\n\\tau_{2} G^{1,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\tau_{2} G^{1}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(\\mathfrak b^{*}_{1},\\mathfrak b_{2})$ for any $\\lambda\\in\\mathbb{R}\\backslash{e}(A_{B_{1}})$.\n\\begin{remark}\\label{rem3.6} By $\\tau_{2}G^{1}_{z}=\\tau_{2} (\\tau_{1}R_{\\bar z})^{*}=(\\tau_{1} (\\tau_{2}R_{ z})^{*})^{*}=(\\tau_{1}G^{2}_{\\bar z})^{*}$, hypothesis (H7.4) entails the existence in $\\mathscr B(\\mathfrak b_{2},\\mathfrak b_{1}^{*})$, for any $\\lambda\\in\\mathbb{R}\\backslash{e}(A_{B_{1}})$, of the limits \n\\mathfrak b\n\\tau_{1} G^{2,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\tau_{1} G^{2}_{\\lambda\\pm i\\epsilon}\\,.\n\\end{equation}\n\\end{remark}\n\\begin{remark} Whenever one strengthens hypotheses (H7.2) as in (H5), then, by the same kind of proof that leads to the existence of the limit \\eqref{limLambda} (see \\cite[Lemma 3.6]{JMPA}), one gets the existence of the limits requested in hypotheses (H7.3).\n\\end{remark}\n\\begin{lemma}\\label{rmH7} Suppose that (H1)-(H5) and (H7) hold. Then\n\\begin{equation}\\label{rmH7-1}\nR_{\\lambda}^{B_{1},\\pm}=R^{\\pm}_{\\lambda}+G_{\\lambda}^{{1},\\pm}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}(G_{\\lambda}^{{1},\\mp})^{*}\\,;\n\\end{equation}\n\\begin{equation}\\label{Gz2}\nG^{2}_{z}\\in \\mathscr B(\\mathfrak h_{2} ^{*},L^{2}_{\\varphi}(M))\\,,\\qquad z\\in\\varrho(A_{B_{1}})\\cap\\varrho(A)\\,,\n\\end{equation} \nthe limits\n\\begin{equation}\\label{limG2}\nG_{\\lambda}^{{2},\\pm}:=\\lim_{\\epsilon\\searrow 0}G^{{2}}_{\\lambda\\pm i\\epsilon}\n\\end{equation}\nexist in $\\mathscr B(\\mathfrak h_{2} ^{*},L^{2}_{\\varphi^{-1}}(M))$ for any $\\lambda\\in\\mathbb{R}\\backslash{e}(A_{B_{1}})$ and\n\\begin{equation}\\label{limGB1}\nG_{\\lambda}^{B_{1},\\pm}=G_{\\lambda}^{2,\\pm}+G_{\\lambda}^{1,\\pm}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}\\tau_{1}G_{\\lambda}^{2,\\pm}\\,;\n\\end{equation}\nthe limits \n$$\n\\Lambda_{\\lambda}^{\\!\\mathsf B,\\pm}:=\n\\lim_{\\epsilon\\searrow 0}\\Lambda_{\\lambda\\pm i\\epsilon}^{\\!\\mathsf B}\n$$\nexist in $\\mathscr B(\\mathfrak h_{1}\\oplus \\mathfrak b_{2},\\mathfrak h_{1}^{*}\\oplus \\mathfrak b_{2}^{*})$ and\n\\begin{align}\n\\Lambda_{\\lambda}^{\\!\\mathsf B,\\pm}=&\\begin{bmatrix}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}+\n\\Lambda^{\\!B_{1},\\pm}_{\\lambda}\\tau_{1}G^{2,\\pm}_{\\lambda}\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\tau_{2}G^{1,\\pm}_{\\lambda}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}&\n\\Lambda^{\\!B_{1},\\pm}_{\\lambda}\\tau_{1}G^{2,\\pm}_{\\lambda}\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\\\\n\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\tau_{2}G^{1,\\pm}_{\\lambda}\\Lambda_{\\lambda}^{\\!B_{1},\\pm}&\n\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\n\\end{bmatrix}\\label{LBpm}\\\\\n=&\\left(1+\\begin{bmatrix}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}&0\\\\0&\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\n\\end{bmatrix}\\begin{bmatrix}\\tau_{1}G^{2,\\pm}_{\\lambda}\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\tau_{2}G^{1,\\pm}_{\\lambda}&\n\\tau_{1}G^{2,\\pm}_{\\lambda}\\\\\n\\tau_{2}G^{1,\\pm}_{\\lambda}&0\n\\end{bmatrix}\\,\\right)\\begin{bmatrix}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}&0\\\\0&\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\n\\end{bmatrix}\\,.\\label{LBpm2}\n\\end{align}\n\\end{lemma}\n\\begin{proof} The relation \\eqref{rmH7-1} is an immediate consequence of \\eqref{res1.1} and (H7.1)-(H7.3). By \\eqref{GB1}, \n$$\nG_{z}^{2}=G^{B_{1}}_{z}-G_{z}^{1}\\Lambda^{\\!B_{1}}_{z}\\tau_{1}G_{z}^{2}\n$$\nand \\eqref{Gz2} follows from (H4.2) and (H7.2). Then, Remark \\ref{rem3.6}, (H.5) and (H7.3) entail \n\\eqref{limG2} and \\eqref{limGB1}. Finally, \\eqref{LBpm} and \\eqref{LBpm2} are consequence of \\eqref{LB-new}, \\eqref{LB-new2}, Corollary \\ref{limbb}, (H7.3), Remark \\ref{rem3.6} and (H7.4). \n\\end{proof}\nBefore stating the next results, let us notice the relations \n\\begin{equation}\\label{RR}\n\\left(-R_{\\mu}+z\\right)^{-1}=\\frac1z\\,\\left(1+\\frac1{z}\\,R_{\\mu-\\frac1z}\\right)\\,,\n\\quad \\left(-R_{\\mu}^{\\mathsf B}+z\\right)^{-1}=\\frac1z\\,\\left(1+\\frac1{z}\\,R^{\\mathsf B}_{\\mu-\\frac1z}\\right)\\,,\n\\end{equation}\nTherefore, by (H7.1) and Theorem \\ref{LAP}, the limits \n\\begin{equation}\\label{RRlim1}\n\\left(-R_{\\mu}+(\\lambda\\pm i0)\\right)^{-1}:=\\lim_{\\epsilon\\searrow 0}\\left(-R_{\\mu}+(\\lambda\\pm i\\epsilon)\\right)^{-1}\\,,\\quad \\lambda\\not=0\\,,\\ \\mu-\\frac1\\lambda\\in\\mathbb{R}\\backslash{e}(A)\\,,\n\\end{equation} \n\\begin{equation}\\label{RRlim2}\n\\left(-R_{\\mu}^{\\mathsf B}+(\\lambda\\pm i0)\\right)^{-1}:=\n\\lim_{\\epsilon\\searrow 0}\\left(-R_{\\mu}^{\\mathsf B}+(\\lambda\\pm i\\epsilon)\\right)^{-1}\\,,\n\\quad \\lambda\\not=0\\,,\\ \\mu-\\frac1\\lambda\\in\\mathbb{R}\\backslash{e}(A_{\\mathsf B})\\,,\n\\end{equation}\nexist in $\\mathscr B(L^{2}_{\\varphi}(M),L^{2}_{\\varphi^{-1}}(M))$.\n\\begin{theorem}\\label{BK}\nSuppose that hypotheses (H1)-(H7) hold. Then the strong limits \n\\begin{equation}\\label{WR}\nW_{\\pm}(R^{\\mathsf B}_{\\mu},R_{\\mu})\n:=\\text{s-}\\lim_{t\\to\\pm\\infty}e^{itR^{\\mathsf B}_{\\mu}}e^{-itR_{\\mu}}P^{\\mu}_{ac}\n\\end{equation}\nexist everywhere in $L^{2}(M)$.\nMoreover, for any $\\lambda\\not=0$ such that $\\mu-\\frac1\\lambda\\in \\sigma_{ac}(A)\\cap(\\mathbb{R}\\backslash{e}(A_{\\mathsf B}))$, one has\n\\begin{equation}\\label{S1}\n{\\mathcal S}^{\\mathsf B,\\mu}_{\\lambda}=1-2\\pi i\\,\\mathcal L^{\\mu}_{\\lambda}\n\\Lambda^{\\!\\mathsf B}_{\\mu}\\big(1+G^{*}_{\\mu}\\big(-R_{\\mu}^{\\mathsf B}+(\\lambda+ i0)\\big)^{-1}G_{\\mu}\\Lambda^{\\!\\mathsf B}_{\\mu}\\big)\n(\\mathcal L^{\\mu}_{\\lambda})^{*}\\,,\n\\end{equation}\nwhere\n\\begin{equation}\\label{SR}\n\\mathcal L^{\\mu}_\\lambda: \\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*}\\to(L^{2}(M)_{ac})_{\\mu-\\frac1\\lambda}\\,,\\quad \n\\mathcal L^{\\mu}_{\\lambda}(\\phi_{1}\\oplus\\phi_{2}):=\\frac1\\lambda(FG_{\\mu}(\\phi_{1}\\oplus\\phi_{2}))_{\\mu-\\frac1\\lambda}\\,.\n\\end{equation}\n\\end{theorem} \n\\begin{proof}\nBy \\eqref{resolvent}, one has $R_{\\mu}^{\\mathsf B}-R_{\\mu}=G_{\\mu}\\Lambda^{\\! B}_{\\mu}G^{*}_{\\mu}$ and we can use \\cite[Theorem 4', page 178]{Y} (notice that the maps there denoted by $G$ and $V$ corresponds to our $G_{\\mu}^{*}$ and $\\Lambda^{\\! B}_{\\mu}$ respectively). Let us check that the hypotheses there required are satisfied. Since $G^{*}_{\\mu}\\in \\mathscr B(L^{2}(M),\\mathfrak h_{1}\\oplus\\mathfrak h_{2})$, the operator $G_{\\mu}$ is $|R_{\\mu}|^{1\/2}$-bounded. By (H7.2) and \\eqref{Gz2}, one has $G_{z}\\in\\mathscr B(\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*}, L^{2}_{\\varphi}(M))$ for any $z\\in\\varrho(A_{B_{1}})\\cap\\varrho(A)\\supset[\\lambda_{1},+\\infty)\\ni\\mu$. Therefore, by \\eqref{RRlim1}, \\eqref{RRlim2}, (H7.1), Theorem \\ref{LAP} and (H4), the limits \n$$\n\\lim_{\\epsilon\\searrow 0}\\, G^{*}_{\\mu}(-R _{\\mu}+(\\lambda\\pm i\\epsilon))^{-1}\\,,\n$$\n$$\n\\lim_{\\epsilon\\searrow 0}\\, G^{*}_{\\mu}(-R^{\\mathsf B}_{\\mu}+(\\lambda\\pm i\\epsilon))^{-1}\\,,\n$$\n$$\n\\lim_{\\epsilon\\searrow 0}\\, G^{*}_{\\mu}(-R^{\\mathsf B}_{\\mu}+(\\lambda\\pm i\\epsilon))^{-1}G_{\\mu}\n$$\nexist. Therefore, to get the thesis we need to check the validity of the remaining hypothesis in \n\\cite[Theorem 4', page 178]{Y}: $G^{*}_{\\mu}$ is weakly-$R _{\\mu}$ smooth, i.e., by \\cite[Lemma 2, page 154]{Y}, \n\\begin{equation}\\label{in1.1}\n\\sup_{0<\\epsilon<1}\\epsilon\\,{\\|}G^{*}_{\\mu} (-R _{\\mu}+(\\lambda\\pm i\\epsilon))^{-1}{\\|}_{L^{2}(M),\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}^{2}\\le c_{\\lambda}<+\\infty\\,,\\quad\\text{a.e. $\\lambda$}\\,.\n\\end{equation}\nBy \\eqref{RR}, this is consequence of\n\\begin{equation}\\label{in2.2}\n\\sup_{0<\\epsilon<1}\\epsilon\\,{\\|}G^{*}_{\\mu}R_{\\mu-\\frac1\\lambda\\pm i\\epsilon}{\\|}_{L^{2}(M),\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}^{2}\\le C_{\\lambda}<+\\infty\\,,\\quad\\text{a.e. $\\lambda$}\\,.\n\\end{equation}\nBy \\cite[(3.16)]{JMPA},\n\\begin{align*}\n&\\epsilon\\,{\\|}G_{\\lambda\\pm i\\epsilon}{\\|}_{\\mathfrak h^{*}_{1}\\oplus\\mathfrak h^{*}_{2},L^{2}(M)}^{2}\\\\\n\\le& \n\\frac12\\,(|\\mu-\\lambda|+\\epsilon)\\, {\\|}G_{\\mu}{\\|}_{\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*},L_{\\varphi}^{2}(M)} \\left(\\|G_{\\lambda- i\\epsilon}\\|_{\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*},L_{\\varphi^{-1}}^{2}(M)} +\n\\|G_{\\lambda+ i\\epsilon}\\|_{\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*},L_{\\varphi^{-1}}^{2}(M)} \\right)\\,.\n\\end{align*}\nThen, \\eqref{in2.2} follows from \\eqref{limG1}, \\eqref{limG2} and the equality\n\\begin{align*}\n&{\\|}G^{*}_{\\mu} R _{z}{\\|}_{L^{2}(M),\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}\n={\\|}\\tau R_{\\mu}R _{z}{\\|}_{L^{2}(M),\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}= \n{\\|}\\tau R _{z}R _{\\mu}{\\|}_{L^{2}(M),\\mathfrak h_{1}\\oplus\\mathfrak h_{2}}\\\\\n=&\n{\\|}R _{\\mu}(\\tau R _{z})^{*}{\\|}_{\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*},L^{2}(M)}\\le\n{\\|}R _{\\mu}{\\|}_{L^{2}(M),L^{2}(M)} {\\|}G_{\\bar z}{\\|}_{\\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*},L^{2}(M)}\\,.\n\\end{align*}\nThus, by \\cite[Theorem 4', page 178]{Y}, \nthe limits \\eqref{WR} exist everywhere in $L^{2}(M)$ and the corresponding scattering matrix is given by \\eqref{S1}, where $\\mathcal L^{\\mu}_{\\lambda}\\phi:=(F^{\\mu}G_{\\mu}\\phi)_{\\lambda}=\\frac1\\lambda(FG_{\\mu}\\phi)_{\\mu-\\frac1\\lambda}$. \n\\end{proof}\n\\begin{theorem}\\label{S-matrix} Suppose that hypotheses (H1)-(H7) hold. Then the scattering matrix of the couple $(A_{\\mathsf B},A)$ has the representation\n\\begin{equation}\\label{S-M}\n{\\mathcal S}^{\\mathsf B}_{\\lambda}=1-2\\pi i\\mathcal L_{\\lambda}\\Lambda^{\\!\\mathsf B,+}_{\\lambda}\\mathcal L_{\\lambda}^{*}\\,,\\quad \\lambda\\in\\sigma_{ac}(A)\\cap(\\mathbb{R}\\backslash{e}(A_{\\mathsf B}))\\,,\n\\end{equation}\nwhere $\\mathcal L_\\lambda: \\mathfrak h_{1}^{*}\\oplus\\mathfrak h_{2}^{*}\\to(L^{2}(M)_{ac})_{\\lambda}$ is the $\\mu$-independent linear operator defined by \n$$\n\\mathcal L_{\\lambda}(\\phi_{1}\\oplus\\phi_{2}):=(\\mu-\\lambda)(FG_{\\mu}(\\phi_{1}\\oplus\\phi_{2}))_{\\lambda}\n$$\nand $\\Lambda^{\\!\\mathsf B,+}_{\\lambda}$ is given in \\eqref{LBpm}.\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref {AC}, Theorem \\ref{BK} and by Birman-Kato invariance principle (see e.g. \\cite[Section II.3.3]{BW}), one has\n$$\nW_{\\pm}(A_{\\mathsf B},A)=W_{\\pm}(R^{\\mathsf B}_{\\mu},R_{\\mu})\n$$\nand so\n$$\nS_{\\mathsf B}=S_{\\mathsf B}^{\\mu}\\,.\n$$\nThus, since $(F^{\\mu}u)_{\\lambda}=\\frac1\\lambda(Fu)_{\\mu-\\frac1\\lambda}$, one obtains (see also \\cite[Equation (14), Section 6, Chapter 2]{Y})\n\\begin{equation}\\label{SS}\n{\\mathcal S}^{\\mathsf B}_{\\lambda}={\\mathcal S}^{\\mathsf B,\\mu}_{(-\\lambda+\\mu)^{-1}}\\,.\n\\end{equation}\nBy \\cite[Lemma 4.2]{JMPA}, for any $z\\not=0$ such that $\\mu-\\frac1z\\in \\varrho(A_{\\mathsf B})\\cap\\varrho(A)$, there holds\n$$\n\\Lambda^{\\!\\mathsf B}_{\\mu}\\left(1+G^{*}_{\\mu}\\left(-R_{\\mu}^{\\mathsf B}+z\\right)^{-1}G_{\\mu}\\Lambda^{\\! B}_{\\mu}\\right)=\\Lambda^{\\!\\mathsf B}_{\\mu-\\frac1z}\\,.\n$$\nHence, whenever $z=\\lambda\\pm i\\epsilon$ and $\\mu-\\frac1{\\lambda}\\in\\mathbb{R}\\backslash{e}(A_{\\mathsf B})$, one gets, as $\\epsilon\\downarrow 0$, \n$$\n\\Lambda^{\\!\\mathsf B}_{\\mu}\\big(1+G^{*}_{\\mu}\\left(-R_{\\mu}^{\\mathsf B}+(\\lambda\\pm i0)\\right)^{-1}G_{\\mu}\\Lambda^{\\!\\mathsf B}_{\\mu}\\big)=\\Lambda^{\\!\\mathsf B,\\pm}_{\\mu-\\frac1{\\lambda}}\\,.\n$$\nThe proof is then concluded by Theorem \\ref{BK}, by \\eqref{SS} and by setting \n$\\mathcal L_{\\lambda}:=\\mathcal L^{\\mu}_{{(-\\lambda+\\mu)^{-1}}}$. The operator $\\mathcal L_{\\lambda}$ is $\\mu$-independent by invariance principle (see the proof in \\cite[Corollary 4.3]{JMPA} for an explicit check).\n\\end{proof}\n\\begin{remark}\nBy \\eqref{LBpm}, \n$$\n\\Lambda^{\\!\\mathsf B,\\pm}_{\\lambda}=\\begin{bmatrix}\\Lambda^{\\!B_{1},\\pm}_{z}&0\\\\0&0\\end{bmatrix}+\\widetilde \\Lambda^{\\!\\mathsf B,\\pm}_{\\lambda}\\,,\n$$\nwhere\n$$\n\\widetilde \\Lambda^{\\!\\mathsf B,\\pm}_{\\lambda}:=\n\\begin{bmatrix}\n\\Lambda^{\\!B_{1},\\pm}_{\\lambda}\\tau_{1}G^{2,\\pm}_{\\lambda}\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\tau_{2}G^{1,\\pm}_{\\lambda}\\Lambda^{\\!B_{1},\\pm}_{\\lambda}&\n\\Lambda^{\\!B_{1},\\pm}_{\\lambda}\\tau_{1}G^{2,\\pm}_{\\lambda}\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\\\\n\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\\tau_{2}G^{1,\\pm}_{\\lambda}\\Lambda_{\\lambda}^{\\!B_{1},\\pm}&\n\\widehat \\Lambda^{\\mathsf B,\\pm}_{\\lambda}\n\\end{bmatrix}\\,.\n$$\nTherefore, defining \n$$\n\\mathcal L^{1}_{\\lambda}\\phi_{1}:=\\mathcal L_{\\lambda}(\\phi_{1}\\oplus 0)\\,,\n$$ \none gets\n$$\n{\\mathcal S}^{\\mathsf B}_{\\lambda}={\\mathcal S}^{B_{1}}_{\\lambda}-2\\pi i\\mathcal L_{\\lambda}\\widetilde\\Lambda^{\\!\\mathsf B,+}_{\\lambda}\\mathcal L_{\\lambda}^{*}\\,,\n$$\nwhere\n\\begin{equation}\\label{SB1}\n{\\mathcal S}^{B_{1}}_{\\lambda}=1-2\\pi i\\mathcal L^{1}_{\\lambda}\\widetilde\\Lambda^{\\!B_{1},+}_{\\lambda}(\\mathcal L^{1}_{\\lambda})^{*}\n\\end{equation}\nis the scattering matrix relative to the couple $(A_{B_{1}},A)$. Moreover, in the case $B_{1}=0$, \ndefining \n$$\n\\mathcal L^{2}_{\\lambda}\\phi_{2}:=\\mathcal L_{\\lambda}(0\\oplus\\phi_{2})\\,,\n$$ \none gets the following representation formula for the scattering couple $(A_{B_{0},B_{2}},A)$ (compare with \\cite[Corollary 4.3]{JMPA}):\n$$\n{\\mathcal S}^{B_{0},B_{2}}_{\\lambda}=1-2\\pi i\\mathcal L^{2}_{\\lambda}\\Lambda^{\\!B_{0},B_{2},+}_{\\lambda}(\\mathcal L^{2}_{\\lambda})^{*}\\,.\n$$\nLet us further notice that, whenever $A$ is the free Laplacian in $L^{2}(\\mathbb{R}^{3})$ and $B_{1}$ corresponds to a perturbation by a regular potential as in Section 5 below, then \\eqref{SB1} gives the usual formula for the scattering matrix for a short-range potential (see, e.g., \\cite[Section 8]{Y-LNM}).\n\\end{remark}\n\n\\section{Kato-Rellich perturbations and their layers potentials}\n\n\\subsection{\\label{Sec_V}Potential perturbations}\n\nIn this section we suppose that the real-valued potential $\\mathsf v$ is of Kato-Rellich type, i.e., $\\mathsf v\\in L^{2}(\\mathbb{R}^{3})+L^{\\infty}(\\mathbb{R}^{3})$, equivalently,\n\\begin{equation}\n\\mathsf v=\\mathsf v_{2}+\\mathsf v_{\\infty} \\,,\\qquad \\mathsf v_{2}\\in L^{2}( \\mathbb{R}^{3}) \\,,\\qquad \\mathsf v_{\\infty}\\in L^{\\infty}\n(\\mathbb{R}^{3}) \\,. \\label{K-R}%\n\\end{equation}\nWe use the same simbol $\\mathsf v$ to denote both the potential function and the corresponding multiplication operator $u\\mapsto \\mathsf v u$. \n\\par\nGiven $\\Omega\\subset\\mathbb{R}^{3}$, open and bounded with a Lipschitz boundary $\\Gamma$, we define $H^{s}(\\mathbb{R}^{3}\\backslash\\Gamma)\\hookleftarrow H^{s}(\\mathbb{R}^{3})$ by \n$$\nH^{s}(\\mathbb{R}^{3}\\backslash\\Gamma):=H^{s}(\\Omega)\\oplus H^{s}(\\Omega_{\\rm ex})\\,,\\qquad s\\ge 0\\,.\n$$\nWe refer to \\cite[Chapter 3]{McL} for the definition of the Sobolev spaces $H^{s}(\\mathbb{R}^{3})$, $H^{s}(\\Omega)$ and $H^{s}(\\Gamma)$. One has \n$$\nH^{s}(\\mathbb{R}^{3}\\backslash\\Gamma)=H^{s}(\\mathbb{R}^{3})\\,,\\qquad 0\\le s<1\/2\\,.\n$$\nSince (see \\cite[Theorems 3.29 and 3.30]{McL}),\n$$\nH^{s}( \\mathcal O) ^{\\ast}\n=H_{\\overline{\\mathcal O}}^{-s}( \\mathbb{R}^{3}) \\,,\\qquad\ns\\in\\mathbb{R}\\,,\n$$\n$H_{\\overline{\\mathcal O}}^{-s}( \\mathbb{R}^{3})$ denoting the set of distributions $H^{-s}( \\mathbb{R}^{3})$ with support in ${\\overline{\\mathcal O}}$,\none has\n\\begin{equation}\nH^{s}( \\mathbb{R}^{3}\\backslash\\Gamma)^{\\ast}=H^{s}( \\Omega)^{\\ast}\n\\oplus H^{s}( \\mathbb{R}^{3}\\backslash\\overline\\Omega)^{\\ast}=\nH^{-s}_{\\overline\\Omega}(\\mathbb{R}^{3})\\oplus H^{-s}_{\\Omega^{c}}(\\mathbb{R}^{3})\\hookrightarrow H^{-s}(\\mathbb{R}^{3})\n\\,.\n\\label{dual}%\n\\end{equation}\nLet us notice that\n\\begin{equation}\\label{B}\n\\mathscr B(H^{s}( \\mathbb{R}^{3}\\backslash\\Gamma),H^{t}( \\mathbb{R}^{3}\\backslash\\Gamma)^{\\ast})\n\\hookrightarrow\n\\mathscr B(H^{s}( \\mathbb{R}^{3}),H^{-t}( \\mathbb{R}^{3}))\\,,\\qquad s,t\\ge 0\\,,\n\\end{equation}\nand\n\\begin{equation}\\label{B*}\n\\mathscr B(H^{-s}( \\mathbb{R}^{3}),H^{t}( \\mathbb{R}^{3}))\\hookrightarrow\\mathscr B(H^{s}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{t}( \\mathbb{R}^{3}\\backslash\\Gamma))\n\\,,\\qquad s,t\\ge 0\\,.\n\\end{equation}\n\\begin{lemma}\\label{v}\n\\begin{equation}\\label{v-sob}\n\\mathsf v\\in{\\mathscr B}( H^{1+s}( \\mathbb{R}^{3}\\backslash\\Gamma) ,H^{1-s}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,,\\qquad -1\\le s\\le 1 \\,. \n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nGiven $u= u_{\\rm in}\\oplus u_{\\rm ex}\\in H^{2}( \\mathbb{R}^{3}\\backslash\\Gamma) \n$ one has%\n\\[\n\\|\\mathsf v_{\\infty}u\\| _{L^{2}(\\mathbb{R}^{3})}\\leq\\|\\mathsf v\\|_{L^{\\infty}(\\mathbb{R}^{3})}\\| u\\| _{L^{2}(\\mathbb{R}^{3})}\\leq\\|\\mathsf v\\|_{L^{\\infty}(\\mathbb{R}^{3})}\\| u\\|_{H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma) }\\,.\n\\]\nand\n\\begin{align*}\n\\|\\mathsf v_{2}u\\|_{L^{2}(\\mathbb{R}^{3})}=&\\|\\mathsf v_{2}\\|_{L^{2}(\\Omega)}\\|u_{\\rm in}\\|_{L^{\\infty}(\\Omega)}+\\|\\mathsf v_{2}\\|_{L^{2}(\\mathbb{R}^{3}\\backslash\\overline\\Omega)}\n\\|u_{\\rm ex}\\|_{L^{\\infty}(\\mathbb{R}^{3}\\backslash\\overline\\Omega)}\\\\\n\\lesssim &\n\\|\\mathsf v_{2}\\|_{L^{2}(\\Omega)}\\|u_{\\rm in}\\|_{H^{2}(\\Omega)}+\n\\|\\mathsf v_{2}\\|_{L^{2}(\\mathbb{R}^{3}\\backslash\\overline\\Omega)}\n\\|u_{\\rm ex}\\|_{H^{2}(\\mathbb{R}^{3}\\backslash\\overline\\Omega)}\\\\\n\\lesssim &\\|\\mathsf v_{2}\\|_{L^{2}(\\mathbb{R}^{3})}\n\\|u\\|_{H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)}\\,.\n\\end{align*}\nHence $\\mathsf v\\in{\\mathscr B}( H^{2}( \\mathbb{R}^{3}\\backslash\\Gamma) ,L^{2}( \\mathbb{R}^{3}))$. Then, for any $u,v\\in H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)$, one has\n\\begin{align*}\n&\\big|\\langle \\mathsf v u,v\\rangle_{H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)}\\big|\n=\\big|\\langle \\mathsf v u,v\\rangle_{L^{2}(\\mathbb{R}^{3})}\\big|\\\\\n=&\n\\big|\\langle u,\\mathsf v v\\rangle_{L^{2}(\\mathbb{R}^{3})}\\big|\n\\le \\|\\mathsf v\\|_{H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma),L^{2}(\\mathbb{R}^{3})}\\|u\\|_{L^{2}(\\mathbb{R}^{3})}\n\\|v\\|_{H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)}\n\\end{align*}\nand so $u\\mapsto\\mathsf v u$ extends to a map in $\\mathscr B(L^{2}( \\mathbb{R}^{3}),H^{2}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*} )$. The proof is then concluded by interpolation.\n\\end{proof}\nIn the following, $R_{z}$ denotes the resolvent of the free Laplacian, i.e.,\n\\begin{equation}\\label{free}\nR_{z}:=\\left( -\\Delta+z\\right) ^{-1}\\in\\mathscr B(H^{s}(\\mathbb{R}^{3}),H^{s+2}(\\mathbb{R}^{3}))\\,,\\quad s\\in\\mathbb{R}\\,.\n\\end{equation}\nSince $\\mathsf v$ is of Rellich-Kato type, one has (see, e.g., \\cite[Section 3, $\\S$5, Chap. V]{Kato}): \n\\begin{theorem}\\label{KR}\n$\\Delta+\\mathsf v:H^{2}( \\mathbb{R}^{3})\\subset L^{2}( \\mathbb{R}^{3})\\to L^{2}( \\mathbb{R}^{3}) $ is\nself-adjoint and semi-bounded from above. Moreover, for $z\\in\\mathbb{C}$ sufficiently far away from $[0,+\\infty)$, $\\|\\mathsf v R_{z}\\|_{L^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3})}<1$, \nand\n\\begin{equation}\nR_{z}^{\\mathsf v}:=(-(\\Delta+\\mathsf v)+z)^{-1}=R_{z}+R_{z}( 1- \\mathsf v R_{z}) ^{-1}\\mathsf v R_{z}\n\\,,\\label{Rvz}\n\\end{equation}\n\\begin{equation}\n( 1-\\mathsf v R_{z}) ^{-1}=\\sum_{k=0}^{+\\infty}\\left( \\mathsf v R_{z}\\right) ^{k}\n\\in{\\mathscr B}( L^{2}(\\mathbb{R}^{3}))\\,.\n\\label{L-v-est}%\n\\end{equation}\n\\end{theorem}\n\\begin{remark}\\label{tpt} Let us notice that the Kato-Rellich theorem could be obtained by Corollary \\ref{cor1} by taking $\\tau_{1}u:=u$ and $B_{1}=\\mathsf v$. Hence, \\eqref{Rvz} holds for any $z$ in $\\varrho(\\Delta+\\mathsf v)\\cap\\mathbb{C}\\backslash(-\\infty,0]$ and $( 1+\\mathsf v R_{z}) ^{-1}\\in {\\mathscr B}( L^{2}(\\mathbb{R}^{3}))$ there.\n\\end{remark}\n\\begin{remark}\\label{sa} \nBy \\eqref{free}, \\eqref{Rvz}, \\eqref{L-v-est}, \\eqref{v-sob} and \\eqref{B}, one has $R^{\\mathsf v}_{\\bar z}\\in\n\\mathscr B(L^{2}( \\mathbb{R}^{3}),H^{2}( \\mathbb{R}^{3}))$ and hence $(R_{\\bar z}^{\\mathsf v})^{*}\\in \\mathscr B(H^{-2}( \\mathbb{R}^{3}),L^{2}( \\mathbb{R}^{3}))$. Since $(\\Delta+\\mathsf v)$ is self-adjoint in $L^{2}(\\mathbb{R}^{3})$, $(R_{\\bar z}^{\\mathsf v})^{*}|L^{2}(\\mathbb{R}^{3})=\nR^{\\mathsf v}_{z}$. Therefore, $R^{\\mathsf v}_{z}:L^{2}(\\mathbb{R}^{3})\\subset H^{-2}(\\mathbb{R}^{3})\\to L^{2}(\\mathbb{R}^{3})$ extends to an operator in $\\mathscr B(H^{-2}( \\mathbb{R}^{3}),L^{2}( \\mathbb{R}^{3}))$ which, by abuse of notation, we still denote by $R_{ z}^{\\mathsf v}$ and which coincides with $(R_{\\bar z}^{\\mathsf v})^{*}$. Then, by interpolation, one gets\n\\begin{equation}\\label{Rvz-int}\nR_{z}^{\\mathsf v}\\in \\mathscr B(H^{s-1}( \\mathbb{R}^{3}),H^{s+1}( \\mathbb{R}^{3}))\\,,\\qquad -1\\le s\\le 1\\,.\n\\end{equation}\n\\end{remark}\n\\begin{remark} By \\eqref{Rvz}, \n\\begin{equation}\\label{btr}\n( 1- \\mathsf v R_{z}) ^{-1}\\mathsf v=(-\\Delta+z)R^{\\mathsf v}_{z}(-\\Delta+z)-(-\\Delta+z)\\,.\n\\end{equation}\nHence, by \\eqref{Rvz-int}, $( 1- \\mathsf v R_{z}) ^{-1}\\mathsf v\\in \\mathscr B(H^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3}))$ extends to a map \n\\begin{equation}\\label{L-int}\n\\Lambda^{\\!\\mathsf v}_{z}\\in \\mathscr B(H^{s+1}( \\mathbb{R}^{3}),H^{s-1}( \\mathbb{R}^{3}))\\,,\n\\qquad -1\\le s\\le 1 \n\\end{equation}\nWith such a notation, $R_{z}^{\\mathsf v}$ in \\eqref{Rvz-int} has the representation\n\\begin{equation}\\label{RF-int}\nR_{z}^{\\mathsf v}=R_{z}+R_{z}\\Lambda^{\\!\\mathsf v}_{z}R_{z}\\,,\\qquad \n\\Lambda^{\\!\\mathsf v}_{z}|H^{2}(\\mathbb{R}^{3})=( 1- \\mathsf v R_{z}) ^{-1}\\mathsf v\\,.\n\\end{equation}\n\\end{remark}\n\\begin{remark}\\label{Lt} Since $\\|R_{z}\\mathsf v\\|_{L^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3})}=\\|(R_{z}\\mathsf v)^{*}\\|_{L^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3})}=\\|\\mathsf v R_{\\bar z}\\|_{L^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3})}\n<1$ whenever $z\\in\\mathbb{C}$ is sufficiently far away from $[0,+\\infty)$, one has\n\\begin{equation}\\label{Lt2}\n( 1-R_{z}\\mathsf v) ^{-1}=\\sum_{k=0}^{+\\infty}\\left( R_{z}\\mathsf v\\right) ^{k}\n\\in{\\mathscr B}( L^{2}(\\mathbb{R}^{3}))\n\\end{equation}\nand\n\\begin{equation}\\label{Lt-2}\n\\mathsf v( 1- R_{z}\\mathsf v) ^{-1}\\in\\mathscr B(L^{2}(\\mathbb{R}^{3}),H^{-2}(\\mathbb{R}^{3}))\\,.\n\\end{equation}\nThen, \n$$\n\\big(( 1- R_{z}\\mathsf v) ^{-1}\\mathsf v\\big)^{*}=\n\\big(\\mathsf v( 1- R_{z}\\mathsf v)^{-1}\\big)^{*}=\\mathsf v(( 1- R_{z}\\mathsf v)^{*})^{-1}=\\mathsf v( 1- R_{\\bar z}\\mathsf v) ^{-1\n$$\nand so \n$$\n\\mathscr B(H^{-2}( \\mathbb{R}^{3}),L^{2}( \\mathbb{R}^{3}))\\ni(R^{\\mathsf v}_{z})^{*}=R_{\\bar z}+R_{\\bar z}\\mathsf v( 1- R_{\\bar z}\\mathsf v)^{-1}R_{\\bar z}=\nR^{\\mathsf v}_{\\bar z}=R_{\\bar z}+R_{\\bar z}\\Lambda_{\\bar z}^{\\!\\mathsf v}R_{\\bar z}\\,.\n$$\nTherefore \n\\begin{equation}\\label{LtL2}\n\\Lambda_{z}^{\\!\\mathsf v}|L^{2}(\\mathbb{R}^{3})=\\mathsf v( 1- R_{z}\\mathsf v) ^{-1}\\,.\n\\end{equation}\n\\end{remark}\n\\begin{lemma}\n\\begin{equation}\\label{Lvz-int}\n\\Lambda_{z}^{\\!\\mathsf v}\\in{\\mathscr B}( H^{1+s}(\\mathbb{R}^{3}\\backslash\\Gamma),\nH^{1-s}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})\n\\,, \\qquad -1\\le s\\le 1\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} By Lemma \\ref{v} and by \\eqref{L-v-est}, one has $\\Lambda_{z}^{\\!\\mathsf v}=( 1+\\mathsf v R_{z}) ^{-1}\\mathsf v\\in{\\mathscr B}( H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma),L^{2}(\\mathbb{R}^{3}))$. By Lemma \\ref{v}, \\eqref{Lt2} and \\eqref{LtL2},\n$\\Lambda_{z}^{\\!\\mathsf v}\\in{\\mathscr B}( L^{2}(\\mathbb{R}^{3}), H^{2}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})$.\nThe proof is then concluded by interpolation.\n\\end{proof}\nBy $H^{1-s}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\hookrightarrow H^{s-1}(\\mathbb{R}^{3})$ and \\eqref{free} one has\n\\begin{corollary}\n\\begin{equation}\\label{RLvz}\nR_{z}\\Lambda_{z}^{\\!\\mathsf v}\\in{\\mathscr B}( H^{s}(\\mathbb{R}^{3}\\backslash\\Gamma),\nH^{s}(\\mathbb{R}^{3}))\n\\,, \\qquad 0\\le s\\le 2\\,.\n\\end{equation}\n\n\\end{corollary}\nIn later proofs we will need the estimate provided in the following:\n\\begin{lemma}\nThere exist $c_{1}>0$, $c_{2}>0$ such that, for any $u\\equiv u_{\\rm in}\\oplus u_{\\rm ex}\\in H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)$ and for any $\\varepsilon>0$,\nthere holds\n\\begin{equation}\n\\big|\\langle \\mathsf v u,u\\rangle _{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}\\big| \\leq\nc_{1}\\epsilon\\left(\\|\\nabla u_{\\rm in}\\|^{2}_{L^{2}(\\Omega_{\\rm in})}+\\|\\nabla u_{\\rm ex}\\|^{2}_{L^{2}(\\Omega_{\\rm ex})}\\right)+c_{2}(1+\\epsilon^{-3})\\|u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\,. \\label{V_est}%\n\\end{equation}\n\\end{lemma}\n\\begin{proof} By $H^{1}(\\Omega_{\\rm in\/\\rm ex}) \\hookrightarrow H^{3\/4}(\\Omega_{\\rm in\/\\rm ex}) \\hookrightarrow L^{4}(\\Omega_{\\rm in\/\\rm ex})$, by the Gagliardo-Niremberg \ninequalities (see \\cite{BM} for the interior case and \\cite{CM} for the exterior one)\n$$\n\\|u_{\\rm in}\\|_{L^{4}(\\Omega_{\\rm in})}\\lesssim\\|u_{\\rm in}\\|_{H^{3\/4}(\\Omega_{\\rm in})}\\lesssim\\|u_{\\rm in}\\|^{3\/4}_{H^{1}(\\Omega_{\\rm in})}\\|u_{\\rm in}\\|^{1\/4}_{L^{2}(\\Omega_{\\rm in})}\\,,\n$$ \n$$\n\\|u_{\\rm ex}\\|_{L^{4}(\\Omega_{\\rm ex})}\\lesssim\\|\\nabla u_{\\rm ex}\\|^{3\/4}_{L^{2}(\\Omega_{\\rm ex})}\\|u_{\\rm ex}\\|^{1\/4}_{L^{2}(\\Omega_{\\rm ex})}\n$$ \nand by Young's inequality\n$$\nxy\\le \\frac1{\\alpha}\\left({\\epsilon}\\ x^{\\alpha}+{(\\alpha-1)}\\,{\\epsilon^{-1\/(\\alpha-1)}}\n\\ y^{\\frac{\\alpha-1}{\\alpha}}\\right)\\,,\\qquad x,y,\\epsilon>0,\\, \\alpha>1\\,,\n$$\none gets\n$$\n\\|u\\|_{L^{4}(\\mathbb{R}^{3})}^{2}\\lesssim\n\\epsilon\\left(\\|\\nabla u_{\\rm in}\\|^{2}_{L^{2}(\\Omega_{\\rm in})}+\\|u\\|^{2}_{L^{2}(\\Omega_{\\rm in})}+\\|\\nabla u_{\\rm ex}\\|^{2}_{L^{2}(\\Omega_{\\rm ex})}\\right)+\\frac{1}{3}\\,\\epsilon^{-3}\\|u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\,.\n$$\nThe proof is then concluded by\n$$\n\\big|\\langle \\mathsf v u,u\\rangle _{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}\\big| \\leq\n\\|\\mathsf v_{2}\\|_{L^{2}(\\mathbb{R}^{3})}\\|u\\|^{2}_{L^{4}(\\mathbb{R}^{3})}+\n\\|\\mathsf v_{\\infty}\\|_{L^{\\infty}(\\mathbb{R}^{3})}\\|u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\,.\n$$\n\\end{proof}\n\\begin{lemma}\\label{vH-1} For any $z\\in\\mathbb{C}$ sufficiently far away from $[0,+\\infty)$, one has \n$$\\|\\mathsf v R_{z}\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\!,H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}<1$$ and so\n\\begin{equation}\n( 1-\\mathsf v R_{z}) ^{-1}=\\sum_{k=0}^{+\\infty}\\left( \\mathsf v R_{z}\\right) ^{k}\n\\in{\\mathscr B}( H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} By \\eqref{V_est} and by the polarization identity, for any $u\\in H^{2}(\\mathbb{R}^{3})$ and $v$ in $H^{1}(\\mathbb{R}^{3})$ one has\n\\begin{align*}\n\\big|\\langle \\mathsf v u,v\\rangle_{L^{2}(\\mathbb{R}^{3})}\\big|\n \\leq&\n\\frac14\\Big(c_{1}\\epsilon\\,\\big|\\langle \\nabla u,\\nabla v\\rangle _{L^{2}(\\mathbb{R}^{3})}\\big|\n+c_{2}(1+\\epsilon^{-3})\\big|\\langle u,v\\rangle _{L^{2}(\\mathbb{R}^{3})}\\big|\\Big) \\\\\n =&\n\\frac14\\Big(c_{1}\\epsilon\\,\\big|\\langle -\\Delta u,v\\rangle _{L^{2}(\\mathbb{R}^{3})}\\big|\n+c_{2}(1+\\epsilon^{-3})\\big|\\langle u,v\\rangle _{L^{2}(\\mathbb{R}^{3})}\\big|\\Big)\\,. \n\\end{align*}\nBy the density of $L^{2}(\\mathbb{R}^{3})$ in $H^{1}(\\mathbb{R}^{3})$, the above inequality (here we refer to the second line) holds for any $v\\in L^{2}(\\mathbb{R}^{3})$. Then, by considering the supremum over the set of functions in $H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)$ with unitary norm, one gets\n\\begin{align*}\n\\|\\mathsf v u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}\n\\le&\\frac14\\Big( c_{1}\\epsilon\\,\\|-\\Delta u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}+\n\\big(c_{1}\\epsilon\\,|z|+c_{2}(1+\\epsilon^{-3})\\big)\\|u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}\\Big)\\\\\n\\le&\\frac14\\Big( c_{1}\\epsilon\\,\\|(-\\Delta+z) u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}+\n\\big(c_{1}\\epsilon\\,|z|+c_{2}(1+\\epsilon^{-3})\\big)\\|u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}\\Big)\\,.\n\\end{align*}\nTherefore\n$$\n\\|\\mathsf v R_{z}u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}\\le\n\\frac14\\Big( c_{1}\\epsilon\\,\\|u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}+\n\\big(c_{1}\\epsilon\\,|z|+c_{2}(1+\\epsilon^{-3})\\big)\\|R_{z}u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}\\Big)\\,.\n$$\nTo conclude the proof it suffices to show that \n$$\n\\|R_{z}\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\!,H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}}\\lesssim{d^{-\\gamma}_{z}}\\,,\n$$\nequivalently\n\\begin{equation}\\label{eqv}\n\\|R_{z}\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma),H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}\\lesssim\n{d^{-\\gamma}_{z}}\\,,\n\\end{equation}\nwhere $\\gamma>0$ and $d_{z}$ is the distance of $z$ from $[0,+\\infty)$.\n\\par\nLet $u\\equiv u_{\\rm in}\\oplus u_{\\rm ex}\\in H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)$; then $u= 1_{\\Omega_{\\rm in}}\\widetilde u_{\\rm in}+1_{\\Omega_{\\rm ex}}\\widetilde u_{\\rm ex}$, where $\\widetilde u_{\\rm in\/\\rm ex}\\in H^{1}(\\mathbb{R}^{3})$ is such that $\\widetilde u_{\\rm in\/\\rm ex}|\\Omega_{\\rm in\/\\rm ex}=u_{\\rm in\/\\rm ex}$. One has\n\\begin{align*}\n&\\|R_{z}u\\|^{2}_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}=\\|R_{z}u\\|^{2}_{H^{1}(\\mathbb{R}^{3})}\n=\\|R_{z}u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}+\n \\|\\nabla R_{z}u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\n =\\|R_{z}u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}+\n \\|R_{z}\\nabla u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\n\\\\\n\\le&\\|R_{z}u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}+\n\\|R_{z}(1_{\\Omega_{\\rm in}}\\nabla\\widetilde u_{\\rm in}+1_{\\Omega_{\\rm ex}}\\nabla\\widetilde u_{\\rm ex}\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\n+\\|R_{z}(\\widetilde u_{\\rm in}\\nabla 1_{\\Omega_{\\rm in}}+\\widetilde u_{\\rm ex}\\nabla 1_{\\Omega_{\\rm ex}}\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\\\\n\\le&\\|R_{z}\\|^{2}_{L^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3})}\\|u\\|^{2}_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}\n+\\|S\\!L_{z}\\nu[\\gamma_{0}]u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\,,\n\\end{align*}\nwhere $\\nu$ is the exterior normal to $\\Gamma$ and $S\\!L_{z}$ is the single-layer operator, i.e., $S\\!L_{z}:=(\\gamma_{0}R_{\\bar z})^{*}=R_{z}\\gamma_{0}^{*}$ and $[\\gamma_{0}]$ denotes the jump of the Dirichlet trace across $\\Gamma$. Here we made use the distributional derivative of $\\nabla 1_{\\Omega_{\\rm in\/\\rm ex}}$, which, by the divergence theorem, gives, for any test function $\\varphi$,\n$$\n\\nabla 1_{\\Omega_{\\rm in\/\\rm ex}}(\\varphi)=\\mp\\int_{\\Omega_{\\rm in\/\\rm ex}}\\nabla\\varphi(x)\n\\,dx=\\mp\\int_{\\Gamma}\\nu(x)\\gamma_{0}\\varphi(x)\\,d\\sigma(x)=\\mp\\gamma_{0}^{*}\\nu\\gamma_{0}\\varphi\\,.\n$$\nSince \n$$\n\\|S\\!L_{z}\\nu[\\gamma_{0}]\\|_{L^{2}(\\mathbb{R}^{3})}\\le\n\\|S\\!L_{z}\\|_{H^{1\/2}(\\Gamma),L^{2}(\\mathbb{R}^{3})}\n\\|[\\gamma_{0}]\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma),H^{1\/2}(\\Gamma)}\\|u\\|_{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}\\,,\n$$\n\\begin{align*}\n&\\|S\\!L_{z}\\|_{H^{1\/2}(\\Gamma),L^{2}(\\mathbb{R}^{3})}\\le\\|S\\!L_{z}\\|_{H^{-1\/2}(\\Gamma),L^{2}(\\mathbb{R}^{3})}\n\\|S\\!L_{\\bar z}^{*}\\|_{L^{2}(\\mathbb{R}^{3}),H^{1\/2}(\\Gamma)}\\\\\n=&\n\\|\\gamma_{0}R_{z}\\|_{L^{2}(\\mathbb{R}^{3}),H^{1\/2}(\\Gamma)}\n\\le\\|\\gamma_{0}\\|_{H^{1}(\\mathbb{R}^{3}),H^{1\/2}(\\Gamma)}\n\\|R_{z}\\|_{L^{2}(\\mathbb{R}^{3}),H^{1}(\\mathbb{R}^{3})}\n\\end{align*}\nand\n$$\n\\|R_{z}\\|_{L^{2}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3})}\\le d_{z}^{-1}\\,,\n\\qquad\\|R_{z}\\|_{L^{2}(\\mathbb{R}^{3}),H^{1}(\\mathbb{R}^{3})}\\le d_{z}^{-1\/2}\\,,\n$$\n\\eqref{eqv} follows and the proof is concluded.\n\\end{proof}\n\\begin{remark} By Lemma \\ref{vH-1}, \n$$\n\\Lambda^{\\!\\mathsf v}_{z}|H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)=( 1- \\mathsf v R_{z}) ^{-1}\\mathsf v\\,.\n$$\n\\end{remark}\n\\begin{remark} By the same kind of proof (indeed a shorter one) as in Lemma \\ref{vH-1}, one gets \n$$\n( 1-\\mathsf v R_{z}) ^{-1}=\\sum_{k=0}^{+\\infty}\\left( \\mathsf v R_{z}\\right) ^{k}\n\\in{\\mathscr B}( H^{-1}(\\mathbb{R}^{3}))\n$$\nand, by \\eqref{v-sob}, $\\mathsf v\\in{\\mathscr B}( H^{1}(\\mathbb{R}^{3}), H^{-1}(\\mathbb{R}^{3}))$. Therefore, $( 1- \\mathsf v R_{z}) ^{-1}\\mathsf v\\in {\\mathscr B}( H^{1}(\\mathbb{R}^{3}), H^{-1}(\\mathbb{R}^{3}))$ and, by \\eqref{L-int} and \\eqref{RF-int},\n\\begin{equation}\\label{LvHs1}\n\\Lambda^{\\!\\mathsf v}_{z}|H^{s}(\\mathbb{R}^{3})=( 1- \\mathsf v R_{z}) ^{-1}\\mathsf v\\,,\\qquad 1\\le s\\le 2\\,.\n\\end{equation}\nBy duality, similarly to Remark \\ref{Lt}, $( 1-R_{z}\\mathsf v) ^{-1}\\in{\\mathscr B}( H^{1}(\\mathbb{R}^{3}))$ and \n\\eqref{LtL2} improves to\n\\begin{equation}\\label{LvHs2}\n\\Lambda^{\\!\\mathsf v}_{z}|H^{s}(\\mathbb{R}^{3})=\\mathsf v( 1- \\mathsf v R_{z}) ^{-1}\\,,\\qquad 0\\le s\\le 1\\,.\n\\end{equation}\n\\end{remark}\n\\subsection{\\label{Sec_Layer}Boundary layer operators.}\nWe introduce the interior\/exterior Dirichlet and Neumann trace operators\n$$\n\\gamma_{0}^{\\rm in\/\\rm ex}:H^{s+1\/2}(\\Omega_{\\rm in\/ex})\\to B_{2,2}^{s}(\\Gamma)\\,,\\qquad s>0\\,, \n$$\n$$\n\\gamma_{1}^{\\rm in\/\\rm ex}:H^{s+3\/2}(\\Omega_{\\rm in\/ex})\\to B_{2,2}^{s}(\\Gamma)\\,,\\qquad s>0\\,,\n$$\nwhere $\\Omega_{\\rm in}\\equiv\\Omega$ and $\\Omega_{\\rm ex}:=\\Omega_{\\rm ex}$. The Besov-like trace spaces $B_{2,2}^{s}( \\Gamma ) $ \nidentify with $H^{s}(\\Gamma) $ when $|s|\\le k+1$ and\n$\\Gamma$ is of class $\\mathcal{C}^{k,1}$ (see \\cite{JoWa}). Then, we define the bounded linear operators \n\\begin{equation}\\label{g0}\n\\gamma_{0}:H^{s+1\/2}(\\mathbb{R}^{3}\\backslash\\Gamma)\\to B_{2,2}^{s}(\\Gamma)\\,,\\quad\\gamma_{0}u:=\\frac12\\,(\\gamma_{0}^{\\rm in}(u|\\Omega_{\\rm in})+\\gamma_{0}^{\\rm ex}(u|\\Omega_{\\rm ex}))\\,,\\qquad s>0\\,,\n\\end{equation}\n\\begin{equation}\\label{g1}\n\\gamma_{1}:H^{s+3\/2}(\\mathbb{R}^{3}\\backslash\\Gamma)\\to B_{2,2}^{s}(\\Gamma)\\,,\\quad\\gamma_{1}u:=\\frac12\\,(\\gamma_{1}^{\\rm in}(u|\\Omega_{\\rm in})+\\gamma_{1}^{\\rm ex}(u|\\Omega_{\\rm ex}))\\,,\\qquad s>0\\,.\n\\end{equation}\nThe corresponding trace jump bounded operators are defined by\n\\begin{equation}\n[\\gamma_{0}]:H^{s+1\/2}( \\mathbb{R}^3\\backslash\\Gamma ) \\rightarrow B_{2,2}^{s}(\\Gamma)\\,,\\quad[\n\\gamma_{0}]u:=\\gamma_{0}^{\\-}(u|\\Omega_{\\rm in})-\\gamma_{0}^{\\+}(u|\\Omega_{\\rm ex})\\,,\n\\end{equation}%\n\\begin{equation}\n[\\gamma_{1}]:H^{s+3\/2}( \\mathbb{R}^3\\backslash\\Gamma )\\rightarrow B_{2,2}^{s}(\\Gamma)\\,,\\quad[\n\\gamma_{1}]u:=\\gamma_{1}^{\\-}(u|\\Omega_{\\rm in})-\\gamma_{1}^{\\+}(u|\\Omega_{\\rm ex})\\,.\n\\end{equation}\nBy \\cite[Lemma 4.3]{McL}, the trace maps $\\gamma_{1}^{\\-\/\\+}$ can be extended to the spaces $$H^{1}_{\\Delta}(\\Omega_{\\-\/\\+}):=\\{u_{\\-\/\\+}\\in H^{1}(\\Omega_{\\-\/\\+}):\\Delta_{\\Omega_{\\-\/\\+}}u_{\\-\/\\+}\\in L^{2}(\\Omega_{\\-\/\\+})\\}$$ \nas $H^{-1\/2}(\\Gamma)$-valued bounded operators:\n$$\n\\gamma_{1}^{\\-\/\\+}: H^{1}_{\\Delta}(\\Omega_{\\-\/\\+})\\to H^{-1\/2}(\\Gamma)\\,.\n$$\nThis gives the extensions of the maps $\\gamma_{1}$ and $[\\gamma_{1}]$ defined on $H^{1}_{\\Delta}(\\mathbb{R}^{3}\\backslash\\Gamma):=H^{1}_{\\Delta}(\\Omega_{\\-})\\oplus H^{1}_{\\Delta}(\\Omega_{\\+})$ with values in $H^{-1\/2}(\\Gamma)$. \\par\nThen, for any $z\\in\\mathbb{C}\\backslash(-\\infty,0]$,\none defines the single and double-layer operators\n\\begin{equation}\\label{SL}\nS\\!L_{z}:=(\\gamma_{0}R_{\\bar z})^{*}=R_{z}\\gamma_{0}^{*}\\in\\mathscr B(B_{2,2}^{-s}(\\Gamma),H^{3\/2-s}(\\mathbb{R}^{3}))\\,,\n\\qquad s>0\\,,\n\\end{equation}\n\\begin{equation}\\label{DL}\nD\\!L_{z}:=(\\gamma_{1}R_{\\bar z})^{*}=R_{z}\\gamma_{1}^{*}\\in\\mathscr B(B_{2,2}^{-s}(\\Gamma),H^{1\/2-s}(\\mathbb{R}^{3}))\\,,\\qquad s>0\\,.\n\\end{equation}\nBy \\eqref{g0}, one has\n\\begin{equation}\\label{BSL}\nS_{z}:=\\gamma_{0}S\\!L_{z}\\in \\mathscr B((H^{s-1\/2}(\\Gamma),H^{s+1\/2}(\\mathbb{R}^{3})))\\,,\\qquad -1\/2\\sup\\sigma(\\Delta+\\mathsf v)$ such that $[\\lambda_{\\mathsf v},+\\infty)\\subset Z^{\\circ}_{\\mathsf v,d}\\cap Z^{\\circ}_{\\mathsf v,n}$; moreover, $Z^{\\circ}_{\\mathsf v,d}\\cap Z^{\\circ}_{0,d}\\not=\\varnothing $, $Z^{\\circ}_{\\mathsf v,n}\\cap Z^{\\circ}_{0,n}\\not=\\varnothing $, and both $Z^{\\circ}_{\\mathsf v,d}$ and $Z^{\\circ}_{\\mathsf v,n}$ can be chosen to be symmetric with respect to the real axis.\n\\end{lemma}\n\\begin{proof} At first, let us notice that it suffices to show that the bounded inverses exist for any real $\\lambda \\ge \\lambda_{\\mathsf v}$ for some $\\lambda_{\\mathsf v}>\\sup\\sigma(\\Delta+\\mathsf v)$. Then, by the continuity of the maps $z\\mapsto S^{\\mathsf v}_{z}$ and $z\\mapsto D^{\\mathsf v}_{z}$, the bounded inverses exist in a complex open neighbourhood of $[\\lambda_{\\mathsf v},+\\infty)$.\\par\nWe proceed as in the proof of \\cite[Lemma 3.2]{JDE16}. By $(-(\\Delta+\\mathsf v)+\\lambda)S\\!L^{\\mathsf v}_{\\lambda}|\\Omega_{\\rm in\/\\rm ex}=0$, by Green's formula and by \\eqref{jumpv0}, one gets, for any $\\phi\\in H^{-1\/2}(\\Gamma)$,\n\\begin{align*}\n0=&\\|\\nabla S\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\n-\\langle \\mathsf v S\\!L^{\\mathsf v}_{\\lambda}\\phi,S\\!L^{\\mathsf v}_{\\lambda}\\phi\\rangle _{H^{-1}(\\mathbb{R}^{3}),H^{1}(\\mathbb{R}^{3})}\n+\\lambda\\,\\|S\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\\\\n&+\\langle [ \\gamma_{1}] S\\!L^{\\mathsf v}_{\\lambda}\\phi ,\\gamma_{0}S\\!L_{\\lambda}\\phi\\rangle _{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\\\\\n=&\\|\\nabla S\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\n-\\langle \\mathsf v S\\!L^{\\mathsf v}_{\\lambda}\\phi,S\\!L^{\\mathsf v}_{\\lambda}\\phi\\rangle _{H^{-1}(\\mathbb{R}^{3}),H^{1}(\\mathbb{R}^{3})}\n+\\lambda\\,\\|S\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\\\\n&-\\langle \\phi ,S^{\\mathsf v}_{\\lambda}\\phi\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\\,.\n\\end{align*}\nThen, by \\eqref{V_est},\n\\[\n\\langle \\phi ,\\gamma_{0}S^{\\mathsf v}_{\\lambda}\\phi\\rangle _{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\\geq( 1-c_{1}\\varepsilon) \\|\\nabla S\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\n+(\\lambda-c_{2}(1+\\varepsilon^{-3}))\\|S\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\,.\n\\]\nChoosing $\\varepsilon>0$ such that $c_{1}\\varepsilon<1$ and then $\\lambda\\in\\varrho(\\Delta+\\mathsf v)$ such that $\\lambda>c_{2}(1+\\varepsilon^{-3})$ (this is always possible since $\\Delta+\\mathsf v$ in bounded from above), one gets\n\\[\n\\langle \\phi,S_{\\lambda}^{\\mathsf v}\\phi\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\n\\gtrsim\\|S\\!L_{\\lambda}^{\\mathsf v}\\phi \\|_{H^{1}( \\mathbb{R}^{3}) }^{2}\\,.\n\\]\nBy Green's formula again, one has, for any $u_{\\rm in\/\\rm ex},v_{\\rm in\/\\rm ex}\\in H^{1}(\\Omega_{\\rm in\/\\rm ex})$, $(-(\\Delta+\\mathsf v)+\\lambda)u_{\\rm in\/\\rm ex}\\in L^{2}(\\Omega_{\\rm in\/\\rm ex})$,\n\\begin{align}\\label{Gf}\n&\\langle(-(\\Delta+\\mathsf v)+\\lambda)u_{\\rm in\/\\rm ex},v_{\\rm in\/\\rm ex}\\rangle_{L^{2},H^{1}(\\Omega_{\\rm in\/\\rm ex})}=\n\\langle\\nabla u_{\\rm in\/\\rm ex},\\nabla v_{\\rm in\/\\rm ex}\\rangle_{L^{2}(\\Omega_{\\rm in\/\\rm ex})}\\nonumber\\\\\n&-\\langle \\mathsf v u_{\\rm in\/\\rm ex},v_{\\rm in\/\\rm ex}\\rangle_{H^{-1}(\\Omega_{\\rm in\/\\rm ex}),\nH^{1}(\\Omega_{\\rm in\/\\rm ex})}+\\lambda\\,\\langle u_{\\rm in\/\\rm ex},v_{\\rm in\/\\rm ex}\\rangle_{L^{2}(\\mathbb{R}^{3})}\\\\\n&\n\\pm \\langle \\gamma_{1}^{\\rm in\/\\rm ex}u_{\\rm in\/\\rm ex},\\gamma^{\\rm in\/\\rm ex}_{0} v_{\\rm in\/\\rm ex}\n\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\\,.\\nonumber\n\\end{align}\nBy \n$$\n\\big|\\langle \\mathsf v u_{\\rm in\/\\rm ex},v_{\\rm in\/\\rm ex}\\rangle_{H^{-1}(\\Omega_{\\rm in\/\\rm ex}),\nH^{1}(\\Omega_{\\rm in\/\\rm ex})}\\big|\\lesssim \\|u_{\\rm in\/\\rm ex}\\|_{H^{1}(\\Omega_{\\rm in\/\\rm ex})}\n\\|v_{\\rm in\/\\rm ex}\\|_{H^{1}(\\Omega_{\\rm in\/\\rm ex})}\\,,\n$$\n\\eqref{Gf} gives,\n\\begin{align*}\n&\\big|\\langle \\gamma_{1}^{\\rm in\/\\rm ex}u_{\\rm in\/\\rm ex},\\gamma^{\\rm in\/\\rm ex}_{0} v_{\\rm in\/\\rm ex}\n\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\\big|\\\\\n\\lesssim\n&\\big(\\|u_{\\rm in\/\\rm ex}\\|_{H^{1}(\\Omega_{\\rm in\/\\rm ex})}+\\|(-(\\Delta+\\mathsf v)+\\lambda)u_{\\rm in\/\\rm ex}\\|_{H^{-1}(\\Omega_{\\rm in\/\\rm ex})}\\big)\n\\|v_{\\rm in\/\\rm ex}\\|_{H^{1}(\\Omega_{\\rm in\/\\rm ex})}\\,.\n\\end{align*}\nSince $\\gamma^{\\rm in\/\\rm ex}_{0}:H^{1}(\\Omega_{\\rm in\/\\rm ex})\\to H^{1\/2}(\\Gamma)$ is surjective, finally one gets \n\\begin{equation}\\label{cmp}\n\\|\\gamma_{1}^{\\rm in\/\\rm ex}u_{\\rm in\/\\rm ex}\\|_{H^{-1\/2}(\\Gamma)}\\lesssim \n\\|u_{\\rm in\/\\rm ex}\\|_{H^{1}(\\Omega_{\\rm in\/\\rm ex})}+\\|(-(\\Delta+\\mathsf v)+\\lambda)u_{\\rm in\/\\rm ex}\\|_{H^{-1}(\\Omega_{\\rm in\/\\rm ex})}\\,.\n\\end{equation}\nThen, proceeding as in \\cite[Lemma 3.2]{JDE16} (compare (3.31) there with \\eqref{cmp} here), this\nyields\n\\begin{equation*}\n\\langle \\phi,S_{\\lambda}^{\\mathsf v}\\phi\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\n\\gtrsim \\|\\phi\\|_{H^{-1\/2}(\\Gamma)}^{2}\\label{SL_coer}%\n\\end{equation*}\nand so $(S_{\\lambda}^{\\mathsf v}) ^{-1}\\in{\\mathscr B}( H^{1\/2}(\n\\Gamma ) ,H^{-1\/2}(\\Gamma) )$ by the Lax-Milgram theorem.\\par\nAs regards $D_{\\lambda}^{\\mathsf v}$, the proof is almost the same. By $(-(\\Delta+\\mathsf v)+\\lambda)D\\!L^{\\mathsf v}_{\\lambda}|\\Omega_{\\rm in\/\\rm ex}=0$, by Green's formula and by \\eqref{jumpv1}, one gets, for any $\\phi\\in H^{1\/2}(\\Gamma)$,\n\\begin{align*}\n0=&\\|\\nabla D\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\Omega_{\\rm in})}+\n\\|\\nabla D\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\Omega_{\\rm ex})}\n-\\langle \\mathsf v D\\!L^{\\mathsf v}_{\\lambda}\\phi,D\\!L^{\\mathsf v}_{\\lambda}\\phi\\rangle _{H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)}\n+\\lambda\\,\\|D\\!L^{\\mathsf v}_{\\lambda}\\phi\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\\\\n&+\\langle D^{\\mathsf v}_{\\lambda}\\phi ,\\phi\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\\,.\n\\end{align*}\nwhich leads to \n\\[\n-\\langle D_{\\lambda}^{\\mathsf v}\\phi,\\phi\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\n\\gtrsim\\|D\\!L_{\\lambda}^{\\mathsf v}\\phi \\|_{H^{1}( \\mathbb{R}^{3}\\backslash\\Gamma) }^{2}\\,.\n\\]\nThen, proceeding as in \\cite[Lemma 3.2]{JDE16}, by \\eqref{cmp}, this\nyields\n$$\n-\\langle D_{\\lambda}^{\\mathsf v}\\phi,\\phi\\rangle_{H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma)}\n\\gtrsim \\|\\phi\\|_{H^{1\/2}(\\Gamma)}^{2}\n$$\nand so $(D_{\\lambda}^{\\mathsf v}) ^{-1}\\in{\\mathscr B}( H^{-1\/2}(\n\\Gamma ) ,H^{1\/2}(\\Gamma) )$ by the Lax-Milgram theorem.\n\\end{proof}\n\n\n\\section{Laplacians with regular and singular perturbations}\nHere we apply the abstract results in Section \\ref{Sec_Krein}, presenting various examples were \nthe self-adjoint operator $A$ is the free Laplacian $\\Delta:H^{2}(\\mathbb{R}^{3})\\subset L^{2}(\\mathbb{R}^{3})\\to L^{2}(\\mathbb{R}^{3})$ and $A_{B_{1}}=\\Delta+\\mathsf v$. All over this section we consider a Kato-Rellich potential $\\mathsf v=\\mathsf v_{2}+\\mathsf v_{\\infty}$ of short-range type (however, see next Remark \\ref{suff}), i.e.,\n\\begin{equation}\\label{short}\n\\mathsf v_{2}\\in L^{2}(\\mathbb{R}^{3}),\\quad \\text{supp}(\\mathsf v_{2})\\ \\text{bounded}, \\qquad\n\\ |\\mathsf v_{\\infty}(x)|\\lesssim\\, (1+|x|\\,)^{-\\kappa(1+\\varepsilon)}\\,,\\quad\\kappa\\ge 1\\,,\\quad\\varepsilon>0\\,.\n\\end{equation}\nIn the next Lemmata we show that all our hypotheses but (H3) hold whenever $\\kappa=1$, while hypothesis (H3) holds whenever $\\kappa=2$; we conjecture that all our results hold true with $\\kappa=1$ and that the requirement $\\kappa=2$ is merely of technical nature. \\par\nWe take \n$$\n\\mathfrak h_{1}=H^{2}(\\mathbb{R}^{3})\\hookrightarrow \\mathfrak b_{1}=H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)\n\\hookrightarrow\\mathfrak h_{1}^{\\circ}=L^{2}(\\mathbb{R}^{3})\n\\,,\n$$\nand, introducing the multiplication operator $\\langle x\\rangle$ by $\\langle x\\rangle u: x\\mapsto(1+|x|^{2})^{1\/2}u(x)$, we define\n\\begin{equation}\\label{tau1}\n\\tau_{1}:H^{2}(\\mathbb{R}^{3})\\to H^{2}(\\mathbb{R}^{3})\\,,\\quad \\tau_{1}u:=\\langle x\\rangle^{-s}u\\,,\\qquad\n1<2s< 1+\\varepsilon\\,,\n\\end{equation}\nand\n$$\nB_{1}u:=\\langle x\\rangle^{2s}\\mathsf v u\\,.\n$$\nBy our hypotheses on $\\mathsf v$ and $s$, one has $\\langle x\\rangle^{2s}\\mathsf v\\in L^{2}(\\mathbb{R}^{3})+L^{\\infty}(\\mathbb{R}^{3})$ and so, by Lemma \\ref{v},\n$$\nB_{1}\\in\\mathscr B(H^{1}( \\mathbb{R}^{3}\\backslash\\Gamma),H^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,.\n$$\nConsidering the weight $\\varphi(x)=(1+|x|^{2})^{w\/2}$, $w\\in\\mathbb{R}$, we use the notation $L^{2}_{\\varphi}(\\mathbb{R}^{3})\\equiv L^{2}_{w}(\\mathbb{R}^{3})$; $H^{k}_{w}(\\mathbb{R}^{3})$, $H^{k}_{w}(\\mathbb{R}^{3}\\backslash\\Gamma)$ denotes the corresponding scales of weighted Sobolev spaces. \\par \nSince \n$$\n\\langle x\\rangle^{w}\\in\\mathscr B(H^{1}_{w'}(\\mathbb{R}^{3}\\backslash\\Gamma),H^{1}_{w'+w}(\\mathbb{R}^{3}\\backslash\\Gamma))$$ and, by duality, $$\\langle x\\rangle^{w}\\in\\mathscr B(H^{1}_{w'}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{1}_{w'-w}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,,$$ \none gets \n\\begin{equation}\\label{v-w}\n\\langle x\\rangle^{-w-2s}B_{1}\\langle x\\rangle^{w}=\\mathsf v\\in \\mathscr B(H^{1}_{w}(\\mathbb{R}^{3}\\backslash\\Gamma),H^{1}_{-w-2s}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,.\n\\end{equation} \nSince \n\\begin{equation}\\label{R-w}\nR_{z}\\in\\mathscr B(H^{-1}_{w}(\\mathbb{R}^{3}),H^{1}_{w}(\\mathbb{R}^{3}))\\hookrightarrow \\mathscr B(H^{1}_{-w}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H^{1}_{w}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,,\n\\end{equation}\none has \n$$\\tau_{1}G^{1}_{z}=\\langle x\\rangle^{-s}R_{z}\\langle x\\rangle^{-s}\\in\\mathscr B(H_{-w}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*},H_{w+2s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,.\n$$ \nIn particular, one has $\\tau_{1}G^{1}_{z}\\in\\mathscr B(H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}),H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma))$.\\par\nSince $(1-\\mathsf v R_{z})$ is invertible, \n$$\nM_{z}^{B_{1}}=1-B_{1}\\tau_{1}G^{1}_{z}=1-\\langle x\\rangle^{s}\\mathsf v R_{z}\\langle x\\rangle^{-s}=\n\\langle x\\rangle^{s}(1-\\mathsf v R_{z})\\langle x\\rangle^{-s\n$$ \nis invertible as well an\n\\begin{equation}\\label{LbLv}\n\\Lambda_{z}^{B_{1}}=(M_{z}^{B_{1}})^{-1}B_{1}=\\langle x\\rangle^{s}(1-\\mathsf v R_{z})^{-1}\\langle x\\rangle^{s}\\mathsf v=\\langle x\\rangle^{s}\\Lambda_{z}^{\\!\\mathsf v}\\langle x\\rangle^{s}\n\\,.\n\\end{equation}\n\\begin{lemma} Let $\\mathsf v$ be as in \\eqref{short}, with $\\kappa=1$. Then, for any $s$ such that $1< 2s<1+\\varepsilon$,\n$$\n\\Lambda_{z}^{\\!\\mathsf v}\n\\in{\\mathscr B}(H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma),H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,.\n$$\n\\end{lemma}\n\\begin{proof} By \\eqref{v-w} and by $(1-R_{z}\\mathsf v)^{-1}\\in \\mathscr B(H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})$ (see Lemma \\ref{vH-1}), one has\n$$\nR_{z}\\Lambda^{\\mathsf v}_{z}=(1-\\mathsf v R_{z}\\mathsf v)^{-1}\\mathsf v\\in \\mathscr B(H_{-2s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma), H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma))\\,.\n$$\nBy interpolation (using \\cite{CE}), \n$$\nR_{z}\\Lambda^{\\mathsf v}_{z}\\in \\mathscr B(H_{-s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma), H_{s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma))\n$$\nand so \n$$\n\\Lambda^{\\mathsf v}_{z}\\in \\mathscr B(H_{-s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma), H_{-s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,.\n$$\n\\end{proof}\nBy the previous Lemma and \\eqref{LbLv}, \n$$\n\\Lambda_{z}^{B_{1}}\\in{\\mathscr B}(H^{1}( \\mathbb{R}^{3}\\backslash\\Gamma),H^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*})\\,.\n$$ \nThen, by Theorem \\ref{KR} and Corollary \\ref{cor1} (see Remark \\ref{tpt}), $Z_{B_{1}}=\\varrho(\\Delta+\\mathsf v)\\cap\\mathbb{C}\\backslash(-\\infty,0]$ and\n\\begin{align*}\n(-A_{B_{1}}+z)^{-1}=&R_{z}^{B_{1}}=R_{z}+R_{z}\\langle x\\rangle^{-s}\\Lambda_{z}^{B_{1}}\\langle x\\rangle^{-s}R_{z}\\qquad z\\in\\varrho(\\Delta+\\mathsf v)\\cap\\mathbb{C}\\backslash(-\\infty,0]\\\\\n=&\nR_{z}+R_{z}\\Lambda_{z}^{\\mathsf v}R_{z}\\\\\n=&R^{\\mathsf v}_{z}=(-(\\Delta+\\mathsf v)+z)^{-1}\\,.\n\\end{align*}\n\\begin{remark}\\label{suff} By the above relation, the self-adjoint operator provided in Corollary \\ref{cor1} by the choice $\\tau_{1}u=\\langle x\\rangle^{-s}u$, $B_{1}=\\langle x\\rangle^{2s}\\mathsf v$, coincides with the one corresponding to $\\tau_{1}u=u$, $B_{1}=\\mathsf v$. The first choice is dictated by the need to obtain LAP and a representation formula for the scattering couple $(A_{\\mathsf B}, A)$; whenever one is only interested in providing a resolvent formula for $A_{\\mathsf B}$, then the second choice is preferable: in this case, one does not need to work with a short-range potential $\\mathsf v$: a Kato-Rellich potential suffices.\n\\end{remark}\nNow, we prove the validity of the hypotheses (H1)-(H7); here and below we refer to the weighted spaces $L_{s}^{2}(\\mathbb{R}^{3})$ with $1<2s<1+\\varepsilon$. \\par\n\\begin{lemma} Let $\\mathsf v$ be short-range as in \\eqref{short}, with $\\kappa=1$.\nThen hypotheses (H1), (H2), (H6), (H7.1), (H7.2), (H7.3) hold true. \n\\end{lemma}\n\\begin{proof} By \\cite[Lemma 1, page 170]{ReSi IV}, $R_{z}=(-\\Delta+z)^{-1}\\in\\mathscr B(L^{2}_{s}(\\mathbb{R}^{3}))$ for any $z\\in \\mathbb{C}\\backslash(-\\infty,0]$. Therefore, by the resolvent identity $R^{\\mathsf v}_{z}=R_{z}(1-\\mathsf v R^{\\mathsf v}_{z})$, $z\\in\\varrho(-(\\Delta+\\mathsf v))$, and by $R^{\\mathsf v}_{z}\\in \\mathscr B(L_{s}^{2}(\\mathbb{R}^{3}), H^{2}(\\mathbb{R}^{3}))$, hypothesis (H1) is consequence of $\\mathsf v=\\mathsf v_{2}+\\mathsf v_{\\infty}\\in\\mathscr B(H^{2}(\\mathbb{R}^{3}),L_{s}^{2}(\\mathbb{R}^{3}))$. Since $\\mathsf v_{2}$ has a compact support, $\\mathsf v_{2}\\in \\mathscr B(H^{2}(\\mathbb{R}^{3}), L_{s}^{2}(\\mathbb{R}^{3}))$ by Lemma \\ref{v}. As regards $\\mathsf v_{\\infty}$, one has\n\\begin{align*}\n\\|\\mathsf v_{\\infty} u\\|^{2}_{L_{s}^{2}(\\mathbb{R}^{3})}=\\int_{\\mathbb{R}^{3}}|\\mathsf v_{\\infty} u|^{2}(1+|x|^{2})^{s}dx\\le c\\int_{\\mathbb{R}^{3}}(1+|x|)^{-2(1+\\varepsilon)}(1+|x^{2}|)^{s}\n|u|^{2}dx\n\\le \\,c\\,\\|u\\|^{2}_{L^{2}(\\mathbb{R}^{3})}\\,.\n\\end{align*} \nBy \\cite[Theorem 4.1]{Agmon}, LAP holds for $A=\\Delta$; hence (H7.1) is satisfied. By the sort-range hypothesis on $\\mathsf v$ and by \\cite[Theorem 4.2]{Agmon}, LAP holds for $A_{B_{1}}\\equiv\\Delta+\\mathsf v$ as well and, by \\cite[Theorems 6.1 and 7.1]{Agmon} asymptotic completeness holds for the scattering couple $(A_{B_{1}},A)\\equiv(\\Delta+\\mathsf v,\\Delta)$. Hence hypotheses (H1), (H2)\\footnote{ here ${e}(A_{B_{1}})$ is a discrete set in $(-\\infty,0)$ by \\cite[Theorem 4.2]{Agmon}; it is given by the (possibly empty) set of negative (embedded) eigenvalues of $\\Delta+\\mathsf v$. It is known that there are no negative eigenvalue either if $|\\mathsf v(x)|\\lesssim (1+|x|)^{-(1+\\varepsilon)}$ (see \\cite{Vak}) or if $\\mathsf v\\in L^{3\/2}(\\mathbb{R}^{3})$, i.e., if $\\kappa=2$ in \\eqref{short} (see \\cite{IJ}).} and (H6) are verified. \\par \nBy $R_{z}\\in\\mathscr B(L^{2}_{-s}(\\mathbb{R}^{3}),H^{2}_{-s}(\\mathbb{R}^{3}))$, one gets $G_{z}^{1*}=\\langle x\\rangle^{-s}R_{z}\\in\\mathscr B(L^{2}_{-s}(\\mathbb{R}^{3}),H^{2}(\\mathbb{R}^{3}))$ and so, by duality, $G_{z}^{1}\\in\\mathscr B(H^{-2}(\\mathbb{R}^{3}), L^{2}_{s}(\\mathbb{R}^{3}))$; moreover, by $R^{\\pm}_{\\lambda}\\in\\mathscr B(L^{2}_{s}(\\mathbb{R}^{3}),H^{2}_{-s}(\\mathbb{R}^{3}))$ and by a similar duality argument, one gets $G_{\\lambda}^{1,\\pm}\\in\\mathscr B(H^{-2}(\\mathbb{R}^{3}), L^{2}_{-s}(\\mathbb{R}^{3}))$. Thus hypothesis (H7.2) holds.\\par \nBy $\\Lambda_{z}^{\\!B_{1}}=\\langle x\\rangle^{s}\\Lambda_{z}^{\\!\\mathsf v}\\langle x\\rangle^{s}$, hypothesis (H7.3) is equivalent to the existence in $\\mathscr B(H_{-s}^{2}(\\mathbb{R}^{3}),H_{s}^{-2}(\\mathbb{R}^{3}))$ of $\\lim_{\\epsilon\\searrow 0}\\Lambda_{\\lambda\\pm i\\epsilon}^{\\!\\mathsf v}=\\lim_{\\epsilon\\searrow 0}(1-\\mathsf v R_{\\lambda\\pm i\\epsilon})^{-1}\\mathsf v$. By \\eqref{short}, $\\mathsf v\\in \\mathscr B(H_{-s}^{2}(\\mathbb{R}^{3}),L_{s}^{2}(\\mathbb{R}^{3}))$. Then, $\\lim_{\\epsilon\\searrow 0}(1-\\mathsf v R_{\\lambda\\pm i\\epsilon})^{-1}$ exists in $\\mathscr B(L^{2}_{s}(\\mathbb{R}^{3}))$ (see \\cite[proof of Theorem XIII.33, page 177]{ReSi IV}) and so (H7.3) holds.\n\\end{proof}\n\\begin{lemma}\\label{5.4} Let $\\mathsf v$ be short-range as in \\eqref{short}, with $\\kappa=2$.\nThen hypothesis (H3) holds true. \n\\end{lemma}\n\\begin{proof} The proof is the same as the one for \\cite[Lemma 4.5]{JDE18}, once one proves that \n\\begin{equation}\\label{once}\n\\mathsf v R^{\\mathsf v,\\pm}_{\\lambda}\\in\\mathscr B(L^{2}_{2s}(\\mathbb{R}^{3}))\\,.\n\\end{equation}\nSince $R^{\\mathsf v,\\pm}_{\\lambda}\\in\\mathscr B(L^{2}_{2s}(\\mathbb{R}^{3}), H^{2}_{-2s}(\\mathbb{R}^{3}))$, \n\\eqref{once} is consequence of \n\\begin{equation}\\label{vw}\n\\mathsf v=\\mathsf v_{2}+\\mathsf v_{\\infty}\\in \\mathscr B(H^{2}_{-2s}(\\mathbb{R}^{3}), L^{2}_{2s}(\\mathbb{R}^{3}))\\,.\n\\end{equation}\nLemma \\ref{v} entails $\\mathsf v_{2}\\in \\mathscr B(H^{2}(\\mathbb{R}^{3}), L^{2}(\\mathbb{R}^{3}))$ and so, since $\\mathsf v_{2}$ has a compact support, one gets that $\\mathsf v_{2}$ satisfies \\eqref{vw}. As regards $\\mathsf v_{\\infty}$, one has, by $1<2s<1+\\varepsilon$,\n\\begin{align*}\n\\|\\mathsf v_{\\infty} u\\|^{2}_{L^{2}_{2s}(\\mathbb{R}^{3})}=&\\int_{\\mathbb{R}^{3}}|\\mathsf v_{\\infty} u|^{2}(1+|x|^{2})^{2s}dx\\le c\\int_{\\mathbb{R}^{3}}(1+|x|)^{-4(1+\\varepsilon)}(1+|x|^{2})^{4s}\n|u|^{2}(1+|x|^{2})^{-2s}dx\\\\\n\\le& \\,c\\,\\|u\\|^{2}_{L^{2}_{-2s}(\\mathbb{R}^{3})}\\le c\\,\\|u\\|^{2}_{H^{2}_{-2s}(\\mathbb{R}^{3})}\n\\end{align*} \nand so $\\mathsf v_{\\infty}$ satisfies \\eqref{vw} as well.\n\\end{proof}\nIn order to check the validity of the remaining hypotheses, we need to specify the map \n$\\tau_{2}$. In the following examples, according to the case, we take either \n\\begin{equation}\\label{td}\n\\tau_{2}=\\gamma_{0}:H^{2}(\\mathbb{R}^{3})\\to\\mathfrak h_{2}=B^{3\/2}_{2,2}(\\Gamma)\\hookrightarrow\\mathfrak b_{2}=H^{s_{\\circ}}(\\Gamma)\\,,\\quad 00$, one gets $\\Lambda^{\\!\\mathsf v}_{z}\\in \\mathscr B(L^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}),L^{2}(\\mathbb{R}^{3}))$. Hence, by the resolvent formula \\eqref{RF-int} and by $R_{z}\\in \\mathscr B(L^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}),H^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}))$, one gets $R_{z}^{\\mathsf v}\\in \\mathscr B(L^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}),H^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}))$. This entails $\\gamma_{0}R_{z}^{B_{1}}=\\gamma_{0}R_{z}^{\\mathsf v}=\\gamma_{0}\\chi R_{z}^{\\mathsf v}\\in \\mathscr B(L^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}),B^{2}_{2,2}(\\Gamma))$ and $\\gamma_{1}R_{z}^{B_{1}}=\\gamma_{1}R_{z}^{\\mathsf v}=\\gamma_{1}\\chi R_{z}^{\\mathsf v}\\in \\mathscr B(L^{2}_{-(2s+\\gamma)}(\\mathbb{R}^{3}),H^{1\/2}(\\Gamma))$. Then, by duality, one gets $G_{z}^{B_{1}}\\in \\mathscr B(\\mathfrak h_{2}^{*},L^{2}_{2s+\\gamma}(\\mathbb{R}^{3}))$. This shows that (H4.2) holds. \\par\nBy \\cite[Theorem 4.2]{Agmon}, the map $(\\mathbb{R}\\backslash{e}(A_{B_{1}}))\\cup\\mathbb{C}_{\\pm}\\ni z\\mapsto R^{B_{1},\\pm}_{z}=R_{z}^{\\mathsf v,\\pm}\\in \\mathscr B(L^{2}_{s}(\\mathbb{R}^{3}),H^{2}_{-s}(\\mathbb{R}^{3}))$ is continuos. Hence, $z\\mapsto\\gamma_{0}R^{B_{1},\\pm}_{z}=\\gamma_{0}R_{z}^{\\mathsf v,\\pm}=\\gamma_{0}\\chi R_{z}^{\\mathsf v,\\pm}$ and $z\\mapsto\\gamma_{1}R^{B_{1},\\pm}_{z}=\\gamma_{1}R_{z}^{\\mathsf v,\\pm}=\\gamma_{1}\\chi R_{z}^{\\mathsf v,\\pm}$ are continuos as $\\mathscr B(L^{2}_{s}(\\mathbb{R}^{3}),B^{3\/2}_{2,2}(\\Gamma))$-valued and $\\mathscr B(L^{2}_{s}(\\mathbb{R}^{3}),H^{1\/2}(\\Gamma))$-valued maps respectively. Then, by duality, $z\\mapsto G^{B_{1},\\pm}$ is continuos on $(\\mathbb{R}\\backslash{e}(A_{B_{1}}))\\cup\\mathbb{C}_{\\pm}$ as a $\\mathscr B(\\mathfrak h_{2}^{*}, L^{2}_{-s}(\\mathbb{R}^{3}))$-valued map. Since both $\\gamma_{0}:H^{2}(\\mathbb{R}^{2})\\to B^{3\/2}_{2,2}(\\Gamma)$ and $\\gamma_{1}:H^{2}(\\mathbb{R}^{2})\\to H^{1\/2}(\\Gamma)$ are surjective, $G^{B_{1},\\pm}_{z}\\in \\mathscr B(\\mathfrak h_{2}^{*}, L^{2}_{-s}(\\mathbb{R}^{3}))$ is the adjoint of a surjective map and hence is injective. Thus we proved that (H5) holds.\n\\end{proof}\nIn conclusion, we proved that all our hypotheses, except (H4.1), hold true without the need to specify the operators $B_{0}$ and $B_{2}$. The validity of hypothesis \n(H4.1), i.e. the semi-boundedness of $A_{\\mathsf B}$, will be checked case by case in the next examples. Before turning to such examples, we give a result that makes explicit the map $\\mathcal L_{\\lambda}$ appearing in Theorem \\ref{S-matrix}. \n\\begin{lemma}\\label{Llambda} Let $\\tau_{1}$ be as in \\eqref{tau1} and let $\\tau_{2}$ be either as in \\eqref{td} or as in \\eqref{tn}. \nThen $$L_{\\lambda}:=-\\mathcal L_{\\lambda}\\langle x\\rangle^{s}$$\nis $L^{2}({\\mathbb S}^{2})$-valued, where ${\\mathbb S}^{2}$ denotes the 2-dimensional unitary sphere in $\\mathbb{R}^{3}$, and\n$$\nL_{\\lambda}(u\\oplus\\phi)=\\frac{|\\lambda|^{\\frac{1}4}}{2^{\\frac12}}\\,\\left(L^{1}_{\\lambda}u+L^{2}_{\\lambda}\\phi\\right)\\,,\n$$\n$$\nL^{1}_{\\lambda}u(\\xi):=\\widehat{u}\\,(|\\lambda|^{1\/2}\\xi)\n$$\n$$\nL^{2}_{\\lambda}\\phi(\\xi):=\\frac1{(2\\pi)^{\\frac32}}\\,\\begin{cases}\\langle u^{\\xi}_{\\lambda}|\\Gamma,\\phi\\rangle_{H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma)}\\,,&\\tau_{2}=\\gamma_{0}\\,,\\\\\ni\\,|\\lambda|^{\\frac12}\\nu\\!\\cdot\\! \\xi\\,\\langle u^{\\xi}_{\\lambda}|\\Gamma,\\phi\\rangle_{L^{2}(\\Gamma)}\\,,&\\tau_{2}=\\gamma_{1}\\,.\n\\end{cases}\n$$\nHere $u^{\\xi}_{\\lambda}$ is the plane wave with direction $\\xi\\in{\\mathbb S}^{2}$ and wavenumber $|\\lambda|^{\\frac12}$, i.e., $u^{\\xi}_{\\lambda}(x)=e^{i\\,|\\lambda|^{\\frac12}\\xi\\cdot x}$, $\\nu$ is the normal to $\\Gamma$ and $\\widehat u$ denotes the Fourier transform.\n\\end{lemma} \n\\begin{proof} The unitary \nmap $F:L^{2}(\\mathbb{R}^{n})\\to \\int^{\\oplus}_{(-\\infty,0)}L^{2}({\\mathbb S}^{2})\\,d\\lambda\\equiv L^{2}((-\\infty,0);L^{2}({\\mathbb S}^{2}))$ diagonalizing $A=\\Delta$ is given by \n\\begin{equation}\\label{fourier}\n(Fu)_\\lambda(\\xi):=-\\frac{|\\lambda|^{\\frac{1}4}}{2^{\\frac12}}\\, \\widehat u(|\\lambda|^{1\/2}\\xi)\\,.\n\\end{equation} \nTherefore, by $(\\mu-\\lambda)\\widehat{ R_{\\mu}f}(|\\lambda|^{1\/2}\\xi)=-\\widehat f(|\\lambda|^{1\/2}\\xi)$,\none gets \n\\begin{align*}\n(\\mu-\\lambda)(FR_{\\mu}\\tau_{1}^{*}\\langle x\\rangle^{s}u)_{\\lambda}(\\xi)=-\n\\frac{|\\lambda|^{\\frac{1}4}}{2^{\\frac12}}\\,\\widehat{u}\\,(|\\lambda|^{1\/2}\\xi)\n\\,.\n\\end{align*}\nAs regards $L^{2}_{\\lambda}$, the computation was given in \\cite[Theorem 5.1]{JMPA}. \n\\end{proof}\n\\begin{remark} Let us notice that, whenever $u\\in L^{2}_{s'}(\\mathbb{R}^{3})$, $s'>3\/2$,\n$$\nL^{1}_{\\lambda}u(\\xi)=\\frac1{(2\\pi)^{\\frac32}}\\,\\langle u^{\\xi}_{\\lambda},u\\rangle_{L^{2}_{-s'}(\\mathbb{R}^{3}),L^{2}_{s'}(\\mathbb{R}^{3})}\n$$\nand so $L^{1}_{\\lambda}$ and $L^{2}_{\\lambda}$ have a similar structure.\n\\end{remark}\n\\subsection{\\label{Sec_delta} Short-range potentials and semi-transparent boundary conditions\nof $\\delta_{\\Gamma }$-type} \nHere we take \n$$\n\\mathfrak h_{2}= B^{3\/2}_{2,2}(\\Gamma)\\hookrightarrow\\mathfrak b_{2}=\\mathfrak b_{2,2}= H^{s_{\\circ}}(\\Gamma)\n\\hookrightarrow\\mathfrak h_{2}^{\\circ}=L^{2}(\\Gamma)\\,,\\quad 0< s_{\\circ}<1\/2\\,,\n$$\n$$\n\\tau_{2}=\\gamma_{0}:H^{2}(\\mathbb{R}^{3})\\to B^{3\/2}_{2,2}(\\Gamma)\\,,\\qquad\nB_{0}=1\\,\n\\qquad B_{2}=\\alpha\\,,\n$$\nwhere \n$$\n\\alpha\\in\\mathscr B(H^{s_{\\circ}}(\\Gamma),H^{-s_{\\circ}}(\\Gamma))\\,,\\quad \\alpha^{*}=\\alpha\\,.\n$$ \nLet us notice (see \\cite[Remark 2.6]{JDE18}) that in the case $\\alpha$ is the multiplication operator associated to a real-valued function \n$\\alpha$, then $\\alpha\\in L^{p}(\\Gamma)$, $p>2$, fulfills our hypothesis. \\par\nFor any $z\\in\\mathbb{C}\\backslash(-\\infty,0]$, one has\n\\begin{equation}\nM_{z}^{\\mathsf B\n=1-\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v & 0\\\\\n0 & \\alpha%\n\\end{bmatrix}\n\\begin{bmatrix}\n\\langle x\\rangle^{-s}R_{z}\\langle x\\rangle^{-s} & \\langle x\\rangle^{-s}R_{z}\\gamma_{0}^{*}\\\\\n\\gamma_{0}R_{z} \\langle x\\rangle^{-s}& \\gamma_{0}R_{z}\\gamma_{0}^{*}%\n\\end{bmatrix}\n=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{M}_{z}^{\\mathsf v,\\alpha}\\begin{bmatrix}\\langle x\\rangle^{-s}&0\\\\0&1\\end{bmatrix}\\,,\n\\end{equation}\n\\begin{equation}\n{M}_{z}^{\\mathsf v,\\alpha}:=%\n\\begin{bmatrix}\n1-\\mathsf v R_{z} & -\\mathsf v S\\!L_{z}\\\\\n-\\alphaS\\!L_{\\bar z}^{*} & 1-\\alpha S_{z}%\n\\end{bmatrix}\n\\,\n\\end{equation}\nBy the mapping properties provided in Sections \\ref{Sec_V} and \\ref{Sec_Layer}, by \\eqref{v-w} and \\eqref{R-w} with $w=-s$, one gets\n$$\nM_{z}^{\\mathsf v,\\alpha}\\in{\\mathscr B}(H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{-s_{\\circ}}(\\Gamma))\\,.\n$$\nAccording to \n\\cite[Lemma 5.8]{JMPA}, for any $z\\in \\mathbb{C}\\backslash((-\\infty,0]\\cup\\sigma_{\\alpha})$, where $\\sigma_{\\alpha}\\subset (0,+\\infty)$ is finite, one has\n\\begin{equation}\n(M_{z}^{B_{0},B_{2}})^{-1}=(M_{z}^{\\alpha})^{-1}:=\n( 1-\\alpha S_{z}) ^{-1}\\in{\\mathscr B}( H^{-s_{\\circ}}(\n\\Gamma ) ) \\,.\\label{S-alpha}%\n\\end{equation}\nThus $$\nZ_{B_{0},B_{2}}=Z_{\\alpha}:=\\{z\\in \\mathbb{C}\\backslash(-\\infty,0]:(M_{w}^{\\alpha})^{-1}\\in{\\mathscr B}( H^{-s_{\\circ}}(\n\\Gamma ) ) ,\\ w=z,\\bar z\\}\\supseteq \\mathbb{C}\\backslash((-\\infty,0]\\cup\\sigma_{\\alpha})\n$$ \nand \n$$ \n\\Lambda_{z}^{B_{0},B_{2}}=(M_{z}^{B_{0},B_{2}})^{-1}B_{2}=\\Lambda_{z}^{\\!\\alpha}:=(1-\\alpha S_{z})^{-1}\\alpha\\in{\\mathscr B}( H^{s_{\\circ}}(\n\\Gamma ) , H^{-s_{\\circ}}(\\Gamma ) ) \\,.\n$$\nBy \\cite[Corollary 2.4]{JDE18}, for any $z\\in \\varrho(\\Delta+\\mathsf v)\\backslash\\sigma_{\\mathsf v,\\alpha}$, where $\\sigma_{\\mathsf v,\\alpha}\\subset\\mathbb{R}$ is finite, \n\\begin{equation}\n(\\widehat M_{z}^{\\mathsf B})^{-1}=(\\widehat M_{z}^{\\mathsf v,\\alpha})^{-1}:=\n( 1-\\alpha S_{z}^{\\mathsf v}) ^{-1}\n\\in{\\mathscr B}( H^{-s_{\\circ}}(\\Gamma ) ) \n\\,.\\label{Sv-alpha}%\n\\end{equation}\nThus \n$$\n\\widehat Z_{\\mathsf B}=\\widehat Z_{\\mathsf v,\\alpha}:=\\{z\\in \\varrho(\\Delta+\\mathsf v):(\\widehat M_{w}^{\\mathsf v,\\alpha})^{-1}\\in{\\mathscr B}( H^{-s_{\\circ}}(\n\\Gamma ) ) ,\\ w=z,\\bar z\\}\\supseteq \\varrho(\\Delta+\\mathsf v)\\backslash\\sigma_{\\mathsf v,\\alpha}\n$$ \nand\n$$\n\\widehat\\Lambda^{\\mathsf B}_{z}=(\\widehat M_{z}^{\\mathsf B})^{-1}B_{2}=\n\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}:=(1-\\alpha S^{\\mathsf v}_{z} )^{-1}\\alpha\n\\in{\\mathscr B}(H^{s_{\\circ}}(\\Gamma ), H^{-s_{\\circ}}(\\Gamma ) )\\,.\n$$\nHence, \n$$\n\\Lambda_{z}^{\\mathsf B}=\n\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}({M}_{z}^{\\mathsf v,\\alpha})^{-1}\\begin{bmatrix}\\langle x\\rangle^{-s}&0\\\\0&1\\end{bmatrix}\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v & 0\\\\\n0 & \\alpha%\n\\end{bmatrix}\n=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,\\alpha}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\\,,\n$$\nwhere, by \\eqref{LB-new} and \\eqref{LB-new2},\n\\begin{align*}\n{\\Lambda}_{z}^{\\!\\mathsf v,\\alpha}:=&\n\\begin{bmatrix}\n\\Lambda_{z}^{\\!\\mathsf v}+\\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}S\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v}& \\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}\n\\\\\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}S\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v} & \n\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}\n\\end{bmatrix}\\\\\n=&\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\nS\\!L_{z}\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}S\\!L_{\\bar z}^{*}& S\\!L_{z}\n\\\\ S\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}\\end{bmatrix}\n\\,.\n\\end{align*}\nOne has \n\\begin{equation}\\label{Lbbvalpha}\n{\\Lambda}_{z}^{\\!\\mathsf v,\\alpha}\n\\in{\\mathscr B}(H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)\\oplus H^{s_{\\circ}}(\\Gamma),H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{-s_{\\circ}}(\\Gamma))\\,.\n\\end{equation}\nBy Theorems \\ref{Th_Krein} and \\ref{Th-alt-res}, there follows \n\\begin{align}\nR_{z}^{\\mathsf v,\\alpha}=&R_{z}+\n\\begin{bmatrix}R_{z}\\langle x\\rangle^{-s}&S\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,\\alpha}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{2s}\\mathsf v \\langle x\\rangle^{-s}R_{z}\\\\ \\alphaS\\!L_{\\bar z}^{*}\\end{bmatrix}\n\\label{Rv-alpha-0}\n\\\\ \n=&R_{z}+\n\\begin{bmatrix}R_{z}&S\\!L_{z}\\end{bmatrix}\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\nS\\!L_{z}\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}S\\!L_{\\bar z}^{*}& S\\!L_{z}\n\\\\ S\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\alpha}\\end{bmatrix}\n\\begin{bmatrix}R_{z}\\\\ S\\!L_{\\bar z}^{*}\\end{bmatrix}\n\\label{Rv-alpha-1}\\\\ \n=&R_{z}^{\\mathsf v}+S\\!L_{z}^{\\mathsf v}{\\widehat\\Lambda}_{z}^{\\mathsf v,\\alpha}{S\\!L_{\\bar z}^{\\mathsf v}}^{*}\\,.\n\\label{Rv-alpha-2}\n\\end{align}\nis the resolvent of a self-adjoint operator $\\Delta^{\\!\\mathsf v,\\delta,\\alpha}$; \\eqref{Rv-alpha-0} holds for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,\\delta,\\alpha})\\cap\\mathbb{C}\\backslash(-\\infty,0]$, both\n\\eqref{Rv-alpha-1} and \\eqref{Rv-alpha-2} hold for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,\\delta,\\alpha})\\cap\\varrho(\\Delta+\\mathsf v)$. \n\\par\nBy Theorem \\ref{Th-add},\n$$\n\\Delta^{\\!\\mathsf v,\\delta,\\alpha}u=\\Delta u+\\mathsf v u+(\\alpha\\gamma_{0}u)\\delta_{\\Gamma}\\,.\n$$\nBy \\eqref{Rv-alpha-2} and by the mapping properties of $S\\!L^{\\mathsf v}_{z}$, one has \n$$\n\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,\\delta,\\alpha})\\subseteq H^{3\/2-s_{\\circ}}(\\mathbb{R}^{3})\\,.\n$$ \nMoreover, by $R^{\\mathsf v}_{z}u\\in H^{2}(\\mathbb{R}^{3})$, so that $[\\gamma_{1}]R^{\\mathsf v}_{z}u=0$, and by \\eqref{jumpv0}, one gets $[\\gamma_{1}]R^{\\mathsf v,\\alpha}_{z}u=-{\\widehat\\Lambda}_{z}^{\\mathsf v,\\alpha}{S\\!L_{\\bar z}^{\\mathsf v}}^{*}u=-\\widehat\\rho_{\\mathsf B}(R^{\\mathsf v,\\alpha}_{z}u)$. Hence, by Lemma \\ref{alt-abc}, \n$$\nu\\in\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,\\delta,\\alpha})\\quad\\Longrightarrow\\quad\\alpha\\gamma_{0}u+[\\gamma_{1}]u=0\\,.\n$$\nSince $\\widehat Z_{\\mathsf v,\\alpha}$ contains a positive half-line, $\\Delta^{\\!\\mathsf v,\\delta,\\alpha}$ is bounded from above and hypothesis (H4.1) holds. The scattering couple $(\\Delta^{\\!\\mathsf v,\\delta,\\alpha},\\Delta)$ is asymptotically complete and the corresponding scattering matrix is given by \n$$\n{\\mathcal S}_{\\lambda}^{\\mathsf v,\\alpha}=1-2\\pi iL_{\\lambda}\\Lambda^{\\mathsf v,\\alpha,+}_{\\lambda}L_{\\lambda}^{*}\\,,\\quad \\lambda\\in(-\\infty,0]\\backslash(\\sigma^{-}_{p}(\\Delta+\\mathsf v)\\cup \\sigma^{-}_{p}(\\Delta^{\\!\\mathsf v,\\delta,\\alpha}))\\,,\n$$\nwhere $L_{\\lambda}$ is given in Corollary \\ref{Llambda} and $\\Lambda^{\\mathsf v,\\alpha,+}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\Lambda^{\\mathsf v,\\alpha}_{\\lambda+i\\epsilon}$. This latter limit exists by Lemma \\ref{rmH7}; in particular, by \\eqref{LBpm2}, \n\\begin{align*}\n\\Lambda^{\\mathsf v,\\alpha,+}=&\\left(1+\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&(1-\\alpha S^{\\mathsf v,+}_{\\lambda})^{-1}\\alpha\\end{bmatrix}\\begin{bmatrix}\nS\\!L^{+}_{\\lambda}\n(1-\\alpha S^{\\mathsf v,+}_{\\lambda})^{-1}\\alpha(S\\!L^{-}_{\\lambda})^{*}& S\\!L^{+}_{\\lambda}\n\\\\(S\\!L^{-}_{\\lambda})^{*}& 0\n\\end{bmatrix}\\right)\\times\\\\\n&\\times\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&(1-\\alpha S^{\\mathsf v,+}_{\\lambda})^{-1}\\alpha\\end{bmatrix}\\,,\n\\end{align*}\nwhere\n$$\nR^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}R_{\\lambda\\pm i\\epsilon}\\,,\\qquadS\\!L^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}S\\!L_{\\lambda\\pm i\\epsilon}\\,,\\qquad \nS^{\\mathsf v,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\gamma_{0}S\\!L^{\\mathsf v}_{\\lambda\\pm i\\epsilon}\\,.\n$$\n\\subsection{\\label{Sec_dirichlet} Short-range potentials and Dirichlet boundary conditions.} \nHere we take \n$$\n\\mathfrak h_{2}= B^{3\/2}_{2,2}(\\Gamma)\\hookrightarrow\\mathfrak b_{2}=H^{1\/2}(\\Gamma)\n\\hookrightarrow\\mathfrak h_{2}^{\\circ}=L^{2}(\\Gamma)\\hookrightarrow \\mathfrak b_{2,2}=\\mathfrak b_{2}^{*}= H^{-1\/2}(\\Gamma)\\,,\n$$\n$$\n\\tau_{2}=\\gamma_{0}:H^{2}(\\mathbb{R}^{3})\\to B^{3\/2}_{2,2}(\\Gamma)\\,,\\qquad\nB_{0}=0\\,,\\qquad B_{2}=1\\,.\n$$\nFor any $z\\in\\mathbb{C}\\backslash(-\\infty,0]$, one has\n$$\nM_{z}^{\\mathsf B\n=\\begin{bmatrix}\n1 & 0\\\\\n0 & 0%\n\\end{bmatrix}-\n\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v & 0\\\\\n0 & 1%\n\\end{bmatrix}\n\\begin{bmatrix}\n\\langle x\\rangle^{-s}R_{z}\\langle x\\rangle^{-s} & \\langle x\\rangle^{-s}R_{z}\\gamma_{0}^{*}\\\\\n\\gamma_{0}R_{z}\\langle x\\rangle^{-s} & \\gamma_{0}R_{z}\\gamma_{0}^{*}%\n\\end{bmatrix}=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}M_{z}^{\\mathsf v,d}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\\,,\n$$\n$$\nM_{z}^{\\mathsf v,d}:=%\n\\begin{bmatrix}\n1-\\mathsf v R_{z} & -\\mathsf v S\\!L_{z}\\\\\n-S\\!L_{\\bar z}^{*} & -S_{z}%\n\\end{bmatrix}\n\\,.\n$$\nBy the mapping properties provided in Sections \\ref{Sec_V} and \\ref{Sec_Layer}, by \\eqref{v-w} and \\eqref{R-w} with $w=-s$, one gets\n$$\nM_{z}^{\\mathsf v,d}\\in{\\mathscr B}(H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{-1\/2}(\\Gamma),H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{1\/2}(\\Gamma))\\,.\n$$\nBy Lemma \\ref{coerc} with $\\mathsf v=0$, for any $z\\in Z^{\\circ}_{0,d}\\not=\\varnothing $, \n$$\n(M_{z}^{B_{0},B_{2}})^{-1}=\\Lambda_{z}^{B_{0},B_{2}}=\n(M_{z}^{d})^{-1}=\\Lambda_{z}^{\\!d}:=-S_{z}^{-1}\\in\\mathscr B(H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\,.\n$$\nThus,\n$$\nZ_{B_{0},B_{2}}=Z_{d}:=\\{z\\in \\mathbb{C}\\backslash(-\\infty,0]: (M_{z}^{d})^{-1}\\in\\mathscr B(H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\}\\supseteq Z^{\\circ}_{0,d}\\,.\n$$\nBy Lemma \\ref{coerc} again, for any $z\\in Z^{\\circ}_{\\mathsf v,d}\\not=\\varnothing $, \n$$\n(\\widehat M_{z}^{B_{0},B_{2}})^{-1}=(\\widehat \\Lambda_{z}^{B_{0},B_{2}})^{-1}=(\\widehat M_{z}^{\\mathsf v,d})^{-1}=\\widehat \\Lambda_{z}^{\\mathsf v,d}:=-(S^{\\mathsf v}_{z})^{-1}\\in\\mathscr B(H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\,.\n$$\nThus,\n$$\n\\widehat Z_{\\mathsf B}=\\widehat Z_{\\mathsf v,d}:=\\{z\\in \\varrho(\\Delta+\\mathsf v): (\\widehat M_{z}^{\\mathsf v,d})^{-1}\\in\\mathscr B(H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\}\\supseteq Z^{\\circ}_{\\mathsf v,d}\\,.\n$$\nHence,\n\\begin{align*}\n\\Lambda^{\\! \\mathsf B}_{z}=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}(M_{z}^{\\mathsf v,d})^{-1}\n\\begin{bmatrix}\\langle x\\rangle^{-s}&0\\\\0&1\\end{bmatrix}\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v & 0\\\\\n0 & 1%\n\\end{bmatrix}=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,d}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\\,,\n\\end{align*}\nwhere, by \\eqref{LB-new} and \\eqref{LB-new2},\n\\begin{align*}\n{\\Lambda}_{z}^{\\!\\mathsf v,d}:=&\\begin{bmatrix}\n\\Lambda_{z}^{\\!\\mathsf v}-\\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}(S^{\\mathsf v}_{z})^{-1}S\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v}& -\\Lambda_{z}^{\\!\\mathsf v}S\\!L_{z}(S^{\\mathsf v}_{z})^{-1}\\\\\n-(S^{\\mathsf v}_{z})^{-1}S\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v} & -(S^{\\mathsf v}_{z})^{-1}\n\\end{bmatrix}\\\\\n=&\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&-(S^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\n-S\\!L_{z}(S^{\\mathsf v}_{z})^{-1}S\\!L_{\\bar z}^{*}& S\\!L_{z}\n\\\\ S\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&-(S^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\end{align*}\nOne has \n\\begin{equation}\\label{Lbbv-dir}\n{\\Lambda}_{z}^{\\!\\mathsf v,d}\n\\in{\\mathscr B}(H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)\\oplus H^{1\/2}(\\Gamma),H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{-1\/2}(\\Gamma))\\,.\n\\end{equation}\nBy Theorems \\ref{Th_Krein} and \\ref{Th-alt-res}, there follows that\n\\begin{align}\n&R_{z}^{\\mathsf v,d}=R_{z}+\n\\begin{bmatrix}R_{z}\\langle x\\rangle^{-s}&S\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,d}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{2s}\\mathsf v\\langle x\\rangle^{-s} R_{z}\\\\S\\!L^{*}_{\\bar z}\\end{bmatrix}\\label{Rv-dir-0}\n\\\\\n=&R_{z}+\n\\begin{bmatrix}R_{z}&S\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&-(S^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\n-S\\!L_{z}(S^{\\mathsf v}_{z})^{-1}S\\!L_{\\bar z}^{*}& S\\!L_{z}\n\\\\ S\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&-(S^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\begin{bmatrix}R_{z}\\\\S\\!L^{*}_{\\bar z}\\end{bmatrix}\\label{Rv-dir-1}\n\\\\\n=&R_{z}^{\\mathsf v}-S\\!L^{\\mathsf v}(S_{z}^{\\mathsf v})^{-1}{S\\!L_{\\bar z}^{\\mathsf v}}^{*}\n\\label{Rv-dir-2}\n\\end{align}\nis the resolvent of a self-adjoint operator $\\Delta^{\\!\\mathsf v,d}$; \\eqref{Rv-dir-0} holds for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,d})\\cap\\mathbb{C}\\backslash(-\\infty,0]$, both \n\\eqref{Rv-dir-1} and \n\\eqref{Rv-dir-2} hold for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,d})\\cap\\varrho(\\Delta+\\mathsf v)$.\n\\par\nBy Theorem \\ref{Th-add} and by $[\\gamma_{1}]u=-\\widehat\\rho_{\\mathsf B}u$ for any $u\\in\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,d})$,\n$$\n\\Delta^{\\!\\mathsf v,d}\\,u=\\Delta u+\\mathsf v u-([\\gamma_{1}]u)\\delta_{\\Gamma}\\,.\n$$\nBy \\eqref{Rv-alpha-2} and by the mapping properties of $S\\!L^{\\mathsf v}_{z}$, one has \n$$\n\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,d})\\subseteq H^{1}(\\mathbb{R}^{3})\\,.\n$$ \nMoreover, by Lemma \\ref{alt-abc}, \n$$\nu\\in\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,d})\\quad\\Longrightarrow\\quad\\gamma_{0}u=0\\,.\n$$\nTherefore, $\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,d})\\subseteq H_{0}^{1}(\\Omega_{\\rm in})\\oplus H_{0}^{1}(\\Omega_{\\rm ex})$.\nSince $\\widehat Z_{\\mathsf v,\\alpha}$ contains a positive half-line, $\\Delta^{\\!\\mathsf v,d}$ is bounded from above and hypothesis (H4.1) holds. The scattering couple $(\\Delta^{\\!\\mathsf v,d},\\Delta)$ is asymptotically complete and the corresponding scattering matrix is given by \n$$\n{\\mathcal S}_{\\lambda}^{\\mathsf v,d}=1-2\\pi iL_{\\lambda}\\Lambda^{\\mathsf v,d,+}_{\\lambda}L_{\\lambda}^{*}\\,,\\quad \\lambda\\in(-\\infty,0]\\backslash(\\sigma^{-}_{p}(\\Delta+\\mathsf v)\\cup \\sigma^{-}_{p}(\\Delta^{\\!\\mathsf v,d}))\\,,\n$$\nwhere $L_{\\lambda}$ is given in Corollary \\ref{Llambda} and $\\Lambda^{\\mathsf v,d,+}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\Lambda^{\\mathsf v,d}_{\\lambda+i\\epsilon}$. This latter limit exists by Lemma \\ref{rmH7}; in particular, by \\eqref{LBpm2}, \n\\begin{align*}\n\\Lambda^{\\mathsf v,d,+}=&\\left(1+\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&-(S^{\\mathsf v,+}_{\\lambda})^{-1}\\end{bmatrix}\\begin{bmatrix}\n-S\\!L^{+}_{\\lambda}\n(S^{\\mathsf v,+}_{\\lambda})^{-1}(S\\!L^{-}_{\\lambda})^{*}& S\\!L^{+}_{\\lambda}\n\\\\(S\\!L^{-}_{\\lambda})^{*}& 0\n\\end{bmatrix}\\right)\\times\\\\\n&\\times\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&-(S^{\\mathsf v,+}_{\\lambda})^{-1}\\end{bmatrix}\\,,\n\\end{align*}\nwhere\n$$\nR^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}R_{\\lambda\\pm i\\epsilon}\\,,\\qquadS\\!L^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}S\\!L_{\\lambda\\pm i\\epsilon}\\,,\\qquad \nS^{\\mathsf v,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\gamma_{0}S\\!L^{\\mathsf v}_{\\lambda\\pm i\\epsilon}\\,.\n$$\n\\subsection{\\label{Sec_neumann} Short-range potentials and Neumann boundary conditions.} \nHere we take \n$$\n\\mathfrak h_{2}=\\mathfrak b_{2}^{*}=\\mathfrak b_{2,2}= H^{1\/2}(\\Gamma)\\hookrightarrow\\mathfrak h_{2}^{\\circ}=L^{2}(\\Gamma)\\hookrightarrow\\mathfrak b_{2}=\\mathfrak h^{*}_{2}=\\mathfrak b_{2,2}^{*}=H^{-1\/2}(\\Gamma)\n\\,,\n$$\n$$\n\\tau_{2}=\\gamma_{1}:H^{2}(\\mathbb{R}^{3})\\to H^{1\/2}(\\Gamma)\\,,\\qquad\nB_{0}=0\\,,\\qquad B_{2}=1\\,.\n$$\nFor any $z\\in\\mathbb{C}\\backslash(-\\infty,0]$, one has\n$$\nM_{z}^{\\mathsf B\n=\\begin{bmatrix}\n1 & 0\\\\\n0 & 0%\n\\end{bmatrix}-\n\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v & 0\\\\\n0 & 1%\n\\end{bmatrix}\n\\begin{bmatrix}\n\\langle x\\rangle^{-s}R_{z}\\langle x\\rangle^{-s} & \\langle x\\rangle^{-s}R_{z}\\gamma_{1}^{*}\\\\\n\\gamma_{1}R_{z}\\langle x\\rangle^{-s} & \\gamma_{1}R_{z}\\gamma_{0}^{*}%\n\\end{bmatrix}\n=\\begin{bmatrix}\\langle x\\rangle^{s} & 0\\\\\n0 & 1\n\\end{bmatrix}M_{z}^{\\mathsf v,n}\n\\begin{bmatrix}\\langle x\\rangle^{s} & 0\\\\\n0 & 1%\n\\end{bmatrix}\\,,\n$$\n$$\nM_{z}^{\\mathsf v,n}:=%\n\\begin{bmatrix}\n1-\\mathsf v R_{z} & -\\mathsf v D\\!L_{z}\\\\\n-D\\!L_{\\bar z}^{*} & -D_{z}%\n\\end{bmatrix}\n\\,.\n$$\nBy the mapping properties provided in Sections \\ref{Sec_V} and \\ref{Sec_Layer}, by \\eqref{v-w} and \\eqref{R-w} with $w=-s$, one gets\n$$\nM_{z}^{\\mathsf v,n}\\in{\\mathscr B}(H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{1\/2}(\\Gamma),\nH_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{-1\/2}(\\Gamma))\\,.\n$$\nBy Lemma \\ref{coerc} with $\\mathsf v=0$, for any $z\\in Z^{\\circ}_{0,n}\\not=\\varnothing $, \n$$\n(M_{z}^{B_{0},B_{2}})^{-1}=\\Lambda_{z}^{B_{0},B_{2}}=\n(M_{z}^{n})^{-1}=\\Lambda_{z}^{\\!n}:=-D_{z}^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma))\\,.\n$$\nThus,\n$$\nZ_{B_{0},B_{2}}=Z_{n}:=\\{z\\in \\mathbb{C}\\backslash(-\\infty,0]: (M_{z}^{n})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma))\\}\\supseteq Z^{\\circ}_{0,n}\\,.\n$$\nBy Lemma \\ref{coerc} again, for any $z\\in Z^{\\circ}_{\\mathsf v,n}\\not=\\varnothing $, \n$$\n(\\widehat M_{z}^{B_{0},B_{2}})^{-1}=(\\widehat \\Lambda_{z}^{B_{0},B_{2}})^{-1}=(\\widehat M_{z}^{\\mathsf v,n})^{-1}=\\widehat \\Lambda_{z}^{\\mathsf v,n}:=-(D^{\\mathsf v}_{z})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma))\\,.\n$$\nThus,\n$$\n\\widehat Z_{\\mathsf B}=\\widehat Z_{n}:=\\{z\\in \\varrho(\\Delta+\\mathsf v): (\\widehat M_{z}^{\\mathsf v,n})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{1\/2}(\\Gamma))\\}\\supseteq Z^{\\circ}_{\\mathsf v,n}\\,.\n$$\nHence, \n\\begin{align*}\n\\Lambda^{\\! \\mathsf B}_{z}=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}(M_{z}^{\\mathsf v,n})^{-1}\n\\begin{bmatrix}\\langle x\\rangle^{-s}&0\\\\0&1\\end{bmatrix}\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v & 0\\\\\n0 & 1%\n\\end{bmatrix}=\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,n}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\\,,\n\\end{align*}\nwhere, by \\eqref{LB-new} and by \\eqref{LB-new2}\n\\begin{align*}\n{\\Lambda}_{z}^{\\!\\mathsf v,n}:=&\\begin{bmatrix}\n\\Lambda_{z}^{\\!\\mathsf v}-\\Lambda_{z}^{\\!\\mathsf v}D\\!L_{z}(D^{\\mathsf v}_{z})^{-1}D\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v}& -\\Lambda_{z}^{\\!\\mathsf v}D\\!L_{z}(D^{\\mathsf v}_{z})^{-1}\\\\\n-(D^{\\mathsf v}_{z})^{-1}D\\!L_{\\bar z}^{*}\\Lambda_{z}^{\\!\\mathsf v} & -(D^{\\mathsf v}_{z})^{-1}\n\\end{bmatrix}\\\\\n=&\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&-(D^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\n-D\\!L_{z}(D^{\\mathsf v}_{z})^{-1}D\\!L_{\\bar z}^{*}& D\\!L_{z}\n\\\\ D\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&-(D^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\end{align*}\nOne has \n\\begin{equation}\\label{Lbbv-neu}\n{\\Lambda}_{z}^{\\!\\mathsf v,n}\n\\in{\\mathscr B}(H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)\\oplus H^{-1\/2}(\\Gamma),H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{1\/2}(\\Gamma))\\,.\n\\end{equation}\nBy Theorems \\ref{Th_Krein} and \\ref{Th-alt-res}, there follows that \n\\begin{align}\n&R_{z}^{\\mathsf v,n}\n=R_{z}+\n\\begin{bmatrix}R_{z}\\langle x\\rangle^{-s}&D\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,n}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\\begin{bmatrix}\\langle x\\rangle^{2s}\\mathsf v \\langle x\\rangle^{-s}R_{z}\\\\ D\\!L^{*}_{\\bar z}\\end{bmatrix}\\label{Rv-neu-0}\\\\ \n=&R_{z}+\n\\begin{bmatrix}R_{z}&D\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\!\\!\\!-(D^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\n-D\\!L_{z}(D^{\\mathsf v}_{z})^{-1}D\\!L_{\\bar z}^{*}& D\\!L_{z}\n\\\\ D\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\!\\!\\!-(D^{\\mathsf v}_{z})^{-1}\\end{bmatrix}\n\\begin{bmatrix}R_{z}\\\\ D\\!L^{*}_{\\bar z}\\end{bmatrix}\\label{Rv-neu-1}\\\\ \n=&R_{z}^{\\mathsf v}-D\\!L_{z}^{\\mathsf v}(D_{z}^{\\mathsf v})^{-1}{D\\!L^{\\mathsf v}_{\\bar z}}^{*}\n\\label{Rv-neu-2}\n\\end{align}\nis the resolvent of a self-adjoint operator $\\Delta^{\\!\\mathsf v,n}$; \\eqref{Rv-neu-0} holds for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,n})\\cap\\mathbb{C}\\backslash(-\\infty,0]$, both \\eqref{Rv-neu-1} and \\eqref{Rv-neu-2} hold for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,n})\\cap\\varrho(\\Delta+\\mathsf v)$.\n\\par \nBy Theorem \\ref{Th-add} and by $[\\gamma_{0}]u=\\widehat\\rho_{\\mathsf B}u$ for any $u\\in\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,n})$,\n$$\n\\Delta^{\\!\\mathsf v,n}\\,u=\\Delta u+\\mathsf v u+([\\gamma_{0}]u)\\delta'_{\\Gamma}\\,.\n$$\nBy \\eqref{Rv-alpha-2} and by the mapping properties of $D\\!L^{\\mathsf v}_{z}$, one has \n$$\n\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,n})\\subseteq H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)\\,.\n$$ \nMoreover, by Lemma \\ref{alt-abc}, \n$$\nu\\in\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,n})\\quad\\Longrightarrow\\quad\\gamma_{1}u=0\\,.\n$$\nSince $\\widehat Z_{\\mathsf v,n}$ contains a positive half-line, $\\Delta^{\\!\\mathsf v,n}$ is bounded from above and hypothesis (H4.1) holds. The scattering couple $(\\Delta^{\\!\\mathsf v,n},\\Delta)$ is asymptotically complete and the corresponding scattering matrix is given by \n$$\n{\\mathcal S}_{\\lambda}^{\\mathsf v,n}=1-2\\pi iL_{\\lambda}\\Lambda^{\\mathsf v,n,+}_{\\lambda}L_{\\lambda}^{*}\\,,\\quad \\lambda\\in(-\\infty,0]\\backslash(\\sigma^{-}_{p}(\\Delta+\\mathsf v)\\cup \\sigma^{-}_{p}(\\Delta^{\\!\\mathsf v,n}))\\,,\n$$\nwhere $L_{\\lambda}$ is given in Corollary \\ref{Llambda} and $\\Lambda^{\\mathsf v,n,+}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\Lambda^{\\mathsf v,n}_{\\lambda+i\\epsilon}$. This latter limit exists by Lemma \\ref{rmH7}; in particular, by \\eqref{LBpm2}, \n\\begin{align*}\n\\Lambda^{\\mathsf v,n,+}=&\\left(1+\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&-(D^{\\mathsf v,+}_{\\lambda})^{-1}\\end{bmatrix}\\begin{bmatrix}\n-D\\!L^{+}_{\\lambda}\n(D^{\\mathsf v,+}_{\\lambda})^{-1}(D\\!L^{-}_{\\lambda})^{*}& D\\!L^{+}_{\\lambda}\n\\\\(D\\!L^{-}_{\\lambda})^{*}& 0\n\\end{bmatrix}\\right)\\times\\\\\n&\\times\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&-(D^{\\mathsf v,+}_{\\lambda})^{-1}\\end{bmatrix}\\,,\n\\end{align*}\nwhere\n$$\nR^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}R_{\\lambda\\pm i\\epsilon}\\,,\\qquadD\\!L^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}D\\!L_{\\lambda\\pm i\\epsilon}\\,,\\qquad \nD^{\\mathsf v,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\gamma_{1}D\\!L^{\\mathsf v}_{\\lambda\\pm i\\epsilon}\\,.\n$$\n\\subsection{\\label{Sec_delta'} Short-range potentials and semi-transparent boundary conditions\nof $\\delta'_{\\Gamma }$-type} \nHere we take \n$$\n\\mathfrak h_{2}=\\mathfrak b_{2}^{*}=\\mathfrak b_{2,2}= H^{1\/2}(\\Gamma)\\hookrightarrow\\mathfrak h_{2}^{\\circ}=L^{2}(\\Gamma)\\hookrightarrow \\mathfrak b_{2}=\\mathfrak h^{*}_{2}=\\mathfrak b_{2,2}^{*}=H^{-1\/2}(\\Gamma)\n\\,,\n$$\n$$\n\\tau_{2}=\\gamma_{1}:H^{2}(\\mathbb{R}^{3})\\to H^{1\/2}(\\Gamma)\\,,\\qquad\nB_{0}=\\theta\\,,\\qquad B_{2}=1\\,,\n$$\nwhere \n$$ \n\\theta\\in\\mathscr B(H^{s_{\\circ}}(\\Gamma),H^{-s_{\\circ}}(\\Gamma))\\,,\\quad \n02$, fulfills our hypothesis. \nLet us also remark that $\\mathscr B(H^{s_{\\circ}}(\\Gamma),H^{-s_{\\circ}}(\\Gamma))\\subseteq\n\\mathscr B(H^{1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))=\\mathscr B(\\mathfrak b_{2}^{*},\\mathfrak b_{2,2}^{*})$.\\par\nFor any $z\\in\\mathbb{C}\\backslash(-\\infty,0]$, one has\n$$\nM_{z}^{\\mathsf B\n=\\begin{bmatrix}\n 1 & 0\\\\\n0 & \\theta%\n\\end{bmatrix}\n-\n\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v & 0\\\\\n0 & 1%\n\\end{bmatrix}\n\\begin{bmatrix}\n\\langle x\\rangle^{-s}R_{z}\\langle x\\rangle^{-s} & \\langle x\\rangle^{-s}R_{z}\\gamma_{1}^{*}\\\\\n\\gamma_{1}R_{z}\\langle x\\rangle^{-s} & \\gamma_{1}R_{z}\\gamma_{1}^{*}%\n\\end{bmatrix}=\n\\begin{bmatrix}\\langle x\\rangle^{s} & 0\\\\\n0 & 1\\end{bmatrix} \nM_{z}^{\\mathsf v,\\theta}\n\\begin{bmatrix}\\langle x\\rangle^{-s} & 0\\\\\n0 & 1\\end{bmatrix}\\,,\n$$\n$$\nM_{z}^{\\mathsf v,\\theta}:=%\n\\begin{bmatrix}\n1-\\mathsf v R_{z} & -\\mathsf v D\\!L_{z}\\\\\n-D\\!L_{\\bar z}^{*} & \\theta-D_{z}\n\\end{bmatrix}\\,.%\n$$\nBy the mapping properties provided in Sections \\ref{Sec_V} and \\ref{Sec_Layer}, by \\eqref{v-w} and \\eqref{R-w} with $w=-s$, one gets\n$$\nM_{z}^{\\mathsf v,\\theta}\\in{\\mathscr B}(H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{1\/2}(\\Gamma),H_{-s}^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{-1\/2}(\\Gamma))\\,.\n$$\n\\begin{lemma}\\label{LLt} \n$$\n\\forall z\\in \\widehat Z^{\\circ}_{\\mathsf v,n}:=Z^{\\circ}_{\\mathsf v,n}\\cap\\mathbb{C}\\backslash\\mathbb{R}\\,,\\qquad (1-\\theta (D^{\\mathsf v}_{z})^{-1})^{-1}\\in \\mathscr B( H^{-1\/2}( \\Gamma)) \\,.\n$$\n\\end{lemma}\n\\begin{proof}\nWe follow the same the arguments as in the proof of \\cite[Lemma 5.4]{JMPA}. Since, by the compact embedding $H^{-s_{\\circ}}(\\Gamma)\\hookrightarrow H^{-1\/2}(\\Gamma)$, $\\theta( D_{z}^{\\mathsf v})^{-1}\\in{\\mathscr B}(H^{-1\/2}(\\Gamma))$ is compact, \nby the Fredholm alternative, $1-\\theta( D_{z}^{\\mathsf v}) ^{-1}$ has a bounded inverse if and only if it has trivial kernel. Let $\\varphi\\in H^{-1\/2}(\\Gamma)$ be such that $D_{z}^{\\mathsf v}\\varphi=\\theta\\varphi$; using the self-adjointness of $\\theta$, we get%\n\\[\n( D_{z}^{\\mathsf v}-D_{\\bar z}^{\\mathsf v}) \\varphi=0\\,.\n\\]\nBy the resolvent identity,\n\\[\n\\text{Im}(z)\\gamma_{1}R_{\\bar z}^{\\mathsf v}R_{z}^{\\mathsf v}%\n\\gamma_{1}^{\\ast}\\varphi=0\\,.\n\\]\nThis gives \n\\begin{equation}\n\\|R_{z}^{\\mathsf v}\\gamma_{1}^{\\ast}\\varphi\\| _{L^{2}(\\mathbb{R}^{3})}=0\\,.\n\\end{equation}\nSince $(R_{z}^{\\mathsf v}\\gamma_{1}^{\\ast})^{\\ast}=\\gamma_{1}R_{\\bar\n{z}}^{\\mathsf v}\\in{\\mathscr B}( L^{2}( \\mathbb{R}^{3}),H^{1\/2}(\\Gamma)) $ is surjective, then\n$R_{z}^{\\mathsf v}\\gamma_{1}^{\\ast}\\in{\\mathscr B}( H^{-1\/2}( \\Gamma) ,L^{2}( \\mathbb{R}^{3})) $ has closed\nrange by the closed range theorem and, by \\cite[Theorem 5.2, p. 231]{Kato},\n\\[\n\\|R_{z}^{\\mathsf v}\\gamma_{1}^{\\ast}\\varphi\\|_{L^{2}(\n\\mathbb{R}^{3}) }\\gtrsim\\|\\varphi\\|_{H^{-1\/2}(\\Gamma) }\\,.\n\\]\nThus $\\text{\\rm ker}(1-\\theta (D_{z}^{\\mathsf v})^{-1})=\\{0\\}$ and the proof is done.\n\\end{proof}\nAccording to Lemma \\ref{LLt} with $\\mathsf v=0$, for any $z\\in \\widehat Z^{\\circ}_{0,n}\\not=\\varnothing $,\n$$\n(M_{z}^{B_{0},B_{2}})^{-1}=(M_{z}^{\\theta})^{-1}=\\Lambda_{z}^{\\!\\theta}:=(\\theta-D_{z})^{-1}=\n-D_{z}^{-1}(1-\\theta D_{z}^{-1})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\,.\n$$\nThus\n$$\nZ_{B_{0},B_{2}}=Z_{\\theta}:=\\{z\\in \\mathbb{C}\\backslash(-\\infty,0]:(M_{z}^{\\theta})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\}\\supseteq\\widehat Z^{\\circ}_{0,n}\\,.\n$$\nAccording to Lemma \\ref{LLt} again, for any $z\\in \\widehat Z^{\\circ}_{\\mathsf v,n}\\not=\\varnothing $,\n$$\n(\\widehat M_{z}^{B_{0},B_{2}})^{-1}=(\\widehat M_{z}^{\\mathsf v,\\theta})^{-1}=\\widehat \\Lambda_{z}^{\\mathsf v,\\theta}:=(\\theta-D_{z}^{\\mathsf v})^{-1}=\n-(D^{\\mathsf v}_{z})^{-1}(1-\\theta (D^{\\mathsf v}_{z})^{-1})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\,.\n$$\nThus\n$$\n\\widehat Z_{\\mathsf B}=\\widehat Z_{\\mathsf v,\\theta}:=\\{z\\in \\varrho(\\Delta+\\mathsf v):(\\widehat M_{z}^{\\mathsf v,\\theta})^{-1}\\in\\mathscr B(H^{-1\/2}(\\Gamma),H^{-1\/2}(\\Gamma))\\}\\supseteq\\widehat Z^{\\circ}_{\\mathsf v,n}\\,.\n$$\nHence, \n$$\n\\Lambda_{z}^{\\mathsf B}=\\begin{bmatrix}\\langle x\\rangle^{s} & 0\\\\\n0 & 1\\end{bmatrix} \n(M_{z}^{\\mathsf v,\\theta})^{-1}\n\\begin{bmatrix}\\langle x\\rangle^{-s} & 0\\\\\n0 & 1\\end{bmatrix}\\begin{bmatrix}\n\\langle x\\rangle^{2s}\\mathsf v & 0\\\\\n0 & 1\n\\end{bmatrix}=\\begin{bmatrix}\\langle x\\rangle^{s} & 0\\\\\n0 & 1\\end{bmatrix} \n\\Lambda_{z}^{\\!\\mathsf v,\\theta}\n\\begin{bmatrix}\\langle x\\rangle^{s} & 0\\\\\n0 & 1\\end{bmatrix}\\,,\n$$\nwhere, by \\eqref{LB-new} and by \\eqref{LB-new2},\n\\begin{align*}\n\\Lambda_{z}^{\\!\\mathsf v,\\theta}:=&\\begin{bmatrix}\n\\Lambda_{z}^{\\!\\mathsf v,\\theta}\\Lambda^{\\!\\mathsf v}_{z}+\\Lambda^{\\!\\mathsf v}_{z} D\\!L_{z}\\widehat\\Lambda^{\\mathsf v,\\theta}_{z}D\\!L_{\\bar z}^{*}\\Lambda^{\\!\\mathsf v}_{z}\n& \\Lambda^{\\mathsf v}_{z} D\\!L_{z}\\widehat\\Lambda^{\\mathsf v,\\theta}_{z}\\\\\n\\widehat\\Lambda^{\\mathsf v,\\theta}_{z}D\\!L_{\\bar z}^{*}\\Lambda^{\\!\\mathsf v}_{z}& \\widehat\\Lambda^{\\mathsf v,\\theta}_{z}\\end{bmatrix}\\\\\n=&\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\theta}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\nD\\!L_{z}\\widehat\\Lambda_{z}^{\\mathsf v,\\theta}D\\!L_{\\bar z}^{*}& D\\!L_{z}\n\\\\ D\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\theta}\\end{bmatrix}\n\\,.\n\\end{align*}\nOne has \n\\begin{equation}\\label{Ltbbteta}\n{\\Lambda}_{z}^{\\!\\mathsf v,\\theta}\n\\in{\\mathscr B}(H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)\\oplus H^{-1\/2}(\\Gamma),H_{-s}^{1}( \\mathbb{R}^{3}\\backslash\\Gamma)^{*}\\oplus H^{1\/2}(\\Gamma))\\,.\n\\end{equation}\nBy Theorems \\ref{Th_Krein} and \\ref{Th-alt-res}, there follows that\n\\begin{align}\nR_{z}^{\\mathsf v,\\theta}\n=&R_{z}+\\begin{bmatrix}R_{z}\\langle x\\rangle^{-s}&D\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}{\\Lambda}_{z}^{\\!\\mathsf v,\\theta}\\begin{bmatrix}\\langle x\\rangle^{s}&0\\\\0&1\\end{bmatrix}\n\\begin{bmatrix}\\langle x\\rangle^{2s}\\mathsf v \\langle x\\rangle^{-s}R_{z}\\\\D\\!L^{*}_{\\bar z}\\end{bmatrix}\n\\label{Rv-teta-0}\\\\\n=&R_{z}+\n\\begin{bmatrix}R_{z}&D\\!L_{z}\\end{bmatrix}\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\theta}\\end{bmatrix}\n\\left(1+\\begin{bmatrix}\nD\\!L_{z}\\widehat\\Lambda_{z}^{\\mathsf v,\\theta}D\\!L_{\\bar z}^{*}& D\\!L_{z}\n\\\\ D\\!L_{\\bar z}^{*} & 0\n\\end{bmatrix}\\right)\n\\begin{bmatrix}\\Lambda^{\\mathsf v}_{z}&0\\\\0&\\widehat\\Lambda_{z}^{\\mathsf v,\\theta}\\end{bmatrix}\n\\begin{bmatrix}R_{z}\\\\D\\!L^{*}_{\\bar z}\\end{bmatrix}\n\\label{Rv-teta-1}\\\\ \n=&R_{z}^{\\mathsf v}+D\\!L_{z}^{\\mathsf v}{\\widehat\\Lambda}_{z}^{\\mathsf v,\\theta}{D\\!L_{\\bar z}^{\\mathsf v}}^{*}\\,.\n\\label{Rv-teta-2}\n\\end{align}\nis the resolvent of a self-adjoint operator $\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta}$; \\eqref{Rv-teta-0} holds for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta})\\cap\\mathbb{C}\\backslash(-\\infty,0]$, both \\eqref{Rv-teta-1} and \\eqref{Rv-teta-2} hold for any \n$z\\in \\varrho(\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta})\\cap\\varrho(\\Delta+\\mathsf v)$.\n\\par\nBy Theorem \\ref{Th-add},\n$$\n\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta}u=\\Delta u+\\mathsf v u+(\\theta\\gamma_{1}u)\\delta'_{\\Gamma}\\,.\n$$\nBy \\eqref{Rv-alpha-2} and by the mapping properties of $D\\!L^{\\mathsf v}_{z}$, one has \n$$\n\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta})\\subseteq H^{1}(\\mathbb{R}^{3}\\backslash\\Gamma)\\,.\n$$ \nMoreover, by $R^{\\mathsf v}_{z}u\\in H^{2}(\\mathbb{R}^{3})$, so that $[\\gamma_{1}]R^{\\mathsf v}_{z}u=0$, and by \\eqref{jumpv1}, one gets $[\\gamma_{0}]R^{\\mathsf v,\\theta}_{z}u={\\widehat\\Lambda}_{z}^{\\mathsf v,\\theta}{D\\!L_{\\bar z}^{\\mathsf v}}^{*}u=\\widehat\\rho_{\\mathsf B}(R^{\\mathsf v,\\theta}_{z}u)$. Hence, by Lemma \\ref{alt-abc}, \n$$\nu\\in\\text{\\rm dom}(\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta})\\quad\\Longrightarrow\\quad\\gamma_{1}u=\\theta[\\gamma_{0}]u\\,.\n$$\nProceeding as in \\cite[Subsection 5.5]{JMPA} (see the proof of Theorem 5.15 there), $\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta}$ is bounded from above and so hypothesis (H4.1) holds. The scattering couple $(\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta},\\Delta)$ is asymptotically complete and the corresponding scattering matrix is given by \n$$\n{\\mathcal S}_{\\lambda}^{\\mathsf v,\\theta}=1-2\\pi iL_{\\lambda}\\Lambda^{\\mathsf v,\\theta,+}_{\\lambda}L_{\\lambda}^{*}\\,,\\quad \\lambda\\in(-\\infty,0]\\backslash(\\sigma^{-}_{p}(\\Delta+\\mathsf v)\\cup \\sigma^{-}_{p}(\\Delta^{\\!\\mathsf v,\\delta'\\!,\\theta}))\\,,\n$$\nwhere $L_{\\lambda}$ is given in Corollary \\ref{Llambda} and $\\Lambda^{\\mathsf v,\\theta,+}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\Lambda^{\\mathsf v,\\theta}_{\\lambda+i\\epsilon}$. This latter limit exists by Lemma \\ref{rmH7}; in particular, by \\eqref{LBpm2}, \n\\begin{align*}\n\\Lambda^{\\mathsf v,\\theta,+}=&\\left(1+\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&(\\theta-D^{\\mathsf v,+}_{\\lambda})^{-1}\\end{bmatrix}\\begin{bmatrix}\nD\\!L^{+}_{\\lambda}\n(\\theta-D^{\\mathsf v,+}_{\\lambda})^{-1}\\alpha(D\\!L^{-}_{\\lambda})^{*}& D\\!L^{+}_{\\lambda}\n\\\\(D\\!L^{-}_{\\lambda})^{*}& 0\n\\end{bmatrix}\\right)\\times\\\\\n&\\times\\begin{bmatrix}\n(1-\\mathsf v R^{+}_{\\lambda})^{-1}\\mathsf v&0\\\\\n0&(\\theta-D^{\\mathsf v,+}_{\\lambda})^{-1}\\alpha\\end{bmatrix}\\,,\n\\end{align*}\nwhere\n$$\nR^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}R_{\\lambda\\pm i\\epsilon}\\,,\\qquadD\\!L^{\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}S\\!L_{\\lambda\\pm i\\epsilon}\\,,\\qquad \nD^{\\mathsf v,\\pm}_{\\lambda}:=\\lim_{\\epsilon\\searrow 0}\\gamma_{0}D\\!L^{\\mathsf v}_{\\lambda\\pm i\\epsilon}\\,.\n$$\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThere is a commonly accepted idea that the sunspot activity is produced\nby the large-scale toroidal magnetic field which is generated inside\nthe convection zone by means of the differential rotation \\citep{P55}.\nThe theory explains the 11-year solar cycle as a result of the large-scale\ndynamo operating in the solar interior, where, in addition to the\nmagnetic fields generated by the differential rotation, the helical\nconvective motions transforms the energy of the toroidal magnetic\nfields to poloidal. The effect of meridional circulation on the large-scale\ndynamo is not well understood. It is the essential part of the flux-transport\ndynamo model scenario \\citep{choud95,cd99} to explain the equatorward\ndrift of the toroidal magnetic field in the solar cycle. Here, it\nassumed that the toroidal field at the bottom of the convection zone\nforms sunspot activity. Feasibility of this idea can be questioned\nboth the observational and theoretical arguments \\citep{b05}. The\ndistributed dynamo models can be constructed with \\citep{2002AA...390..673B,2008AA...483..949J,2014AA563A18P}\nand without \\citep{moss00M,pip13M} effect of meridional transport\nof the large-scale magnetic field.\n\nRecent results of helioseismology reveal the double-cell meridional\ncirculation structure \\citep{Zhao13m,2017ApJ845.2B}. It demolishes\nthe previously accepted scenario of the flux-transport models \\citep{2014ApJ782.93H,2016MNRAS.456.2654W,2016ApJ832.9H}.\nContrary, \\citet{PK13} showed the distributed dynamo models can reproduce\nobservations with regards to the subsurface rotational shear layer\nand the double-cell meridional circulation. In their model, the double-cell\nmeridional circulation was modeled in following to results of helioseismology\nof \\citet{Zhao13m}. The effect of the multi-cell meridional circulation\non the global dynamo was also studied in the direct numerical simulations\n\\citep{kap2012,2016ApJ819.104G,2018AA609A..51W}. There were no attempts\nto construct the non-kinematic mean-field dynamo models with regards\nto the multi-cell meridional circulation.\n\nThe standard mean-field models of the solar differential rotation\npredict a one-cell meridional circulation per hemisphere. This contradicts\nto the helioseismology inversions and results of direct numerical\nsimulations. In the mean-field theory framework, the differential\nrotation of the Sun is explained as a result of the angular momentum\ntransport by the helical convective motions. Similarly to a contribution\nof the $\\alpha$ effect in the mean-electromotive force, i.e., \n\\[\n\\mathbf{\\mathcal{E}}=\\left\\langle \\mathbf{u}\\times\\mathbf{b}\\right\\rangle =\\hat{\\alpha}\\circ\\left\\langle \\mathbf{B}\\right\\rangle +\\dots,\n\\]\nwhere $\\mathbf{u}$ is the turbulent velocity $\\mathbf{u}$, and $\\mathbf{b}$\nis the turbulent magnetic field, the $\\Lambda$-effect, (e.g., \\citealp{1989drsc.book.....R})\nappears as the non-dissipative part of turbulent stresses\\textbf{\n\\[\n\\hat{T}_{ij}=\\left\\langle u_{i}u_{j}\\right\\rangle =\\Lambda_{ijk}\\Omega_{k}+\\dots\n\\]\n}where $\\boldsymbol{\\Omega}$ is the angular velocity. The structure\nof the meridional circulation is determined by directions of the non-diffusive\nangular momentum transport due to the $\\Lambda$ effect \\citep{2017ApJ835.9B}.\nIn particular, the vertical structure of the meridional circulation\ndepends on the sign of the radial effect. It was found that the double-cell\nmeridional circulation can be explained if the of $\\Lambda$-effect\nchanges sign in the depth of the convection zone. \\citet{2018ApJ854.67P}\nshowed that this effect can result from the radial inhomogeneity of\nthe convective turnover timescale. It was demonstrated that if this\neffect is taken into account then the solar-like differential rotation\nand the double-cell meridional circulation are both reproduced by\nthe mean-field model .\n\nIn this paper, we apply the meridional circulation profile, which\nis calculated from the solution of the angular momentum balance to\nthe nonkinematic dynamo models. Previously, the similar approach was\napplied by \\citet{1992AA...265..328B} and \\citet{2006ApJ...647..662R}\nin the distributed and the flux-transport models with one meridional\ncirculation cell as the basic stage in the non-magnetic case.\n\nOur main goal is to study how the double-cell meridional circulation\naffects the nonlinear dynamo generation of the large-scale magnetic\nfield. The magnetic feedback on the global flow can result in numerous\nphysical phenomena such as the torsional oscillations \\citep{1982SoPh...75..161L,2011JPhCS271a2074H},\nthe long-term variability of the magnetic activity \\citep{sok1994AA,2014JGRA119.6027F}\netc. The properties of the nonlinear evolution depend on the dynamo\ngoverning parameters such as amplitude of turbulent generation of\nthe magnetic field by the $\\alpha$ effect, as well as the other nonlinear\nprocesses involved in the dynamo, i.e., the dynamo quenching by the\nmagnetic buoyancy effect \\citep{kp93,tob98} and the magnetic helicity\nconservation \\citep{kleruz82}. We study if the long-term variation\nof magnetic activity can result from the increasing level of turbulent\ngeneration of magnetic field by the $\\alpha$ effect. The increasing\nof the $\\alpha$ effect results to an increase of the magnetic helicity\nproduction. This affects the large-scale magnetic field generation\nby means of the magnetic helicity conservation. Hence, the magnetic\nhelicity balance has to be taken into account. \n\n{It is hardly possible to consider in full all the goals within\none paper. From our point of view, the most important tasks includes:\nconstruction of the solar-type dynamo model with the multi-cell meridional\ncirculation and studying the principal nonlinear dynamo effects. The\nlatter includes the magnetic helicity conservation and the nonkinematic\neffects due to the magnetic feedback on the large-scale flow. Accordingly,\nthe paper is organized as follows.} Next Section describes the hydrodynamic,\nthermodynamic and magnetohydrodynamic parts of the model. Then, I\npresent an attempt to construct the solar-type dynamo model and discuss\nthe effect of the turbulent pumping on the properties of the dynamo\nsolution. {The next subsections consider result for the principal\nnonlinear dynamo effects. They deals with the global flows variations,\nthe Grand activity cycles and the magnetic cycle variations of the\nthermodynamic parameters in the model.} The paper is concluded with\na discussion of the main results using results of other theoretical\nstudies and results of observations.\n\n\\section{Basic equations.}\n\n\\subsection{The angular momentum balance\\label{subsec:am}}\n\nWe consider the evolution of the axisymmetric large-scale flow, which\nis decomposed into poloidal and toroidal components: $\\mathbf{\\overline{U}}=\\mathbf{\\overline{U}}^{m}+r\\sin\\theta\\Omega\\hat{\\mathbf{\\boldsymbol{\\phi}}}$,\nwhere $\\boldsymbol{\\hat{\\phi}}$ is the unit vector in the azimuthal\ndirection. The mean flow satisfies the stationary continuity equation,\n\\begin{equation}\n\\boldsymbol{\\nabla}\\cdot\\overline{\\rho}\\mathbf{\\overline{U}}=0,\\label{eq:cont}\n\\end{equation}\nDistribution of the angular velocity inside convection zone is determined\nby conservation of the angular momentum \\citep{1989drsc.book.....R}:\n\\begin{eqnarray}\n\\frac{\\partial}{\\partial t}\\overline{\\rho}r^{2}\\sin^{2}\\theta\\Omega & = & -\\boldsymbol{\\nabla\\cdot}\\left(r\\sin\\theta\\overline{\\rho}\\left(\\hat{\\mathbf{T}}_{\\phi}+r\\sin\\theta\\Omega\\mathbf{\\overline{U}^{m}}\\right)\\right)\\label{eq:angm}\\\\\n & + & \\boldsymbol{\\nabla\\cdot}\\left(r\\sin\\theta\\frac{\\overline{\\mathbf{B}}\\overline{B}_{\\phi}}{4\\pi}\\right).\\nonumber \n\\end{eqnarray}\nTo determine the meridional circulation we consider the azimuthal\ncomponent of the large-scale vorticity , $\\omega=\\left(\\boldsymbol{\\nabla}\\times\\overline{\\mathbf{U}}^{m}\\right)_{\\phi}$\n, which is governed by equation: \n\\begin{eqnarray}\n\\frac{\\partial\\omega}{\\partial t}\\!\\!\\! & \\negthinspace\\!=\\!\\!\\!\\! & r\\sin\\theta\\boldsymbol{\\nabla}\\cdot\\left(\\frac{\\hat{\\boldsymbol{\\phi}}\\times\\boldsymbol{\\nabla\\cdot}\\overline{\\rho}\\hat{\\mathbf{T}}}{r\\overline{\\rho}\\sin\\theta}-\\frac{\\mathbf{\\overline{U}}^{m}\\omega}{r\\sin\\theta}\\right)\\!\\!+r\\sin\\theta\\frac{\\partial\\Omega^{2}}{\\partial z}\\label{eq:vort}\\\\\n & +\\!\\!\\! & \\frac{1}{\\overline{\\rho}^{2}}\\left[\\boldsymbol{\\nabla}\\overline{\\rho}\\times\\boldsymbol{\\nabla}\\overline{p}\\right]_{\\phi}\\!\\!\\nonumber \\\\\n & + & \\!\\frac{1}{\\overline{\\rho}^{2}}\\left[\\!\\!\\boldsymbol{\\nabla}\\overline{\\rho}\\times\\left(\\!\\!\\boldsymbol{\\nabla}\\frac{\\overline{\\mathbf{B}}^{2}}{8\\pi}-\\frac{\\left(\\overline{\\mathbf{B}}\\boldsymbol{\\cdot\\nabla}\\right)\\overline{\\mathbf{B}}}{4\\pi}\\!\\right)\\!\\!\\right]_{\\phi},\\nonumber \n\\end{eqnarray}\nThe turbulent stresses tensor, $\\hat{\\mathbf{T}}$, is written in\nterms of small-scale fluctuations of velocity and magnetic field:\n\\begin{equation}\n\\hat{T}_{ij}=\\left(\\left\\langle u_{i}u_{j}\\right\\rangle -\\frac{1}{4\\pi\\overline{\\rho}}\\left(\\left\\langle b_{i}b_{j}\\right\\rangle -\\frac{1}{2}\\delta_{ij}\\left\\langle \\mathbf{b}^{2}\\right\\rangle \\right)\\right).\\label{eq:stres}\n\\end{equation}\nwhere ${\\partial\/\\partial z=\\cos\\theta\\partial\/\\partial r-\\sin\\theta\/r\\cdot\\partial\/\\partial\\theta}$\nis the gradient along the axis of rotation. The turbulent stresses\naffect generation and dissipation of large-scale flows, and they are\naffected by the global rotation and magnetic field. The magnitude\nof the kinetic coefficients in tensor $\\hat{\\mathbf{T}}$ depends\non the rms of the convective velocity, ${u}'$, the strength of the\nCoriolis force and the strength of the large-scale magnetic field.\nThe effect of the Coriolis force is determined by parameter $\\Omega^{*}=2\\Omega_{0}\\tau_{c}$,\nwhere $\\Omega_{0}=2.9\\times10^{-6}$rad\/s is the solar rotation rate\nand $\\tau_{c}$ is the convective turnover time. The effect of the\nlarge-scale magnetic field on the convective turbulence is determined\nby parameter $\\beta=\\left\\langle \\left|\\mathbf{B}\\right|\\right\\rangle \/\\sqrt{4\\pi\\overline{\\rho}u'^{2}}$.\n\nThe magnetic feedback on the coefficients of turbulent stress tensor\n$\\hat{\\mathbf{T}}$ was studied previously with the mean-field magnetohydrodynamic\nframework \\citep{rob-saw}. In our model we apply analytical results\nof \\citet{1994AN....315..157K}, \\citet{kit-rud-kuk} and \\citet{kuetal96}.\nThe analytical expression for $\\hat{\\mathbf{T}}$ is given in Appendix.\nIt was found that the standard components of the nondissipative of\n$\\hat{\\mathbf{T}}$ ($\\Lambda$-effect) are quenched with the increase\nof the magnetic field strength as $\\beta^{-2}$ and the magnetic quenching\nof the viscous parts is the order of $\\beta^{-1}$. Also, there is a non-trivial\neffect inducing the latitudinal angular momentum flux proportional\nto the magnetic energy \\citep{kit-rud-kuk,kuetal96}. This effect\nis quenched as $\\beta^{-2}$ for the case of the strong magnetic field.\nImplications of the magnetic feedback on the turbulent stress tensor\n$\\hat{\\mathbf{T}}$ were discussed in the models of solar torsional\noscillations and Grand activity cycles \\citep{kit-rud-kuk,kuetal96,p99,1999AA343.977K}.\nThe analytical results of the mean-field theory are in qualitative\nagreement with the direct numerical simulations \\citep{2007AN....328.1006K,kap2011,2017arXiv171208045K}.\n\nProfile of $\\tau_{c}$ (as well as profiles of $\\overline{\\rho}$\nand other thermodynamic parameters) is obtained from a standard solar\ninterior model calculated using the MESA code \\citep{mesa11,mesa13}.\nThe rms velocity, $u'$, is determined in the mixing length approximations\nfrom the gradient of the mean entropy, $\\overline{s}$, \n\\begin{equation}\nu'=\\frac{\\ell}{2}\\sqrt{-\\frac{g}{2c_{p}}\\frac{\\partial\\overline{s}}{\\partial r}},\\label{eq:uc}\n\\end{equation}\nwhere $\\ell=\\alpha_{MLT}H_{p}$ is the mixing length, $\\alpha_{MLT}=2.2$\nis the mixing length theory parameter, and $H_{p}$ is the pressure\nscale height. For a non-rotating star the ${u}'$ profile corresponds\nto results of the MESA code. The mean-field equation for heat transport\ntakes into account effects of rotation and magnetic field \\citep{2000ARep...44..771P}:\n\\begin{equation}\n\\overline{\\rho}\\overline{T}\\left(\\frac{\\partial\\overline{s}}{\\partial t}+\\left(\\overline{\\mathbf{U}}\\cdot\\boldsymbol{\\nabla}\\right)\\overline{s}\\right)=-\\boldsymbol{\\nabla}\\cdot\\left(\\mathbf{F}^{conv}+\\mathbf{F}^{rad}\\right)-\\hat{T}_{ij}\\frac{\\partial\\overline{U}_{i}}{\\partial r_{j}}-\\frac{1}{4\\pi}\\boldsymbol{\\mathcal{E}}\\cdot\\nabla\\times\\boldsymbol{\\overline{B}},\\label{eq:heat}\n\\end{equation}\nwhere, $\\overline{\\rho}$ and $\\overline{T}$ are the mean density\nand temperature, $\\boldsymbol{\\mathcal{E}}=\\left\\langle \\mathbf{u\\times b}\\right\\rangle $\nis the mean electromotive force. The Eq.(\\ref{eq:heat}) includes\nthe thermal energy loss and gain due to generation and dissipation\nof large-scale flows. The last term of the Eq.(\\ref{eq:heat}) takes\ninto account effect of thermal energy exchange because of dissipation\nand generation of magnetic field \\citep{2000ARep...44..771P}. In\nderivation of the mean-field heat transport equation (see, \\citealp{2000ARep...44..771P}),\nit was assumed that the magnetic and rotational perturbations of the\nreference thermodynamic state are small. Also the parameters of the\nreference state are given independently by the MESA code.\n\nFor the anisotropic convective flux we employ the expression suggested\nby \\citet{1994AN....315..157K} (hereafter KPR94), \n\\begin{equation}\nF_{i}^{conv}=-\\overline{\\rho}\\overline{T}\\chi_{ij}\\nabla_{j}\\overline{s}.\\label{conv}\n\\end{equation}\nThe further details about dependence of the eddy conductivity tensor\n$\\chi_{ij}$ from effects of both the global rotation and large-scale\nmagnetic field are given in Appendix. The diffusive heat transport\nby radiation reads, \n\\[\n\\mathbf{F}^{rad}=-c_{p}\\overline{\\rho}\\chi_{D}\\boldsymbol{\\nabla}T,\n\\]\nwhere \n\\[\n\\chi_{D}=\\frac{16\\sigma\\overline{T}^{3}}{3\\kappa\\overline{\\rho}^{2}c_{p}}.\n\\]\nBoth the eddy conductivity and viscosity are determined from the mixing-length\napproximation: \n\\begin{eqnarray}\n\\chi_{T} & = & \\frac{\\ell^{2}}{4}\\sqrt{-\\frac{g}{2c_{p}}\\frac{\\partial\\overline{s}}{\\partial r}},\\label{eq:chi}\\\\\n\\nu_{T} & = & \\mathrm{Pr_{T}}\\chi_{T},\\label{eq:nu}\n\\end{eqnarray}\nwhere $\\mathrm{Pr_{T}}$ is the turbulent Prandtl number. {Note,\nthat in Eq\\eqref{eq:chi} we employ factor $1\/2$ instead of $1\/3$.\nWith this choice the distribution of the mean entropy gradient, which\nresults from solution of the Eq\\eqref{eq:heat} for the nonrotating\nand nonmagnetic case is close to results of the MESA code.} It is\nassumed that $\\mathrm{Pr_{T}}={\\displaystyle 3\/4}$. {This\ncorresponds to the theoretical results of KPR94}. For this choice\nwe have the good agreement with solar angular velocity latitudinal\nprofile. We assume that the solar rotation rate corresponds to rotation\nrate of solar tachocline at 30$^{\\circ}$ latitude, i.e., $\\Omega_{0}\/2\\pi=430$nHz\n\\citep{1997SoPh170.43K}. We employ the stress-free boundary conditions\nin the hydrodynamic part of the problem. For the Eq(\\ref{eq:heat})\nthe thermal flux at the bottom is taken from the MESA code. At the\ntop, the thermal flux from the surface is approximated by the flux\nfrom a blackbody: \n\\begin{equation}\nF_{r}=\\frac{L_{\\odot}}{4\\pi r^{2}}\\left(1+4\\frac{T_{e}}{T_{eff}}\\frac{\\overline{s}}{c_{p}}\\right),\\label{eq:flx}\n\\end{equation}\nwhere where $T_{eff}$ is the effective temperature of the photosphere\nand $T_{e}$ is the temperature at the outer boundary of the integration\ndomain.\n\nFigure \\ref{fig:sun-flow} shows profiles of the angular velocity,\nstreamlines of the meridional circulations and the radial profiles\nof the angular velocity and the meridional flow velocity for a set\nof latitudes. The given results were discussed in details by \\citet{2018ApJ854.67P}.\nThe model shows the double-layer circulation pattern with the upper\nstagnation point at $r=0.88R_{\\odot}$. The amplitude of the surface\npoleward flow is about 15m\/s. The angular velocity profile shows a\nstrong subsurface shear that is higher at low latitudes and it is\nless near poles. Contrary to results of \\citet{Zhao13m} and model\nof \\citet{PK13} the double-cell meridional circulation structure\nextends from equator to pole. This is partly confirmed by the new\nresults of helioseismology by \\citet{2017ApJ849.144C} who also found\nthat the poleward flow at the surface goes close enough to pole. {It\nis important no mention that the current results of the helioseismic\ninversions for the meridional circulation remains controversial For\nexample, \\citet{2015ApJ813.114R} found that the meridional circulation\ncan be approximated by a single-cell structure with the return flow\ndeeper than 0.77R$_{\\odot}$. However, their results indicate an additional\nweak cell in the equatorial region, and contradict to the recent results\nof \\citet{2017ApJ845.2B} who confirmed a shallow return flow at 0.9R$_{\\odot}$.\nAlso, their results indicated that the upper meridional circulation\ncell extends close to the solar pole.}\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{fig1}\n\n\\caption{\\label{fig:sun-flow}a) angular velocity profile, $\\Omega\\left(r,\\theta\\right)\/2\\pi$,\ncontours are in range of 327-454 nHz; b) the radial profiles of the\nangular velocity for latitudes: $\\varphi=0^{\\circ}$, $30^{\\circ}$\nand $60^{\\circ}$; c) streamlines of the meridional circulation; d)\nradial profile of the meridional flow at $\\theta=45^{\\circ}$.}\n\\end{figure}\n\n\n\\subsection{Dynamo equations}\n\nWe model evolution of the large-scale axisymmetric magnetic field,\n$\\overline{\\mathbf{B}}$, by the mean-field induction equation \\citep{KR80},\n\\begin{equation}\n\\partial_{t}\\overline{\\mathbf{B}}=\\boldsymbol{\\nabla}\\times\\left(\\boldsymbol{\\mathcal{E}}+\\mathbf{\\overline{U}}\\times\\overline{\\mathbf{B}}\\right),\\label{eq:mfe-1}\n\\end{equation}\nwhere, $\\boldsymbol{\\mathcal{E}}=\\left\\langle \\mathbf{u\\times b}\\right\\rangle $\nis the mean electromotive force with $\\mathbf{u}$ and $\\mathbf{b}$\nstanding for the turbulent fluctuating velocity and magnetic field\nrespectively.\n\nSimilar to our recent paper (see, \\citep{2014ApJ_pipk,2017MNRAS.466.3007P}),\nwe employ the mean electromotive force in form: \n\\begin{equation}\n\\mathcal{E}_{i}=\\left(\\alpha_{ij}+\\gamma_{ij}\\right)\\overline{B}_{j}-\\eta_{ijk}\\nabla_{j}\\overline{B}_{k}.\\label{eq:EMF-1-1}\n\\end{equation}\nwhere symmetric tensor $\\alpha_{ij}$ models the generation of magnetic\nfield by the $\\alpha$- effect; antisymmetric tensor$\\gamma_{ij}$\ncontrols the mean drift of the large-scale magnetic fields in turbulent\nmedium, including the magnetic buoyancy; the tensor $\\eta_{ijk}$\ngoverns the turbulent diffusion. The reader can find further details\nabout the $\\boldsymbol{\\mathcal{E}}$ in the above cited papers.\n\nThe $\\alpha$ effect takes into account the kinetic and magnetic helicities\nin the following form: \n\\begin{eqnarray}\n\\alpha_{ij} & = & C_{\\alpha}\\eta_{T}\\psi_{\\alpha}(\\beta)\\alpha_{ij}^{(H)}+\\alpha_{ij}^{(M)}\\frac{\\overline{\\chi}\\tau_{c}}{4\\pi\\overline{\\rho}\\ell^{2}},\\label{alp2d-2}\\\\\n\\eta_{T} & = & \\frac{\\nu_{T}}{\\mathrm{Pm_{T}}}\n\\end{eqnarray}\nwhere $C_{\\alpha}$ is a free parameter which controls the strength\nof the $\\alpha$- effect due to turbulent kinetic helicity; tensors\n$\\alpha_{ij}^{(H)}$ and $\\alpha_{ij}^{(M)}$ express the kinetic\nand magnetic helicity parts of the $\\alpha$-effect, respectively;\n$\\mathrm{Pm_{T}}$ is the turbulent magnetic Prandtl number, and $\\overline{\\chi}=\\left\\langle \\mathbf{a}\\cdot\\mathbf{b}\\right\\rangle $\n($\\mathbf{a}$ and $\\mathbf{b}$ are the fluctuating parts of magnetic\nfield vector-potential and magnetic field vector). Both the $\\alpha_{ij}^{(H)}$\nand the $\\alpha_{ij}^{(M)}$ depend on the Coriolis number. Function\n$\\psi_{\\alpha}(\\beta)$ controls the so-called ``algebraic'' quenching\nof the $\\alpha$- effect where $\\beta=\\left|\\overline{\\mathbf{B}}\\right|\/\\sqrt{4\\pi\\overline{\\rho}u'^{2}}$,\n$u'$ is the RMS of the convective velocity. It is found that $\\psi_{\\alpha}(\\beta)\\sim\\beta^{-3}$\nfor $\\beta\\gg1$. The $\\alpha$- effect tensors $\\alpha_{ij}^{(H)}$\nand $\\alpha_{ij}^{(M)}$ are given in Appendix. \n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{fig2}\n\n\\caption{\\label{fig:alp} a) the radial profiles of the total (solid line)\nand anisotropic (dashed line) parts of the eddy diffusivity at $\\theta=45^{\\circ}$;\nb) radial profiles of the kinetic $\\alpha$-effect components at $\\theta=45^{\\circ}$;\nc) the equipartition strength of the magnetic field, $B_{eq}=\\sqrt{4\\pi\\overline{\\rho}u'^{2}}$,\nwhere $u'$ is determined by the equatorial profile of the mean entropy,\nsee the Eq(\\ref{eq:uc}); the dashed line is from results of the reference\nmodel (MESA code) and the solid line is for the rotating convection\nzone,i.e., after solution of the Eq(\\ref{eq:heat}).}\n\\end{figure}\n\nContribution of the magnetic helicity to the $\\alpha$-effect is expressed\nby the second term in Eq.(\\ref{alp2d-2}). The evolution of the turbulent\nmagnetic helicity density, $\\overline{\\chi}=\\left\\langle \\mathbf{a}\\cdot\\mathbf{b}\\right\\rangle $,\nis governed by the conservation law \\citep{pip13M}:\n\n\\begin{eqnarray}\n\\frac{\\partial\\overline{\\chi}}{\\partial t} & = & -2\\left(\\boldsymbol{\\mathcal{E}}\\cdot\\overline{\\bm{B}}\\right)-\\frac{\\overline{\\chi}}{R_{m}\\tau_{c}}+\\boldsymbol{\\nabla}\\cdot\\left(\\eta_{\\chi}\\boldsymbol{\\nabla}\\bar{\\chi}\\right)\\label{eq:hel-1}\\\\\n & & -\\eta\\overline{\\mathbf{B}}\\cdot\\mathbf{\\overline{J}}-\\boldsymbol{\\nabla}\\cdot\\left(\\boldsymbol{\\mathcal{E}}\\times\\overline{\\mathbf{A}}\\right),\\nonumber \n\\end{eqnarray}\nwhere $R_{m}=10^{6}$ is the magnetic Reynolds number and $\\eta$\nis the microscopic magnetic diffusion. In the drastic difference to\nanzatz of \\citet{kleruz82}, the Eq(\\ref{eq:hel-1}) contains the\nterm $\\left(\\boldsymbol{\\mathcal{E}}\\times\\overline{\\mathbf{A}}\\right)$.\nIt consists of the magnetic helicity density fluxes which result from\nthe large-scale magnetic dynamo wave evolution. The given contribution\nalleviates the catastrophic quenching problem \\citep{hub-br12,pip13M}.\nAlso the catastrophic quenching of the $\\alpha$-effect can be alleviated\nwith help of the diffusive flux of the turbulent magnetic helicity,\n$\\boldsymbol{\\boldsymbol{\\mathcal{F}}}^{\\chi}=-\\eta_{\\chi}\\boldsymbol{\\nabla}\\bar{\\chi}$\n\\citep{guero10,chatt11}. The coefficient of the turbulent helicity\ndiffusivity, $\\eta_{\\chi}$, is a parameter in our study. It affects\nthe hemispheric helicity transfer \\citep{mitra10}.\n\nIn the model we take into account the mean drift of large-scale field\ndue to the magnetic buoyancy, $\\gamma_{ij}^{(buo)}$ and the gradient\nof the mean density, $\\gamma_{ij}^{(\\Lambda\\rho)}$: \n\\begin{eqnarray}\n\\gamma_{ij} & = & \\gamma_{ij}^{(\\Lambda\\rho)}+\\gamma_{ij}^{(buo)},\\nonumber \\\\\n\\gamma_{ij}^{(\\Lambda\\rho)} & = & 3C_{pum}\\eta_{T}\\left(f_{1}^{(a)}\\left(\\mathbf{\\boldsymbol{\\Omega}}\\cdot\\boldsymbol{\\Lambda}^{(\\rho)}\\right)\\frac{\\Omega_{n}}{\\Omega^{2}}\\varepsilon_{inj}-\\frac{\\Omega_{j}}{\\Omega^{2}}\\varepsilon_{inm}\\Omega_{n}\\Lambda_{m}^{(\\rho)}\\right)\\label{eq:pump1}\\\\\n\\gamma_{ij}^{(buo)} & = & -\\frac{\\alpha_{MLT}u'}{\\gamma}\\beta^{2}K\\left(\\beta\\right)g_{n}\\varepsilon_{inj},\\nonumber \n\\end{eqnarray}\nwhere $\\mathbf{\\boldsymbol{\\Lambda}}^{(\\rho)}=\\boldsymbol{\\nabla}\\log\\overline{\\rho}$\n; functions $f_{1}^{(a)}$ and $K\\left(\\beta\\right)$ are given in\n\\cite{kp93,2017MNRAS.466.3007P}. The standard choice of the pumping\nparameter is $C_{pum}=1$. In this case the pumping velocity is scaled\nin the same way as the magnetic eddy diffusivity. In the presence\nof the multi-cell meridional circulation, the direction and magnitude\nof the turbulent pumping become critically important for the modelled\nevolution of the magnetic field. It is confirmed in the direct numerical\nsimulations, as well (see, \\cite{2018AA609A..51W}). For the standard\nchoice, the turbulent pumping is about an order of magnitude less than\nthe meridional circulation. For this case, explanation of the latitudinal\ndrift of the toroidal magnetic field near the surface faces a problem\n(cf., \\cite{PK13}). To study the effect of turbulent pumping we\nintroduce this parameter $C_{pum}$.\n\nFor the bottom boundary we apply the perfect conductor boundary conditions:\n$\\mathcal{E}_{\\theta}=0,\\,A=0$. The boundary conditions at the top\nare defined as follows. Firstly, following ideas of \\citet{1992AA256371M}\nand \\citet{pk11apjl} we formulate the boundary condition in the form\nthat allows penetration of the toroidal magnetic field to the surface:\n\\begin{eqnarray}\n\\delta\\frac{\\eta_{T}}{r_{e}}B+\\left(1-\\delta\\right)\\mathcal{E}_{\\theta} & = & 0,\\label{eq:tor-vac}\n\\end{eqnarray}\nwhere $r_{e}=0.99R_{\\odot}$, and parameter $\\delta=0.99$. The magnetic\nfield potential in the outside domain is \n\\begin{equation}\nA^{(vac)}\\left(r,\\mu\\right)=\\sum a_{n}\\left(\\frac{r_{e}}{r}\\right)^{n}\\sqrt{1-\\mu^{2}}P_{n}^{1}\\left(\\mu\\right).\\label{eq:vac-dec}\n\\end{equation}\n\nThe coupled angular momentum and dynamo equations are solved using\nfinite differences for integration along the radius and the pseudospectral\nnodes for integration in latitude. The number of mesh points in radial\ndirection was varied from $100$ to $150$. The nodes in latitude\nare zeros of the Legendre polynomial of degree ${N}$, where N was\nvaried from ${N=64}$ to ${N=84}$. The resolution with 64 nodes in\nlatitude and with 100 points in radius was found satisfactory. The\nmodel employed the Crank-Nicolson scheme, using a half of the time-step\nfor integration in the radial direction and another half for integration\nalong latitude.\n\nTo quantify the mirror symmetry type of the toroidal magnetic field\ndistribution relative to equator we introduce the parity index $P$:\n\\begin{eqnarray}\nP & = & \\frac{E_{q}-E_{d}}{E_{q}+E_{d}},\\label{eq:parity}\\\\\nE_{d} & = & \\int\\left(B\\left(r_{0},\\theta\\right)-B\\left(r_{0},\\pi-\\theta\\right)\\right)^{2}\\sin\\theta d\\theta,\\nonumber \\\\\nE_{q} & = & \\int\\left(B\\left(r_{0},\\theta\\right)+B\\left(r_{0},\\pi-\\theta\\right)\\right)^{2}\\sin\\theta d\\theta,\\nonumber \n\\end{eqnarray}\nwhere $E_{d}$ and $E_{q}$ are the energies of the dipole-like and\nquadruple-like modes of the toroidal magnetic field at $r_{0}=0.9R_{\\odot}$.\nAnother integral parameter is the mean density of the toroidal magnetic\nfield in the subsurface shear layer: \n\\begin{equation}\n\\overline{B^{T}}=\\sqrt{E_{d}+E_{q}}.\\label{eq:bt}\n\\end{equation}\nAnother parameter characterize the mean strength of the dynamo processes\nin the convection zone: \n\\begin{equation}\n\\overline{\\beta}=\\left\\langle \\left|\\overline{\\mathbf{B}}\\right|\/\\sqrt{4\\pi\\overline{\\rho}u'^{2}}\\right\\rangle ,\\label{eq:bet}\n\\end{equation}\nwhere the averaging is done over the convection zone volume. The boundary\nconditions Eq(\\ref{eq:tor-vac}) provide the Poynting flux of the\nmagnetic energy out of the convection zone. Taking into account the\nEq(\\ref{eq:flx}) the variation of the thermal flux at the surface\nare given as follows: \n\\begin{eqnarray}\n\\delta F & = & \\delta F_{c}+\\delta F_{B}\\label{eq:dF}\\\\\n\\delta F_{c} & = & 4\\frac{T_{e}}{T_{eff}}\\frac{\\delta\\overline{s}}{c_{p}}\\label{eq:flxb}\\\\\n\\delta F_{B} & = & \\frac{1}{4\\pi}\\left(\\mathcal{E}_{\\phi}\\overline{B}_{r}-\\mathcal{E}_{\\theta}\\overline{B}_{\\phi}\\right),\\label{eq:fcfb}\n\\end{eqnarray}\nwhere $\\delta\\overline{s}$ is the entropy variation because of the\nmagnetic activity. The second term of the Eq(\\ref{eq:dF}) governs\nthe magnetic energy input in the stellar corona.\n\n\\section{Results}\n\nTo match the solar cycle period we put $\\mathrm{Pm_{T}}=10$ in all\nour models. {The theoretical estimations of \\citet{1994AN....315..157K}\ngives $\\mathrm{Pm_{T}}=4\/3$. This is the long standing theoretical\nproblem of the solar dynamo period \\citep{brsu05}. Currently, the\nsolar dynamo period can be reproduced for $\\mathrm{Pm_{T}}\\gg1$.\nThe issue exists both in the distributed and in the flux-transport\ndynamo. Moreover, the flux-transport dynamo can reproduce the observation\nonly with the special radial profile of the eddy diffusivity (see,\ne.g., \\citep{2006ApJ...647..662R}). In our models we employ the rotational\nquenching the eddy diffusivity coefficients and the high $\\mathrm{Pm_{T}}$.\nFigures \\ref{fig:alp}a and b show the radial profiles of the eddy\ndiffusivity coefficients and components of the $\\alpha_{ij}^{(H)}$\nat latitude $45^{\\circ}$ in our model for $\\mathrm{Pm_{T}}=10$.\nIn the upper part of the convection zone the magnitude of the turbulent\nmagnetic diffusivity is close to estimations of \\citet{1993AA...274..521M}\nbased on observations of the sunspot decay rate. The eddy diffusivity\nis an order of $10^{10}$cm\/s and less near the bottom of the convection\nzone. The diffusivity profile is the same as in our previous paper\n\\citet{2014ApJ_pipk}. }\n\n{The radial profiles of the $\\alpha$ effect for $C_{\\alpha}=C_{\\alpha}^{(cr)}$\nare illustrated in Figure \\ref{fig:alp}b. The $\\alpha$ effect (cf,\nthe above discussion about $\\Lambda$-effect) change the sign near\nthe bottom of the convection zone. This is also found in the direct\nnumerical simulations \\citep{2006AA455.401K}.}\n\nTable \\ref{tab:C} gives the list of our models, their control and\noutput parameters. We sort the models with respect to magnitude of\nthe $\\alpha$ effect using the ratio ${\\displaystyle \\frac{C_{\\alpha}}{C_{\\alpha}^{(cr)}}}$,\nthe magnitude of the eddy-diffusivity of the magnetic helicity density,\n${\\displaystyle \\frac{\\eta_{\\chi}}{\\eta_{T}}}$, where $\\eta_{T}=\\nu_{T}\/\\mathrm{Pm_{T}}$,\nand $\\nu_{T}$ is determined from Eq(\\ref{eq:nu}), and with respect\nof the magnetic feedback on the differential rotation.{ The\n$C_{pum}$ controls the pumping velocity magnitude (see, Eq\\eqref{eq:pump1});\nthe parameter ${\\displaystyle \\frac{\\Delta\\Omega}{\\Omega_{0}}}$ show\nthe relative difference of the surface angular velocity between the\nsolar equator and pole; the strength of the dynamo is characterized\nby the range of the magnetic cycle variations of $\\overline{\\beta}$\n(see, Eq\\eqref{eq:bet}); the dynamo cycle period; the magnitude of\nthe surface meridional circulation. From the Table 1 we see that the\nnonkinematic runs show the magnetic cycle variations of ${\\displaystyle \\frac{\\Delta\\Omega}{\\Omega_{0}}}$\nand the surface meridional circulation. }\n\nFigure \\ref{fig:alp}b shows the radial profiles of the equipartition\nstrength of the magnetic field, $B_{eq}=\\sqrt{4\\pi\\overline{\\rho}u'^{2}}$\nin the solar convection zone for the reference model (non-rotating)\ngiven by MESA code and in the rotating convective zone. In the rotating\nconvection zone, the mean-entropy gradient is larger than in the nonrotating\ncase. {This is because of the rotational quenching of the eddy-conductivity.\nThe magnitude of the convective heat flux is determined by the boundary\ncondition at the bottom of the convection zone and it remains the\nsame for the rotating (our model) and nonrotating (MESA code) cases.\nAssuming that the convective turnover time is not subjected to the\nrotational quenching, the reduction of the eddy conductivity because\nof the rotational quenching is compensated by the increase of the\nmean-entropy gradient.} This results in the increase of the parameter\n$B_{eq}$.\n\n{The increase of the RMS convective velocity in case of the\nrotating convection zone seems to contradict the results of direct numerical\nsimulations of \\cite{2016AA596A.115W}. This is\nlikely because of inconsistent assumptions behind the MLT expression\nfor the RMS convective velocity, see Eq(\\ref{eq:uc}). The given issue\ncan affect the amplitude of the dynamo generated magnetic field near the\nbottom of the convection zone. Our models operate in regimes where\n$\\left|B\\right|\\le B_{eq}$, and the substantial part of the dynamo\nquenching is due to magnetic helicity conservation. Therefore the\ngiven issue does not much affect our results.}\n\n\n\n\\begin{table}\n\\caption{\\label{tab:C}Control and output parameters of the dynamo models.}\n\\begin{tabular}{>{\\centering}p{1cm}>{\\centering}p{2cm}>{\\centering}p{1cm}>{\\centering}p{2cm}>{\\centering}p{1.5cm}>{\\centering}p{2cm}>{\\centering}p{1cm}>{\\centering}p{1cm}}\n\\toprule \nModel & $C_{pum}$ & ${\\displaystyle \\frac{\\eta_{\\chi}}{\\eta_{T}}}$ & ${\\displaystyle \\frac{C_{\\alpha}}{C_{\\alpha}^{(cr)}}}$ & ${\\displaystyle \\frac{\\Delta\\Omega}{\\Omega_{0}}}$ & $\\overline{\\beta}$ & \\begin{centering}\nPeriod \n\\par\\end{centering}\n{[}YR{]} & $\\max U_{\\theta}$\n\n{[}M\/S{]}\\tabularnewline\n\\midrule \nM1 & 1 & $0.1$ & 1.1 & 0.279 & 0.05-0.1 & 6 & 15\\tabularnewline\n\\midrule \nM2 & Pm$_{T}$ & 0.1 & 1.1 & 0.279 & 0.13-0.24 & 10.5 & 15\\tabularnewline\n\\midrule \nM4 & -\/- & 0.3 & -\/- & 0.279 & 0.15-0.3 & 10.5,12.05 & 15\\tabularnewline\n\\midrule \nM5 & -\/- & 0.01 & -\/- & 0.279 & 0.14-0.26 & 9.3 & 15\\tabularnewline\n\\midrule \n & & & Nonkinematic & runs & & & \\tabularnewline\n\\midrule \nM3 & Pm$_{T}$ & 0.1 & 1.1 & 0.263-0.275 & 0.11-0.21 & 10.3 & 15.0\n\n$\\pm0.5$\\tabularnewline\n\\midrule \nM3a2 & -\/- & -\/- & 2 & 0.232-0.253 & 0.36-0.66 & 4.7,265 & 14.0\n\n$\\pm2.1$\\tabularnewline\n\\midrule \nM3a3 & -\/- & -\/- & 3 & 0.22-0.24 & 0.52-0.91 & 4.0 & 14.5 $\\pm2.5$\\tabularnewline\n\\midrule \nM3a4 & -\/- & -\/- & 4 & 0.21-0.245 & 0.68-1.05 & 3.4 & 14.7\n\n$\\pm3.$5\\tabularnewline\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Effects of turbulent pumping}\n\nAs the first step, we consider the kinematic dynamo model with the\nnonlinear $\\alpha$ effect. Results of the model M1 are shown in Figure\n\\ref{pumpa}. The model M1 roughly agree with results of \\citet{PK13}\n(hereafter, PK13). It employs the same mean electromotive force as\nin our previous paper. In particular, the maximum pumping velocity\nis the order of 1m\/s. The effective velocity drift due to the magnetic\npumping and meridional circulation is shown in Figures \\ref{pumpa}(a)\nand (b). Figures \\ref{pumpa}(d) and (e) show the time-latitude variations\nof the toroidal magnetic field at $r=0.9R$ and in the middle of the\nconvection zone. The agreement with the solar observations is worse\nthan in the previous model PK13 because of difference in the meridional\ncirculation structure. The model of PK13 employed the meridional circulation\nprofile provided by results of \\citet{Zhao13m}. In that profile,\nthe near-surface meridional circulation cell is more shallow and it\ndoes not touch the pole as it happens in the present model, see, Figure\n\\ref{fig:sun-flow}. By this reason, the polar magnetic field in model\nM1 is much larger than in results of PK13.\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{m1}\n\n\\caption{\\label{pumpa}a) Direction of pumping velocity of the toroidal magnetic\nfield in model M1; b) the effective velocity drift of the toroidal\nmagnetic field (pumping + meridional circulation); c) the snapshot\nof the toroidal magnetic field distribution (color image) and streamlines\nof the poloidal magnetic field in the Northern hemisphere of the Sun;\nd) the time-latitude diagram of the toroidal magnetic field evolution\n(contours in range of of $\\pm500$G at $r=0.9R$ and radial magnetic\nfield at the surface (color image).}\n\\end{figure}\n\nFor the purpose of our study, it is important to get the properties\nof the dynamo solution as close as possible to results of solar observations.\nTo solve the above issues we increase the turbulent pumping velocity\nmagnitude by factor $\\mathrm{Pm_{T}}$. The results are shown in Figure\n\\ref{pumpb}. The model has the correct time-latitude diagram of the\ntoroidal magnetic field in the subsurface shear layer. The surface\nradial magnetic field evolves in agreement with results of observations\n\\citep{2013AARv2166S}. The magnitude of the polar magnetic field\nis 10 G, which is in a better agreement with observations (e.g., \\cite{2007AAS...210.2405L})\nthan the model M1. Figures \\ref{pumpb} (a) and (b) show the effective\nvelocity drift of the large-scale toroidal magnetic field. The equatorward\ndrift with magnitude the order of 1-2 m\/s operates in major part of the\nsolar convection zone from $0.75R$ to $0.91R$. Interesting that\nthe obtained results are similar to those from the direct numerical\nsimulation of \\citet{2018AA609A..51W}. Note that in the given model\nthe magnitude of the pumping velocity is about factor 2 less than\nin results of \\citet{2018AA609A..51W}. It seems that some of the\nissues in model M1 would be less pronounced if the meridional circulation\npattern was closer to results of helioseismology of \\citet{Zhao13m}\nor \\citet{2017ApJ849.144C}. However results of the direct numerical\nsimulations of \\citep{2016AA596A.115W,2018AA609A..51W} seem to show\nthat the evolution of the large-scale magnetic field inside convection\nzone does not depend much on the meridional circulation. This argues\nfor the strong magnetic pumping effects in the global dynamo. This\nimportant issue can be debated further. This is out of the main scopes\nof this paper. The rest of our models employ the same pumping effect\nas in the model M2 (see, Table \\ref{tab:C}). \n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{m2}\n\n\\caption{\\label{pumpb}a) Direction of pumping velocity of the toroidal magnetic\nfield in model M2; b) the effective velocity drift of the toroidal\nmagnetic field (pumping + meridional circulation); c) the snapshot\nof the toroidal magnetic field distribution (color image) and streamlines\nof the poloidal magnetic field in the Northern hemisphere of the Sun;\nd) the time-latitude diagram of the toroidal magnetic field evolution\n(contours in range of of $\\pm1$kG at $r=0.9R$ and radial magnetic\nfield at the surface (color image).}\n\\end{figure}\n\n\n\\subsection{The global flows oscillations in magnetic cycle}\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{m3}\n\n\\caption{\\label{fig:s-bat}The model M3, a) Time-latitude butterfly diagram\nfor the toroidal field in the upper part of the convection zone (color\nimage) and the surface radial magnetic field shown by contours ($\\pm$5G);\nb) the surface variations of the azimuthal velocity (color image)\nand the meridional velocity (contours in the range of $\\pm0.5$ m\/s). }\n\\end{figure}\n\nFigure \\ref{fig:s-bat} show the time-latitude diagrams of the magnetic\nfield and the global flow variations for the model M3. The torsional\noscillations on the surface are about $\\pm2$m\/s. They are defined\nas follows, $\\delta U_{\\phi}=\\left(\\Omega\\left(r,\\theta t\\right)-\\overline{\\Omega\\left(r,\\theta,t\\right)}\\right)r\\sin\\theta$,\nwhere the averaging is done over the stationary phase of evolution.\nThe torsional wave has both the equator- and poleward branches. In\nthe equatorward torsional wave, the change from the positive to negative\nvariation goes about 2 years ahead of the maxima of the toroidal magnetic\nfield wave. This agrees with results of observations of \\citet{2011JPhCS271a2074H}\nand with direct numerical simulations of \\citet{2016ApJ828L.3G}.\nThe magnitude of the meridional flow variations agrees with results\nof \\citet{2014ApJ789L7Z}. Also, we see that on the surface the meridional\nvelocity variations converge toward the maximum of the toroidal magnetic\nfield wave. This is also in qualitative agreement with the observations.\n\n\\begin{figure}\n\\includegraphics[width=0.95\\columnwidth]{m3s}\\caption{\\label{fig:M3}The model M3: a) snapshots of the magnetic field in\nfour phase of the magnetic cycle, the toroidal magnetic field strength\nis shown by color, contours show streamlines of the poloidal magnetic\nfield; b) color image show variations of the angular velocity, contours\n(range of $\\pm0.5$m\/s) show variations of the meridional flow; c)\ncontours show the azimuthal component of the total (kinetic and magnetic\nhelicity parts)$\\alpha$ -effect, the background image shows the part\nof the $\\alpha_{\\phi\\phi}$ induced by the magnetic helicity conservation\n(see, the second term of Eq(\\ref{alp2d-2})).}\n\\end{figure}\n\nFigure \\ref{fig:M3} shows snapshots of the magnetic field, the global\nflows variations and the azimuthal component of the total (kinetic\nand magnetic helicity parts)$\\alpha$ -effect for a half magnetic\ncycle. The Figure shows that a new cycle starts at the bottom of the\nconvection zone. The main part of the dynamo wave drifts to surface\nequatorward. There is a polar branch which propagates poleward along\nthe bottom of the convection zone. The torsional oscillations, as\nwell as, the meridional flow variations are elongated along the axis\nof rotation. This can be interpreted as a result of mechanical perturbation\nof the Taylor-Proudman balance \\citep{2006ApJ...647..662R}. We postpone\nthe detailed analysis of the torsional oscillation to another paper.\n{Figure \\ref{fig:M3}b shows that maxima of the meridional\nflow variations are located at the upper boundary of the dynamo domain.}\nThis is because the main drivers of the meridional circulation, which\nare the baroclinic forces, have the maximum near the boundaries of\nthe solar convection zone \\citep{rem2005ApJ,2015ApJ...804...67F,2017arXiv170202421P}.\nIn comparing Figures \\ref{fig:M3}c and \\ref{fig:alp} it is seen\nthat the dynamo wave affect the $\\alpha$ -effect. Also, in agreement\nwith our previous model \\citep{pip2013ApJ}, we find that the magnetic\nhelicity conservation results into increasing the $\\alpha$ -effect\nin the subsurface shear layer. It occurs just ahead of the dynamo\nwave drifting toward the top. The given effect support the equatorward\npropagation of the large-scale toroidal field in subsurface shear\nlayer \\citep{kap2012}.\n\nThe increasing the $\\alpha$-effect parameter results in a number\nof consequences for the non-linear evolution of the large-scale magnetic\nfield. The dynamo period is decreasing with the increase of the $\\alpha$-effect\n\\citep{pk11}. The magnitude of the dynamo wave increases with the\nincrease of the parameter $C_{\\alpha}$.{ Therefore, our models\nshow that in the distributed solar-type dynamo the dynamo period can\ndecrease with the increase of the magnetic activity level. This is\nin agreement with the results of the stellar activity observations\nof \\citet{1984ApJ...287..769N,2009AA_strassm,2017PhDT3E}. Here we\nfor the first time demonstrate this effect in the distributed dynamo\nmodel with the meridional circulation. }Figure \\ref{fig:m8} shows\nresults for the model M3a4 with ${\\displaystyle C_{\\alpha}=4C_{\\alpha}^{(cr)}}$.\nThe model shows the solar-like dynamo waves in the subsurface shear\nlayer. The toroidal magnetic field reaches the strength of 3kG in\nthe upper part of the convection zone. Simultaneously, the polar magnetic\nfield has the maximum strength of 100 G. Variations of the zonal and\nmeridional flows on the surface are about of factor 6 larger than\nin the model M3. The model M3a4 show a high level magnetic activity\nwith a strong toroidal magnetic field in the subsurface layer and\nvery strong polar field. {Results of stellar observations show\nthat this is expected on the young solar analogs and late K-dwarfs\nas well (e.g., \\citep{2005LRSP2.8B,2016MNRAS1129S}). However, the\ngiven results can not be consistent with those cases because in our\nmodel the rotation rate is much slower than for the young solar-type\nstars. Results of the linear models show that the internal differential\nrotation and meridional circulation change with increase of the stellar\nangular velocity \\citep{kit11}.}\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{m8}\n\n\\caption{\\label{fig:m8}The model M3a4, a) Time-latitude butterfly diagram\nfor the toroidal field in the upper part of the convection zone (color\nimage) and the surface radial magnetic field shown by contours ($\\pm$100G);\nb) the surface variations of the azimuthal velocity (color image)\nand the meridional velocity (contours in the range of $\\pm3.5$ m\/s). }\n\\end{figure}\n\nFigure \\ref{fig:M3a4dr} shows snapshots of the global flows distributions\nin the solar convection zone. In drastic difference to the model M3,\nthe counter-clockwise meridional circulation cell in the upper part\nof the convection zone is divided into two parts. Also, the stagnation\npoint of the bottom cell is shifted equatorward.\n\n\\begin{figure}\n\\includegraphics[width=0.7\\columnwidth]{m8dr}\n\n\\caption{\\label{fig:M3a4dr}The model M3a4: a) the snapshot of angular velocity\nprofile, $\\Omega\\left(r,\\theta\\right)\/2\\pi$, contours are in range\nof 347-450 nHz; b) the streamlines of the meridional circulation;\nc) the radial profile of the meridional flow at $\\theta=45^{\\circ}$;\nd) the profiles of the meridional flow at the specific depths of the\nsolar convection zone.}\n\\end{figure}\n\nFigure \\ref{fig:M3a4} shows the magnetic cycle variations of the\nglobal flows in the Northern segment of the solar convection zone\nfor the model M3a4. The patterns of these variations are qualitatively\nthe same as in the model M3. The model M3a4 show the strong magnetic\ncycle variations of the $\\alpha$ effect. Similar to the model M3\nwe see that the magnetic helicity conservation results into increasing\nthe $\\alpha$ -effect in the subsurface shear layer. It occurs just\nahead of the dynamo wave drifting toward the top. In the polar regions,\nthe $\\alpha$ -effect inverses the sign during inversion of the polar\nmagnetic field. This is different from the model M3 which has the smaller\nstrength of the polar magnetic field than the model M3a4.\n\n\\begin{figure}\n\\includegraphics[width=0.95\\columnwidth]{mf6}\\caption{\\label{fig:M3a4}The same as Figure \\ref{fig:M3} for the model M3a4.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{f7a}\n\n\\caption{\\label{fig:avar}a) Variations of the equatorial symmetry (parity\nindex, see, Eq(\\ref{eq:parity})); b) the same as (a) for the mean\ndensity of the toroidal magnetic field flux in the subsurface shear\nlayer. The time series were smoothed to filter out the basic magnetic\ncycle.}\n\\end{figure}\n\n\n\\subsection{The long-term dynamo evolution}\n\nFigure \\ref{fig:avar} shows the smoothed time series of evolution\nof the global properties of the dynamo model, such as the equatorial\nsymmetry index, or the parity index $P$, (see, Eq(\\ref{eq:parity}))\nand the mean density of the toroidal magnetic field flux, $\\overline{B^{T}}$,\nin the subsurface shear layer, see, Eq(\\ref{eq:bt}). In each time\nseries, the basic magnetic cycle was filtered out. The set of models\nshown in Figure (\\ref{fig:avar}) illustrates the effect of variations\nof magnitude of the eddy-diffusivity of the magnetic helicity density,\n${\\displaystyle \\frac{\\eta_{\\chi}}{\\eta_{T}}}$. From results of \\citet{mitra10},\nit is expected that ${\\displaystyle \\frac{\\eta_{\\chi}}{\\eta_{T}}}<1$.\nIn our set of models it is $0.01<{\\displaystyle \\frac{\\eta_{\\chi}}{\\eta_{T}}}<0.3$.\nThe magnetic helicity diffusion affects the magnetic helicity exchange\nbetween hemispheres \\citep{2000JGR...10510481B,mitra10}. Therefore\nit affects an interaction of the dynamo waves through the solar equator.\nIt is found that the increasing of ${\\displaystyle \\frac{\\eta_{\\chi}}{\\eta_{T}}}$\nresults into change of the parity index $P$. The model M4 show the\nsymmetric about equator magnetic field. The magnitude of $\\overline{B^{T}}$\nin the model M4 is larger than in the models M3 and M5. Both models\nM3 and M5 operate in a weak nonlinear regime with $0.13<\\overline{\\beta}<0.24$\n(see, Eq(\\ref{eq:bet}) and Table(\\ref{tab:C})). The model M4 has\na slightly higher $\\overline{\\beta}$. This means that the magnetic\nhelicity diffusion affects the strength of the dynamo. This is in\nagreement with \\citet{guero10}. The smoothed time series of $\\overline{B^{T}}$\nin the model M4 show oscillations at the end of the evolution. This\nis because the period of the symmetric dynamo mode ( $P=1$) is about\n12 years that is larger than the period of the basic magnetic cycle\nfor antisymmetric mode $\\left(P=-1\\right)$ . The latter is about\n10.5 years.\n\nIt is interesting to compare the kinematic and nonkinematic dynamo\nmodels, which are the models M2 and M3. The nonkinematic model M3\nhas the smaller parameters $\\overline{B^{T}}$and $\\overline{\\beta}$\nthan the model M2. Also, it is found that in the kinematic model M2\nthe stationary phase of evolution is the mix of the dipole-like and\nquadrupole-like parity, with the mean $P\\approx-0.9$. In the model\nM3 the mean $P\\approx-1$. Therefore the nonkinematic dynamo regimes\naffect the equatorial symmetry of the dynamo solution \\citep{bran89,p99}.\n{Figure \\ref{fig:avar}b shows another interesting difference\nbetween the nonlinear kinematic and nonkinematic runs. The kinematic\nmodels M2, M4 and M5 show a very slow evolution toward the stationary\nstage. The given effect was reported earlier by \\citet{pip2013ApJ}.\nThe high $R_{m}=10^{6}$ and small diffusivity $\\eta_{\\chi}$ result\nto the long time-scale of establishment of the nonlinear balance in\nmagnetic helicity density distributions.}\n\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{f7b}\n\n\\caption{\\label{fig:lt}The same as Figure (\\ref{fig:avar}) for the models\nwith different $C_{\\alpha}$, see Table(\\ref{tab:C}).}\n\\end{figure}\n\nFigure (\\ref{fig:lt}) shows results for the nonkinematic dynamo models\nin a range the $\\alpha$-effect parameter $C_{\\alpha}$. The increasing\nof the $C_{\\alpha}$ results into increasing the nonlinearity of the\ndynamo model. The parameter $\\overline{\\beta}$ grows from $0.2$\nto $1$ with the increasing of $C_{\\alpha}$by factor 4. The model\nM3a2 shows the long-term periodic variations of the parity index and\nthe magnitude of the toroidal magnetic field $\\overline{B^{T}}$.\nThese long-term cycles are likely due to the parity breaking because\nof the hemispheric magnetic helicity exchange. We made the separate\nrun where the magnetic helicity conservation was ignored and did not\nfind the long-term cycles solution. These cycles are not robust against\nchanges of $C_{\\alpha}.$ For the case $\\eta_{\\chi}=0.1\\eta_{T}$,\nthey exist in the range $1.5C_{\\alpha}^{(cr)}3C_{\\alpha}^{(cr)}$}. This coincides\nwith the formation of the second meridional circulation cell near\nthe equator. Currently, it is not clear if both phenomena are tightly\nrelated or this is an accident. This will be studied further.\n\nWe show the first results about effects of the large-scale magnetic\nactivity on the heat transport and the heat energy flux from the dynamo\nregion. This was previously discussed in papers of \\citet{1992AA...265..328B}\nand \\citet{2000ARep...44..771P}. In the mean-field framework, the\nmajor contributions of the large-scale magnetic field on the heat\nenergy balance inside the convection zone are caused by the magnetic\nquenching of the eddy heat conductivity and the energy expenses (associating\nwith the heat energy loss and gain) on the large-scale dynamo. These\nprocesses are modeled by the mean-field heat transport equations.\nThe magnetic perturbations of the heat flux in the model M3 are an\norder of $10^{-3}$ of the background value. It is an order of magnitude\nless at the surface because of the screening effect and the smaller\nstrength of the large-scale magnetic field in the upper layer of the\nconvection zone. The heat perturbation screening effect is due to\nthe huge heat capacity of the solar convection zone \\citep{stix:02}.\nResults of the model M3a4 illustrate it better than the model M3.\nThe model M3a4 shows the strong toroidal magnetic field in the bulk\nof the convection zone (see Figure \\ref{fig:m34}b). In the upper\nlayer of the convection zone, the strength of the toroidal field exceeds\nthe equipartition level. Besides this, the heat flux perturbations\nare efficiently smoothed out toward the top of the dynamo region.\nAnother interesting feature is that the weakly nonlinear model M3\nshows the increasing mean heat flux at the maximum of the magnetic\ncycle. In the model with the overcritical $\\alpha$ effect, we find\nthe opposite situation. The solar observations show the increasing\nluminosity during the maximum of the solar cycles \\citep{1999GeoRL26.3613W}.\nThe variation of the photometric brightness of solar-type stars tends\nto inverse the sign with the increasing level of the magnetic activity\n\\citep{2016ASPC504.273Y}. From the point of view of our model, this\nmeans that the effect of the magnetic shadow become dominant when\nthe total magnetic activity is increased. This is a preliminary conclusion.\nAlso, the relationship between the magnetic shadow effect in the large-scale\ndynamo and the stellar surface darkening because of starspots is not\nstraightforward.\n\nFinally, our results can be summarized as follows: \n\\begin{enumerate}\n\\item We constructed the nonkinematic solar-type dynamo model with the double-cell\nmeridional circulation. The role of the turbulent pumping in the dynamo\nmodel should be investigated. This requires a better theoretical and\nobservational knowledge of the solar meridional circulation. \n\\item The torsional oscillations are explained as a result of the magnetic\nfeedback on the angular momentum transport by the turbulent stresses,\nthe effect of the Lorentz force and the magneto-thermal perturbations\nof the Taylor-Proudman balance. The increasing level of the magnetic\nactivity results in separation of the upper meridional circulation\ncell for two parts. \n\\item The model shows the decrease of the dynamo period with the increase\nof the magnetic cycle amplitude. The shape of the strong magnetic\ncycle is more asymmetric than the shape of the weak cycles. \n\\item The magnetic helicity density diffusion and the increase turbulent\ngeneration of the large-scale magnetic field results in the increasing\nhemispheric magnetic helicity exchange, the magnetic parity breaking\nand the Grand activity cycles. The Grand activity cycles exists in\nthe intermediate range of the $\\alpha$ - effect parameter, when $1.5C_{\\alpha}^{(cr)}