{"text":"\\section{Introduction}\\label{sec.intro}\nOne of the most important tasks in the field of multi-agent systems (MAS) is reaching agreement. Distributed protocols guaranteeing the agents reach agreement appear in many different fields, including robotics \\cite{Chopra2006}, sensor networks \\cite{OlfatiSaber2007cdc}, and distributed computation \\cite{Xiao2004}. A natural generalization is the \\emph{cluster agreement} problem, which seeks to drive agents into groups, so that agents within the same group reach an agreement. The clustering problem appears in social networks \\cite{Lancichinetti2012}, neuroscience \\cite{Schnitzler2005}, and biomimetics \\cite{Passino2002}.\nClustering has been studied using different approaches, e.g. network optimization \\cite{Burger2011TAC}, pinning control \\cite{Qin2013}, inter-cluster nonidentical inputs \\cite{Han2013}, and exploiting the structural balance of the underlying graph \\cite{Altafini2013}. We tackle the cluster agreement problem by using \\emph{symmetries} within the MAS. Recent works on MAS apply graph symmetries to study various problems, e.g., controllability and observability of MAS \\cite{Rahmani2007, Chapman2014, Chapman2015, Notarstefano2013TAC}. \n\nOur recent work \\cite{Sharf2019b} introduced the notion of the \\emph{weak automorphism group of a MAS}, combining the two concepts of the automorphism group for graphs and weak equivalence of dynamical systems.\nThe former summarizes all symmetries for a given graph, while the latter characterizes similarities between achievable steady-states of heterogeneous (dynamical) agents. Therefore, the weak automorphisms of a MAS can be  understood as permutations of the nodes in the underlying graph that preserve both graph symmetries and certain input-output properties of the corresponding agents. More specifically, \\cite{Sharf2019b} focused on clustering for \\emph{diffusively coupled networks},  and showed that under appropriate passivity assumptions, these diffusively coupled networks converge to a clustered steady-state solution, where two agents are in the same cluster if and only if there exists a weak automorphism mapping one to the other. Thus, the clustering of the MAS can be understood by studying the action of the weak automorphism group on the underlying interaction graph. \n\nWe focus in this paper on homogeneous networks, i.e., networks where the agent dynamics are all identical, noting that the weak automorphism group is identical to the automorphism group of the underlying graph in this case. We wish to design graphs ensuring the MAS will converge to a prescribed cluster configuration, i.e., specifying the number of clusters and the number of agents within each cluster. \\textcolor{black}{Our previous work, \\cite{Sharf2022a},} applied tools from group theory to prescribe an algorithm for constructing an oriented graph such that the action of the automorphism group on the graph has orbits of prescribed sizes. \\textcolor{black}{It also} provided upper and lower bounds on the number of edges needed to construct such graph. \\textcolor{black}{This work extends the previous work \\cite{Sharf2022a} in two ways. First, we further explore the bounds on the number of edges, by understanding the reason for the difference between the upper and the lower bounds (the example in Remark \\ref{rem.TreeOrPath}) as well as the scaling properties of the upper bound (Theorem \\ref{thm.GlobalUpperBound} and Remark \\ref{rem.WorstCaseSparse}). Furthermore, we study the robustness of such graphs to agent malfunctions, and alter the synthesis procedure to guarantee the most extensive possible robustness of the clustering possible (Subsection \\ref{subsec.Robust}).}\n\nThe rest of paper is organized as  follows. Section \\ref{sec.background} reviews basic concepts related to network systems and group theory required to define a notion of symmetry for MAS. Section \\ref{sec.cluster} presents the main results about cluster assignment, as well as a numerical study to demonstrate the theory. Finally, some concluding remarks are offered in Section \\ref{sec.conclusion}.\n\n\\paragraph*{Notations}\nThis work employs basic notions from graph theory \\cite{Godsil2001}.  An undirected graph $\\mathcal{G}=(\\mathbb{V},\\mathbb{E})$ consists of finite sets of vertices $\\mathbb{V}$ and edges $\\mathbb{E} \\subset \\mathbb{V} \\times \\mathbb{V}$.  We denote the edge with ends $i,j\\in \\mathbb{V}$ as $e=\\{i,j\\}$. For each edge $e$, we pick an arbitrary orientation and denote $e=(i,j)$ when $i\\in \\mathbb{V}$ is the \\emph{head} of edge and $j \\in \\mathbb{V}$ is its \\emph{tail}. A path is a sequence of distinct nodes $v_1, v_2,\\ldots , v_n$ such that $\\{v_i, v_{i+1}\\} \\in \\mathbb{E}$ for all $i$. A cycle is path $v_1,\\ldots,v_n,v_1$. A simple cycle is a cycle whose vertices are all distinct. A graph is connected if there is a path between any two vertices, and a tree if it is connected but contains no simple cycles.\nThe incidence matrix of $\\mathcal{G}$, denoted $\\mathcal{E}\\in\\mathbb{R}^{|\\mathbb{E}|\\times|\\mathbb{V}|}$, is defined such that for any edge $e=(i,j)\\in \\mathbb{E}$, $[\\mathcal{E}]_{ie} =+1$, $[\\mathcal{E}]_{je} =-1$, and $[\\mathcal{E}]_{\\ell e} =0$ for $\\ell \\neq i,j$.\nMoreover, the greatest common divisor of two positive integers $m,n$ is denoted by $\\gcd(m,n)$, and their least common multiple is denoted by $\\mathrm{lcm}(m,n)$. Note that $\\mathrm{lcm}(m,n) = \\frac{mn}{\\gcd(m,n)}$ always holds. Two integers are relatively prime (or coprime) if there is no integer greater than one that divides them both. The cardinality of a finite set $A$ is denoted by $|A|$.\n\n\\vspace{-15pt}\n\\section{Symmetries in Networked Systems}\\label{sec.background}\nIn this section, we provide background on the notion of symmetries for multi-agent systems originally proposed in~\\cite{Sharf2019b}.\n\n\\vspace{-15pt}\n\\subsection{Diffusively Coupled Networks}\nThis section describes the structure of the MAS studied in \\cite{Burger2014, Sharf2019b}. Consider a set of agents interacting over a network $\\mathcal{G}=(\\mathbb{V},\\mathbb{E})$. Each node $i\\in \\mathbb{V}$ is assigned a dynamical system $\\Sigma_i$, and the edges $e\\in\\mathbb{E}$ are assigned controllers $\\Pi_e$, having the following form:\n\\begin{align} \\label{Dynamics}\n\\Sigma_i: \n\\begin{cases}\n\\dot{x}_i = f_i(x_i,u_i), ~\ny_i = h_i(x_i,u_i),\n\\end{cases} \\quad\n\\Pi_e: \n\\begin{cases}\n\\dot{\\eta}_e= \\phi_e(\\eta_e,\\zeta_e), ~\n\\mu_e = \\chi_e(\\eta_e,\\zeta_e)\n\\end{cases}.\n\\end{align}\n\n\\begin{figure}\n        \\centering\n        \\includegraphics[scale=0.7]{Figures\/nds.pdf}\n        \\vspace{-10pt}\n        \\caption{A diffusively coupled network.}\n        \\vspace{-10pt}\n        \\label{ClosedLoop}\n\\end{figure}\n\nWe consider the stacked vectors $u=[u_1^T,...,u_{|\\mathbb{V}|}^T]^T$ and similarly for $y,\\zeta$ and $\\mu$.  \nThe MAS is diffusively coupled with the controller input described by $\\zeta = \\mathcal{E}^Ty$, and the control input is $u = -\\mathcal{E}\\mu$, where $\\mathcal{E}$ is the incidence matrix of the graph $\\mathcal{G}$.  This structure is denoted by the triplet $(\\mathcal{G},\\Sigma,\\Pi)$, and is illustrated in Fig. \\ref{ClosedLoop}.\nIn \\cite{Burger2014} it was shown that \nthe network converges to a steady-state satisfying the interconnection constraints if the agents and controllers are (output-strictly) maximum equilibrium independent passive (MEIP). \nThe details of this definition and related definitions are not essential for the development of this work, and the interested reader is referred to \\cite{Burger2014, Sharf2017} for more details. \nIn this work, we focus on homogeneous networks, i.e., where all the agent dynamics and control dynamics are identical.\nMoreover, we assume one of the following two alternatives (Assumption \\ref{assump.1}).  If this is not the case, see \\cite{Jain2018, Sharf2019a, Sharf2019c} for plant augmentation techniques. \n\n\\begin{assump}\\label{assump.1}\nThe agents $\\Sigma_i$ are output-striclty MEIP and the controllers $\\Pi_e$ are MEIP, or vice versa.\n\\end{assump}\n\n\nA final technical definition is needed to characterize the steady-states of the network. Indeed, we implicitly assume that each agent and controller converges to a steady-state output given a constant input. This allows us to define a relation between constant inputs and constant outputs called the \\emph{steady-state relation} of a system; see \\cite{Burger2014}. We denote the steady-state relations of node $i$ and edge $e$ by $k_i$ and $\\gamma_e$, respectively. For example, for an agent $i$, we say that $(\\mathrm{u}_i,\\mathrm{y}_i)$ is a steady-state input\/output pair if $\\mathrm{y}_i \\in k_i(\\mathrm{u})$.  We now introduce the notion of weak equivalence for dynamical systems.\n\n\\begin{defn}[\\hspace{0.01pt}\\cite{Sharf2019b}]\nTwo systems $\\Sigma_1$ and $\\Sigma_j$ are \\emph{weakly equivalent} if their steady-state relations are identical.\n\\end{defn}\nWe refer the reader \\cite{Sharf2019b} for a more thorough study and examples of weakly equivalent systems.\n\n\\vspace{-15pt}\n\\subsection{Group Theory, Graph Automorphisms, and Symmetric MAS}\nOur approach for clustering will hinge on symmetry, which is modelled by the mathematical theory of groups \\cite{Dummit2004}. The notion of a group can be defined in various ways, but we opt for the most concrete one.\n\\begin{defn}\nLet $X$ be a set, and let $\\mathbb{G}$ be a collection of invertible functions $X\\to X$. The collection $\\mathbb{G}$ is called a \\emph{group} if for any $\\mathbb{G} \\ni f,g: X\\to X$, both the composite function $f\\circ g$ and the inverse function $f^{-1}$ belong to $\\mathbb{G}$.\n\\end{defn}\nColloquially, the group $\\mathbb{G}$ defines a collection of symmetries of the set $X$, and its action on $X$ allows us to identify certain elements of $X$ which are symmetric. \nOf interest in this work is the automorphism group of a (oriented) graph, which encodes structural symmetries of a graph. \n\\begin{defn}\nAn automorphism of a \\textcolor{black}{(directed or undirected)} graph $\\mathcal{G} = (\\mathbb{V},\\mathbb{E})$ is a permutation $\\psi:\\mathbb{V}\\to\\mathbb{V}$ such that $i\\in\\mathbb{V}$ is connected to $j\\in\\mathbb{V}$ if and only if $\\psi(i)$ is connected to $\\psi(j)$. We denote the automorphism group of $\\mathcal{G}$ by $\\mathrm{Aut}(\\mathcal{G})$. \n\\end{defn}\nWe slightly abuse  notation and say that $\\mathrm{Aut}(\\mathcal{G})$ acts on $\\mathcal{G}$ (rather than on $\\mathbb{V}$).\n\n\\begin{defn}\nLet $\\mathbb{G}$ be a group of functions $X\\to X$. We say $x,y\\in X$ are \\emph{exchangeable} (under the action of $\\mathbb{G}$) if there is some $f\\in \\mathbb{G}$ such that $f(x)=y$. The \\emph{orbit} of $x\\in X$ is the set of elements which are exchangeable with $X$.\n\\end{defn}\nExchangeability was considered in \\cite{Sharf2019b} to describe the clustering behavior of a MAS. Namely, the different clusters corresponded to the different orbits of the weak automorphism group of the MAS.   The following result ensures that we can consider orbits, and consequently different clusters in an MAS, as disjoint sets.\n\\begin{prop}[\\cite{Dummit2004}]\nLet $\\mathbb{G}$ be a group of functions $X\\to X$. Then $X$ can be written as the union of disjoint orbits. \\textcolor{black}{In particular, any element of $X$ belongs to exactly one orbit.}\n\\end{prop}\n\nFinally, we combine the notions of graph automorphisms and diffusively-coupled MAS comprised of weakly equivalent agents.  \n\n\\begin{defn}[\\cite{Sharf2019b, Sharf2022a}]\\label{weakMASaut}\nLet $(\\mathcal{G},\\Sigma,\\Pi)$ be a diffusively coupled MAS. A \\emph{weak automorphism of a MAS} is a map $\\psi:\\mathbb{V}\\to\\mathbb{V}$ with the following properties:\n$1$) $\\psi$ is an automorphism of the graph $\\mathcal{G}$, and preserves edge orientations; \n$2$) for any $i\\in\\mathbb{V}$, $\\Sigma_i$ and $\\Sigma_{\\psi(i)}$ are weakly equivalent; and\n$3$) for any $e \\in\\mathbb{E}$, $\\Pi_e$ and $\\Pi_{\\psi(e)}$ are weakly equivalent. \nWe denote the collection of all weak automorphisms of $(\\mathcal{G},\\Sigma,\\Pi)$ by $\\mathrm{Aut}(\\mathcal{G},\\Sigma,\\Pi)$. \n\\end{defn}\nNaturally, the weak automorphism of a MAS is a subgroup of the group of automorphisms $\\mathrm{Aut}(\\mathcal{G})$ of the graph $\\mathcal{G}$.\n\\vspace{-20pt}\n\\section{Cluster Assignment in MAS}\\label{sec.cluster}\n\nWe now consider the clustering problem for MAS. Specifically, we focus on the case where the agents are homogeneous, i.e., they have the exact same model, and restrict ourselves by requiring the edge controllers \\eqref{Dynamics} are also homogeneous.\nThe paper \\cite{Sharf2019b} established a link between the clustering behaviour of a MAS and certain symmetries it has, using Definition \\ref{weakMASaut}. The main result from \\cite{Sharf2019b} is summarized below.\n\n\\begin{thm}[\\hspace{0.01pt}\\cite{Sharf2019b}] \\label{thm.Symmetry}\nTake a diffusively-coupled MAS $(\\mathcal{G},\\Sigma,\\Pi)$ where Assumption \\ref{assump.1} holds. Then for any steady-state $\\mathrm y=\\begin{bmatrix} \\mathrm y_1 &  \\cdots & \\mathrm y_{|\\mathbb{V}|}\\end{bmatrix}^T$ of the closed-loop and any weak automorphism $\\psi\\in\\mathrm{Aut}(\\mathcal{G},\\Sigma,\\Pi)$, we have $P_\\psi \\mathrm y = \\mathrm y$, where $P_\\psi$ is the permutation matrix representation  of $\\psi$.\n\\end{thm}\n\nThis result can in fact be used to show that the network converges to a clustering configuration, where the clusters are given by the orbits of the weak automorphism group. Namely, one considers diffusively-coupled MAS $(\\mathcal{G},\\Sigma,\\Pi)$ satisfying Assumption \\ref{assump.1}, for which the closed-loop is known to converge, and the invariance properties of the steady-state limit are studied. Focusing on homogeneous networks, \\cite{Sharf2019b} identified the value of $\\gamma_e(0)$, the steady-state relation for the controller on the $e$th edge,  as indicative of clustering. Namely, it shows that if $0\\in \\gamma_e(0)$ for all $e\\in \\mathbb{E}$, then the MAS $(\\mathcal{G},\\Sigma,\\Pi)$ converges to consensus, and otherwise it displays a clustering behavior. Namely, for homogeneous MAS, two nodes are in the same cluster if they are exchangable under the action of $\\mathrm{Aut}(\\mathcal{G})$, and the converse is almost surely true.\n\nAlthough \\cite{Sharf2019b} presented a strong link between symmetry and clustering in MAS, it did not consider a synthesis procedure for solving the clustering problem:\n\\begin{prob} \\label{prob.cluster}\nConsider a collection of $n$ homogeneous agents $\\{\\Sigma_i\\}_{i\\in \\mathbb{V}}$, and let $r_1,\\ldots,r_k$ be positive integers which sum to $n$. Find a directed graph $\\mathcal{G} = (\\mathbb{V},\\mathbb{E})$ and homogeneous edge controllers $\\{\\Pi_e\\}_{e\\in \\mathbb{E}}$ such that the closed loop MAS converges to a clustered steady-state, with a total of $k$ clusters of sizes $r_1,\\ldots,r_k$.\n\\end{prob}\n\nThe goal of this section is use the tools of \\cite{Sharf2019b} to solve Problem \\ref{prob.cluster}. As described above, this can be achieved in two steps. We first make the following assumption about the controllers:\n\\begin{assump}\\label{assump.3}\nThe homogeneous MEIP controllers are chosen so that $0\\not\\in\\gamma_e(0)$ for any edge $e\\in \\mathbb{E}$.\n\\end{assump}\nSecond, given the desired cluster sizes $r_1,\\ldots,r_k$, we seek an oriented graph $\\mathcal{G} = (\\mathbb{V},\\mathbb{E})$ such that the action of $\\mathrm{Aut}(\\mathcal{G})$ on $\\mathcal{G}$ has orbits of sizes $r_1,\\ldots,r_k$. Moreover, we require $\\mathcal{G}$ to be weakly connected\\footnote{Recall that a directed graph is weakly connected if its unoriented counterpart is connected.} to assure a flow of information throughout the corresponding diffusively-coupled network. If we find such a graph and Assumption \\ref{assump.3} holds, the results of \\cite{Sharf2019b} guarantee that the desired clustering behavior is achieved almost surely. We make the following definition for the sake of brevity, and define the corresponding problem:\n\\begin{defn}\nThe oriented graph $\\mathcal{G}$ is said to be of type OS$(r_1,\\ldots,r_k)$ if it is weakly connected and the action of $\\mathrm{Aut}(\\mathcal{G})$ on $\\mathcal{G}$ has orbits of sizes $r_1,\\ldots,r_k$.\\footnote{OS stands for \"orbit structure.\"}\n\\end{defn}\n\\begin{prob}\nGiven positive integers $r_1,\\ldots,r_k$, determine if an oriented graph of type OS$(r_1,\\ldots,r_k)$ exists, and if so, construct it.\n\\end{prob}\n\n\\textcolor{black}{Before moving on to the solving this problem, we present a tool we apply later in the proofs, called the graph quotient.}\n\\textcolor{black}{\n\\begin{defn}\nLet $\\mathcal{G} = (\\mathbb{V},\\mathbb{E})$ be a (directed or undirected) graph, and let $V_1,V_2,\\ldots, V_k$ be a partition of $\\mathbb{V}$ to disjoint sets. The \\emph{quotient of $\\mathcal{G}$}, according to the partition $V_1,V_2,\\ldots, V_k$, is a graph $\\mathcal{C}$ with the following properties:\n\\begin{itemize}\n    \\item[i)] The nodes of $\\mathcal{C}$ are denoted by $1,2,\\ldots,k$.\n    \\item[ii)] For any $l_1,l_2 \\in \\{1,2,\\ldots,k\\}$, There is an edge $l_1\\to l_2$ in the quotient graph $\\mathcal{C}$ if and only if there is at least one edge between elements of $V_{l_1}$ and $V_{l_2}$.\n\\end{itemize}\n\\end{defn}\nIn other words, the quotient graph is achieved by grouping the nodes of $\\mathcal{G}$ by the sets $V_1,V_2,\\ldots, V_k$, removing edges within the same set, and identifying all edges going between the same two groups $V_i$ and $V_j$. An illustration can be seen in Fig. \\ref{fig.Condensation}. It is easy to see that if $\\mathcal{G}$ is connected, then so is its quotient, and this fact will play a vital role later.\n}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width = 0.7\\textwidth]{Figures\/Graph_Condensation.PNG}\n    \\vspace{-15pt}\n    \\caption{An example of graph quotient. The original graph $\\mathcal{G}$ is depicted on left, and the sets $V_1,V_2$ of the partition are marked in blue and red, respectively. The corresponding quotient graph can be seen on the right.}\n    \\vspace{-15pt}\n    \\label{fig.Condensation}\n\\end{figure}\n\n\\vspace{-15pt}\n\\subsection{Construction and Sparsity Bounds on OS-type Graphs}\nIn this subsection, we exhibit a construction for OS-type graphs, as well as bounds on the sparsity of such graphs. Running the system requires means to implement the corresponding interconnections, ergo graphs with fewer edges are desirable. The following theorem provides a lower bound on the number of edges required to construct OS-type graphs.\n\n\\begin{thm}\\label{thm.LowerBound}\nLet $r_1,\\ldots r_k$ be any positive integers. Any directed graph of type OS$(r_1,\\ldots,r_k)$ has at least $m$ edges, where \n\\begin{align} \\label{eq.LowerBound}\nm = \\min_{\\mathcal{T}\\text{\\emph{ tree on $k$ vertices}}} \\sum_{\\{i,j\\}\\in\\mathcal{T}} \\mathrm{lcm}(r_i,r_j).\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nLet  $\\mathcal{G}$ be a graph of type OS($r_1,\\ldots,r_k$), and $V_1,\\ldots,V_k$ be the orbits of $\\mathrm{Aut}(\\mathcal{G})$ in $\\mathcal{G}$, corresponding to the different clusters. \\textcolor{black}{The proof will consist of two steps. First, we show that if there exists at least one edge between $V_i$ and $V_j$, then there are at least $\\mathrm{lcm}(r_i,r_j)$ edges between $V_i$ and $V_j$. This will follow from the fact that the automorphism group of $\\mathcal{G}$ can map any two nodes in each $V_i$ to one another. Second, we consider the quotient graph by the partition $V_1,\\ldots,V_k$, which must be connected as $\\mathcal{G}$ is connected. Hence, it must have a spanning tree, which can be used to determine which pairs $V_i,V_j$ have edges between them. These two facts together will allow us to establish the lower bound \\eqref{eq.LowerBound}.}\n\nWe start with the former claim. Suppose that there is an edge between elements of $V_i,V_j$ for some $i,j\\in\\{1,\\ldots,k\\}$. Let $\\mathcal{G}_{ij}$ be the induced bi-partite subgraph on the vertices $V_i\\cup V_j$, i.e., only edgess between $V_i$ and $V_j$ appear in $\\mathcal{G}_{ij}$. We recall that $V_j$ is invariant under $\\mathrm{Aut}(\\mathcal{G})$, meaning that for any node $v \\in V_i$ and any automorphism $\\psi \\in \\mathrm{Aut}(\\mathcal{G})$, $x$ and $\\psi(x)$ are linked to the same number of nodes in $V_j$. As all the nodes in $V_i$ can be mapped to one another using graph automorphisms, we conclude that they all have the same $\\mathcal{G}_{ij}$-degree, denoted $d_i$. By reversing the roles of $V_i$ and $V_j$, we conclude that all nodes in $V_j$ have the same $\\mathcal{G}_{ij}$ degree as well, which will be denoted by $d_j$. As each edge in $\\mathcal{G}_{ij}$ touches one node from $V_i$ and one node from $V_j$, we conclude that the total number of edges in $\\mathcal{G}_{ij}$ is equal to $d_ir_i = d_jr_j$, as the sets $V_i,V_j$ have $r_i,r_j$ nodes, respectively.  We note that by assumption, there is at least one edge between $V_i$ and $V_j$, hence $d_ir_i = d_jr_j \\ge 1$. In particular, all the numbers $d_i,r_i,d_j$ and $r_j$ are positive integers.\n\nThe rest of the proof of this part of the claim follows from basic number theory and some algebra - the equation $d_ir_i = d_jr_j$, together with the fact that $d_i,d_j,r_i$ and $r_j$ are all positive integers, imply that $r_j$ divides the product $r_i d_i$, and hence the integer $\\frac{r_j}{\\gcd(r_i,r_j)}$ must divide $\\frac{r_i}{\\gcd(r_i,r_j)} d_i$. However, by definition of the greatest common divisor, $\\frac{r_j}{\\gcd(r_i,r_j)}$ and $\\frac{r_i}{\\gcd(r_i,r_j)}$ are disjoint ,meaning that $\\frac{r_j}{\\gcd(r_i,r_j)}$ must divide $d_i$. In particular, as $d_i$ is a positive integer, we conclude that $d_i \\ge \\frac{r_j}{\\gcd(r_i,r_j)}$. Thus, the number of edges in $\\mathcal{G}_{ij}$ is at least $r_i d_i \\ge \\frac{r_i r_j}{\\gcd(r_i,r_j)} = \\mathrm{lcm}(r_i,r_j)$, as claimed.\n\nNow, we move to the second part of the proof. We consider the quotient $\\mathcal{C}$ of $\\mathcal{G}$ by the partition $V_1,\\ldots,V_k$. As $\\mathcal{G}$ is (weakly) connected, so is the quotient $\\mathcal{C}$. In particular, there exists a spanning tree $\\mathcal{T}$ for $\\mathcal{C}$. By the definition of the quotient, for any two nodes $i,j$ connected in $\\mathcal{T}$ (and hence in $\\mathcal{C}$), there is at least one edge between elements of $V_{i}$ and $V_{j}$ in $\\mathcal{G}$. However, by the first part of the proof, we conclude that there are at least $\\mathrm{lcm}(r_i,r_j)$ edges between them. By summing over all connected pairs of nodes in $\\mathcal{T}$, we conclude that the graph $\\mathcal{G}$ has at least $ \\sum_{\\{i,j\\}\\in\\mathcal{T}} \\mathrm{lcm}(r_i,r_j) \\ge m$ edges. This concludes the proof.\n\\end{proof}\n\nBeside giving a lower bound on the number of edges in an OS-type graph, Theorem \\ref{thm.LowerBound} also highlights the role of the quotient base graph $\\mathcal{T}$ in the construction of such graphs. Namely $\\mathcal{T}$, which was taken as a tree, determines which clusters are connected in $\\mathcal{G}$. The following algorithm, Algorithm \\ref{alg.BuildingGraphs}, uses this idea to construct OS-type graphs when $\\mathcal{T}$ is taken as a path graph.\n\n\\begin{algorithm} [!h]\n\\caption{Building sparse OS-type graphs}\n\\label{alg.BuildingGraphs}\n{\\bf Input:} A collection $r_1,\\ldots,r_k$ of positive integers summing to $n$, and a path $\\mathcal{T}$ on $k$ vertices.\\\\\n{\\bf Output:} A graph $\\mathcal{G}$ of type OS$(r_1,\\ldots,r_k)$.\n\\begin{algorithmic}[1]\n\\State Let $\\mathcal{G} = (\\mathbb{V},\\mathbb{E})$ be an empty graph.\n\\State For any $j=1,\\ldots,k$ and $p=1,\\ldots,r_j$, add a node with label $v^j_p$ to $\\mathbb{V}$.\n\\State For any edge $\\{i,j\\}$ in $\\mathcal{T}$ and any $p=1,\\ldots,\\mathrm{lcm}(r_i,r_j)$, add the edge $v^i_{p ~\\mathrm{mod}~ r_i} \\to v^j_{p ~\\mathrm{mod}~ r_j}$ to $\\mathbb{E}$\n\\State Compute $i^\\star = \\arg\\min\\{r_i\\}$. If $r_{i^\\star} = 1$, go to step 6\n\\State For any $p=1,\\ldots,r_{i^\\star}$, add the edge $v_p^{i^\\star} \\to v_{(p+1) ~\\mathrm{mod}~ r_{i^\\star}}^{i^\\star}$ to $\\mathbb{E}$.\n\\State {\\bf Return} $\\mathcal{G} = (\\mathbb{V},\\mathbb{E})$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\\textcolor{black}{Algorithm \\ref{alg.BuildingGraphs} essentially tries to reverse the quotient process described in the proof of Theorem \\ref{thm.LowerBound}. It starts with the quotient graph, given by the tree $\\mathcal{T}$, and it constructs the original graph $\\mathcal{G}$ be assigning nodes to each element of the partition. More precisely, the algorithm assigns to each node $j$ in the tree a set of $r_j$ nodes in $\\mathcal{G}$, denoted by $V_j = \\{v_p^j\\}_{p=1}^{r_j}$. In step 3, the algorithm populates the graph $\\mathcal{G}$ with edges corresponding to those found in the quotient $\\mathcal{T}$, and it does so with the minimal possible amount of edges guaranteeing symmetry (as seen in the proof of Theorem \\ref{thm.LowerBound}). An illustration of this step can be seen in Fig. \\ref{fig.Step3}. In step 5, the algorithm adds a few more edges to guarantee the constructed graph $\\mathcal{G}$ is connected, but in a way that does not effect the quotient process, as all new edges are between nodes in the same set $V_{i^\\star}$ of the partition.} As the following theorem shows, choosing any path $\\mathcal{T}$ will result in a graph of type OS$(r_1,\\ldots,r_k)$. However, the number of edges in the graph depends on the path $\\mathcal{T}$. We note that $\\mathcal{T}$ must be a path rather than a general tree, see Remark \\ref{rem.TreeOrPath} for further discussion.\n\n\\begin{figure}[b]\n    \\centering\n    \\vspace{-15pt}\n    \\includegraphics[width = 0.7\\textwidth]{Figures\/Synthesis_Graph_Example.PNG}\n    \\vspace{-10pt}\n    \\caption{An illustration of step 3 in Algorithm \\ref{alg.BuildingGraphs} with $r_i = 4$ (in red) and $r_j = 2$ (in blue). The algorithm starts with the edge between $v_1^i$ and $v_1^j$. It then moves along the green dashed lines, adding an edge after each one step. This step of the algorithm terminates when the algorithm tries to add an already existing edge, resulting in the black edges depicted in the figure.}\n    \\label{fig.Step3}\n\\end{figure}\n\n\\begin{thm}\\label{thm.UpperBound}\nLet $r_1,\\ldots r_k$ be positive integers summing to $n$. For any path $\\mathcal{T}$, Algorithm \\ref{alg.BuildingGraphs} outputs a graph $\\mathcal{G}$ of type OS$(r_1,\\ldots,r_k)$ having $\\sum_{\\{i,j\\}\\in\\mathcal{T}} \\mathrm{lcm}(r_i,r_j) + \\min_i r_i$ edges. Thus, there is a graph of type OS$(r_1,\\ldots,r_k)$ having $M$ edges, where\n\\begin{align} \\label{eq.UpperBound}\nM = \\min_{\\mathcal{T}\\text{\\emph{ path on $k$ vertices}}} \\sum_{\\{i,j\\}\\in\\mathcal{T}} \\mathrm{lcm}(r_i,r_j) + \\min_{i} r_i.\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\nWe assume, without loss of generality and for the benefit of neater notation, that the path $\\mathcal{T}$ is of the form $1 \\to 2 \\to \\ldots \\to k$, and let $i^\\star$ be the vertex chosen in step 5, i.e. the node at which $r_i$ is minimized. By construction, the number of edges in $\\mathcal{G}$ is equal to $\\sum_{\\{i,j\\}\\in\\mathcal{T}} \\mathrm{lcm}(r_i,r_j)$, plus $r_{i^\\star}$ if $r_{i^\\star} \\ge 2$. Thus, it suffices to show the orbits of the action of $\\mathrm{Aut}(\\mathcal{G})$ on $\\mathcal{G}$ are given by $V_1,\\ldots, V_k$, where $V_j = \\{v_j^p\\}_{p=1}^{r_j}$, and that $\\mathcal{G}$ is weakly connected. We start with the former.\n\nRegarding the orbits, we must show that all nodes in $V_j$ are exchangeable, and that $V_j$ are invariant under $\\mathrm{Aut}(\\mathcal{G})$, for any $j=1,\\ldots,k$. For the first claim, we consider the map $\\psi:\\mathbb{V}\\to\\mathbb{V}$ defined by sending each vertex $v^j_p$ to $v^j_{(p+1)~\\mathrm{mod}~ r_j}$, illustrated by the dashed green edges in Fig. \\ref{fig.Step3}. By construction, this map is an automorphism of $\\mathcal{G}$. Moreover, iterating $\\psi$ enough times would move $v^j_p$ to any vertex in $V_j$, hence any two nodes in $V_j$ are exchangable.\n\nSecond, we show that each $V_j$ is invariant under $\\mathrm{Aut}(\\mathcal{G})$, implying that  the orbits of $\\mathrm{Aut}(\\mathcal{G})$ in $\\mathcal{G}$ are exactly $V_1,\\ldots V_k$. This is obvious if $k=1$, as $V_1 = \\mathbb{V}$. If $k\\ge 2$, then either $i^\\star\\neq 1$ or $i^\\star\\neq k$ (or both). We assume $i^\\star\\neq k$ without loss of generality. Graph automorphisms preserve all graph properties, and in particular, they preserve the out-degree of vertices. As all edges are oriented from $V_j$ to $V_{j+1}$ or from $V_i^\\star$ to itself. Therefore, the vertices in $V_k$ have an out-degree of $0$, and they are the only ones with this property. Thus $V_k$ must be invariant under $\\mathrm{Aut}(\\mathcal{G})$. The only vertices with edges to $V_k$ are in $V_{k-1}$, so $V_{k-1}$ is also invariant under $\\mathrm{Aut}(\\mathcal{G})$. Iterating this argument, we conclude that $V_1,\\cdots V_k$ must all be invariant under the action of $\\mathrm{Aut}(\\mathcal{G})$, as claimed.\n\nNow, we show that $\\mathcal{G}$ is weakly connected. First, note the induced subgraph on $V_{i^\\star}$ is weakly connected, due to the construction made in step 5. Indeed, this is clear if $r_{i^\\star}=1$, and in the case $r_{i^\\star} \\ge 2$, the cycle $v^{i^\\star}_1 \\to v^{i^\\star}_2 \\to v^{i^\\star}_3 \\to \\cdots$ eventually passes through all the nodes in $V_{i^\\star}$. However, by the construction in step 3 of the algorithm, any two nodes $v^i_p$ and $v^j_p$ can be connected by a path. Thus any two arbitrary vertices $v^{j_1}_{p_1}$ and $v^{j_2}_{p_2}$ can be linked as follows - first, go from $v^{j_1}_{p_1}$ to $v^{i}_{p_1~\\mathrm{mod}~ r_i}$ using the edges added in step 3; then, move to $v^{i}_{p_2~\\mathrm{mod}~ r_{i}}$ using the edges added in step 5; lastly, continue from $v^{i}_{p_2~\\mathrm{mod}~ r_{i}}$ to $v^{j_2}_{p_2}$ using the edges added to the graph in step 3. Hence, $\\mathcal{G}$ is weakly connected, and the proof is complete.\n\\end{proof}\n\n\\begin{rem} \\label{rem.TreeOrPath}\nThe lower bound considers all possible trees, but the upper bounds only considers path graphs. It can be seen in the proof of Theorem \\ref{thm.UpperBound} that the fact that $\\mathcal{T}$ is a path is only used to prove that each $V_j$ is invariant under the action of $\\mathrm{Aut}(\\mathcal{G})$. This might be false if $\\mathcal{T}$ is any tree, as the following example shows.  Consider Algorithm \\ref{alg.BuildingGraphs} with 4 clusters of size $1$ and a tree $\\mathcal{T}$ as depicted in Fig. \\ref{fig,Tree}. In this case, the graph $\\mathcal{G}$ is equal to the tree $\\mathcal{T}$. However, the permutation switching the nodes $2$ and $3$ is a graph automorphism, so there is a cluster of size at least $2$, hence $\\mathcal{G}$ is not OS$(1,1,1,1)$. Nevertheless, one should notice that the upper and lower bounds coincide when the number of clusters $k$ is at most $3$, as any tree on at most $3$ nodes must be a path.\n\\end{rem}\n\n\\begin{figure}[b]\n        \\centering\n        \\vspace{-10pt}\n        \\includegraphics[width=0.4\\textwidth]{ASJC_Paper\/Tree.PNG}\n        \\vspace{-10pt}\n        \\caption{The tree graph discussed in Remark \\ref{rem.TreeOrPath}.}\n        \\label{fig,Tree}\n\\end{figure}\n\n\\begin{rem}\nThe lower bound \\eqref{eq.LowerBound} can be found using Kruskal's algorithm, which finds a minimal spanning tree in polynomial time \\cite{Cormen2009}. However, the upper bound \\eqref{eq.UpperBound} requires one to solve a variant of the traveling salesman problem, which is {\\bf NP}-hard \\cite{Cormen2009}. \n\\end{rem}\n\n\nAlgorithm \\ref{alg.BuildingGraphs} solves the general cluster assignment problem, as it constructs graphs of type OS$(r_1,\\ldots,r_k)$ for any cluster sizes $r_1,\\ldots,r_k$. Unfortunately, the bound \\eqref{eq.UpperBound} is implicit in terms of the number of nodes and clusters. We elucidate it by applying it to more specific cases, resulting in concrete bounds on the number of edges needed for clustering in these cases.\n\\begin{cor}\nSuppose all cluster sizes $r_1,\\ldots,r_k > 1$ are equal. Then there exists a graph of type OS($r_1,\\ldots,r_k$) with at most $n = r_1+\\cdots+r_k$ edges, and this number of edges is minimal.\n\\end{cor}\n\n\\begin{proof}\nLet $r$ be the size of all clusters, so that $\\mathrm{lcm}(r,r) = r$ and the number of clusters is $k=n\/r$. Thus, the graph $\\mathcal{G}$  outputted by Algorithm \\ref{alg.BuildingGraphs} (for an arbitrary $\\mathcal{T}$) has exactly $n$ edges, as the summation over the edges has $k-1$ elements. It remains to show that no graph of type OS($r_1,\\ldots,r_k$) can have fewer than $n$ edges. As any graph with fewer than $n-1$ edges is not weakly connected \\cite{Godsil2001}, it suffices to prove that such a graph cannot have exactly $n-1$ edges. \n\nFirst, note that the out-degree is preserved by the action of $\\mathrm{Aut}(\\mathcal{G})$, meaning that vertices in the same cluster have the same out-degree. Denoting the out-degree of nodes in the $i$-th cluster by $d_i$, the total number of edges is equal to the sum of the out-degree over all nodes, i.e. to $r(d_1+\\cdots+d_k)$. In particular, the number of edges, $n-1$ is divisible by $r$. As $n=kr$, $n$ is also divisible by $r$, which together implies that $r$ divides $1$, which is absurd. Thus, no such graph on $n-1$ edges can exist.\n\\end{proof}\n\n\\begin{cor}\nLet $r_1,\\ldots,r_k$ be positive integers such that $k\\ge 2$ and that for every $j,l$, either $r_j$ divides $r_l$ or vice versa. Then there exists a graph of type OS($r_1,\\ldots,r_k$) with $n = r_1+\\cdots+r_k$ edges.\n\\end{cor}\n\n\\begin{proof}\nWe reorder the numbers $r_1,\\cdots, r_k$ so that $r_l$ divides $r_j$ for $l\\le j$. We note that if $r_l$ divides $r_j$, then $\\mathrm{lcm}(r_l,r_j) = r_l$. Thus, running Algorithm \\ref{alg.BuildingGraphs} with $\\mathcal{T} = 1\\to 2\\to\\ldots\\to k$ gives a graph type OS($r_1,\\ldots,r_k$) with the following number of edges:\n$$\n\\sum_{j = 1}^{k-1} \\mathrm{lcm}(r_j,r_{j+1}) + r_1 = \\sum_{j=1}^{k-1} r_{j+1} + r_1=\\sum_{j=1}^k r_j = n.\n$$\n\\end{proof}\n\n\\begin{cor}\nLet $r_1,\\ldots,r_k$ be positive integers such that $r_j \\le q$ for all $j$, and let $n = r_1+\\cdots+r_k$. Then there exists a graph of type OS($r_1,\\ldots,r_k$) with at most $n+O(q^3)$ edges.\n\\end{cor}\n\n\\begin{proof}\nWe assume without loss of generality that $r_1\\le r_2\\le \\cdots \\le r_k$. Let $m_l$ be the number of clusters of size $l$ for $l=1,2,\\ldots,q$, and let $\\mathcal{G}$ be the graph constructed by Algorithm \\ref{alg.BuildingGraphs} for the path $\\mathcal{T} = 1\\to 2\\to\\cdots\\to k$. If $r_j=r_{j+1}$ then $\\mathrm{lcm}(r_j,r_{j+1}) = r_j$, and $\\mathrm{lcm}(r_j,r_{j+1}) \\le r_j r_{j+1}$ otherwise. Thus, the number of edges in $\\mathcal{G}$ is given by:\n\\begin{align*}\n\\sum_{j =1}^{k-1} \\mathrm{lcm}(r_j,r_{j+1}) +r_1 \\le  \\hspace{-7pt}\\sum_{\\substack{l\\in\\{1,\\ldots,q\\},\\\\ m_l \\neq 0}} \\hspace{-7pt}(m_l-1)l + \\sum_{l=1}^{q-1} l(l-1) + r_1.\n\\end{align*}\nIndeed, for each $l\\in\\{1,\\ldots,q\\}$, if there's at least one cluster of size $l$, then there are $m_l -1$ edges in the path $\\mathcal{T}$ that touch two clusters of size $l$. The second term bounds the number of edges between clusters of different sizes. We note that $n = \\sum_{l=1}^q l m_l$, so the first term is bounded by $n$. As for the second term, we write $l(l-1) \\le l^2$ and use the formula $\\sum_{l=1}^{q-1} l^2 = \\frac{(q-1)q(2q-1)}{6}$.\nLastly, the last term $r_1$ is bounded by $q$. This completes the proof.\n\\end{proof}\n\nTheorem \\ref{thm.GlobalUpperBound} will show that the upper bound \\eqref{eq.UpperBound} is bounded by $\\frac{(k-1)n^2}{k^2}$ for any cluster sizes, and it will also give a heuristic for choosing the path $\\mathcal{T}$ for Algorithm \\ref{alg.BuildingGraphs} guaranteeing the number of edges does not exceed this upper bound. \n\n\\vspace{-15pt}\n\\subsection{Robust OS-type graphs} \\label{subsec.Robust}\nThe previous section presented a solution to the problem of cluster assignment. Namely, given a collection of homogeneous agents and the desired cluster sizes, the designer constructs the interconnection graph using Algorithm \\ref{alg.BuildingGraphs}, and then chooses a controller following Assumption \\ref{assump.3}. The analysis depicted in \\cite{Sharf2019b} shows that the closed-loop network would then converge to the desired clustered steady-state. However, there are no guarantees on what happens if the network changes abruptly, either due to hardware or software malfunction, a cyber attack, or both. In these cases, one (or more) of the agents effectively become disconnected from the rest of the network, and are effectively removed from the dynamical system and the interaction graph. For this reason, we wish to explore the robustness of OS-type graphs to changes. Ideally, when one (or more) agents are removed from the system due to a malfunction, all other agents should still cluster as before. More specifically, two non-compromised agents that were previously in the same cluster, should still belong to the same cluster. We thus make the following definition:\n\n\\begin{defn}\nThe oriented graph $\\mathcal{G} =(\\mathbb{V},\\mathbb{E})$ with $n$ nodes is said to be \\emph{$s$-robustly OS$(r_1,\\ldots,r_k)$} for a positive integer $s$ (called the clustering robustness parameter) if the following conditions hold: i) $\\mathcal{G}$ is weakly connected; ii) The orbits $V_1,\\ldots,V_k$ of the action of $\\mathrm{Aut}(\\mathcal{G})$ on $\\mathcal{G}$ have sizes $r_1,\\ldots,r_k$ respectively; and iii) for any set $A\\subseteq \\mathcal{V}$ such that $|A| \\ge n - s$ (comprised of the non-compromised agents), denoting the induced subgraph with node set $A$ as $\\tilde{\\mathcal{G}}$, the action of $\\mathrm{Aut}(\\mathcal{\\tilde G})$ on $\\mathcal{\\tilde G}$ has orbits $V_1\\cap A, \\ldots, V_k \\cap A$. Moreover, we say that the oriented graph $\\mathcal{G}$ is \\emph{totally robustly OS$(r_1,\\ldots,r_k)$} if it is $s$-robustly OS$(r_1,\\ldots,r_k)$ for $s = n$. \n\\end{defn}\n\n\\begin{prop} \\label{prop.robust}\nLet $r_1,\\ldots,r_k$ be positive integers, and let $\\mathcal{G}= (\\mathbb{V},\\mathbb{E})$ be a $1$-robustly OS$(r_1,\\ldots,r_k)$ with clusters $V_1,\\ldots,V_k$. Then for any $i\\neq j \\in \\mathcal{V}$, the induced bi-partite subgraph $\\mathcal{G}_{ij}$ is either empty or complete.\n\\end{prop}\n\\begin{proof}\nSuppose that $\\mathcal{G}_{ij}$ is non-empty, and let $v_i \\in V_i$, $v_j\\in V_j$ be two connected nodes in it. We claim all vertices in $V_j$ are connected to $v_i$. Indeed, consider the graph $\\tilde{\\mathcal{G}} \\setminus \\{v_i\\}$ corresponding to the case in which $v_i$ malfunctions. By assumption, both the actions of $\\mathrm{Aut}(\\mathcal{G})$ and $\\mathrm{Aut}(\\tilde{\\mathcal{G}})$ can send any node in $V_j$ to any other node in $V_j$. In particular, all nodes in $V_j$ share both their $\\mathcal{G}$-degree and their $\\tilde{\\mathcal{G}}$-degree. However, the $\\tilde{\\mathcal{G}}$-degree of $v_j$ is smaller than its $\\mathcal{G}$-degree by one, so this must also be the case for any other node in $V_j$, meaning that node must be connected to $v_i$. Reiterating this argument (while changing $i$ with $j$ and the edge $\\{v_i,v_j\\}$ with the edge $\\{v_i,v_j^\\prime\\}$ for $v_j^\\prime \\in V_j$) shows that any node in $V_i$ is connected to any node in $V_j$. \n\\end{proof}\n\nProposition \\ref{prop.robust} suggests a necessary update to Algorithm \\ref{alg.BuildingGraphs} to get robust OS-type graphs. Indeed, the modulo-based construction of edges between different clusters is replaced by taking all possible edges. The adapted Algorithm \\ref{alg.BuildingRobustGraphs} is given below.\n\n\\begin{algorithm} [t!]\n\\caption{Building totally robust OS-type graphs}\n\\label{alg.BuildingRobustGraphs}\n{\\bf Input:} A collection $r_1,\\ldots,r_k$ of positive integers summing to $n$, and a path $\\mathcal{T}$ on $k$ vertices.\\\\\n{\\bf Output:} A graph $\\mathcal{G}$ of type OS$(r_1,\\ldots,r_k)$.\n\\begin{algorithmic}[1]\n\\State If $k=1$, return the complete graph on $n$ nodes. Otherwise, continue.\n\\State Let $\\mathcal{G} = (\\mathbb{V},\\mathbb{E})$ be an empty graph.\n\\State For any $j=1,\\ldots,k$ and $p=1,\\ldots,r_j$, add a node with label $v^j_p$ to $\\mathbb{V}$.\n\\State For any edge $\\{i,j\\}$ in $\\mathcal{T}$, $p_i = 1,\\ldots,r_i$ and $p_j = 1\\ldots,r_j$, add the edge $v^i_p \\to v^j_q$  to $\\mathbb{E}$.\n\\State {\\bf Return} $\\mathcal{G} = (\\mathbb{V},\\mathbb{E})$.\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{thm}\\label{thm.robust}\nLet $r_1,\\ldots r_k$ be positive integers, and let $n = r_1+\\cdots+r_k$. For any path $\\mathcal{T}$, Algorithm \\ref{alg.BuildingRobustGraphs} outputs a graph $\\mathcal{G}$ which is totally robustly OS$(r_1,\\ldots,r_k)$, having $\\sum_{\\{i,j\\}\\in\\mathcal{T}} r_ir_j$ edges.\n\\end{thm}\n\n\\begin{proof}\nWe denote $V_i = \\{v_p^i\\}_{p=1}^{r_i}$. We first prove $\\mathcal{G}$ is an OS-type graph. It is obviously weakly connected as $r_i \\ge 1$ for all $i$, and one can prove that the sets $V_i$ are $\\mathrm{Aut}(\\mathcal{G})$-invariant similarly to the proof of Theorem \\ref{thm.UpperBound}. Moreover, one can see that the combination of any permutations on $V_i$ for $i=1,\\ldots,k$ gives an automorphism of $\\mathcal{G}$, meaning that all nodes inside $V_i$ are equivalent under $\\mathrm{Aut}(\\mathcal{G})$. In particular, we conclude that $\\mathcal{G}$ is an OS$(r_1,\\ldots,r_k)$ graph.\nNow, let $A$ be a set of non-compromised agents, and let $c_i = |V_i \\setminus A|$ be the number of compromised agents in cluster $i$. We observe that the induced subgraph $\\tilde{\\mathcal{G}}$ on $A$ is identical to the output of the algorithm for the path $\\mathcal{T}$ and cluster sizes $r_1-c_1,\\ldots,r_k-c_k$, and in particular, the first part of this proof shows that the clusters in $\\tilde{\\mathcal{G}}$ are exactly $V_1\\cap A,\\ldots, V_k\\cap A$. As the number of edges in $\\mathcal{G}$ is obviously $\\sum_{\\{i,j\\}\\in\\mathcal{T}} r_ir_j$, the proof is complete.\n\\end{proof}\n\nRegrading sparsity, Proposition \\ref{prop.robust} and Algorithm \\ref{alg.BuildingRobustGraphs} show that the number of edges in $s$-robust (or totally robust) OS-type graphs can be bounded from above by $M^\\prime$ and from below by $m^\\prime$, where:\n\\begin{align} \\label{eq.RobustBounds}\n    m^\\prime = \\min_{\\mathcal{T}\\text{\\emph{ tree on $k$ vertices}}} \\sum_{\\{i,j\\}\\in\\mathcal{T}} r_ir_j, \\quad M^\\prime = \\min_{\\mathcal{T}\\text{\\emph{ path on $k$ vertices}}} \\sum_{\\{i,j\\}\\in\\mathcal{T}} r_ir_j.\n\\end{align}\n\nFurthermore, the relation $ab = \\gcd(a,b)\\mathrm{lcm}(a,b)$ gives a connection between \\eqref{eq.RobustBounds}, \\eqref{eq.LowerBound} and \\eqref{eq.UpperBound}. Namely, if $\\rho = \\max_{i,j} \\gcd(r_i,r_j)$, then $m\\le m^\\prime \\le \\rho m$ and $M \\le M^\\prime \\le \\rho M$. We note that $\\rho \\le \\max_i r_i$, meaning the number of edges required to get totally robust OS-type graphs isn't much larger than the number of edges required to get OS-type graph, at least when there are no large clusters.\n\nBefore moving on to a numerical example, we wish to provide one more result relating the upper bound $M^\\prime$ (and hence $M$) to the number of nodes and the number of clusters.\n\\begin{thm} \\label{thm.GlobalUpperBound}\nLet $r_1,\\ldots,r_k$ be positive integers summing to $n$. Suppose without loss of generality that $r_1 \\ge r_2 \\ge \\cdots \\ge r_k$, and let $\\mathcal{T}$ be the path $1\\to k\\to 2\\to (k-1)\\to\\cdots$ on $k$ nodes. The graph $\\mathcal{G}$ outputted by Algorithm \\ref{alg.BuildingRobustGraphs} has no more than $\\frac{(k-1)n^2}{k^2}$ edges.\n\\end{thm}\n\n\\begin{proof}\nWe first consider the case in which $k$ is odd, i.e., $k = 2\\ell+1$ for some integer $\\ell$. There are two types of edges in the path $\\mathcal{T}$ - edges of the form $i\\to (k+1)-i$ for $i=1,2,\\ldots, \\ell$, and edges of the form $(k+1)-i \\to i+1$ for $i=1,2,\\ldots,\\ell$. Thus, by Theorem \\ref{thm.robust}, the number of edges in $\\mathcal{G}$ is no larger than the value of the following (continuous) optimization problem:\n\\begin{align*}\n    \\nu = \\max \\left\\{\\sum_{\\{i,j\\}\\in\\mathcal{T}} x_ix_j = \\sum_{i=1}^\\ell x_i x_{(k+1-i)} + \\sum_{i=1}^{\\ell-1} x_{i+1} x_{(k+1-i)} : \\sum_{i=1}^k x_i = n,~x_1 \\ge x_2 \\ge \\ldots \\ge x_k \\right\\}.\n\\end{align*}\n\nWe let $x_1^\\star,\\ldots,x_k^\\star$ be the optimal solution to the problem above. If we show that all $x_i^\\star$ are equal to each other (and hence to $n\/k$), this would imply that the number of edges in $\\mathcal{G}$ is bounded by the value of the cost function at $x_1^\\star,\\ldots,x_k^\\star$, which is equal to $\\nu = \\frac{(k-1)n^2}{k^2}$, as claimed. Suppose, for example and heading toward contradiction, that $x_2^\\star < x_3^\\star$ and that $\\ell \\ge 3$. The derivative of the cost function $F$ satisfies the following inequality:\n\\begin{align*}\n    \\frac{\\partial F}{\\partial x_2}|_{x^\\star} = x_k^\\star + x_{k-1}^\\star \\le x_{k-1}^\\star + x_{k-2}^\\star = \\frac{\\partial F}{\\partial x_3}|_{x^\\star},\n\\end{align*}\nwhere the inequality follows from the constraints of the optimization problem. Thus, slightly reducing $x_2^\\star$ and increasing $x_3^\\star$ by the same amount results in a feasible solution with at least the same value of the cost function. This is contradictory to the manner in which $x^\\star$ was chosen, meaning that $x_2^\\star = x_3^\\star$. A similar argument shows that in fact, $x_1^\\star = x_2^\\star = \\cdots = x_\\ell^\\star$, and that $x_{\\ell+1}^\\star = \\cdots = x_k^\\star$. Finally, if the value of the former is different from the value of the latter, one similarly shows that reducing all $x_1^\\star,\\ldots,x_\\ell^\\star$ by some small $\\epsilon > 0$ and simultaneously increasing all $x_{\\ell+1}^\\star,\\ldots,x_k^\\star$ by the same $\\epsilon$ gives a feasible solution with at least the same value of the cost function. Therefore, all the $x_i^\\star$-s must be equal, and the bound is achieved. The proof for an even number of clusters $k$ is analogous, and is omitted for the sake of brevity and due to lack of space.\n\\end{proof}\n\n\\begin{rem} \\label{rem.WorstCaseSparse}\nThe proof of Theorem \\ref{thm.GlobalUpperBound} also gives the worst case scenario for Algorithm \\ref{alg.BuildingRobustGraphs}, namely, the maximal number of edges is achieved when all clusters are equally-sized. \nThe upper bound in Theorem \\ref{thm.GlobalUpperBound} also applies to the non-robust graphs outputted by Algorithm \\ref{alg.BuildingGraphs}. In fact, this upper bound is tight, at least in order of magnitude. Indeed, a graph with $O\\left(\\frac{(k-1)n^2}{k^2}\\right)$ edges can be found by taking $r_1,\\ldots,r_k$ as the $(L+1),\\ldots,(L+k)$-th prime numbers, where the $L$ is chosen as an integer satisfying $L\\log L \\approx \\frac{n}{k}$.\n\\end{rem}\n\n\\section{Numerical Example}\nWe consider a collection of identical agents, all of the form $\\dot{x} = -x+u+\\alpha ,y=x$ where $\\alpha$ is a log-uniform random variable between $0.1$ and $1$, identical for all agents. In all experiments described below, we considered identical controllers, equal to the static nonlinearity of the form\n$$\n\\mu = a_1 + a_2 (\\zeta + \\cos(\\zeta)),\n$$\nwhere $a_1$ was chosen as a Gaussian random variable with mean $0$ and standard deviation $10$, and $a_2$ was chosen as a log-uniform random variable between $0.1$ and $10$. We note that the agents are indeed output-strictly MEIP and the controllers are MEIP. Moreover, the network is homogeneous, so $\\mathrm{Aut}(\\mathcal{G},\\Sigma,\\Pi) = \\mathrm{Aut}(\\mathcal{G})$. Thus, we can use the graphs constructed Algorithm \\ref{alg.BuildingGraphs}, to force a clustering behavior.\n\nWe first consider a network of $n=15$ agents and tackle the cluster synthesis problem with five equally-sized clusters, i.e., $r_1=r_2=r_3=r_4=r_5=3$. One possible graph forcing these clusters, as constructed by Algorithm  \\ref{alg.BuildingGraphs} for a path $\\mathcal{T}$ of the form $1\\to 2\\to 3\\to 4\\to 5$, can be seen in Fig. \\ref{fig.ClusterSynthesisTheoremGraph1}, along with the agents' trajectories for the closed-loop system. \n\nSecond, we consider a network of $n=10$ agents with desired cluster sizes $r_1=1,r_2=2,r_3=3,r_4=4$. We build a graph forcing these cluster sizes by using Algorithm  \\ref{alg.BuildingGraphs} for a path $\\mathcal{T}$ of the form $4\\to 2\\to 1\\to 3$, which is the minimizer in \\eqref{eq.UpperBound}. The graph can be seen in Fig. \\ref{fig.ClusterSynthesisTheoremGraph1}, along with the agents' trajectories for the closed-loop system. \n \\begin{figure}[t!]\n \\begin{center}\n \t\\hspace{.5cm}\\subfigure[Graph forcing cluster sizes $r_1=r_2=r_3=r_4=r_5=3$. Nodes with the same color will be in the same cluster.] {\\scalebox{.80}{\\includegraphics{Figures\/Graph33333.pdf}}} \\hspace{0.5cm}\n\t\\subfigure[Agent's trajectories for the closed-loop system. Colors correspond to node colors in the graph.] {\\scalebox{.5}{\\includegraphics{Figures\/GraphSynthesis33333-eps-converted-to.pdf}}} \n   \\caption{First example of graphs solving the cluster synthesis problem, achieved by running Algorithm \\ref{alg.BuildingGraphs}.}\\hspace{.5cm}\n\t\\label{fig.ClusterSynthesisTheoremGraph1}\n\t\\vspace{-25pt}\n \\end{center}\n \\end{figure}\n \\begin{figure}[t!]\n \\begin{center}\n \t\\hspace{0.5cm}\\subfigure[Graph forcing cluster sizes $r_1=1,r_2=2,r_3=3,r_4=4$. Nodes with the same color will be in the same cluster.] {\\scalebox{.80}{\\includegraphics{Figures\/Graph4312.pdf}}}\\hspace{0.5cm} \\vspace{0.2cm}\n\t\\subfigure[Agent's trajectories for the closed-loop system. Colors correspond to node colors in the graph.] {\\scalebox{.5}{\\includegraphics{Figures\/GraphSynthesis4213-eps-converted-to.pdf}}} \\hspace{0.5cm}\n   \\caption{First example of graphs solving the cluster synthesis problem, achieved by running Algorithm \\ref{alg.BuildingGraphs}.}\n\t\\label{fig.ClusterSynthesisTheoremGraph2}\n\t\\vspace{-15pt}\n \\end{center}\n \\end{figure}\n\n\\section{Conclusions}\\label{sec.conclusion}\nThis work explored the problem of cluster assignment for homogeneous diffusively-coupled multi-agent systems. We relied upon the clustering analysis results of \\cite{Sharf2019b} to exhibit synthesis procedures that guarantee a clustering behaviour, no matter the desired number of clusters nor their size. This was done by prescribing graph synthesis algorithms which have certain symmetry properties that reflect the desired clustering assignment. When such graphs are used in a network comprised of weakly equivalent agent and controller dynamics, the network converges to a cluster configuration. We further studied the robustness of such MAS to node malfunctions, and presented a graph synthesis procedure which guarantees the clustering structure is robust to any number of agent malfunctions. The results were demonstrated in a numerical example.\n\n\\vspace{-15pt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{\\label{sec:intro} Introduction} \nThe Very Long Baseline Interferometry (VLBI) technique allows us to establish with great accuracy the instantaneous spin axis of the Earth relative to an inertial frame built by observing radio signals from distant quasars. We also know with good accuracy the external torques acting on our spinning planet, mainly from the Sun and the Moon. Thus, knowing both the forcing and the planet's response, and assisted with other valuable data from seismology and geomagnetism, it is possible to infer reasonably accurate models of the Earth and its interior. This indeed is the case, see e.g. \\citet{mathews2002}. The model's parameters, known as the Basic Earth Parameters (BEP), are obtained by fitting the VLBI observations to the models. Mantle elasticity, ocean loading and atmospheric effects make those models highly sophisticated and also successful in explaining most, but not all of the observations, for example as noted by  \\citet{koot2010}. This might be related to the fact that the treatment of the liquid core flow is not nearly as sophisticated as other elements in the model. In fact, the core flow is modelled simply as an inviscid, incompressible flow with uniform vorticity, which for some specific purposes might be sufficient. On the other hand, there is a vast amount of experimental and numerical work on rapidly rotating fluids, where the motion of the bounding surface is prescribed (as steady rotation, including precession or libration), that has revealed surprising features and instabilities related to inertial modes (i.e. modes restored by the Coriolis force), see  \\citet{lebars2015} for a review.\n\nOne of the problems of modelling planets with inviscid fluids is that although viscous effects might not be important a priori, the limit when viscosity tends to zero is not well behaved and inviscid solutions might become singular, a fact already noted by  \\citet{rieutord2001}. In a spherical shell, like the Earth's fluid core, the only inertial modes that stay regular in the inviscid limit are the purely toroidal ones, also called planetary or Rossby modes. In astrophysical or geophysical applications the small but finite viscosity regularizes any singularities, which appear instead as detached shear layers  \\citep{hollerbach1995,rieutord1997,rieutord2001}. There is consequently additional energy dissipation caused by these shear layers, but more importantly, in the case electrically conducting fluids, the ohmic dissipation is large in these layers and can become the main energy dissipation mechanism as the inviscid limit is approached, see  \\citet{buffett2010} and  \\citet{lin2017}.\n\nAnother aspect is the possible interaction between inertial modes and the rotational modes of the Earth as studied by  \\citet{rogister2009}. This study is quite suggestive because it shows that avoided crossings can take place under certain conditions. However the limited resolution they could achieve precluded them to identify unambiguously their eigenmodes as inertial modes, apart from the fact that the fluid core is assumed inviscid, therefore limited to \\emph{regular} (i.e. purely toroidal) inertial modes in their spherical shell configuration. The present study has a similar aim, namely to understand how the rotational modes like the Free Core Nutation (FCN) and the Chandler Wobble (CW) interact with inertial modes. \n\nAs a first step towards a better fluid dynamical treatment of the problem we consider a rigid spheroidal mantle and a completely fluid core, including viscosity. Instead of considering eigendisplacements as done by  \\citet{rogister2009} we consider the Navier-Stokes equation (linear in velocity) to model with high resolution the core flow, including the motion of the mantle through the Coriolis and Poincar\\'e forces. Simultaneously, we compute the motion of the mantle by considering the torques exerted by the fluid on the mantle (Euler equations). To deal with the spheroidal shape of the CMB, we adopt the same Taylor expansion method that we developed in  \\citet{rekier2018}. We discuss in detail the model in Section \\ref{sec:model} and the numerical method in Section \\ref{sec:num}. Sections \\ref{sec:res} through \\ref{sec:disc} are devoted to the presentation and discussion of the results. In Section \\ref{sec:end} we end with a summary and perspectives for future work. The appendix presents important technical details.\n\n\n\\section{The coupled inertial-rotational model}\n\\label{sec:model}\nWe consider a two-layer model planet with a completely fluid core enclosed by a rigid, oblate and axisymmetric ellipsoidal mantle. The basic (unperturbed) state of the planet is that of uniform angular speed $\\Omega_0$ throughout and around the short axis of symmetry (the $\\hat {\\bm z}$ axis). We allow the mantle's spin $\\vb{\\Omega}(t)$, as referred to an inertial frame, to respond freely to torques exerted by the fluid core. We assume small perturbed motions compared to the basic state. \n\n\\subsection{The fluid core}\nWe model the core as an incompressible, homogeneous fluid with kinematic viscosity $\\nu$ and density $\\rho_f$. We describe the flow, $\\vb{u}$, as observed from a reference frame attached rigidly to the mantle, where the boundary conditions can be prescribed more conveniently. With this choice, the dimensionless, \\emph{linearized} Navier-Stokes equation describing the fluid's momentum balance reads\n\\begin{equation}\n\\partial_t \\vb{u} + 2\\,\\vu{z} \\cross \\vb{u} + \\partial_t \\vb{\\Omega} \\cross \\vb{r} = -\\grad p+ E\\,\\nabla^2 \\vb{u},\n\\label{eq:ns}\n\\end{equation}\nwhere the unit time is $1\/\\Omega_0$, the unit length is the semimajor axis $a$ of the spheroidal CMB, $E$ is the Ekman number defined as $E=\\nu\/\\Omega_0 a^2$, and $p$ is the \\emph{reduced} pressure related to the physical pressure $P$ through\n\\begin{equation}\n\\label{eq:redp}\np = \\frac{P}{\\rho_f}-\\left(\\vb{\\hat z}\\cross\\vb{r}\\right)\\cdot\\left(\\vb{M}\\cross\\vb{r}\\right)+\\Phi,\n\\end{equation}\nwhere the total angular velocity of the mantle is $\\vb{\\Omega}=\\vb{\\hat z}+\\vb{M}$, see Eq.~({\\ref{eq:Mdef}}) further below, and $\\Phi$ is the gravitational potential. The term $\\partial_t \\vb{\\Omega} \\cross \\vb{r}$ in Eq.~(\\mbox{\\ref{eq:ns}}) is known as the \\emph{Poincar\\'e} or \\emph{Euler} force, a ficticious force arising due to the mantle's unsteady rotation.\nNow, to have a well defined problem we need to determine the mantle's angular velocity $\\vb{\\Omega}$ by considering the torque balance. The total torque exerted by the fluid core on the mantle is the sum of the pressure (i.e. topographic) and viscous torques (the gravitational torque vanishes since there is no mass redistribution):\n\\begin{equation}\n\\vb*{\\Gamma} = \\int \\vb{r} \\cross \\grad P \\dd{V} - \\rho_f E \\int \\vb{r} \\cross \\nabla^2 \\vb{u} \\dd{V} \\equiv \\rho_f \\vb*{\\gamma},\n\\end{equation}\nwhere the volume integrals extend over the whole fluid domain and $\\vb*{\\gamma}$ is dimensionless. The torque $\\vb*{\\Gamma}$ becomes itself dimensionless once we choose an appropriate unit for mass, which we do in such a way that $\\rho_f=1$. The pressure torque involves the physical pressure, however it is desirable to have an expression for the torque that only involves $\\vb{u}$. This can be accomplished once the poloidal-toroidal decomposition of the flow is introduced, see Section~\\ref{sec:num}. The Ekman number $E$ constitutes one of the control parameters of the system.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{diagram_twolayer.pdf}\n\\caption{Schematic diagram of the two-layer planet model. The fluid core has density $\\rho_f$ and the mantle has density $\\rho_m$. We choose the vertical $z$ axis in the mantle frame to coincide with the mantle's vertical figure axis. In our choice of units the semimajor axis $a$ of the axisymmetric CMB has unit length. The semimajor axis of the mantle's outer surface, $r_o$, together with the density ratio $\\rho_m\/\\rho_f$ determines the control parameter $q$, see Eq.~\\ref{eq:q}.}\n\\label{fig:schem}\n\\end{figure}\n\n\n\\subsection{The rigid mantle}\nFor simplicity we assume that the flattening $\\alpha$ (defined as $(a-b)\/a$ where $a$ and $b$ are the semimajor and semiminor axes, respectively) of the core-mantle boundary is small and coincides with the flattening of the mantle's outer surface. See Fig.~\\ref{fig:schem} for a schematic. If the mantle's moment of inertia around the short axis of symmetry is represented by $\\rho_m C$, and the moment of inertia around an equatorial axis by $\\rho_m A$ ($\\rho_m$ being the mantle's density) then, up to third order in $\\alpha$, the \\emph{dynamical} flattening is\n\\begin{equation}\nf(\\alpha)\\equiv\\frac{C-A}{A}=\\alpha+\\frac{1}{2}\\,\\alpha^2-\\frac{1}{4}\\,\\alpha^3+\\order{\\alpha^4}.\n\\end{equation} \nThe mantle's spin $\\bm \\Omega$ is a vector relative to an inertial reference frame, however for our purposes it is convenient to express its components using the mantle's body axes, namely\n\\begin{equation}\n\\label{eq:Mdef}\n\\vb{\\Omega} = M_x\\,\\vu{x}+ M_y\\,\\vu{y}+ (M_z+1)\\,\\vu{z}=\\vb{M} + \\vu{z}\n\\end{equation}\nwith $M_x,\\,M_y,\\,M_z$ being time-dependent and small compared to one. The Liouville equations of motion representing the angular momentum balance in an inertial reference frame reduce to the Euler equations for the motion of the rigid mantle. To first order in $\\vb{M}$ they are\n\\begin{align}\n\\partial_t{M_x} + f(\\alpha)\\, M_y & = \\frac{\\rho_f}{\\rho_m A}\\gamma_x  \\label{eulx}\\\\\n\\partial_t{M_y} - f(\\alpha)\\, M_x & = \\frac{\\rho_f}{\\rho_m A}\\gamma_y  \\label{euly}\\\\\n\\partial_t{M_z} &= \\frac{\\rho_f}{\\rho_m C}\\gamma_z.\\label{eulz}\n\\end{align}\n\nThe moment of inertia $\\rho_m A$ depends on the outer semimajor axis of the mantle $r_0$ and also on the flattening $\\alpha$ (the inner semimajor axis of the mantle being unity in our choice of units). To third order in $\\alpha$ we can write\n\\begin{equation}\n\\frac{\\rho_f}{\\rho_m A}=\\frac{15}{16\\pi(r_0^5-1)}\\frac{\\rho_f}{\\rho_m}\\left(2+4\\alpha+5\\alpha^2+5\\alpha^3\\right),\n\\end{equation}\nconsequently, we define a control parameter $q$ as\n\\begin{equation}\n\\label{eq:q}\nq\\equiv\\frac{1}{(r_0^5-1)}\\frac{\\rho_f}{\\rho_m},\n\\end{equation}\nwhich is essentially the inverse of the mantle's mean moment of inertia.\nUsing recent data for Mercury as an example, we estimate $q\\approx0.269$. For Enceladus' icy crust $q\\approx 1.6$ and for Earth's mantle $q\\approx 0.026$. We have then a total of three control parameters: $E,\\,q$ and $\\alpha$. Setting $q=0$ will render the torques $\\gamma_x,\\gamma_y,\\gamma_z$ ineffective and as a result the mantle's rotation becomes steady in time. In this case the spectrum of eigenmodes would correspond to the spectrum of inertial modes of a steadily rotating spheroidal cavity. If $\\alpha=0$ the boundaries are spherical and therefore only the pressure torque would vanish.\n\n\\subsection{Boundary conditions}\n\nWe employ a no-slip boundary condition at the CMB and require regularity at the center. Given the near-spherical shape of the CMB it is possible to prescribe the no-slip condition using a Taylor expansion  \\citep{rekier2018} as follows:\n\\begin{equation}\n\t\\label{bc}\n\t\\left.\\vb{u}\\right|_\\textsc{\\tiny CMB} = \\left.\\vb{u}\\right|_{r=1} + \n    (r_\\textsc{\\tiny CMB}-1)\\,  \\left. \\partial_r   \\vb{u}\\right|_{r=1} +\n\t{\\frac{1}{2}}(r_\\textsc{\\tiny CMB}-1)^2\\,\\left. \\partial_r^2 \\vb{u}\\right|_{r=1} +\n\t{\\frac{1}{6}}(r_\\textsc{\\tiny CMB}-1)^3\\,\\left. \\partial_r^3 \\vb{u}\\right|_{r=1} = 0,\n\\end{equation}\nwhere $r_\\textsc{\\tiny CMB}$ describes the shape of the spheroidal CMB. Up to third order in $\\alpha$, it satisfies\n\\begin{equation}\n\t\\label{eq:cmb1}\n\tr_\\textsc{\\tiny CMB}(\\theta,\\phi)-1 = a_0(\\alpha)+a_2(\\alpha)\\,Y_2^0+a_4(\\alpha)\\,Y_4^0+a_6(\\alpha)\\,Y_6^0,\n\\end{equation}\nwhere $Y_l^m$ are the familiar spherical harmonics and $a_i(\\alpha)$ are polynomials of order three in $\\alpha$ (see the appendix for their explicit form).\nThe advantage of this approach is that it is compatible with spherical harmonic expansions for the flow field $\\vb{u}$. However, this technique is inadequate if the flattening is comparable or larger than the typical thickness $\\sqrt{E}$ of the viscous boundary layer where the components of $\\vb{u}$ undergo large spatial variations. In such a situation, the spherical surface $r=1$ would be too far from the Ekman boundary layer for a polynomial series expansion to describe accurately the exponentially decaying flow near the CMB. Therefore we limit our study to combinations of $E$ and $\\alpha$ such that $\\alpha\\ll\\sqrt{E}$. This is probably not an issue if stress-free boundary conditions are used since in that case the viscous boundary layers would be mostly suppressed.\nWe have chosen to extend the Taylor series up to $\\order{\\alpha^3}$ in order to allow a range as large as possible in the flattening $\\alpha$ for a given Ekman number. The resolution of the Taylor expansion is $\\sim (\\alpha\/\\sqrt{E})^k$ and should be $\\ll 1$ For comparison, a first order calculation ($k=1$) would only allow us to use flattening only up to $\\alpha\\sim\\sqrt{E}\/20$, while a third order calculation ($k=3$) allows us to go confidently up to $\\alpha\\sim\\sqrt{E}\/5$. A fourth order calculation would only bring a very small increase from this range while greatly increasing the computational cost.\n\nThe regularity condition at the origin is better described using the symmetry characteristics of the spherical harmonics, which we introduce below.\n\n\n\\section{Numerical method}\n\\label{sec:num}\n\n\\subsection{Radial and angular discretization}\nWe deal with oscillatory and possibly damped motions, therefore we write the time dependence as\n\\begin{equation}\n\\vb{u}(\\vb{r},t) =\\vb{u}_0(\\vb{r})\\text{e}^{\\lambda t},\\,\\vb{M}(t) = \\vb{M}_0\\text{e}^{\\lambda t},\n\\end{equation}\nwhere $\\vb{u}_0,\\,\\vb{M}_0$ and $\\lambda=\\sigma+i\\omega$ are complex valued. We assume an incompressible fluid, hence a divergenceless flow. A poloidal-toroidal decomposition for $\\vb{u}_0$ is then adequate:\n\\begin{equation}\n\\vb{u}_0(\\vb{r}) = \\curl{\\curl{\\mathcal P\\vb{r}}}+\\curl{\\mathcal T\\vb{r}}.\n\\end{equation}\nThe scalar functions $\\mathcal P,\\mathcal T$ are in turn expanded into spherical harmonics:\n\\begin{align}\n\\label{eq:poltor}\n\\mathcal P(r,\\theta,\\phi) &= \\sum_{\\ell=1}^{\\ell_\\text{max}} \\sum_{m=-\\ell}^{\\ell}  P_{\\ell m}(r)\\,Y_\\ell^m(\\theta,\\phi), \\\\\n\\mathcal T(r,\\theta,\\phi) &= \\sum_{\\ell=1}^{\\ell_\\text{max}} \\sum_{m=-\\ell}^{\\ell}  T_{\\ell m}(r)\\,Y_\\ell^m(\\theta,\\phi).\n\\end{align}\n \n\nFor the radial discretization of the components $P_{\\ell m}$ and $T_{\\ell m}$ we employ a fast spectral method devised by  \\citet{olver2013}.  This method represents the unknown variables using Chebyshev polynomials and Gegenbauer polynomials for their derivatives, its most distinguishing feature is that the resulting matrices are \\emph{sparse} as opposed to common spectral collocation methods where the resulting matrices are dense. Explicitly, the Chebyshev expansions for a given spherical harmonic component are \n\\begin{align}\nP_{\\ell m} = \\sum_{k=0}^N P_{\\ell m}^k\\,t_k(r),\\\\\nT_{\\ell m} = \\sum_{k=0}^N T_{\\ell m}^k\\,t_k(r),\n\\end{align}\nwhere $t_k$ is the Chebyshev polynomial of degree $k$. The coefficients $P_{\\ell m}^k,\\,T_{\\ell m}^k$ constitute the unknowns specifying the core flow once the pressure has been eliminated, see section 3.3. The total number of such coefficients is determined by the truncation levels $N$ and $\\ell_\\text{max}$. \n\n\\subsection{Symmetries}\nTo comply with the regularity condition at the origin we extend first the radial physical domain from $r\\in[0,1]$ to $r\\in[-1,1]$ in order to match the natural domain of the Chebyshev polynomials. There is a consideration that must be taken care of, however; for any given radius $r$, $T_{\\ell m}(r)$ and $T_{\\ell m}(-r)$ are connected by a reflection through the origin and consequently they should possess the same symmetry (parity) as the corresponding spherical harmonic $Y_\\ell^m$, i.e.\n\\begin{equation}\nY_\\ell^m(\\pi-\\theta,\\pi+\\phi)=(-1)^\\ell Y_\\ell^m(\\theta,\\phi).\n\\end{equation}\nTherefore we use only even or odd Chebyshev polynomials depending on whether $\\ell$ is even or odd. Obviously the same consideration goes for $P_{\\ell m}$.\n\nThe problem specified by Eq.~(\\ref{eq:ns}) cleanly decouples in the azimuthal wave number $m$ after carrying out the expansions explained above, which is a consequence of the CMB being axisymmetric. This is not the case for different angular degrees $\\ell$ that end up coupled by the Coriolis force and by the boundary condition at the CMB.\n\nThe solutions for $\\vb{u}$ possess a well defined equatorial symmetry: they are either equatorially symmetric, in which case the poloidal scalars $P_{\\ell m}$ are such that $\\ell-m=$ even while the toroidal scalars $T_{\\ell m}$ fulfil $\\ell+m=$ odd; or vice versa if the solutions are equatorially antisymmetric. This, together with the symmetry requirements for the Chebyshev polynomials, reduces the size of the problem to one fourth of the original.\nIn this study we focus mainly with flows associated with nutations, which are equatorially antisymmetric with $m=1$.\n\n\\subsection{Pressure torque as a function of flow velocity}\nWe still need a way to compute the topographic torque as a function of $\\vb{u}_0$ in order to solve simultaneously the motion of the fluid core and the rigid mantle. The goal is to find an expression for the spherical harmonic component $\\Pi_{\\ell m}(\\vb{r})$ of the physical pressure involving only $M_0$ as well as the poloidal and toroidal scalars $\\mathcal P(\\vb{r})$ and $\\mathcal T(\\vb{r})$, which may contain spherical harmonic components additional to $\\ell$ due to the coupling induced by the Coriolis force. This can be accomplished by taking the \\emph{consoidal} $\\ell, m$ component (i.e. proportional to $\\grad Y_l^m$) of Eq.~(\\ref{eq:ns}) after expanding all terms in spherical harmonics and then solving for $\\Pi_{\\ell m}(\\vb{r})$. Since these steps are algebraically very cumbersome, we perform them with the help of \\classname{TenGSHui}, a symbolic tensor calculus package developed by  \\citet{trinh2018}. The resulting expressions are listed in Appendix \\ref{sec:apb}.  \n\n\\subsection{The generalized eigenvalue problem}\nOur unknowns are the set of coefficients $P_{\\ell m}^k,\\,T_{\\ell m}^k$ and the components of $\\vb{M}_0$. For the fluid core part we take first the curl of both sides of Eq.~(\\ref{eq:ns}), compute the projection along the position vector $\\vb{r}$, and multiply by $r^2$ (to avoid singularities at the origin). Reorganizing the terms a bit this is\n\\begin{equation}\n\\label{eq:curl1}\nr^2\\,\\vb{r}\\cdot \\curl{\\left( E\\nabla^2\\vb{u}_0 -2\\vu{z}\\cross\\vb{u}_0   \\right)} = \\lambda\\,r^2\\,\\vb{r}\\cdot \\curl{\\left(\\vb{u}_0+\\vb{M}_0\\cross\\vb{r}\\right)}.\n\\end{equation}\nOnce the spherical harmonic expansion is carried out the expression above will involve the toroidal coefficients $T_{\\ell m}^k$ for the most part plus some poloidal coefficients brought in by the Coriolis force. Taking the curl twice and projecting along $r^4\\vb{r}$ brings us to\n\\begin{equation}\n\\label{eq:curl2}\nr^4\\,\\vb{r}\\cdot\\curl\\curl\\left(E\\nabla^2\\vb{u}_0 -2\\vu{z}\\cross\\vb{u}_0\\right) = \\lambda\\,r^4\\,\\vb{r}\\cdot\\curl\\curl\\left(\\vb{u}_0\\right),\n\\end{equation}\ninvolving mostly the poloidal coefficients $P_{\\ell m}^k$ and some toroidal ones. Lastly, we write the angular momentum balance more conveniently as\n\\begin{align}\nM_{+}\\left[\\lambda^\\dagger +i\\,f(\\alpha)\\right] &= \\frac{\\rho_f}{\\rho_m A}\\gamma_{+},\\\\\nM_{-}\\left[\\lambda -i\\,f(\\alpha)\\right] &= \\frac{\\rho_f}{\\rho_m A}\\gamma_{-},\\label{eq:lio}\n\\end{align}\nwhere $M_\\pm\\equiv M_x\\pm i\\,M_y$, $\\gamma_\\pm\\equiv \\gamma_x\\pm i\\,\\gamma_y$, and $(^\\dagger)$ denotes the complex conjugate. The reason for this choice is that the quantity $\\gamma_+$ depends only on the $m<0$ components of $\\vb{u}_0$, therefore if we restrict to $m>0$ we need to consider $M_-$ and $\\gamma_-$ only. The torque $\\gamma_z$ vanishes as long as $m\\ne0$, and since we are focusing on antisymmetric $m=1$ modes, we do not need to consider $M_z$.\nThe vector of unknowns $\\vb{x}$ is then composed with the set of coefficients $\\{P_{\\ell m}^k\\}$, $\\{T_{\\ell m}^k\\}$ together with $M_-$. The problem represented by Eqs.~(\\ref{eq:curl1}), (\\ref{eq:curl2}) and (\\ref{eq:lio}) becomes a generalized eigenvalue problem of the form\n\\begin{equation}\n\\label{eq:evp}\n\\vb{A}\\,\\vb{x} = \\lambda\\,\\vb{B}\\,\\vb{x}\n\\end{equation}\nwhere $\\vb{A}$ and $\\vb{B}$ are complex square matrices. The boundary condition represented by Eq.~(\\ref{bc}) is included by replacing appropiate rows in both $\\vb{A}$ and $\\vb{B}$, see  \\citet{olver2013} for further details. We solve Eq.~(\\ref{eq:evp}) numerically using the open-source packages SLEPc  \\citep{dalcin2011,slepc-toms,slepc-manual} and MUMPS  \\citep{MUMPS01,MUMPS02} employing a \\emph{shift and invert} strategy. \n\n\n\n\n\n\\section{Overview of the results}\n\\label{sec:res}\n\n\\subsection{The mode spectrum}\nWe begin by presenting an overview of the spectrum of the least damped equatorially antisymmetric eigenmodes with $m=1$ when $q=0.1$, $\\alpha=10^{-4}$ and $E=10^{-6}$. Figure~\\ref{fig:first} shows the kinetic energy of the mantle compared to the total kinetic energy of the fluid core as observed from a reference frame that rotates with the mean rotation rate of the planet, i.e. the steadily rotating frame (SRF).\nIn Appendix \\ref{sec:apc} we provide details on how to transform the flow field as seen from the mantle frame (the frame used in the eigenvalue problem) to the flow field as seen from the SRF.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=1.15\\linewidth]{firstfig2.pdf}\n\\caption{Ratio of mantle's kinetic energy to core kinetic energy in the steadily rotating frame (SRF). Horizontal axis is the mode frequency $\\omega={\\mathcal Im}(\\lambda)$, the color scale corresponds to the damping $\\sigma={\\mathcal Re}(\\lambda)$ scaled by $\\sqrt{E}$.\nThe spectrum of our two-layer planet model exhibits naturally inertial modes, including the Free Core Nutation (FCN) and the Chandler Wobble (CW). The latter being the least-damped mode of all.}\n\\label{fig:first}\n\\end{figure}\n\\noindent Two eigenmodes stand out, the Chandler Wobble (CW) and the Free Core Nutation (FCN). The CW, with a small prograde (i.e. negative in our convention) frequency $\\omega$, has most of its kinetic energy as an oscillation of the mantle's spin axis and a small fraction of the energy in the fluid core flow. It is by several order of magnitudes the least damped mode of all and it has no counterpart in the spectrum of purely inertial eigenmodes. The FCN on the other hand, is retrograde with frequency $\\omega\\sim 1$ and about 10\\% of its total kinetic energy is taken by the mantle's motion. This fraction is dependent primarily on $q$ as we discuss further below. In an inertial reference frame the FCN and the CW can be described as mostly a uniformly rotating core flow with an axis slightly different from the mantle's spin axis. The FCN mode resembles the well-known `spin-over' inertial mode with the main difference being that the mantle here is executing an oscillatory motion in addition to steady rotation. The rest of the modes have only a very small fraction of their energy in the mantle's spin oscillation, particularly the prograde ones. All modes, with the exception of the CW, have a counterpart in the spectrum of the purely inertial eigenmodes of a steadily rotating spheroid.\n\nOther features evident in Fig.~\\mbox{\\ref{fig:first}}, such as the `voids' in the frequency distribution of the eigenmodes (e.g. near $\\omega\\sim\\pm \\sqrt{2}$), or their semingly ordered distribution (e.g. around the frequency band $-0.25<\\omega<0.25$), are most likely related to the 'quasi-regular' character of some modes. These features, albeit very interesting in themselves, involve many subtleties that are well beyond the scope of our study. Interested readers are encouraged to consult the recent and very detailed discussion by \\mbox{\\citet{rieutord2018}}.\n\n\n  \n\n\\begin{figure}\n\\centering\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{omega_qramp2.pdf}}\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{sigma_qramp2.pdf}}\\\\\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{kin_ratio_qramp.pdf}}\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{k11_qramp.pdf}}\\\\\n\\caption{As the inverse moment of inertia represented by $q$ is varied, the FCN induces a profound rearrangement of neighbouring modes. The FCN's frequency grows approximately linearly with $q$ when $q\\lesssim 1$ as seen in panel (a), while it grows more damped as shown in (b). To ease mode identification, the curves across all panels are colored according to the scaled damping $\\sigma\/\\sqrt{E}$ shown in (b). Here the Ekman number is $E=10^{-6}$ and the flattening is $\\alpha=10^{-4}$. See main text for more details.}\n\\label{fig:main}\n\\end{figure}\n\n\\subsection{Mode interactions}\nNow we want to explore the behavior of the modes in the vicinity of the FCN as the control parameter $q$ is varied. We keep the same flattening and Ekman number as before and vary $q$ starting from zero as shown in Fig.~\\ref{fig:main}. When $q\\sim 0$ the FCN is essentially identical to the spin-over mode and is the second least-damped mode, with the CW being the least-damped one. The toroidal $\\ell=1,m=1$ component $T_{11}$ of the core flow is dominant as evidenced by Fig.~\\ref{fig:main}(d), which is a typical feature of a flow undergoing mostly a solid body rotation. Both the FCN and the CW exhibit this characteristic. As $q$ continues increasing the FCN's frequency goes up at the same time that it becomes more damped. From Fig.~{\\ref{fig:main}}(b) we see that at $q\\sim 2$ the FCN is already more damped than the mode at $\\omega\\approx1.017$, which in turn becomes the second least damped mode after the CW. The fraction of the total energy involved in the FCN's mantle oscillation increases and reaches a maximum around $q\\sim 6$ as Fig.~\\ref{fig:main}(c) shows.\nInterestingly, around $q\\sim 8$, the FCN begins interacting with nearby modes and it undergoes `avoided crossings' in frequency while `crossing' in damping, or vice versa. At $q\\sim 11$ the toroidal $T_{11}$ component of the FCN is not anymore dominant and therefore the mode cannot be described any longer as a solid body rotation flow. In the range $10<q<30$ other modes undergo similar interactions. In this range frequency crossings take place, while avoided crossings in the damping occur that could rather be described as consecutive `mode bumpings'. The FCN mutates into a mode that is not affected by changes in $q$ any more. Around $q\\sim 26$ a mode crossing in damping takes place (near $\\sigma\/\\sqrt{E}\\sim -80$) along with a corresponding avoided crossing in frequency. It is possible that similar dynamics can take place again at higher $q$ and more negative damping values.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.65\\textwidth]{complex.pdf}\n\\caption{Mode trajectories in the $(\\omega,\\sigma)$ plane as $q$ is varied (indicated by the color scale). It is possible to establish a path from the FCN mode at $q=0$ leading to the most damped mode of Fig.~\\ref{fig:main}(b) linking intermediate damping modes and varying $q$ back and forth. Modes performing loops do not lose their identity.}\n\\label{fig:complex}\n\\end{figure}\n\nThe FCN mode induces a substantial rearrangement of the modes as $q$ is varied. At the same time its own identity is lost in the process and when $q\\sim15$ it has taken the place of the mode that initially had $\\sigma\/\\sqrt{E}\\sim -30$ when $q=0$. This latter mode in turn morphed into the one that had initially $\\sigma\/\\sqrt{E}\\sim -36$ when $q=0$. This exchange appears to continue successively as the modes `bump' each other into more negative damping values. In Fig.~\\ref{fig:complex} we show the trajectory that the modes describe in the $(\\omega,\\sigma)$ plane as $q$ changes. Interestingly, it is indeed possible to follow a path connecting the FCN mode at $q=0$ to the most damped mode visible in Fig.~\\ref{fig:main}(b) at $q\\sim6$. The modes that form isolated loops in Fig.~\\ref{fig:complex} are the ones that do not lose their identity throughout the interactions induced by $q$. We also note that even though the modes at $q=0$ seem to match the `morphed' modes at large $q$, the eigenmodes are not identical since the mantle's kinetic energy is different as Fig.~\\ref{fig:main}(c) shows. Only their fluid part match.\n\nAs already mentioned, the FCN mode at $q=0$ corresponds to the viscous spin-over mode in a steadily rotating spheroid. Apart from the viscous boundary layer and internal shear layers, this mode corresponds to the analytical (inviscid) mode in a full sphere with wavenumbers $(n=2,k=1,m=1)$ using Greenspan's notation  \\citep{greenspan1968}. As it turns out, the other modes (at $q=0$) appearing in Fig.~\\ref{fig:main}(a) can be linked also to their spherical inviscid counterpart where their $n$ wavenumber differ by \\emph{six} between two consecutive modes. This is, they form a family for which $n=6j+2$ with $j=0,1,2,\\dots$ and $m=1$ ($m$ being the azimuthal wavenumber) and $k$ is such that the frequency $\\omega$ is closest to one, see also Figs.~\\ref{fig:Eramp} and \\ref{fig:schm}. As $n$ increases, the mode's structure becomes more complex and hence more damped. Interestingly enough, the first three of these modes ($n=8,14,20$) are precisely the ones studied by  \\citet{schmitt2006}. According to that study, as the flattening increases the spin-over frequency and the modes in question can undergo frequency crossings, resulting in a similar viscous interaction as in our case.\n\n\\begin{figure}\n\\centering\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{omega_Eramp2.pdf}}\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{sigma_Eramp2.pdf}}\n\\caption{The frequency (a) and scaled damping (b) of eigenmodes near the FCN as the Ekman number is varied. Here $\\alpha=10^{-5}$ and $q=5$. Colors indicate the scaled damping $\\sigma\/\\sqrt{E}$. The black horizontal lines in (a) mark the frequency of the theoretical inviscid modes of a sphere for the family of modes with $n=2+6j,\\,j=1,2,\\dots$ (See main text for further details). The FCN on panel (a) starts near $\\omega\\sim1.000$ at $E=10^{-8}$ and on panel (b) it starts near $\\sigma\/\\sqrt{E}\\sim -16$ at $E=10^{-8}$. As $E$ increases the FCN undergoes a crossing in frequency and an avoided crossing in the damping near $E\\sim 10^{-5}$.}\n\\label{fig:Eramp}\n\\end{figure}\n\n\\subsection{Reducing the Ekman number}\nIf we reduce the Ekman number, the correspondence explained above can be made more evident since we expect the solutions to approach their inviscid value. We do so in Fig.~\\ref{fig:Eramp} where we reduce the Ekman number down to about $E=10^{-7.9}$. However, we need to reduce as well the flattening in order to stay in the range of validity of the spheroidal boundary condition in Eq.~\\ref{bc}, therefore we set $\\alpha=10^{-5}$ and also set $q=5$. As we discuss later in Section \\ref{sec:FCN}, the slope $\\partial\\omega\/\\partial q$ of the FCN reduces as $E$ decreases and therefore mode interactions will take place at comparatively higher $q$ values. For instance in Fig.~\\ref{fig:Eramp}, where $q=5$, we see already a frequency crossing (with a corresponding avoided crossing in damping) near $E\\sim 10^{-5}$. Taking the flattening aside momentarily, this is to be compared with the example at $E=10^{-6}$ in Fig.~\\ref{fig:main} where modes begin interacting at $q\\sim 8$. Therefore we do not expect mode interactions of this kind to occur at realistic values of $q$ ($q\\lesssim 1$) when considering more astrophysically relevant Ekman numbers ($E\\lesssim10^{-12}$).\nAlready at $E=10^{-8}$ the FCN is essentially noninteracting even though $q=5$. In fact, at that Ekman number the frequencies of the $n=2+6j$ inviscid mode family are reassuringly in good agreement with the frequencies of our model as Fig.~\\ref{fig:Eramp}(a) shows.\n\nAs opposed to the FCN, the CW mode does not appear to interact with any nearby modes in frequency because its damping is very different from its neighbors', which are several orders of magnitude more damped. As we have seen, interactions occur when modes get close enough to each other in the $(\\omega,\\sigma)$ plane. In fact, although the CW's frequency does change with $q$ and crosses other modes' frequencies, its damping remains small even for moderate $q$, so no interaction occurs. We present a more detailed study of the CW's frequency and damping as functions of $q,\\,\\alpha$ and $E$ in Section \\ref{sec:CW}.\n\n\n\n\n\\section{The Free Core Nutation (FCN) mode}\n\\label{sec:FCN}\n\nWe turn now to the dependence of the FCN mode on the parameters $q,\\,\\alpha$ and $E$ away from the interaction region, which is more relevant for planets. The frequency $\\omega_\\textsc{\\tiny fcn}$ and damping $\\sigma_\\textsc{\\tiny fcn}$ are approximately linear with $q$ as long as there are no other modes with similar $(\\omega,\\sigma)$ that could interact with the FCN. This translates to $q\\lesssim 0.1$ for Ekman numbers in the range $10^{-8}<E<10^{-4}$, and flattenings in the range $10^{-7}<\\alpha<10^{-4.6}$. Within those limits we perform linear fits against $q$ following:\n\\begin{equation}\n\\begin{split}\n\\omega_\\textsc{\\tiny fcn}-\\omega_* &= b_\\omega (\\alpha,E) + q\\,m_\\omega(\\alpha,E),\\\\\n\\sigma_\\textsc{\\tiny fcn} &= b_\\sigma (\\alpha,E)+ q\\,m_\\sigma(\\alpha,E),\n\\end{split}\n\\label{eq:fcn1}\n\\end{equation}\nwhere $\\omega_*$ is the theoretical inviscid spin-over mode frequency in a spheroid with flattening $\\alpha$ (computed to third order in $\\alpha$). The intercepts $b_\\omega$ and $b_\\sigma$ represent the frequency and the damping of the viscous spin-over mode, therefore suitable for direct comparison with the viscous corrections computed by  \\citet{zhang2004}.\n\nIn turn, the slopes and intercepts $m_\\omega, b_\\omega$ and $m_\\sigma, b_\\sigma$ appear to be either linear or quadratic in $\\alpha$ if $E$ is fixed,\n so we perform the following additional fits:\n\\begin{equation}\n\\begin{split}\nm_\\omega(\\alpha,E) &= m_{0,\\omega}(E) + \\alpha\\,m_{1,\\omega}(E) + \\alpha^2\\,m_{2,\\omega}(E),\\\\\nb_\\omega(\\alpha,E) &= b_{0,\\omega}(E) + \\alpha\\,b_{1,\\omega}(E),\n\\end{split}\n\\label{eq:fcn2}\n\\end{equation}\ntogether with\n\\begin{equation}\n\\begin{split}\nm_\\sigma(\\alpha,E) &= m_{0,\\sigma}(E) + \\alpha\\,m_{1,\\sigma}(E) + \\alpha^2\\,m_{2,\\sigma}(E),\\\\\nb_\\sigma(\\alpha,E) &= b_{0,\\sigma}(E) + \\alpha\\,b_{1,\\sigma}(E).\n\\end{split}\n\\label{eq:fcn3}\n\\end{equation}\nLastly, we establish approximate scalings of the coefficients $m_{i,\\omega},b_{i,\\omega}$ and $m_{i,\\sigma},b_{i,\\sigma}$ $(i=0,1,2)$ with\nrespect to the Ekman number $E$. The resulting scaling laws for $\\omega_\\textsc{\\tiny fcn}$ and $\\sigma_\\textsc{\\tiny fcn}$ are shown\nin Fig.~\\ref{fig:omega_fcn} and Fig.~\\ref{fig:sigma_fcn}, respectively.\n\n\\begin{figure}\n\\centering\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{fcn_omega_m.pdf}}\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{fcn_omega_b.pdf}}\\\\\n\\caption{Fit coefficients for $m_\\omega$ (a) and $b_\\omega$ (b) corresponding to the FCN mode frequency $\\omega_\\textsc{\\tiny fcn}$, see Eqs.(\\ref{eq:fcn1}) and (\\ref{eq:fcn2}). Dots represent the numerical results (after performing linear fits against $q$) and dashed lines are the power law fits against $E$. Inset panel in (a) is a zoom to reveal more detail on $m_{1,\\omega}$. The continuous lines in panel (b) represent the theoretical viscous corrections of a steadily rotating spheroid as computed by  \\citet{zhang2004}.}\n\\label{fig:omega_fcn}\n\\end{figure}\n\nA few comments are in order. First, from Fig.~\\ref{fig:omega_fcn}(a), we note that $m_{0,\\omega}(E)$ follows approximately a power law once $E$ is below $\\sim10^{-6}$, therefore we perform a power law fit (quoted in the figure) for $E$ in the range $10^{-8}<E<10^{-6}$. In Fig,~\\ref{fig:omega_fcn}(a), $m_{1,\\omega}$ seems to level off at a value very close to unity, which is expected as we discuss in the appendix. The behaviour of $m_{2,\\omega}$ is rather surprising since it scales with a negative power of the Ekman number, which means that the magnitude of $m_{2,\\omega}$ \\emph{increases} as the $E$ decreases, while we expected $m_{2,\\omega}$ to at least level off to a constant value near $-1\/2$ for asymptotically small $E$. This scaling is likely valid in only the limited range we used for the flattening and the Ekman number as we discuss further below. The power law fits for $b_\\omega$ shown in Fig.~\\ref{fig:omega_fcn}(b) are in reasonable agreement with the asymptotically vanishing viscosity correction of the spin-over frequency computed by  \\citet{zhang2004}. Summarizing, we approximate the FCN's frequency through $m_\\omega$ and $b_\\omega$ as\n\\begin{equation}\n\\begin{split}\n\\omega_\\textsc{\\tiny FCN} &=m_\\omega\\,q + b_\\omega,\\,\\textrm{where}\\\\\nm_\\omega &\\approx 0.599\\,E^{0.543}+\\alpha-0.427\\,\\alpha^2\\,E^{-0.499},\\,\\textrm{and}\\\\\nb_\\omega &\\approx 0.558\\,E^{0.539}+2.915\\,\\alpha\\,E^{0.527}.\n\\label{eq:omega_fcn}\n\\end{split}\n\\end{equation}\n\n\\begin{figure}\n\\centering\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{fcn_sigma_m.pdf}}\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{fcn_sigma_b.pdf}}\\\\\n\\caption{Fit coefficients for $m_\\sigma$ in panel (a) and $b_\\sigma$ in panel (b) corresponding to the FCN mode damping $\\sigma_\\textsc{\\tiny fcn}$, see Eqs.(\\ref{eq:fcn1}) and (\\ref{eq:fcn3}). Dots represent the numerical results (after performing linear fits against $q$) and dashed lines are the power law fits against $E$. In panel (b) the continuous lines represent the expectations from  \\citet{zhang2004}, only with the sign changed. Our results for $b_{0,\\sigma}$ are in excellent agreement with the theoretical expectation while $b_{1,\\sigma}$ only begins approaching the theoretical scaling towards the lower Ekman numbers we explored.}\n\\label{fig:sigma_fcn}\n\\end{figure}\n\n\nWe move on now to the linear fits of the damping $\\sigma_\\textsc{\\tiny FCN}$ against $q$. We show the resulting slopes $m_\\sigma$ and intercepts $b_\\sigma$ in Fig.~\\ref{fig:sigma_fcn}. As there is no prediction of what the damping of the FCN mode might be, theoretically or otherwise, we can only compare the intercepts $b_\\sigma$ as they should reduce to the damping of the spin-over mode in a spheroid.  Regarding $m_\\sigma$, in Fig.~\\ref{fig:sigma_fcn}(a), we find a similar situation as before, the $m_{2,\\sigma}$ coefficient seems to follow an inverse power law scaling with $E$, albeit with a smaller magnitude exponent and amplitude compared to $m_{2,\\omega}$. Below $E\\sim10^{-7}$ we start seeing the effects of numeric round-off errors, which affect more high-order quantities such as $m_{2,\\sigma}$ compared to $m_{0,\\sigma}$, for example. For this reason we perform fits for $m_{1,\\sigma}$ and $m_{2,\\sigma}$ in the range $10^{-7.5}<E<10^{-6}$ only. In Fig.~\\ref{fig:sigma_fcn}(b) the fit for $b_{0,\\sigma}$ matches the theoretical asymptotic expectation rather satisfactorily, however the observed power law for $b_{1,\\sigma}$ only begins to approach the asymptotic expectation at the lowest end of our Ekman number range. At the higher end, above $E\\sim10^{-4.5}$, our numerical result for $b_{1,\\sigma}$ has the opposite sign compared to expectation, which seems to be also the case in the study by  \\citet{schmitt2006}, see their Figure 10(a). In sum, we find that the damping of the FCN is approximately\n\\begin{equation}\n\\begin{split}\n\\sigma_\\textsc{\\tiny FCN} &=  m_\\sigma\\,q + b_\\sigma,\\,\\textrm{where}\\\\\nm_\\sigma &\\approx -2.705\\,E^{0.502}+11.58\\,\\alpha\\,E^{0.553} +0.088\\,\\alpha^2\\,E^{-0.37},\\,\\textrm{and}\\\\\nb_\\sigma &\\approx -2.630\\,E^{1\/2}-25.40\\,E + \\alpha\\,\\left(-0.566\\,E^{1\/2}+299.3\\,E\\right).\n\\label{eq:sigma_fcn}\n\\end{split}\n\\end{equation}    \n\n\n\n\n\\section{The Chandler Wobble (CW) mode}\n\\label{sec:CW}\nThere appears to be no mode interactions between the CW and other modes as $q$ is varied. In fact, there are other modes with similarly low frequencies as the CW, but the only ones with potentially low damping that could interact happen to be equatorially symmetric, as can be seen in Fig.~\\ref{fig:schm}(a), hence completely decoupled from the CW. The frequency and damping of the CW appear to be directly proportional to the flattening $\\alpha$ if $q$ and $E$ are fixed, at least for the parameter range we tested, so it is convenient to perform linear fits against $\\alpha$ first:\n\\begin{equation}\n\\begin{split}\n\\omega_\\textsc{\\tiny cw} &= m_\\omega (q,E)\\,\\alpha,\\\\\n\\sigma_\\textsc{\\tiny cw} &= m_\\sigma (q,E)\\,\\alpha.\n\\end{split}\n\\label{eq:cw1}\n\\end{equation}\nThe proportionality constants $m_\\omega$ and $m_\\sigma$ themselves turn out to be approximately linear or quadratic in $q$: \n\\begin{equation}\n\\begin{split}\nm_\\omega(q,E) &= m_{0,\\omega}(E) + q\\,m_{1,\\omega}(E),\\\\\nm_\\sigma(q,E) &= q\\,m_{1,\\sigma}(E) + q^2\\,m_{2,\\sigma}(E).\n\\end{split}\n\\label{eq:cw2}\n\\end{equation}\nHere we do analogously to the FCN case by establishing the approximate scalings of the coefficients $m_{i,\\omega}$ and $m_{i,\\sigma}$ $(i=0,1,2)$ appearing in Eq.~(\\ref{eq:cw2}) with respect to the Ekman number $E$. Figure \\ref{fig:m_cw} shows the results.\n\n\\begin{figure}\n\\centering\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{cw_omega.pdf}}\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{cw_sigma.pdf}}\\\\\n\\caption{Fits for $m_\\omega$ (a) and $m_\\sigma$ (b) corresponding to the Chandler Wobble mode frequency $\\omega_\\textsc{\\tiny CW}$ and damping $\\sigma_\\textsc{\\tiny CW}$, respectively. See Eqs.(\\ref{eq:cw1}) and (\\ref{eq:cw2}). In panel (b) $m_{2,\\sigma}$ and its corresponding fit have been deliberately multiplied by $1\/2$ in order to distinguish it visually from $m_{1,\\sigma}$.  }\n\\label{fig:m_cw}\n\\end{figure}\n\nAs the Ekman number decreases the CW frequency approaches the theoretical inviscid value, i.e., the proportionality constants $m_{0,\\omega}$ and $m_{1,\\omega}$ get both close to $-1$ following approximately a familiar $E^{1\/2}$ scaling. The same scaling applies to the damping as Figure~\\ref{fig:m_cw}(b) shows.\nSummarizing, we find that the Chandler Wobble frequency is\n\\begin{equation}\n\\begin{split}\n\\omega_\\textsc{\\tiny CW}=\\alpha &\\left[-1-1.02\\cdot10^{-5}-4.75\\cdot10^{-3}\\,E^{1\/2}+0.237\\,E \\right. \\\\\n                                      &\\,\\, \\left. +q\\left(-1+1.21\\cdot10^{-5}+3.292\\,E^{1\/2}+12.53\\,E \\right) \\right],\n\\label{eq:omega_cw}\n\\end{split}\n\\end{equation}\nand its damping is\n\\begin{equation}\n\\sigma_\\textsc{\\tiny CW}=\\alpha\\,q\\,\\left[-3.579\\,E^{1\/2}-17.37\\,E+q\\left(-3.579\\,E^{1\/2}+7.511\\,E \\right) \\right].\n\\label{eq:sigma_cw}\n\\end{equation}\n\n\n\n\\section{Discussion}\n\\label{sec:disc}\n\nBy allowing the mantle to react to torques exerted by the fluid we extend the spectrum of purely inertial eigenmodes to include the Chandler Wobble (CW), which induces a flow in the bulk of the fluid core, however small in magnitude compared to the mantle's motion. Additionally, the Free Core Nutation (FCN) appears as an extension of the spin-over mode as soon as the mantle's moment of inertia ($1\/q$) is considered finite. The FCN has a smaller mantle\/fluid kinetic energy ratio than the CW, while other modes have even smaller ratios. As $q$ is varied the FCN may interact with the $n=8,14,20,\\dots$ family of modes when they become close enough in the $(\\omega,\\sigma)$ plane, in a conceptually similar way as they interact with the spin-over mode as shown by  \\citet{schmitt2006}, the difference being that in our case the relevant parameter is $q$ while in Schmitt's case it is the flattening. Figure~\\ref{fig:schm} helps understanding why these modes are prone to either kind of interaction. In Fig.~\\ref{fig:schm}(a) we show the inviscid eigenfrequency distribution for modes with azimuthal wave number $m=1$ for a spheroid with flattening $\\alpha=0.0202$. We see that the equatorially antisymmetric $n=8,14,20,\\dots$ modes are indeed the closest in frequency to the spin-over mode. Note that we can always find modes arbitrarily close in frequency to the spin-over mode, given that the inviscid spectrum is dense. However, those modes will have very complex structure, i.e. they will have high $n$ numbers, and would be very damped in the viscous case preventing them effectively from any interaction with the spin-over mode. In our case, with the FCN replacing the spin-over mode, increasing $q$ from zero causes the FCN frequency to go up at the same time that the damping increases in magnitude, resulting in complex interactions as shown in Fig.~\\ref{fig:main}. In Schmitt's case, increasing the flattening causes the spin-over frequency to cross with the $n=8,14,20,\\dots$ modes frequencies as well, although their damping is affected somewhat differently, \\citep[see][their Figure 4]{schmitt2006}.\n\n\\begin{figure}\n\\centering\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{schmitt2.pdf}}\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{schmitt3.pdf}}\\\\\n\\caption{The spin-over mode in a spheroid can interact with nearby modes if the flattening is large enough. Panel (a) shows the distribution of theoretical (inviscid) eigenfrequencies at $\\alpha=0.0202$), the red dashed line indicating the spin-over ($n=2$) mode frequency. Panel (b) shows the distance in the frequency axis between the spin-over mode and other nearby modes (antisymmetric ones indicated by continuous lines) as the flattening is varied. For details on the ensuing viscous interaction see  \\citet{schmitt2006}.}\n\\label{fig:schm}\n\\end{figure}\n\n\nFigure~\\ref{fig:schm}(a) also explains why the CW does not interact with any nearby modes. The only modes with small enough prograde frequencies (i.e. negative in our convention) like the CW, and possibly small enough damping, are equatorially symmetric, thus unable to interact.\n\nNothing stops both effects from acting simultaneously if the mantle is free to react to torques and the flattening is $\\alpha\\gtrsim 0.08$, in which case an astrophysically small Ekman number would not prevent the FCN to interact with the other modes as it does in our case. This is indeed an interesting perspective, however such a large flattening is beyond our reach given the Taylor expansion method that we use to implement the spheroidal boundary condition.\n\nThe width of the viscous boundary layer scales as $E^{1\/2}$, so we expect that our Taylor expansion method is valid as long as the flattening is such that $\\alpha\/E^{1\/2}\\ll 1$. This might help a bit in understanding the somewhat surprising scaling of the FCN's $m_{2,\\omega}$ and $m_{2,\\sigma}$ coefficients. We can rearrange slightly the expression for the FCN's $m_\\omega$ given by Eq.~(\\ref{eq:omega_fcn}) as\n\\begin{equation}\nm_\\omega \\approx E^{1\/2}\\left[ 0.599\\,E^{0.043} + \\left(\\frac{\\alpha}{E^{1\/2}}\\right) -0.427 \\left(\\frac{\\alpha}{E^{1\/2}}\\right)^2 E^{0.001}\\right].\n\\end{equation}\nWe see that the powers of $\\alpha\/E^{1\/2}$ within the square brackets form a second order power series with coefficients that are either independent of $E$ or decrease slightly as $E$ decreases. Clearly, this series would not converge if the flattening is comparable or larger than the width of the viscous boundary layer.\n\nThe damping of the FCN presents a similar but slightly milder issue. The coefficient $m_{\\sigma}$ shown in Eq.~\\ref{eq:sigma_fcn} can be written\n\\begin{equation}\nm_{\\sigma} \\approx E^{1\/2}\\left[ -2.705\\,E^{0.002} + 11.58\\,\\left(\\frac{\\alpha}{E^{1\/2}}\\right)\\,E^{0.553} +0.088 \\left(\\frac{\\alpha}{E^{1\/2}}\\right)^2 E^{0.13} \\right],\n\\end{equation} \nwhich shows the coefficient of $\\alpha^2\/E$ within the square brackets decreasing faster as $E$ decreases compared to the same coefficient in $m_\\omega$'s case, i.e., $E^{0.13}$ vs. $E^{0.001}$. These `anomalous' scalings effectively prevent us from extrapolating the FCN's frequency and damping to larger flattenings. On the other hand, the Chandler Wobble appears free of such a complication and there is in principle no obstacle to extrapolate Eqs.~(\\ref{eq:omega_cw}) and (\\ref{eq:sigma_cw}) to higher values of the flattening. This is also true for the FCN's $b_\\omega$ and $b_\\sigma$ (that reduce to the spin-over mode frequency and damping respectively) where the scalings we found are not restricted to small $\\alpha$, of course up to where Schmitt's mode interactions start becoming noticeable.\n\n\\begin{figure}\n\\centering\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{Po_dis_ratio.pdf}}\n\\subfloat[]{\\includegraphics[width=0.5\\textwidth]{torque_ratio.pdf}}\n\\caption{In panel (a) we can see that the power $D_P=|\\partial_t\\vb{M}\\cdot\\vb{L}|$ associated with the Poincar\\'e force can be many times larger than the viscous dissipation $D_\\text{kin}=|\\mathcal{D}_\\nu|$, thus dominating the overall damping rate. The pressure torque to viscous torque ratio in panel (b) shows that the viscous torque is dominant over the range of $q$ that we explored ($0\\le q \\le 30$). In both panels the parameters are identical to those of Fig.~\\ref{fig:main}, color scale indicate the scaled damping $\\sigma\/\\sqrt{E}$.}\n\\label{fig:ratios}\n\\end{figure}\n\n\nThe magnitude increase of the FCN's damping rate with $q$ is closely related to the fact that not all the damping is due to viscous dissipation processes. In fact, the Poincar\\'e force is capable of performing work (as opposed to the Coriolis force) and such work must be taken into account in the overall power balance of the system. Starting from the linear momentum balance in Eq.~(\\mbox{\\ref{eq:ns}}), taking the dot product with $\\vb{u}$ and integrating over the whole fluid volume it is straightforward to show that the damping rate $\\sigma$ of any given mode fulfills\n\\begin{equation}\n2\\,\\sigma\\,K_f = \\mathcal{D}_\\nu - \\left(\\partial_t \\vb{M}\\right) \\cdot \\vb{L},\n\\end{equation}\nwhere $K_f$ is the total kinetic energy of the fluid (in the mantle frame), $\\vb{L}$ is the total angular momentum of the fluid (relative to the mantle frame) and $\\mathcal{D}_\\nu$ is the viscous dissipation computed as\n\\begin{equation}\n\\mathcal{D}_\\nu=E\\int \\vb{u}\\cdot \\nabla^2\\vb{u}\\dd{V}\n\\end{equation}\nover the whole fluid volume. For the FCN, as $q$ increases the mantle's motion also goes up (because its moment of inertia decreases), see Fig.~\\ref{fig:main}(c), along with the ratio $|\\partial_t\\vb{M}\\cdot\\vb{L}|\/|\\mathcal{D}_\\nu|$ as shown in Fig.~\\ref{fig:ratios}(a). Near the maximum at $q\\sim8$ the power associated with the Poincar\\'e force is almost seven times as large as the power associated to viscosity. Beyond this value of $q$ mode interactions dominate and the FCN loses its character as a mainly rotational (toroidal) motion.\n\nWe should remark that the mode coupling we witness in this system is essentially viscous in nature for the sole reason that pressure torques exerted on the spheroidal mantle are much smaller than the viscous torques. Figure~{\\ref{fig:ratios}}(b) shows that pressure torque to viscous torque ratio sits near 4{\\%} when the mode interactions take place. That being said, it is the mixing among poloidal and toroidal components of different angular degrees induced by the torques that is at the heart of the mode interaction mechanism. However different in magnitude, both viscous and pressure torques induce similar mixing among components and so there is nothing in principle preventing pressure and gravitational torques, if they were to be dominant over viscous ones, from leading to similar resonant coupling among modes. This would be the case for a more realistic Earth model, where the Ekman number is very small and the pressure torques dominate. Note that additionally, a relatively large flattening such as Earth's, would bring the FCN's mode frequency closer to the region where it might interact resonantly with neighbouring modes regardless of the pressure torques, just as is the case for the spin-over mode as shown by \\mbox{\\citet{schmitt2006}}. Unfortunately, our Taylor expansion method prevents us from exploring such regime.\n\nLastly, other dissipation mechanisms such as Joule heating (in an electrically conducting core) might lead to similar resonant interactions. In such case magnetic torques would play a similar role as the viscous torque. This is particularly relevant for the Earth where we know that pressure, gravitational and magnetic torques dominate over viscous torques. It is then perfectly possible that resonant mode interaction, conceptually similar to the one presented in this study, might take place using more realistic (Earth or planetary) parameters.  \n\n\n\n\n\n\n \n\\section{Summary and outlook}\n\\label{sec:end}\nWe have implemented an efficient spectral method  \\citep{olver2013} to obtain numerically the eigenmodes of a rapidly rotating two-layer planet model in which the rigid mantle is free to react to torques exterted by the viscous, incompressible fluid core. The aim of this model is to understand at least qualitatively the relationship between inertial modes as studied in the fluid dynamics literature, where the rotational motion of the container (the `mantle') is prescribed, and the rotational modes of a planet (Free Core Nutation, Chandler Wobble) as studied in Geodesy where the motion of the fluid core is generally taken as a simple Poincar\\'e flow. We can accomodate a flattened core-mantle boundary by means of a Taylor expansion technique as long as the flattening is smaller than the typical width of the Ekman boundary layer. In the special case of $q=0$ (i.e. a steadily rotating mantle) our model reproduces satisfactorily the viscous inertial eigenmodes in rotating spheroids as reported by previous studies  \\citep{schmitt2006,zhang2004}.\n\nWhen $q\\ne0$ we find a rich system where the Free Core Nutation, which reduces to the spin-over mode when $q=0$, interacts with a family of nearby modes ($n=8,14,20,\\dots$) and induces a substantial reordering. This family of modes that interact with the FCN include, incidentally, the three modes that  \\citet{schmitt2006} describes as entering in resonance with the spin-over mode for large enough values of the flattening. We thus uncover a common underlying mechanism in which a varying parameter can bring modes' eigenvalues close enough in the complex plane, i.e. the $(\\omega,\\sigma)$ plane, leading to avoided crossings. If a crossing in frequency occurs, an avoided crossing in the damping takes place and vice versa.\n\nThe spin-over mode transitions into the FCN mode, increasing its frequency and becoming more damped as $q$ is increased. The other modes in the $n=8,14,20,\\dots$ family are basically immune to changes in $q$ as long as the frequency and damping of the FCN is still away from the frequency and damping of any member of that family. The FCN, as $q$ keeps going up, gradually morphes into one of the members of the family, losing its own identity. The mode the FCN morphes into, with its identity taken away by the FCN, transforms itself into the next more damped mode of the family, and so on `bumping' successively each other into more negative values of $\\sigma$. A clear explanation for this remarkable reordering still evades us.\n\nThe Chandler Wobble, just as the FCN, is affected by varying $q$, with the difference that the CW is not able to interact with other modes, the reason being that the very small damping of the CW is a difficult match for any low-order, equatorially antisymmetric modes with similar frequencies. The only modes that could match the CW's damping and frequency are low-order equatorially symmetric ones, but their interaction is precluded by their differing symmetry.\n\nAway from the interaction region, when $q\\lesssim0.1$, the FCN's frequency and damping seem to be approximatly linear with $q$. In the case of the CW, the frequency and damping appeared directly proportional to the flattening $\\alpha$. We established the frequency and damping of both modes in this regime on the three control parameters $q,\\alpha$ and $E$. We find that the theoretical inviscid expectation for the FCN frequency, to first order in $\\alpha$, is recovered when $E\\to 0$ but only as long as $\\alpha E^{-1\/2}\\ll1$. We recover as well the inviscid CW frequency when $E\\to 0$ this time without restrictions on $\\alpha$.\n\nThe model we present in this study can be readily modified to include magnetic fields, in which case magnetic torques and ohmic dissipation are likely to be important if the Ekman number is reduced  \\citep{buffett2010,lin2017}. Additionally, the inclusion of an inner core together with gravitational torques will bring two new modes, namely the Free Inner Core Nutation (FICN) and the Inner Core Wobble (ICW). The mere presence of an inner core will also impact significantly the modes and will probably increase the computational cost.\n\nWe acknowledge that our study is far removed from Earth's parameters and therefore it is not able to address quantitatively the shortcomings of current nutation models in relation to observations. In a speculative realm, our model might be of some relevance in the dynamics of rapidly rotating neutron neutron stars where rigid crusts are very thin (and with low density compared to the dense fluid core) and thus their $q$ parameter might be larger than unity, leading possibly to resonant mode interactions. In a less exotic note, however, our work provides a proof of concept indicating that inertial modes are relevant for Earth's nutations and that their study, including an inner core and magnetic fields, is well worth the effort. These additional features will render our model much more realistic in relation to the Earth or other planets and moons. They are relatively straighforward to implement and will certainly add surprises to the rich dynamics we already discovered.\n\n\n\n\n\n\\begin{acknowledgments}\nWe would like to thank Rapha\\\"el Laguerre, Ping Zhu and J\\'er\\'emy Vidal for many useful conversations and suggestions. We thank as well Ankit Barik for proofreading the article and giving constructive feedback. Also we express our gratitude to the PETSc and SLEPc team for their prompt help using their solver. The funding for this research is provided by the European Research Council through the Advanced ERC Grant `Rotation and Nutation of a Wobbly Earth' (RotaNut) No. 670874.\n\\end{acknowledgments}\n\n\\bibliographystyle{gji}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\n\n\n\\textbf{Context.} \nA map is  a combinatorial object describing the embedding up to homeomorphism of a multigraph on a compact orientable surface (see  \\cref{sec:def} for precise definitions).\nMap enumeration has been an important research topic for many years now. Tutte first enumerated planar maps and found simple formulas \\cite{Tutte}, raising the natural concern of finding bijections explaining those formulas. Such bijections have since been found, mostly thanks to the family of bijections between maps and decorated trees (blossoming trees and mobiles) \\cite{Cori1981,theseSch,BDG,AP,BF}. \nSome of these bijections have recently been extended to maps on surfaces of higher genus (see for instance \\cite{Schaeffer}) and on non orientable surfaces \\cite{CD}. These bijections also opened the way to the study of random maps and their scaling limits (see e.g. \\cite{CS,Miermont,LG})\n\n\n\n\n\nAnother powerful tool for map enumeration is the KP hierarchy. The Kadomtsev-Petviashvili hierarchy is an infinite set of PDE's on functions with a infinite number of variables $(p_i)_{i\\geq 1}$, which arose from mathematical physics (see \\cite{Solitons},\\cite{Ok} and references therein). The first equation of the hierarchy is\n\\[{F_{3,1}=\\frac{1}{12}F_{1,1,1,1}+F_{2,2}+\\frac{1}{2}(F_{1,1})^2},\\]\nwhere the indices indicate partial derivatives (e.g. $F_{3,1}=\\frac{\\partial^2}{\\partial p_1\\partial p_3}F$). \nThe link between integrable hierachies and maps was observed in mathematical physics through matrix integrals (see e.g. \\cite{LZ} and references therein).\nIn 2008, Goulden and Jackson \\cite{GJ} showed that certain generating functions for maps (with $p_i$ counting vertices of degree $i$) are solutions to the KP hierarchy. It allowed them to derive a very simple recurrence formula\n\\begin{equation}\n\\begin{split}\n(n+1)T(n,g)=&4n(3n-2)(3n-4)T(n-2,g-1)+4(3n-1)T(n-1,g)\\\\\n&+4\\sum_{\\substack{i+j=n-2\\\\i,j\\geq 0}}\\sum_{\\substack{g_1+g_2=g\\\\mathbf{g}_1,g_2\\geq 0}}  (3i+2)(3j+2)T(i,g_1)T(j,g_2)+2\\mathbb{1}_{n=g=1},\n\\end{split}\n\\label{GJ}\n\\end{equation}\nwhere $T(n,g)$ is the number of rooted cubic maps of genus $g$ with $3n$ edges (equivalently, triangulations of genus $g$ with $3n$ edges). Then, using similar methods, Carrell and Chapuy \\cite{CC} proved a recurrence formula on general (i.e. non necessarily cubic nor triangular) maps\n\\begin{equation}\n\\begin{split}\n(n+1)Q_g(n,f)=&2(2n-1)Q_g(n-1,f)+2(2n-1)Q_g(n-1,f-1)\\\\\n&+(2n-3)(n-1)(2n-1)Q_{g-1}(n-2,f)\\\\\n&+3\\sum_{\\substack{i+j=n-2\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}\\sum_{\\substack{g_1+g_2=g\\\\mathbf{g}_1,g_2\\geq 0}}(2i+1)(2j+1)Q_{g_1}(i,f_1)Q_{g_2}(j,f_2),\n\\end{split}\n\\label{CC}\n\\end{equation}\n\nwhere $Q_g(n,f)$ is the number of rooted maps of genus $g$ with $n$ edges and $f$ faces. Taking $f=1$ in \\eqref{CC}, one recovers the famous Harer-Zagier recurrence \\cite{HZ}.\n\nFinding bijections for formulas arising from the KP hierarchy on maps would allow us to understand maps in greater depth, but for now it is still mainly an open problem. The only special case of \\eqref{CC} and \\eqref{GJ} for which a bijective proof is known is the case of one-faced maps \\cite{CFF}.\n\n\\textbf{Contributions of this article.} \nIn this paper, we present bijective proofs for the planar case (g=0) of Goulden-Jackson and Carrell-Chapuy formulas \\eqref{CC} and \\eqref{GJ}. Note that contrarily to the one-faced case which is linear, the planar formulas are, as in the general case, quadratic.\n\nWe actually prove a more general formula on precubic maps (see \\cref{sec:def} for a definition) which implies Goulden-Jackson formula \\eqref{GJ}. The Carrell-Chapuy formula \\eqref{CC} comes from two separate (but somehow related by their bijective proofs) formulas that were not predicted by the KP hierarchy, one of them being a generalization of the famous R\\'emy bijection on trees \\cite{Remy} to all planar maps (see formulas \\eqref{cut-slide} and \\eqref{remy}).\n\n \n\nOur bijections rely on a particular exploration of the map and on a \"cut and slide\" operation. Although non-local, this operation allows us to keep track of the degrees of the vertices (see \\cref{sec:deg}). We also take a first step towards the higher genus case by proving the Goulden-Jackson formula for two-faced maps in any genus (see \\cref{sec:twofaces}).\n\n\\textbf{Comments and related works.} \nThe bijective study of planar maps is a well understood topic, especially thanks to bijections with tree-like objects. However, as far as we know, our bijection is not equivalent to those bijections (although certain similarities can be observed, such as a search of the dual map, DFS in our case, BFS in the bijections above). A related cut and slide operation appeared in \\cite{Bettinelli}, but it is different from ours (see Remark \\ref{rem} for more details).  \n\n\n\n\n\nIt is a natural question to try to unify bijections in this paper (that apply to planar maps with arbitrary number of faces) and bijections in \\cite{CFF} (that apply to unicellular maps wiyth arbitrary genus) in order to prove \\eqref{GJ} and \\eqref{CC} in full generality. These two approaches seem different at first sight, but they both consist of taking a map with a marked special half edge, and \"cutting\" at this special half edge in order to obtain one or two maps with less edges, faces, or lower genus. In our case, those special half edges are called \\textit{discoveries}, and there are $f-1$ of them in a map with $f$ faces ; in \\cite{CFF}, they are called \\textit{trisections}, and there are $2g$ of them in a map of genus $g$. Slightly rewriting \\eqref{GJ} suggests that in the general case these special half-edges still exist and there would be $2g+f-1$ of them. They would thus be a common generalization of trisections and discoveries. We already found a bijective proof of the precubic recurrence formula for two faced maps (a sketch of proof is included in \\cref{sec:twofaces}), but it is already quite complicated and involves several cases. That is why putting it all together seems to be a challenge of its own. In another direction, we can also look at the second KP equation (which is also quadratic), which yields a recurrence formula (very similar to \\eqref{precubic}) for planar maps with vertices of degrees $1$ or $4$, and this formula is also proven by our bijection (it is a specialization of the more general formula obtained in \\cref{sec:deg}). This suggests that this exploration+cut-and-slide scheme is maybe somehow underlying in the whole KP hierarchy for maps. \n\n\\textbf{Structure of the paper.} \nIn \\cref{sec:def}, we will give some definitions on maps and state the main results. The bijections will be described in \\cref{sec:bij}, with some proofs postponed to \\cref{sec:proof}. \\cref{sec:deg} will present refined formulas with control over the degrees of the vertices. In \\cref{sec:calcul}, we prove the Carrell-Chapuy formula using formulas \\eqref{cut-slide} and \\eqref{remy}. In \\cref{sec:twofaces}, we give a sketch of proof of the precubic recurrences for two-faced maps in any genus.\n\n\n\\section{Definitions and main results}\n\\label{sec:def}\n\\label{sec:results}\n\\begin{definition}\nA map $M$ is the data of a connected multigraph (multiple edges and loops are allowed) $G$ (called the underlying graph) embedded in a compact connected oriented surface $S$, such that $S\\setminus G$ is homeomorphic to a collection of disks. The connected components of $S\\setminus G$ are called the \\textit{faces}. $M$ is defined up to homeomorphism. Equivalently, $M$ is the data of $G$ and a \\textbf{rotation system} which describes the clockwise cyclic order of the half-edges around each vertex.\nThe \\textbf{genus} $g$ of $M$ is the genus of $S$ (the number of \"handles\" in $S$). A \\textbf{corner} of $M$ is an angular sector between two consecutive half-edges around a vertex. A \\textbf{rooted map} is a map with a distinguished corner. A small arrow is placed in the distinguished corner, thus splitting the corner in two separate corners (left and right of the arrow).\nIf a rooted map $M$ of genus $g$ has $n$ edges, $v$ vertices and $f$ faces, the Euler formula links those quantities: $v-n+f=2-2g$. $M$ has $2n+1$ corners.\n\nA \\textbf{planar map} is a rooted map of genus $0$. It can be drawn on the plane with the root lying on the outer face. A \\textbf{precubic map} is a map with vertices of degree $1$ or $3$ only, rooted on a vertex of degree $1$. A \\textbf{leaf} is a vertex of degree $1$ that is not the root.\n\\end{definition}\n\nFrom now on, all the maps we consider will be rooted, and planar unless stated otherwise.\n\n\n\\begin{figure}[!h]\n\\captionsetup{width=0.8\\textwidth}\n\\center\n\\includegraphics[scale=0.5]{map.pdf}\n\\caption{A planar map}\n\\label{map}\n\\end{figure}\n\nIn \\cref{sec:bij}, we will introduce the concept of discovery, which are special edges obtained by an exploration of the map. An important feature is that a map with $f$ faces has $f-1$ discoveries\n\n\\begin{thm}\n\\label{cs}\nThere is a bijection between planar maps $M$ with a marked discovery and pairs of planar maps $(M_1,M_2)$ such that $M_1$ has a marked vertex and $M_2$ has a marked leaf. This bijection preserves the total number of edges and faces. It gives the following formula on on planar maps\n\\begin{equation}\n(f-1)Q(n,f)=\\sum_{\\substack{i+j=n-1\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}v_1Q(i,f_1)(2j+1)Q(j,f_2),\n\\label{cut-slide}\n\\end{equation}\nwhere $Q(n,f)$ is the number of planar maps with $n$ edges and $f$ faces, and $v_1$ counts the number of vertices in the first map (i.e. $v_1=2+i-f_1$)\n\nThis bijection adapts to precubic maps, and the marked vertex of $M_1$ is now a marked leaf. This yields\n\\begin{equation}\n(f-1)\\alpha(n,f)=\\sum\\limits_{i+j=n}\\sum\\limits_{f_1+f_2=f}\\alpha^{(1)}(i,f_1)\\alpha^{(1)}(j,f_2),\n\\label{precubic}\n\\end{equation}\nwhere $\\alpha(n,f)$ counts the number of (planar) precubic maps with $n$ edges and $f$ faces, and $\\alpha^{(1)}(n,f)$ counts the number of precubic maps with $n$ edges and $f$ faces and a marked leaf. \n\\end{thm}\n\nRetracting a leaf into a corner (see \\cref{feuille}) is a classical operation that gives a bijection between planar maps $M$ with a marked leaf and planar maps $M'$ with a marked corner, such that $M'$ has as many faces and one edge less than $M$. This operation, along with a more subtle retraction operation on internal nodes, is at the core of the R\u00e9my bijection on plane trees \\cite{Remy}. We generalize the operation on internal nodes to all planar maps in the following :\n\\begin{thm}[Generalized R\\'emy bijection]\n\\label{re}\n\nThere is a bijection between planar maps $M$ with a marked node (i.e. a vertex that is not a leaf) and the union of planar maps $M'$ with a marked corner and pairs of planar maps $M_1$ and $M_2$ such that they both have a marked vertex. The total number of faces is preserved, and the total number of edges decreases by one.\n\nTogether with the operation of retracting a leaf into a corner, it yields\n\\begin{equation}\nvQ(n,f)=2(2n-1)Q(n-1,f)+\\sum_{\\substack{i+j=n-1\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}v_1Q(i,f_1)v_2Q(j,f_2),\n\\label{remy}\n\\end{equation}\nwhere the \"$v$-variables\" count the number of vertices (i.e $v=2+n-f$, $v_1=2+i-f_1$ and $v_2=2+j-f_2$). This holds for $n>0$.\n\\end{thm}\nTaking $n=3m+2$ and $f=m+2$ in \\eqref{precubic}, one recovers\n\\begin{cor}[Goulden-Jackson \\eqref{GJ} planar case]\n\\label{cor}\n\\[(n+1)T(n)=4(3n-1)T(n-1)+4\\sum_{\\substack{i+j=n-2\\\\i,j\\geq 0}}  (3i+2)(3j+2)T(i)T(j),\\]\nwhere $T(n)$ counts the number of planar cubic maps with $3n$ edges.\n\\end{cor}\n\nCombining  \\eqref{cut-slide} and \\eqref{remy} and doing some manipulations, one recovers\n\\begin{cor}[Carrell-Chapuy \\eqref{CC} planar case]\n\\begin{equation}\n\\label{planar}\n\\begin{split}\n\\begin{split}\n(n+1)Q(n,f)=&2(2n-1)Q(n-1,f)+2(2n-1)Q(n-1,f-1)\\\\\n&+3\\sum_{\\substack{i+j=n-2\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(2i+1)(2j+1)Q(i,f_1)Q(j,f_2).\n\\end{split}\n\\end{split}\n\\end{equation}\n\\end{cor}\nFormula \\eqref{planar} is not straightforwardly derived from \\eqref{cut-slide} and \\eqref{remy}. The calculations to recover \\eqref{planar} from there are displayed in \\cref{sec:proof}.\n\\section{The bijections}\n\\label{sec:bij}\nIn this section, we will define the exploration of a planar map and the notion of discoveries that result from it, then we will explain our bijections.\n\n\\subsection{The exploration}\n\\label{explo}\n\\begin{definition}\nThe \\textbf{exploration} of a planar map (see \\cref{exploration}) is defined iteratively in the following way: starting from the root, go along the edges, keeping the edges on the right (progress in clockwise order). When an edge that is at the interface of the current face and a face not yet discovered is found, open this edge into a bud (an outgoing half-edge) and a stem (an ingoing half edge), and continue the process, thus entering the new face. Continue until the root is reached again.\n\nEach edge that has been opened during the process is called a \\textbf{discovery}, and the vertex attached to the bud is called a \\textbf{discovery vertex}. If there are $f$ faces , there are $f-1$ discoveries (note that several discoveries can share the same discovery vertex).\n\nThe exploration is actually equivalent to a DFS of the dual, with a \"right first\" priority. Thus for each face but the outer face we can define its \\textbf{previous face} as the face that is its parent in the covering tree of the dual found by the exploration. The notion of \\textbf{previous discovery} can be similarly defined. It also defines an order on the corners (resp. half edges) incident to each vertex, according to the order in which they were visited during the exploration. \n\nLet $e$ be a discovery, incident to faces $f_1$ and $f_2$, such that $f_1$ is the previous face of $f_2$. We say $e$ \\textbf{leaves} $f_1$ and \\textbf{enters} $f_2$.\n\\end{definition}\n\n\\begin{rem}\nThe exploration is a dynamic process that modifies the map along the way, but in the end, once the exploration is over and the discoveries have been found, we will deal with the original, unmodified map, with its original edges and faces. It is as if we did the exploration then closed the map back. Alternatively, one can think of an exploration that doesn't open the discoveries but just crosses them.\n\\end{rem}\n\\begin{figure}[!h]\n\\captionsetup{width=0.8\\textwidth}\n\\label{exploration}\n\\centering\n\n\\includegraphics[scale=0.7]{exploration.pdf}\n\\caption{The exploration of a planar map. The buds are the outgoing arrows, the stems are the ingoing arrows. Left: the original map. Center: the opened map. Right: The original map, with its discoveries and discovery vertices in purple. The red tree describes the partial order among the faces. The corners are labeled in the order they were found during the discovery}\n\\end{figure}\n\n\\begin{lem}\n\\label{clockwise}\nAround each vertex, the order of the corners as defined by the exploration agrees with the clockwise order.\n\\end{lem}\n\n\\begin{proof}\nThe order between the corners of the map is exactly the same as the order of the corners of the blossoming tree (a blossoming tree is a tree with some buds and stems attached) obtained by opening the discoveries. The exploration of a tree is just a tour of the unique face, and it is clear that the corners of each vertex are in clockwise order.\n\\end{proof}\n\nWe can now relate the bijections and the formulas. In a map with $f$ faces, there are $f-1$ discoveries, so there are $(f-1)Q(n,f)$ maps with $n$ edges, $f$ faces and a marked discovery. A marked leaf can be retracted into a marked corner (see \\cref{feuille} left), such that there are $(2i+1)Q(i,f_1)$ maps with $i+1$ edges, $f_1$ faces and a marked leaf. There are $v_2Q(j,f_2)$ maps with $j$ edges, $f_2$ faces and a marked vertex. So \\eqref{cut-slide} and \\eqref{remy} are indeed consequences of \\cref{cs,re}. In a precubic map, one can retract a leaf into a marked side-edge losing two edges (see \\cref{feuille} right), so a precubic map with no leaf is equivalent to a cubic map, and we fin\n\n\\begin{figure}[!h]\n\\captionsetup{width=0.8\\textwidth}\n\\centering\n\\includegraphics[scale=0.5]{feuille.pdf}\n\\caption{Retracting a leaf: in a general map (left), in a precubic map (right)}\n\\label{feuille} \n\\end{figure}\n\\subsection{Cut and slide bijection}\n\nWe will describe the bijection of \\cref{cs} between maps $M$ with a marked discovery (see \\cref{sec:bij} for a precise definition) and pairs of planar maps $(M_1,M_2)$ such that $M_1$ has a marked vertex and $M_2$ has a marked leaf. It will then be straightforward to see that restricting the bijection to precubic maps gives \\eqref{precubic}.\n\\begin{definition}\nA discovery is said to be \\textbf{disconnecting} if the corner preceding the discovery and the last corner (in the order defined by the exploration) around the discovery vertex lie in the same face. \n \n\\label{disco}\n\\end{definition}\n\\begin{spli}\nAny map with a marked disconnecting discovery can be (bijectively) split into two maps, one with a marked vertex, the other with a marked leaf in the outer face (see \\cref{split}).\n\\end{spli}\n\\begin{figure}[!h]\n\\captionsetup{width=0.8\\textwidth}\n\\centering\n\\includegraphics[scale=1.5]{splitting.pdf}\n\\caption{Splitting a map at a disconnecting discovery. Here we only see what happens locally around the disconnecting discovery. On the left, the discovery and its discovery vertex are in purple, $c$ is the corner preceding the discovery, and $c^*$ is the last corner around the discovery vertex}\n\\label{split}\n\\end{figure}\n\nThe splitting operation of a disconnecting discovery describes our bijection in the case where the marked discovery is disconnecting, and the reverse bijection in the case where the marked leaf lies in the outer face.\n\nIn general, discoveries are not disconnecting. In order to still split the map in two, given a discovery $e$, we will need to find a disconnecting discovery $e'$ that is \"canonically\" related to $e$. Conversely, the marked leaf in $M_2$ is not always in the outer face, we will need to \"propagate\" the leaf all the way up to the outer face. The notion of previous discovery will help us with that. This will be the general construction of the bijection that is detailed below.\n\n\\begin{lem}\nIf a discovery leaves the outer face, then it is disconnecting\n\\label{lemma}\n\\end{lem}\n\n\\begin{proof}\nWe will show the following stronger result, which directly implies \\cref{lemma} : if a vertex has a corner in the outer face, then its last corner lies in the outer face.\n\nIn a map $M$, let $v$ be a vertex with one of its corners lying in the outer face. If $v$ is the root vertex, then it is obvious that its last corner lies in the outer face. Else, let $e$ be the first edge around $v$, as defined by the exploration (i.e. the edge we see in the exploration just before the first occurrence of $v$). Let $c$ be one corner around $v$ that lies in the outer face. Suppose it is not the last corner around $v$. Let $c_0$ (resp. $c^*$) be the first (resp. the last) corner around $v$, and let $F_0$ (resp. $F^*$) be the face in which $c_0$ (resp. $c^*$) lies. Suppose $F^*$ is not the outer face. (See \\cref{corners})\n\n\\begin{figure}[!h]\n\\captionsetup{width=0.8\\textwidth}\n\\centering\n\\includegraphics[scale=1]{corners.pdf}\n\\caption{}\n\\label{corners}\n\\end{figure}\n\nIf a face $F'$ is deeper (in terms of the partial order defined by the exploration) than a face $F$ (and $F\\neq F'$), then during the exploration (ignoring all other faces), we first see a part of $F$, then all of $F'$, then the rest of $F$. This would mean that during the exploration of $M$, at the time when we see $c$, $F^*$ hasn't been discovered yet. But that would mean that at the time when we see $e$ (which comes before $c$ in the exploration), $F^*$ hasn't been discovered yet. But then $e$ would be a discovery, and thus $c^*$ would be seen before $c_0$, contradiction\n\\end{proof}\n\n\\begin{bij}\n\\label{CS1}\nThe general process is iterative (see \\cref{bij} for an example).\n\n\\begin{figure}[!h]\n\\captionsetup{width=0.8\\textwidth}\n\\centering\n\\includegraphics[scale=0.5]{bijection.pdf}\n\\caption{The bijection (above) and its inverse (below)}\n\\label{bij}\n\\end{figure}\n\n\\textbf{Cut process:} Start from a map $M$ with a marked discovery $e$, let $v$ be its discovery vertex. If the discovery is disconnecting, then split $M$ at $v$ as described in the splitting operation.\n\nOtherwise, open $e$ into a bud $b_0$ and a stem $s_0$, and consider its previous discovery $e_1$ (in the order defined above). If it is disconnecting, then split it, otherwise open it (into $b_1$ and $s_1$) and consider the previous discovery $e_2$, and so on until a splitting operation is made. Note that a discovery that leaves the outer face is always disconnecting (because of \\cref{lemma}), so the algorithm terminates. One ends up with two maps $M_1$ and $M_2'$, such that $M_1$ has a marked vertex and $M_2'$ has a marked leaf $l$ and (possibly) some buds and stems, all lying in the outer face.\n\n\\textbf{Slide process:} We will not modify $M_1$. If there are no buds and stems in $M_2'$, we are done. Else, consider $s_0$, and make it a marked leaf. Then consider $l$, and make it a stem. Finally, glue back the buds and stems together canonically: starting from the root of $M_2'$, taking a clockwise tour of the outer face, one encounters a certain number of buds, then the same number of stems. There is only one way to match each bud with each stem such that the map remains planar. Equivalently, if there are $k+1$ buds and $k+1$ stems, match $b_0$ with $s_1$, and so on, until $b_k$ is matched with $l$. We obtain a map $M_2$ with a marked leaf, together with the map $M_1$ with a marked vertex.\n\\end{bij}\nWe can now describe the inverse bijection :\n\\begin{bij}\n\\label{CS2}\nStarting from $M_2$ with a marked leaf $l$ and $M_1$ with a marked vertex, consider $M_2$. $l$ lies in a certain face $F$, and if $F$ is not the outer face, there is a certain discovery $e_0$ that enters $F$. Open it into a bud $b_0$ and a stem $s_0$, then open the previous discovery $e_1$, and repeat the process until a discovery that leaves the outer face has been opened (in that case there is no previous discovery to open). One ends up with a map $M_2'$ with a marked leaf $l$ and possibly some buds and stems, all lying in the outer face. If there are some buds and stems, let $s$ be the stem that was created last in the process. Make $s$ a marked leaf $l^*$, and make $l$ a marked stem $s^*$, then close the map canonically. One now has a map $M_2^*$ with a marked leaf on the outer face and (possibly) a marked edge $e$ (that comes from the closure of $s^*$). This marked edge is actually a discovery in $M_2^*$ (and will be a discovery in the final map). If $e$ doesn't exist, let $l^*=l$, and mark the edge adjacent to $l^*$ (and call it $e$). Now do the inverse of the splitting operation: glue $l^*$ to the root vertex of $M_2^*$ at its first corner, and then glue the root of the resulting map at the last corner of the marked vertex of $M_1$ to obtain a map $M$ with a marked discovery $e$.\n\\end{bij}\n\\begin{rem}\nThis operation restricts to precubic maps, and in this case discovery vertices are always of degree $3$, so that when split, the marked vertex of $M_1$ and the root of $M_2$ are both of degree $1$.\n\\end{rem}\n\n\nThose two applications are inverse of each other because of the following property (that will be proved in \\cref{sec:proof}) :\n\\begin{lem}\nIn the bijection and its inverse, the closure of the buds and stems are discoveries in the resulting map. Moreover, in the bijection, if $e_1$,\u2026$e_k$ are the discoveries created by closing buds and stems, $e_k$ leaves the outer face, and for all $i<k$, $e_{i+1}$ is the previous discovery of $e_i$. In the inverse bijection, if $e_1$,\u2026$e_{k+1}$ are the discoveries created by closing buds and stems, $e_{k+1}$ is disconnecting, and for all $i<k$, $e_{i+1}$ is the previous discovery of $e_i$.\n\\end{lem}\n\\begin{rem}\n\\label{rem}\nA similar but different cut and slide operation also appeared in \\cite{Bettinelli} (and in some sense also in \\cite{slice}). However there are significant differences : the cut path is geodesic (leftmost BFS), and both endpoints of the path need to be specified. Whereas here, the cut path is defined by a leftmost DFS, and only one endpoint of the path (the marked discovery) needs to be specified, the other endpoint (the disconnecting discovery) is uniquely determined. What's more, the bijections in \\cite{Bettinelli} imply linear formulas, contrarily to our quadratic formulas.\n\\end{rem}\n\\subsection{Generalized R\\'emy bijection}\n\\label{sec:Remy}\nWe will now describe a generalized R\\'emy bijection on planar maps, that also relies on the cut-and-slide operation. \n\nWe recall R\\'emy's bijection for plane tree that proves the formula $(n+1)Cat(n)=2(2n-1)Cat(n-1)$ where $Cat(n)$ counts the number of rooted plane trees with $n$ edges (see \\cref{tree} for an example). Start from a tree $T$ with $n$ edges and a marked vertex $v$. If $v$ a leaf, retract it (as in \\cref{feuille}) to obtain a tree $T'$ with $n-1$ edges, with a marked corner. Otherwise, $v$ has a last son $v'$, that is its son that is found last during a clockwise tour of the unique face. We can then contract the edge between $v$ and $v'$, and mark a corner around the merging vertex to remember where to grow the edge back (as in \\cref{tree}). The reverse bijection is then straightforward.\n\n\\begin{rem}\nLeaves are defined as vertices of degree $1$ \\textbf{that are not the root vertex}. Thus, if the root vertex has degree $1$, one should apply the second operation to it\n\\end{rem}\n\\begin{figure}[!h]\n\\captionsetup{width=0.8\\textwidth}\n\n\\centering\n\\includegraphics[scale=0.5]{remy.pdf}\n\\caption{R\\'emy's bijection for plane trees}\n\\label{tree}\n\\end{figure}\n\nIn a general planar map, let us define precisely the operations of edge contraction and edge growing :\n\nLet $v$ be a non-leaf vertex, it has a last corner $c_2$ (as defined by the exploration), let $e$ be the edge that comes just before $c_2$ in clockwise order around $v$. We call $e$ the last edge around $v$. Let $v'$ be the other end of $e$ (note that it is possible that $v=v'$). $v'$ is called the last son of $v$. We will define the contraction operation only in the case $v\\neq v'$.  $e$ is adjacent to two corners of $v$ : $c_2$ and another corner $c_1$. $e$  is also adjacent to two corners of $v'$, $c'$ and $c''$ such that $c'$ appears before $c''$ in the exploration.\n$c\"$ is the last corner around $v'$\n\nThe contraction operation consists in contracting $e$, merging $v$ and $v'$ into a vertex $v^*$, merging $c_1$ and $c'$ into $c^*$ and $c_2$ and $c''$ into $c$. $c$ is the last corner around $v^*$, and $c^*$ is the marked corner.\n\nConversely, starting with a marked corner $c^*$ adjacent to a vertex $v^*$ whose last corner is $c$, one can grow an edge $e$, splitting $v^*$ into $v$ and $v'$, $c^*$ into $c_1$ and $c'$, and $c$ into $c^*$ and $c_2$. $c_2$ is the last corner around $v$ and $c''$ the last corner around $v$. $v$ is the marked vertex, and $v'$ is its last son. This defines the growing operation.\n\n\\begin{figure}[!h]\n\\captionsetup{width=0.8\\textwidth}\n\\centering\n\\includegraphics[scale=0.7]{growing.pdf}\n\\caption{The contraction and growing operations}\n\\label{growing}\n\\end{figure}\n\n\\begin{lem}\nLet $M$ be a map with a marked corner $c^*$, and let $M'$ be the map obtained after a growing operation is performed at $c^*$. Let $v$ be the marked vertex and $v'$ its last son. Then $v$ appears before $v'$ in the exploration\n\\end{lem}\n\\begin{proof}\nThe growing operation is local, so both explorations of $M$ and $M'$ look exactly the same before $c^*$ (resp. $c_1$) is found. Let $p$ be the first corner around $v^*$ in $M$, after the growing operation it is adjacent to $v$, and it is reached before any other corner around $v$ or $v'$, thus $v$ appears before $v'$ in the exploration (the special case $p=c^*$ works the same).\n\\end{proof}\n\n\\begin{lem}\nLet $v$ be a non-leaf vertex, $e$ its last edge, and $v'$ its last son. If $v'$ is found before $v$ in the exploration (including the case $v=v'$), then $e$ is a discovery.\n\\end{lem}\n\\begin{proof}\nIf $v=v'$, then $e$ is a loop, thus a discovery (since a loop separates a planar map in two, it has to be opened during the exploration.\n\nOtherwise, let $c_1,c_2$ (resp. $c',c''$) be the corners of $v$ (resp. $v'$) that are adjacent to $e$. Note that $c_2$ is the last corner of $v$, and assume $e$ is not a discovery. Thus, it is not opened during the exploration, such that $c''$ comes before $c_2$, and $c_1$ before $c'$ (see \\cref{growing}). $c'$ cannot be the first corner of $v'$, otherwise $v$ would be seen before $v'$ in the exploration. Thus, because of \\cref{clockwise}, $c'$ comes after $c''$. But that would mean the order among the four corners in the exploration is $c''$, then $c_2$, then $c_1$, then $c'$, which is impossible since $c_2$ is the last corner of $v$.\n\\end{proof}\nNow we are ready to explain the generalized R\\'emy bijection :\n\n\\begin{bij}\n\\label{RM}\nTake a planar map $M$ and mark a vertex $v$. If it is a leaf, contract it to mark a corner. Else, let $v'$ be its last son, and $e$ be its last edge. If $v'$ is seen after $v$ in the exploration, then contract $e$ and mark the corresponding corner. The inverse operation is the growing operation as described above.This gives the first term of the RHS of \\eqref{remy}.\nOtherwise, $e$ is a discovery, apply the cut-and-slide operation at $e$. One ends up with two maps $(M_1,M_2)$. $M_1$ has a marked vertex, and $M_2$ has a marked leaf $l$ that is the last son of its neighbor $w$. Contract $l$ to mark $w$. To go backwards, grow $l$ out of the last corner around $w$, then apply the inverse of the cut and slide operation. This gives the second term of the RHS of \\eqref{remy}.\n\n\n\\begin{figure}[!h]\n\\captionsetup{width=0.8\\textwidth}\n\\centering\n\\includegraphics[scale=0.5]{remy2.pdf}\n\\caption{The generalized R\\'emy bijection. The case similar to trees (above), and the cut-and-slide case (below)}\n\\label{remy2}\n\\end{figure}\n\\end{bij}\n\n\\begin{rem}\nThere is also a generalization of R\\'emy's bijection on binary trees to precubic maps, that works exactly the same, we leave it as an exercise to the reader.\n\\end{rem}\n\n\n\\section{Proofs}\n\n\\label{sec:proof}\n\n\n\nIn this section we will present a proof that \\cref{CS1} and \\cref{CS2} are inverse to each other, thus implying that the cut and slide operation is a bijection. Then, together with the lemmas presented in \\cref{sec:Remy}, it will be enough to prove that the generalized R\u00e9my bijection (\\cref{RM}) is indeed a bijection. To do so, we will describe what happens to the different faces of the map during the operation (see \\cref{faces}).\n\nThanks to the exploration, we can consider faces as words on the alphabet of side-edges (each face is described by the list of the side edges that lie inside it, ordered as they appear during the exploration). For any side-edge $e$, we will call $\\overline{e}$ its opposite side-edge. By abuse of notation we may also refer to the edge $e$ that is the edge made of the side edges $e$ and $\\overline{e}$, but we will stress on the fact that we are referring to an edge.\n\nFix a planar map $M$. Let $d_0$ be a discovery, $d_1$ its previous discovery (if it exists), and so on, $d_{i+1}$ being the previous discovery of $v_i$, as long as it exists. For all $i$, the discovery $d_i$ discovers the face $F_i$ while leaving face $F_{i+1}$. For all $i$, let $v_i$ be the discovery vertex corresponding to $d_i$, and let us differentiate both side edges of the discovery : $d_i$ sits in $F_{i+1}$ and $\\overline{d_i}$ sits in $F_i$. Let $k$ be the smallest $i$ such that $d_i$ is a disconnecting discovery. For all $i$ between $1$ and $k$, we can set $F_i=\\overline{d_i}A_id_{i-1}B_i$, that is $A_i$ are the side-edges of $F_i$ encountered before $d_{i-1}$ in the exploration, and $B_i$ are the the side-edges of $F_i$ encountered after $d_{i-1}$ in the exploration. We also set $F_0=\\overline{d_0}A_0$ and $F_{k+1}=\\overline{d_{k+1}}Cd_kB_{k+1}C'$ where $B_{k+1}$ is the word of side-edges of $F_{k+1}$ that appear between $d_k$ and the last corner around $v_k$.\\\\\nNow let us describe the cut and slide operation when $d_0$ is marked. First, for all $i<k$, all \\textbf{edges} $d_i$ are opened into a bud $b_i$ and an stem $s_i$. The \\textbf{edge} $d_k$ becomes a marked leaf $s_k$. Then $s_k$ becomes a stem, and $s_0$ becomes the marked leaf. Then $b_i$ is matched with $s_{i+1}$ to create and edge $e_i$. The map has been split into two distinct maps $M_1$ and $M_2$, now let us see what happened to the faces. In $M_1$, there is only one face that has been modified, it is the face of the marked vertex $v^*$, that we call $G^*$. It is straightforward to see that $G^*=\\overline{d_{k+1}}CC'$ where the last corner of $v^*$ appears exactly between the end of $C$ and the beginning of $C'$. In $M_2$, the modified faces will be called $G_i$. We have $G_0=\\overline{e_0}A_0s_0\\overline{s_0}B_1$, $G_i=\\overline{e_i}A_ie_{i-1}B_{i+1}$ and $G_k=A_ke_{k-1}B_{k+1}$. $G_k$ is the outer face of the second map, and for all $i$, $G_{i+1}$ is the previous face of $G_i$.\\\\\n\\begin{figure}[!h]\n\\captionsetup{width=0.8\\textwidth}\n\\centering\n\\includegraphics[scale=1]{faces.pdf}\n\\caption{What happens to the faces during the cut and slide operation. Marked objects are in purple, other discoveries involved are in gold. Blue contour materializes the $A_i$'s and red contour, the $B_i$'s}\n\\label{faces}\n\\end{figure}\\\\\nWe must now prove that, for all $i<k$, the \\textbf{edge} $e_i$ is a discovery that leaves $G_{i+1}$ and discovers $G_i$. We already now that $e_i$ lies in $G_{i+1}$ and $\\overline{e_i}$ lies in $G_i$, we just have to prove that the \\textbf{edge} $e_i$ is a discovery, i.e $e_i$ is the first side edge $e$ of $G_{i+1}$ such that $\\overline{e}$ lies in $G_i$. Assume the contrary. There must be a side edge $f$ in $A_{i+1}$ such that $\\overline{f}$ lies in $G_i$. There are two possibilities. Either $\\overline{f}$ belongs to $A_i$, but then in the original map, in $F_{i+1}$, $f$ comes before $d_i$ and $\\overline{f}$ lies in $F_i$, thus $d_i$ could not be the discovery, contradiction. Or $\\overline{f}$ belongs to $B_{i+1}$, but then in the original map, in $F_{i+1}$ we see the side edges $f$, then $d_i$, then $\\overline{f}$ in that order. But since we deal with planar maps, an edge which has both sides in the same face is a cut-edge (removing it disconnects the map). But then in that configuration, $d_i$ must be a disconnecting discovery (see \\cref{moche}). Indeed, let $M^*$ be the connected component containing $d_i$ that we obtain after removing $e$. Then $d_i$ would be a discovery in the map $M^*$ that leaves the outer face, thus disconnecting (because of \\cref{lemma}). This  is a contradiction, because $i<k$. We proved what we wanted. \\\\ \n\\begin{figure}[!h]\n\\captionsetup{width=0.8\\textwidth}\n\\centering\n\\includegraphics[scale=0.5]{cutedge.pdf}\n\\caption{The impossibility of a cut-edge \"surrounding\" a non disconnecting-discovery. In red, the cut-edge, in purple the discovery}\n\\label{moche}\n\\end{figure}\\\\\nKnowing exactly what happens to the faces and that the discoveries are in some sense \"preserved\" in enough to prove that both the cut and slide operation and its inverse work fine and are actually inverse applications of each other.\n\n\\section{Controling the degrees of the vertices}\n\\label{sec:deg}\nHere we present an analogue of \\eqref{cut-slide} with control over the degrees of the vertices. If $\\textbf{v}=(v_i)_{i\\in \\mathbb{N}}$ is a sequence of integers, for any $j>0$, we set $\\delta_j(\\textbf{v})=\\textbf{w}$ where $w_j=v_j+1$ and $w_i =v_i$ for $i\\neq j$, and $\\delta_{-j}(\\textbf{v})=\\textbf{w'}$ where $w_j'=v_j-1$ and $w_i '=v_i$ for $i\\neq j$. Finally, we set $\\delta(\\textbf{v},j_1,\\hdots,j_k)=\\delta_{j_1}\\circ \\hdots\\circ \\delta_{j_k}(\\textbf{v})$.\n\nLet $M(r,f,\\textbf{v})$ the number of planar maps with $f$ faces, with root of degree $r$, with $\\textbf{v}=(v_i)_{i\\in \\mathbb{N}}$ s.t. there are $v_i$ vertices of degree $i$ (root included). The cut-and-slide operation only modifies the degrees at the marked leaf and the splitting vertex, so we can immediately derive this more precise formula\n\\begin{thm} \n\\begin{align*}\n(f-1)M(r,f,\\textbf{v})=&\\sum_{\\substack{j,k\\geq 1}}\\sum_{\\textbf{u}+\\textbf{w}=\\delta(\\textbf{v},1,j,k,-(j+k+1))}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(u_j-\\mathbb{1}_{j=r})M(r,f_1,\\textbf{u})(w_1-\\mathbb{1}_{k=1})M(k,f_2,\\textbf{w})\\\\\n&+\\sum_{\\substack{j+k=r-1}}\\sum_{\\textbf{u}+\\textbf{w}=\\delta(\\textbf{v},1,-r,k,j)}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}M(j,f_1,\\textbf{u})(w_1-\\mathbb{1}_{k=1})M(k,f_2,\\textbf{w}).\n\\end{align*}\n\\end{thm}\n\n\\begin{proof}\nNotice that, in the cut-slide operation, very few of the vertex degrees are modified : there is the leaf that is created in $M_2$, and the disconnecting discovery vertex that is split in three. Other than that, for all other vertices, although some of their adjacent edges might be split, they are reattached somewhere else so their degree doesn't change.\n\nLet $v$ be the vertex that is split in three : it gives birth to $v_1$, the marked vertex in $M_1$, $v_2$, the root vertex of $M_2$, and $v_3$, that after some possible transformation \"becomes\" the marked leaf in $M_2$. Say $v$ is of degree $j+k+1$, with $deg(v_1)=j$ and $deg(v_2)=k$.\n\nThe $(w_1-\\mathbb{1}_{k=1})$ term means that the marked leaf in $M_2$ cannot be the root. Finally, there are two possible cases, depending on whether $v$ is the root of $M$ or not, each implying one term in the RHS.\n\\end{proof}\n\n\\begin{rem}\n$\\delta_j(\\textbf{v})$ means there is one more vertex of degree $j$, and $\\delta_{-j}(\\textbf{v})$ means there is one less vertex of degree $j$. This notation is somehow complicated but avoids dealing with special cases, e.g. in $\\delta(\\textbf{v},j,k)$ there is no problem if $j=k$, contrarily to saying something like $v_j'=v_j+1$, $v'_k=v_k+1$ and $v_i'=v_i$ for $i\\neq j,k$.\n\\end{rem}\n Note that this recurrence formula allows us to compute the number of maps with bounded vertex degrees, i.e. it can be specialized to maps with vertex degrees $\\leq d$ for some $d$. Taking $d=3$, one recovers \\eqref{precubic}.\\\\\n\nFormula \\eqref{remy} also has an analog where the degrees are recorded, but it splits into three different cases, according to whether the marked vertex is a leaf, the root, or another node. If it is a leaf, it is the trivial bijection described in \\cref{feuille}. If it is the root vertex, the formula is actually Tutte's formula. We will only include the formula corresponding to the case where the marked vertex is a node. This formula alone suffices to calculate all the terms by induction.\nAs before, there are extra terms because it depends on whether the modification affects the root or not.\n\nFor $p\\neq 1$:\n\\begin{align*}\n(v_p-\\mathbb{1}_{p=r})M(r,f,\\textbf{v})=&\\sum_{\\substack{j\\geq 1}}u_{p+j-2}M(r,f,\\textbf{u}=\\delta(\\textbf{v},-j,-p,j+p-2))\\\\\n&+\\sum_{\\substack{j,k\\geq 0}}\\sum_{\\substack{\\textbf{u}+\\textbf{w}=\\delta(\\textbf{v},-p,p-1,j,k,-(j+k+1))}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(u_j-\\mathbb{1}_{j=r})M(r,f_1,\\textbf{u})w_{p-1}M(k,f_2,\\textbf{w})\\\\\n&+\\sum_{k=0}^{r-1} \\sum_{\\substack{\\textbf{u}+\\textbf{w}=\\delta(\\textbf{v},-p,p-1,-r,k,r-k-1)}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}M(r-1-k,f_1,\\textbf{u})w_{p-1}M(k,f_2,\\textbf{w}).\n\\end{align*}\n\n\\section{The proof of the Carrell-Chapuy formula in the planar case}\n\\label{sec:calcul}\n\nWe are now ready to prove the Carrell-Chapuy formula  in the planar case \\eqref{planar}. We will prove it by induction on $n$, only knowing that $Q(0,f)=\\mathbb{1}_{f=1}$.\\\\\nTaking the dual of \\eqref{cut-slide}, we obtain the following formula, which will be helpful for the proof\n\\begin{cor}\n\n\\begin{equation}\n(v-1)Q(n,f)=\\sum_{\\substack{i+j=n-1\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f+1\\\\f_1,f_2\\geq 1}}(2i+1)Q(i,f_1)f_2Q(j,f_2)\n\\label{dual}\n\\end{equation}\n\\end{cor}\n\nStarting with \\eqref{cut-slide}, and applying \\eqref{remy}, then in a second time applying \\eqref{cut-slide} backwards, we obtain\n\n\\begin{align*}\n(f-1)Q(n,f)=&(2n-1)Q(n-1,f-1)+2(2n-1)Q(n-1,f)\\\\&+2\\sum_{\\substack{i+j=n-2\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(2i+1)Q(i,f_1)(2j+1)Q(j,f_2)\\\\&+\\sum_{\\substack{i+j+k=n-2\\\\i,j,k\\geq 0}}\\sum_{\\substack{f_1+f_2+f_3=f\\\\f_1,f_2,f_3\\geq 1}}(2i+1)Q(i,f_1)v_2Q(j,f_2)v_3Q(k,f_3)\n\\\\&=(2n-1)Q(n-1,f-1)+2\\sum_{\\substack{i+j=n-2\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(2i+1)Q(i,f_1)(2j+1)Q(j,f_2)\\\\&+\\sum_{\\substack{i+j=n-1\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(f_1-1)Q(i,f_1)v_2Q(j,f_2).\n\\end{align*}\nSo summing \\eqref{remy} to this\n\\begin{align*}\n(n+1)Q(n,f)=&(2n-1)Q(n-1,f-1)+2\\sum_{\\substack{i+j=n-2\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(2i+1)Q(i,f_1)(2j+1)Q(j,f_2)\\\\&+\\sum_{\\substack{i+j=n-1\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(i+1)Q(i,f_1)v_2Q(j,f_2).\n\\end{align*}\nLet \\[S=\\sum_{\\substack{i+j=n-1\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(i+1)Q(i,f_1)v_2Q(j,f_2).\\]\nWe want to prove \n\\[S=(2n-1)Q(n-1,f-1)+\\sum_{\\substack{i+j=n-2\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(2i+1)Q(i,f_1)(2j+1)Q(j,f_2).\\]\n We apply the recursion hypothesis\n\\begin{align*}\nS&=\\sum_{\\substack{i+j=n-1\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(2(2i-1)Q(i-1,f_1)+2(2i-1)Q(i-1,f_1-1))v_2Q(j,f_2)\\\\&+3\\sum_{\\substack{i+j+k=n-3\\\\k,l\\geq 0}}\\sum_{\\substack{f_1+f_2+f_3=f\\\\f_1,f_2,f_3\\geq 1}}(2i+1)Q(i,f_1)(2j+1)Q(j,f_2)v_3Q(k,f_3)\\\\&+vQ(n-1,f-1).\n\\end{align*}\nbut, according to \\eqref{cut-slide}, \\[\\sum_{\\substack{i+j=n-1\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(2(2i-1)Q(i-1,f_1-1))v_2Q(j,f_2)=2(f-2)Q(n-1,f-1),\\]\nand \n\\begin{align*}\n&\\sum_{\\substack{i+j+k=n-3\\\\k,l\\geq 0}}\\sum_{\\substack{f_1+f_2+f_3=f\\\\f_1,f_2,f_3\\geq 1}}(2i+1)Q(i,f_1)(2j+1)Q(j,f_2)v_3Q(k,f_3)\\\\&=\\sum_{\\substack{i+j=n-2\\\\i,j\\geq 0}}\\sum_{\\substack{f_1+f_2=f\\\\\\sf_1,f_2\\geq 1}}(2i+1)Q(i,f_1)(f_2-1)Q(j,f_2).\n\\end{align*}\nSo\n\\begin{align*}\nS=&(2(f-2)+v)Q(n-1,f-1)+\\sum_{\\substack{i+j=n-2}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(2i+1)Q(i,f_1)(2j+1)Q(j,f_2)\\\\&+\\sum_{\\substack{i+j=n-2}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(2i+1)Q(i,f_1)f_2Q(j,f_2).\n\\end{align*}\nBut using \\eqref{dual}, we have \n\\[\\sum_{\\substack{i+j=n-2}}\\sum_{\\substack{f_1+f_2=f\\\\f_1,f_2\\geq 1}}(2i+1)Q(i,f_1)f_2Q(j,f_2)=(v-1)Q(n-1,f-1),\\]\nwhich finishes the proof.\n\n\\begin{rem}\nThe proof above is not straightforward, and since it uses duality, our method cannot be applied to finding an extended Carrell-Chapuy formula with control over the degrees (if it even exists). It cannot be used either for proving the planar case of the analogue formula on bipartite maps found by Kazarian and Zograf \\cite{KZ}.\n\\end{rem}\n\n\\section{A bijection for precubic maps with two faces}\n\\label{sec:twofaces}\nIn this section, we will give a sketch the proof of a first step towards uniting higher genus and multiple faces : the case of two-faced precubic maps.\n\nIf we restrict \\eqref{precubic} to $f=2$, we obtain the following formula\n\\begin{equation}\n\\label{deuxfaces}\n(2g+1)\\alpha_g(n,2)=\\alpha^{(3)}_{g-1}(n-6,2)+\\sum\\limits_{g_1+g_2=g} \\sum\\limits_{i+j=n}\\alpha^{(1)}_{g_1}(i,1)\\alpha^{(1)}_{g_2}(j,1)\n\\end{equation}\nwhere $\\alpha_g(n,f)$ counts the number of precubic maps with $n$ edges and $f$ faces, of genus $g$, and $\\alpha^{(k)}_g(n,f)$ counts the number of those maps with $k$ marked leaves.\n\nWe can define the exploration of a precubic map with two faces in the following way :\n\\begin{definition}\nWe will describe an exploration of the map as a canonical labeling of all its corners (see Fig. \\ref{twofac}).\n\nAs in \\cref{explo}, we can define the discovery as the first edge adjacent to both faces that is encountered in a clockwise tour of the root face starting from the root, and the discovery vertex as the vertex that appears right before the discovery in this tour (there is only one discovery since there are 2 faces). Opening the discovery into a bud and a stem creates a blossoming (i.e. with a bud and a stem) unicellular map. In this map, it is possible to label all the corners in their order during the tour of the unique face. Thus, there is a labeling of all the corners of the map, and a discovery vertex (we won't need the discovery itself in what follows).\n\\end{definition}\nThe discovery vertex is obviously of degree $3$ (because a leaf is adjacent to only one face).  We can now introduce special vertices and \\textit{trisections}.\n\\begin{definition}\nA trisection is a vertex whose corner labels are in counterclockwise order around this vertex. A vertex is said to be \\textit{special} if it is a trisection  of if it is the discovery vertex.\n\\end{definition}\nA trisection of the map is exactly a trisection of the unicellular (blossoming) map. In \\cite{unicellular,trisections}, it is proven that in a unicellular map of genus $g$, there are $2g$ trisections. What's more, we can easily verify that the discovery vertex is not a trisection thus\n\\begin{lem}\nThere are $2g+1$ special vertices in a two-faced precubic map of genus $g$ (see Fig. \\ref{twofac} right).\n\\end{lem}\n\n\\begin{figure}[!h]\n\n\\captionsetup{width=0.8\\textwidth}\n\n\\centering\n\\includegraphics[scale=1]{twoface.pdf}\n\\caption{A precubic map (left), its exploration (center), and its special points (right, discovery vertex in red, trisections in purple)}\n\\label{twofac}\n\\end{figure}\n\nThe operation we will consider is fairly simple : take a map of genus $g$ with a marked special vertex, and split it.\nThere are two possible cases :\\begin{enumerate}\n\n\n\\item the map is disconnected into two maps with a marked leaf each, such that the genii, number of edges and faces add up,\n\\item the resulting map is of genus $g-1$ and has $3$ marked leaves, along with information on how to glue back the leaves together (given $3$ leaves, there are $2$ possible ways of gluing them together).\n\\end{enumerate}\n\n\\begin{rem}\nThe first case can only appear if the special vertex is the discovery vertex. In the second, case, the special vertex can be either the discovery vertex or a trisection.\n\\end{rem}\nThere is a bijection between maps in case (1) and pairs of maps with a marked leaf each, the inverse operation being the same as the case of disconnecting discoveries in the planar case : given two maps with a marked leaf each, it is possible to glue them back together as in \\cref{split}, and one obtains a map with a marked discovery vertex. \n\nCase (2) is more complicated : given a map of genus $g-1$ with $3$ marked leaves, that we will call a \\textit{tripod}, there are two ways to glue back the leaves. A gluing is said to be \\textit{valid} if the resulting map has genus $g$ and the gluing vertex is a special vertex. Tripods can have $0$, $1$ or $2$ valid gluings, and we will provide a classification of tripods with respect to their number of gluings. In the following, we will refer to \"marked leaves\" as just \"leaves\" as there is no risk of ambiguity. In order to prove \\eqref{deuxfaces}, we will then have to prove there is a bijection between tripods with respectively $0$ and $2$ valid gluings.\n\n\\begin{definition}\nIn a given map, let $v$ be the discovery vertex. It has two corners in the root face, and is thus seen twice in the tour of the root face. We can decompose the root face $F$ as a word on side edges as in \\cref{sec:proof}, starting from the root, as $F=TvOv\\overline{T}$ (see Fig. \\ref{tripod} left).\n\nTake a given tripod $M$, with exactly two leaves in the root face, both in $\\overline{T}$. Glue those two leaves to split the root face in two. Let $F'$ be the face thus obtained that is not the root face. We say that $M$ is in the special case if no side edge of $F'$ has its opposite side edge in $T$  (see Fig. \\ref{tripod} right).\n\\end{definition}\n\\begin{figure}[!h]\n\\captionsetup{width=0.8\\textwidth}\n\n\\center\n\\includegraphics[scale=1]{tripod.pdf}\n\\caption{The schematic decomposition of the root face (left), and the special case (right, with the leaves in purple)}\n\\label{tripod}\n\\end{figure}\n\n\\begin{lem} The following classification holds :\n\\begin{itemize}\n\\item A tripod with all leaves in the same face or in the special case or with two leaves in $O$ has $1$ valid gluing.\n\\item A tripod having exactly one leaf in the root face has $2$ valid gluings.\n\\item A tripod with exactly two leaves in the root face, at least one of which in $T$ or $\\overline{T}$ (except for the special case), has no valid gluings.\n\\end{itemize}\n\\end{lem}\nThe proof of this lemma is done by carefully considering the cases implied (we omit it here).\n\nTo finish the proof, one needs to find a bijection between tripods with $0$ valid gluings and tripods with $2$ valid gluings. The bijection consists of cleverly \"unplugging\" the discovery and plugging it somewhere else to \"transfer\" leaves between the faces. The proof involves several cases and is a bit technical, thus we prefer to omit it. This shows that in average, a tripod has one valid gluing, implying \\eqref{deuxfaces}.\n\n\\section*{Acknowledgements}\nThe author wishes to thank Guillaume Chapuy for suggesting the problem and for useful discussions.\n\\section*{Funding}\nThis work was supported by ERC-2016-STG 716083 \"CombiTop\".\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nSubstantial progress has been made in the last decade or so in our\nunderstanding of the evolution of disk galaxies in a cosmological\ncontext, both with analytical models (Mo, Mao \\& White 1998;\nEfstathiou 2000) and with numerical simulations, getting to higher\nresolution and including more and more physical ingredients: dark\nmatter profiles and assembly, baryon accretion histories, gas\ncooling and supernova feedback, multi-phase interstellar medium\n(e.g. Cole et al. 2000; Samland \\& Gerhard 2003; Monaco 2004; Kang\net al. 2005; Heller et al. 2007; Guo \\& White 2008; Ro\\v{s}kar et\nal. 2008). However, the complexity of the relevant baryonic physics\nis not yet fully understood and no satisfactory disk model exists at\npresent, although the most recent simulations with high resolution\nand improved stellar feedback start producing  disks resembling the\nobserved ones (Governato et al. 2007; Mayer et al. 2008).\n\nBecause of those difficulties, the simple phenomenological models\ndeveloped in the 80ies and 90ies may still be of considerable help.\nIn those models, the complex processes related to disk formation\nthrough gas accretion (i.e. merging histories, gas cooling, stellar\nfeedback, etc.) are simply described by an analytical infall law.\nSuch models do not construct the galaxy ``ab initio'' but rely on\nthe observed present-day features of a galaxy in order to infer its\npast history, thus they have been characterized as ``backwards''\nmodels (e.g. Ferreras \\& Silk 2001). They have been widely used in\nstudies of the chemical evolution of the Milky Way (e.g. Matteucci\n\\& Francois 1989; Prantzos \\& Aubert 1995; Chiappini et al. 1997;\nPrantzos \\& Silk 1998; Boissier \\& Prantzos 1999; Chang et al. 1999,\n2002; Cescutti et al. 2007 etc.) allowing important progress towards\nour understanding of our Galaxy's evolution. Indeed, some\nconvergence has been reached among the various groups concerning\ne.g. i) the necessity of substantial infall on long time scales (to\nexplain the local G-dwarf metallicity distribution and the small\ndegree of astration of deuterium); ii) the necessity of  radially\nvarying time scales for the infall and the star formation rate (to\nobtain the observed profiles of metallicity, gas fraction and\ncolors), and iii) no need for varying IMF or strong galactic winds.\n\nAs one of the three disk galaxies in the Local Group, Andromeda\n(M31, or NGC224) provides a unique opportunity for testing theories\nof galaxy formation and evolution (Klypin et al. 2002; Widrow et al.\n2003; Widrow \\& Dubinski 2005; Geehan et al. 2006; Tamm et al. 2007;\nTempel et al. 2007). The wealth of available data can be used to\nconstrain models of the evolution of the disk, bulge and halo of\nM31. However, due to its size, proximity and big bulge, most of the\nwork has been done on the stellar and kinematic properties of M31\nhalo and globular clusters (Beasley et al. 2004; Burstein et al.\n2004; Durrell et al. 2004; Chapman et al. 2006; Font et al. 2008;\nKoch et al. 2008; Lee et al. 2008), outer disk (Ibata et al. 2005;\nIrwin et al. 2005; Worthey et al 2005; Brown et al. 2006,2007,2008;\nRichardson et al. 2008) and the central bulge (Jacoby \\& Ciardullo\n1999; Salow \\& Statler 2004; Sarajedini \\& Jablonka 2005; Davidge et\nal. 2006; Olsen et al. 2006).\n\nStar formation (SF) histories in various regions of the M31 disk and\nhalo have also been measured with the Hubble Space Telescope (HST)\nand ground based large telescopes. For example, Williams (2003a,\n2003b) has measured the star formation history in several regions of\nthe M31 disk from the KPNO\/CTIO Local Group Survey and  found that\nthe total mean star formation rate for the disk is about 1 M$_{\\odot}$\nyr$^{-1}$. With deep HST photometry, Bellazzini et al. (2003) and\nWorthey et al. (2005) have studied the stellar abundance\ndistributions and star formation history in many locations of the\ndisk (see also Ferguson \\& Johnson 2001; Sarajedini \\& van Duyne\n2001; Williams 2002; Brown et al. 2006; Olsen et al. 2006). Those\nobservations have revealed that the M31 disk has a mean disk age\naround 6-8 Gyr and mean metallicity of [Fe\/H]$\\sim-0.2$, albeit with\nsubstantial spread in both cases.\n\nCompared with the Milky Way, M31 appears to have been more active in\nthe past, although its current star formation rate is smaller than\nthat of our Galaxy. Based on a survey of spiral properties, Hammer\net al. (2007) suggested that the Milky Way is a rather quiescent\ngalaxy, untypical of its class, while the M31 may be closer to a\ntypical spiral. Using detailed two components (disk+halo) chemical\nevolution models, Renda et al. (2005) have compared some chemical\nproperties between M31 and MW disk\/halo, and conclude that  M31 must\nhave a higher star formation efficiency and\/or shorter infall time\nscale. Deep photometry of the M31 halo shows that it hosts\npopulations of old and metal-poor stars, along with younger and of\nhigher metallicity ones, pointing to a prolonged period of active\nmerging. (Brown 2009 and references therein). The two ring-like\nstructures observed in M31 (Block et al. 2006) are interpreted as\nthe result of a recent ($<$200 Myr ago) collision with a companion\ngalaxy (Block et al. 2006) and give support to the idea of recent\nmerging activity of M31.\n\nIn this paper, we attempt a comparative study of the chemical\nevolution of MW and M31, by constraining our model with a more\nextended data set than in any previous work. Our data include global\nproperties and radial profiles of gas, stars, gas fraction, star\nformation rate, and oxygen abundances, as well as stellar\nmetallicity distributions along the disk of M31 (Sec. 2). We find\nthat, when the radial profiles are expressed in terms of the\ncorresponding scale lengths of the stellar disks, the MW and M31\npresent some interesting similarities (Sec. 2.4) encouraging us to\nadopt a single phenomenological model for the description of both\ngalaxies (Sec. 3). The model describes fairly well all of the key\nproperties of MW and most of M31 (Sec. 4), provided the star\nformation efficiency is twice as large in the latter case. We\ndiscuss the successes and failures of the model, and we compare to\nprevious work in Sec. 5. Sec. 6 summarizes our results.\n\n\n\\section{Observational Properties }\n\n\\subsection{Stellar disks: Scale lengths and masses }\n\nThe stellar disks of Milky Way and M31 are well described by\nexponential surface density profiles, given by:\n\\begin{equation}\\label{eq:sigma}\n    \\Sigma_{*}(r,t_g)=\\Sigma(r_0,t_g)e^{-(r-r_0)\/r_d}\n\\end{equation}\nwhere $r_d$ is the disk scale length and $\\Sigma(r_0,t_g)$ is the\nlocal surface density at some distance $r_0$ from the galactic\ncenter at the present time $t_g$=13.5 Gyr. In the case of the Milky\nWay, the reference distance is obviously Sun's distance of\n$r_0=R_{\\odot MW}$=8 kpc, where the local stellar surface density is\nevaluated to $\\Sigma(r_0,t_g)$=37 ~M$_{\\odot}$~pc$^{-2}$ \\ (Flynn et al. 2006). The\ntotal stellar mass of the disk is then given by\n\\begin{equation}\\label{eq:diskmass}\nM_d \\ = \\ \\int_{r_b}^{r_2}   2 \\pi \\ r \\ \\Sigma_{*}(r,t_g) dr\n\\end{equation}\nwhere $r_b$=2.5 kpc  is the bulge radius and $r_2$ the outer disk radius.\n\nObserved disk scale lengths  are obtained from measurements  of\nsurface brightness profiles in various wavelength bands and they are\nwavelength dependent. $B$ band scale length reflects mostly the SFR\nprofile in the past $\\sim$ 1 Gyr, while  $K$ or $R$ scale lengths\nreflect the total stellar population, cumulated over the age of the\ndisk.\n\nFor the Milky Way disk, we adopt the mean value of $r_d = 2.3\\pm0.6$\nkpc (from measurements in the $R$ or $I$ bands), derived from the\ncompilation of Hammer et al. (2007). The total  mass up to 15 kpc is\nthen $\\sim$~3$\\times$10$^{10}$ M$_{\\odot}$.\n\nBy adding $\\sim$0.7$\\times$10$^{10}$ M$_{\\odot}$~for the gaseous disk mass\n(as estimated in the next section) one gets a total baryonic disk\nmass of $\\sim$3.7$\\times$10$^{10}$ M$_{\\odot}$, in good agreement with mass\nmodels of the Milky Way (e.g. Dehnen \\& Binney 1998, Naab and\nOstriker 2006).\n\nBased on the observed disk surface brightness of M31, Walterbos \\&\nKennicutt (1987, 1988) have measured the disk scale length in\ndifferent wavelengths. They obtained $r_d$ = 6.8, 5.8, 5.3, and 5.2\nkpc in the $U,B,V,$ and $R$ bands, respectively. Recently, Worthey\net al.(2005) obtained 5.6 kpc in the $I$ band, while for $K$ band\nHiromoto et al. (1983) find  $r_d$ = 4.2 kpc. With the IRAC on board\nthe SPITZER space telescope, Barmby et al. (2006) measured, for the\nfirst time, the mid infrared surface brightness profile of M31 and\nfound a scale length of 6.08 kpc in the $L$ band. Note that\ndifferent authors adopt different distance scale of M31. In Table 1,\nwe list all the available observed disk scale lengths and scale them\nto the same distance of 785 kpc (McConnachie et al. 2005). Overall,\nthe values are consistent for different bands, except for the\nshorter wavelengths which are likely to be affected by dust\nextinction. In this paper we adopt an averaged value from four\nobserved values from three bands ($R, I, K$), which is $r_d$=5.5\nkpc. This value is within the range of $r_d$=5.8$\\pm$0.4 kpc found\nin Hammer et al. (2007).\n\nThe total mass of M31 disk is obtained through observational data\nand  mass models. In their disk-bulge-halo model, Widrow et al.\n(2003) find  that their best model requires the M31 disk mass (stars\n+ gas) to be about 7 $\\times$ 10$^{10}$ M$_{\\odot}$. Recent mass model of\nGeehan et al. (2006) also gives a similar disk mass value  $\\sim$7.2\n$\\times$ 10$^{10}$ M$_{\\odot}$, by adopting a disk mass-to-light ratio of\n3.3. In this paper, we adopt then the M31 total disk mass to be\nM$_{tot}$ = 7$\\times$10$^{10}$M$_{\\odot}$. By subtracting $\\sim$6 $\\times$\n10$^9$ M$_{\\odot}$ \\ for the gas (see next section) we obtain a total disk\nstellar mass of 5.9 $\\times$ 10$^{10}$ M$_{\\odot}$ \\ for Andromeda.\n\nIn  summary, the M31 disk is about 2 times as massive and 2.4 times\nas large as the Milky Way disk.\n\n\n\\begin{table*} [!t]\n\n\\begin{center}\n\n{\\bf Table 1} Observation of Milky Way and M31 (re-scaled to 785\nkpc)\n\\begin{tabular}{lllll}\n     \\hline  \\hline\n     Observable                  & Milky Way     & reference  & M31       & reference \\\\\n    \\hline \\hline\n    \\textbf{Global properties} \\\\\n    \\hline\n     Type                         & SbcI-II      & 1   & SbI-II           & 1\\\\\n     $K$-band                       & $M_K=-24.02$ & 2   & $M_K=-24.70$     & 3 \\\\\n     Total luminosity             &              &     &                  &   \\\\\n      \\qquad $L_B$ ($10^{10}L_{B\\odot}$) &  1.8  & 4   & 3.3              & 5 \\\\\n      \\qquad $L_V$ ($10^{10}L_{V\\odot}$) &  2.1  & 6   & 2.6$\\sim$2.7     & 1 \\\\\n      \\qquad $L_K$ ($10^{10}L_{K\\odot}$) &  5.5  & 7   & 6 $\\sim$ 12$^a$  & 8 \\\\\n     Total color ($B-V$)                 &  0.84 & 4   & 0.81             & 9\\\\\n     Mass                                &       &     &                  &  \\\\\n      \\qquad Total ($10^{10}$M$_{\\odot}$)        & 40-55 & 10, 11 & $107-140$     & 12\\\\\n      \\qquad Visible ($10^{10}$M$_{\\odot}$)      & 5.0   & 3      & $5.9-8.7$     & 5\\\\\n     Rotational Curve (~km~s$^{-1}${})             &       &        &                &  \\\\\n      \\qquad Flat velocity               & 220   & 1      & 226            & 13 \\\\\n    \\hline \\hline\n     \\textbf{Bulge}                      &       &        &                &   \\\\\n    \\hline\n      \\qquad Stellar mass ($10^{10}$M$_{\\odot}$) & 1-2   & 14     & 3.2            & 15  \\\\\n      \\qquad Effective radius (kpc)      & 2.5   & 1      & 2.6            & 1   \\\\\n    \\hline \\hline\n     \\textbf{Disk}          &                    &        &                & \\\\\n    \\hline\n     Scale length (kpc)       &                    &        &                &  \\\\\n      \\qquad $U$              &                    &        &  7.7           & 9 \\\\\n      \\qquad $B$              & $4.0\\sim5.0$       & 7      &  6.6           & 9 \\\\\n      \\qquad $V$              & $2.5\\sim3.5$       & 6, 16  &  6.0           & 9 \\\\\n      \\qquad $R$              &  2.3               & 3      &  5.9           & 9 \\\\\n      \\qquad $I$              &                    &        & 5.7            & 17 \\\\\n      \\qquad $K$              & $2.3\\sim2.8$       & 18     & 4.8           & 19 \\\\\n      \\qquad $L$              &                    &        & 6.08          & 8 \\\\\n     Total SFR (M$_{\\odot}$ $yr^{-1}$)  & $\\sim$1-5        & 20,21   & 0.35 - 1.0  & 1, 8, 21, 22\\\\\n     Infall rate (M$_{\\odot}$ $yr^{-1}$)& $0.5\\sim5$     & 21, 23 &               & \\\\\n     Total mass of          &                    &        &                  & \\\\\n      \\qquad disk($10^{10}$M$_{\\odot}$) & 3.5 $^b$       & 24     & $\\sim$7       & 15, 25, 26\\\\\n      \\qquad star($10^{10}$M$_{\\odot}$) & 3.0 $^c$       & 16     & $\\sim$6   & 5\\\\\n      \\qquad gas($10^{10}$M$_{\\odot}$)  & $\\sim$0.7  & 27, 28 & $\\sim$0.6             & 29\\\\\n      \\qquad HI($10^{10}$M$_{\\odot}$)   & 0.4            & 1      & $\\sim$0.5   & 1, 29, 30\\\\\n      \\qquad H$_2$($10^{10}$M$_{\\odot}$)& 0.11           & 31     & $\\sim$0.02-0.04 & 29, 30, 31\\\\\n     Gas fraction               & $\\sim$0.15-0.2 & 20, This paper & $\\sim$ 0.09  & This paper \\\\\n     Abundance gradient         &                  &            &                    & \\\\\n      \\qquad [O\/H] (~dex~kpc$^{-1}$)  & $-0.04 \\sim -0.07$ & 32, 33, 34 & $-0.018 \\sim -0.027$ & 35, 36\\\\\n    Color gradient                &                &            &                    & \\\\\n      \\qquad $B-V$ (mag kpc$^{-1}$) &              &            & 0.016              & 9\\\\\n    \\hline\n    \\hline\n\\end{tabular} \\\\\n\\end{center}\nNote:\\\\\na: assuming M\/$L_K$ = 1.15 (M\/$L_K)_{\\odot}$ and  M31 mass is taken\nto be (7-14)$\\times$ $10^{10}$ $M_\\odot$ \\\\\nb: derived based on the disk scale length $r_d$=2.3 kpc and total\ndisk surface density at the solar neighborhood $\\Sigma_{tot}$ =\n50 M$_{\\odot}$ $pc^{-2}$ \\\\\nc: derived based on the stellar disk scale length $r_d$=2.3 kpc and\nstellar surface density at the solar neighborhood $\\Sigma_{*}$ = 37\nM$_{\\odot}$ $pc^{-2}$ \\\\\n\\noindent Reference: (1) van den Bergh 1999; (2) Drimmel \\& Spergel\n2001; (3) Hammer et al. 2007; (4) van der Kruit 1986; (5) Tamm et\nal. 2007; (6) Sackett 1997; (7) Kent et al. 1991; (8) Barmby et al.\n2006; (9) Walterbos \\& Kennicutt 1988; (10) Xue et al. 2008; (11)\nSakamoto et al. 2003; (12) Tempel et al. 2007; (13) Carignan et al.\n2006; (14) Dehnen \\& Binney 1998; (15) Geehan et al. 2006; (16)\nZheng et al. 2001; (17) Worthey et al. 2005; (18) Freudenreich 1998;\n(19) Hiromoto et al. 1983; (20) Boissier \\& Prantzos 1999; (21)\nFraternali 2009; (22) Williams 2003a; (23) Blitz et al. 1999; (24)\nHolmberg \\& Flynn 2004; (25) Klypin et al. 2002; (26) Widrow et al.\n2003; (27) Dame 1993; (28) Kulkarni \\& Heiles 1987; (29) Nieten et\nal. 2006; (30) Dame et al. 1993; (31) Koper et al. 1991; (32)\nDeharveng et al. 2000; (33) Daflon \\& Cunha 2004; (34) Rudolph et\nal. 2006; (35) Smartt et al. 2001; (36) Trundle et al. 2002.  \\\\\n\n\\end{table*}\n\n\n\\subsection{Gas and SFR Profiles}\n\nThe present-day profiles of gas and star formation  provide strong\nconstraints on models of the chemical evolution of a galactic disk.\nIn the case of the Milky Way, relevant observational data have been\ncollected in Boissier \\& Prantzos (1999) and we adopt those data in\nthis work (Fig. 1, left panels). The gaseous profile is\ncharacterized by a broad peak at a galactocentric distance $\\sim4-5$\nkpc (due to the ``molecular ring'' present at this distance) and the\nSFR profile is also concentrated towards the inner disk. The total\ngas mass is estimated to $\\sim$7$\\times$10$^9$ M$_{\\odot}$ and the total\nstar formation rate is SFR$\\sim1-3$~M$_{\\odot}$~yr$^{-1}$ (e.g. Boissier \\& Prantzos\n1999, and references therein).\n\nFor M31, the observed radial profiles for HI and H$_2$ gas surface\ndensities (Berkhuijsen 1977; Walterbos 1986; Koper et al. 1991;\nLoinard et al. 1999) allow us to establish the radial gas profile\ndisplayed in Fig.~\\ref{Fig:GasSFRobserved} (right top). It is also\ncharacterized by a broad peak, located at a galactocentric distance\ntwice as large as in the case of the MW. The HI profile from these\nstudies is also consistent with the one recently measured by Chemin\net al. (2009).\n\nThe star formation rate in several regions of M31 disk has been\ncarefully measured by both ground base photometry and Hubble Space\nTelescope (Bellazzini et al. 2003; Williams 2002, 2003a,b; Brown et\nal. 2006; Olsen et al. 2006). The current total SFR for M31 disk is\nestimated to be 0.4 $\\sim$1 ~M$_{\\odot}$~yr$^{-1}$ (Williams 2003a,b; Barmby et al.\n2006), i.e. less than half of the value in the Milky Way disk;\nthis shows that M31 is currently a rather quiescent galaxy.\n\nThanks to the GALEX UV satellite, it is now possible to obtain SFR\nradial profiles for a number of local galaxies derived not from\nH$\\alpha$ data, but from the UV continuum (Boissier et al. 2007). In\nFig.~\\ref{Fig:GasSFRobserved}, we show  the adopted gas (upper\npanels) and SFR (lower panels) profiles of the Milky Way and M31\ndisks. From this figure, we see that the two spirals show quite\ndifferent properties in their gas and SFR profiles in the inner part\nof the disk (between 3 and 7 kpc). The Milky Way has more gas in\nthis region, while M31 has most of its gas outside that region.\n\nThe total gas mass of the disks is obtained by integrating the gas\nprofiles from Fig.~\\ref{Fig:GasSFRobserved} for both galaxies,\nstarting from the inner disk boundary (assumed to be at the bulge\nradius $r_b\\sim2.5$ kpc for the MW and $r_b\\sim5$ kpc for M31)\noutwards. We find M$_{gas,MW} \\sim$ 7$\\times$10$^9$ M$_{\\odot}$ and\nM$_{gas,M31} \\sim$ 6$\\times$10$^9$ M$_{\\odot}$ (average value, taking\nuncertainties into account). Since the M31 disk is twice as massive\nas the Milky Way disk, its global gas fraction is about 1\/2 of that\nof the Milky Way disk (0.09 vs 0.19, respectively). This implies\nthat  M31 disk had an overall higher star formation efficiency than\nthe Milky Way (assuming that they have similar ages).\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{f1.eps}\\\\\n  \\caption{Observed profiles of the gas and star formation rate for\n  the Milky Way and M31 disks. Upper panel: current surface density\n  profiles of HI, molecular gas and total gas in the Milky Way\n  and M31. Lower panel: observed estimations of SFR of the Milky Way\n  (filled symbols, from Boissier \\& Prantzos 1999) and M31 disks\n  (open symbols from Boissier et al. 2007). The lines in the lower\n  panels are the results calculated according to the star formation\n  law given in the right panel.}\n  \\label{Fig:GasSFRobserved}\n\\end{figure}\n\n\nUV studies of star formation with GALEX showed that, in general, the\ncorrelation between SFR and gas surface density is compatible with\nempirical Kennicutt (1998a,b) SFR laws, with some scatter in the low\nsurface density side. But as Boissier et al. (2007) show, this\ncorrelation fails for some individual galaxies, and especially for\nM31 (see their Fig.6). In the lower two panels of\nFig.~\\ref{Fig:GasSFRobserved}, we also show the expected behaviour\nof two different SFR laws. The first depends only on gas surface\ndensity, according to:\n\\begin{equation}\\label{eq:sfrKennLaw}\n    \\Psi(r)=0.25~\\Sigma_{gas}^{1.4}(r)\n\\end{equation}\nfrom obsrvational data of Kennicutt (1998b, hereafter KS Law). The\nsecond depends on both gas surface density and radius and is\nmotivated by the idea that star formation is induced by spiral waves\nmoving around a rotating disk (e.g. Wyse \\& Silk 1989; see also Sec.\n3.2):\n\\begin{eqnarray} \\label{eq:sfrBP00}\n\\Psi(r)\\propto\\frac{\\Sigma_{gas}^{n}(r)}{r}\n\\end{eqnarray}\n\n\nIt turns out that, in the case of the MW disk, observed gas and SFR\nprofiles fit well both SFR laws; as a result, the total star SFR is\nalso readily reproduced. But for M31, none of the SFR laws brings\nagreement between observed gas and SFR profiles: there is a great\ndifference between theoretical expectations and observations in the\ninner part of the disk (with little gas but, curiously,  high\nobserved SFR). As a result, any attempt to fit the SFR of the inner\ndisk with one of the aforementioned SFR laws will lead to an\noverestimate of the SFR in the outer disk.\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=0.48\\textwidth]{f2.eps}\\\\\n    \\caption{Relationship between observed gas surface density and\n    star formation rate surface density for M31 (filled circles)\n    and Milky Way disks (filled triangles).\n    Full line is the classical Kennicutt law (equation(\\ref{eq:sfrKennLaw})).\n    Regions between two dashed lines indicate the results for a number\n    of local galaxies observed by GALEX (from Boissier et al. 2007).\n    The M31 shows an untypical path in its inner region, while the Milky\n    Way shows a rather normal behaviour.}\n  \\label{Fig:SFRbyGas}\n\\end{figure}\n\nIn order to further demonstrate the different behavior of local star\nformation rate in M31 and Milky Way disks, we plot the relationship\nbetween the observed gas surface density and star formation rate\nsurface density in Figure~\\ref{Fig:SFRbyGas}. Also we show the\ngeneral trend of the SFR with gas surface density (within  dashed\ncurves) for a number of nearby galaxies (from Boissier et al. 2007).\nIt is clear that M31 has a peculiar behaviour compared to both the\nMilky Way and other local galaxies, especially in its inner region,\nwhere high star formation rate corresponds to low gas amounts. In\nthe range 7-11 kpc the  SFR decreases when the gas surface density\nincreases, contrary to the classical SFR law. Then, beyond 11 kpc,\nboth SFR and gas amount decrease with radius, roughly following the\nKennicutt law.\n\nIf observations of SFR in M31 are not heavily distorted by incorrect\nextinction corrections, then one concludes that the current SFR in\nthat galaxy does not obey one of the classical star formation laws\n(Schmidt, Kennicutt, or some modified form of them). Perhaps, star\nformation in M31 is (or has been) perturbed by some external event,\ne.g. a major recent encounter with a galaxy of the Local group.\nIndeed, observations of a two-ring-like structures by Block et al.\n(2006) are interpreted as due to a  nearly central head-on encounter\nwith a companion galaxy (probably M32) about 200 Myr ago. If this is\nindeed the case, then the present day SFR profile of M31 cannot be\nused as a constraint on the chemical evolution model, since the\nperturbation induced in the gaseous disk by the collision most\nprobably affected for (at least) one orbital time the SFR in M31.\nTime-intergrated observables, like e.g. the total stellar profile or\nthe abundance profile (and, to a lesser extent, the gaseous profile)\ncertainly remain valid constraints.\n\n\n\\subsection{Disk Abundance Gradients}\n\nAbundance gradients are an essential ingredient in an accurate\npicture of galaxy formation and evolution (Boissier \\& Prantzos\n1999; Hou et al. 2000; Chiappini et al. 2001; Hou et al. 2002;\nCescutti et al. 2007; Magrini et al. 2009; Fu et al. 2009). The\nexistence of abundance gradients along the MW disk has been\nestablished in the past twenty years using different tracers (Hou \\&\nChang 2001). However, the magnitude of that gradient is still\nsubject to debate. Thus, oxygen or\/and iron abundance gradients of\nabout $-0.06 \\sim -0.07$ ~dex~kpc$^{-1}$ are obtained by using tracers as\nHII regions and B stars (Rudolph et al. 2006 and references\ntherein), planetary nebulae (Maciel et al. 2006; Maciel \\& Costa\n2008) and open clusters (Chen et al. 2003,2008). However, values\nabout 40\\% smaller are obtained by using those same tracers, e.g.\nDeharveng et al. (2000, with HII regions), Daflon \\& Cunha (2004,\nusing several tracers) and Andrievsky et al. (2004, with Cepheids).\n\nThe situation of the abundance gradient in the disk of M31 is also\nfar from clear. Early observations (Dennefeld \\& Kunth 1981; Blair\net al. 1982) used HII regions and supernova remnants. A value of\ndlog(O\/H)\/dr= $-0.06\\pm 0.034$ ~dex~kpc$^{-1}$  was derived using nebular\nemission line ratios by Galarza et al. (1999).  The main uncertainty\ncomes from the empirical calibrations which are used to derive the\nelectronic temperatures in the nebular phase. By re-analyzing\nearlier data from various authors, Trundle et al. (2002) find\nsmaller values for the oxygen abundance gradient, ranging from\n$-$0.027 ~dex~kpc$^{-1}$ down to $-$0.013 ~dex~kpc$^{-1}$. Furthermore, their\nanalysis of  five B-type supergiants covering the galactocentric\ndistance of 5-12 kpc leads to a negligible oxygen abundance gradient\nof $-0.006 \\pm 0.02$ ~dex~kpc$^{-1}$; In contrast, they find a slightly more\nsignificant gradient for Mg, of $-0.023 \\pm 0.02$ ~dex~kpc$^{-1}$.\n\nIn summary, the abundance gradient in M31 is very poorly known at\npresent. We adopt here a value of $-$0.017 ~dex~kpc$^{-1}$, i.e. the mean\nbetween the most extreme values of $-$0.027 ~dex~kpc$^{-1}$ and $-$0.006\n~dex~kpc$^{-1}$ found in the various analysis of Trundle et al. (2002). We\nnote that this value is substantially smaller (factor 3-4) than that\nof the Milky Way disk ($-0.07$ ~dex~kpc$^{-1}$), but only by a factor of 2\nif the value of $-$0.04 ~dex~kpc$^{-1}$ \\ is adopted for our Galaxy.\nFinally, if we express the abundance gradients in terms of\ncorresponding scale lengths (dex\/$r_d$), then the scaled gradient of\nM31 disk is found to be two times smaller (equal) to the one of the\nMW disk for the cases of $-$0.07 ~dex~kpc$^{-1}$ \\  ($-$0.04 ~dex~kpc$^{-1}$).\n\n\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[height=9cm,width=9cm]{f3.eps}\\\\\n    \\caption{Observed oxygen abundance gradient in the Milky Way (top,\n    data from Rudolph et al. 2006; Deharveng et al. 2000) and M31 (bottom,\n    data from Dennefeld \\& Kunth 1981; Blair 1982; Trundle et al. 2002).\n    In the top panel, the two commonly referred values of $-$0.07 ~dex~kpc$^{-1}$\n    and $-$0.04 ~dex~kpc$^{-1}$ appear as {\\it dashed} and {\\it solid} line respectively.\n    }\n  \\label{Fig:OxygenGradient}\n\\end{figure}\n\nFigure~\\ref{Fig:OxygenGradient} displays  the observed oxygen\nabundance profiles in the Milky Way and M31 disks. In the case of\nMW, two values for the gradient are shown, corresponding to $-$0.07\n~dex~kpc$^{-1}$ \\ (circles) and $-$0.04 ~dex~kpc$^{-1}$ \\ (solid line),\nrespectively.\n\n\n\\subsection{A Unified Description of the Milky Way and M31}\n\nTable 1 summarizes the main observational features of the MW and M31\ndisks. The different sizes of the two major galaxies of the Local\ngroup make difficult a direct comparison between their radial\nprofiles, thus giving no hints as to the physical ingredients\nrequired for a successful simultaneous description of both disks. In\norder to have a coherent picture, we attempt in this work a more\nphysical description, expressing all distances in terms of the\ncorresponding disk scale lengths.\n\n\\begin{figure*}[!t]\n  \\centering\n  \\includegraphics[height=14cm,width=18cm]{f4.eps}\\\\\n    \\caption{Observed profiles for Milky Way and M31 disks, with radius\n    expressed in units of the corresponding scalelengths $r_d$. For the\n    abundance gradient of the Milky Way disk, two sets of values are\n    plotted, one is $-$0.07 ~dex~kpc$^{-1}$ (bottom right panel), the other is\n    $-$0.04 ~dex~kpc$^{-1}$ (bottom left panel). In the latter case, MW and M31\n    have similar abundance gradients, when expressed in dex\/$r_d$. Shaded\n    areas are the typical observed scatter. }\n  \\label{Fig:scaleobs}\n\\end{figure*}\n\nIn Fig.~\\ref{Fig:scaleobs}, we plot the radial profiles of gas,\nstar, SFR, gas fraction, and the oxygen abundance gradients for the\ntwo disks, using their scale lengths as distance units. It can be\nseen that:\n\n(1) The gaseous profiles (top left) are rather similar, in the sense\nthat they both display a broad peak at $\\sim$2 scale lengths from\ntheir centers. The MW has a more extended gaseous profile (in terms\nof scale length).\n\n(2) The Milky Way disk is more compact than M31, since it has a\nhigher stellar surface density at a given $r_d$ value (middle left).\n\n(3) The profiles of scaled gas fractions (middle right) of the two\ngalaxies are quite similar in the inner disks. However, the overall\ngas fractions of the two disks are quite different, with the MW\nhaving a gas fraction twice higher than  that of M31.\n\n(4) The scaled abundance gradients between two disks are similar if\nwe adopt the smaller reported value  for the MW disk (bottom left).\n\nThus, when the observed profiles are expressed in terms of scale\nlengths, the two disks show some similarities in their properties.\nOne may hope then to describe both disks with a single chemical\nevolution model, by varying as few as possible of the relevant\nparameters. We describe such a model in the next section.\n\n\n\\section{The model}\n\nIn the case of the Milky Way, models with radially dependent infall\nand star formation laws, forming the disk inside-out, are\ngenerically used (Prantzos \\& Aubert 1995; Boissier \\& Prantzos\n1999; Hou et al. 2000; Chiappini et al. 2001; Magrini et al. 2009).\nSuch models  reproduce several of the salient observational features\nof the MW disk (including the abundance gradients), albeit with\ndifferent levels of success.\n\nIn this section, we present briefly our chemical evolution model\nwhich is similar to the one adopted successfully in the past for the\nMilky Way disk (see details in Boissier \\& Prantzos 1999, 2000; Hou\net al. 2000).\n\n\\subsection{IMF and Stellar Yields}\n\nThe initial mass function (IMF) $\\Phi(m)$ describes the mass\ndistribution of newborn stars and can be inferred from the observed\nluminosity function on the basis of the mass-to-light ratio for\nstars. Similar to our previous works, we adopt the IMF from the work\nof Kroupa, Tout \\& Gilmore (1993, KTG93), where some complex factors\n(like stellar binarity, ages and metallicities, as well as\nmass-luminosity and color-magnitude relationships) are explicitly\ntaken into account (Boissier \\& Prantzos 1999; 2000; Hou et al.\n2000).\n\n\nStellar yields are taken from Woosley \\& Weaver (1995, WW95) for\nmassive stars, and from van den Hoek \\& Groenewegen (1997, vdHG97)\nfor low and intermediate mass stars (mass from 0.8 to 8 M$_{\\odot}$). They\nare all metallicity dependent.\n\n\nIn order to account for the additional source of Fe-peak elements,\nrequired to explain the observed decline of O\/Fe abundance ratio in\nthe Milky Way disk (Goswami \\& Prantzos 2000), we utilize the yields\nof SNIa from the exploding Chandrashekhar-mass CO white dwarf models\nW7 and W70 of Iwamoto et al. (1999). These are updated versions of\nthe original W7 model of Thielemann et al. (1986), calculated for\nmetallicities Z = Z$_{\\odot}$ (W7) and Z = 0 (W70), respectively.\n\n\\subsection{Infall Rate and Timescale}\n\nWe assume that the MW and M31 disks are progressively built up by\ninfall of primordial gas cooling down from their dark haloes. The\nform of the time dependence of the infall rate is unknown at\npresent. In Prantzos \\& Silk (1998), an asymmetric Gaussian infall\nrate was assumed, on the basis of dynamical arguments. However,\nusually simpler parametrizations are adopted, i.e. infall rate is\nexponentially decreasing in time:\n\n\n\\begin{equation} \\label{eq:infall}\nf(t,r) \\  = \\ A(r) \\ e^{-t\/\\tau(r)}\n\\end{equation}\nwhere $A(r)$ is a normalizing function and can be obtained by:\n\\begin{equation}\\label{eq:Ar}\n    \\int_0^{t_g}A(r)\\cdot e^{-t\/\\tau(r)}dt=\\Sigma_{tot}(r,t_g)\n\\end{equation}\nwhere $\\Sigma_{tot}(r,tg)$ is the current total mass profile and\n$\\tau(r)$ is the infall time scale which is radially dependent. In\nthe Milky Way disk, the characteristic infall time scale in the\nsolar neighborhood ($R_{\\odot MW}$ = 8~kpc)is $\\sim$7 Gyr (Chiappini\net al. 1997; Boissier \\& Prantzos 1999; Chang et al. 1999, 2002), in\norder to reproduce the local G-dwarf metallicity distribution.\n\nThe radial dependence of the infall time scale for the MW disk is\ngiven by $\\tau_{MW}(r)=b~r\/r_d$, where $r_d$ is the scale length and\n$b$ is a free parameter. Positive values of $b$ imply an inside-out\nformation of the disk and we adopt here $b$=2.5, which leads to\nformation time scales of $\\sim$2 Gyr for the  inner disk and\n$\\sim$10 Gyr for the outer disk.\n\nIn the case of M31, we adopt the prescription used in Boissier \\&\nPrantzos (2000), according to which the infall time scale is assumed\nto be correlated with both surface density and galaxy mass:\n\\begin{equation}\\label{eq:tau}\n    \\tau^{-1}(r)=\\tau_{MW}^{-1}(r)+0.4(1.0-\\frac{V_{C}}{220})\n\\end{equation}\nwhere $V_{C}$ (in km~s$^{-1}$) is the flat rotational velocity for\nthe galaxy disk and $\\tau_{MW}(r)$ (in Gyr) is the infall time scale\nfor the Milky Way disk.  According to Hammer et al. (2007), $V_{C}$\nfor M31 is about 226 km~s$^{-1}$, i.e. the same as that of Milky Way\ndisk. Therefore, our adopted prescription leads to similar infall\ntime scale laws for both disks.\n\n\n\n\\subsection{Star Formation}\n\nThe star formation rate  remains the major unknown in chemical\nevolution studies. Kennicutt (1998a,b) found that the global SFR of\ndisks and circumnuclear starburst galaxies is correlated with the\nlocal gas density  over a large range in surface density and SFR per\nunit area, spanning 5 orders in magnitude. Over that range, the\nempirical SFR vs gas surface density relation can be fitted by a\nsimple power law with index $n \\sim 1.4$. Kennicutt (1998a,b) also\nfound that the data can be fitted equally well as a function of the\nlocal dynamical time scale, $\\tau_{dyn}$: $\\Psi\n\\propto\\frac{\\Sigma_{gas}}{\\tau_{dyn}} \\propto\\Sigma_{gas}\n \\Omega$, where $\\Omega$ is the rotation speed of the gas. Since\n$\\Omega \\sim V(r)\/r$, the SFR could be expressed as:\n\\begin{equation}\n    \\Psi(r) \\ \\propto \\  \\Sigma_{gas}\\ {{V(r)}\\over{r}}\n\\end{equation}\nwhere $V(r)$ is the circular velocity at radius $r$. Since spiral\ngalaxies display $V(r)\\sim$ constant, one gets a modified\nKennicutt-Schmidt law (hereafter M-KS law), as suggested on\ntheoretical grounds in Wyse \\& Silk (1989, see also Prantzos \\&\nAubert 1995).\n\nBoissier \\& Prantzos (1999) adopted the index $n$ of the M-KS SFR\nlaw to be $n=1.5$ on an empirical basis, in order to fit the present\nday profiles of the MW SFR (Fig. 1, bottom left). They also adopted\nthis M-KS law in subsequent models for external spirals, which can\nsuccessfully reproduce most of the chemical and photometric\nproperties of disk galaxies (Boissier \\& Prantzos 2000; Boissier et\nal. 2001) and in particular the observed abundance gradients\n(Prantzos \\& Boissier 2000). In a recent study, Fu et al. (2009),\n have used both KS law and M-KS law to predict the time\nevolution of Galactic disk abundance gradient. By comparing the\nmodel predictions with the observed results from open clusters and\nplanetary nebulae with different ages, it is concluded that by\nadopting the M-KS law, model results are more consistent with the\nobserved evolution of abundance gradient. Therefore, we will adopt\nthis M-KS law for Milky Way and M31 disks:\n\\begin{equation} \\label{eq:sfrB}\n    \\Psi(r) = \\alpha~\\Sigma_{gas}^{1.5}(\\frac{r_{eq\\odot}}{r})\n\\end{equation}\nThe coefficient $\\alpha$ is related to the star formation\nefficiency. All other things being equal, it appears that the star\nformation efficiency in M31 has to be at least twice as high as in\nthe MW, since its observed gas fraction is twice as small (Table 1\nand discussion in Sec. 2.2). We shall see indeed in the next section\nthat such a larger $\\alpha$ is required in order to fit the M31\ndata.\n\n\\begin{table}[!t]\n\\noindent Table 2. Model parameters \\\\\n\\begin{tabular}{lll}\n\\hline \\hline\n General                       &   Prescription               &  Parameter \\\\\n\\hline\n IMF                           &   KTG1993                    &            \\\\\n Mass limits                   &   (0.1-100) M$_{\\odot}$              &            \\\\\n SFR                           &   $\\alpha$$\\Sigma_{gas}^{1.5}(r_{eq\\odot}\/r)$ &  $\\alpha$  \\\\\n Stellar yields                &   vdHG97,WW95                &            \\\\\n Metallicity of infall gas     &   Z$_{f}$ = 0                &            \\\\\n Infall time scale             &   $\\tau(r) = b \\ (r\/r_d)$    &   $b$        \\\\\n Age of disk (Gyr)             &   13.5                       &            \\\\\n\\hline\nIndividual                     &   Milky Way                  &   M31         \\\\\n\\hline\n Scale length $r_{d}$ (kpc)     &   2.3                       &   5.5           \\\\\n Equivalent $r_{eq\\odot}$(kpc)  &   8.0                       &   19.0    \\\\\n Total disk mass ($10^{10}$M$_{\\odot}$) &  5.0                        &   7.0           \\\\\n V$_{rot}$(~km~s$^{-1}${})                &  220                        &   226           \\\\\n\\hline \\hline\n\\end{tabular} \\\\\n\\end{table}\n\n\n\\begin{figure*}[!t]\n  \\centering\n  \\includegraphics[angle=-90,width=18cm]{f5.eps}\\\\\n  \\caption{Current profiles of gas, stars, SFR, gas fraction and oxygen abundance\n  (from top to bottom) for MW and M31. In the {\\it left} and {\\it middle} panels,\n  profiles for MW and M31, respectively, are expressed in terms of physical radius\n  $r$ (in kpc); in the {\\it right} panels, profiles for both disks are expressed\n   in terms of normalised radius $r\/r_d$. Observations are presented\n   as yellow shaded ( for MW with $-0.04 dex\/kpc$ ) or blue shaded ( for M31 )\n   areas and model results by solid (for the MW) and dashed (for M31) curves,\n   respectively.\n  }\n  \\label{Fig:Profiles}\n\\end{figure*}\n\n\\section{Model Results and Comparison with the Milky Way and M31 Disks}\n\nWe run our simulations with the parameters of Table 2 for MW and M31\ndisks. Notice that we adopt  the same model parameters for both\ngalaxies (hence, attempting to describe them in a unified\nframework), except for: (i) the small difference in the infall rate\nfrom Eq. (\\ref{eq:tau}), which makes M31 slightly older than the MW\nand (ii) the star formation efficiency parameter $\\alpha$, assumed\nhere to be twice as large for M31 than for the MW, i.e. $\\alpha_{MW}\n= 0.1$ and $\\alpha_{M31} = 0.2$.\n\n\n\\subsection{Radial Profiles}\n\nIn Fig.~\\ref{Fig:Profiles}, we show the model predictions of the\nradial profiles for gas, stars, SFR, gas fraction and oxygen at time\n$t$=13.5 Gyr and we compare them with observational data. The first\ntwo columns display results for MW and M31, respectively, as a\nfunction of radius $r$ expressed in kpc. The third column presents\nthe same results in a common scale of normalized radius $r\/r_d$ for\nboth galaxies; this allows to better visualize the similarities and\ndifferences between the two disks.\n\nThe main results of the comparison with observations can be\nsummarized as follows:\n\n1) In both cases, exponential disk profiles are obtained by\nconstruction, since most of the infalling gas (the radial profile of\nwhich is normalised through Eq.(~\\ref{eq:Ar})) is  turned into\nstars.\n\n2) The model gaseous profiles go through a broad maximum, obtained\nat the observed position, approximately at two scale lengths from\nthe galactic centers. This maximum is obtained in the models through\nthe radial dependence of the SF efficiency (being greater in the\ninner disk, it produces a gas fraction profile $f_{gas}(r)$\nincreasing with radius, see next paragraph) and the total surface\ndensity profile $\\Sigma_{tot}(r)$ which decreases with radius (by\nconstruction). The gaseous profile being the product of the two\n($\\Sigma_{gas}(r)=f_{gas}(r) \\Sigma_{tot}(r)$), the resulting curve\ngoes through a maximum, and within our unified scheme this happens\nat $\\sim$2 $r_d$.\n\n3) The gas fraction profile decreases monotonically inwards, in\nperfect agreement with observations for the MW  and in fair\nagreement for the inner disk of M31. Only for the outer disk of M31\nthe model predicts slightly higher than observed gas fractions. We\nnotice that, in terms of normalized radius $r\/r_d$, the gas fraction\nprofiles of the two disks are very similar, which explains their\nsuccessful description by our unified model. Notice that, in terms\nof {\\it physical radius}, M31 has a smaller gas fraction than the MW\nat a given $r$, which explains the need for a higher SF efficiency\nin that case. The situation is less satisfactory in the outer disk\nof M31, where the gas fraction is overestimated by our model.\n\n4) Our model predicts correctly the present day SFR profile of MW,\nbut fails completely in the case of M31, and in particular in the\nouter disk of M31. As already noticed (Sec. 2.2) the observed SFR vs\ngas relationship in M31 cannot be fit by any form of the KS laws.\nOur result reflects just this impossibility. As already argued (last\nparagraph of Sec. 2.2), we believe that the observed SFR is affected\nby recent perturbations of the gaseous disk of M31, e.g. the\ncollision with a nearby galaxy suggested by Block et al. (2006).\n\n5) The resulting abundance gradients are compatible with\nobservations for both MW and M31. In fact, the predicted abundance\nprofile of MW is somewhat steeper than in the case of M31. This is\nnot a surprise since the two disks do not have the same scale. At\nthe same distance from the galactic center, they have different gas\n  amounts and SFR. As a result of its larger scalelength, the current\n  gas is more widely spread in M31 than in the Milky Way, with a\n  resulting gas fraction rising less steeply in Andromeda, and\n  correspondingly a flatter abundance gradient.  When we express the\n  model abundance gradient in terms of their scale length, we obtain\n  similar value for the two disks. In any case, taking into account\nall the uncertainties mentioned in Sec. 2.3, we consider the overall\nagreement as satisfactory. Further observations will hopefully\nestablish the true abundance profiles of MW and M31 with greater\naccuracy, perhaps pointing to some different prescriptions for our\nunified model.\n\n\n\\subsection{Infall and Star Formation History }\n\nIn Fig.~\\ref{Fig:InfallSFRtime}  we show the evolution of the total\namount of gas, stars, SFR and infall rate, and of the gas fraction\nfor the disks of MW and M31. Reasonable agreement with observations\nis obtained for all those quantities in the case of MW disk; this\nagreement results from the adopted normalization of the total disk\nmass and the adopted star formation efficiency. In the case of M31,\nthe model predicts current global SFR$\\sim$2.0 M$_{\\odot}$ $yr^{-1}$, which\nis substantially larger than  observational estimates (Williams\n2003a,b; Barmby et al. 2006). In view of the discussion in Sec. 2.2,\nwe do not consider this discrepancy as significant: recent star\nformation in M31 may have been considerably perturbed by external\neffects (i.e. collision with another galaxy), unaccounted for in our\nmodel. We also notice that the current infall rate is poorly\nconstrained in MW and virtually unconstrained in M31. In the case of\nM31, Thilker et al. (2004) found that there exists an extensive\npopulation of HI clouds in the outskirts of the galaxy. The values\ndisplayed in Fig. 6 are in the range of 0.2-2 M$_{\\odot}$ $yr^{-1}$, the\nformer being the typical value inferred from observations of\naccreting cold gas in disks (Sancisi et al. 2008) and the latter\nfrom a simple theoretical argument, namely that such values are\nrequired to maintain a quasi-constant SFR over disk history for\nMW-size disks.\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[height=10cm,width=9cm]{f6.eps}\\\\\n  \\caption{Time evolution of global gas, star, gas fraction, infall\n  rate and SFR in M31 and Milky Way disks. The disk parameters are\n  given in Table 2. And the coefficients $\\alpha$ of SFR in Table 2\n  for the Milky Way and M31 disks are $\\alpha_{MW} = 0.1$, $\\alpha_{M31} = 0.2$,\n  respectively. Infall time scale is $\\tau(r) = 2.5~r\/r_d$.\n  Bar in the right of each plot gives the observed estimations.\n  It can be seen that the model predicts two much present SFR\n  for M31 disk. }\n  \\label{Fig:InfallSFRtime}\n\\end{figure}\n\n\n\\subsection{Metallicity Distributions}\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[height=10cm,width=9cm]{f7.eps}\\\\\n  \\caption{The metallicity distribution functions of various regions\n  in the Milky Way and M31 disks. The observed data in the solar neighborhood\n  of the Milky Way disk come from Holmberg et al. (2007) and Chang et al.\n  (1999), and the data of M31 come from Worthey et al. (2005).\n  The model predictions are plotted as smooth curves,\n  after convolution with a Gaussian error function with $\\sigma$ = 0.10 dex\n  (full lines) and 0.3dex (dashed lines). For M31, we construct\n  metallicity distributions for the age-range of RGB stars observed by\n  Worthey et al.(2005) (stellar ages between 6 and 12 Gyr). }\n  \\label{Fig:MDFs}\n\\end{figure}\n\nIn this subsection, we compare the model metallicity distribution\nfunctions (MDFs) with currently available observations in various\nregions of the Milky Way and M31 disks. The model results correspond\nto main sequence stars with lifetimes $\\tau>$10 Gyr, and they have\nbeen convolved with Gaussian error functions with $\\sigma$ = 0.1 dex\n(Fig.~\\ref{Fig:MDFs}).\n\nFor the Milky Way disk, the observed data in the solar neighborhood\nare from the GK survey (Holmberg et al. 2007, who revised estimates\nin Nordstr\\\"{o}m et al. (2004)), which includes ages, metallicity\nand kinematic properties for about 14000 F and G dwarfs. MDFs in\nother regions are not available at present, but future surveys e.g.\nSDSS\/SEGUE (Ivezi\\'{c} et al. 2008) and China's LAMOST project (Zhao\net al. 2006), are expected to provide information on those regions\nas well. As expected (from the adopted infall rate)  our model fits\nrather well, albeit not perfectly, the local MDF. Notice that we\ncompare to the data of Holmberg et al. (2007) corrected for the\nscale height of sellar populations (dashed curve in their Fig. 22,\nright panel): indeed, our results concern the full extent of the\nso-called ``solar cylinder'' at 8 kpc from the Galactic center,\nwhile local surveys are complete only within a limited volume\ncentered on the Sun. However, the corrections in Holmberg et al.\n(2007) are made after some assumptions are made about the star\nformation history of the local disk, which is not necessarily the\nsame as the SF history in our model. Thus, it should not be\nsurprising that the fit is not perfect.\n\nIn the case of M31, data are available for the MDF in various places\nalong its disk (Bellazzini et al. 2003; Worthey et al. 2005; Chapman\net al. 2006). Right panel histograms are results from Worthey et al.\n(2005), who observed 11 regions from the inner regions to the outer\ndisk along the major axis of M31. The median abundances in each\nobserved field increase steadily from the inner to the outer disk.\nThe mean stellar metallicity is $\\sim$0.2 dex lower than the\ngas-phase abundance (Fig.~\\ref{Fig:OxygenGradient}) in the same\nlocation.\n\nIn the right panels of Fig.~\\ref{Fig:MDFs}, we present also the\nmodel predictions for the MDFs in the same  radial positions as the\ndata available for M31. Our results are in broad agreement with\nobservations and they reproduce the decrease of mean stellar\nmetallicity with radius, as a consequence of the star formation and\ninfall schemes adopted for the disk. We conclude then that, to a\nfirst approximation, M31 evolved inside-out, as expected for a\nnormal spiral. Despite this, rather satisfactory agreement, between\nthe data and our model, we would like to emphasize the need for more\ndata on the MDFs of M31 as a function of radial position in order to\nfurther constrain the evolution of its disk.\n\nIt should be noted that when we plot the model predicted MDFs for\nM31 disk, we have assumed an error about 0.1dex based on the Worthey\net al. (2005). This assumption is self-consistent with observations\nsince we have used the data from Worthey et al. (2005). But this\nadopted error on the photometric metallicity may be too small as we\nknow that even for the situation of the halo, where the age spread\nshould be smaller than in the disk (and associated uncertainties\nlower) a comparison of spectroscopic and photometric metallicities\nfor RGB stars in M31 should scatter about $\\pm$0.3 dex (see Kalirai\net al. 2008). Therefore, we also plot the model MDFs for M31 disk\nwith photometric error of 0.3dex by dashed lines in Fig. 7. As\nexpected, the model predicted MDFs are wider than the observed\ndistributions in this case. But the peak position is roughly the\nsame.\n\nWe emphasis that while calculating the models, the results are for\ndisk evolution with full star formation history, that is, includes\nall stars in the disk with all ages. Worthey et al.\\,(2005) did not\ndiscuss the age spread in details, however they claimed that the age\nspread for their RGB stars is about 6-12Gyr (section 2, last\nparagraph in Worthey et al.). Therefore, we construct the\nmetallicity distributions for M31 disk with the age-range of RGB\nstars observed by Worthey et al. (stellar ages between 6 and 12 Gyr)\nand compare them to their observations in Fig. 7.\n\nOn the other hand, Koch et al. (2005) found that in the Carina dSph,\nthere is a larger age spread ( from 2Gyr to more than 11Gyr ), while\nits color-magnitude diagram shows a narrow RGB distribution. The\nlarge age spread means that Carina dSph must have undergone various\nepisodes of star formation process. This is different from the\nsmaller age spread reported by Worthey et al. (2005). We think this\nuncertainty calls for a more work on the observational side about\nthe metallicity distribution of M31 disk.\n\n\n\\section{Discussion}\n\nEarly works on simultaneous modeling of MW and M31 (Diaz \\& Tosi\n1984) found some similarities between the evolutionary properties of\nthe two disks, but they were performed at an epoch where scarce\nobservational data provided little constraint to the models (for\ninstance, Diaz \\& Tosi 1984 compared their model to M31 data\navailable only in the 5-11 kpc region). A more detailed comparison\nto observations is made in Molla et al. (1996), who use a\nmulti-parameter model and reproduce successfully several features of\nthe M31 disk.\n\nThe recent work of Renda et al. (2005) focuses on MW and M31,\nbenefits from a larger data set and presents some similarities to\nour work. The disks are constructed inside-out by slow infall and\nthe adopted SFR is $\\Psi = \\alpha \\Sigma_{Gas}^2\/r$, i.e. with an\nexponent $n$=2 instead of 1.5 in our case. By adopting exactly the\nsame SFR law for MW and M31, Renda et al. (2005) find that the\ngaseous profile is over predicted in M31 (their model M31a), hence\nthe need to increase their SF efficiency $\\alpha$ by a factor of 2\nin order to improve the fit to the data (their model M31b). Had they\nnoticed the lower gas fraction in M31, they would have anticipated\nthe problem (see our discussion in Sec. 3.3).  Our  models agree\nboth in the conclusion for a higher SF efficiency in M31 compared to\nthe MW as well as on the resulting abundance gradients (smaller in\nthe case of M31), when radius in all radially dependent terms is\nexpressed in e.g. kpc. For some unclear reason, our model fits\nbetter the gaseous profile of M31 (perhaps, because of our smaller\nexponent $n$=1.5 in the adopted SFR). Finally, both our model and\ntheirs fail in the outer disk of M31. It is hard to push the\ncomparison further, since Renda et al. (2005) do not provide SFR and\nstellar or  gas fraction profiles. The latter are in fact, mandatory\nin any work on chemical evolution, since they constrain, more than\nanything else the combined history of star formation and infall.\n\nDespite their simplicity (independently evolving rings, no\ncosmological framework), models such as the one presented here can\nprovide some interesting physical insights to the evolution of MW\nand M31, based both on their successes and their failures. The\nsuccess in reproducing simultaneously the profiles of gas, SFR and\nmetallicity in the MW, as well as its global properties (gas\nfraction, total SFR and colours, the latter being discussed in\nBoissier \\& Prantzos (1999)) implies that the overall history of the\nMilky Way cannot have been very different from the one found here,\ni.e. a slow, inside-out disk formation. However, similar solutions\nmay, perhaps, be obtained by some other combinations of SFR and\ninfall rate, i.e. the problem may well be degenerate, thus no firm\nconclusions can be drawn on each one of those two key ingredients.\n\nIn the case of M31, it is clear that a higher star formation\nefficiency is required, as deduced from its gas fraction, smaller by\na factor of $\\sim$2 than in the MW. This was already found by Renda\net al. (2005), while Hammer et al. (2007) went one step further, to\nsuggest that the MW is a particularly ``quiescent'' disk galaxy (for\nits mass) and M31 may be closer to an average large spiral. This\n``quiescence'' of the MW may be due to its relative isolation, while\nM31 may have undergone a larger number of (and\/or more important)\ninteractions with neighboring  galaxies. Such a picture is in line\nwith the finding of Block et al. (2006), namely that M31 has\nundergone a major interaction about 200 Myr ago; no such interaction\nappears to have occurred in the case of the MW over the last\nbillions of years.\n\nOur formalism allows us to describe in a unified framework the\nproperties of both the MW and M31, by using the same expression for\nthe radial dependence of the SFR in both cases. Such a description\nis demanded by the similarity in the radial profiles of those two\ndisks, when they are expressed in terms of their respective scale\nlengths (Sec. 2.4). However, it is not clear whether the higher SF\nefficiency of M31 is due to an external factor (i.e. more\nfrequent\/important interactions of that galaxy) or to an internal\none (e.g. its mass, as argued in Boissier \\& Prantzos 2000).\nApplying this formalism to other disk galaxies for which large data\nbasis are available (work in progress) will help to clarify the\nsituation.\n\nOn the other hand, the failure of both this work and Renda et al\n(2005)  to reproduce satisfactorily the gaseous profile of M31, and\nthe fact that we over predict the global SFR of M31, as well as its\nouter SFR profile, suggests that those properties are considerably\naffected by recent interactions. Thus, they cannot be predicted by\nsuch simple models (unless if more parameters are introduced). If\nthis is true, and if M31 is really closer to a typical disk (as\nHammer et al. 2007 suggest), then the cosmological framework will be\nmandatory for the description of galactic disks; simple models, like\nthis one, will be able to describe successfully only the most\nquiescent disks, such as the MW.\n\n\\section{Summary}\n\nIn this work, we study the chemical evolution of the disk of M31,\nusing a  model already applied to the study of the  Milky Way\n(Boissier \\& Prantzos 1999, Hou et al. 2000). We use an extensive\ndata set of M31 properties, including radial profiles of gas surface\ndensity, gas fraction, star formation rate, oxygen abundances, as\nwell as metallicity distribution functions at different regions of\nthe disk.  In particular, the star formation profile of M31  is from\nrecent UV data of GALEX (Boissier et al. 2007). Our main purpose is\nto see whether a simple chemical evolution model can successfully\ndescribe the radial and global properties of both disks.\n\nWe first summarize and compare the  observational data (Sec. 2) for\nthe two galaxies. The disk of M31 is about 2.4 times larger  and 2\ntimes more massive than the Milky Way disk, while its gas fraction\nis approximately half of the one of the MW. All other things being\nequal,this implies a higher average star formation efficiency for\nM31. We find that the SF radial profile of MW is well described by\n``standard'' SF laws, but not the one of M31 (Sec. 2.2). We\nattribute the latter to a recent major perturbation of M31 by a\nnearby galaxy, in line with the findings of Block et al. (2006). We\nconclude that our model (which adopts such ``standard'' SF laws)\nwill fail to reproduce the observed SF profile of M31, and perhaps\nalso the gas profile.\n\nWe find that, when radii are expressed in terms of the corresponding\nscale lengths, the two disks display very interesting similarities\nin their radial profiles (Sec. 2.4). This concerns, in particular,\nthe gas fraction, the profile of which is quasi-identical inside the\ninnermost two scale lengths (Fig.~\\ref{Fig:scaleobs}). Also, the\nscaled abundance gradients of the two disks are quite similar {\\it\nif} we adopt for the MW the lower range of reported values (e.g.\nDeharveng et al. 2000; Daflon \\& Cunha 2004; Andrievsky et al. 2004;\nChen et al. 2008). Such a similarity was found in a sample of\nexternal spirals and successfully described by the models of\nBoissier \\& Prantzos (2001), which cover a much larger range of\ngalaxian properties than the two disks studied here. We stress,\nhowever, that the status of the MW abundance gradient, especially in\nthe outer part, is still very controversial: observations show that\nit may not be described by a simple exponential (see e.g. Yong et\nal. 2005 and Carraro et al. 2007 for open cluster abundances; and\nAndrievsky et al. 2004 and Lemasle et al. 2008 for Cepheids). {\\it\nAssuming that} the scaled abundance gradients are similar in MW and\nM31, we seek then a description of the radial properties of the two\ndisks within the framework of a single model, which we present in\nSec. 3.\n\nDetailed calculations show that our unified model describes fairly\nwell all the main properties of the MW disk and most of those of\nM31, provided its SF efficiency is adjusted to be twice as large in\nthe latter case (as anticipated from the lower gas fraction of M31).\nThe radial profiles of both MW and M31 are well described, albeit\nless successfully in the case of M31. In particular, the model fails\nto match the present SFR in M31, producing too large values in the\nouter disk and globally. We attribute this failure to the fact that\nM31 has been perturbed recently by a major encounter, as already\nanticipated by the fact that the observed SFR profile of M31 does\nnot seem to follow any form of the Kennicutt-Schmidt star formation\nlaw. On the other hand, the stellar metallicity distributions\nmeasured along the disk of M31 reflect the integrated star formation\nduring the whole disk history and should not be affected by recent\nevents. Our model, where the bulk of Fe originates in SNIa,\nreproduces rather well those distributions, from 6 to 21 kpc.\n\nThe unified description that we propose here for MW and M31, by\nexpressing their radial profiles in terms of the ``natural units''\n(the corresponding disk scale lengths), offers valuable insights\ninto the evolution of those two disk galaxies and this may also be\nthe case for other spirals as well (work is in progress).\n\n\\acknowledgements This work is supported by the National Science\nFoundation of China No.10573028, the Key Project No.10833005, the\nGroup Innovation Project No.10821302, and by 973 program No.\n2007CB815402.\n\n\\defCJAA{CJAA}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nBelyi's theorem says that any non-singular algebraic curve over a number \nfield is a covering of the complex projective line $P^1(\\mathbf{C})$ ramified at \nthe three points $\\{0,1,\\infty\\}$   [Belyi 1979] \\cite[Theorem 4]{Bel1}. \nThe aim of our note is an analog of Belyi's theorem for the algebraic \nsurfaces based on an  approach of  \\cite{Nik1}. \nNamely, we associate to the countable division ring  an   avatar,\nsee definition \\ref{def1.1}.  \nIf the ring  is commutative (non-commutative, resp.),  then\nits avatar is an algebraic curve (an algebraic surface, resp.) defined over the field \n$\\mathbf{C}$.   \nFor example, an avatar of the ring of rationals   (rational quaternions, resp.) \nis the complex projective line  $P^1(\\mathbf{C})$  (complex projective plane  $ P^2(\\mathbf{C}) $, resp.) \n Belyi's theorem follows from an embedding of rationals into the field of algebraic numbers\n \\cite[Section 4]{Nik1}   and its analog for surfaces from an embedding of rational quaternions into a quaternion algebra,\n see Section 4 in below.  An extension of Belyi's theorem to  complex surfaces was studied independently in [Gonz\\'alez-Diez 2008]\n\\cite{Gon1}.\n\n\n \n \n \n Recall that  an analogy between  the number fields \n  and  complex algebraic curves is well  known  [Eisenbud \\& Harris 1999] \\cite[p. 83]{EH}.\n The  Grothendieck's  theory of schemes  cannot explain  this relation\n [Manin 2006] \\cite[Section 2.2]{Man1}. The problem can be solved in terms of the \n$C^*$-algebras  \\cite{Nik1}. To explain the solution, let $R$ be a (discrete) associative ring,\nlet $M_2(R)$ be the matrix ring over $R$\nand  let \n\\begin{equation}\n\\rho: M_2(R)\\to \\mathscr{B}(\\mathcal{H})\n\\end{equation}\nbe a self-adjoint representation of $M_2(R)$  by the bounded linear operators \n on a Hilbert space $\\mathcal{H}$.  Taking the norm-closure of $\\rho(M_2(R))$ \n in the strong operator topology,  one gets a $C^*$-algebra  $\\mathscr{A}_R$. \n Likewise, let $B(V, \\mathcal{L}, \\sigma)$ be the twisted \nhomogeneous coordinate ring of a complex projective variety $V$, where $\\mathcal{L}$ is an invertible sheaf and $\\sigma$\nis an automorphism of $V$ [Stafford \\& van ~den ~Bergh 2001]  \\cite[p. 173]{StaVdb1}. \nRecall that the  Serre $C^*$-algebra, $\\mathscr{A}_V$,   is the norm closure of a\nself-adjoint representation of the ring  $B(V, \\mathcal{L}, \\sigma)$\nin $\\mathscr{B}(\\mathcal{H})$  \\cite[Section 5.3.1] {N}. \nFinally, let $\\mathscr{K}$ be the $C^*$-algebra of all compact operators on the Hilbert space \n $\\mathcal{H}$. \n We refer the reader to \\cite{Nik1} for the motivation and examples illustrating the following\n definition.  \n\n\\begin{definition}\\label{def1.1}\n The complex projective variety $V$ is called an {\\it avatar}\n of the ring $R$,   if there exists  a $C^*$-algebra homomorphism: \n\n \\begin{equation}\\label{eq1.2}\n  \\mathscr{A}_V\\to\\mathscr{A}_R\\otimes\\mathscr{K}.\n \\end{equation}\n\n\\end{definition}\n\n\n\\begin{theorem}\\label{thm1.2}\n{\\bf (\\cite{Nik1})}\nLet $\\overline{\\mathbf{Q}}$ be the algebraic closure of the field  $\\mathbf{Q}$.\nThen:\n\n\n\\medskip\n(i) $P^1(\\mathbf{C})$ is an avatar of the field $\\mathbf{Q}$;\n\n\n\\smallskip\n(ii) the avatar of a field $K\\subset \\overline{\\mathbf{Q}}$\nis a non-singular algebraic curve $C(\\overline{\\mathbf{Q}})$; \n\n\n\\smallskip\n(iii)  the field extension $\\mathbf{Q}\\subset K$ defines a covering\n$C(\\overline{\\mathbf{Q}})\\to P^1(\\mathbf{C})$ ramified at the  points $\\{0,1, \\infty\\}$. \n\\end{theorem}\n \n \\medskip\n\n\\begin{remark}\nBelyi's theorem follows from item (iii) of theorem \\ref{thm1.2}. Roughly speaking, \nit can be shown that each non-singular algebraic curve $C(\\overline{\\mathbf{Q}})$\nis  the avatar of a field $K\\subset \\overline{\\mathbf{Q}}$. \nWe refer the reader to \\cite[Section 4]{Nik1} for the details.  \n\\end{remark}\n\n\n\\medskip\nTo formalize our results, we use the following notation.\nDenote by $\\left({a,b\\over F}\\right)$ a quaternion algebra, i.e \nthe algebra over a field $F$,    such that $\\{1,i.j, ij\\}$ is a basis for\nthe algebra and \n  \\begin{equation}\n  i^2=a, \\quad j^2=b, \\quad ji=-ij\n  \\end{equation}\n \n   for some $a,b\\in F^{\\times}$ [Voight 2021] \\cite[Section 2.2]{V}. \n   The quaternion algebra with $a=b=-1$ will be written as $\\mathbb{H}(F)$. \n  The algebra  $\\mathbb{H}(\\mathbf{R})$ corresponds   to the Hamilton  quaternions and \n   the algebra  $\\mathbb{H}(\\mathbf{Q})$  corresponds to the rational quaternions.  \n  Our main result is a generalization of theorem \\ref{thm1.2} to the division rings $\\left({a,b\\over K}\\right)$,\n where $K\\subset\\overline{\\mathbf{Q}}$. \n  \n\\begin{theorem}\\label{thm1.4}\nLet $\\left({a,b\\over K}\\right)$ be a division ring, such that $K\\subset\\overline{\\mathbf{Q}}$. \n Then:\n\n\n\\medskip\n(i) $P^2(\\mathbf{C}) $ is an avatar of the division ring $\\mathbb{H}(\\mathbf{Q})$;\n\n\n\\smallskip\n(ii)  the avatar of a division ring  $\\left({a,b\\over K}\\right)$ is a non-singular\nalgebraic surface  $S(\\overline{\\mathbf{Q}})$; \n\n\n\n\n\\smallskip\n(iii)  the field extension $\\mathbf{Q}\\subset K$ defines a covering\n$S(\\overline{\\mathbf{Q}})\\to P^2(\\mathbf{C})$ ramified at \n  three  knotted two-dimensional spheres\n$P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})$.\n\\end{theorem}\n\n\n\\medskip\n\\begin{remark}\nA relation between algebraic surfaces and division rings follows from \\cite[Section 7.2]{N}. \nIndeed, each non-singular algebraic surface is a smooth 4-dimensional manifold and\nthe arithmetic topology relates such manifolds to the cyclic division algebras\n\\cite[Theorem 7.2.1]{N}.  \n\\end{remark}\n\\begin{remark}\nThe knotting type of $P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})$  depends  \non the arithmetic of the field $K$ \nand extends the Grothendieck's theory of {\\it dessin d'enfant} to the case of algebraic surfaces.   \n\\end{remark}\n\n\n\n\\medskip\nThe paper is organized as follows.  A brief review of the preliminary facts is \ngiven in Section 2. Theorem \\ref{thm1.4} is proved in Section 3. \nAn analog of Belyi's theorem for the algebraic surfaces is  proved\nin Section 4. \n\n\n\n\n\n\n\\section{Preliminaries}\nThis section is a brief review of the quaternion and Serre $C^*$-algebras.\nWe refer the reader to [Voight 2021] \\cite[Section 2.2]{V} and \\cite[Section 5.3.1]{N} \nfor a detailed account. \n\\subsection{Quaternion algebras}\n\\begin{definition}\nThe algebra  $\\left({a,b\\over F}\\right)$ over a field $F$ is called a quaternion algebra\n  if there exists $i,j\\in \\left({a,b\\over F}\\right)$ such that $\\{1,i.j, ij\\}$ is a basis for $\\left({a,b\\over F}\\right)$ and \n  \\begin{equation}\n  i^2=a, \\quad j^2=b, \\quad ji=-ij\n  \\end{equation}\n \n   for some $a,b\\in F^{\\times}$. \n\\end{definition}\n\\begin{example}\nIf $F\\cong\\mathbf{R}$ and $a=b=-1$, then the quaternion algebra \n$\\left({-1,-1\\over \\mathbf{R}}\\right)$ consists  of the Hamilton quaternions $\\mathbb{H}(\\mathbf{R})$;\nhence the notation.  \nIf $F\\cong\\mathbf{Q}$, then the quaternion algebra \n$\\left({-1,-1\\over \\mathbf{Q}}\\right)$ consists  of the rational quaternions $\\mathbb{H}(\\mathbf{Q})$. \n\\end{example}\n\n\\medskip\nA $\\ast$-involution on $\\left({a,b\\over F}\\right)$ is defined by the formula\n$(1,i,j,k)\\mapsto (1, -i,-j,-k)$.  The norm $N(u)=uu^*$ of an element $u=x_0+xi+yj+ zk\\in\\left({a,b\\over F}\\right)$ \nis a quadratic form: \n\\begin{equation}\\label{eq2.2} \nN(x_0+xi+yj+ zk)=x_0^2-ax^2-by^2+abz^2. \n\\end{equation}\n\n\n\\medskip\nSince $N(1)=1$ and $N(uv)=N(u)N(v)$, one concludes that the  $\\left({a,b\\over F}\\right)$ is a division algebra if \nand only if   quadratic form (\\ref{eq2.2}) vanishes only at the zero element $u=0$. \nThus  (\\ref{eq2.2}) must be a positive form \nfor all $x_0, x, y, z\\in F$. \n\n\\medskip\nIt is easy to see, that the form   (\\ref{eq2.2}) admits non-trivial zeroes if and only if there are \nsuch zeroes for the ternary quadratic form: \n\\begin{equation}\\label{eq2.3} \n\\mathcal{Q}(x,y,z)=-ax^2-by^2+abz^2. \n\\end{equation}\n\n\n\\medskip\nThe substitution $a'={1\\over b}, ~b'={1\\over a}$ maps the zeroes of (\\ref{eq2.3})  to the $F$-points of \na conic surface given by the equation: \n\\begin{equation}\\label{eq2.4} \nz^2=ax^2+by^2. \n\\end{equation}\n\n\n\\medskip\nThe following classification of the quaternion algebras is well known. \n\\begin{theorem}\\label{thm2.3}\n{\\bf (\\cite[Theorem 5.1.1]{V})}\nThe formula\n\\begin{equation}\n\\left({a,b\\over F}\\right)\\mapsto \\mathcal{Q}(x,y,z)\n \\end{equation}\n\nmaps  isomorphic quaternion algebras to the similar ternary quadratic forms. \nEquivalently, the quaternion algebras are classified by the isomorphism classes\nof the conic surfaces (\\ref{eq2.4}). \n\\end{theorem}\n\n\n\n\n\\subsection{Serre $C^*$-algebras}\nLet $V$ be an $n$-dimensional complex  projective variety endowed with an automorphism $\\sigma:V\\to V$  \n and denote by $B(V, \\mathcal{L}, \\sigma)$   its  twisted homogeneous coordinate ring  [Stafford \\& van ~den ~Bergh 2001]  \\cite{StaVdb1}.\nLet $R$ be a commutative  graded ring,  such that $V=Proj~(R)$.  Denote by $R[t,t^{-1}; \\sigma]$\nthe ring of skew Laurent polynomials defined by the commutation relation\n$b^{\\sigma}t=tb$  for all $b\\in R$, where $b^{\\sigma}$ is the image of  $\\left({a,b\\over \\mathbf{Q}}\\right)$ under automorphism \n$\\sigma$.  It is known, that $R[t,t^{-1}; \\sigma]\\cong B(V, \\mathcal{L}, \\sigma)$.\n\n\n\nLet $\\mathcal{H}$ be a Hilbert space and   $\\mathscr{B}(\\mathcal{H})$ the algebra of \nall  bounded linear  operators on  $\\mathcal{H}$.\nFor a  ring of skew Laurent polynomials $R[t, t^{-1};  \\sigma]$,  \n consider a homomorphism: \n\\begin{equation}\\label{eq2.1}\n\\rho: R[t, t^{-1};  \\sigma]\\longrightarrow \\mathscr{B}(\\mathcal{H}). \n\\end{equation}\nRecall  that  $\\mathscr{B}(\\mathcal{H})$ is endowed  with a $\\ast$-involution;\nthe involution comes from the scalar product on the Hilbert space $\\mathcal{H}$. \nWe shall call representation (\\ref{eq2.1})  $\\ast$-coherent,   if\n(i)  $\\rho(t)$ and $\\rho(t^{-1})$ are unitary operators,  such that\n$\\rho^*(t)=\\rho(t^{-1})$ and \n(ii) for all $b\\in R$ it holds $(\\rho^*(b))^{\\sigma(\\rho)}=\\rho^*(b^{\\sigma})$, \nwhere $\\sigma(\\rho)$ is an automorphism of  $\\rho(R)$  induced by $\\sigma$. \nWhenever  $B=R[t, t^{-1};  \\sigma]$  admits a $\\ast$-coherent representation,\n$\\rho(B)$ is a $\\ast$-algebra.  The norm closure of  $\\rho(B)$  is   a   $C^*$-algebra\n   denoted  by $\\mathscr{A}_V$.  We  refer to  $\\mathscr{A}_V$  as   the   Serre $C^*$-algebra\n of the complex projective variety $V$. \n\n\n\n\n\\section{Proof}\n\\subsection{Part I} \nLet us prove item (i) of theorem \\ref{thm1.4}. \nDenote by $\\mathcal{O}$ the ring of integers of the quaternion algebra  $\\left({a,b\\over \\mathbf{Q}}\\right)$. \nConsider a polynomial ring:\n\\begin{equation}\\label{eq3.1}\n\\mathfrak{R}=\\mathbf{Z}[x,y,z]\/[\\mathcal{Q}],\n\\end{equation}\nwhere $[\\mathcal{Q}]$ is an ideal generated by the quadratic form $\\mathcal{Q}(x,y,z)$ given by formula \n(\\ref{eq2.3}).  The proof of item (i) is based on the following lemma.\n\n\n\n\\begin{lemma}\\label{lm3.1}\nThe  matrix rings $M_2(\\mathcal{O})$ and  $M_2(\\mathfrak{R})$  are isomorphic. \n\\end{lemma}\n\\begin{proof}\nRoughly speaking, lemma \\ref{lm3.1}  follows from the classification of the quaternion algebras given by  Theorem \\ref{thm2.3}. \nNamely, the quaternion algebras are classified by the conic surfaces defined by (\\ref{eq2.4}) or, equivalently,  by their coordinate rings  \n $\\mathfrak{R}$. The same is true for the corresponding matrix rings.  Let us pass do a detailed argument. \n\n\n\\begin{figure}[h]\n\\begin{picture}(300,110)(-70,0)\n\\put(20,70){\\vector(0,-1){35}}\n\\put(130,70){\\vector(0,-1){35}}\n\\put(45,23){\\vector(1,0){60}}\n\\put(45,83){\\vector(1,0){60}}\n\\put(0,20){$\\mathcal{Q}(x,y,z)$}\n\\put(55,30){${\\sf similarity}$}\n\\put(45,90){${\\sf isomorphism}$}\n\\put(117,20){$\\mathcal{Q}'(x,y,z)$}\n\\put(7,80){$\\left({a,b\\over \\mathbf{Q}}\\right)$}\n\\put(115,80){$\\left({a',b'\\over \\mathbf{Q}}\\right)$}\n\\put(0,50){$F$}\n\\put(140,50){$F$}\n\\end{picture}\n\\caption\n}\n\\end{figure}\n\n\n\n\n\\medskip\n(i) One can recast theorem \\ref{thm2.3} as a commutative diagram in Figure 1.\nWe wish  to upgrade the map $F$  to a ring isomorphism. \nThe simplest non-commutative ring attached  to the ternary quadratic form $\\mathcal{Q}(x,y,z)$\nis the matrix ring  $M_2(\\mathfrak{R})$ over the ring $\\mathfrak{R}$  defined by (\\ref{eq3.1}).   \nOn the other hand, the quaternion algebra  $\\left({a,b\\over \\mathbf{Q}}\\right)$ is simple and, therefore, cannot be isomorphic\nto  $M_2(\\mathfrak{R})$. However, the ring of integers $\\mathcal{O}$ of the algebra $\\left({a,b\\over \\mathbf{Q}}\\right)$ admits non-trivial \ntwo-sided ideals.  Here again, the ring $\\mathcal{O}$  cannot be isomorphic to  $M_2(\\mathfrak{R})$,\nsince $\\mathcal{O}$ is a domain while  $M_2(\\mathfrak{R})$ admits the zero divisors, e.g. the  projections. \nThus we must consider the matrix ring $M_2(\\mathcal{O})$ as a candidate for the required \nring isomorphism.  Let us show that  $M_2(\\mathcal{O})\\cong M_2(\\mathfrak{R})$  whenever $ \\mathcal{Q}(x,y,z)=F\\left({a,b\\over \\mathbf{Q}}\\right).$ \n\n\n\n\\bigskip\n(ii)  Indeed, it follows from  [Voight 2021] \\cite[Corollary 5.5.2]{V}  that the similar  \nquadratic forms $\\mathcal{Q}(x,y,z)$ and $ \\mathcal{Q}'(x,y,z)$ correspond to \n the isomorphic   conic surfaces  (\\ref{eq2.4}) and, therefore, to \nthe isomorphic rings  $\\mathfrak{R}$ and   $\\mathfrak{R}'$. \nSince $\\mathcal{O}\\subset \\left({a,b\\over \\mathbf{Q}}\\right)$, we conclude that an isomorphism between  $\\left({a,b\\over \\mathbf{Q}}\\right)$ and \n$\\left({a',b'\\over \\mathbf{Q}}\\right)$\ninduces an isomorphism between  $\\mathcal{O}$ and  $\\mathcal{O}'$.  In other words,  the\ndiagram in Figure 2 must be  commutative. \n\n\n\\begin{figure}[h]\n\\begin{picture}(300,110)(-70,0)\n\\put(20,70){\\vector(0,-1){35}}\n\\put(130,70){\\vector(0,-1){35}}\n\\put(45,23){\\vector(1,0){60}}\n\\put(45,83){\\vector(1,0){60}}\n\\put(15,20){$\\mathfrak{R}$}\n\\put(45,30){${\\sf isomorphism}$}\n\\put(45,90){${\\sf isomorphism}$}\n\\put(125,20){$\\mathfrak{R}'$}\n\\put(17,80){$\\mathcal{O}$}\n\\put(125,80){$\\mathcal{O}'$}\n\\put(0,50){$F$}\n\\put(140,50){$F$}\n\\end{picture}\n\\caption\n}\n\\end{figure}\n\n\n\n\n\\bigskip\n(iii) The tensor product  with the matrix ring $M_2$ in Figure 2 gives us\n $\\mathcal{O}\\otimes M_2\\cong M_2(\\mathcal{O})$ and  $\\mathfrak{R}\\otimes M_2\\cong M_2(\\mathfrak{R})$. \nThe functor $F$ extends to the tensor product and  one gets  a \ncommutative diagram in Figure 3. \n\n\n\n\n\\bigskip\n(iv) It remains to show that the map $F$ defines a ring isomorphism\n  $M_2(\\mathcal{O})\\cong M_2(\\mathfrak{R})$. \n  Indeed,  let $x_0\\in M_2(\\mathcal{O})$. The left multiplication\n  $y\\mapsto x_0y$ (addition $y\\mapsto x_0+y$, resp.) \n  defines a morphism $\\phi_{x_0}^{mult}$ ($\\phi_{x_0}^{add}$, resp.)\n  of the ring   $M_2(\\mathcal{O})$.  Since the map $F$ preserves morphisms,\n  we conclude that $F(\\phi_{x_0}^{mult})$  ($F(\\phi_{x_0}^{add})$, resp.) \n  is a morphism of the ring  $M_2(\\mathfrak{R})$. \n  Namely, the morphism $F(\\phi_{x_0}^{mult})$   ($F(\\phi_{x_0}^{add})$, resp.) \n  acts  by the formula $F(y)\\mapsto F(x_0)F(y)$ \n  ($F(y)\\mapsto F(x_0)+F(y)$, resp.) \n  Thus one gets $F(x_0y)=F(x_0)F(y)$ and  $F(x_0+y)=F(x_0)+F(y)$\n  for all $x_0,y\\in M_2(\\mathcal{O})$. In other words, the map $F$ defines \n  an an isomorphism  between the rings \n  $M_2(\\mathcal{O})$ and $M_2(\\mathfrak{R})$. \n \n \n \\bigskip \n  Lemma \\ref{lm3.1} is proved. \n   \\end{proof}\n\n\n\n\n\\begin{lemma}\\label{lm3.2}\nConic surface  (\\ref{eq2.4}) is an avatar of the quaternion algebra  $\\left({a,b\\over \\mathbf{Q}}\\right)$. \n\\end{lemma}\n\\begin{proof}\n(i) \nAccording to definition \\ref{def1.1},\nwe  must consider a self-adjoint representation\n$\\rho$ of the rings $M_2(\\mathfrak{R})\\cong M_2(\\mathcal{O})$\nby the bounded linear operators\non a Hilbert space $\\mathcal{H}$.  We take the norm-closure \nof $\\rho$ in the strong operator topology.  Lemma \\ref{lm3.1}\nimplies the following isomorphism of the $C^*$-algebras:\n\\begin{equation}\\label{eq3.3}\n\\rho(M_2(\\mathfrak{R}))\\otimes\\mathscr{K}\\cong \\rho(M_2(\\mathcal{O}))\\otimes\\mathscr{K}. \n\\end{equation}\n\n\n\n\\bigskip\n(ii) On the other hand, from the definition of the Serre $C^*$-algebra $\\mathcal{A}_V$ one gets\nthe following isomorphisms: \n\\begin{equation}\\label{eq3.4} \n\\begin{cases} \n\\rho(M_2(\\mathfrak{R}))\\cong \\mathcal{A}_{V(\\mathbf{Q})}&\\cr\n\\rho(M_2(\\mathfrak{R}))\\otimes\\mathscr{K}\\cong \\mathcal{A}_{V(\\mathbf{C})}, &\n\\end{cases}\n\\end{equation}\n where  $V(\\mathbf{Q})$ ($V(\\mathbf{C})$, resp.) is the conic surface (\\ref{eq2.4})\n over the field of rational numbers $\\mathbf{Q}$ (complex numbers $\\mathbf{C}$, resp.) \nThus the LHS of (\\ref{eq3.3}) is the Serre $C^*$-algebra of the  conic surface  (\\ref{eq2.4}). \n\n\n\n\n\\bigskip\n(iii) It is immediate that the $\\rho(M_2(\\mathcal{O}))$ at the RHS of  (\\ref{eq3.3}) is the $C^*$-algebra $\\mathscr{A}_R$ of \nthe  ring  $R\\cong \\left({a,b\\over \\mathbf{Q}}\\right)$.  \n\n\n\\bigskip\n(iv)  Using (ii) and (iii), we can write (\\ref{eq3.3}) in the form: \n\\begin{equation}\\label{eq3.5}\n\\mathcal{A}_{V(\\mathbf{C})} \\cong \\mathscr{A}_R\\otimes\\mathscr{K}. \n\\end{equation}\n\n\n\\bigskip\n(v) It remains to compare (\\ref{eq3.5}) and  the definition \\ref{def1.1},  \nwhere the connecting map in (\\ref{eq1.2}) is an isomorphism between the  $C^*$-algebras. \nWe conclude that the conic surface  (\\ref{eq2.4}) \nis an avatar of the quaternion algebra  $\\left({a,b\\over \\mathbf{Q}}\\right)$. \n\n\\bigskip\nLemma \\ref{lm3.2} is proved.\n\\end{proof}\n\n\n\n\n\\begin{figure}[h]\n\\begin{picture}(300,110)(-70,0)\n\\put(20,70){\\vector(0,-1){35}}\n\\put(130,70){\\vector(0,-1){35}}\n\\put(45,23){\\vector(1,0){60}}\n\\put(45,83){\\vector(1,0){60}}\n\\put(0,20){$M_2(\\mathfrak{R})$}\n\\put(45,30){${\\sf isomorphism}$}\n\\put(45,90){${\\sf isomorphism}$}\n\\put(115,20){$M_2(\\mathfrak{R}')$}\n\\put(0,80){$M_2(\\mathcal{O})$}\n\\put(117,80){$M_2(\\mathcal{O}')$}\n\\put(0,50){$F$}\n\\put(140,50){$F$}\n\\end{picture}\n\\caption\n}\n\\end{figure}\n\n\n\n\n\\begin{lemma}\\label{lm3.3}\nThe  (\\ref{eq2.4}) is a rational surface over the field $k\\cong\\mathbf{Q}(\\sqrt{-1}, \\sqrt{-a}, \\sqrt{-b})$. \nIn particular, the complex points of (\\ref{eq2.4}) define a simply connected 4-dimensional manifold. \n\\end{lemma}\n\\begin{proof}\n(i) Let us show that the conic  (\\ref{eq2.4}) is a rational surface over the field $\\mathbf{Q}(\\sqrt{-1}, \\sqrt{-a}, \\sqrt{-b})$. \nIndeed, the reader can verify that a parametrization  $(u,v)\\mapsto (x,y, z)$  of  (\\ref{eq2.4})\nis given by the formulas:\n\\begin{equation}\\label{eq3.6} \n\\begin{cases} \nx={u^2-v^2\\over\\sqrt{-a}}&\\cr\ny={2uv\\over\\sqrt{-b}}&\\cr\nz=\\sqrt{-1}(u^2+v^2) &\n\\end{cases}\n\\end{equation}\nWe conclude that  the conic  (\\ref{eq2.4}) is a rational surface $P^2(k)$ \ndefined over the field  $k\\cong \\mathbf{Q}(\\sqrt{-1}, \\sqrt{-a}, \\sqrt{-b})$. \n\n \n\n\\bigskip\n(ii) It is well known that the rational complex projective variety is  simply connected as a manifold. \nBy item (i) surface (\\ref{eq2.4}) is rational and therefore the underlying\n4-dimensional manifold is simply connected. \n\n\\bigskip\nLemma \\ref{lm3.3} is proved.\n\\end{proof}\n\n\n\n\\begin{remark}\nNotice that  $k\\ne \\mathbf{Q}$.  For otherwise \nthe ternary quadratic form  (\\ref{eq2.3}) admits (infinitely many) non-trivial  zeroes\nand  $\\left({a,b\\over \\mathbf{Q}}\\right)$ is no longer a division\nring, see Section 2.1. \n \\end{remark}\n\n\n\n\\begin{corollary}\\label{cor3.5}\n $P^2(\\mathbf{C}) $ is an avatar of the division ring $\\mathbb{H}(\\mathbf{Q})$.\n\\end{corollary}\n\\begin{proof}\nOur proof is based on the result  [Piergallini 1995] \\cite{Pie1} which\nsays that for each smooth 4-dimensional manifold $M^4$\n  there exists a transverse immersion $X\\hookrightarrow S^4$ of a 2-dimensional surface $X$\n  into the 4-dimensional sphere $S^4$,  such that $M^4$ is the 4-fold PL  cover of  $S^4$ branched at the points of $X$.  \n  We pass to a detailed argument.\n  \n  \n  \n  \\bigskip\n  (i)  Recall that if  $J$ is the complex conjugation,  then  $P^2(\\mathbf{C})\/J\\cong S^4$.  \n  Using Piergallini's Theorem,  we conclude that $J$  acts on the conic  surface  (\\ref{eq2.4})\n  so that it becomes a branched cover of  $P^2(\\mathbf{C})$. \n  In view of lemma \\ref{lm3.3},  the  conic surface is rational over the field \n $k\\cong \\mathbf{Q}(\\sqrt{-1}, \\sqrt{-a}, \\sqrt{-b})$. \nThus there exists a regular map:\n\\begin{equation}\nP^2(k)\\to \\mathbb{P}^2(k_0),\n\\end{equation}\nwhere $k_0\\subset k$ is the minimal non-trivial subfield of $k$. \n\n\n\\bigskip\n(ii) But the minimal non-trivial subfield of $k\\cong \\mathbf{Q}(\\sqrt{-1}, \\sqrt{-a}, \\sqrt{-b})$\nindependent of the parameters $a$ and  $b$ coincides with the field  $k_0\\cong \\mathbf{Q}(\\sqrt{-1})$. \nClearly, the  $k_0$ corresponds to the case $a=b=-1$.\nAll other possible combinations $a=\\pm 1, ~b=\\pm 1$ must be  \nexcluded since the ternary quadratic form (\\ref{eq2.3}) must be  positive-definite.\n\n\\bigskip\n(iii) It remains to notice that the quaternion algebra with  $a=b=-1$ corresponds to the \nrational quaternions $\\mathbb{H}(\\mathbf{Q})$.\n\n\\bigskip\nCorollary \\ref{cor3.5} is proved. \\end{proof}\n\n\\smallskip\nItem (i) of theorem \\ref{thm1.4} follows from corollary \\ref{cor3.5}. \n\n\n\n\n\\bigskip\n\\subsection{Part II} \nLet us prove item (ii) of theorem \\ref{thm1.4}.  We proceed with construction of  an algebraic \nsurface $S(\\overline{\\mathbf{Q}})$ from the quaternion algebra  $\\left({a,b\\over K}\\right)$\nwith $K\\subset \\overline{\\mathbf{Q}}$. \nFrom Part I  if  $K\\cong\\mathbf{Q}$,    then  $S(\\overline{\\mathbf{Q}})$ is given \nby the equation $\\mathcal{Q}(x,y,z)=0$,  where \n\\begin{equation}\\label{eq3.8}\n\\mathcal{Q}(x,y,z)=-ax^2-by^2+abz^2. \n\\end{equation}\n\n\n\n\\bigskip\n(i) Let $K\\subset \\overline{\\mathbf{Q}}$ be a number field and let $a,b\\in K$. Denote by $p\\in\\mathbf{Z}[u]$ and  $q\\in\\mathbf{Z}[w]$  the minimal \npolynomials of $a$ and  $b$, respectively.  We  set $a=h, ~b=w$ and  we write (\\ref{eq3.8}) in the form:\n\\begin{equation}\\label{eq3.9}\nF(x,y,z, u, w)=-ux^2-wy^2+uwz^2\\in \\mathbf{Z}[x,y,z,u,w].\n\\end{equation}\n\n\n\\bigskip\n(ii)  Solving the equation $F(x,y,z, u, w)=0$,   one gets:\n\\begin{equation}\\label{eq3.10}\nu={wy^2\\over wz^2-x^2}, \\quad w={ux^2\\over uz^2-y^2}. \n\\end{equation}\n\n\\bigskip\n(iii) The required algebraic surface  $S(\\overline{\\mathbf{Q}})$ is defined  as an intersection of two  hyper-surfaces \ngiven  the equations: \n\\begin{equation}\\label{eq3.11} \n\\begin{cases} \np\\left( {wy^2\\over wz^2-x^2}  \\right)=0,&\\cr\nq\\left( {ux^2\\over uz^2-y^2}\\right)=0. &\n\\end{cases}\n\\end{equation}\n\n\\bigskip\n(iv) The reader can verify that the surface (\\ref{eq3.11}) is defined over the field $\\overline{\\mathbf{Q}}$\nand coincides with the conic surface  (\\ref{eq3.8}) when $K\\cong \\mathbf{Q}$. \n\n\n\\bigskip\nItem (ii) of theorem \\ref{thm1.4} is proved. \n\n\n\n\n\n\\bigskip\n\\subsection{Part III} Let us prove item (iii) of theorem \\ref{thm1.4}.\nFor the sake of clarity, we  consider the case $K\\cong \\mathbf{Q}$ first, \nand then the general case $K\\subset\\overline{\\mathbf{Q}}$. \n\n\\bigskip\n\\subsubsection{Case $K\\cong \\mathbf{Q}$}\n(i)  Lemma \\ref{lm3.2} says that  conic surface  (\\ref{eq2.4}) is an avatar of the quaternion algebra  $\\left({a,b\\over \\mathbf{Q}}\\right)$. On the other hand, it is known that (\\ref{eq2.4}) \nis a rational surface $P^2(k)$ over the number field $k\\cong\\mathbf{Q}(\\sqrt{-1}, \\sqrt{-a}, \\sqrt{-b})$, see lemma \\ref{lm3.3}.  \n\n\n\\bigskip\n(ii)  Let $k_1\\cong\\mathbf{Q}(\\sqrt{-1}), ~k_2\\cong\\mathbf{Q}(\\sqrt{-a})$ and\n$k_3\\cong\\mathbf{Q}(\\sqrt{-b})$. \nConsider  a regular map $P^2(k_i)\\to P^2(\\mathbf{Q})$\nbetween the rational surfaces   $P^2(k_i)$ and  $P^2(\\mathbf{Q})$\nshown at the lower level in Figure 4. \n\n\n\n\\begin{figure}[h]\n\\begin{picture}(300,140)(-70,0)\n\n\\put(63,100){$P^2(k)$}\n\n\\put(75,90){\\vector(0,-1){20}}\n\\put(55,90){\\vector(-1,-1){20}}\n\\put(95,90){\\vector(1,-1){20}}\n\n\n\\put(20,50){$P^2(k_1)$}\n\\put(60,50){$P^2(k_2)$}\n\\put(110,50){$P^2(k_3)$}\n\n\\put(75,40){\\vector(0,-1){20}}\n\\put(35,40){\\vector(1,-1){20}}\n\\put(115,40){\\vector(-1,-1){20}}\n\n\\put(60,0){$P^2(\\mathbf{Q})$}\n\n\\end{picture}\n\\caption\n}\n\\end{figure}\n\n\n\n\n\\bigskip\n(iii) Using  [Piergallini 1995] \\cite{Pie1} (see proof of corollary \\ref{cor3.5}),  \nwe conclude that each $P^2(k_i)\\to P^2(\\mathbf{Q})$\nis a covering map ramified over an embedded 2-dimensional surface $X$. \nTo determine the genus of $X$, recall that the group of deck transformations \nof the covering $P^2(k_i)\\to P^2(\\mathbf{Q})$ is isomorphic to the Galois group \n$Gal~(k_i | \\mathbf{Q})\\cong \\mathbf{Z}\/2\\mathbf{Z}$ of the field $k_i$. \nIn particular,  the ramification set $X$ is fixed by the deck transformations\nand, therefore, corresponds to the field $\\mathbf{Q}$ fixed by the group  \n$Gal~(k_i | \\mathbf{Q})$. But the avatar of $\\mathbf{Q}$ is a \nprojective line $P^1(\\mathbf{C})\\cong X$,  see item (i) of theorem \\ref{thm1.2}. \nWe conclude that the surface $X$ has genus zero. \n\n\n\n\n\\bigskip\n(iv)   Since $k\\cong\\mathbf{Q}(\\sqrt{-1}, \\sqrt{-a}, \\sqrt{-b})$,\none gets for each $i=1,2, 3$  a regular map $P^2(k)\\to P^2(k_i)$ as shown at the upper level of diagram in Figure 4. Composing these  maps  with the maps $P^2(k_i)\\to P^2(\\mathbf{Q})$, one concludes the algebraic surface $P^2(k)$\nis a covering of  $ P^2(\\mathbf{C}) $ ramified over three  knotted two-dimensional spheres\n$P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})$ embedded in $ P^2(\\mathbf{C})$.  \n\\begin{remark}\nThe knotting type of $P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})$  depends  on the arithmetic of the fields $k_i$ \nand extends the Grothendieck's theory of {\\it dessin d'enfant} to the case of algebraic surfaces.   \n\\end{remark}\n\n\n\n\n\\bigskip\n\\subsubsection{Case  $K\\subset\\overline{\\mathbf{Q}}$}\n(i)  Let  $S(\\overline{\\mathbf{Q}})$ is an avatar of the quaternion algebra \n$\\left({a,b\\over K}\\right)$ constructed in Part II.\nIt follows from formulas (\\ref{eq3.9})-(\\ref{eq3.11}) that there exists \na regular map:\n\\begin{equation}\\label{eq3.12}\nf:  S(\\overline{\\mathbf{Q}})\\to P^2(k).\n\\end{equation}\n\n\n\\bigskip\n(ii)  Let $f^{-1}(P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})\\cup P^1(\\mathbf{C}))$\nbe the pre-image of the three knotted two-dimensional spheres embedded in $P^2(\\mathbf{C})$\nunder the map (\\ref{eq3.12}).  It is not hard to see, that such a pre-image \nconsists again of  three  spheres $P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})$ but knotted differently when compared  \nto the case of the surface  $P^2(k)$. \n\n\\bigskip\n(iii)  Since our surface is an avatar of the quaternion algebra \n$\\left({a,b\\over K}\\right)$, it  is defined over $\\overline{\\mathbf{Q}}$, see item (ii) \nof theorem \\ref{thm1.4}.  This argument finishes the proof of item (iii) \nof theorem \\ref{thm1.4}. \n\n\n\\bigskip\nTheorem \\ref{thm1.4} is proved. \n\n\n\n\n\n\\section{Belyi's theorem for algebraic surfaces}\nThe aim of this section is an analog of Belyi's theorem for algebraic surfaces. \nOur approach is geometric and  follows from  theorem \\ref{thm1.4}. \nWe refer the reader to  [Gonz\\'alez-Diez 2008] \\cite{Gon1}\nfor an analytic treatment to this problem. \n\\begin{theorem}\\label{thm4.1}\nA non-singular algebraic surface is defined over a number field $K$\nif and only if it is a covering  of the complex projective plane $P^2(\\mathbf{C}) $\nramified at three knotted two-dimensional spheres\n$P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})\\cup P^1(\\mathbf{C})$.\n\\end{theorem}\n\\begin{proof}\nOur proof is based on item (iii) of  theorem \\ref{thm1.4} and the following lemma.\n\\begin{lemma}\\label{lm4.2}\nFor each algebraic surface $S(\\overline{\\mathbf{Q}})$\n there exists a quaternion algebra  $\\left({a,b\\over K}\\right)$\n such that the avatar of $\\left({a,b\\over K}\\right)$ is isomorphic to  $S(\\overline{\\mathbf{Q}})$. \n\\end{lemma}\n\\begin{proof}\n(i) In view of [Piergallini 1995] \\cite{Pie1} (see proof of corollary \\ref{cor3.5}), \nthe algebraic surface  $S(\\overline{\\mathbf{Q}})$ is a covering of the \nprojective plane $P^2(\\mathbf{C}) $ ramified over a knotted 2-dimensional surface  $X\\subset P^2(\\mathbf{C})$. \nIn particular, there exists a regular map $\\phi: S(\\overline{\\mathbf{Q}})\\to  P^2(\\mathbf{C}) $. \n\n\\bigskip\n(ii) Recall that the  $ P^2(\\mathbf{C}) $ is an avatar  of the division ring $\\left({-1,-1\\over \\mathbf{Q}}\\right)$,\nsee item (i) of theorem \\ref{thm1.4}.  Moreover, the field  inclusion\n$\\mathbf{Q}\\subset\\overline{\\mathbf{Q}}$ gives rise to a regular map \n$\\psi_0: S_0(\\overline{\\mathbf{Q}})\\to  P^2(\\mathbf{C}) $, where   $S_0(\\overline{\\mathbf{Q}})$\nis an avatar of the quaternion algebra $\\left({-1, -1\\over \\overline{\\mathbf{Q}}}\\right)$. \n\n\n\n\\begin{figure}[h]\n\\begin{picture}(300,140)(-70,0)\n\n\\put(35,100){$S(\\overline{\\mathbf{Q}})$}\n\\put(90,100){$S_0(\\overline{\\mathbf{Q}})$}\n\n\\put(65,103){\\vector(1,0){20}}\n\\put(50,90){\\vector(0,-1){70}}\n\\put(105,90){\\vector(0,-1){20}}\n\n\\put(100,90){\\vector(-2,-3){40}}\n\n\n\\put(85,50){ $\\left({-1, -1\\over K}\\right)$}\n\n\n\\put(105,40){\\vector(0,-1){20}}\n\n\\put(30,0){$P^2(\\mathbf{C})$}\n\\put(90,0){$\\left({-1,-1\\over \\mathbf{Q}}\\right)$}\n\\put(65,3){\\vector(1,0){20}}\n\n\\put(35,50){$\\phi$}\n\\put(70,110){$\\psi$}\n\\put(70,70){$\\psi_0$}\n\\end{picture}\n\\caption\n}\n\\end{figure}\n\n\n\n\\bigskip\n(iii)  One gets a regular map $\\psi:  S(\\overline{\\mathbf{Q}})\\to S_0(\\overline{\\mathbf{Q}})$\nby closing arrows of the commutative diagram in Figure 5. Notice that $\\psi$ is a finite covering of \nthe surface $S_0(\\overline{\\mathbf{Q}})$, since $\\phi$ and $\\psi_0$ are mappings of finite degree. \n\n\n\\bigskip\n(iv) On the other hand, it follows from  equations (\\ref{eq3.9})-(\\ref{eq3.11}) \nthat each finite covering   $S(\\overline{\\mathbf{Q}})$ of \nthe avatar $S_0(\\overline{\\mathbf{Q}})$\nof the algebra  $\\left({-1, -1\\over K}\\right)$ must \nbe  avatar of an algebra  $\\left({a, b\\over K}\\right)$\nfor some $a,b\\in K$. Thus  there exists\na quaternion algebra   $\\left({a, b\\over K} \\right)$,\nsuch that  algebraic surface $S(\\overline{\\mathbf{Q}})$ is the \navatar of  $\\left({a, b\\over K}\\right)$. \n\n\nLemma \\ref{lm4.2} is proved. \n\\end{proof}\n\n\n\\bigskip\nReturning to the proof of theorem \\ref{thm4.1},  one combines lemma \\ref{lm4.2} with the conclusion of \nitem (iii) of theorem \\ref{thm1.4}.  This argument finishes the proof of theorem \\ref{thm4.1}. \n\\end{proof}\n\n\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}