diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzznnnl" "b/data_all_eng_slimpj/shuffled/split2/finalzznnnl"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzznnnl"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\nFirst let us introduce some terminology and notations. Unless other\nstated, we follow [2] for undefined terms on hypergraphs. Let\n$H=(V,E)$ be a hypergraph on $n$ vertices with vertex set $V$ and\nedge set $E$. We say that $H$ is an $r$-uniform hypergraph if every\nedge contains $r$ vertices, and that $H$ is a complete $r$-uniform\nhypergraph, denoted by $K_n^r$, if $E$ consists of all $r$-subsets\nof $V$. $H$ is simple if there are no repeated edges. Thus, a simple\n2-uniform hypergraph is a simple graph. For a vertex $v$ in\nhypergraph $H$, the degree of $v$, denoted $d_H(v)$ (or simply\n$d(v)$ when $H$ is understood) is the number edges of $H$ that\ncontain $v$.\n\nLet $d=(d_1, d_2, \\cdots, d_n)$ be a sequence of nonnegative\nintegers with $d_1\\le d_2\\le \\cdots \\le d_n$. We say that $d$ is\n$r$-uniform hypergraphic if it is the degree sequence of a simple\n$r$-uniform hypergraph $H$ on $n$ vertices, and such an $r$-uniform\nhypergraph $H$ is referred to as a realization of $d$. When $r=2$,\nwe will simply say that $d$ is graphic. The degree sequences for\nsimple graphs have been studied for many years and and by several\nauthors, including the celebrated work of Erd\\\"{o}s and Gallai [8].\nBased on this, Sierksma and Hoogeveen [12] listed seven criteria and\n Cai et al.[5] gave it a generalization.\n\nIn this article, we mainly study sufficient degree conditions for a\nsimple hypergraph to be $k$-edge-connected and super edge-connected.\nWe say that a hypergraph $H=(V,E)$ is connected if for any pair of\nvertices $u, v\\in V$ there exists a path from $u$ to $v$; otherwise\nwe say $H$ is disconnected. For any $E_0\\subseteq E$, if $H-E_0=H\n(V, E-E_0)$ is disconnected, we say that $E_0$ is an edge-cut. The\nedge connectivity of $H$, denoted by $\\lambda (H)$, is the least\ncardinality of all edge-cuts of $H$. A hypergraph $H$ is\n$k$-edge-connected if $\\lambda(H)\\ge k$, where $1\\le k\\le \\delta(H)$\nwith $\\delta(H)$ denoting the minimum degree of $H$. If\n$\\lambda(H)=\\delta(H)$, we say that $H$ is maximally edge-connected.\nMoreover, a hypergraph $H$ is super edge-connected (or,\nsuper-$\\lambda$) if every minimum edge-cut consists of edges\nincident with one vertex with minimum degree. There are several\ndegree sequence conditions for maximally edge-connected graphs and\nsuper edge-connected graphs. For example, Dankelmann and Meierling\n[7] and Zhao et al. [15], etc.\n\n\nIf $d=(d_1, d_2,\\cdots,d_n)$ and $d'=(d'_1, d'_2,\\cdots,d'_n)$ are\ntwo sequences of nonnegative integers, we say that $d'$ majorizes\n$d$, denoted $d'\\ge d$, if $d'_j\\ge d_j$ for $1\\le j\\le n$. Let $P$\ndenote an $r$-uniform hypergraphic property and $d$ be an\n$r$-uniform hypergraphic sequence. If every realization  of $d$ has\nproperty $P$, we say that $d$ is forcibly $P$. Historically,\nsufficient conditions for a graphic sequence to have a certain\nproperty, e.g., $k$-connected [3,4], $k$-edge-connected [1,10,14],\nhamiltonian [6], clique size [13], etc, have been studied. We call a\nforcibly $P$ degree condition $\\chi$ is monotone increasing if\n$d',d$ are hypergraphic, $d'\\ge d$ and $d$ satisfies $\\chi$ implies\nthat $d'$ satisfies $\\chi$. Furthermore, $\\chi$ is said to be a\nstrongest degree condition for property $P$ if whenever it does not\nguarantee that $d$ is forcibly $P$, then $d$ is majorized by an\n$r$-uniform hypergraphic sequence $d'$ which has a realization\nwithout property $P$. In 1969, Bondy [4] gave a strongest degree\ncondition for a graphic sequence to be forcibly $k$-connected. In\n1972, Chv$\\acute{a}$tal [6] gave a strongest degree condition for a\ngraphic sequence to be forcibly hamiltonian.  Recently, Frosini et\nal. [9] gave new sufficient conditions on the degree sequences of\nuniform hypergraphs. Liu et al. [11] gave a simple sufficient degree\nconditions for a uniform hypergraph to be $k$-edge-connected or\nsuper edge-connected and the strongest monotone increasing degree\nconditions for a uniform hypergraph to be $k$-edge-connected when\n$k=1, 2, 3$. Inspired by this, we give the strongest sufficient\ndegree conditions for an $r$-uniform hypergraph to be\n$k$-edge-connected for all $k\\ge 1$ and super edge-connected. These\nare in section 2 and 3. Additionally, Yin et al. gave a sufficient\ndegree condition for a graphic sequence to be forcibly\n$k$-edge-connected. We shall extend the result to hypergraphs in\nsection 4.\n\n\n\n\\section{The strongest conditions for $k$-edge-connected hypergraphs}\n\nThe following theorem due to Liu, Meng and Tian [11] gave simple\nsufficient degree conditions for an $r$-uniform hypergraph to be\n$k$-edge-connected. To present the result, they utilized a parameter\n$g$ and define $g=g(k, r)(\\ge r)$ to be the smallest integer such\nthat $gk\\le g {g-1\\choose r-1}+(r-1)(k-1)$, i.e., $k\\le {g\\choose\nr-1}-{r-1 \\choose g-r+1}$, where $r\\ge 2$ and $k\\ge 2$.\n\n\n \\begin{theorem} ([11])\\label{21t}\nFor two integers $r\\ge2$ and $k\\ge 2$. Let $d=(d_1, d_2, \\cdots,\nd_n)$ be an $r$-uniform hypergraphic sequence with $d_1\\le\nd_2\\le\\cdots\\le d_n$. If $d$ satisfies all of the following\nconditions, then $d$ is forcibly $k$-edge connected:\n\n\\begin{itemize}\n\\item [$(1)$]  $d_1\\ge k$,\n\\item [$(2)$] $d_{j-(r-1)(k-1)}\\le {j-1\\choose r-1}$ and $d_j\\le {j-1\\choose r-1}+k-1$ implies $d_n\\ge{n-j-1\\choose r-1}+k$ for $g(k, r)\\le j\\le \\lfloor {n\\over 2}\\rfloor$.\n\\end{itemize}\n\\end{theorem}\n\n\nIn addition to Theorem \\ref{21t}, Liu et al. also established the\nstrongest monotone increasing conditions for an $r$-uniform\nhypergraph be be $k$-edge-connected when $k=1, 2, 3$. Now comes the\ncase for $k=2$, which differs a little from one of the corollaries\nof Theorem 2.3. In fact, however, they are equivalent.\n\n\n \\begin{theorem} ([11])\\label{22t}\nGiven integer $r\\ge2$. Let $d=(d_1, d_2, \\cdots, d_n)$ be an\n$r$-uniform sequence with $d_1\\le d_2\\le\\cdots\\le d_n$. If $d$\nsatisfies all of the following conditions, then $d$ is forcibly\n$2$-edge connected:\n\n\n\\begin{itemize}\n\\item [$(1)$]  $d_1\\ge 2$,\n\\item [$(2)$] $d_{j-t_1}\\le {j-1\\choose r-1}$ and $d_j\\le {j-1\\choose r-1}+1$ implies\n $d_{n-r+t_1}\\ge{n-j-1\\choose r-1}+1$ or $d_{n}\\ge{n-j-1\\choose r-1}+2$\n for $r+1\\le j< {n\\over 2}$ and $1\\le t_1\\le r-1$.\n\n\\item [$(3)$] $d_{n\\over 2}\\le {{n\\over 2}-1\\choose r-1}$ and $d_{n-r}\\le {{n\\over 2}-1\\choose r-1}$\nimplies $d_{n}\\ge{{n\\over 2}-1\\choose r-1}+2$ for even $n\\ge 2r+2$.\n\n\\end{itemize}\n\\end{theorem}\n\nInspired by these results, we first give  the strongest monotone\ndegree condition\n for an $r$-uniform hypergraph to be $k$-edge-connected for all $k\\ge 1$, in which four conditions are contained, where (2)-(4) are \"Bondy-Chv$\\acute{a}$tal type\" conditions.\nBefore\n presenting it, we introduce some parameters.\n\nGiven integers $r\\ge 2$ and $k\\ge 2$, $j^\\ast=j^\\ast (k,r)(r+1\\le\nj^\\ast <{n\\over 2})$ is defined to be the biggest integer such that\n${j^\\ast-1\\choose r-1}+k-1\\le {n-j^\\ast-1\\choose r-1}$. Let\n$t_1,\\cdots,t_{k-1},s_1,\\cdots,$  $s_{k-1}$ be nonnegative integers\nwhich satisfied:\n$$[t_1+2t_2+\\cdots+(k-1)t_{k-1}]+[s_1+2s_2+\\cdots+(k-1)s_{k-1}]=(k-1)r,$$\n $$k-1\\le t_1+s_1\\le (k-1)r,$$\n $$0\\le t_i+s_i\\le \\lfloor{k-1\\over i}\\rfloor(r-1),2\\le i \\le k-1,$$\n$$1\\le t_1+t_2+\\cdots+t_{k-1}\\le \\min\\{(k-1)r-(k-1), j\\}, k\\ge\n2,$$\n$$0\\le t_i+\\cdots+t_{k-1}\\le \\lfloor{k-1\\over i}\\rfloor(r-1), k\\ge 3,2\\le i\\le k-2,$$\n$$1\\le s_1+s_2+\\cdots+s_{k-1}\\le \\min\\{(k-1)r-(k-1), n-j\\}, k\\ge 2,$$\n$$0\\le s_i+\\cdots+s_{k-1}\\le \\lfloor{k-1\\over i}\\rfloor(r-1), k\\ge 3,2\\le i\\le k-2.$$\n\n\n\n\n\n\n\n\n \\begin{theorem} \\label{23t}\n    For integers $r\\ge2$ and $k\\ge 1$. Let $d=(d_1, d_2, \\cdots, d_n)$ be an\n    $r$-uniform hypergraphic sequence with $d_1\\le d_2\\le\\cdots\\le d_n$.\n    If $d$ satisfies all of the following conditions, then $d$ is forcibly $k$-edge-connected:\n\n\n    \\begin{itemize}\n        \\item [$(1)$]  $d_1\\ge k$,\n        \\item [$(2)$] $d_{j-t_1-t_2-\\cdots-t_{k-1}}\\le {j-1\\choose r-1}$, $d_{j-t_2-\\cdots-t_{k-1}}\n        \\le {j-1\\choose r-1}+1$,\n        $\\cdots,$ $d_{j-t_{k-1}}\\le {j-1\\choose r-1}+k-2$ and $d_j\\le {j-1\\choose r-1}+k-1$\n        implies $d_{n-s_1-\\cdots-s_{k-1}}\\ge{n-j-1\\choose r-1}+1$ or\n        $d_{n-s_2-\\cdots-s_{k-1}}\\ge{n-j-1\\choose r-1}+2$, $\\cdots$, or\n        $d_{n-s_{k-1}}\\ge{n-j-1\\choose r-1}+k-1$ or $d_{n}\\ge{n-j-1\\choose r-1}+k$\n         for $r+1\\le j\\le j^\\ast$,\n\\item [$(3)$] $d_{j-t_1-\\cdots-t_{k-1}}\\le\n         {n-j-1\\choose r-1}-\\vartriangle$, $d_{j-t_2-\\cdots-t_{k-1}}\n         \\le {n-j-1\\choose r-1}+1-\\vartriangle$,$\\cdots$,\n         $d_{j-t_\\vartriangle-\\cdots-t_{k-1}} \\le {n-j-1\\choose r-1}-1$,\n         $\\cdots$, $d_{j-t_{k-1}} \\le {n-j-1\\choose\n         r-1}+k-2-\\vartriangle$ and $d_j \\le {n-j-1\\choose\n         r-1}+k-1-\\vartriangle$ implies\n          $d_{n-t_{\\vartriangle+1}-\\cdots-t_{k-1}-s_1-\\cdots-s_{k-1}} \\ge {n-j-1\\choose\n          r-1}+1$, $\\cdots$, or\n          $d_{n-t_{k-1}-s_{k-1-\\vartriangle}-\\cdots-s_{k-1}} \\ge {n-j-1\\choose r-1}+k-1-\\vartriangle$\nor\n          $d_{n-s_{k-\\vartriangle}-\\cdots-s_{k-1}} \\ge {n-j-1\\choose\n          r-1}+k-\\vartriangle $, $\\cdots$, or $d_{n-s_{k-1}} \\ge {n-j-1\\choose\n          r-1}+k-1$ or $d_n \\ge {n-j-1\\choose\n          r-1}+k$  for  $j^\\ast<j<{n\\over 2}$,\n          where $\\vartriangle={n-j-1\\choose  r-1}-{j-1\\choose\n          r-1}$.\n\n\n        \\item [$(4)$] $d_{n-t_1-\\cdots-t_{k-1}-s_1-\\cdots-s_{k-1}}\\le\n         {{n\\over 2}-1\\choose r-1}$, $d_{n-t_2-\\cdots-t_{k-1}-s_2-\\cdots-s_{k-1}}\n         \\le {{n\\over 2}-1\\choose r-1}+1$,$\\cdots$, and $d_{n-t_{k-1}-s_{k-1}}\n         \\le {{n\\over 2}-1\\choose r-1}+k-2$ implies $d_n\n         \\ge {{n\\over 2}-1\\choose r-1}+k$ for even $n\\ge 2r+2$.\n\n    \\end{itemize}\n\\end{theorem}\n\n\n\\pf By induction on $k$. The theorem is trivial for $k=1, 2, 3$\n(refer to Corollary 2.4-2.6). Assume the theorem holds for all $k\\le\nm-1$. Now suppose that $d$ satisfies $(1)-(4)$ for $k=m\\ge 4$, but\n$d$ is not forcibly $m$-edge-connected. Namely, there exists a\nnon-$m$-edge-connected $r$-uniform realization $H$ consisting of two\nsubhypergraphs $C_1$ and $C_2$ joined by $m-1$ hyperedges $e_1, e_2,\n\\cdots, e_{m-1}$, where $V(C_2)=V(H)\\setminus V(C_1)$ and\n$|V(C_1)|\\le |V(C_2)|$. Denote $E_0=\\{e_1, e_2, \\cdots, e_{m-1}\\}$\nand let $|V(C_1)|=j$. One can see that $r+1\\le j\\le \\lfloor {n\\over\n2}\\rfloor$, since $j\\le r$ implies there exists a vertex of degree\nless than $m$. Let $U_i(V_i)$ be the set of vertices in\n$V(H_1)(V(H_2))$, any one of which belong to exactly $t_i(s_i)$\nhyperedges of $e_1, e_2,\\cdots, e_{m-1}$, where $1\\le i\\le m-1$.\nThen we can see that\n$[t_1+2t_2+\\cdots+(m-1)t_{m-1}]+[s_1+2s_2+\\cdots+(m-1)s_{m-1}]=(m-1)r$,\n$m-1\\le t_1+s_1\\le (m-1)r$, $0\\le t_i+s_i\\le \\lfloor{m\\over\ni}\\rfloor(r-1)$ for $i=2,\\cdots,m-1$, $1\\le\nt_1+t_2+\\cdots+t_{m-1}\\le \\min\\{(m-1)r-(m-1), j\\}$,  $0\\le\nt_2+\\cdots+t_{m-1}\\le \\lfloor{m-1\\over 2}\\rfloor(r-1),\\cdots,0\\le\nt_{m-2}+t_{m-1}\\le \\lfloor{m-1\\over m-2}\\rfloor(r-1)$, $1\\le\ns_1+s_2+\\cdots+s_{m-1}\\le \\min\\{(m-1)r-(m-1), n-j\\}$, $0\\le\ns_2+\\cdots+s_{m-1}\\le \\lfloor{m-1\\over 2}\\rfloor(r-1)$, $\\cdots$,\n$0\\le s_{m-2}+s_{m-1}\\le \\lfloor{m-1\\over m-2}\\rfloor(r-1)$.\n\nIt is not difficult to see that there are at most\n$t_1+t_2+\\cdots+t_{m-1}$ vertices in $C_1$ of degree larger than\n$j-1 \\choose r-1$, at most $t_2+\\cdots+t_{m-1}$ vertices in $C_1$ of\ndegree larger than ${j-1 \\choose r-1}+1$, at most\n$t_3+\\cdots+t_{m-1}$ vertices in $C_1$ of degree larger than ${j-1\n\\choose r-1}+2$, $\\cdots$, and at most  $t_{m-1}$ vertices in $C_1$\nof degree larger than ${j-1 \\choose r-1}+m-2$, while the degree of\neach vertex in $C_1$ is at most ${j-1 \\choose r-1}+m-1$. Thus we\nhave $d_{j-t_1-t_2-\\cdots-t_{m-1}}\\le {j-1\\choose r-1}$ if $j>\nt_1+t_2+\\cdots+t_{m-1}$, $d_{j-t_2-\\cdots-t_{m-1}}\\le {j-1\\choose\nr-1}+1$ if $j> t_2+\\cdots+t_{m-1}$, $\\cdots$, $d_{j-t_{m-1}}\\le\n{j-1\\choose r-1}+m-2$ if $j>t_{m-1}$ and $d_{j}\\le {j-1\\choose\nr-1}+m-1$. We distinguish the following cases.\n\n{\\bf Case  1.}  $r+1\\le j<{n\\over 2}$.\n\n{\\bf Case  1.1.} $r+1\\le j\\le j^\\ast .$\n\n In this case, each vertex in\n$C_1$ has degree at most ${n-j-1\\choose r-1}$\n according to  the definition of $j^\\ast$ described above.\nAdditionally, there are at most $s_1+ \\cdots + s_{m-1}$ vertices in\n$C_2$ of degree larger than ${n-j-1\\choose r-1}$, at most  $s_2+\n\\cdots + s_{m-1}$ vertices in $C_2$ of degree larger than\n${n-j-1\\choose r-1}+1, \\cdots,$ and at most  $s_{m-1}$ vertices in\n$C_2$ of degree larger than ${n-j-1\\choose r-1}+m-2$. Thus,\n$d_{n-s_1-\\cdots-s_{m-1}}\\le{n-j-1\\choose r-1}$,\n$d_{n-s_2-\\cdots-s_{m-1}}\\le{n-j-1\\choose r-1}+1$, $\\cdots$,\n$d_{n-s_{m-1}}\\le{n-j-1\\choose r-1}+m-2$ and $d_{n}\\le{n-j-1\\choose\nr-1}+m-1$. Therefore the consequent in $(2)$ fails, while the\nantecedent in $(2)$ is satisfied, yielding a contradiction.\n\n{\\bf Case  1.2.} $j^\\ast<j <{n\\over 2}.$\n\nObviously, ${j-1\\choose r-1}<{n-j-1\\choose r-1}$. Write\n$\\vartriangle={n-j-1\\choose r-1}-{j-1\\choose r-1}$, then $1\\le\n\\vartriangle <m-1$ since ${j-1\\choose r-1}+m-1>{n-j-1\\choose r-1}$\nwhen $j>j^\\ast$. Note that the degree of each vertex in $V(C_1)-E_0$\nis at most ${j-1\\choose r-1}={n-j-1\\choose r-1}-\\vartriangle$, the\ndegree of each vertex in $ U_1$ is at most ${j-1\\choose\nr-1}+1={n-j-1\\choose r-1}+1-\\vartriangle$, the degree of each vertex\nin $ U_2$ is at most ${j-1\\choose r-1}+2={n-j-1\\choose\nr-1}+2-\\vartriangle$, $\\cdots$, the degree of each vertex in $\nU_\\vartriangle$ is at most ${j-1\\choose\nr-1}+\\vartriangle={n-j-1\\choose r-1}$, the degree of each vertex in\n $ U_{\\vartriangle +1}$ is at most ${j-1\\choose\nr-1}+\\vartriangle+1={n-j-1\\choose r-1}+1$, $\\cdots$, the degree of\neach vertex in $U_{m-1}$ is at most ${j-1\\choose\nr-1}+m-1={n-j-1\\choose r-1}+m-1-\\vartriangle$, and so\n$d_{j-t_1-\\cdots-t_{m-1}}\\le\n         {n-j-1\\choose r-1}-\\vartriangle$, $d_{j-t_2-\\cdots-t_{m-1}}\n         \\le {n-j-1\\choose r-1}+1-\\vartriangle$, $d_{j-t_3-\\cdots-t_{m-1}}\n         \\le {n-j-1\\choose r-1}+2-\\vartriangle$, $\\cdots$,\n         $d_{j-t_\\vartriangle-\\cdots-t_{m-1}} \\le {n-j-1\\choose r-1}-1$,\n         $d_{j-t_{\\vartriangle+1}-\\cdots-t_{m-1}} \\le {n-j-1\\choose\n         r-1}$, $\\cdots$, $d_{j-t_{m-1}} \\le {n-j-1\\choose\n         r-1}+m-2-\\vartriangle$ and $d_j \\le {n-j-1\\choose\n         r-1}+m-1-\\vartriangle$. Furthermore, the degree of each\n         vertex in $V_i$ is at most ${n-j-1\\choose\n         r-1}+i$, where $1\\le i\\le m-1$. Thus,\n         $d_{n-t_{\\vartriangle+1}-\\cdots-t_{m-1}-s_1-\\cdots-s_{m-1}} \\le {n-j-1\\choose\n          r-1}$, $d_{n-t_{\\vartriangle+2}-\\cdots-t_{m-1}-s_2-\\cdots-s_{m-1}} \\le {n-j-1\\choose\n          r-1}+1$, $\\cdots$,\n          $d_{n-t_{m-1}-s_{m-1-\\vartriangle}-\\cdots-s_{m-1}} \\le {n-j-1\\choose\n          r-1}+m-2-\\vartriangle$,\n          $d_{n-s_{m-\\vartriangle}-\\cdots-s_{m-1}} \\le {n-j-1\\choose\n          r-1}+m-1-\\vartriangle $, $\\cdots$,  $d_{n-s_{m-1}} \\le {n-j-1\\choose\n          r-1}+m-2$ and $d_n \\le {n-j-1\\choose\n          r-1}+m-1$, this contradicts (3).\n\n\n{\\bf Case  2.} $j={n\\over 2}$ for even $n.$\n\n Note that the degree of each\nvertex in $V(H)-E_0$ is at most ${n\\over 2}-1 \\choose r-1$.\nMoreover, there are at most $t_2+\\cdots+t_{m-1}+s_2+\\cdots+s_{m-1}$\nvertices in $H$ of degree larger than ${{n\\over 2}-1 \\choose r-1}+1,\n\\cdots$, and at most $s_{m-1}+t_{m-1}$ vertices in $H$ of degree\nlarger than ${{n\\over 2}-1 \\choose r-1}+m-2$, while the degree of\neach vertex in $H$ is at most ${{n\\over 2}-1 \\choose r-1}+m-1$.\nThus, $d_{n-t_1-t_2-\\cdots-t_{m-1}-s_1-\\cdots-s_{m-1}}\\le {{n\\over\n2}-1\\choose r-1}$, $d_{n-t_2-\\cdots-t_{m-1}-s_2-\\cdots-s_{m-1}}\\le\n{{n\\over 2}-1\\choose r-1}+1, \\cdots$, $d_{n-t_{m-1}-s_{m-1}}\\le\n{{n\\over 2}-1\\choose r-1}+m-2$ and $d_n \\le {{n\\over 2}-1\\choose\nr-1}+m-1$, contrary to $(4)$.\n\nTherefore, the hypothesis is not valid, and so Theorem 2.3 follows\nby the principle of induction. \\qed\n\nIf $k=1, 2, 3$ in Theorem \\ref{23t},\n then we can obtain the strongest monotone increasing degree conditions for\n  an $r$-uniform hypergraph to be $k$-edge-connected when $k=1, 2, 3$.\n\n \\begin{corollary}\\label{24c}([11])\nFor integers $r\\ge2$ and $k\\ge 1$. Let $d=(d_1, d_2, \\cdots, d_n)$\nbe an $r$-uniform hypergraphic sequence with $d_1\\le d_2\\le\\cdots\\le\nd_n$. If $d_1\\ge 1$ and $d_j \\le {j-1 \\choose r-1}$ implies $d_n \\ge\n{n-j-1 \\choose r-1}+1$ for $r+1\\le j\\le \\lfloor {n\\over 2}\\rfloor$,\nthen $d$ is forcibly connected.\n\\end{corollary}\n\n\n \\begin{corollary}\\label{25c}\n    For integers $r\\ge2$. Let $d=(d_1, d_2, \\cdots, d_n)$ be an $r$-uniform\n    hypergraphic sequence with $d_1\\le d_2\\le\\cdots\\le d_n$. If $d$ satisfies all of the\n    following conditions, then $d$ is forcibly $2$-edge-connected:\n\n    \\begin{itemize}\n        \\item [$(1)$]  $d_1\\ge 2$,\n        \\item [$(2)$] $d_{j-t_1}\\le {j-1\\choose r-1}$ and $d_j\\le {j-1\\choose r-1}+1$ implies $d_{n-s_1}\\ge{n-j-1\\choose r-1}+1$ or $d_{n}\\ge{n-j-1\\choose r-1}+2$ for $r+1\\le j< {n\\over 2}$, $1\\le t_1\\le r-1$ and $t_1+s_1=r$.\n\n        \\item [$(3)$] $d_{n-r}\\le {{n\\over 2}-1\\choose r-1}$ implies $d_{n}\\ge{{n\\over 2}-1\\choose r-1}+2$ for even $n\\ge 2r+2$.\n\\end{itemize}\n\n\\end{corollary}\n\n\nNote that ${n\\over 2}< n-r$ since $n\\ge 2r+2$, then $d_{n\\over 2}\\le\nd_{n-r}$. Therefore, Corollary \\ref{25c} and Theorem \\ref{22t} are\nequivalent.\n\n \\begin{corollary}\\label{26c}([11])\n    For integers $r\\ge3$. Let $d=(d_1, d_2, \\cdots, d_n)$ be an $r$-uniform hypergraphic sequence with $d_1\\le d_2\\le\\cdots\\le d_n$. If $d$ satisfies all of the following conditions, then $d$ is forcibly $3$-edge-connected:\n\n    \\begin{itemize}\n        \\item [$(1)$]  $d_1\\ge 3$,\n        \\item [$(2)$] $d_{j-t_1-t_2}\\le {j-1\\choose r-1}$, $d_{j-t_2}\\le {j-1\\choose r-1}+1$ and\n         $d_j\\le {j-1\\choose r-1}+2$ implies $d_{n-s_1-s_2}\\ge{n-j-1\\choose r-1}+1$ or\n         $d_{n-s_2}\\ge{n-j-1\\choose r-1}+2$ or $d_{n}\\ge{n-j-1\\choose r-1}+3$ for\n         $r+1\\le j< {n\\over 2}$, $2\\le t_1+s_1\\le 2r$, $0\\le t_2+s_2\\le r-1$,\n         $1\\le t_1+t_2\\le min\\{2r-2, j\\}$, $1\\le s_1+s_2\\le min\\{2r-2, n-j\\}$,\n         $t_1+2t_2+s_1+2s_2=2r$ and $t_i, s_i\\ge 0$ for $1\\le i\\le\n         2$,\n\n        \\item [$(3)$] $d_{n-t_1-t_2-s_1-s_2}\\le {{n\\over 2}-1\\choose r-1}$, $d_{n-t_2-s_2}\\le\n         {{n\\over 2}-1\\choose r-1}+1$ implies $d_{n}\\ge{{n\\over 2}-1\\choose r-1}+3$ for even $n\\ge 2r+2$, where $t_i$ and $s_i (1\\le i\\le 2)$ are defined as shown in $(2)$.\n    \\end{itemize}\n\n\\end{corollary}\n\n\n \\begin{theorem}\\label{27b}\nThe degree conditions in Theorem \\ref{23t} are strongest.\n\\end{theorem}\n\n\\pf In order to very that the degree conditions in Theorem 2.3 are\nstrongest, we now show that if any of (1)-(4) fails for $d$, then\n$d$ is majorized by an $r$-uniform hypergraphic sequence $d'$ having\na realization $H'$ which is not $k$-edge-connected.\n\n\n Let $H_{t_1, \\cdots, t_{k-1}, s_1, \\cdots, s_{k-1}}(j, n-j)$\ndenote an $r$-uniform hypergraph consisting of two disjoint $K_j^r$\nand $K_{n-j}^r$ joined by $k-1$ edges $e_1, e_2, \\cdots, e_{k-1}$,\nwhere $t_i$ and $s_i$ $(1\\le i\\le k-1)$ are defined as shown in\nTheorem 2.3.\n\n\nIf $(1)$ fails, we take $d^\\prime=d$ and $H^\\prime=H$, where $H$ is\na non-$k$-edge-connected $r$-uniform realization of $d$ with minimum\ndegree less than $k$.\n\n If $(2)$ fails for $t_1, \\cdots, t_{k-1}, s_1, \\cdots, s_{k-1}$ and $j$ with $r+1\\le j <j^\\ast$,\n we take $d^\\prime=({j-1 \\choose r-1}^{j-t_1-\\cdots-t_{k-1}}\\big({j-1\\choose r-1}+\n 1\\big)^{t_1}\\cdots \\big({j-1\\choose r-1}\n +(k-1)\\big)^{t_{k-1}}\n {n-j-1 \\choose r-1}^{n-j-s_1-\\cdots-s_{k-1}}\n \\big({n-j-1\\choose r-1}\n +1\\big)^{s_1}\\big({n-j-1\\choose r-1}+2\\big)^{s_2}\\cdots \\big({n-j-1\\choose r-1}\n +(k-2)\\big)^{s_{k-2}}\\big({n-j-1\\choose r-1}+(k-1)\\big)^{s_{k-1}})$ and\n $H^\\prime=H_{t_1, \\cdots, t_{k-1}, s_1,\\cdots,s_{k-1}}(j, n-j)$,\n where the symbol $x^y$ in $d'$ stands for $y$ consecutive terms,\n each equals to $x$.\n\n\nIf $(3)$ fails for $t_1,\\cdots,t_{k-1}, s_1,\\cdots,s_{k-1}$ and $j$\nwith $j^\\ast<j<{n\\over 2}$, we may take $d'=(({n-j-1 \\choose\nr-1}-\\vartriangle)^{j-t_1-\\cdots-t_{k-1}}({n-j-1 \\choose\nr-1}+1-\\vartriangle)^{t_1}\\cdots ({n-j-1 \\choose\nr-1}+k-1-\\vartriangle)^{t_{k-1}} {n-j-1 \\choose\nr-1}^{n-j-t_{\\vartriangle +1}-\\cdots-t_{k-1}-s_1-\\dots-s_{k-1}}\n({n-j-1 \\choose r-1}+1)^{t_{\\vartriangle+1}}\\cdots ({n-j-1 \\choose\nr-1}+k-1-\\vartriangle)^{t_{k-1}}({n-j-1 \\choose\nr-1}+k-\\vartriangle)^{s_{k-\\vartriangle}}\\cdots ({n-j-1 \\choose\nr-1}+k-1)^{s_{k-1}})$ and  $H^\\prime=H_{t_1,\\cdots,t_{k-1},\ns_1,\\cdots,s_{k-1}}$ $(j, n-j)$,\n\n\n\n\n\nIf $(4)$ fails for $t_1, \\cdots, t_{k-1}, s_1, \\cdots, s_{k-1}$, we\nmay take $d^\\prime=({{n\\over 2}-1 \\choose\nr-1}^{n-t_1-\\cdots-t_{k-1}-s_1-\\cdots-s_{k-1}} \\big({{n\\over\n2}-1\\choose r-1}+1\\big)^{t_1+s_1} \\big({{n\\over 2}-1\\choose\nr-1}+2\\big)^{t_2+s_2} \\cdots \\big({{n\\over 2}-1\\choose\nr-1}+(k-2)\\big)^{t_{k-2}+s_{k-2}} \\big({{n\\over 2}-1\\choose\nr-1}+(k-1)\\big)^{t_{k-1}+s_{k-1}})$ and $H^\\prime=H_{t_1, \\cdots,\nt_{k-1}, s_1, \\cdots, s_{k-1}}$ $({n\\over 2}, {n\\over 2})$. \\qed\n\n\n\n\n\\begin{corollary}\\label{28c}\n    For integers $r\\ge 2$. Let $d=(d_1, d_2, \\cdots, d_n)$ be an\n    $r$-uniform hypergraphic sequence with $\\delta=d_1\\le d_2\\le\\cdots\\le d_n$.\n    If $d$ satisfies $(2)-(4)$ in Theorem \\ref{23t} for $k=\\delta$, then $d$ is forcibly maximally edge-connected.\n\\end{corollary}\n\nTaking $t_1=(r-1)(k-1)$, $t_2=t_3=\\cdots=t_{k-1}=0$, $s_1=\ns_2=\\cdots=s_{k-2}=0$ and $s_{k-1}=1$ in Theorem \\ref{23t}, we\nimmediately get the following conclusion.\n\n\n \\begin{corollary}\\label{29c}\n    For integers $r\\ge 2$. Let $d=(d_1, d_2, \\cdots, d_n)$ be an $r$-uniform hypergraphic sequence with $d_1\\le d_2\\le\\cdots\\le d_n$. If $d$ satisfies each of the following, then $d$ is forcibly $k$-edge-connected:\n\n    \\begin{itemize}\n        \\item [$(1)$]  $d_1\\ge k$,\n        \\item [$(2)$] $d_{j-(r-1)(k-1)}\\le {j-1\\choose r-1}$ and $d_{j}\\le {j-1\\choose r-1}+k-1$ implies  $d_{n-1}\\ge{n-j-1\\choose r-1}+k-1$ or $d_{n}\\ge{n-j-1\\choose r-1}+k$ for $(r-1)(k-1)\\le j \\le \\lfloor{n\\over 2}\\rfloor$.\n    \\end{itemize}\n\\end{corollary}\n\nRecall that the parameter $g(k,r)\\ge (r-1)(k-1)$ in Theorem\n\\ref{21t} since the index $j-(r-1)(k-1)>0$. Therefore, Corollary\n\\ref{29c} is a generalization of Theorem \\ref{21t}.\n\n\\section{Super edge-connected $r$-uniform hypergraphs}\n\nIn this section, We provide the strongest degree conditions for an\n$r$-uniform hypergraphic sequence with minimum degree $3$ or $k$ to\nbe forcibly super edge-connected.\n\n\\begin{theorem}\\label{31t}\n    For integer $r\\ge 4$. Let $d=(d_1, d_2, \\cdots, d_n)$ be an\n    $r$-uniform hypergraphic sequence with $3=d_1\\le d_2\\le\\cdots\\le d_n$.\n    If $d$ satisfies each of the following, then $d$ is forcibly super edge-connected:\n\n    \\begin{itemize}\n    \\item [$(1)$] $d_{j-t_1-t_2-t_3}\\le {j-1\\choose r-1}$,\n    $d_{j-t_2-t_3}\\le {j-1\\choose r-1}+1$, $d_{j-t_3}\\le {j-1\\choose r-1}+2$ and\n    $d_j\\le {j-1\\choose r-1}+3$ implies $d_{n-s_1-s_2-s_3}\\ge{n-j-1\\choose r-1}+1$\n    or $d_{n-s_2-s_3}\\ge{n-j-1\\choose r-1}+2$ or $d_{n-s_3}\\ge{n-j-1\\choose r-1}+3$\n    or $d_{n}\\ge{n-j-1\\choose r-1}+4$ for $r\\le j< {n\\over 2}$,\n\n    \\item [$(2)$] $d_{n-t_1-t_2-t_3-s_1-s_2-s_3}\\le {{n\\over 2}-1\\choose r-1}$,\n     $d_{n-t_2-t_3-s_2-s_3}\\le {{n\\over 2}-1\\choose r-1}+1$,\n     $d_{n-t_3-s_3}\\le {{n\\over 2}-1\\choose r-1}+2$ implies $d_{n}\\ge{{n\\over 2}-1\\choose r-1}+4$\n     for even $n\\ge 2r+2$,\n where $t_1+2t_2+3t_3+s_1+2s_2+3s_3=3r$,\n    $3\\le t_1+s_1\\le 3r$,\n     $0\\le t_2+s_2\\le 2(r-1)$, $0\\le t_3+s_3\\le r-1$,\n     $1\\le t_1+t_2+t_3\\le min\\{3r-3, j\\}$,\n     $1\\le s_1+s_2+s_3\\le min\\{3r-3, n-j\\}$,\n     $0\\le t_2+t_3\\le r-1$, $0\\le s_2+s_3\\le r-1$ and\n $t_i(s_i)\\ge 0$ for $1\\le i\\le 3$.\n    \\end{itemize}\n\\end{theorem}\n\n\n\\pf If $d$ satisfies $(1)$ and $(2)$, then any $r$-uniform\nrealization of $d$ is maximally edge-connected by Corollary\n\\ref{28c} corresponding to $k=3$. Now suppose that $d$ satisfies\n$(1)$ and $(2)$ and there exists a non-super-$\\lambda$ $r$-uniform\nrealization $H$ consisting of two subhypergraphs $C_1$ and $C_2$\njoined by three edges $e_1, e_2$ and $e_3$, where\n$V(C_2)=V(H)\\setminus V(C_1)$ and $|V(C_1)|\\le |V(C_2)|$. Denote\n$E_0=\\{e_1, e_2,e_3\\}$ and let $|V(C_1)|=j$. Note that $r\\le j \\le\n\\lfloor{n\\over 2}\\rfloor$. Let $U_i(V_i)$ be the set of vertices in\n$V(H_1)(V(H_2))$, any one of which belong to exactly $t_i(s_i)$\nhyperedges of $e_1, e_2,\\cdots, e_{m-1}$, where $1\\le i\\le m-1$.\nThen we know that $t_1+2t_2+3t_3+s_1+2s_2+3s_3=3r$, $3\\le t_1+s_1\\le\n3r$, $0\\le t_2+s_2\\le 2(r-1)$, $0\\le t_3+s_3\\le r-1$, $1\\le\nt_1+t_2+t_3\\le min\\{3r-3, j\\}$, $1\\le s_1+s_2+s_3\\le min\\{3r-3,\nn-j\\}$, $0\\le t_2+t_3\\le r-1$, $0\\le s_2+s_3\\le r-1$ and $t_i(\ns_i)\\ge 0$ for $1\\le i\\le 3$.\n\nWe can see that the degree of each vertex in $V(C_1)-E_0$ is at most\n$j-1 \\choose r-1$, the degree of each vertex in $U_1$ is at most\n${j-1 \\choose r-1}+1$, the degree of each vertex in $U_2$ is at most\n${j-1 \\choose r-1}+2$ and the degree of each vertex in $U_3$ is at\nmost ${j-1 \\choose r-1}+3$. So we have $d_{j-t_1-t_2-t_3}\\le\n{j-1\\choose r-1}$, $d_{j-t_2-t_3}\\le {j-1\\choose r-1}+1$,\n$d_{j-t_3}\\le {j-1\\choose r-1}+2$ and $d_j\\le {j-1\\choose r-1}+3$.\n\nIf $j<{n\\over 2}$, the degree of each vertex in $C_1$ is at most\n${n-j-1\\choose r-1}$ since ${j-1\\choose r-1}+3\\le {n-j-2\\choose\nr-1}+3\\le {n-j-2\\choose r-1}+{n-j-3\\choose r-2}+2 \\le {n-j-2\\choose\nr-1}+{n-j-3\\choose r-2}+{n-j-3\\choose r-3}={n-j-2\\choose\nr-1}+{n-j-2\\choose r-2}={n-j-1\\choose r-1}$ for $r\\ge 4$. Moreover,\nthe degree of each vertex in $V(C_2)-E_0$ is at most ${n-j-1\\choose\nr-1}$, the degree of each vertex in $V_1$ is at most ${n-j-1\\choose\nr-1}+1$, the degree of each vertex in $V_2$ is at most\n${n-j-1\\choose r-1}+2$ and the degree of each vertex in $V_3$ is at\nmost ${n-j-1\\choose r-1}+3$. Then we have\n$d_{n-s_1-s_2-s_3}\\le{n-j-1\\choose r-1}$,\n$d_{n-s_2-s_3}\\le{n-j-1\\choose r-1}+1$, $d_{n-s_3}\\le{n-j-1\\choose\nr-1}+2$ and $d_{n}\\le{n-j-1\\choose r-1}+3$, contrary to $(1)$.\n\nIf $j={n\\over 2}$ for even $n$, the degree of each vertex in\n$V(H)-E_0$ is at most ${n\\over 2}-1 \\choose r-1$, the degree of each\nvertex in $V_1\\cup U_1$ is at most ${{n\\over 2}-1 \\choose r-1}+1$,\nthe degree of each vertex in $V_2\\cup U_2$ is at most ${{n\\over 2}-1\n\\choose r-1}+2$, and the degree of each vertex in $V_3\\cup U_3$ is\nat most ${{n\\over 2}-1 \\choose r-1}+3$. Therefore,\n$d_{n-t_1-t_2-t_3-s_1-s_2-s_3}\\le {{n\\over 2}-1\\choose r-1}$,\n$d_{n-t_2-t_3-s_2-s_3}\\le {{n\\over 2}-1\\choose r-1}+1$,\n$d_{n-t_3-s_3}\\le {{n\\over 2}-1\\choose r-1}+2$ and $d_{n}\\le{{n\\over\n2}-1\\choose r-1}+3$, contrary to $(2)$.\\qed\n\n\n\\begin{theorem}\\label{32t}\n    For integers $r\\ge 2$ and $\\delta \\ge 1$. Let $d=(d_1, d_2, \\cdots, d_n)$ be an $r$-uniform hypergraphic sequence with $\\delta=d_1\\le d_2\\le\\cdots\\le d_n$. If $d$ satisfies each of the following, then $d$ is forcibly super edge-connected:\n\n    \\begin{itemize}\n\\item [$(1)$] $d_{j-t_1-t_2-\\cdots-t_{\\delta}}\\le {j-1\\choose r-1}$, $d_{j-t_2-\\cdots-t_{\\delta}}\n        \\le {j-1\\choose r-1}+1$,\n        $\\cdots$, $d_{j-t_{\\delta}}\\le {j-1\\choose r-1}+\\delta-1$ and $d_j\\le {j-1\\choose r-1}+\\delta$\n        implies $d_{n-s_1-\\cdots-s_{\\delta}}\\ge{n-j-1\\choose r-1}+1$ or\n        $d_{n-s_2-\\cdots-s_{\\delta}}\\ge{n-j-1\\choose r-1}+2$,$\\cdots$, or\n        $d_{n-s_{\\delta}}\\ge{n-j-1\\choose r-1}+\\delta$ or $d_{n}\\ge{n-j-1\\choose r-1}+\\delta+1$\n         for $r+1\\le j\\le j^\\ast$.\n\\item [$(2)$] $d_{j-t_1-\\cdots-t_{\\delta}}\\le\n         {n-j-1\\choose r-1}-\\vartriangle$, $d_{j-t_2-\\cdots-t_{\\delta}}\n         \\le {n-j-1\\choose r-1}+1-\\vartriangle$,$\\cdots$,\n         $d_{j-t_\\vartriangle-\\cdots-t_{\\delta}} \\le {n-j-1\\choose r-1}-1$,\n         $\\cdots$, $d_{j-t_{\\delta}} \\le {n-j-1\\choose\n         r-1}+\\delta-1-\\vartriangle$ and $d_j \\le {n-j-1\\choose\n         r-1}+\\delta-\\vartriangle$ implies\n          $d_{n-t_{\\vartriangle+1}-\\cdots-t_{\\delta}-s_1-\\cdots-s_{\\delta}} \\ge {n-j-1\\choose\n          r-1}+1$, $\\cdots$, or\n          $d_{n-t_{\\delta}-s_{\\delta-\\vartriangle}-\\cdots-s_{\\delta}} \\ge {n-j-1\\choose r-1}+\\delta-\\vartriangle$\nor\n          $d_{n-s_{\\delta+1-\\vartriangle}-\\cdots-s_{\\delta}} \\ge {n-j-1\\choose\n          r-1}+\\delta+1-\\vartriangle $, $\\cdots$, or $d_{n-s_{\\delta}} \\ge {n-j-1\\choose\n          r-1}+\\delta$ or $d_n \\ge {n-j-1\\choose\n          r-1}+\\delta+1$  for  $j^\\ast<j<{n\\over 2}$,\n          where $\\vartriangle={n-j-1\\choose  r-1}-{j-1\\choose\n          r-1}$.\n\n\n        \\item [$(3)$] $d_{n-t_1-\\cdots-t_{\\delta}-s_1-\\cdots-s_{\\delta}}\\le\n         {{n\\over 2}-1\\choose r-1}$, $d_{n-t_2-\\cdots-t_{\\delta}-s_2-\\cdots-s_{\\delta}}\n         \\le {{n\\over 2}-1\\choose r-1}+1$,$\\cdots$, and $d_{n-t_{\\delta}-s_{\\delta}}\n         \\le {{n\\over 2}-1\\choose r-1}+\\delta-1$ implies $d_n\n         \\ge {{n\\over 2}-1\\choose r-1}+\\delta+1$ for even $n\\ge 2r+2$,\nwhere $(t_1+2t_2+\\cdots+\\delta t_{\\delta})+(s_1+2s_2+\\cdots+\\delta\ns_{\\delta})=\\delta r$,\n        $\\delta\\le t_1+s_1\\le \\delta r$,\n        $0\\le t_2+s_2\\le \\lfloor{\\delta\\over 2}\\rfloor(r-1)$, $\\cdots$, $0\\le t_{\\delta}+s_{\\delta}\\le \\lfloor{\\delta\\over {\\delta}}\\rfloor(r-1)$,\n        $1\\le t_1+t_2+\\cdots+t_{\\delta}\\le \\min\\{\\delta r-\\delta, j\\}$,     $1\\le s_1+s_2+\\cdots+s_{\\delta}\\le \\min\\{\\delta r-\\delta, n-j\\}$,   $0\\le t_2+\\cdots+t_{\\delta}\\le \\lfloor{\\delta\\over 2}\\rfloor(r-1)$ if $\\delta\\ge 2$,\n        $0\\le t_3+\\cdots+t_{\\delta}\\le \\lfloor{\\delta\\over 3}\\rfloor(r-1),\\cdots$,\n        $0\\le t_{\\delta-1}+t_{\\delta}\\le \\lfloor{\\delta\\over \\delta-1}\\rfloor(r-1)$,\n        $0\\le s_2+\\cdots+s_{\\delta}\\le \\lfloor{\\delta\\over 2}\\rfloor(r-1)$ if $\\delta\\ge 2$,\n        $0\\le s_3+\\cdots+s_{\\delta}\\le \\lfloor{\\delta\\over 3}\\rfloor(r-1)$ if $\\delta\\ge 3$,$\\cdots$,\n        $0\\le s_{\\delta-1}+s_{\\delta}\\le \\lfloor{\\delta\\over \\delta-1}\\rfloor(r-1)$ and\n        $t_i( s_i)\\ge 0$ for $1\\le i\\le \\delta$.\n\n        \\end{itemize}\n\\end{theorem}\n\n\n\n\\pf If $d$ satisfies $(1)$ and $(3)$, then $d$ is forcibly maximally\nedge-connected\n by Corollary \\ref{28c}. Suppose that $d$ is not forcibly super edge-connected.\n Then there exists a non-super edge-connected $r$-uniform realization $H$,\n with $\\lambda(H)=\\delta$, consisting of two subhypergraphs $C_1$ and $C_2$\n joined by $\\delta$ edges $e_1, \\cdots, e_{\\delta}$,\n where $V(C_2)=V(H)\\setminus V(C_1)$ and $|V(C_1)|\\le |V(C_2)|$.\n Let $|V(C_1)|=j$. Note that $r\\le j \\le \\lfloor{n\\over 2} \\rfloor$.\nLet $U_i(V_i)$ be the set of vertices in $V(H_1)(V(H_2))$, any one\nof which belong to exactly $t_i(s_i)$ hyperedges of $e_1, e_2,\n\\cdots, e_{\\delta}$, where $1\\le i\\le \\delta$, then we know that\n$(t_1+2t_2+\\cdots+\\delta t_{\\delta})+(s_1+2s_2+\\cdots+\\delta\ns_{\\delta})=\\delta r$, $\\delta\\le t_1+s_1\\le \\delta r$, $0\\le\nt_2+s_2\\le \\lfloor{\\delta\\over 2}\\rfloor(r-1)$, $\\cdots$, $0\\le\nt_{\\delta}+s_{\\delta}\\le \\lfloor{\\delta\\over {\\delta}}\\rfloor(r-1)$,\n$1\\le t_1+t_2+\\cdots+t_{\\delta}\\le \\min\\{\\delta r-\\delta, j\\}$,\n$1\\le s_1+s_2+\\cdots+s_{\\delta}\\le \\min\\{\\delta r-\\delta, n-j\\}$,\n$0\\le t_2+\\cdots+t_{\\delta}\\le \\lfloor{\\delta\\over 2}\\rfloor(r-1),\n\\cdots$, $0\\le t_{\\delta-1}+t_{\\delta}\\le \\lfloor{\\delta\\over\n\\delta-1}\\rfloor(r-1)$, $0\\le s_2+\\cdots+s_{\\delta}\\le\n\\lfloor{\\delta\\over 2}\\rfloor(r-1), \\cdots$, $0\\le\ns_{\\delta-1}+s_{\\delta}\\le \\lfloor{\\delta\\over\n\\delta-1}\\rfloor(r-1)$ and $t_i( s_i)\\ge 0$ for $1\\le i\\le \\delta$.\n\nWe can see that there are at most  $t_1+t_2+\\cdots+t_{\\delta}$\nvertices in $C_1$ of degree larger than $j-1 \\choose r-1$, at most\n$t_2+\\cdots+t_{\\delta}$ vertices in $C_1$ of degree larger than\n${j-1 \\choose r-1}+1$,$\\cdots$, and at most  $t_{\\delta}$ vertices\nin $C_1$ of degree larger than ${j-1 \\choose r-1}+\\delta-1$, while\nthe degree of each vertex in $C_1$ is at most ${j-1 \\choose\nr-1}+\\delta$. Thus we have $d_{j}\\le {j-1\\choose r-1}+\\delta$,\n$d_{j-t_{\\delta}}\\le {j-1\\choose r-1}+\\delta-1$, $\\cdots$,\n$d_{j-t_2-\\cdots-t_{\\delta}}\\le {j-1\\choose r-1}+1$ and\n$d_{j-t_1-t_2-\\cdots-t_{\\delta}}\\le {j-1\\choose r-1}$.\n\n\n\n\nIf $j<{n\\over 2}$, by similar argument in the proof of Theorem\n\\ref{23t}, we can obtain some inequalities, which contradicts $(1)$\nand $(2)$, respectively.\n\n\nIf $j={n\\over 2}$ for even $n$, analogously,\n we have  $d_{n-t_1-t_2-\\cdots-t_{\\delta}-s_1-\\cdots-s_{\\delta}}\\le {{n\\over 2}-1\\choose r-1}$,\n  $d_{n-t_2-\\cdots-t_{\\delta}-s_2-\\cdots-s_{\\delta}}\\le {{n\\over 2}-1\\choose r-1}+1,\\cdots,$\n   $d_{n-t_{\\delta}-s_{\\delta}}\\le\n   {{n\\over 2}-1\\choose r-1}+\\delta-1$ and\n   $d_n \\le {{n\\over 2}-1\\choose r-1}+\\delta$, contrary to $(3)$. \\qed\n\nWhen $t_1=(r-1)\\delta, t_2=t_3=\\cdots=t_{\\delta}=0$, $s_1=\ns_2=\\cdots=s_{\\delta-1}=0$ and $s_{\\delta}=1$ in Theorem \\ref{32t},\nwe obtain the following immediate consequence for $r$-uniform\nhypergraphic sequence with minimum $\\delta$ to be super\nedge-connected, which generalize the relevant result of Liu et al.\n[11] since $g(\\delta, r)\\ge (r-1)\\delta$.\n\n\\begin{corollary}\\label{33c}\n    For integers $r\\ge2$ and $\\delta \\ge 1$. Let $d=(d_1, d_2, \\cdots, d_n)$ be an\n    $r$-uniform hypergraphic sequence with $\\delta=d_1\\le d_2\\le\\cdots\\le d_n$.\nIf $d_{j-(r-1)\\delta}\\le {j-1\\choose r-1}$ and $d_j\\le {j-1\\choose\nr-1}+\\delta$ implies $d_{n-1}\\ge{n-j-1\\choose r-1}+\\delta$ or\n$d_{n}\\ge{n-j-1\\choose r-1}+\\delta+1$ for $(r-1)\\delta\\le j \\le\n\\lfloor{n\\over 2}\\rfloor$, then $d$ is forcibly super\nedge-connected.\n\\end{corollary}\n\n\nIf $r=2$ in Corollary \\ref{33c}, then we obtain the following result for a graphic sequence\nwith minimum element at least $1$ to be forcibly super edge-connected.\n\n\n\\begin{corollary} ([11])\\label{34c}\n    For integer $\\delta \\ge 1$. Let $d=(d_1, d_2, \\cdots, d_n)$ be a graphic sequence with\n     $\\delta=d_1\\le d_2\\le\\cdots\\le d_n$.\nIf  $d_{j-\\delta}\\le {j-1}$ and $d_j\\le j-1+\\delta$ implies\n$d_{n-1}\\ge n-j-1+\\delta$ or $d_{n}\\ge n-j+\\delta$ for $\\delta\\le\nj\\le \\lfloor{n\\over 2}\\rfloor$, then $d$ is forcibly super\nedge-connected.\n\\end{corollary}\n\n\\section{Another sufficient degree condition for forcibly k-edge-connected hypergraph}\n\nIn [14], Yin and Guo gave a sufficient degree condition for a\ngraphic sequence to be forcibly k-edge-connected. We shall extend\nthis result to hypergraphs. That is Theorem 4.2, which contains five\n\"Bondy-Chv$\\acute{a}$tal type\" conditions.\n\n\n\\begin{theorem}([14])\\label{31t}\n    Let $d=(d_1,d_2,\\cdots,d_n)$ be a non-decreasing graphic\n    sequence, and let $k\\ge 2$ be an integer. If $d$ satisfies each\n    of the following, then $d$ is forcibly $k$-edge-connected:\n\n\n    \\begin{itemize}\n    \\item [$(1)$] $d_1\\ge k$,\n\n    \\item [$(2)$] for any integers $x,z,y$ and $j$ with $1\\le x\\le\n    k-1$, $\\lceil {k-1\\over x}\\rceil\\le z\\le k-x$,  $ z\\le y\\le k-1 $ and  $k+1\\le j\\le\n    {n-z\\over 2}$, we have that $d_{j-x}\\le j-1$,\n    $d_{j-x+1}\\le \\lfloor {k-1\\over x}\\rfloor+ j-1$ and\n    $d_j\\le z+j-1$ implies $d_{n-y}\\ge n-j$ or $d_{n-y+1}\\ge n-j+\\lfloor {k-1\\over\n    y}\\rfloor$ or $d_n\\ge n-j+min\\{x,k-y\\}$,\n\n    \\item [$(3)$] for $n\\ge 2k+2$ and $ {n-k+1\\over 2}<j\\le \\lfloor{n \\over\n    2}\\rfloor$, we have that $d_{j-1}\\le j-1$ implies\n$d_{n-k}\\ge n-j$ or $d_{n-1}\\ge n-j+1$ or $d_n\\ge j-1+k$,\n\n    \\item [$(4)$] for $n\\ge 2k+2$, $1\\le y\\le\n    k-1$ and $n$ even, we have that $d_{{n\\over 2}-k+1}\\le {n\\over 2}-1$\nimplies $d_{n-(k-1+y)}\\ge {n\\over 2}$ or\n    $d_{n-y}\\ge {n\\over 2}+1$ or $d_{n-y+1}\\ge max\\{{n\\over 2}+1,{n\\over 2}+\\lfloor {k-1\\over\n    y}\\rfloor\\}$ or $d_n\\ge {n\\over 2}+k-y$,\n\n    \\item [$(5)$] for $n\\ge 2k+2$ and any integers $x,z,y$ and $j$\n    with $2\\le x\\le k-2$, $\\lceil {k-1\\over x}\\rceil\\le z\\le k-x$, $z\\le y\\le\n    k-1$ and $ {n-z\\over 2}<j\\le \\lfloor{n \\over\n    2}\\rfloor$,\n\n    \\item [$(5.1)$] if $j<{n-z+min\\{x,k-y\\}\\over 2}$, then we have that\n    $d_{j-x}\\le j-1$ implies\n    $d_{n-(x+y)}\\ge n-j$ or $d_{n-(x+y)+1}\\ge max\\{n-j,j+\\lfloor{k-1\\over x}\\rfloor\\}$\nor $d_{n-y+1}\\ge max\\{z+j,n-j+\\lfloor{k-1\\over y}\\rfloor\\}$ or\n$d_n\\ge n-j+min\\{x,k-y\\}$,\n\n    \\item [$(5.2)$] if $j>{n-z+min\\{x,k-y\\}\\over 2}$,\n    then we have that $d_{j-x}\\le j-1$ implies\n    $d_{n-(x+y)}\\ge n-j$ or $d_{n-(x+y)+1}\\ge max\\{n-j,j+\\lfloor{k-1\\over x}\\rfloor\\}$\nor $d_{n-x+1}\\ge max\\{n-j+min\\{x,k-y\\},j+\\lfloor{k-1\\over\nx}\\rfloor\\}$ or $d_n\\ge z+j$,\n\n\n    \\item [$(5.3)$] if $j={n-z+min\\{x,k-y\\}\\over 2}$, then we have that $d_{j-x}\\le j-1$ implies\n    $d_{n-(x+y)}\\ge n-j$ or $d_{n-(x+y)+1}\\ge max\\{n-j,j+\\lfloor{k-1\\over x}\\rfloor\\}$\nor $d_{n-(x+y)+2}\\ge max\\{n-j+\\lfloor{k-1\\over\ny}\\rfloor,j+\\lfloor{k-1\\over x}\\rfloor\\}$ or $d_n\\ge z+j$.\n\n\n\n\n    \\end{itemize}\n\\end{theorem}\n\n\n\n\n\n\n\n\nFor integers $z\\ge 1, r\\ge 2$ and $r+1\\le j\\le \\lfloor {n\\over\n2}\\rfloor$, we define $j_{0}=j(z,r)(\\ge r+1)$ to be the biggest\ninteger such that $z+{j_0-1\\choose r-1}\\le {n-j_0-1\\choose r-1}$.\nClearly, $r+1\\le j_{0}< {n \\over 2}$. Otherwise, when $j_{0}={n\n\\over 2}$, the contradiction $z\\le 0$ is derived.\n\n\n\\begin{theorem}\\label{31t}\n    Given integers $r\\ge 2$ and $k\\ge 2$. Let $d=(d_1, d_2, \\cdots, d_n)$\n    be an $r$-uniform hypergraphic sequence with $d_1\\le d_2\\le\\cdots\\le d_n$.\n    If $d$ satisfies each of the following, then $d$ is forcibly k-edge-connected:\n\n    \\begin{itemize}\n    \\item [$(1)$] $d_1\\ge k$,\n\n    \\item [$(2)$] for any integers $x,z,y$ and $j$ with $1\\le x\\le\n    (k-1)(r-1)$, $\\lceil {k-1\\over q}\\rceil\\le z\\le k-q$, $x\\equiv\n    q(mod(k-1))$, $1\\le q\\le k-1, z\\le y\\le (k-1)(r-1)$ and $r+1\\le j\\le\n    j_{0}$, we have that $d_{j-x}\\le {j-1\\choose r-1}$,\n    $d_{j-x+1}\\le \\lfloor {k-1\\over q}\\rfloor+ {j-1\\choose r-1}$ and\n    $d_j\\le z+{j-1\\choose r-1}$ implies $d_{n-y}\\ge {n-j-1\\choose\n    r-1}+1$ or $d_{n-y+1}\\ge {n-j-1\\choose r-1}+\\lfloor {k-1\\over\n    R}\\rfloor+1$ or $d_n\\ge {n-j-1\\choose r-1}+min\\{q,k-R\\}+1$,\n    where $y\\equiv R(mod(k-1))$ and $1\\le R\\le k-1$.\n\n    \\item [$(3)$] for $n\\ge 2r+2$ and any integers $x$ and $j$ with\n$r-1\\le x\\le (k-1)(r-2)+1$ and $ j_{0}<j\\le \\lfloor{n \\over\n    2}\\rfloor$,\nwe have that $d_{j-x}\\le {j-1\\choose r-1}$ implies\n    $d_{n-(x+k-1)}\\ge {n-j-1\\choose r-1}+1$ or $d_{n-x}\\ge {n-j-1\\choose\n    r-1}+2$ or $d_n\\ge {j-1\\choose r-1}+k$.\n\n\\item [$(4)$] for $n\\ge 2r+2$ and any integers $x$, $y$ and $j$ with\n$1\\le x\\le (k-1)(r-2)$, $k-1\\le y\\le (k-1)(r-1)$ and $ j_{0}<j\\le\n\\lfloor{n \\over\n    2}\\rfloor$,\nwe have that $d_{j-x}\\le {j-1\\choose r-1}$ implies\n    $d_{n-(x+y)}\\ge {n-j-1\\choose r-1}+1$ or\n    $d_{n-x}\\ge {n-j-1\\choose\n    r-1}+2$ or $d_n\\ge {j-1\\choose r-1}+k$.\n\n\n\n\n\n\n\n\n    \\item [$(5)$] for $n\\ge 2r+2$, $k-1\\le x\\le (k-1)(r-1)$, $k-1\\le y\\le\n    (k-1)(r-1)$ and $n$ even, we have that $d_{{n\\over 2}-x}\\le {{n\\over 2}-1\\choose\n    r-1}$ implies $d_{n-(x+y)}\\ge {{n\\over 2}-1\\choose r-1}+1$ or\n    $d_{n-y}\\ge {{n\\over 2}-1\\choose\n    r-1}+2$ or $d_{n-y+1}\\ge max\\{{{n\\over 2}-1\\choose r-1}+2,{{n\\over 2}-1\\choose r-1}+\\lfloor {k-1\\over\n    R}\\rfloor+1\\}$ or $d_n\\ge {{n\\over 2}-1\\choose r-1}+k-R+1$.\n\n    \\item [$(6)$] for $n\\ge 2r+2$ and any integers $x,z,y$ and $j$\n    with $2\\le z\\le k-2$, $1\\le x\\le (k-1)(r-1)$, $z\\le y\\le\n    (k-1)(r-1)$ and $ j_{0}<j\\le \\lfloor{n \\over\n    2}\\rfloor$,\n\n    \\item [$(6.1)$] if $z+{j-1\\choose r-1}<{n-j-1\\choose\n    r-1}+min\\{q,k-R\\}$, then we have that\n    $d_{j-x}\\le {j-1\\choose r-1}$ implies\n    $d_{n-(x+y)}\\ge {n-j-1\\choose r-1}+1$ or $d_{n-(x+y)+1}\\ge max\\{{n-j-1\\choose\n    r-1},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}+1$ or $d_{n-y+1}\\ge max\\{{j-1\\choose\n    r-1}+z,{n-j-1\\choose r-1}+\\lfloor{k-1\\over R}\\rfloor\\}+1$ or\n     $d_n\\ge {n-j-1\\choose r-1}+min\\{q,k-R\\}+1$.\n\n    \\item [$(6.2)$] if $z+{j-1\\choose r-1}>{n-j-1\\choose\n    r-1}+min\\{q,k-R\\}$, then we have that $d_{j-x}\\le {j-1\\choose r-1}$ implies\n    $d_{n-(x+y)}\\ge {n-j-1\\choose r-1}+1$ or $d_{n-(x+y)+1}\\ge max\\{{n-j-1\\choose\n    r-1},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}+1$ or\n$d_{n-x+1}\\ge max\\{{n-j-1\\choose\n    r-1}+min\\{q,k-R\\},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}+1$ or\n$d_n\\ge {j-1\\choose r-1}+z+1$.\n\n    \\item [$(6.3)$] if $z+{j-1\\choose r-1}={n-j-1\\choose\n    r-1}+min\\{q,k-R\\}$, then we have that $d_{j-x}\\le {j-1\\choose r-1}$ implies\n    $d_{n-(x+y)}\\ge {n-j-1\\choose r-1}+1$\n    or $d_{n-(x+y)+1}\\ge max\\{{n-j-1\\choose r-1},\n{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}+1$ or\n$d_{n-(x+y)+2}\\ge max\\{{j-1\\choose r-1}+\\lfloor{k-1\\over\nq}\\rfloor,{n-j-1\\choose r-1}+\\lfloor{k-1\\over R}\\rfloor\\}+1$ or\n$d_n\\ge {j-1\\choose r-1}+z+1$.\n\n    \\end{itemize}\n\\end{theorem}\n\n\\pf Suppose $d$ satisfies (1)-(6) of Theorem 4.2, but is not\nforcibly k-edge-connected. Then $d$ has a realization $H$ with an\nedge-cut $E_0$ of $k-1$ hyperedges joining two subhypergraphs $H_1$\nand $H_2$, with $|V(H_1)|=j$, $|V(H_2)|=n-j$ and $|V(H_1)|\\le\n|V(H_2)|$. Note that $r+1\\le j\\le \\lfloor {n\\over 2}\\rfloor$ and\n$n\\ge 2r+2$, since $j\\le r$ implies there exists a veretex of degree\nless than 2, contradicting $d_1\\ge k\\ge 2$.\n\n\n\n\n By $E_H(V(H_1),V(H_2))$ we\ndenote the set of edges of $H$ which connect a vertex from $V(H_1)$\nto one of $V(H_2)$. By $\\partial _H V(H_1)$ we denote the set of\nvertices of $H$ from $V(H_1)$ which have at least one neighbor\noutside $V(H_1)$. Let $F=(X\\cup Y,E_H(X,Y))$, $X=\\partial _H V(H_1)$\nand $Y=\\partial _H V(H_2)$. Denote $x=|X|,y=|Y|$ and $z=max\\{d_F\n(v)|v\\in X\\}$. Then $1\\le x\\le\n    (k-1)(r-1)$, $\\lceil {k-1\\over q}\\rceil\\le z\\le k-q$\n    and  $ z\\le y\\le (k-1)(r-1)$, where $x\\equiv\n    q(mod(k-1))$ and $1\\le q\\le k-1$. We consider two cases.\n\n{\\bf Case  1.} $ r+1\\le j\\le j_{0}$.\n\nLet $z'=min\\{d_F (v)|v\\in X\\}$, then $z'\\le \\lfloor {k-1\\over\nq}\\rfloor$. At most $x$ vertices in $H_1$ can have degree larger\nthan ${j-1\\choose r-1}$ and at most $x-1$ vertices in $H_1$ can have\ndegree larger than ${j-1\\choose r-1}+z'$, and so $d_{j-x}\\le\n{j-1\\choose r-1}$,\n    $d_{j-x+1}\\le \\lfloor {k-1\\over q}\\rfloor+ {j-1\\choose r-1}$.\n    Also, no vertex in $H_1$ can have\ndegree larger than ${j-1\\choose r-1}+z$, and so $d_j\\le {j-1\\choose\nr-1}+z$. No vertex in $H_2 \\backslash Y$ has degree at most\n${n-j-1\\choose r-1}$, and since ${j-1\\choose r-1}+z\\le {n-j-1\\choose\nr-1}$, we have $d_{n-y}\\le {n-j-1\\choose r-1}$.\n\nLet $w=max\\{d_F (v)|v\\in Y\\}$ and $w'=min\\{d_F (v)|v\\in Y\\}$, then\n$w\\le min\\{q,k-R\\}$ and $w'\\le \\lfloor {k-1\\over R}\\rfloor$, where\n$y\\equiv R(mod(k-1))$ and $1\\le R\\le k-1$. Thus, at most $y-1$\nvertices in $H_2$ can have degree larger than ${n-j-1\\choose\nr-1}+w'$ and each vertex in $H_2$ can has degree at most\n${n-j-1\\choose r-1}+w$. Thus $d_{n-y+1}\\le {n-j-1\\choose\nr-1}+\\lfloor {k-1\\over R}\\rfloor$ and $d_n\\le {n-j-1\\choose\nr-1}+min\\{q,k-R\\}$. This contradicts (2).\n\n{\\bf Case  2.} $ j_{0}<j\\le \\lfloor{n \\over 2}\\rfloor$.\n\nBy the definition of $j_0$, we know that ${n-j-1\\choose r-1}<\n{j-1\\choose r-1}+z $. Now three cases arise.\n\n{\\bf Subcase 2.1.}  $z=1$.\n\n Note that $k-1\\le x\\le (k-1)(r-1)$, $k-1\\le y\\le\n    (k-1)(r-1)$. It follows clearly from ${j-1\\choose r-1}\\le\n{n-j-1\\choose r-1} $ and ${n-j-1\\choose r-1}< {j-1\\choose r-1}+1 $\nthat $n$ is even and $j={n\\over 2}$.\n\n\n\nAt most $x$ vertices in $H_1$ can have degree larger than ${{n\\over\n2}-1\\choose r-1}$ and at most $y$ vertices in $H_2$ can have degree\nlarger than ${{n\\over 2}-1\\choose r-1}$, so $d_{{n\\over\n2}-x}\\le{{n\\over 2}-1\\choose r-1}$ and $d_{n-(x+y)}\\le{{n\\over\n2}-1\\choose r-1}$. No vertex in $H_1$ can have degree larger than\n${{n\\over 2}-1\\choose r-1}+1$, and at most $y$ vertices in $H_2$ can\nhave degree larger than ${{n\\over 2}-1\\choose r-1}+1$ and at most\n$y-1$ vertices in $H_2$ can have degree larger than ${{n\\over\n2}-1\\choose r-1}+\\lfloor {k-1\\over R}\\rfloor$, and so\n$d_{n-y}\\le{{n\\over 2}-1\\choose r-1}+1$ and $d_{n-y+1}\\le\nmax\\{{{n\\over 2}-1\\choose r-1}+1,{{n\\over 2}-1\\choose r-1}+\\lfloor\n{k-1\\over R}\\rfloor\\}$. No vertex in $H_2$ can have degree larger\nthan ${{n\\over 2}-1\\choose r-1}+k-R$, and so $d_n\\le {{n\\over\n2}-1\\choose r-1}+k-R$. Thus (5) fails, a contradiction.\n\n{\\bf Subcase 2.2.} $z=k-1$.\n\n{\\bf Subsubcase 2.2.1.} $y=z$.\n\nNote that $r-1\\le x\\le (r-2)(k-1)+1$. There are at most $x$ vertices\nin $H_1$ can have degree larger than ${j-1\\choose r-1}(\\le\n{n-j-1\\choose r-1})$, at most $k-1$ vertices in $H_2$ can have\ndegree larger than ${n-j-1\\choose r-1}$, and so $d_{j-x}\\le\n{j-1\\choose r-1}$ and $d_{n-k+1-x}\\le {n-j-1\\choose r-1}$. No vertex\nin $H_1$ can have degree larger than ${j-1\\choose r-1}+k-1$, and no\nvertex in $H_2$ can have degree larger than ${n-j-1\\choose r-1}+1$.\nSince $z+{j-1\\choose r-1}>{n-j-1\\choose r-1}$, $k-1+{j-1\\choose\nr-1}\\ge {n-j-1\\choose r-1}+1$, and so $d_{n-x}\\le {n-j-1\\choose\nr-1}+1$ and $d_n\\le {j-1\\choose r-1}+k-1$. Thus (3) fails, a\ncontradiction.\n\n{\\bf Subsubcase 2.2.2.} $z<y\\le (r-1)(k-1)$.\n\nNote that $1\\le x\\le (r-2)(k-1)$. At most $x$ vertices in $H_1$ can\nhave degree larger than ${j-1\\choose r-1}(\\le {n-j-1\\choose r-1})$,\nat most $y(>k-1)$ vertices in $H_2$ can have degree larger than\n${n-j-1\\choose r-1}$, and so $d_{j-x}\\le {j-1\\choose r-1}$,\n$d_{n-x-y}\\le {n-j-1\\choose r-1}$, $d_{n-x-y+1}\\le {n-j-1\\choose\nr-1}+1$ and $d_{n-x}\\le {n-j-1\\choose r-1}+1$. No vertex in $H_1$\ncan have degree larger than ${j-1\\choose r-1}+k-1$, and no vertex in\n$H_2$ can have degree larger than ${n-j-1\\choose r-1}+1(\\le\n{n-j-1\\choose r-1}+k-1)$. So $d_n\\le {j-1\\choose r-1}+k-1$. Thus (4)\nfails, a contradiction.\n\n\n\n\n\n\n\n\n{\\bf Subcase 2.3.}    $2\\le z\\le k-2.$\n\n{\\bf Subsubcase 2.3.1.}  $z+{j-1\\choose r-1}<{n-j-1\\choose\nr-1}+min\\{q,k-R\\}$.\n\n\nThere are at most $x$ vertices in $H_1$ can have degree larger than\n${j-1\\choose r-1}(\\le {n-j-1\\choose r-1})$, and at most $y$ vertices\nin $H_2$ can have degree larger than $ {n-j-1\\choose r-1}$, and so\n$d_{n-(x+y)}\\le {n-j-1\\choose r-1}$. At most $x-1$ vertices in $H_1$\ncan have degree larger than $max\\{{n-j-1\\choose\n    r-1},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$ and at most $y$ vertices in\n$H_2$ can have degree larger than $max\\{{n-j-1\\choose\n    r-1},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$, and so\n$d_{n-(x+y)+1}\\le max\\{{n-j-1\\choose\n    r-1},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$.\n    No vertex in $H_1$ can have\ndegree larger than $max\\{{j-1\\choose\n    r-1}+z,{n-j-1\\choose r-1}+\\lfloor{k-1\\over R}\\rfloor\\}$, and at most $y-1$ vertices in\n$H_2$ can have degree larger than $max\\{{j-1\\choose\n    r-1}+z,{n-j-1\\choose r-1}+\\lfloor{k-1\\over R}\\rfloor\\}$, and so $d_{n-y+1} \\le max\\{{j-1\\choose\n    r-1}+z,{n-j-1\\choose r-1}+\\lfloor{k-1\\over R}\\rfloor\\}$.\nNo vertex in $H_1$ can have degree larger than ${j-1\\choose r-1}+z$\nand no vertex in $H_2$ can have degree larger than ${n-j-1\\choose\nr-1}+min\\{q,k-R\\}$, and so $d_n\\le {n-j-1\\choose r-1}+min\\{q,k-R\\}$,\ncontrary to (6.1).\n\n\n\n{\\bf Subsubcase 2.3.2.} $z+{j-1\\choose r-1}>{n-j-1\\choose\nr-1}+min\\{q,k-R\\}$.\n\nAt most $x$ vertices in $H_1$ can have degree larger than\n${j-1\\choose r-1}(\\le {n-j-1\\choose r-1})$, and at most $y$ vertices\nin $H_2$ can have degree larger than $ {n-j-1\\choose r-1}$, and so\n$d_{n-(x+y)}\\le {n-j-1\\choose r-1}$. At most $x-1$ vertices in $H_1$\ncan have degree larger than $max\\{{n-j-1\\choose\n    r-1},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$ and at most $y$ vertices in\n$H_2$ can have degree larger than $max\\{{n-j-1\\choose\n    r-1},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$, and so\n$d_{n-(x+y)+1}\\le max\\{{n-j-1\\choose\n    r-1},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$.\n No vertex in $H_2$ can have\ndegree larger than $max\\{{n-j-1\\choose\n    r-1}+min\\{q,k-R\\},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$, and at most $x-1$ vertices in\n$H_1$ can have degree larger than $max\\{{n-j-1\\choose\n    r-1}+min\\{q,k-R\\},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$,  and so $d_{n-x+1}\\le max\\{{n-j-1\\choose\n    r-1}+min\\{q,k-R\\},{j-1\\choose r-1}+\\lfloor{k-1\\over\n    q}\\rfloor\\}$.\nNo vertex in $H_1$ can have degree larger than ${j-1\\choose r-1}+z$\nand no vertex in $H_2$ can have degree larger than ${n-j-1\\choose\nr-1}+min\\{q,k-R\\}$, and so $d_n\\le {j-1\\choose r-1}+z$, contrary to\n(6.2).\n\n\n{\\bf Subsubcase 2.3.3.} $z+{j-1\\choose r-1}={n-j-1\\choose\nr-1}+min\\{q,k-R\\}$.\n\nThere are at most $x$ vertices in $H_1$ can have degree larger than\n${j-1\\choose r-1}(\\le {n-j-1\\choose r-1})$, and at most $y$ vertices\nin $H_2$ can have degree larger than $ {n-j-1\\choose r-1}$, and so\n$d_{n-(x+y)}\\le {n-j-1\\choose r-1}$. At most $x-1$ vertices in $H_1$\ncan have degree larger than $max\\{{n-j-1\\choose\n    r-1},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$ and at most $y$ vertices in\n$H_2$ can have degree larger than $max\\{{n-j-1\\choose\n    r-1},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$, and so\n$d_{n-(x+y)+1}\\le max\\{{n-j-1\\choose\n    r-1},{j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$.\nAt most $x-1$ vertices in $H_1$ can have degree larger than\n$max\\{{n-j-1\\choose\n    r-1}+\\lfloor{k-1\\over R}\\rfloor, {j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$ and at most $y-1$ vertices in\n$H_2$ can have degree larger than $max\\{{n-j-1\\choose\n    r-1}+\\lfloor{k-1\\over R}\\rfloor, {j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$, and so\n$d_{n-(x+y)+2}\\le max\\{{n-j-1\\choose\n    r-1}+\\lfloor{k-1\\over R}\\rfloor, {j-1\\choose r-1}+\\lfloor{k-1\\over q}\\rfloor\\}$.\nNo vertex in $H_1$ can have degree larger than ${j-1\\choose\n    r-1}+z$, and no vertex in\n$H_2$ can have degree larger than ${n-j-1\\choose r-1}+min\\{q,k-R\\}$,\n     and so $d_n\\le {j-1\\choose r-1}+z$,\ncontrary to (6.3). \\qed\n\nWhen $k=2$ in Theorem 4.2, we obtain $(z,q,R)=(1,1,1)$ and the\nfollowing conclusion.\n\n\\begin{corollary} \\label{34c}\n    Given integer $r \\ge 2$. Let $d=(d_1, d_2, \\cdots, d_n)$ be an\n    r-uniform hypergraphic sequence with $d_1\\le d_2\\le\\cdots\\le d_n$.\nIf $d$ satisfies the following conditions, then $d$ is forcibly\n2-edge-connected:\n\n\n \\begin{itemize}\n    \\item [$(1)$] $d_1\\ge 2$,\n\n    \\item [$(2)$] for any integers $x,y$ and $j$ with $1\\le x\\le r-1,1\\le y\\le\n    r-1$ and $r+1\\le j< {n\\over 2}$, we have that $d_{j-x}\\le {j-1\\choose\n    r-1}$ and $d_j\\le {j-1\\choose r-1}+1$ implies $d_{n-y}\\ge {n-j-1\\choose\n    r-1}+1$ or $d_n\\ge {n-j-1\\choose r-1}+2$,\n\n\n    \\item [$(3)$]  for $n\\ge 2r+2$ and $n$ even, we have that\n     $d_{{n\\over 2}}\\le {{n\\over 2}-1\\choose r-1}$ implies\n$d_{n-2}\\ge {{n\\over 2}-1\\choose r-1}+1$ or $d_n\\ge {{n\\over\n2}-1\\choose r-1}+2$.\n\n\n\n\n\n    \\end{itemize}\n\n\n\n\n\\end{corollary}\n\n\nIf $k=3$, then from condition (2) of Theorem 4.2, we obtain\n$(q,R,z)=(1,2,2),(2,2,1)$. Thus Theorem 4.2 reduces to Corollary\n4.4.\n\n\n\n\n\n\\begin{corollary} \\label{34c}\n    Given integer $r \\ge 2$. Let $d=(d_1, d_2, \\cdots, d_n)$ be an\n    r-uniform hypergraphic sequence with $d_1\\le d_2\\le\\cdots\\le d_n$.\nIf $d$ satisfies the following conditions, then $d$ is forcibly\n3-edge-connected:\n\n\n \\begin{itemize}\n    \\item [$(1)$] $d_1\\ge 3$,\n\n    \\item [$(2)$] for any integers $x,y$ and $j$ with $1\\le x\\le 2(r-1),2\\le y\\le\n    2(r-1)$ and $r+1\\le j\\le j_0$, we have that $d_{j-x}\\le {j-1\\choose\n    r-1}$ and $d_j\\le {j-1\\choose r-1}+2$ implies $d_{n-y}\\ge {n-j-1\\choose\n    r-1}+1$ or $d_n\\ge {n-j-1\\choose r-1}+2$,\n\n\n\\item [$(3)$] for any integers $x,y$ and $j$ with $1\\le x\\le 2(r-1),1\\le y\\le\n    2(r-1)$ and $r+1\\le j\\le j_0$, we have that $d_{j-x}\\le {j-1\\choose\n    r-1}$ and $d_j\\le {j-1\\choose r-1}+1$ implies $d_{n-y}\\ge {n-j-1\\choose\n    r-1}+1$ or $d_n\\ge {n-j-1\\choose r-1}+2$,\n\n\\item [$(4)$] for  $n\\ge 2r+2$ and   $j_0< j\\le \\lfloor {n\\over 2}\\rfloor$,\nwe have that $d_j\\le {j-1\\choose r-1}$ implies $d_{n-3}\\ge\n{n-j-1\\choose\n    r-1}+1$ or $d_{n-1}\\ge\n{n-j-1\\choose\n    r-1}+2$ or $d_n\\ge {n-j-1\\choose r-1}+3$,\n\n\n\n    \\item [$(5)$]  for $n\\ge 2r+2$ and $n$ even,\n$2\\le x\\le 2(r-1),2\\le y\\le\n    2(r-1)$,\nwe have that\n     $d_{{n\\over 2}-x}\\le {{n\\over 2}-1\\choose r-1}$ implies\n$d_{n-(x+y)}\\ge {{n\\over 2}-1\\choose r-1}+1$ or $d_n\\ge {{n\\over\n2}-1\\choose r-1}+2$.\n\n\n\n\n\n    \\end{itemize}\n\n\n\n\n\\end{corollary}\n\n\n\n\n\n\n\n\n\n\n\n\n    ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Modeling and problem definition}\n\\label{sec_modeling}\n\nIn this section, we introduce our stochastic fluid model for highway bottlenecks with mixed traffic consisting of both CAVs and non-CAVs (Section~\\ref{sub_modeling}) and formally define the problems that we study in the rest of this paper (Section~\\ref{sub_preliminaries}).\n\n\\subsection{Stochastic fluid queuing model}\n\\label{sub_modeling}\n\nConsider a highway section with a downstream bottleneck and an off-ramp, as illustrated in Fig.~\\ref{capacities}.  We model the highway as a tandem-link fluid queuing system in Fig.~\\ref{fig_tandem}.\n\\begin{figure}[hbt]\n\\centering\n\\subfigure[2-link highway section and main parameters.]{\n\\centering\n\\includegraphics[width=0.4\\textwidth]{.\/Images\/capacities}\n\\label{capacities}\n}\n\\subfigure[Fluid model for system in Fig.~\\ref{capacities}.]{\n\\centering\n\\includegraphics[width=0.3\\textwidth]{.\/Images\/fig_tandem_control.png}\n\\label{fig_tandem}\n}\n\\caption{Fluid queuing system for highway section with CAVs and non-CAVs.}\n\\end{figure}\nLink 1 has a \\emph{mainline capacity} $F$ (veh\/hr) and an \\emph{off-ramp capacity} $R$ (veh\/hr). Link 2 has an on-ramp with capacity $R$, which creates a bottleneck with capacity $F-R$.\\footnote{For simplicity, we assume the on-ramp capacity to be identical to the off-ramp capacity.}\nTo model the effect of coordinated platooning operations (see Fig.~\\ref{cartoon_control}), we introduce a ``virtual link'' 0 upstream to link 1, which we refer as the \\emph{gate}. In our model, this gate can temporarily hold platoons and control their rate of release in order to regulate the downstream traffic flow. Furthermore, the storage space in link 2 (i.e., the maximum queue length that it can admit before the traffic spills over to link 1) is finite, modeled as a \\emph{buffer} with space $\\Theta$ (veh). On the other hand, links 0 and 1 are assumed to have an infinite buffer space. \nHowever, the model can be extended to other typical road configurations \\cite{qian2012system}, which allows extension of our approach to more general settings. Note that our model does not account for details such as car following and lane changing.\n\nThe highway section is subject to a total demand $a>0$ (veh\/hr). This demand comprises of $\\rho a$ amount of mainline  demand that is discharged to link 2, and $(1-\\rho)a$ amount of demand exiting through the off-ramp. Out of the $\\rho a$ mainline demand, CAVs traveling in platoons amount to a fraction $\\eta\\in[0,1]$, and the remaining $(1-\\eta)$ fraction is comprised of non-CAV traffic. In this demand pattern, CAVs only contribute to the mainline traffic (with demand $\\eta\\rho a$), and the off-ramp demand $(1-\\rho)a$ is entirely comprised of non-CAVs.\\footnote{Our setup can be extended to a more general case of CAV platoons that are bound to different destinations.} We call $\\rho$ the \\emph{mainline ratio} and $\\eta$ the \\emph{platooning ratio}. \nWe refer to the total non-CAV demand $(1-\\eta\\rho)a$ as \\emph{background traffic}.\nFor CAV platoons, the mean inter-arrival time is much greater than minimal inter-arrival times, so we consider platoons as discrete arrivals. For non-CAVs, however, the mean and the minimal inter-arrival times are in the same order, so we consider background traffic as continuous fluid.\nWe consider a Poisson process rather than a general renewal process, since Poisson processes are standard models for random arrivals in transportation systems \\cite{newell13} and ensure that the model is Markovian \\cite{benaim15}.\n\nCAV platoons arrive according to a Poisson process of rate $\\lambda$ (platoons per hr), which is given by\n\\begin{align}\n    \\lambda:=\\frac{\\eta\\rho a}{l},\n    \\label{eq_lambda}\n\\end{align}\nwhere $l$ is the number of CAVs in each platoon. Typical values of $l$ are between 2 and 10 \\cite{bess+16procieee}.\nFor ease of presentation, we consider homogeneous platoon lengths. Non-homogeneous platoon lengths can be modeled as jumps with randomized magnitudes, and our Lyapunov function-based approach (see Appendices 1--3) is still valid.\nThe Poisson process captures the randomness of the arrival of platoons.\n\n\nWe are now ready to introduce the stochastic fluid model. \n\\begin{dfn} \\textbf{The stochastic fluid model} is defined as the tuple $\\langle\\mathcal Q,\\mathcal U,G,\\lambda,\\mathcal V,S\\rangle$, where\n\\begin{itemize}\n    \\item[-] $\\mathcal Q:=\\mathbb R_{\\ge0}^3\\times[0,\\Theta]$ is the state space as well as the set of initial conditions,\n    \\item[-] $\\mathcal U\\subseteq\\mathbb R_{\\ge0}$ is the set of control inputs to the vector field $G$ (called the ``gate discharge''),\n    \\item[-] $G:\\mathcal Q\\times\\mathcal U\\to\\mathbb R^4$ is a vector field such that $(d\/dt)Q(t)=G(Q(t))$,\n    \\item[-] $\\lambda\\in\\mathbb R_{\\ge0}$ is the Poisson rate at which resets occur (called the ``arrival rate''),\n    \\item[-] $\\mathcal V\\subseteq\\mathbb R^3$ is the set of control inputs to the reset mapping $S$ (called the ``allocation''),\n    \\item[-] $S:\\mathcal Q\\times\\mathcal V\\to\\mathcal Q$ is a mapping that resets the state when a platoon arrives.\n\\end{itemize}\n\\end{dfn}\n\nIn our model, queuing happens due to (i) sudden increases in queues, which occur at rate $\\lambda$ and according to the reset mapping $S$ , and (ii) interaction between queues in various links, which is captured by the vector field $G$.\nWe consider the control inputs $(u,v)$ to be determined by a \\emph{control policy} $(\\mu,\\nu)$, i.e. $(u,v)=(\\mu(q),\\nu(q))$:\n\n\\begin{dfn}[Control policy]\n    A control policy $(\\mu,\\nu)$ is specified by functions $\\mu:\\mathcal Q\\to\\mathcal U$ and $\\nu:\\mathcal Q\\to\\mathcal V$.\n   \n\\end{dfn}\n\\noindent\nWe will describe how control policies for the stochastic fluid model can be translated to platoon coordination strategies in Section~\\ref{sec_control}.\n\nGiven a control policy $(\\mu,\\nu)$, the model's dynamics can be expressed via the \\emph{infinitesimal generator}\n\\begin{align}\n    &\\mathscr Lg(q)=G^T\\Big(q;\\mu(q)\\Big)\\nabla_qg(q)+\\lambda\\bigg(g\\Big(S(q;\\nu(q))\\Big)-g(q)\\bigg),\\nonumber\\\\\n    &\\qquad q\\in\\mathcal Q,\n\\end{align}\nwhere $g$ is a differentiable function \\cite{davis84}. In the above, the first term on the right-hand side results from the fluid dynamics governed by the vector field $G$, and the second term results from the resets governed by the reset mapping $S$.\n\nIn summary, our model (as well as the subsequent analysis) focuses on the impact due to the following parameters:\n\\begin{enumerate}[(i)]\n    \\item Total demand $a$,\n    \\item Platooning ratio $\\eta$ (or equivalently, platoon arrival rate $\\lambda$),\n    \\item Platoon size $l$,\n    \\item Buffer size $\\Theta$.\n\\end{enumerate}\n\nThe rest of this subsection is devoted to specifying the elements in the tuple $\\langle\\mathcal Q,\\mathcal U,G,\\lambda,\\mathcal V,S\\rangle$.\n\n\\subsubsection{State space $\\mathcal Q$}\n\nWe use $q_0^m$ to denote the CAVs held in the gate, $q_1^m$ and $q_2^m$ to denote the queues of mainline traffic in links 1 and 2, respectively, and $q_1^o$ to denote the queue of off-ramp traffic in link 1.\nThe \\emph{state} of the stochastic fluid model is $q=[q_0^m\\ q_1^m\\ q_1^o\\ q_2^m]^T\\in\\mathcal Q$.\nNote that $q_0^m$ consists of only CAVs, $q_1^m$ and $q_2^m$ consist of both CAVs and non-CAVs, and $q_1^o$ consists of only non-CAVs.\nA key characteristic of platooning is the reduced inter-vehicle spacing.\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=0.2\\textwidth]{.\/Images\/gamma.png}\n\\caption{$H$ is the spacing between ordinary vehicles, while $h$ is the spacing within a platoon; $\\gamma=H\/h$.}\n\\label{gamma}\n\\end{figure}\nWe model this by scaling down the CAV part of $q_k^m$ by a factor $\\gamma>1$: a platoon of $l$ CAVs are roughly equivalent to $l\/\\gamma$ non-CAVs in terms of the occupied road space, where $\\gamma$ is the ratio between the inter-non-CAV spacing $H$ and the inter-CAV spacing $h$; see Fig.~\\ref{gamma}. A typical value for $\\gamma$ is 2 \\cite{bess+16procieee,santini2016consensus}.\nHence, $q$ is the vector of \\emph{effective} queue lengths with the CAV part scaled down, which is in general smaller than the \\emph{nominal} queue lengths.\nThroughout this paper, we use $Q(t)=[Q_0^m(t)\\ Q_1^m(t)\\ Q_1^o(t)\\ Q_2^m(t)]^T$ to denote the vector of queues at time $t$ and $q$ to denote a particular state.\n\n\\subsubsection{Gate discharge $\\mathcal U$}\n\nThe rate at which the gate discharges traffic to link 1 is $u\\in\\mathcal U$.\nWe consider $u$ to be controlled by a \\emph{gate discharge policy} $\\mu:\\mathcal Q\\to\\mathbb R_{\\ge0}$ satisfying the following:\n\\begin{asm}\n\\label{asm_mu}\nThe gate discharge policy $\\mu$ satisfies the following:\n\\begin{enumerate}[(i)]\n    \\item $\\mu(q)$ is non-negative, bounded, and piecewise-continuous in $q$;\n    \\item $\\mu(q)=0$ for $q$ such that $q_0^m=0$;\n    \\item $\\mu(q)$ is non-increasing in $q_1^m,q_1^o,q_2^m$.\n\\end{enumerate}\n\\end{asm}\n\\noindent\nIn the above, (i) ensures regularity to facilitate analysis.\n(ii) means that the gate discharge must vanish if $q_0^m=0$.\n(iii) means that CAVs in the gate will be discharged slower if the downstream queues are longer.\nWe use $\\mathscr U$ to denote the set of {gate discharge policies} satisfying the above assumption.\nThis assumption basically ensures that $Q(t)$ is bounded and piecewise-continuous in $t$.\n\n\\subsubsection{Vector field $G$}\n\\label{subsub_G}\n\nThe vector field $G$ specifies the model's dynamics between resets.\nSpecifically, the inflow of mainline background traffic is $(1-\\eta)\\rho a$, and the inflow of off-ramp background traffic is $(1-\\rho)a$.\nIf the queue in link 2 is less than the buffer size, then the queue in link 1 is discharged at the link's capacity $F$; otherwise, the queue propagates to link 1 and reduces the off-ramp flow (called \\emph{spillback}, see Fig.~\\ref{fig_spillback}). \n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=0.35\\textwidth]{.\/Images\/fig_spillback.png}\n\\caption{Two examples of trajectories of $Q_2^m(t)$ with (left) and without spillback (right).}\n\\label{fig_spillback}\n\\end{figure}\nThe flows in Fig.~\\ref{fig_tandem} are given by\n\\begin{subequations}\n\\begin{align}\n&f_0(q;\\mu(q)):=\\mu(q),\\\\\n&f_1(q;\\mu(q)):=\\nonumber\\\\\n&\\left\\{\\begin{array}{ll}\n\\min\\{(1-\\eta)\\rho a+\\mu(q),F\\} & \\mbox{if $q_1^m=0,q_2^m<\\Theta$}, \\\\\nF & \\mbox{if $q_1^m>0,q_2^m<\\Theta$}, \\\\\n\\min\\{(1-\\eta)\\rho a+\\mu(q),F-R\\} & \\mbox{if $q_1^m=0,q_2^m=\\Theta$}, \\\\\nF-R & \\mbox{if $q_1^m>0,q_2^m=\\Theta$},\n\\end{array}\\right.\\label{eq_f1}\\\\\n&f_{2}(q;\\mu(q)):=\\left\\{\\begin{array}{ll}\n\\min\\{f_1(q;\\mu(q)),F-R\\} & \\mbox{if $q_2^m=0$}, \\\\\nF-R & \\mbox{if $q_2^m>0$}\n\\end{array}\\right.\\\\\n&{r}(q;\\mu(q)):=\\nonumber\\\\\n&{\\footnotesize\\left\\{\\begin{array}{ll}\n\\min\\{(1-\\rho)a,F-f_1(q;\\mu(q)),R\\} & \\mbox{if $q_1^o=0,q_2^m<\\Theta$}, \\\\\n\\min\\{F-f_1(q;\\mu(q)),R\\} & \\mbox{if $q_1^o>0,q_2^m<\\Theta$}, \\\\\n\\min\\{(1-\\rho)a,F-R-f_1(q;\\mu(q)),R\\} & \\mbox{if $q_1^o=0,q_2^m=\\Theta$}, \\\\\n\\min\\{F-R-f_1(q;\\mu(q)),R\\} & \\mbox{if $q_1^o>0,q_2^m=\\Theta$}.\n\\end{array}\\right.}\\label{eq_r}\n\\end{align}\n\\end{subequations}\n\\eqref{eq_f1} and \\eqref{eq_r} indicate that spillback happens if $q_2^m=\\Theta$: whenever the threshold is attained, the upstream capacity is dropped.\nTo focus on the impact of capacity, we assume that traffic queues are always discharged at the rate of $F$ or $F-R$.\\footnote{Simulation results show that this simplified flow model is largely consistent with more sophisticated models such as the CTM; see Section Section~\\ref{sub_ctm}.}\nThen, the fluid dynamics is specified by the vector field $G:\\mathcal Q\\times\\mathcal U\\to\\mathbb R^4$ defined as\n\\begin{subequations}\n\\begin{align}\n&G_0^m(q;\\mu(q)):=-\\mu(q),\\label{eq_G0}\\\\\n&G_1^m(q;\\mu(q)):=(1-\\eta)\\rho a-(f_1(q;\\mu(q))+{r}(q;\\mu(q))),\\\\\n&G_1^o(q;\\mu(q)):=(1-\\rho)a-{r}(q;\\mu(q)),\\\\\n&G_2^m(q;\\mu(q)):=f_1(q;\\mu(q))-f_2(q;\\mu(q)).\\label{eq_G2}\n\\end{align}\n\\end{subequations}\nTo emphasize that the vector field depends on the control policy, we use the notation $G(q,\\mu(q))$.\nNote that for an admissible gate discharge policy $\\mu\\in\\mathscr U$, $G$ is bounded and piecewise-continuous in $q$.\nSince we focus on the aggregate behavior of both traffic classes, our results hold for a variety of capacity-sharing models (see e.g. \\cite{jin2019behavior}). In addition, microscopic maneuvers such as overtaking are implicitly captured by the flow dynamics.\n\n\\subsubsection{Arrival rate $\\lambda$}\nCAV platoons arrive according to a Poisson process of rate $\\lambda$ (per hr), which is given by \\eqref{eq_lambda}.\n$\\lambda$ specifies the rate at which the continuous state $Q(t)$ is reset.\nRandomness of the platoon arrival process can be attributed to the process of platoon formation \\cite{xiong2019analysis,xiong2020optimizing}.\n\n\\subsubsection{Allocation $\\mathcal V$}\nPlatoon control is modeled by a vector $v=[v_0\\ v_1\\ v_2]^T\\in\\mathcal V$. Arriving platoons are allocated to each link according to $v$: for example, $v=[l,0,0]^T$ means that a platoon is allocated to the gate. Recall that $v$ is determined by a mapping $\\nu:\\mathcal Q\\to\\mathcal V$.\nWe assume the following for $\\nu$:\n\\begin{asm}\n\\label{asm_nu}\nThe allocation policy $\\nu$ satisfies the following:\n\\begin{enumerate}[(i)]\n    \\item $\\sum_{k=0}^2\\nu_k(q)=0$, $0\\le\\nu_0(q)\\le l\/\\gamma$, $-l\/\\gamma\\le\\nu_1(q)\\le0$, $\\max\\{-l\/\\gamma,q_2^m-\\Theta\\}\\le\\nu_2(q)\\le0$,\n    \\item $\\nu_2(q)=0$ for $q$ such that $q_1^m>0$, and $\\nu_1(q)=\\nu_2(q)=0$ for $q$ such that $q_0^m>0$,\n    \\item $\\nu_k(q)$ is non-increasing in $q_k^m$ and non-decreasing in $q_j^m$ for $j\\neq k$, and $\\nu_k(q)$ is non-increasing in $q_1^o$ for $k=1$ and non-decreasing in $q_1^o$ for $k\\neq 1$.\n\\end{enumerate}\n\\end{asm}\n\\noindent\nIn the above, (i) means that $\\nu$ only {distributes} but does not {creates} traffic. \n(ii) results from the ``first-come-first-served'' principle: a platoon cannot be allocated to link $k$ if there is a non-zero queue in link $k-1$.\n(iii) means that more traffic is allocated to a link with a shorter queue.\nWe use $\\mathscr V$ to denote the set of {gate discharge policies} satisfying the above assumption.\n\n\\subsubsection{Reset mapping $S$}\n\\label{subsub_S}\n\nArrivals of platoons lead to sudden increases in the state $Q(t)$, and the reset mapping is given by\n\\begin{subequations}\n\\begin{align}\n&S_0^m(q;v):= q_0^m+v_0,\\label{eq_T0}\\\\\n&S_1^m(q;v):= q_1^m+(q_2^m+l\/\\gamma-\\Theta)_++v_1,\\label{eq_T1}\\\\\n&S_1^o(q;v):= q_1^o,\\\\\n&S_2^m(q;v):= \\min\\{\\Theta,q_2^m+l\/\\gamma\\}+v_2.\\label{eq_T2}\n\\end{align}\n\\end{subequations}\nIn particular, $S(q;0)$ represents the reset mapping if no control is applied (i.e. platoons are not coordinated).\\footnote{If $v=0$, then no CAVs will ever be allocated to the gate; hence the gate discharge has no impact.}\nFor $v=0$, no platoons will be allocated to the gate upon arrival; instead, every platoon will be allocated to link 2 unless $Q_2^m(t)$ attains the buffer size $\\Theta$.\nIf a platoon arrives at time $t$, then the state is reset according to\n\\begin{align*}\n    Q(t)=S\\Big(Q(t_-);v\\Big),\n\\end{align*}\nwhere $Q(t_-)$ is the vector of queues immediately before the arrival.\n\n\n\n\n\n\n\\subsection{Problem definition}\n\\label{sub_preliminaries}\n\nThe main questions that we study are\n\\begin{enumerate}[(i)]\n    \\item Given model parameters and a control policy, how to determine whether the queues are bounded (in expectation), and how to compute or estimate the model's throughput?\n    \\item How to design the control policy to ensure bounded queues and to improve throughput and travel time?\n\\end{enumerate}\nTo study the above questions, we introduce the following definitions.\n\nFirst, following \\cite{dai1995stability}, we define stability as follows:\n\\begin{dfn}[Stability]\nThe stochastic fluid model is {stable} if there exists $Z<\\infty$ such that for each initial condition $q\\in\\mathcal Q$\n\\begin{align}\\label{eq_bounded}\n\\limsup_{t\\to\\infty}\\frac1t\\int_{s=0}^t{\\mathrm E}\\left[|Q(s)|\\right]ds\\le Z.\n\\end{align}\n\\end{dfn}\n\\noindent \nPractically, stability means that expected queue size is bounded, and hence the probability of long queues is small.\n\nSecond, given a control policy $(\\mu,\\nu)$, the stochastic fluid model typically admits an \\emph{invariant set}, which is defined as follows:\n\\begin{dfn}[Invariant set]\nGiven a control policy $(\\mu,\\nu)$, a compact set $\\mathcal M_{\\mu,\\nu}\\subseteq\\mathcal Q$ is an invariant set if\n\\begin{enumerate}[(i)]\n    \\item $\\lim_{t\\to\\infty}\\Pr\\{Q(t)\\in\\mathcal M_{\\mu,\\nu}|Q(0)=q\\}=1,\n    \\quad\\forall q\\in\\mathcal Q,$\n    \\item $Q(t)\\in\\mathcal M_{\\mu,\\nu}, \\quad\\forall Q(0)=q\\in\\mathcal M_{\\mu,\\nu}.$\n\\end{enumerate}\n\\end{dfn}\n\\noindent The interpretation of an invariant set is that (i) for each initial condition, the process $\\{Q(t);t\\ge0\\}$ enters the set $\\mathcal M_{\\mu,\\nu}$ almost surely (a.s.), and (ii) if the process $\\{Q(t);t\\ge0\\}$ starts within $\\mathcal M_{\\mu,\\nu}$, then it never leaves $\\mathcal M_{\\mu,\\nu}$.\nSince stability is defined for an infinite time horizon in \\eqref{eq_bounded}, we can focus on the model's evolution over $\\mathcal M_{\\mu,\\nu}$ rather than over $\\mathcal Q$; this simplifies the analysis.\nAlso note that $\\mathcal M_{\\mu,\\nu}$ depends on $(\\mu,\\nu)$ and is thus typically determined based on characteristics of $(\\mu,\\nu)$.\n\nThe theoretical tool that we use to establish stability is the \\emph{Foster-Lyapunov criterion}, which is a sufficient condition for \\eqref{eq_bounded}:\n\n\\noindent{\\bf Foster-Lyapunov criterion \\cite{meyn93}}.\n\\emph{Consider a Markov process with an invariant set $\\mathcal Y$ and infinitesimal generator $\\mathscr A$. If there exist a Lyapunov function $W:\\mathcal Y\\to\\mathbb R_{\\ge0}$ and constants $c>0$, $d<\\infty$ satisfying\n\\begin{align}\n    \\mathscr AW(y)\\le-cg(y)+d,\n    \\quad\\forall y\\in\\mathcal Y,\n    \\label{eq_AW}\n\\end{align}\nthen for each initial condition $y\\in\\mathcal Y$\n\\begin{align}\n    \\limsup_{t\\to\\infty}\\frac1t\\int_{\\tau=0}^t\\mathrm E[g(Y(t))]d\\tau\\le d\/c.\n\\end{align}\n}%\nIn the above, \\eqref{eq_AW} is called the ``drift condition'' \\cite{meyn93}.\nVerifying the drift condition is in general challenging, since it requires (i) finding an effective Lyapunov function, which is not straightforward for a nonlinear system as our stochastic fluid model, and (ii) checking the inequality \\eqref{eq_AW} over a possibly unbounded set $\\mathcal Y$, which involves a non-convex optimization.\nIn Section~\\ref{sec_analysis}, we argue how we address these challenges.\n\nThird, with the notion of stability, we define the stochastic fluid model's \\emph{throughput} as follows:\n\\begin{dfn}[Throughput]\nGiven a control policy $(\\mu,\\nu)\\in\\mathscr U\\times\\mathscr V$, the throughput of the stochastic fluid model is\n\\begin{align*}\n    \\bar a_{\\mu,\\nu}:=&\\sup a\\\\\n&\\mbox{s.t.} \\mbox{ fluid model is stable under }(\\mu,\\nu)\\in\\mathscr U\\times\\mathscr V.\n\\end{align*}\n\\end{dfn}\n\\noindent \n$\\bar a_{\\mu,\\nu}$ is defined as supremum rather than maximum, since the stability constraint may lead to strict inequalities.\nAs indicated in the above definition, the key to throughput analysis is to develop stability conditions for the stochastic fluid model, which we discuss in Section~\\ref{sec_analysis}.\nWe have the following preliminary result for throughput:\n\\begin{lmm}[Nominal throughput]\n\\label{lmm_maximum}\nFor any control policy $(\\mu,\\nu)\\in\\mathscr U\\times\\mathscr V$, throughput $\\bar a_{\\mu,\\nu}$ of the stochastic fluid model is upper-bounded by\n\\begin{align}\n\\bar a_{\\mu,\\nu}\\le a^*:=\\min\\Big\\{\n\\frac{R}{1-\\rho},\n\\frac{F-R}{(\\eta\/\\gamma+1-\\eta)\\rho}\n\\Big\\}.\n\\label{eq_nominal}\n\\end{align}\n\\end{lmm}\nWe call $a^*$ as defined in \\eqref{eq_nominal} the \\emph{nominal throughput}.\nTo interpret the expression for $a^*$, note that the first (resp. second) term in $\\min\\{\\cdot\\}$ results from the capacity constraint of the off-ramp (resp. the mainline bottleneck).\nImportantly, we will show that the nominal throughput cannot always be attained due to the interaction between CAV and non-CAV traffic and due to spillback of traffic queues. \n\nFinally, for control design, we consider the following formulation:\n\\begin{align*}\n\\mbox{decision:}&\\quad (\\mu,\\nu)\\in\\mathscr U\\times\\mathscr V\\nonumber\\\\\n\\mbox{(P)}\\quad\\mbox{objective:}&\\quad\\max\\quad \\bar a_{\\mu,\\nu}\\\\\n&\\quad s.t.\\quad \\mbox{stability}.\n\\end{align*}\nThat is, the first objective is stabilization, and the second objective is throughput maximization or queue minimization.\nNote that the objective value of (P) is upper-bounded by $a^*$ in Lemma~\\ref{lmm_maximum}.\n\\section{Introduction}\n\\label{sec_intro}\n\n\nPlatooning of connected and autonomous vehicles (CAVs) has the potential for significant throughput improvement~\\cite{hor+var00,litman2017autonomous}. The idea of automatically regulating strings of vehicles is well-known~\\cite{lev+ath66,swaroop1996string}, and several experimental studies in real-world traffic conditions have been conducted in the past decades~\\cite{chang1993automated,naus+10,bess+16procieee,tsugawa16}. \nImportant progress has been made in vehicle-level control design~\\cite{naus+10,coogan15interconnected,bess+16procieee} and system-level simulations \\cite{talebpour16}.\nHowever, we still lack both link- or network-level models for evaluating the impact of platooning on highway traffic and coordinating the movement of platoons on mixed-traffic highways.\nIn particular, although platooned CAVs have smaller inter-vehicle spacing, uncoordinated and randomly arriving CAV platoons may act as ``moving obstacles'' and result in recurrent local congestion, especially at bottlenecks; see Fig.~\\ref{cartoon_nocontrol} for illustration. \nAppropriate inter-platoon coordination, such as regulating the headways between platoons and managing the platoon sizes, is essential for realizing the full benefits of platooning in mixed-traffic conditions.\n\\begin{figure}[hbt]\n\\centering\n\\subfigure[Without inter-platoon coordination, the exiting traffic can be blocked by local congestion induced by platoons.]{\n\\centering\n\\includegraphics[width=0.4\\textwidth]{.\/Images\/cartoon_nocontrol}\n\\label{cartoon_nocontrol}\n}\n\\subfigure[Inter-platoon coordination can help mitigate the ``moving-obstacle'' effect and maintain free flow.]{\n\\centering\n\\includegraphics[width=.4\\textwidth]{.\/Images\/cartoon_control}\n\\label{cartoon_control}\n}\n\\caption{Traffic scenarios without (top) and with (bottom) inter-platoon coordination.}\n\\end{figure}\n\nIn this paper, we consider a platoon coordination problem over a generic highway section as shown in Fig.~\\ref{cartoon_nocontrol}.\nThe highway section has a downstream on-ramp, which forms a geometric bottleneck.\nIf more traffic is arriving at the bottleneck than the bottleneck can discharge, then a queue would be growing there.\nThe bottleneck may impede free flow of traffic, and consequently uncoordinated platoons may start inducing congestion (via local queuing), thereby impacting the upstream off-ramp.\nOne of the objectives of platoon coordination is ensure that the potential throughput gain due to platooning is not limited by such local congestion.\nInter-platoon coordination can be achieved by (i) regulating the times at which platoons arrive at the bottleneck by specifying the average or reference speed of platoons over the highway section (called \\emph{headway regulation}) and (ii) maintaining desirable platoon sizes through split or merge maneuvers (called \\emph{size management}).\nImplementation of such coordination strategies can be facilitated by traffic sensors that collect real-time traffic information and road-side units that send coordination instructions to each platoon (Fig.~\\ref{cartoon_control}); such capabilities are already available in modern transportation systems \\cite{duret2019hierarchical}.\n\nTo model the interaction between platoons and background traffic, we present a multiclass tandem-link fluid model.\nFluid models are standard for highway bottleneck analysis \\cite[Ch. 2]{newell13} and allow tractable analysis \\cite{kulkarni97}.\nOur model specifically captures two important features of platooning operations.\nFirst,  platoons travel in a clustered manner (i.e. with small between-vehicle spacings and relatively large inter-platoon headways), whereas non-CAVs do not follow such a configuration. \nIn our model, random platoon arrivals are modeled as Poisson jumps in the traffic queue at the bottleneck.\nThis model captures the inherent randomness in platooning operations \\cite{bess+16procieee}. Second, platoons share highway capacity with the background traffic and may act as ``moving obstacles''; our model captures this interaction by constraining the sum of discharge rates of platoons and background traffic at the bottleneck with an overall capacity.\n\n\nFor comparison, Table~\\ref{tab_existing} lists various models that researchers have studied for traffic flow with CAV platoons under various contexts.\n\\begin{table}[hbt]\n\\caption{Models for traffic with platooning.}\n\\label{tab_existing}\n\\centering\n\\begin{tabular}{@{}cclc@{}}\n\\toprule\nScale                  & Model & Control actions & References      \\\\ \\midrule\nVehicle & \n\\begin{tabular}{@{}l@{}}Vehicle dynamic,\\\\ car following\\end{tabular}  \n& \\begin{tabular}{@{}l@{}}Throttle, break,\\\\ steering\\end{tabular} \n&\\begin{tabular}{@{}c@{}} \\cite{li2017dynamical,smith2019balancing}\t\\\\\n\\cite{zhou2019stabilizing,duret2019hierarchical}\n\\end{tabular}    \\\\ \\hline\nIntersection & Discrete queuing  & \n \\begin{tabular}{@{}l@{}}Signal timing,\\\\ CAV coordination\\end{tabular}\n &\\cite{lioris17,miculescu2019polling}          \\\\ \\hline\nSegment              &  \n\\begin{tabular}{@{}l@{}}Partial differential\\\\\nequation\\end{tabular} \n& \n\\begin{tabular}{@{}l@{}}Speed regulation,\\\\ platoon merge\/split, \\\\  lane management\\end{tabular} & \\cite{keim+17,vcivcic2019coordinating}      \\\\ \\hline\nLink            & Fluid  & \n\\begin{tabular}{@{}l@{}}Headway regulation,\\\\ platoon merge\/split, \\\\  \\end{tabular} & this paper \n \\\\ \\hline\nNetwork        &   Static     & Routing  & \\cite{lazar2018routing,mehr2018can}      \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\nFluid models are particularly suitable for link-level analysis and control design due to the following advantages.\nCompared with discrete queuing models, fluid models do not do not track individual vehicles; instead, only the aggregate flow is required. This considerably simplifies the state space and the system dynamics. Compared with PDE models, fluid models entail smaller computational requirements and enable tractable analysis. Compared with static models, fluid models allow real-time control design rather than long-term decisions such as day-to-day traffic assignment.\n\nTwo major differences are also worth noting here.\nFirst, the behavior of discrete queuing models (e.g. M\/M\/1) is closely related to their fluid counterparts \\cite{dai1995stability}. \nAs opposed to fully discrete queuing models, our model considers platoon as discrete Poisson arrivals and non-CAVs as a continuous inflow.\nThis is motivated by practical situations where the arrival rate of CAV platoons (less than five per minute) is much lower than that of non-CAVs (50--100 per minute).\nSecond, fluid models share some common features with PDE models \\cite{yu2019traffic} or their discretization (e.g. cell transmission model \\cite{cicic2018traffic}). \nIn contrast to PDE models, the fluid model retains the queuing delay due to demand-capacity imbalance but does not capture the evolution of congestion waves.\nStill, both the traffic fluid model and classical traffic flow models are based on conservation laws. Some authors showed that these two types models lead to equivalent results in traffic network optimization \\cite{qian2012system}. This paper also demonstrates the consistency between the fluid model and more detailed models via simulation (Section~\\ref{sub_simulation}).\n\nUsing the fluid model, we study the throughput of the highway section with uncoordinated platoons\\footnote{In this paper, ``coordination'' refers to the link-level coordination of multiple platoons. We acknowledge that ``coordination'' is also used to refer to the CACC applied to individual vehicles within one platoon \\cite{vandehoef2018fuel}.}.\nWe utilize an M\/D\/1 queuing characterization of the fluid model and establish a Foster-Lyapunov drift condition for stability of the fluid model.\nThis leads to an easy-to-check sufficient condition for bounded queues of the uncontrolled system (Theorem~\\ref{thm_sufficient}) which relies on the general stability\/convergence theory of Markov processes \\cite{meyn1993stability}.\nWe also derive explicit lower and upper bounds for throughput in the uncoordinated scenario (Theorem~\\ref{thm_bounds}).\nThese results also contribute to the literature on fluid queuing systems \\cite{mitra1988stochastic,kulkarni97,cassandras02,kroese2001joint,jin2018stability}\n\n\nWe also design a class of platoon coordination strategies that realize the full potential of platooning for throughput improvement.\nThe control actions that we consider include regulating inter-platoon headways and splitting platoons into shorter platoons.\nIn terms of the fluid model, these regulation strategies are formulated as control laws regulating the arrival process of platoons with the knowledge of the arrival times of previous platoons. In practice, such knowledge can be obtained by tracking the movement of existing platoons on the highway section via vehicle-to-infrastructure communications \\cite{vandehoef2018fuel}.\nWe prove that a set of control laws (Theorem~\\ref{thm_optimal}) stabilize the system in a fairly strong sense (bounded moment-generating function and exponential convergence to steady-state distribution) if and only if the total inflow is less than the capacity.\nThus, they are optimal in the sense of throughput maximization and delay minimization.\nIntuitively, these control laws coordinate the movement of platoons so that they arrive at the bottleneck with relatively evenly distributed headways;\nthus, queuing delay is absorbed en-route, congestion at the bottleneck does not build up, and spillback is eliminated.\n\n\nNote that the control actions considered in this paper are related to but different from a class of longitudinal\/lateral Cooperative Adaptive Cruise Control (CACC) capabilities. We focus on link-level decision variables including (i) the reference speed or the average speed that a traffic manager would recommend a platoon to take over a highway section and (ii) the decision whether to maintain or split a platoon over a highway section, both of which are concerned with a typical scale of 10 km or 10 minutes. In the context of vehicle-level CACC, however, the real-time speed is dynamically regulated to maintain string stability \\cite{li2017dynamical} or to mitigate local stop-and-go behavior \\cite{stern2018dissipation,wu2018stabilizing,zhou2019stabilizing}, which involve much finer space and time resolutions. The objective of CACC design is to optimize microscopic driving behavior and regulate congestion waves, while our control objective is to reduce the congestion due to randomness in platoon arrivals and develops coordination strategies that work effectively in the presence of such randomness.\n\nFinally, we discuss how the fluid model-based results can be translated to implementable actions for actual CAV platoons and validate the proposed coordination strategies in standard simulation environments.\nWe implement the proposed strategies in a macroscopic (cell transmission model, CTM \\cite{daganzo94}) and a microscopic (Simulation for Urban Mobility, SUMO \\cite{krajzewicz2002sumo}) simulation model.\nResults indicate that the proposed strategy effectively and consistently improves travel times in both simulation models.\nIn spite of multiple simplifications that our modeling approach makes, simulation results suggest that the theoretically optimal headway regulation strategy attains more than 80\\% of the improvement attained by the simulation-optimal strategies.\n\nThe main contributions of this paper include:\n\\begin{enumerate}[(i)]\n\\item A novel fluid model a highway with randomly arriving platoons and constant inflow of background traffic that captures the queuing and throughput loss due to interaction between various traffic classes;\n\\item A set of easily checkable conditions for stability of the fluid model derived by combining ideas from queuing theory (mainly M\/D\/1 process) and the theory of stability of continuous-time, continuous-state Markov processes. These conditions enable quantitative analysis of throughput of mixed-autonomy highways;\n\\item A set of platooning strategies that regulate movement of platoons to attain maximum throughput as well as minimum delay;\n\\item Validation of the fluid model-based analysis and design results via simulation of macroscopic and microscopic models.\n\\end{enumerate}\n\nThe rest of this article is organized as follows.\nSection~\\ref{sec_modeling} introduces the fluid model.\nSection~\\ref{sec_analysis} presents throughput analysis based on stability conditions.\nSection~\\ref{sec_control} discusses a class of optimal control strategies.\nSection~\\ref{sec_ctm} presents the implementation and validation in simulation environments.\nSection~\\ref{sec_conclude} summarizes the main results and mentions several directions for future work.\n\\section{Implementation, simulation, and discussion}\n\\label{sec_ctm}\n\nIn this section, we translate the control laws discussed in the previous section to platoon coordination instructions that can be implemented in practice (Section~\\ref{sub_implementation}).\nWe also validate the optimality of the headway regulation strategy via two standard simulation environments, viz. CTM and SUMO (Section~\\ref{sub_simulation}).\n\n\\subsection{Implementation of proposed control policies}\n\\label{sub_implementation}\n\nWe now discuss how the two platoon coordination strategies presented in Section~\\ref{sec_control}, viz. headway regulation and size management, can be translated to implementable instructions for platoons.\nFig.~\\ref{cartoon} illustrates the implementation.\n\\begin{figure}[hbt]\n\\centering\n\\subfigure[Headway regulation.]{\n\\centering\n\\includegraphics[width=0.4\\textwidth]{.\/Images\/cartoon_speed}\n\\label{cartoon_speed}\n}\n\\subfigure[Platoon size management.]{\n\\centering\n\\includegraphics[width=.4\\textwidth]{.\/Images\/cartoon_split}\n\\label{cartoon_split}\n}\n\\caption{Two practical platoon coordination strategies.}\n\\label{cartoon}\n\\end{figure}\nThese strategies is enabled by modern vehicle-to-infrastructure (V2I) communications technologies \\cite{papadimitratos2009vehicular}.\nWe do not explicitly consider lower-level control actions such as longitudinal and lateral control; instead, we assume that the platoons are equipped with adequate lower-level controllers that can implement the instructions from the operator.\n\n\n\\subsubsection{Headway regulation}\nTo regulate headways, the operator sends a recommended speed to each platoon when it enters the highway, and no more instructions need to be sent to this platoon; see Fig.~\\ref{cartoon_speed}. The decision variable is the average speed for each platoon over the highway section, or, equivalently, the time at which a platoon is scheduled to arrive at the bottleneck.\nIn practice, let $L_1$ and $L_2$ be the lengths of links 1 and 2, respectively, as in Fig.~\\ref{capacities}, and let $v_0$ be the nominal speed for the highway section. \nThe recommended speed for an incoming platoon is\n\\begin{align}\n    v^{hr}=\\frac{L_1+L_2}{\\frac{L_1+L_2}{v_0}+W^{hr}},\n    \\label{eq_vhr}\n\\end{align}\nwhere $W^{hr}$ is given by \\eqref{eq_Whr}.\n\n\n\n\n\\subsubsection{Size management}\nTo manage the size of platoons, when a platoon enters the highway, the coordination strategy first predicts the traffic condition if the platoon arrives at the bottleneck without any intervention; then, if congestion is predicted at the bottleneck, the strategy will split the platoon into shorter platoons; see Fig.~\\ref{cartoon_split}.\nThe fluid model can be used to make such predictions.\nIn practice, suppose a platoon enters the highway at time $t$ and let $T$ be the solution to \\eqref{eq_T}.\nSpecifically, the coordination decision is made in two steps:\n\\begin{enumerate}\n\\item Whether to split: The platoon is instructed to split if $Q_2^m(T)\\ge\\Theta-\\frac{l}{2\\gamma}$ and not to split otherwise; if the platoon is splitting, then the separation (i.e. headway) between the short platoons will be $l\/(2\\gamma(F-R-(1-\\eta)\\rho a))$.\n\\item When to arrive at bottleneck: If the platoon is not splitting, then it will travel at the nominal speed and no intervention will be needed. If the platoons are splitting into two short platoons, the leading short platoon will travel at the speed\n$$\nv_1^{sm}=\\frac{L_1+L_2}{\\frac{L_1+L_2}{v_0}+W_1^{sm}}\n$$\nand the following short platoon will travel at the speed\n$$\nv_2^{sm}=\\frac{L_1+L_2}{\\frac{L_1+L_2}{v_0}+W_2^{sm}}\n$$\nwhere $W^{sm}_1$ and $W^{sm}_2$ are given by \\eqref{eq_W1}.\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Simulation-based validation}\n\\label{sub_simulation}\n\nThe purpose of the simulations is to show that the optimal headway regulation strategy designed using the fluid model-based approach is consistent with the simulation-optimal values.\nWe use two standard simulation models, viz. the cell transmission model (CTM \\cite{daganzo94}) and the Simulation for Urban Mobility (SUMO \\cite{krajzewicz2002sumo}).\nThe CTM is a macroscopic traffic flow model, which evolves according to (i) the conservation law and (ii) the flow-density relation (also called ``fundamental diagram'' by transportation researchers \\cite{daganzo94}).\nThe CTM accounts for the spatial distribution of traffic and the detailed flow-density relation, which are not captured by the fluid model.\nThe SUMO is a microscopic simulation model, which evolves according to vehicle-following and lane-changing behavior models for individual drivers.\nSuch microscopic details are captured by neither the CTM nor the fluid model.\n\nFig.~\\ref{fig_compare} shows the simulation results with parameters in Table~\\ref{tab_parameters}. The mainline demand is 2500 veh\/hr, and the off-ramp demand is 1400 veh\/hr.\nThe theoretical optimal headway $W^{hr}$ (36 sec) is close to the simulation-obtained value (30 sec).\n\\begin{figure}[hbt!]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{.\/Images\/fig_compare}\n\\caption{Simulated performance metrics for various values of minimal inter-platoon headways.}\n\\label{fig_compare}\n\\end{figure}\nAs expected, the theoretical optimum $W^{hr}$ is greater than the simulated one. The main reason is that in the fluid model decelerating a platoon only affects the platoon itself and does not directly impact the neighboring traffic.\nIn both the CTM and the SUMO, however, a decelerated platoon will induce local congestion. Hence, both CTM and SUMO prefer less deceleration than the fluid model.\n\nNext, we provide more details and discussion of the simulations.\n\n\\subsubsection{Macroscopic simulation (CTM)}\n\\label{sub_ctm}\n\nWe consider the cell transmission model (CTM, \\cite{daganzo94}) for the highway section in Fig.~\\ref{capacities}.\nIn particular, we consider CAVs and non-CAVs as multiple traffic classes in the CTM.\nMore details of the multi-class CTM is available in \\cite{cicic2019coordinating}.\n\nThe parameters of the macroscopic model are chosen in accordance with the ones given in Table~\\ref{tab_parameters}, i.e. the capacity of the bottleneck will be set to $F-R$, and platoon and buffer lengths chosen appropriately. Note that in order for platoons to be properly represented in this framework, we need the spatial and temporal discretization steps to be fairly short, with the platoon length spanning at least two cells. In this work, the physical platoon length is taken to be 0.1 miles, or 10 cells.\n\n\n\n\n\n\n\\begin{figure}[hbt!]\n\\centering\n\\subfigure[Without coordination.]{\n\\centering\n\\includegraphics[width=0.4\\textwidth]{.\/Images\/offramb_bneck_noctrl}\n\\label{fig_sim_noctrl}\n}\n\\subfigure[With coordination.]{\n\\centering\n\\includegraphics[width=0.4\\textwidth]{.\/Images\/offramb_bneck_ctrl.png}\n\\label{fig_sim_ctrl}\n}\n\\caption{Traffic density contour plots for CTM simulation. The vertical red (resp. black) dashed line indicates the location of the bottleneck (resp. off-ramp). Color indicates traffic density in veh\/mi.}\n\\label{fig_exm2}\n\\end{figure}\n\n\nFig.~\\ref{fig_sim_noctrl} shows the simulated traffic evolution without inter-platoon coordination using color-coded traffic density.\nThe location of the bottleneck is shown in dashed red line, and the location of the off-ramp is outlined in dashed black line.\nThe streaks of brighter color represent the increased traffic density near the moving platoons.\nThe congestion from the bottleneck propagates upstream and the off-ramp cell becomes congested approximately one hour into the simulation.\nBecause of the congestion, the off-ramp is partially blocked, preventing vehicles from exiting the highway and further degrading the traffic situation.\nAlthough the total demand $a$ is lower than the capacity of the bottleneck, randomness of the platoon arrivals may still create local congestion that disrupt the traffic flow.\nSuch disruptions produce congestion at the bottleneck and block the off-ramp (i.e. the bright-color area in the interval indicated by black dashed lines near distance $x=24.4$ miles), as shown in Fig.~\\ref{fig_sim_noctrl} from $t \\approx 1.25$ hours to $t \\approx 1.5$ hours.\n\n\n\n\nWe apply the recommended speed \\eqref{eq_vhr} to coordinate platoons in the CTM simulation.\nFig.~\\ref{fig_sim_ctrl} shows the traffic evolution under headway regulation of platoons.\nThe simulation run considers the same situation as in the uncoordinated case (Fig.~\\ref{fig_sim_noctrl}).\nWhereas in the uncoordinated case the congestion from the bottleneck blocked the off-ramp, in the coordinated case we are able to spread the arrival of platoons more evenly, thus avoiding causing spillback.\nFig.~\\ref{fig_compare} further illustrates how close the theoretical optimal strategy (36 sec) is closed to the simulated one (30 sec).\n\n\n\n\n\\begin{figure*}[hbt!]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{.\/Images\/sumo}\n\\caption{Micro-simulation environment. Red, cyan, and yellow vehicles represent mainline traffic. on-ramp\/off-ramp traffic, and CAV platoons, respectively.}\n\\label{fig_sumo}\n\\end{figure*}\n\n\\subsubsection{Microscopic simulation (SUMO)}\nWe also implement the headway regulation strategy introduced in Section~\\ref{sub_implementation} in a micro-simulation model; see Fig.~\\ref{fig_sumo}.\nWe use the TraCI (Traffic Control Interface) to customize the simulation and realize the functions required for this specific experiment. The TraCI features 13 individual modules varying from simulation, vehicle type, vehicle. We use Python to code the route and runner files. Some variables that we control in particular include platoon speed, total simulation time, platooning ratio, platoon length, and platooning state. We also customized the lane-changing function to prevent platoons from breaking apart at the bottleneck. The coordination instructions are realized by the runner script.\n\nThe simulation results are shown in Fig.~\\ref{fig_compare}. The simulations lead to a simulation-optimal inter-platoon headway (30 sec), which is close to the value $W^{hr}$ given by the fluid model and computed via \\eqref{eq_Whr} (36 sec).\nA prominent pattern to be noted in Fig.~\\ref{fig_compare} is that as the coordinated headway increases, the travel time in SUMO grows more slowly than the VHT in the CTM.\nA reasonable explanation is that the CTM that we simulated assumes the first-come-first-serve principle: if a platoon is decelerated, then all the traffic behind will be simultaneously decelerated.\nHowever, overtaking is allowed in SUMO, which is implemented according to SUMO's internal over-taking algorithm.\nConsequently, the impact of decelerating platoons is less significant in SUMO than in the CTM.\n\n\\subsection{Further discussion}\n\nThe stochastic fluid model that we consider focuses on (i) the capacity-sharing between platoons and background traffic and (ii) the throughput gain due to reduced inter-vehicle spacing in platoons, and (iii) the impact of congestion propagation (spillback).\nFrom a practical perspective, our model is based on the following simplifications.\n\\begin{enumerate}\n\\item The interaction between platoons and background traffic only occurs at the link boundaries. This is actually a characteristic of any queuing model. Consequently, our model does not account for interactions occurring over a distance, such as the impact of speed difference between platoons and background traffic.\n\n\\item At the interface between links 1 and 2, mainline traffic (i.e. demand $a$ and $b$) are prioritized for discharging. This is reflected in the definition of the off-ramp flow \\eqref{eq_r}. Alternative models for discharging exist \\cite{wright2017node}, which can be incorporated in our model as well.\n\n\\item Each link's capacity is independent of speed difference between CAVs and non-CAVs. In practice, headway regulation modifies CAVs' speeds. Consequently, highway capacity is dependent of the speed difference between CAVs and non-CAVs as well as the traffic mixture, i.e. the percentage of CAVs. In our model, the impact of heterogeneous speed can be modeled as state-dependent link capacities.\n\n\\end{enumerate}\nNote that the CTM and SUMO simulations that we conducted are not restricted by the above simplifications.\nThe simulation results indicate that the fluid-model approach is adequate in spite of the above simplifications.\n\\begin{table}[hbt]\n\\centering\n\\caption{Improvement attained by various coordination strategies.}\n\\label{tab_simulation}\n\\begin{tabular}{|l|c|c|c|c|}\n\\hline\nStrategy                                                        & \\begin{tabular}[c]{@{}l@{}}VHT,\\\\ CTM\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Improve-\\\\ ment (VHT)\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Traverse time,\\\\ SUMO {[}min{]}\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Improve-\\\\ ment {[}min{]}\\end{tabular} \\\\ \\hline\n\\begin{tabular}[c]{@{}l@{}}No\\\\ coordination\\end{tabular}       & 3471                                               & 0                                                             & 36.26                                                                   & 0                                                                 \\\\ \\hline\n\\begin{tabular}[c]{@{}l@{}}Theoretically\\\\ optimal\\end{tabular} & 3466                                               & 5.3                                                           & 33.27                                                                   & 2.99                                                              \\\\ \\hline\n\\begin{tabular}[c]{@{}l@{}}Simulation\\\\ optimal\\end{tabular}   & 3465                                               & 6.2                                                           & 32.58                                                                   & 3.68                                                               \\\\ \\hline\n\\end{tabular}\n\\end{table}\nAs Table~\\ref{tab_simulation} shows, compared to the baseline scenario without headway regulation, the theoretically optimal strategy attains 85\\% (resp. 81\\%) of the improvement attained by the simulation optimum in CTM (resp. SUMO).\n\n\\section{Stability and throughput}\n\\label{sec_analysis}\n\nIn this section, we study the stability and throughput of the stochastic fluid model, which provides insights about the efficiency of the mixed-traffic highway.\n\nThe first main result gives a criterion to check the stability of the stochastic fluid model under a given control policy $(\\mu,\\nu)$.\n\\begin{thm}[Stability criterion]\n\\label{thm_sufficient}\nSuppose that the stochastic fluid model admits an invariant set $\\mathcal M_{\\mu,\\nu}\\subseteq\\mathcal Q$.\nThe stochastic fluid model is stable if\n\\begin{subequations}\n\\begin{align}\n    & a<a^*,\\quad\\mbox{and}\n    \\label{eq_a<a*}\\\\\n    %\n    &\\max_{\\substack{\\xi\\in\\mathcal M_{\\mu,\\nu}:\\\\\\xi_0^m=\\xi_1^m=\\xi_1^o=0}}\n    \\frac{\\xi_2^m}{\\Theta}\\Big((1-\\eta)\\rho a-(F-R)\\Big)\\nonumber\\\\\n    &\\hspace{1.5cm}+\\frac{\\eta\\rho a}{l} \\bigg(S^m_0(\\xi;\\nu(\\xi))+S^m_1(\\xi;\\nu(\\xi))\\nonumber\\\\\n    &\\hspace{1.5cm}+\\frac{1}{2\\Theta}\\Big(S^m_2(\\xi;\\nu(\\xi))\\Big)^2-\\frac1{2\\Theta}(\\xi_2^m)^2\\bigg)\\nonumber\\\\\n    &\\quad<\\frac{R-(1-\\rho)a}{(1-\\rho)a}\\Big((F-R)-(\\eta\/\\gamma+1-\\eta)\\rho a\\Big).\\label{eq_max}\n\\end{align}\n\\end{subequations}\n\\end{thm}\n\nOne can interpret the stability criterion as follows.\n\\eqref{eq_a<a*} results from the nominal upper bound in Lemma~\\ref{lmm_maximum}.\n\\eqref{eq_max} essentially results from the interaction between the mainline and the off-ramp traffic.\n\\eqref{eq_max} also captures the influence of the control policy $(\\mu,\\nu)$.\nAlthough the complexity of the maximization on the left-hand side of \\eqref{eq_max} depends on the control policy $(\\mu,\\nu)$, the decision variable $\\xi$ can only vary in the direction of $\\xi_2^m$; all the other components of $\\xi$ must be zero.\nHence, the maximization involves essentially only one decision variable $\\xi_2^m$, which takes values from a compact interval $[0,\\Theta]$, and is thus not hard to solve numerically.\nThis is a significant refinement of the Foster-Lyapunov criterion, which, in its general form, does not give a ready-to-use stability criterion for our stochastic fluid model.\nTheorem~\\ref{thm_sufficient} can be used for throughput analysis by finding the largest demand $a$ that satisfies \\eqref{eq_a<a*}--\\eqref{eq_max}.\nSince Theorem~\\ref{thm_sufficient} is a sufficient condition for stability, it leads to a lower bound for throughput.\n\nWe prove Theorem~\\ref{thm_sufficient} by considering a quadratic Lyapunov function and establishing the Foster-Lyapunov criterion \\cite{meyn1993stability} for the queuing process. The main technique is to relate the fluid queuing process to an underlying M\/D\/1 process, which we discuss below. The detailed proof is in Appendix 1.\n\\vspace{-12pt}\n\\begin{figure}[hbt]\n\\centering\n\\subfigure[Trajectories.]{\n\\centering\n\\includegraphics[width=0.3\\textwidth]{.\/Images\/md1}\n\\label{md1}\n}\n\\subfigure[Steady-state CDFs.]{\n\\centering\n\\includegraphics[trim=0 2cm 0 0,clip,width=0.15\\textwidth]{.\/Images\/cdf.png}\n\\label{cdf}\n}\n\\caption{Relation between M\/D\/1 process $N(t)$ and fluid process $\\sum_{k=0}^2Q_k^m(t)$.}\n\\end{figure}\n\n\\begin{figure*}[hbt]\n\\centering\n\\subfigure[Throughput vs. fraction.]{\n\\centering\n\\includegraphics[width=0.31\\textwidth]{.\/Images\/Rbar_eta.png}\n\\label{Rbar_eta}\n}\n\\subfigure[Throughput vs. platoon size.]{\n\\centering\n\\includegraphics[width=.31\\textwidth]{.\/Images\/Rbar_l.png}\n\\label{Rbar_l}\n}\n\\subfigure[Throughput vs. buffer size.]{\n\\centering\n\\includegraphics[width=0.31\\textwidth]{.\/Images\/p_bounds.png}\n\\label{fig_bounds}\n}\n\\caption{Relation between throughput and model parameters. Bounds are computed using Theorem~\\ref{thm_bounds}. ``Nominal maximum'' refers to $a^*$ in Lemma~\\ref{lmm_maximum}. In Fig.~\\ref{Rbar_eta} (resp. Fig.~\\ref{Rbar_l}), $\\eta=0$ (resp. $l=1$) means no platooning.}\n\\end{figure*}\n\nIn particular, if no control is applied, then \\eqref{eq_max} can be manually solved, and explicit lower and upper bounds for throughput can be derived. \nTo see this, consider the process\n\\begin{align}\\label{eq_N}\nN(t):=\\Big\\lceil\\frac{\\gamma}l\\Big(\\sum_{k=0}^2Q_k^m(t)\\Big)\\Big\\rceil\n\\quad t\\ge0\n\\end{align}\nwhich satisfies the following:\n\\begin{lmm}\nThe process $\\{N(t);t\\ge0\\}$ is an M\/D\/1 process with arrival rate $\\lambda=\\eta\\rho a\/l$ and service time $s:=\\frac{l}{\\gamma(F-R-(1-\\eta)\\rho a))}$. Furthermore, \n\\begin{align}\n    \\Big((l\/\\gamma)N(t)-1\\Big)_+\\le \\sum_{k=0}^2Q_k^m(t)\\le(l\/\\gamma)N(t),\n    \\quad \\forall t\\ge0.\n\\end{align}\n\\end{lmm}\n\\noindent\nThat is, we can use $N(t)$ to bound $\\sum_{k=0}^2Q_k^m(t)$; see Fig.~\\ref{md1}.\nThe steady-state probabilities $\\pi_n$ of the M\/D\/1 process are given by a standard result in queuing theory \\cite{shortle2018fundamentals}:\n\\begin{align}\n&\\pi_n=\\left\\{\\begin{array}{l}\n1-\\lambda s \n\\quad n=0,\\\\\n(1-\\lambda s)(e^{\\lambda s} - 1) \n\\quad n=1,\\\\\n(1-\\lambda s)\\Big(e^{n\\lambda s}\n+\\sum_{k=1}^{n-1}e^{k\\lambda s}(-1)^{n-k}\\\\\n\\quad\\times\\left[\\frac{(k\\lambda s)^{n-k}}{(n-k)!}+\\frac{(k\\lambda s)^{n-k-1}}{(n-k-1)!}\\right]\\Big)\n\\quad n\\ge2.\n\\end{array}\\right.\\label{eq_pin}\n\\end{align}\nUsing the above result, we can obtain a lower bound\n\\begin{align}\n    \\omega=1-\\sum_{n=0}^{\\lfloor{\\gamma\\Theta\/l}\\rceil}\\pi_n,\\label{eq_omega}\n\\end{align}\nfor the actual fraction of time $\\omega_0$ that the stochastic fluid model experiences spillback; see Fig.~\\ref{cdf}.\n\nWith the above arguments, we can state the second main result of this section as follows; the proof is in Appendix 2.\n\n\\begin{thm}[Throughput without control]\n\\label{thm_bounds}\nSuppose that the stochastic fluid model is not controlled, i.e. $\\mu(q)=0$ and $\\nu(q)=0$ for all $q\\in\\mathcal Q$.\nThen, the throughput $\\bar a$ of the model is bounded by\n\\begin{align}\n&\\min\\Big\\{\n\\frac{F-R}{\\rho(\\eta\/\\gamma+1-\\eta)},\n\\frac{R}{1-\\rho+\\frac{1}{2}\\Big(\\sqrt{\\zeta^2+\\frac{2\\rho Rl}{\\gamma\\Theta(F-R)}}-\\zeta)\\Big)}\n\\Big\\}\\nonumber\\\\\n&\\quad\n\\le\n\\bar a\\le\n\\min\\Big\\{\n\\frac{F-R}{\\rho(\\eta\/\\gamma+1-\\eta)},\n\\frac{(1-\\omega)R}{(1-\\rho)}\\Big\\}\n\\end{align}\nwhere\n\\begin{align}\n\\zeta=(1-\\rho)-\\rho(\\eta\/\\gamma+1-\\eta)\\frac{R}{F-R}\\label{eq_zeta0}\n\\end{align}\nand $\\omega$ is given by \\eqref{eq_omega}.\n\\end{thm}\n\nUsing the bounds in Theorem~\\ref{thm_bounds}, we can analyze the model's throughput without control.\nFigs.~\\ref{Rbar_eta}--\\ref{fig_bounds} summarize our results for throughput analysis with the nominal parameters in Table~\\ref{tab_parameters}.\nImportantly, due to the interaction between multiple traffic classes and due to lack of coordination between platoons, the nominal throughput given by Lemma~\\ref{lmm_maximum} is not attained.\nSpecific discussions about the figures are as follows:\n\n\n\n\n\\begin{table}[h]\n\\caption{Nominal model parameter values.}\n\\label{tab_parameters}\n\\centering\n\\begin{tabular}{@{}lcc@{}}\n\\toprule\nQuantity                   & Notation & Nominal value       \\\\ \\midrule\nmainline capacity          & $F$       & 4500 veh\/hr \\\\ \\hline\nramp capacity          & $R$       & 1500 veh\/hr \\\\ \\hline\nplatooning ratio & $\\eta$        & 0.2 \\\\ \\hline\nplatoon scaling coefficient & $\\gamma$        & 2 \\\\ \\hline\nplatoon size             & $l$        & 5 veh       \\\\ \\hline\nmainline ratio & $\\rho$        & 0.75 \\\\ \\hline\nbuffer size & $\\Theta$        & 50 veh \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{enumerate}\n    \\item Fig.~\\ref{Rbar_eta} shows that higher fraction of platooning lead to higher throughput, which is consistent with previous results \\cite{lioris17}. The nominal throughput is not attained due to interaction between CAVs and non-CAVs.\n\n    \\item Fig.~\\ref{Rbar_l} shows that platooning does improve throughput, but overly long platoons may result in lower throughput.\nThe reason is that longer platoons have stronger impact on local traffic and cause larger local congestion. For this example, the empirically optimal platoon size is 4 CAVs.\n\n        \\item As shown in Fig.~\\ref{fig_bounds}, when the buffer size $\\Theta$ is small (e.g. less than 20), spillback occurs frequently, and thus an obvious throughput drop (with respect to the nominal throughput) is observed.\nAs $\\Theta$ approaches infinity, spillback hardly occurs, and both bounds approach the nominal throughput given by Lemma~\\ref{lmm_maximum}.\n\\end{enumerate}\n\n\n\n\n\n    \n    \n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Control Design}\n\\label{sec_control}\n\nIn this section, we study a set of control policies that attain the nominal throughput given by Lemma~\\ref{lmm_maximum}.\n\n\nOne can indeed use Theorem~\\eqref{thm_sufficient} as the stability constraint and solve (P).\nHowever, since Theorem~\\ref{thm_sufficient} is a sufficient condition, it  in general leads to a sub-optimal solution.\nThe main result of this section is a sufficient condition for optimality of a given control policy:\n\n\\begin{thm}[Optimality criterion]\n\\label{thm_optimal}\nSuppose that a control policy $(\\mu,\\nu)\\in\\mathscr U\\times\\mathscr V$ satisfies\n\\begin{subequations}\n\\begin{align}\n    &\\mu(q)=0,\n    \\quad\\forall q:q_1^m>0\\mbox{ or } q_2^m=\\Theta,\\label{eq_muq=0}\\\\\n    &\\mu(q)\\le F-((1-\\eta)\\rho+(1-\\rho))a,\n    \\quad\\forall q\\in\\mathcal Q,\\label{eq_muq<F}\\\\\n    &\\mu(q)+(1-\\eta)\\rho a\\ge F-R,\n    \\quad\\forall q:q_0^m>0\\mbox{ and }q_2^m=0,\\label{eq_muq>F-R}\\\\\n    &S_1^m(q;\\nu(q))=0,\\quad\\forall q\\in\\mathcal Q,\\label{eq_S1m}\\\\\n    &S_2^m(q;\\nu(q))<\\Theta,\n    \\quad\\forall q\\in\\mathcal Q.\n    \\label{eq_S2m}\n\\end{align}\n\\end{subequations}\nThen, $(\\mu,\\nu)$ stabilizes the stochastic fluid model if and only if\n\\begin{align}\n    a<a^*,\\label{eq_iff}\n\\end{align}\nwhere $a^*$ is the nominal throughput given by \\eqref{eq_nominal}.\nFurthermore, under \\eqref{eq_iff}, the control $(\\mu,\\nu)$ minimizes the total queue length $|Q(t)|$ among all admissible controls at any time $t\\ge0$, and the time-average queuing delay converges as follows:\n\\begin{align}\n&\\lim_{t\\to\\infty}\\frac1t\\int_{\\tau=0}^t|Q(\\tau)|d\\tau\n\\stackrel{\\footnotesize a.s.}=\\bar Q\n:=\\frac{\\eta\\rho al}{2\\gamma^2(F-R-(1-\\eta)\\rho a)}\\nonumber\\\\\n&\\qquad\\times\\bigg(\\frac{\\eta\\rho a}{\\gamma(F-R-(\\eta\/\\gamma+1-\\eta)\\rho a)}+1\\bigg).\n\\label{eq_Qbar}\n\\end{align}\n\\end{thm}\n\nNote that Theorem 3 addresses not only stabilization, but also throughput maximization and queue minimization.\nThe optimality criterion \\eqref{eq_muq=0}--\\eqref{eq_S1m} can be interpreted as follows. \\eqref{eq_muq=0}, \\eqref{eq_muq<F}, and \\eqref{eq_S1m} ensure no queuing in link 1 (i.e. $Q_1^m(t)=0$).\n\\eqref{eq_muq=0} and \\eqref{eq_S2m} ensure no spillback at link 2 (i.e. $Q_2^m(t)<\\Theta$).\n\\eqref{eq_muq>F-R} ensures that as long as there is a non-zero queue in the gate (i.e. $Q_0^m(t)>0$), link 2 must be discharging traffic at its capacity $F-R$.\n\nWe use $\\mathscr U^*\\times\\mathscr V^*$ to denote the set of control policies satisfying \\eqref{eq_muq=0}--\\eqref{eq_S2m}.\nSince $(\\mu,\\nu)\\in\\mathscr U^*\\times\\mathscr V^*$ stabilizes the stochastic fluid model if and only if the demand is less than the nominal throughput, $(\\mu,\\nu)$ maximizes the throughput.\nIn addition, each control in $\\mathscr U^*\\times\\mathscr V^*$ not only maximizes throughput, but also minimizes queuing delay in a sample path-wise manner.\nFurthermore, an analytical expression for the mean queuing delay is obtained.\nThe proof of Theorem~\\ref{thm_optimal} is in Appendix 3.\n\n\n\nIn the rest of this section, we discuss two concrete control policies with practical interpretations, viz. headway regulation and platoon size management.\nHere we focus on their formulation in the stochastic fluid model.\nIn Section~\\ref{sub_implementation}, we discuss and demonstrate how they can be implemented in practice (e.g. speed).\n\n\\subsection{Headway regulation}\n\\label{exm_ustar}\nUnder this strategy, if a large number of platoons arrive within a short time period, some platoons will be allocated to the gate so that their arrival at the bottleneck is postponed to avoid cumulative congestion at the bottleneck.\nIn the stochastic fluid model, the headway regulation strategy can be formulated as a control policy $(\\mu,\\nu)$ such that:\n\\begin{subequations}\n\\begin{align}\n&\\mu^{hr}(q):=\\left\\{\\begin{array}{ll}\n\\alpha & \\mbox{if }q_0^m>0,q_2^{m}=0,\\\\\n\\alpha & \\mbox{if }\\gamma q_0^m\/l\\notin\\{0,1,2,\\ldots\\},\\\\\n0 & \\mbox{o.w.}\n\\end{array}\\right.\n\\label{eq_ustar}\\\\\n&\\nu^{hr}(q):=\\left[\\begin{array}{c}\n     l\/\\gamma  \\\\\n     q_1^m-S_1(q;0)\\\\\n     q_2^m-S_2(q;0)\n\\end{array}\n\\right],\n\\end{align}\n\\end{subequations}\nwhere $\\alpha$ is the saturation flow rate of CAVs given by\n\\begin{align}\n\\alpha=v_0\/h,\n\\label{eq_alpha}\n\\end{align}\n$v_0$ is the nominal speed of the highway section, and $h$ is the between-vehicle spacing within platoons.\nOne can check that $(\\mu^{hr},\\nu^{hr})\\in\\mathscr U^*\\times\\mathscr V^*$.\n\n\\begin{figure}[hbt]\n\\centering\n\\subfigure[Regulating headway.]{\n\\centering\n\\includegraphics[width=0.2\\textwidth]{.\/Images\/fig_Qt_controlled}\n\\label{fig_Qt_controlled}\n}\n\\subfigure[Splitting platoon.]{\n\\centering\n\\includegraphics[width=0.2\\textwidth]{.\/Images\/fig_Qt_controlled2}\n\\label{fig_Qt_controlled2}\n}\n\\caption{Illustration of how regulating headway $(\\mu^{hr},\\nu^{hr})$ and splitting platoons $(\\mu^{sm},\\nu^{sm})$ avoid spillback; see Fig.~\\ref{fig_spillback} for the uncontrolled case.}\n\\label{fig_Qt}\n\\end{figure}\n\nFig.~\\ref{fig_Qt_controlled} illustrates the idea of this control policy: if two platoons enter the highway with a short inter-arrival time, the following platoon is decelerated so that its arrival at the bottleneck is postponed by $W^{hr}$ amount of time.\nConsequently, the platoons arrive at the bottleneck with sufficient headway in between.\nAs illustrated in Fig.~\\ref{fig_Qt_controlled}, $(\\mu^{hr},\\nu^{hr})$ essentially regulates times at which platoons arrive at the bottleneck so that congestion does not build up or spill back from the bottleneck, and the off-ramp traffic is not blocked.\nTo compute $W^{hr}$, suppose that a platoon enter the highway at time $t$ and the state immediately before the arrival is $Q(t_-)$; then $W^{hr}$ is the solution to the deterministic equation\n\\begin{align}\n    \\int_{s=t}^{W^{hr}}\\mu^{hr}(Q(s))ds=Q_0^m(t_-).\n    \\label{eq_Whr}\n\\end{align}\nNote that $W^{hr}$ is independent of any platoon arrivals after $t$.\n\n\n\n\\subsection{Platoon size management}\n\\label{exm_u2star}\nIf the highway is congested, long platoons will be disadvantageous at bottlenecks due to their sizes. Consequently, the operator can instruct platoons to split into shorter platoons to mitigate local congestion. The decision variable is whether to split or maintain a platoon as it enters the highway.\nIn the stochastic fluid model, splitting a platoon can be modeled by a control policy $(\\mu^{sm},\\nu^{sm})$ defined as follows:\n\\begin{align}\n&{{\\mu}^{sm}}(q):=\\left\\{\\begin{array}{ll}\n\\alpha & \\mbox{if }q_0^m>0,q_2^{a}\\le \\Theta-\\frac{l((\\eta\/\\gamma+1-\\eta)\\rho a-(F-R))}{2\\gamma \\eta\\rho a},\\\\\n\\alpha & \\mbox{if }(\\gamma q^m_0\/l)\\notin\\{0,\\frac12,1,\\frac32,\\ldots\\},\\\\\n0 & \\mbox{o.w.}\n\\end{array}\\right.\n\\label{eq_u2star}\\\\\n&\\nu^{sm}(q):=\\left[\\begin{array}{c}\n     l\/\\gamma  \\\\\n     q_1^m-S_1(q;0)\\\\\n     q_2^m-S_2(q;0)\n\\end{array}\n\\right].\n\\end{align}\nOne can check that $(\\mu^{sm},\\nu^{sm})\\in\\mathscr U^*\\times\\mathscr V^*$.\nAs shown in Fig.~\\ref{fig_Qt_controlled2}, ${\\mu}^{sm}$ opens the gate only if link 2 has sufficient space to accept at least half a platoon.\nFurthermore, if $q_2^{a}$ is close to the buffer size $\\Theta$, ${\\nu}^{sm}$ will split a platoon into two short platoons and allocate the two short platoons to links 0 and 2, respectively, to avoid spillback.\nIn practice, suppose a platoon enters the highway at time $t$ and let $T$ be the solution to \n\\begin{align}\n&\\int_{s=t}^Tf_2(Q(s))ds=Q^m_0(t_-)+Q^m_1(t_-)\\nonumber\\\\\n&\\qquad\\qquad+\\min\\Big\\{\\frac{l}{2\\gamma},\\Big(\\Theta-\\frac{l}{2\\gamma}- Q_2^m(t_-)\\Big)_+\\Big\\},\n\\label{eq_T}\n\\end{align}\nwhere $Q_k^m(t_-)$ is the queue size in the gate immediately before the platoon arrives.\nThen, the delays $W^{sm}_1$ and $W^{sm}_2$ indicated in Fig.~\\ref{fig_Qt_controlled2} are given by\n\\begin{align}\nW_1^{sm}=T,\\\nW_2^{sm}=T+\\frac{l}{2\\gamma(F-R-(1-\\eta)\\rho a)}.\n\\label{eq_W1}\n\\end{align}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Concluding remarks}\n\\label{sec_conclude}\n\nIn this paper, we develop a stochastic fluid model for analysis and design of platoon coordination strategies. The model focuses on the interaction between CAV platoons and non-CAVs, the impact of key platooning parameters (including platooning ratio and platoon size), and road geometry (buffer space). Based on the theory of Markov processes and queuing properties of the model, we derive theoretical bounds for the throughput of the model and identify a set of coordination strategies that maximize throughput as well as minimize delay. We discuss how such strategies can be implemented in practice and validate the fluid model-based results \nusing standard macroscopic and microscopic simulation environments.\nOur results are useful for link-level coordination of CAV platoons.\n\nThis work can be extended in the following directions: (i) evaluation of macroscopic impact due to various vehicle-level controllers, (ii) extension to multiple-section highways, and (iii) integration with network-level scheduling and routing of CAVs.\n\n\\section*{Appendix 1. Proof of Theorem~\\ref{thm_sufficient}}\n\\label{sub_sufficient}\n\nThe stability criterion is obtained by showing that the Lyapunov function\n\\begin{align}\n    V(q):=&\\frac12(q_0^m+q_1^m+q_2^m)^2\\nonumber\\\\\n    &+\\bigg(k\\Big(q_0^m+q_1^m+\\frac1{2\\Theta}(q_2^m)^2\\Big)+q_1^o\\bigg)q_1^o\n    \\label{eq_V}\n\\end{align}\nsatisfies the Foster-Lyapunov criterion if \\eqref{eq_a<a*}--\\eqref{eq_max} hold.\nSpecifically, we show that there exist $k>0$, $c>0$, and $d<\\infty$ such that\n\\begin{align}\n    \\mathscr LV(q)\\le-c|q|+d,\n    \\quad\\forall q\\in{\\mathcal M},\n    \\label{eq_LV}\n\\end{align}\nwhich implies stability via the Foster-Lyapunov criterion.\nTo proceed, we decompose $V$ into\n\\begin{subequations}\\begin{align}\n    &V^m(q):=\\frac12(q_0^m+q_1^m+q_2^m)^2,\\label{eq_Vm}\\\\\n    &V^o(q):=\\bigg(k\\Big(q_1^m+\\frac1{2\\Theta}(q_2^m)^2\\Big)+q_1^o\\bigg)q_1^o\\label{eq_Vo}\n\\end{align}\\end{subequations}\nand show that there exists $c^m,c^o>0$ and $d^m,d^o<\\infty$ such that for all $q\\in\\mathcal M_{\\mu,\\nu}$,\n\\begin{subequations}\n\\begin{align}\n    &\\mathscr LV^m(q)\\le-c^m(q_0^m+q_1^m+q_2^m)+d^m,\\label{eq_LVa}\\\\\n    &\\mathscr LV^o(q)\\le-c^oq_1^o+d^o.\\label{eq_LVb}\n\\end{align}\n\\end{subequations}\nNote that the above implies \\eqref{eq_LV} and hence stability.\nThe rest of this subsection is devoted to the proof of \\eqref{eq_LVa}--\\eqref{eq_LVb}.\n\n\\subsubsection{Proof of \\eqref{eq_LVa}}\nFor $q$ such that $q_2^m=0$, we have\n\\begin{align*}\n    \\mathscr LV^m(q)=\\frac12(\\lambda l\/\\gamma)^2=\\frac{\\eta \\rho a}{2\\gamma}\n\\end{align*}\nFor $q$ such that $q_2^m>0$, we have\n\\begin{align*}\n    \\mathscr LV^m(q)&=((1-\\eta)\\rho a+\\lambda l\/\\gamma-(F-R))(q_0^m+q_1^m+q_2^m)\\\\\n    &=((\\eta\/\\gamma+1-\\eta)\\rho a-(F-R))(q_0^m+_1^m+q_2^m).\n\\end{align*}\nHence, we have \n\\begin{align*}\n    &c^m=(F-R)-(\\eta\/\\gamma+1-\\eta)\\rho a\n    \\stackrel{\\footnotesize\\eqref{eq_a<a*}}>0,\\\\\n    &d^m=\\frac{\\eta \\rho a}{2\\gamma}<\\infty\n\\end{align*}\nthat satisfy \\eqref{eq_LVa}.\n\n\\subsubsection{Proof of \\eqref{eq_LVb}}\n\\label{subsub_LVo}\n\nFor $q$ such that $q_1^o=0$, we have\n\\begin{align}\n    \\mathscr LV^o(q)=0.\n    \\label{eq_LVb0}\n\\end{align}\nFor $q$ such that $q_1^o>0$, we need to consider the following cases:\n\\begin{enumerate}\n    \\item $q_0^m=q_1^m=0$. In this case, we require a $c^o>0$ such that\n    \\begin{align}\n        &\\mathscr LV^o(q)\\nonumber\\\\\n        &= \\bigg(k(q_2^m\/\\Theta)((1-\\eta)\\rho a-(F-R))+\\lambda k\\Big(S^m_0(q;\\nu(q))\\nonumber\\\\\n        &\\quad+S^m_1(q;\\nu(q))+\\frac{1}{2\\Theta}(S^m_2(q;\\nu(q)))^2-\\frac1{2\\Theta}(q_2^m)^2\\Big)\\nonumber\\\\\n        &\\quad+(1-\\rho)a-R\\bigg)q_1^o\\nonumber\\\\\n        &\\le-c^oq_1^o,\n        \\quad \\forall q:q_1^m=0,q_1^o>0.\n        \\label{eq_LVb1}\n    \\end{align}\n    By Assumption~\\ref{asm_nu}, we have\n    \\begin{align*}\n        &S^m_0(q;\\nu(q))+S^m_1(q;\\nu(q))+\\frac{1}{2\\Theta}(S^m_2(q;\\nu(q)))^2\\\\\n        &-\\frac1{2\\Theta}(q_2^m)^2\\le S^m_0(q;\\nu(q))+S^m_1(q;\\nu(q))\\\\\n        &+\\frac{1}{2\\Theta}(S^m_2(q;\\nu(q)))^2-\\frac1{2\\Theta}(q_2^m)^2\\Big|_{q_1^o=0},\n        \\quad\\forall q\\in\\mathcal Q.\n    \\end{align*}\n    Hence, we require a $k>0$ such that\n    \\begin{align}\n        &(q_2^m\/\\Theta)((1-\\eta)\\rho a-(F-R))+\\lambda \\Big(S^m_0(q;\\nu(q))\\nonumber\\\\\n        &+S^m_1(q;\\nu(q))+\\frac{1}{2\\Theta}(S^m_2(q;\\nu(q)))^2-\\frac1{2\\Theta}(q_2^m)^2\\Big|_{q_1^o=0}\\Big)\\nonumber\\\\\n        &<\\frac{R-(1-\\rho)a}k,\n        \\quad \\forall q:q_0^m=q_1^m=0,q_1^o>0.\n        \\label{eq_Db1}\n    \\end{align}\n    \n    \\item $q_0^m>0$ or $q_1^m>0,\\ q_2^m<\\Theta$. In this case, we require $c^0>0$ such that\n    \\begin{align}\n        &\\mathscr LV^o(q)=\\bigg(k\\Big((1-\\eta)\\rho a-F+(q_2^m\/\\Theta)(F\\nonumber\\\\\n        &\\quad-(F-R))\\Big)+k\\lambda(l\/\\gamma)\\Big)+(1-\\rho)a-R\\bigg)q_1^o\\nonumber\\\\\n        &=\\Big(k\\Big((\\frac{\\eta}{\\gamma}+1-\\eta)\\rho a-(F-R)\\Big)+(1-\\rho)a-R\\Big)q_1^o\\nonumber\\\\\n        &\\le-c^oq_1^o,\n        \\quad \\forall q:q_0^m>0\\mbox{ or }q_1^m>0,q_1^o<\\Theta.\n        \\label{eq_LVb2}\n    \\end{align}\n    For all $k>0$, the existence of such a $c^o$ is ensured by \\eqref{eq_a<a*}.\n    \n    \\item $q_0^m>0$ or $q_1^m>0,\\ q_2^m=\\Theta$. In this case,\n    \\begin{align}\n        \\mathscr LV^o(q)&=\\bigg(k\\Big((\\eta\/\\gamma+1-\\eta)\\rho a-(F-R)\\Big)\\nonumber\\\\\n        &\\quad+(1-\\rho)a\\bigg)q_1^o\\le-c^oq_1^o.\n        \\label{eq_LVb3}\n    \\end{align}\n    We require a $k>0$ such that\n    \\begin{align}\n       k\\Big((F-R)-(\\eta\/\\gamma+1-\\eta)\\rho a\\Big)>(1-\\rho)a.\n       \\label{eq_Db3}\n    \\end{align}\n\\end{enumerate}\nNote that \\eqref{eq_max} ensures the existence of $k>0$ that simultaneously satisfies \\eqref{eq_Db1} and \\eqref{eq_Db3}. Hence, there exists $k>0$ and $c^o>0$ such that satisfy \\eqref{eq_LVb1}, \\eqref{eq_LVb2}, and \\eqref{eq_LVb3}, which, together with \\eqref{eq_LVb0}, imply \\eqref{eq_LVb}.\n\n\\section*{Appendix 2. Proof of Theorem~\\ref{thm_bounds}}\n\\label{sub_bounds}\n\n\n\n\\emph{1) Lower bound}\nThe lower bound is the minimum of the following two terms:\n\\begin{align*}\n    &\\underline a_1:=\\frac{F-R}{\\rho(\\eta\/\\gamma+1-\\eta)},\\\\\n    &\\underline a_2:=\n\\frac{R}{1-\\rho+\\frac{1}{2}\\Big(\\sqrt{\\zeta^2+\\frac{2\\rho Rl}{\\gamma\\Theta(F-R)}}-\\zeta)\\Big)},\n\\end{align*}\nwhere $\\zeta$ is given by \\eqref{eq_zeta0}.\nWe prove the lower bound by applying Theorem~\\ref{thm_sufficient}, i.e. verifying that \\eqref{eq_a<a*}--\\eqref{eq_max} hold if $a<\\min\\{\\underline a_1,\\underline a_2\\}$:\n\\begin{enumerate}\n    \\item Since $2\\rho Rl\/(\\gamma\\Theta(F-R))>0$, we have $\\underline a_2\\le R\/(1-\\rho)$. Hence, $a<\\underline a_2$ and $a<\\underline a_1$ ensure that \\eqref{eq_a<a*} holds.\n    \\item For $\\xi\\in\\mathcal Q$ such that $\\xi_0^m=\\xi^m_1=0$,\n\\begin{align*}\n    &\\frac{\\xi_2^m}{\\Theta}\\Big((1-\\eta)\\rho a-(F-R)\\Big)+\\frac{\\eta\\rho a}{l} \\bigg(S^m_0(\\xi;0)\\nonumber\\\\\n    &\\qquad+S^m_1(\\xi;0)+\\frac{1}{2\\Theta}\\Big(S^m_2(\\xi;0)\\Big)^2-\\frac1{2\\Theta}(\\xi_2^m)^2\\bigg)\\\\\n    &=\\Big(\\frac{\\xi_2^m}{\\Theta}((1-\\eta)\\rho a-(F-R)\\Big)\\\\\n    &\\qquad+\\frac{\\lambda }{2\\Theta}(2\\xi_2^m l\/\\gamma+(l\/\\gamma)^2).\n\\end{align*}\nSince $a<\\underline a_1$ and since $\\xi_2^m\\le\\Theta$, the above implies that\n\\begin{align*}\n    &\\max_{\\substack{\\xi\\in\\mathcal Q:\\xi_0^m\\\\=\\xi_1^m=\\xi_1^o=0}}\\frac{\\xi_2^m}{\\Theta}\\Big((1-\\eta)\\rho a-(F-R)\\Big)+\\frac{\\eta\\rho a}{l} \\bigg(S^m_0(\\xi;0)\\nonumber\\\\\n    &\\qquad+S^m_1(\\xi;0)+\\frac{1}{2\\Theta}\\Big(S^m_2(\\xi;0)\\Big)^2-\\frac1{2\\Theta}(\\xi_2^m)^2\\bigg)\\\\\n    &\\le \\Big(\\frac{\\lambda k}{2\\Theta}(2\\Theta l\/\\gamma+(l\/\\gamma)^2).\n\\end{align*}\nWith the above, one can verify that if $a<\\underline a_2$, then \\eqref{eq_max} holds.\n\\end{enumerate}\n\n\\emph{2) Upper bound}\n\nWe prove the upper bound by showing that if $\\{Q(t);t>0\\}$ is stable, then $a\\le\\min\\{\\bar a_1,\\bar a_2\\}$, where\n\\begin{align*}\n    &\\bar a_1:=\\frac{F-R}{\\rho(\\eta\/\\gamma+1-\\eta)},\\\\\n    &\\bar a_2:=\\frac{(1-\\omega)R}{1-\\rho}.\n\\end{align*}\n\n\\begin{enumerate}\n    \\item $a\\le\\bar a_1$ can be obtained from the nominal throughput given by Lemma~\\ref{lmm_maximum}.\n    \\item To show $a\\le a_2$, note that when $a<\\bar a_1$, the M\/D\/1 process $\\{N(t);t>0\\}$ is stable and admits a steady-state distribution $\\{\\pi_n;n=0,1,\\ldots\\}$ defined in \\eqref{eq_pin}.\nHence, there exists $\\omega_0$ and $\\omega$ such that\n\\begin{align*}\n    &\\lim_{t\\to\\infty}\\frac1t\\int_{\\tau=0}^t\\mathbb I_{Q_1^m>0,Q_2^m=\\Theta}d\\tau=\\omega_0\\quad a.s.,\\\\\n    &\\lim_{t\\to\\infty}\\frac1t\\int_{\\tau=0}^t\\mathbb I_{N(t)\\ge\\ceil{\\gamma\\Theta\/l}}d\\tau=\\omega\\quad a.s.\n\\end{align*}\nwhere $\\omega$ is in fact given by \\eqref{eq_omega}.\n\nNext, consider the set $\\mathcal M_{0,0}\\subset\\mathcal Q$ defined by\n\\begin{align}\\label{eq_M}\n{\\mathcal M_{0,0}}=\\Big((\\{0\\}\\times[0,\\Theta])\\cup((0,\\infty)\\times\\{\\Theta\\})\\Big)\\times[0,\\infty).\n\\end{align}\nOne can show that $\\mathcal M_{0,0}$ is an invariant set.\nHence, for each initial condition $q\\in\\mathcal M_{\\mu,\\nu}$, we have $Q(t)\\in\\mathcal M_{\\mu,\\nu}$ for all $t>0$.\nThus we have $Q_2^m(t)=\\Theta$ if $Q_1^m(t)>0$ for sufficiently large $t$. \nHence, if $Q_1^m(t)+Q_2^m(t)>\\Theta$, i.e. if $Q_1^m(t)>0$ and $Q_2^m(t)=\\Theta$, then $N(t)\\ge\\ceil{\\gamma\\Theta\/l}$.\nTherefore, we have\n$\n\\omega_0\\ge\\omega.\n$\nFinally, note that if $|Q(t)|$ is bounded, then \n\\begin{align*}\n    (1-\\rho)a&\\le\\lim_{t\\to\\infty}\\frac1t\\int_{\\tau=0}^tr(Q(\\tau);0)d\\tau\\\\\n    &=\\lim_{t\\to\\infty}\\frac1t\\Big(\\int_{\\tau:Q_1^m(\\tau)+Q_1^m(\\tau)\\le\\Theta}^tr(Q(\\tau);0)d\\tau\\\\\n    &\\quad+\\int_{\\tau:Q_1^m(\\tau)+Q_1^m(\\tau)>\\Theta}^tr(Q(\\tau);0)d\\tau\\Big)\\\\\n    &\\stackrel{a.s.}=(1-\\omega_0)R\n    \\le(1-\\omega)R=\\bar a_2.\n\\end{align*}\n\\end{enumerate}\n\n\\section*{Appendix 3. Proof of Theorem~\\ref{thm_optimal}}\n\\label{sub_optimal}\n\n\\emph{1) Stability}\n\\label{sub_max}\n\nThe necessity of \\eqref{eq_iff} results from Lemma~\\ref{lmm_maximum}.\nTo show the sufficiency, one can indeed use Theorem~\\ref{thm_sufficient} to show that the model is stabilized by $(\\mu,\\nu)\\in\\mathscr U^*\\times\\mathscr V^*$ if $a<a^*$ in the sense of a bounded 1-norm.\nIn this subsection, we use an alternative Lyapunov function\n\\begin{align}\n\\tilde V(q)=e^{\\beta|q|},\n\\quad q\\in\\mathcal Q\n\\label{eq_Vtilde}\n\\end{align}\nand obtain a stronger stability:\n\\begin{align}\n    \\limsup_{t\\to\\infty}\\frac1t\\int_{\\tau=0}^t{\\mathrm E}[e^{\\beta|{Q}(\\tau)|}]d\\tau\\le Z.\n    \\label{eq_MGF}\n\\end{align}\nThat is, the state is bounded in the MGF.\n\nTo proceed, define\n\\begin{align}\n\\zeta_{\\mu,\\nu}:=&\\max\\Big\\{\\max_{q\\in\\mathcal Q}S_2^m(q;\\nu(q)),\\ \\sup\\{\\zeta\\ge0:\\nonumber\\\\\n&(\\forall {q}:q_2^m=\\zeta)\\ {\\mu}(q)\\ge {F-R}-(1-\\eta)\\rho a\\}\\Big\\}.\n\\label{eq_zeta}\n\\end{align}\nWe know from \\eqref{eq_muq=0}--\\eqref{eq_S1m} that $\\zeta_{\\mu,\\nu}<\\Theta$ for all $(\\mu,\\nu)\\in\\mathscr U^*\\times \\mathscr V^*$.\nThen, consider the set\n\\begin{align}\\label{eq_Mtilde}\n\\mathcal M_{\\mu,\\nu}=[0,\\infty)\\times\\{0\\}^2\\times[0,\\zeta_{\\mu,\\nu}].\n\\end{align}\n\n\\begin{lmm}\nThe set $\\mathcal M_{\\mu,\\nu}$ as defined in \\eqref{eq_Mtilde} is an invariant set.\n\\end{lmm}\n\\noindent\\emph{Proof.}\n(i) For all $q\\in\\mathcal M_{\\mu,\\nu}$, we have\n\\begin{align*}\n    G^m_1(q;\\mu(q))&=\\mu(q)+((1-\\eta)\\rho+(1-\\rho))a\\\\\n    &\\quad-\\min\\Big\\{\\Big((1-\\eta)\\rho+(1-\\rho)\\Big)a+\\mu(q),F\\Big\\}\\\\\n    &\\stackrel{\\footnotesize\\eqref{eq_muq<F}}=0.\n\\end{align*}\nHence, $\\{0\\}$ is invariant for $q_1^m$.\nIn addition, for $q\\in\\mathcal M_{\\mu,\\nu}$ such that $q_2^m=\\zeta$, we have\n\\begin{align*}\n    G_2^m(q;\\mu(q))&=f_1(q;\\mu(q))-f_2(q;\\mu(q))\\\\\n    &\\stackrel{\\footnotesize\\eqref{eq_zeta}}<(\\mu(q)+(1-\\eta)\\rho a)-(\\mu(q)+(1-\\eta)\\rho a)\\\\\n    &<0.\n\\end{align*}\nHence, $[0,\\zeta_{\\mu,\\nu}]$ is invariant for $q_2^m$.\n\n(ii) By \\eqref{eq_S1m}, since $S_1^m(q;\\nu(q))=0$ for all $q\\in\\mathcal Q$, $\\{0\\}$ is invariant for $q_1^m$. By \\eqref{eq_zeta}, $S_2^m(q;\\nu(q))\\le\\zeta$ for all $q\\in\\mathcal Q$. Hence, $[0,\\zeta]$ is invariant for $q_2^m$.\n\\hfill$\\square$\n\nIn the rest of this proof, we show that there exists $\\beta>0$, $c>0$, and $\\tilde d<\\infty$ verifying the drift condition\n\\begin{align}\\label{eq_drift2}\n{\\mathcal L}\\tilde V(q)\\le-ce^{\\beta|q|}+ \\tilde d\n\\quad\\forall q\\in\\mathcal M_{\\mu,\\nu},\n\\end{align}\nwhich leads to \\eqref{eq_MGF} by the Foster-Lyapunov criterion.\n\nWe partition the invariant set $\\mathcal M_{\\mu,\\nu}$ into two subsets:\n\\begin{align*}\n&{\\mathcal M_{\\mu,\\nu}^0}=\\{0\\}^3\\times[0,\\zeta_{\\mu,\\nu}],\\\\\n&{\\mathcal M_{\\mu,\\nu}^1}=(0,\\infty)\\times\\{0\\}^2\\times[0,\\zeta_{\\mu,\\nu}],\n\\end{align*}\nFor $q\\in{\\mathcal M_{\\mu,\\nu}^0}$, we have\n\\begin{align*}\n{\\mathcal L}\\tilde V(q)\n&=\\nabla_{{q}}e^{\\beta|q|}G(q;\\mu(q))+\\lambda(e^{\\beta |S(q;\\nu(q))|}-e^{\\beta|q|})\\\\\n&=(\\beta((1-\\eta)\\rho a-\\tilde f_2(q;\\mu(q)))+\\lambda(e^{\\beta l\/\\gamma}-1))e^{\\beta|q|}\\\\\n&\\le\\lambda(e^{\\beta l\/\\gamma}-1)e^{\\beta l\/\\gamma}=\\tilde d^*,\n\\end{align*}\nwhich also defines $\\tilde d^*$.\nFor $q\\in{\\mathcal M_{\\mu,\\nu}^1}$, we have\n\\begin{align*}\n{\\mathcal L}\\tilde V(q)\n&\n=(\\beta((1-\\eta)\\rho a-(F-R))+\\lambda(e^{\\beta l\/\\gamma}-1))e^{\\beta|q|}\\\\\n&=\\phi(\\beta)e^{\\beta|q|},\n\\end{align*}\nwhere the definition of the function $\\phi$ is clear.\nSince\n\\begin{align*}\n\\phi(0)=0,\\\n\\frac{d}{d\\beta}\\phi(\\beta)\\Big|_{\\beta=0}=(1-\\eta)\\rho a-(F-R)+\\eta\\rho a\/\\gamma,\n\\end{align*}\nthere exists $\\beta^*>0$ such that $\\phi(\\beta^*)<0$ if $a<a^*$.\n\nIn conclusion, there exist $\\beta=\\beta^*,$ $c=\\phi(\\beta^*)$, and $\\tilde d=\\tilde d^*$ that verify \\eqref{eq_drift2}. Then, by the Foster-Lyapunov stability criterion, we conclude \\eqref{eq_MGF}.\n\n\n\\emph{2) Queue minimization}\n\nNext, we use a sample path-based method to show that any $(\\mu,\\nu)\\in\\mathscr U^*\\times\\mathscr V^*$ minimizes the total queue size $|Q(t)|$ for all $t$ over all control policies $(\\mu,\\nu)\\in\\mathscr U\\times\\mathscr V$.\n\nLet $\\{{M}(t);t>0\\}$ be the counting process of platoon arrivals.\nFor a given sample path $\\{{m}(t);t>0\\}$ of the counting process and a given initial condition $q\\in{\\mathcal Q}$, let $\\{{q}(t);t>0\\}$ and $\\{\\psi(t);t>0\\}$ be the corresponding trajectories under a control policy $(\\mu^*,\\nu^*)\\in\\mathscr U^*\\times\\mathscr V^*$ and under a control policy $(\\mu,\\nu)\\in\\mathscr U\\times\\mathscr V$, respectively.\nTo show the optimality of $(\\mu^*,\\nu^*)$, it suffices to show $|{q}(t)|\\le|\\psi(t)|$ for any $t\\ge0$.\nWithout loss of generality, we consider zero initial condition. \nWe prove this by contradiction as follows.\n\nAssume by contradiction that there exists $(\\mu,\\nu)\\in\\mathscr U\\times\\mathscr V$ such that\n\\begin{align}\n\\exists t_1>0,\\ |q(t_1)|>|\\psi(t_1)|.\\label{eq_contradiction}\n\\end{align}\nBetween resets, the continuity of ${q}(t)$ and $\\psi(t)$ follows from Assumption~\\ref{asm_mu}.\nTherefore, there must exist a ``crossing time'' $t_0\\in[0,t)$ such that\n\\begin{align}\n|{q}(t_0)|=|\\psi(t_0)|,\\\n\\frac d{dt}|q(t)|>\\frac d{dt}|\\psi(t)|.\n\\label{crossing}\n\\end{align}\nNote that the ``crossing'' must happen between resets. To see this, recall that Assumption~\\ref{asm_nu} ensures that if $|{q}(t_-)|=|\\psi(t_-)|$ and if a reset occurs at time $t$, then\n\\begin{align*}\n    |q(t)|=|\\psi(t)|=|q(t_-)|+l\/\\gamma.\n\\end{align*}\n\nSince the system admits the invariant set $\\mathcal M_{\\mu^*,\\nu^*}$ as given in \\eqref{eq_Mtilde} under $(\\mu^*,\\nu^*)$, we have ${q}_1^m(t)={q}_1^o(t)=0$ for all $t\\ge0$.\nHence, a necessary condition for \\eqref{crossing} is that\n\\begin{subequations}\n\\begin{align}\n&{q}_0^m(t_0)+{q}_2^m(t_0)\n\\ge\\psi_0^m(t_0)+\\psi_1^m(t_0)+\\psi_2^m(t_0),\\\\\n&G_0^m(q(t_0);\\mu^*)+G_2^m(q(t_0);\\mu^*)\\nonumber\\\\\n&>G_0^m(\\psi(t_0);\\mu)+G_1^m(\\psi(t_0);\\mu)+G_2^m(\\psi(t_0);\\mu)).\\label{eq_dot*=dotd}\n\\end{align}\n\\end{subequations}\nHowever, one can obtain from \\eqref{eq_G0}--\\eqref{eq_G2} that if ${q}_0^m(t_0)+{q}_2^m(t_0)\n\\ge\\psi_0^m(t_0)+\\psi_1^m(t_0)+\\psi_2^m(t_0)=0$, then\n\\begin{align}\n&G_0^m(q(t_0);\\mu^*)+G_2^m(q(t_0);\\mu^*)\\nonumber\\\\\n&\n=G_0^m(\\psi(t_0);\\mu)+G_1^m(\\psi(t_0);\\mu)+G_2^m(\\psi(t_0);\\mu));\\label{eq_sum=0}\n\\end{align}\nif ${q}_0^m(t_0)+{q}_2^m(t_0)\n\\ge\\psi_0^m(t_0)+\\psi_1^m(t_0)+\\psi_2^m(t_0)>0$, then\n\\begin{subequations}\n\\begin{align}\n&G_0^m(q(t_0);\\mu^*)+G_2^m(q(t_0);\\mu^*)=(1-\\eta)\\rho a-(F-R),\\label{eq_sum>0_1}\\\\\n&G_0^m(\\psi(t_0);\\mu)+G_1^m(\\psi(t_0);\\mu)+G_2^m(\\psi(t_0);\\mu))\\nonumber\\\\\n&\\quad=(1-\\eta)\\rho a-r(\\psi(t_0))\n\\ge (1-\\eta)\\rho a-(F-R).\\label{eq_sum>0_2}\n\\end{align}\n\\end{subequations}\nSince both \\eqref{eq_sum=0} and \\eqref{eq_sum>0_1}--\\eqref{eq_sum>0_2} contradict with \\eqref{eq_dot*=dotd}, we conclude that $(\\mu,\\nu)$ cannot achieve \\eqref{eq_contradiction}. That is, if $|q(t_1)|=|\\psi(t_1)|$, then $|q(t)|$ cannot increase faster (or decrease slower) than $|\\psi(t)|$ at time $t=t_1$. This proves the optimality of $(\\mu^*,\\nu^*)$.\n\n\\emph{3) Mean queue size}\n\\label{sub_queue}\n\nTo show that the 1-norm of the state converges as in \\eqref{eq_Qbar}, we first need to show that the process $\\{Q(t);t\\ge0\\}$ is ergodic. Formally, let $P_t(q)$ be the distribution of $Q(t)$ given the initial condition $Q(0)=q$ for $q\\in\\mathcal Q$. Then, there exists a unique probability measure $P^*$ on $\\mathcal Q$ such that\n\\begin{align}\n    \\lim_{t\\to\\infty}\\|P_t(q)-P^*\\|_{\\mathrm{TV}}=0,\n    \\label{eq_TV}\n\\end{align}\nwhere $\\|\\cdot\\|_{\\mathrm{TV}}$ is the total-variation distance between two probability measures \\cite{benaim15}. Ergodicity ensures convergence of the time average towards the expected value, if the expected value exists.\n\n\\emph{Convergence}:\nThe Foster-Lyapunov criterion ensures the existence of an invariant measure $P^*$ \\cite[Theorem 4.5]{meyn93}.\nWe only need to further show that the invariant measure is unique.\nThis can be shown via the ``coupling'' condition:\n\n\\noindent{\\bf Coupling condition \\cite{meyn93}.}\n\\emph{Let $q,q'\\in\\mathcal Q$ be two initial conditions and $Q(t),Q'(t)$ be the trajectories starting therefrom. Then, there exists $\\delta>0$ and $T<\\infty$ such that\n\\begin{align}\n    \\Pr\\{Q(T)=Q'(t)|Q(0)=q,Q'(0)=q'\\}=\\delta.\n    \\label{eq_coupling}\n\\end{align}\n}\n\nTo show that the stochastic fluid model controlled by $(\\mu,\\nu)\\in\\mathscr U^*\\times\\mathscr V^*$ satisfies the coupling condition, note that for an arbitrary initial condition $q\\in{\\mathcal Q}$, there exists\n$\nT=(\\sum_{k=1}^3q_k^m)\/(F-R)+q_1^o\/R\n$\nsuch that\n\\begin{align*}\n\\Pr\\{Q(T)=0|Q(0)=q\\}\\ge e^{-\\lambda T}>0.\n\\end{align*}\nThen, by \\cite[Theorem 6.1]{meyn1993stability}, the above, together with \\eqref{eq_drift2}, implies convergence in the sense of \\eqref{eq_TV}.\n\n\\begin{rem}\nIn fact, the above argument ensures {exponentially convergent} \\cite{meyn93} in the sense that there exist a constant $\\kappa>0$ and a finite-valued function $U:\\tilde{\\mathcal Q}\\to\\mathbb R_{\\ge0}$ such that\n$$\n\\|P_t(q)-P^*\\|_{\\mathrm{TV}}\\le U(q)e^{-\\kappa t}\n\\quad\\forall t\\ge0.\n$$\n\\end{rem}\n\n\\emph{Queuing delay}:\nThe evolution of the total queue length $|Q(t)|$ can be viewed as the superposition of two subprocesses; see Fig.~\\ref{subprocesses}. \n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=0.3\\textwidth]{.\/Images\/subprocesses}\n\\caption{The controlled fluid process ${Q}(t)$ envelops an M\/D\/1 process $\\tilde N(t)$.}\n\\label{subprocesses}\n\\end{figure}\n\\begin{enumerate}\n    \\item The first process is an M\/D\/1 process $\\{N(t);t\\ge0\\}$ defined by \\eqref{eq_N}.\nBy the Pollazcek-Khinchin formula \\cite[p. 248]{gallager13} and the Little's theorem~\\cite[Theorem 5.5.9]{gallager13}, the mean number of waiting jobs (excluding the one being served) of this process is\n\\begin{align*}\n&\\bar N=\\frac{\\lambda^2 s^2}{2(1-\\lambda s)}\\\\\n&=\\frac{\\eta^2\\rho^2 a^2\/(2\\gamma^2)}{(F-R-(1-\\eta)\\rho a)(F-R-(\\eta\/\\gamma+1-\\eta)\\rho a)}.\n\\end{align*}\n    \\item The second process is the ``services'' each job experiences. As Fig.~\\ref{subprocesses} shows, the cumulative queuing delay during services is given by\n$$\n\\bar R=\\frac{l^2}{2\\gamma^2(F-R-(1-\\eta)\\rho a)}.\n$$\nThus, the total fluid queue length is\n$$\n\\bar Q=\\frac l\\gamma\\bar N+\\frac{\\eta\\rho a}{l}\\bar R,\n$$\nwhich leads to \\eqref{eq_Qbar}.\n\\end{enumerate}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe dynamic programming algorithm for computing optimal control policies has, since its development, been known to suffer from the ``curse of dimensionality'' \\citep{Bellman57}. Its applicability in practice is typically limited to systems with four or five continuous state variables because the number of points required to grid a space of $n$ continuous state variables increases exponentially with the state dimension $n$. This complexity has led to a collection of algorithms for approximate dynamic programming, which scale to systems with larger state dimension but lack the guarantees of global optimality of the solution associated with the original dynamic programming algorithm \\citep{Bellman59, Bertsekas12, Powell07, Powell16}. \n\nIn practice, many real-world systems exhibit symmetries that can be exploited to reduce the complexity of system models. Symmetry reduction has found applications in fields ranging from differential equations \\citep{Clarksonz94, Bluman13} to model checking \\citep{Emerson96, Kwiatkowska06}. In control engineering, symmetries have been exploited to improve control of mechanical systems \\citep{Marsden90, Bloch96, Bullo99}, develop more reliable state estimators \\citep{Barrau14}, study the controllability of multiagent systems \\citep{Rahmani09} and to reduce the complexity of stability and performance certification for interconnected systems \\citep{Arcak16, Rufino17}. Symmetry reduction has also been applied to the computation of optimal control policies for continuous-time systems in \\citep{Grizzle84, Ohsawa13} and Markov decision processes (MDPs) in \\citep{Zinkevich01, Narayanamurthy07}. \n\nIn this paper, we present a theory of symmetry reduction for the optimal control of discrete-time, stochastic nonlinear systems with continuous state variables. This reduction allows dynamic programming to be performed in a lower-dimensional state space. Since the computational complexity of a dynamic programming iteration increases exponentially with state dimension, this reduction significantly decreases computational burden. Further, our proposed method does not rely on an explicit transformation of the state update equations, making the method applicable in situations where a such a transformation is difficult or impossible to find analytically. \n\nWe present two theorems that summarize our method of symmetry reduction. Theorem \\ref{thm:symmetric_cost_and_policy} describes how symmetries of the system dynamics imply symmetries of the optimal cost and optimal policy functions. Theorem \\ref{thm:DP_reduced} then describes a method of computing the cost function based on reduced coordinate system that depends on fewer state variables. \n\nThis paper builds on the work we presented in the conference paper \\citep{Maidens17-ACC}. The most substantial improvement is the additional theoretical results presented in Sections \\ref{sec:moving_frames} and \\ref{sec:reduced_coordinates}. The conference version presented an \\textit{ad hoc} symmetry reduction for a magnetic resonance imaging (MRI) application, but did not provide a general methodology for computing the coordinate reduction. This paper addresses this shortcoming by presenting a general method based on the moving frame formalism, which leads to the general symmetry reduction formula presented in Theorem \\ref{thm:DP_reduced}. Additionally, the MRI example has been reworked to match this new formalism, and the numerical implementation and graphs of the numerical solution have been improved. We have also included two new extensions of this formalism to the case of equivariant costs in Section \\ref{sec:equivariance} and to the synchronization problem of stochastic dynamic systems on matrix groups in Section \\ref{sec:matrix_group}, along with examples to illustrate the algorithm in these contexts. \n\nThis paper is organized as follows: in Section \\ref{sec:dynamic_programming} we introduce notation and provide  background information both on dynamic programming for optimal control, and on the mathematical theory of symmetries. In Section \\ref{Main}, we derive our main theoretical results, that is, we prove that control system symmetries induce symmetries of the optimal cost function and optimal control policy, and then leverage the result to present a general method of performing dynamic programming in reduced coordinates. In Section \\ref{sec:vehicles_cooperative} we apply the algorithm to a cooperative control problem for two Dubins vehicles using a Lie group formulation. In Section \\ref{sec:MRI} we apply symmetry reduction to compute the solution of an optimal control problem arising in dynamic MRI acquisition. Code to reproduce the computational results in this paper is available at \\url{https:\/\/github.com\/maidens\/Automatica-2017}. \n\n\n\n\n\n\n\n\n\n\n\\section{Dynamic Programming and Symmetries}\n\\label{sec:dynamic_programming}\nIn this section, we first recall the main features of dynamic programming for optimal control of stochastic discrete time systems. Then we introduce our problem and provide the reader with a primer on the classical theory of symmetries. We also introduce the notion of invariant control systems with invariant costs. \n\n\\subsection{Dynamic programming for optimal control of stochastic systems}\nWe begin by introducing dynamic programming for finite horizon optimal control following the notation of \\citep{Bertsekas05}. We consider a discrete-time dynamical system\n\\begin{align}\n\tx_{k+1} = f_k(x_k, u_k, w_k), \\quad k = 0, 1, \\dots, N-1\\label{syst_initial:eq}\n\\end{align}\nwhere $x_k \\in \\mathcal{X} \\subseteq \\mathbb{R}^n$ is the system state, $u_k \\in \\mathcal{U} \\subseteq \\mathbb{R}^m$ is the control variable to be chosen at time $k$, $w_k \\in \\mathcal{W} \\subseteq \\mathbb{R}^\\ell$ are independent continuous random variables each with density $p_k$, and $N \\in \\mathbb{Z}_+$ is a finite control horizon. Associated with this system is an additive cost function \n\\[\n    g_N(x_N) + \\sum_{k=0}^{N-1} g_k(x_k, u_k, w_k) \n\\]\nthat we wish to minimize through our choice of $u_k$. We define a \\emph{control system} to be a tuple $\\mathcal{S} = (\\mathcal{X}, \\mathcal{U}, \\mathcal{W}, p, f, g, N)$ where $p = \\prod_{k=0}^{N-1} p_k$ is the joint density of the random variables $w_k$. \n\nWe consider a class of control policies $\\pi = \\{ \\mu_0, \\dots, \\mu_{N-1} \\}$ where $\\mu_k: \\mathcal{X} \\to \\mathcal{U}$ maps observed states to admissible control inputs. Given an initial state $x_0$ and a control policy $\\pi$, we define the expected cost under this policy as\n\\[\n  J_\\pi(x_0) = \\mathbb{E}\\left[g_N(x_N) + \\sum_{k=0}^{N-1} g_k(x_k, \\mu_k(x_k), w_k)\\right].  \n\\]\nAn optimal policy $\\pi^*$ is defined as one that minimizes the expected cost:\n\\[\n  J_{\\pi^*}(x_0) = \\min_{\\pi \\in \\Pi} J_\\pi(x_0) \n\\]\nwhere $\\Pi$ denotes the set of all admissible control policies. The optimal cost function, denoted $J^*(x_0)$, is defined to be the expected cost corresponding to an optimal policy. \n\nAs in \\citep{Bertsekas05}, we use $\\min$ to denote the infimum value regardless of whether there is a policy $\\pi \\in \\Pi$ that achieves a minimum. In the example problems presented in Sections \\ref{sec:vehicles_cooperative} and \\ref{sec:MRI}, the existence of a minimum is guaranteed by compactness or finiteness arguments respectively. In the general case, the optimal cost function $J^*$ can be computed using the dynamic programming algorithm regardless of the existence of minimizers, but the existence of an optimal policy $\\pi^*$ requires that a minimum be achieved for each $x_k \\in \\mathcal{X}$. \n\nWe quote the following result due to Bellman from \\citep{Bertsekas05}: \n\n\\begin{prop}[Dynamic Programming]\nFor every initial state $x_0$, the optimal cost $J^*(x_0)$ is equal to $J_0(x_0)$, given by the last step of the following algorithm, which proceeds backward in time from period $N-1$ to period 0:\n\\begin{equation}\n\\resizebox{.9\\hsize}{!}{$\n\\begin{split}\nJ_N(x_N) &= g_N(x_N) \\\\\nJ_k(x_k) &= \\min_{u_k \\in \\mathcal{U}} \\mathbb{E} \\Bigg[ g_k(x_k, u_k, w_k) + J_{k+1}\\Big(f_k(x_k, u_k, w_k)\\Big)\\Bigg] \\\\\n&  \\quad \\quad k=0, 1, \\dots, N-1, \n\\end{split}\n$}\n\\label{eq:DP}\n\\end{equation}\nwhere the expectation is taken with respect to the probability distribution of $w_k$ defined by the density $p$. Furthermore, if there exists $u_k^*$ minimizing the right hand side of \\eqref{eq:DP} for each $x_k$ and $k$, then the policy $\\pi^* = \\{\\mu_0^*, \\dots, \\mu_{N-1}^*\\}$ where $\\mu_k^*(x_k) = u_k^*$ is optimal. \n\\end{prop}\n\nThe intermediate functions $J_k(x_k)$ for $k > 0$ computed in this manner represent the optimal cost of the tail subproblem beginning at $x_k$. The optimal cost of the entire problem is given by the function $J^*(x_0) = J_0(x_0)$ obtained when this recursion terminates. \n\n\n\n\n\n\n\n\n\n\\subsection{Invariant system with invariant costs} \n\\label{sec:symmetry_reduction} \n\nWe first recall the definition of a transformation group for a control system,  as in \\citep{Martin04, Jakubczyk98, Respondek02}. See  \\cite{olver1999classical} for the more general theory. \n\n\\begin{defn}[Transformation group]\nA transformation group on $\\mathcal{X} \\times \\mathcal{U} \\times \\mathcal{W}$ is set of tuples $h_\\alpha = (\\phi_\\alpha, \\chi_\\alpha, \\psi_\\alpha)$ parametrized by elements $\\alpha$ of a Lie group $\\mathcal{G}$  having dimension $r$, such that the functions $\\phi_\\alpha: \\mathcal{X} \\to \\mathcal{X}$, $\\chi_\\alpha: \\mathcal{U} \\to \\mathcal{U}$ and $\\psi_\\alpha: \\mathcal{W} \\to \\mathcal{W}$ are all $C^1$ diffeomorhpisms and satisfy:\n\\begin{itemize}\n\\item $\\phi_e(x) = x$, $\\chi_e(u) = u$, $\\psi_e(w) = w$ when $e$ is the identity of the group $\\mathcal{G}$ and\n\\item $\\phi_{a * b}(x) = \\phi_a \\circ \\phi_b (x)$, $\\chi_{a * b}(u) = \\chi_a \\circ \\chi_b (u)$, $\\psi_{a * b}(x) = \\psi_a \\circ \\psi_b (x)$ for all $a, b \\in \\mathcal{G}$ where $*$ denotes the group operation and $\\circ$ denotes function composition. \n\\end{itemize} \n\\end{defn}\nTo simplify notation we will sometimes suppress the subscripts $\\alpha$.\nIn the present paper, we will consider the following class of systems and cost functions. \n\\begin{defn}\n{\\bf (Invariant control system with invariant costs)}\nA control system $\\mathcal{S}$ is $\\mathcal{G}$-invariant with $\\mathcal{G}$-invariant costs if for all $\\alpha \\in \\mathcal{G}$, $x_k \\in \\mathcal{X}$, $u_k \\in \\mathcal{U}$ and $w_k \\in \\mathcal{W}$ we have: \n\\[\n\\begin{split}\n  \\phi^{-1} \\circ f_k( \\phi(x_k), \\chi(u_k), \\psi(w_k)) &= f_k(x_k, u_k, w_k), \\\\\n  & \\quad k = 0, 1, \\dots, N-1\\\\\n  g_k(\\phi(x_k), \\chi(u_k), \\psi(w_k)) &= g_k(x_k, u_k, w_k),  \\\\\n  & \\quad k = 0, 1, \\dots, N-1, \\\\\n  g_N(\\phi(x_N)) &= g_N(x_N), \\text{ and } \\\\\n  p_k(\\psi(w_k)) |\\det D\\psi(w_k)| &= p_k(w_k) \\\\\n  & \\quad k = 0, 1, \\dots, N-1 \n\\end{split}\n\\]\nwhere $D\\psi$ denotes the Jacobian of $\\psi$. \n\\end{defn}\nThe rationale is simple: For any fixed $\\alpha\\in\\mathcal{G}$, consider the change of variables $X_k=\\phi_\\alpha(x_k)$, $U_k=\\chi_\\alpha(u_k) $, $W_k=\\psi_\\alpha(w_k)$. Then, we have \n$$\nX_{k+1} = f_k(X_k, U_k, W_k), \\quad k = 0, 1, \\dots, N-1,\n$$and for $ k = 0, 1, \\dots, N-1$ we have also $g_k(X_k, U_k, W_k)=g_k(x_k,u_k,w_k)$. As a result, if $u_1,\\dots,u_{N-1}$ is a series of controls that minimize $J(x_0)$, then one can expect $U_1,\\dots,U_{N-1}$ to minimize $J(X_0)$, under some assumptions on the noise. As a result, the optimal control problem needs only be solved once for all initial conditions belonging to the set $\\{\\phi_\\alpha(x_0)|\\alpha\\in\\mathcal G\\}$, reducing the initial $n$ dimensional problem to a $n-r$ dimensional problem. The present paper derives a theory for such symmetry reduction in dynamic programming, and provides various examples of engineering interest. \n\n\n\n\n\\subsection{Cartan's moving frame method} \n\\label{sec:moving_frames}\n\nTo find a reduced coordinate system in which to perform dynamic programming, we will use the moving frame method of Cartan \\citep{Cartan37}. In general, this method only results in a local coordinate transformation as it relies on the implicit function theorem. In this paper we will focus on a transformation within a single coordinate chart. For many practical problems, including both examples in this paper, the transformation computed using this method extends to all of $\\mathcal{X}$ with the exception of a lower-dimensional submanifold of the state space. In such cases, only a single coordinate chart is required for the purpose of gridding the entire state space for dynamic programming. \n\nWe briefly introduce the moving frame method following the presentation in \\citep{Bonnabel08}. Consider an $r$-dimensional transformation group (with $r \\le n$) acting on $\\mathcal{X}$ via the diffeomorphisms $(\\phi_\\alpha)_{\\alpha \\in \\mathcal{G}}$. Assume we can split $\\phi_\\alpha$ as $(\\phi_\\alpha^a, \\phi_\\alpha^b)$ with $r$ and $n-r$ components respectively so that $\\phi_\\alpha^a$ is an invertible map.   Then, for some $c$ in the range of $\\phi^a$, we define a coordinate cross section to the orbits $\\mathcal{C} = \\{x: \\phi_e^a(x) = c\\}$. This cross section is an $n-r$-dimensional submanifold of $\\mathcal{X}$. Assume moreover that for any point  $x \\in \\mathcal{X}$, there is a unique group element $\\alpha \\in \\mathcal{G}$ such that $\\phi_\\alpha(x) \\in \\mathcal{C}$. Such $\\alpha$ will be denoted $\\gamma(x)$, and the map $\\gamma: \\mathcal{X} \\to \\mathcal{G}$ will be called moving frame.\n\nA moving frame can be computed by solving the normalization equation:\n\\[\n\\phi_{\\gamma(x)}^a(x) = c. \n\\]\nDefine the following map $\\rho: \\mathcal{X} \\to \\mathbb{R}^{n-r}$ as \n\\[\n\\rho(x) = \\phi^b_{\\gamma(x)}(x). \n\\]\nNote that, for all $\\alpha\\in\\mathcal G$ we have $\\rho(\\phi_\\alpha(x))=\\rho(x)$, that is, the components of $\\rho$ are $\\emph{invariant}$ to the group action on the state space. Further, due to our assumptions, the restriction of $\\rho$ to $\\mathcal{C}$ is injective. We denote this restricted function $\\bar \\rho$, and it will serve as a reduced coordinate system to solve the invariant optimal control problem. \n\n\n\n\n\n\\section{Main Results}\\label{Main}\n\nIn order to combat the ``curse of dimensionality'' associated with performing dynamic programming in high-dimensional systems, we describe a method to reduce the system's dimension by exploiting symmetries in the dynamics and stage costs. \n\n\\subsection{Symmetries imply equivalence classes of optimal policies}\n\n\n\\begin{thm}\n{\\bf (Symmetries of the optimal cost and policy)}\n\\label{thm:symmetric_cost_and_policy}\nLet $\\mathcal{G}$ be a group and let $\\mathcal{S}$ be a $\\mathcal{G}$-invariant control system with $\\mathcal{G}$-invariant costs. Then the optimal cost functions $J_k(x_0)$ satisfy the symmetry relations\n\\[\n  J_k  = J_k \\circ \\phi_\\alpha \n\\]\nfor any $k = 0, \\dots, N$ and any $\\alpha \\in \\mathcal{G}$. Furthermore, if $\\pi^* = \\{\\mu_0^*, \\dots, \\mu_{N-1}^*\\}$ is an optimal policy then so is $\\tilde \\pi^* := \\{ \\chi_\\alpha \\circ \\mu_0^* \\circ \\phi^{-1}_\\alpha, \\dots,  \\chi_\\alpha \\circ \\mu_{N-1}^* \\circ \\phi^{-1}_\\alpha \\}$ for any $\\alpha \\in \\mathcal{G}$. \n\\end{thm} \n\n\\begin{pf} We prove this by induction on $k$ proceeding backward from the base case $k=N$. \nFirst, note that \n\\[\nJ_N(x_N) = g_N(x_N) = g_N(\\phi(x_N)) = J_N(\\phi(x_N)). \n\\]\nNow, suppose that for some $k \\in \\{0, \\dots, N-1\\}$ we have $J_{k+1}(x_{k+1}) = J_{k+1}(\\phi(x_{k+1}))$ for all $x_{k+1} \\in \\mathcal{X}$. Then for any $x_k \\in X$, and $u_k \\in \\mathcal{U}$ we have\n\\[\n\\resizebox{1.05\\hsize}{!}{$\n\\begin{split}\n& \\mathbb{E} \\Bigg[ g_k(x_k, u_k, w_k) + J_{k+1}(f_k(x_k, u_k, w_k)) \\Bigg] \\\\\n&= \\int_\\mathcal{W} \\Bigg[ g_k(x_k, u_k, w_k) + J_{k+1}(f_k(x_k, u_k, w_k)) \\Bigg] p_k(w_k) dw_k \\\\\n&= \\int_\\mathcal{W} \\Bigg[ g_k(\\phi(x_k), \\chi(u_k), \\psi(w_k)) + J_{k+1}(\\phi^{-1} \\circ f_k(\\phi(x_k), \\chi(u_k), \\psi(w_k))) \\Bigg] p_k(w_k) dw_k \\\\\n&= \\int_\\mathcal{W} \\Bigg[ g_k(\\phi(x_k), \\chi(u_k), \\psi(w_k)) + J_{k+1}(f_k(\\phi(x_k), \\chi(u_k), \\psi(w_k))) \\Bigg] p_k(w_k) dw_k \\\\\n&= \\int_\\mathcal{W} \\Bigg[ g_k(\\phi(x_k), \\chi(u_k), \\psi(w_k)) + J_{k+1}(f_k(\\phi(x_k), \\chi(u_k), \\psi(w_k))) \\Bigg] p_k(\\psi(w_k)) |\\det D\\psi (x_k)| dw_k \\\\\n\\end{split}\n$}\n\\]\n\\[\n\\resizebox{\\hsize}{!}{$\n\\begin{split}\n&= \\int_{\\psi(\\mathcal{W})} \\Bigg[ g_k(\\phi(x_k), \\chi(u_k), \\tilde w_k) + J_{k+1}(f_k(\\phi(x_k), \\chi(u_k), \\tilde w_k)) \\Bigg] p_k(\\tilde w_k) d\\tilde w_k \\\\\n&= \\int_\\mathcal{W} \\Bigg[ g_k(\\phi(x_k), \\chi(u_k), w_k) + J_{k+1}(f_k(\\phi(x_k), \\chi(u_k), w_k)) \\Bigg] p_k(w_k) dw_k \\\\\n&= \\mathbb{E} \\Bigg[ g_k(\\phi(x_k), \\chi(u_k), w_k) + J_{k+1}(f_k(\\phi(x_k), \\chi(u_k), w_k)) \\Bigg]\n\\end{split}\n$} \n\\]\nwhere the change of variables $\\tilde w_k$ is defined via $\\tilde w_k = \\psi(w_k)$ and the tildes are subsequently dropped. Therefore, from the one-step dynamic programming principle we have \n\\[\n\\resizebox{\\hsize}{!}{$\n\\begin{split}\nJ_k(x_k) &= \\min_{u_k \\in \\mathcal{U}} \\mathbb{E} \\Bigg[ g_k(x_k, u_k, w_k) + J_{k+1}(f_k(x_k, u_k, w_k)) \\Bigg] \\\\\n&= \\min_{u_k \\in \\mathcal{U}} \\mathbb{E} \\Bigg[ g_k(\\phi(x_k), \\chi(u_k), w_k) + J_{k+1}(f_k(\\phi(x_k), \\chi(u_k), w_k)) \\Bigg] \\\\\n&= \\min_{\\tilde u_k \\in \\chi(\\mathcal{U})} \\mathbb{E} \\Bigg[ g_k(\\phi(x_k), \\tilde u_k, w_k) + J_{k+1}(f_k(\\phi(x_k), \\tilde u_k, w_k)) \\Bigg] \\\\\n&= J_k(\\phi(x_k)). \n\\end{split}\n$}\n\\] \nThus $J^* = J^* \\circ \\phi$. Now, if $\\pi^* = \\{\\mu_0^*, \\dots, \\mu_{N-1}^*\\}$ is an optimal policy and we denote $\\tilde x_k = \\phi(x_k)$ then for any $k \\in \\{0, \\dots, N-1\\}$ we have \n\\[\n\\resizebox{\\hsize}{!}{$\n\\begin{split}\nJ_k(\\tilde x_k) &= J_k(x_k) \\\\\n&= \\mathbb{E}\\Bigg[ g(x_k, \\mu_k^*(x_k), w_k) + J_{k+1}(f_k(x_k, \\mu_k^*(x_k), w_k)) \\Bigg] \\\\\n&= \\mathbb{E} \\Bigg[ g_k(\\phi(x_k), \\chi(\\mu_k^*(x_k)), w_k) + J_{k+1}(f_k(\\phi(x_k), \\chi(\\mu_k^*(x_k)), w_k)) \\Bigg] \\\\\n&= \\mathbb{E} \\Bigg[ g_k(\\phi(x_k), \\chi \\circ \\mu_k^* \\circ \\phi^{-1}(\\phi(x_k)), w_k) + J_{k+1}(f_k(\\phi(x_k), \\chi \\circ \\mu_k^* \\circ \\phi^{-1} (\\phi(x_k)), w_k)) \\Bigg] \\\\\n&= \\mathbb{E} \\Bigg[ g_k(\\tilde x_k, \\chi \\circ \\mu_k^* \\circ \\phi^{-1}(\\tilde x_k), w_k) + J_{k+1}(f_k(\\tilde x_k, \\chi \\circ \\mu_k^* \\circ \\phi^{-1} (\\tilde x_k), w_k)) \\Bigg].\n\\end{split}\n$}\n\\]\nThus $\\tilde \\pi^* := \\{ \\chi \\circ \\mu_0^* \\circ \\phi^{-1}, \\dots,  \\chi \\circ \\mu_{N-1}^* \\circ \\phi^{-1} \\}$ is an optimal policy. \\qed\n\\end{pf}\n\n\\subsection{Dynamic programming can be performed using reduced coordinates} \n\\label{sec:reduced_coordinates}\n\n\nTheorem \\ref{thm:symmetric_cost_and_policy} readily implies the problem can be reduced, as all states along an orbit of $\\mathcal{G}$ are equivalent in terms of cost, and that there are equivalence classes of optimal policies. So it suffices to only consider the cost corresponding to a single representative of each equivalence class, and to find a single representative of the optimal policy within each class. This can now easily be done using the injective map $\\bar\\rho: \\mathcal{C}\\to\\mathbb{R}^{n-r}$.\n\nFor $\\bar x \\in \\bar\\rho(\\mathcal{C})\\subset \\mathbb{R}^{n-r}$, let $z\\in\\mathcal C$ be such that $\\bar x=\\bar\\rho(z)$, and define\n\\[\n\\bar J_k(\\bar x) = J_k(z). \n\\]\nThe following result  shows that the functions $J_k$ on the $n$-dimensional space $\\mathcal{X} \\subseteq \\mathbb{R}^n$ are completely determined by the values of $\\bar J_k$ on the subset $ \\bar\\rho(\\mathcal{C})$ of $\\mathbb{R}^{n-r}$. \n\n\\begin{cor}\nFor any $x \\in \\mathcal{X}$ and $k = 0, \\dots, N$, the cost function $J_k$ for the full problem can be computed in terms of the lower-dimensional cost function $\\bar J_k$ as\n\\[\nJ_k(x) = J_k(\\phi_{\\gamma(x)}(x))=\\bar J_k(\\bar x), \n\\]where $\\bar x:=\\bar\\rho(\\phi_{\\gamma(x)}(x))$ is well defined as $\\phi_{\\gamma(x)}(x)\\in\\mathcal C$. \n\\end{cor}\nIt is thus sufficient to have evaluated $\\bar J$ at all points of $\\bar\\rho(\\mathcal{C})\\subset \\mathbb{R}^{n-r}$ to be able to instantly evaluate $J$ at any point of $\\mathcal X$. An optimal policy $\\bar \\pi^* = \\{\\bar \\mu_k^*: \\bar\\rho(\\mathcal{C}) \\to \\mathcal{U}\\}_{k=0}^{N-1}$ can also be lifted to an optimal policy on the original state space via this method:\n\\[\n\\mu_k^*(x) = \\bar \\mu_k^*(\\bar \\rho(\\phi_{\\gamma(x)}(x))). \n\\]\nThis allows optimal trajectories of the system to be computed in the original coordinates $\\mathcal{X}$ via the lifted policy $\\pi^* = \\{\\mu_0^*, \\dots, \\mu_{N-1}^*\\}$. \n\n\\begin{thm}{\\bf (Dynamic programming in reduced coordinates)}\n\\label{thm:DP_reduced}\nThe reduced coordinates are in one to one correspondance with the cross-section $\\mathcal C$. For any $\\bar x \\in \\bar \\rho(\\mathcal{C})$, let $z\\in\\mathcal C$ satistisfy $\\bar\\rho(z)=\\bar x$. Then in the reduced coordinates, the sequence $\\bar J_k$ can be computed recursively via\n\\[\n\\resizebox{\\columnwidth}{!}{$\n\\bar J_k(\\bar x) = \\min_{u_k \\in \\mathcal{U}} \\mathbb{E}\\left[ g_k(z, u_k, w_k) + \\bar J_{k+1}(\\rho(f_k(z, u_k, w_k)))\\right]. \n$}\n\\]\n\\end{thm}\n\\begin{pf}\nWe have\n\\[\n\\resizebox{\\columnwidth}{!}{$\n\\begin{split}\n\\bar J_k(\\bar x) = J_k(z) \n&= \\min_{u_k \\in \\mathcal{U}} \\mathbb{E} \\Bigg[ g_k(z, u_k, w_k) + J_{k+1}\\Big(f_k(z, u_k, w_k)\\Big)\\Bigg] \\\\\n&= \\min_{u_k \\in \\mathcal{U}} \\mathbb{E} \\Bigg[ g_k(z, u_k, w_k) + \\bar J_{k+1}\\Big( \\rho \\circ f_k(z, u_k, w_k)\\Big)\\Bigg]. \\qed\n\\end{split}\n$}\n\\]\n\\end{pf}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Case of equivariant costs}\n\\label{sec:equivariance}\n\nSo far, we have considered the costs to be invariant under transformation. We now briefly discuss how this can be generalized. The cost $g_k$ is said to be equivariant if there exists a family of diffeomorphisms $\\varphi_\\alpha$ such that  $g_k(\\phi_\\alpha(x_k), \\chi_\\alpha(u_k), \\psi_\\alpha(w_k)) = \\varphi_\\alpha\\circ g_k(x_k, u_k, w_k)$. As we want the cost function $J$ to be equivariant too, we will need $\\varphi_\\alpha(\\cdot)$ to be linear. Thus we will simply assume that $\\varphi_\\alpha$ is of the form $\\varphi_\\alpha(J)=l(\\alpha)J$, that is, it is a scaling of the cost, where $l:\\mathbb R_{>0}\\to\\mathbb R_{>0}$. For simplicity's sake, we consider here the problem to be noise free. Along the lines of the preceding sections it is easily proved that\n$$\nJ_k(\\phi(x))=\\varphi\\circ J_k(x)\n$$\nas already noticed in \\citep{alvarez1998dynamic} for the case of homogeneous costs. Symmetry reduction can then be applied. We now give two tutorial examples. \n\n\n\\begin{ex}Consider the linear system \n$$\nx_{x+1}=Ax_k+Bu_k\n$$\nwith quadratic costs $g_k=x_k^TQx_k+u_k^TRu_k$. The system is invariant to scalings,  $\\phi_\\alpha(x)=\\alpha x$, $\\chi_\\alpha(u)=\\alpha u$, and the cost is equivariant letting $\\varphi_\\alpha(J)=\\alpha J$, where $\\alpha\\in\\mathcal G=\\mathbb R_{>0}$. The unit sphere is a cross section to the orbits, and the normalization equation yields $\\gamma(x)={1}\/{||x||}$. Applying the results above, we see that the  controls  that minimize $J(x_0)$ are  $||x_0||u_1^*,\\cdots,||x_0||u_{N-1}^*$, where $u_1^*,\\cdots,u_{N-1}^*$ are those minimizing $J(\\frac{x_0}{||x_0||})$. This agrees with the well known fact that the  optimal controller  for the problem above is the linear quadratic controller, and is indeed of the linear form $u_k=-F_kx_k$. \n\\end{ex}\n\n\n\\begin{ex}Consider the following system and costs\n$$\nx_{x+1}=Ax_k+Bu_k,\\quad g_k=h(x_k)+||u_k||_{1}\n$$\nwhere $||u_k||_{1}$ denotes the $L^1$ norm of $u_k$ and $h$ is a map satisfying $h(ax) = ah(x)$ for $a > 0$. Such costs may arise when one tries to force some controls to zero to create sparsity, a method known as $L^1$ regularization. This problem is challenging, particularly for nonconvex $h$. But according to the theory above, it is sufficient to solve it numerically for initial conditions lying on the unit sphere of the state space. \n\\end{ex}\n\n\\subsection{Optimal formation control on Lie groups} \n\\label{sec:matrix_group}\n\nWe now apply the theory presented in Section \\ref{sec:symmetry_reduction} where the state space  $\\mathcal{X} \\subseteq \\mathbb{R}^n$ is the Cartesian product of matrix Lie groups. Note that, straightforward modifications arise along the way as the state space and noise space are not vector spaces as in the theory above. The methodology is then applied to the synchronization of two non-holonomic cars in the presence of uncertainties.  \n\nWe model the system as a collection of $K$ agents, where the state of each agent evolves on a $r$-dimensional matrix Lie group $\\mathcal{G}$. We assume that the evolution of the state of agent $j$ proceeds according to the equation \n\\begin{equation}\nX_{k+1}^j = X_k^j M(u_k^j) W_k^j \n\\label{eq:Lie_group_dynamics}\n\\end{equation}\nwhere $X_k$, $M(u_k)$, $W_k$ are all square matrices belonging to $\\mathcal{G}$, $u_k$ is a control that lives in some finite dimensional vector space, and $W_k$ is the noise. The control objective is to reach a desired configuration, that is, a desired value for the relative  configurations of the agents $(X^1)^{-1} X^2,\\dots,(X^{K-1})^{-1} X^K$, see e.g.,  \\cite{sarlette2010coordinated} for more information.\n\n\n\n\nSystems of this form are naturally invariant to left multiplication of all $X^j$ by some matrix $A \\in \\mathcal{G}$: \n\\[\n\\phi_A(X) = \\begin{bmatrix}AX^1 \\\\ \\vdots \\\\ AX^K \\end{bmatrix}\n\\]\nwhere $X = (X^1, \\dots, X^k) \\in \\mathcal{G}^K$. Letting $\\chi(u^1,\\dots,u^K)\\equiv (u^1,\\dots,u^K)$,  $\\psi(W^1,\\dots,W^K)\\equiv (W^1,\\dots,W^K)$, and the costs be of the form $\\tilde g((X^1)^{-1} X^2,\\dots,(X^{K-1})^{-1} X^K)+h(u^1,\\dots,u^K)$, we get an invariant system with invariant costs.\n\n\n\n\nOne can define a cross section to the orbits by letting the first agent coordinates be equal to the identity matrix, that is, $\\mathcal{C} = \\{X \\in \\mathcal{G}^K: X^1 = I\\}$. The normalization equation is given by $I = \\phi^a_{\\gamma(X)}(X) = \\gamma(X) X^1$, hence the moving frame is given by $\\gamma(X) = (X^1)^{-1}$. The invariants are computed as \n\\[\n\\rho(X) = \\phi^b_{\\gamma(X)} = \\begin{bmatrix} (X^1)^{-1} X^2 \\\\ \\vdots \\\\ (X^1)^{-1} X^K \\end{bmatrix}.\n\\]The optimal stochastic control problem can  then be solved in the reduced coordinate system defined by $\\rho$,  reducing the state space from dimension $Kr$ to $(K-1)r$.\n\n\n\n\n\n\\section{Application I: Cooperative formation control for two stochastic Dubins vehicles}\n\\label{sec:vehicles_cooperative}\n\nWe consider two identical Dubins vehicles each with dynamics \n\\[\n\\begin{split}\nz_{k+1} &= z_k + v_k \\cos \\theta_k \\\\\ny_{k+1} &= y_k + v_k \\sin \\theta_k \\\\\n\\theta_{k+1} &= \\theta_k + \\frac{1}{L} v_k \\tan s_k + w_k\n\\end{split} \n\\]\nwhere $y_k$ and $z_k$ denote the two-dimensional position of the vehicle, $\\theta_k$ denotes the heading of the vehicle, $v_k$ is a velocity input, $s_k$ is a steering angle input, and $w_k$ is independent, identically-distributed zero-mean Gaussian noise with variance $\\sigma^2$, and $L$ is a parameter that determines the vehicle's steering radius. \n\nThese dynamics can be embedded in the three-dimensional special Euclidean matrix Lie group $\\mathcal{G} = SE(2)$, by defining the state \n\\[\nX_k = \\begin{bmatrix} \\cos \\theta_k & -\\sin \\theta_k & z_k \\\\ \\sin \\theta_k & \\cos \\theta_k & y_k \\\\ 0 & 0 & 1 \\end{bmatrix},\n\\]\ninput matrix\n\\[\nM(v_k, s_k) = \\begin{bmatrix} \\cos(\\frac{1}{L} v_k \\tan s_k) & -\\sin(\\frac{1}{L} v_k \\tan s_k) & v_k \\\\\n\\sin(\\frac{1}{L} v_k \\tan s_k) & \\cos (\\frac{1}{L} v_k \\tan s_k) & 0\\\\ 0 & 0 & 1 \\end{bmatrix},\n\\]\nand noise matrix\n\\[\nW_k = \\begin{bmatrix} \\cos w_k & -\\sin w_k & 0 \\\\ \\sin w_k & \\cos w_k & 0 \\\\ 0 & 0 & 1 \\end{bmatrix}, \n\\]\nwith state update equation of the form \\eqref{eq:Lie_group_dynamics}. \n\n\n\n\nWe wish to compute a control policy for a two-vehicle system, with states $X^1$ and $X^2$, where the controls can only take a finite number of values, and with terminal cost \n\\[\nJ(X_0^1,X_0^2)=\\mathbb E  \\Bigg[g_N\\bigl((X^1_N)^{-1} X^2_N\\bigr)   \\Bigg]\n\\]\nwhere  $\ng_N(X) = \\arccos(X_{11})^2 + |\\sqrt{X_{13}^2 + X_{23}^2} - 1 |$, that is, we want the vehicles to have the same heading, and follow each other at unit distance. Thanks to the theory developed above, the stochastic control problem is reduced from problem with a six dimensional state space to a problem with a three dimensional  state space only. \n\nFor numerical simulations, the cost functions $\\bar J_k$ were computed on a fixed grid of dimension $51 \\times 51 \\times 65$ using turning radius parameter $L = 1$, input sets $v_k \\in \\{-0.1, 0, -0.1\\}$ and $s_k \\in \\{-1, 0, -1\\}$ Globally optimal input and state trajectory sequences corresponding to the initial condition $x_0 = \\begin{bmatrix}0.1 & 0 & \\frac{1}{2} \\pi & -0.1 & 0 & \\frac{3}{2}\\pi\\end{bmatrix}^T$ are shown in Figures \\ref{fig:Dubins_cooperative_input} and \\ref{fig:Dubins_cooperative_state}. These are compared against a deterministic version of the model with $w_k = 0$ in Figures  \\ref{fig:Dubins_cooperative_input_deterministic} and \\ref{fig:Dubins_cooperative_state_deterministic}. \n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{figs\/Dubins_cooperative_input_stochastic.pdf}\n\\end{center}\n\\caption{Optimal input sequence for cooperative stochastic Dubins vehicle model with $\\sigma = 0.3$.}\n\\label{fig:Dubins_cooperative_input} \n\\end{figure}\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{figs\/Dubins_cooperative_state_stochastic.pdf}\n\\end{center}\n\\caption{Optimal state sequence for the cooperative stochastic Dubins vehicle model with $\\sigma = 0.3$. \n}\n\\label{fig:Dubins_cooperative_state} \n\\end{figure}\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{figs\/Dubins_cooperative_input_deterministic.pdf}\n\\end{center}\n\\caption{Optimal input sequence for cooperative deterministic Dubins vehicle model.}\n\\label{fig:Dubins_cooperative_input_deterministic} \n\\end{figure}\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{figs\/Dubins_cooperative_state_deterministic.pdf}\n\\end{center}\n\\caption{Optimal state sequence for the cooperative deterministic Dubins vehicle model. \n}\n\\label{fig:Dubins_cooperative_state_deterministic} \n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\section{Application II: MRI Fingerprinting} \n\\label{sec:MRI}\n\nMagnetic resonance imaging (MRI) has traditionally focused on acquisition sequences that are static, in the sense that sequences typically wait for magnetization to return to equilibrium between acquisitions. Recently, researchers have demonstrated promising results based on dynamic acquisition sequences, in which spins are continuously excited by a sequence of random input pulses, without allowing the system to return to equilibrium between pulses. Model parameters corresponding to $T_1$ and $T_2$ relaxation, off-resonance and spin density are then estimated from the sequence of acquired data. This technique, termed magnetic resonance fingerprinting (MRF), has been shown to increase the sensitivity, specificity and speed of magnetic resonance studies \\citep{Ma13, Davies14}. \n\nThis technique could be further improved by replacing randomized input pulse sequences with sequences that have been optimized for informativeness about model parameters. To this end, we present a model of MR spin dynamics that describes the measured data as a function of $T_1$ and $T_2$ relaxation rates and the sequence of radio-frequency (RF) input pulses, used to excite the spins. \n\nThe following model was introduced in the conference paper \\citep{Maidens17-ACC}. In this paper, an optimal control was computed via dynamic programming on a very sparse six-dimensional grid. Now using our symmetry reduction technique, we exploit symmetry reduction to provide a much more accurate optimal input sequence computed on a finer five-dimensional grid. \n\nWe model the spin dynamics via the equations \n\\begin{equation}\n\\mathbf{x}_{k+1} = U_k \\begin{bmatrix} \\theta_2 & 0 & 0 \\\\ 0 & \\theta_2 & 0 \\\\ 0 & 0 & \\theta_1 \\end{bmatrix} \\mathbf{x}_k + \\begin{bmatrix} 0 \\\\ 0 \\\\ 1 - \\theta_1 \\end{bmatrix}\n\\label{eq:MRI} \n\\end{equation} \nwhere the states $x_{1, k}$ and $x_{2, k}$ describe the transverse magnetization (orthogonal to the applied magnetic field) and $x_{3, k}$ describes the longitudinal magnetization (parallel to the applied magnetic field). To simplify the presentation, off-resonance is neglected in this model. Control inputs $U_k \\in$ SO(3) describe flip angles corresponding to RF excitation pulses that rotate the state about the origin. Between acquisitions, transverse magnetization decays according to the parameter $\\theta_2 = e^{- \\Delta t\/T_2}$ and the longitudinal magnetization recovers to equilibrium (normalized such that the equilibrium is $x_0 = [0 \\ \\ 0  \\ \\ 1]^T$) according to the parameter $\\theta_1 = e^{- \\Delta t\/T_1}$ where $\\Delta t$ is the sampling interval. \n\\begin{figure*}[ht!]\n\\normalsize\n\\begin{equation}\n\\begin{split}\nf_k(x_k, U_k, w_k) &=\n\\begin{bmatrix} \n         &         &         &    0    &    0    &    0     \\\\\n         &   U_k   &         &    0    &    0    &    0    \\\\ \n         &         &         &    0    &    0    &    0    \\\\ \n    0    &    0    &    0    &         &         &         \\\\ \n    0    &    0    &    0    &         &   U_k   &          \\\\ \n    0    &    0    &    0    &         &         &      \n\\end{bmatrix}\n\\begin{bmatrix} \n\\theta_2 &    0    &    0    &    0    &    0    &    0      \\\\\n    0    &\\theta_2 &    0    &    0    &    0    &    0      \\\\ \n    0    &    0    &\\theta_1 &    0    &    0    &    0      \\\\ \n    0    &    0    &    0    &\\theta_2 &    0    &    0      \\\\ \n    0    &    0    &    0    &    0    &\\theta_2 &    0      \\\\ \n    0    &    0    &    1    &    0    &    0    &\\theta_1  \n\\end{bmatrix}\nx_k\n+ \n\\begin{bmatrix} 0 \\\\ 0 \\\\ 1-\\theta_1 \\\\ 0 \\\\ 0 \\\\ -1 \\end{bmatrix} \\\\\ng_k(x_k, U_k, w_k) &= \n-x_k^T \\begin{bmatrix} \n    0    &    0    &    0    &    0    &    0    &    0   \\\\\n    0    &    0    &    0    &    0    &    0    &    0   \\\\ \n    0    &    0    &    0    &    0    &    0    &    0   \\\\ \n    0    &    0    &    0    &\\frac{1}{\\gamma}&    0    &    0    \\\\ \n    0    &    0    &    0    &    0    &\\frac{1}{\\gamma}&    0    \\\\ \n    0    &    0    &    0    &    0    &    0    &    0    \n\\end{bmatrix}\nx_k\n\\end{split}\n\\label{eq:f_and_g}\n\\end{equation}\n\\hrulefill\n\\vspace*{4pt}\n\\end{figure*}\n\n\nWe assume that data are acquired immediately following the RF pulse, allowing us to make a noisy measurement of the transverse magnetization. We also assume that the measured data are described by a multivariate Gaussian random variable \n\\[\n\\mathbf y_k = \\begin{bmatrix} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\end{bmatrix}\\mathbf x_k  + v_k \n\\]\nwhere $v_k$ is a zero-mean Gaussian noise with covariance $\\begin{bmatrix} \\gamma & 0 \\\\ 0 & \\gamma \\end{bmatrix}$. This model results from a time discretization of the Bloch equations \\citep{Bloch46, Nishimura10} under a time scale separation assumption that specifies that the RF excitation pulses act on a much faster time scale than the relaxation time constants $T_1$ and $T_2$. A simplified two-state version of this model was considered in \\citep{Maidens16-CDC}, where the transverse magnetization was modelled using a single state describing the magnitude of $[x_{1, k}, x_{2, k}]^T$. \n\nWe see from the model \\eqref{eq:MRI} that magnetization in the transverse direction decays while magnetization in the longitudinal direction grows. However only the transverse component of the magnetization can be measured. Thus there is a trade-off between making measurements (which leads to loss of magnetization) and magnetization recovery. This is the trade-off that we hope to manage through the optimal design of an input sequence $U_k$. \n\nWe wish to quantify the informativeness of an acquisition sequence based on the information about the $T_1$ relaxation parameter $\\theta_1$ that is contained in the resulting data set. More formally, we wish to choose $U_k \\in $ SO(3) to maximize the Fisher information about $\\theta_1$ contained in the joint distribution of $Y = (\\mathbf y_0, \\dots, \\mathbf y_N)$. The Fisher information $\\mathcal{I}$ can be expressed as a quadratic function of the sensitivities of $\\mathbf x_k$ with respect to $\\theta_1$:\n\\[\n\\mathcal I = \\sum_{k=0}^N \n \\frac{\\partial}{\\partial \\theta_1} \\mathbf{x}_k^T\n\\begin{bmatrix} 1\/\\gamma & 0 & 0 \\\\ 0 & 1\/\\gamma & 0 \\\\ 0 & 0 & 0 \\end{bmatrix} \n\\frac{\\partial}{\\partial \\theta_1} \\mathbf{x}_k\n\\]\nwhere the sensitivities $ \\frac{\\partial}{\\partial \\theta_1} \\mathbf{x}_k$ satisfy the following sensitivity equations: \n\\[\n\\resizebox{\\hsize}{!}{$\n\\frac{\\partial}{\\partial \\theta_1} \\mathbf{x}_{k+1} = U_k \\begin{bmatrix} \\theta_2 & 0 & 0 \\\\ 0 & \\theta_2 & 0 \\\\ 0 & 0 & \\theta_1 \\end{bmatrix} \\frac{\\partial}{\\partial \\theta_1} \\mathbf{x}_k + U_k \\begin{bmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 1 \\end{bmatrix} \\mathbf{x}_k + \\begin{bmatrix}0 \\\\ 0 \\\\ -1\\end{bmatrix}. \n$}\n\\]\n\nIt should be noted that for system \\eqref{eq:MRI}, the objective function $\\mathcal{I}$ has many local optima as a function of the input sequence $U_k$. Thus, in contrast with \\citep{Maidens16-TMI} which consider optimal experiment design for hyperpolarized MRI problems, for this model, local search methods provide little insight into what acquisition sequences are good. In contrast with the MRI model presented in \\citep{Maidens16-ACC}, where global optimal experiment design heuristics are developed for linear dynamical systems, in this model the decision variables $U_k$ multiply the state vector $\\mathbf x_k$, making the output $\\mathbf y_k$ a nonlinear function of the sequence $U = (U_0, \\dots U_{k-1})$. Thus we must use dynamic programming to find a solution. \n\n\n\\subsection{Model} \n\nTo present this problem in the formalism we have introduced, we define an augmented state vector \n\\[\nx_k = \\begin{bmatrix} \\mathbf{x}_k \\\\ \\frac{\\partial}{\\partial \\theta_1} \\mathbf{x}_k \\end{bmatrix} \\in \\mathbb{R}^6.\n\\]\nWe can write the dynamics of the augmented state as a control system with $f$ and $g$ defined in Equation \\eqref{eq:f_and_g}.  \nThis system has a one-dimensional group of symmetries defined by \n\\[\n\\resizebox{\\hsize}{!}{$\n\\begin{split}\n\\phi_\\alpha(x_k) &= \n\\begin{bmatrix} \n\\cos(\\alpha)&-\\sin(\\alpha)&    0    &    0    &    0    &    0    \\\\\n\\sin(\\alpha)& \\cos(\\alpha)&    0    &    0    &    0    &    0     \\\\ \n    0    &    0    &    1    &    0    &    0    &    0     \\\\ \n    0    &    0    &    0    &\\cos(\\alpha)&-\\sin(\\alpha)&    0     \\\\ \n    0    &    0    &    0    &\\sin(\\alpha)&\\cos(\\alpha)&     0     \\\\ \n    0    &    0    &    0    &    0    &    0    &    1       \n\\end{bmatrix} x_k \\\\\n\\chi_\\alpha(U_k) &= \\begin{bmatrix} \\cos(\\alpha) & -\\sin(\\alpha) & 0 \\\\\n                             \\sin(\\alpha) &  \\cos(\\alpha) & 0 \\\\\n                             0 & 0 & 1 \n             \\end{bmatrix} \n             U_k\n             \\begin{bmatrix} \\cos(\\alpha) & \\sin(\\alpha) & 0 \\\\\n                             -\\sin(\\alpha) &  \\cos(\\alpha) & 0 \\\\\n                             0 & 0 & 1 \n             \\end{bmatrix}\\\\\n\\psi_\\alpha(w_k) &= w_k\n\\end{split}\n$}\n\\]\nfor any $\\alpha \\in \\mathbb{R}\/2\\pi\\mathbb{Z}$. \n\n\\subsection{Dynamic programming in reduced coordinates} \n\\label{sec:reduced}\n\nTo perform dynamic programming in a reduced coordinate system, we begin by defining the cross-section $\\mathcal{C} = \\{x: x_1 = 0, x_2 > 0\\}$, and computing the moving frame $\\gamma(x)$. To do so, we solve \n\\[\n0 = \\phi^a_{\\gamma(x)} (x) = x_1 \\cos \\gamma(x) - x_2 \\sin \\gamma(x). \n\\]\nIsolating $\\gamma$ yields \n\\[\n\\gamma(x) = \\operatorname{atan2}(x_1, x_2)\n\\]\nwhere $\\operatorname {atan2}$ denotes the multi-valued inverse tangent function\n\\[\n\\operatorname {atan2} (y,x)={\\begin{cases}\\arctan({\\frac {y}{x}})&{\\text{if }}x>0,\\\\\\arctan({\\frac {y}{x}})+\\pi &{\\text{if }}x<0{\\text{ and }}y\\geq 0,\\\\\\arctan({\\frac {y}{x}})-\\pi &{\\text{if }}x<0{\\text{ and }}y<0,\\\\+{\\frac {\\pi }{2}}&{\\text{if }}x=0{\\text{ and }}y>0,\\\\-{\\frac {\\pi }{2}}&{\\text{if }}x=0{\\text{ and }}y<0,\\\\{\\text{undefined}}&{\\text{if }}x=0{\\text{ and }}y=0.\\end{cases}}\n\\]\n\nNext, we compute the invariants $\\rho(x)$ using\n\\begin{equation*}\n\\resizebox{\\columnwidth}{!}{$\n\\begin{split}\n\\rho(x) &= \\phi^b_{\\gamma(x)} \\\\\n&= \n\\begin{bmatrix} \n\\sin(\\operatorname{atan2}(x_1, x_2))& \\cos(\\operatorname{atan2}(x_1, x_2))&    0    &    0    &    0    &    0     \\\\ \n    0    &    0    &    1    &    0    &    0    &    0     \\\\ \n    0    &    0    &    0    &\\cos(\\operatorname{atan2}(x_1, x_2))&-\\sin(\\operatorname{atan2}(x_1, x_2))&    0     \\\\ \n    0    &    0    &    0    &\\sin(\\operatorname{atan2}(x_1, x_2))&\\cos(\\operatorname{atan2}(x_1, x_2))&     0     \\\\ \n    0    &    0    &    0    &    0    &    0    &    1       \n\\end{bmatrix} x \\\\\n&= \\begin{bmatrix} \\sqrt{x_1^2 + x_2^2} \\\\ \nx_3 \\\\ \n\\frac{1}{\\sqrt{x_1^2 + x_2^2}} (x_2 x_4 - x_1 x_5) \\\\\n\\frac{1}{\\sqrt{x_1^2 + x_2^2}} (x_1 x_4 + x_2 x_5) \\\\\nx_6 \\end{bmatrix}. \n\\end{split}\n$}\n\\end{equation*}\n\n\nFurther, $\\rho$ restricted to the cross-section $\\mathcal{C}$ is injective with inverse $\\bar \\rho^{-1}: \\mathbb{R}_+ \\times \\mathbb{R}^4 \\to \\mathcal{C}$ given by \n$\n\\bar \\rho^{-1}(\\bar x) = \\begin{bmatrix} 0 & \\bar x_1 & \\bar x_2 & \\bar x_3 & \\bar x_4 & \\bar x_5 \\end{bmatrix}^T. \n$ The theory above tells us we can thus solve the optimal stochastic control problem in a 5 dimensional state space, reducing the original 6 dimensional problem of 1 dimension. \n\nNote that this reduction can be computed using only the state transformation group $\\phi$, without reference to the state update equation $f$. Thus the same reduction can be applied to any system with the same symmetries. \n\n\\subsection{Results} \n\nTo implement this algorithm, we discretize the reduced five-dimensional state space and two-dimensional input space via grids of size $6 \\times 10 \\times 15 \\times 15 \\times 15$ and $16 \\times 8$ respectively. The code was written in the Julia language and parallelized to allow evaluation of $J_k$ in parallel across grid points \\citep{Maidens16-CDC}. The implementation is publicly available at \\url{https:\/\/github.com\/maidens\/Automatica-2017}. \n\nOptimal input and state trajectories for the model corresponding to the initial condition at the equilibrium $x_0 = [0 \\ \\  0 \\ \\ 1\\ \\  0\\ \\   0\\ \\   0]^T$ are plotted in Figures \\ref{fig:MRI_optimal_input} and \\ref{fig:MRI_optimal_states}.\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{figs\/MRI_optimal_input.png}\n\\end{center}\n\\caption{Optimal input sequence for the MR fingerprinting model. The angles $\\alpha$, $\\beta$ and $\\delta$ represent rotations about the $z$, $y$ and $x$ axes respectively, resulting in an control input $U_k = R_z(\\alpha_k) R_y(\\beta_k) R_x(\\gamma_k)$.}\n\\label{fig:MRI_optimal_input} \n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{\\columnwidth}\n        \\includegraphics[width=\\textwidth]{figs\/MRI_optimal_magnetization.png}\n        \\caption{Magnetizations}\n        \\label{fig:MRI_optimal_magnetization}\n    \\end{subfigure}\n\n    \\begin{subfigure}[b]{\\columnwidth}\n        \\includegraphics[width=\\textwidth]{figs\/MRI_optimal_sensitivity.png}\n        \\caption{Sensitivities}\n        \\label{fig:MRI_optimal_sensitivity}\n    \\end{subfigure}\n    \\caption{Optimal state sequence for the MR fingerprinting model. Here we have plotted the longitudinal and transverse components of both the magnetization (states $x_1$, $x_2$, and $x_3$) and the sensitivities (states $x_4$, $x_5$, and $x_6$) where the transverse component is computed as the Euclidean norm of the vectors $(x_1, x_2)$ and $(x_4, x_5)$ respectively. }\\label{fig:MRI_optimal_states}\n\\end{figure}\n\nIn contrast with the results from \\citep{Maidens16-CDC} where we considered a simplified version of the model, for this full model we no longer find that the optimal flip angle sequence converges to a cyclic pattern, rather it appears irregular. However, state sequence of longitudinal magnetizations and transverse magnetization magnitudes appears to converge to a constant sequence. This is likely because in this work we assumed Gaussian noise in the inputs in contrast with the Rician noise assumed in the previous work, therefore it is no longer necessary to conserve magnetization across multiple time steps before generating a reliable measurement. \n\n\n\n\n\n\\section{Conclusion} \n\\label{sec:conclusion} \n\nWe have presented a method of reducing the complexity of dynamic programming for systems in which the state dynamics, stage costs and transition probabilities are invariant under a group of symmetries. This allows us to compute globally optimal control policies for systems of moderate state dimension. We have applied this technique to compute globally optimal trajectories to a six-dimensional original MRI model with a one-dimensional group of symmetries and for a six-dimensional stochastic Dubins vehicle model with a three-dimensional group of symmetries  by reducing the dimension of the state space to five and three dimensions respectively. Since computation time for dynamic programming depends exponentially on the state space dimension, this technique enables the computation of optimal control policies for systems in which it was previously infeasible. \n\n\\begin{ack}                           \nResearch supported in part by the National Science Foundation under grant ECCS-1405413. \n\\end{ack}\n\n\\bibliographystyle{abbrvnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe analytic behaviour of scattering amplitudes has been a subject of great interest for decades \\cite{Eden:1966dnq}. Recent developments in the theory of amplitudes have led to the application of an array of mathematical ideas to their calculation. The study of poles in tree-level amplitudes led to the BCFW recursion relations \\cite{Britto:2005fq}, that of cuts of integrals to the unitarity approach \\cite{Bern:1994zx}. The combination of these ideas has fed into new constructions of loop integrands for many amplitudes \\cite{ArkaniHamed:2012nw,Arkani-Hamed:2013jha}. The study of polylogarithmic iterated integrals \\cite{Chen:1977oja,Remiddi:1999ew,Goncharov:2005sla,Brown:2009qja} has led to a much greater understanding of loop integrals and motivated a greater push to classify and understand more general functions of elliptic type and beyond \\cite{BrownLevin,CaronHuot:2012ab,Bloch:2013tra,Bourjaily:2017bsb,Bourjaily:2018ycu,Broedel:2018qkq}. These developments have inspired recent advances \\cite{Henn:2013pwa} in the well-studied subject of differential equations for loop integrals \\cite{Chetyrkin:1981qh,Kotikov:1990kg,Bern:1993kr,Remiddi:1997ny} which have been applied to processes of interest for QCD or gauge theories in general. It is clear that the greater understanding we have of the role of singularities in field theory amplitudes the greater our ability is to calculate them and the deeper our understanding of field theory becomes.\n\nHere we will focus on the study of poles and branch cuts in perturbative amplitudes and the algebraic and geometrical structures which govern their appearance. A very helpful toy model in this regard is the planar limit of $\\mathcal{N}=4$ super Yang-Mills theory where many approaches can be taken to calculate amplitudes. In perturbation theory an analytic bootstrap programme has been employed for certain amplitudes, allowing the construction of explicit data for many loop orders \\cite{Dixon:2011pw,Dixon:2011nj,Dixon:2013eka,Dixon:2014voa,Dixon:2014iba,Drummond:2014ffa,Dixon:2015iva,Caron-Huot:2016owq,Dixon:2016nkn}. A different technique relying on the relation of the planar amplitudes with light-like Wilson loops \\cite{Alday:2007hr,Drummond:2007aua,Brandhuber:2007yx,Drummond:2007cf,Bern:2008ap,Drummond:2008aq} is based on multiple expansions in a near-collinear OPE limit \\cite{Alday:2010ku,Basso:2013vsa,Basso:2013aha,Basso:2014koa,Basso:2014nra}, much like correlation functions of local operators in conformal field theories. The interplay of these techniques has revealed surprising structures at the heart of scattering amplitudes.\n\nAn important observation about the perturbative amplitudes came with the work of \\cite{Golden:2013xva} where a link was made between the locations of branch point singularities in scattering amplitudes and certain coordinates (`$\\mathcal{A}$-coordinates') of cluster algebras \\cite{1021.16017,1054.17024}. In \\cite{Drummond:2017ssj} we extended this connection to the interplay of such singularities with each other. Specifically we noticed that the cluster algebras also control the possible sequences of such branch cut singularities; a non-trivial analytic continuation around a given singularity may only be followed by certain others. The set of which singularities are visible on any given Riemann sheet is dictated by the clusters themselves. We refer to this property of amplitudes as `cluster adjacency'. The adjacency relations we find encompass the Steinmann relations \\cite{Steinmann,Steinmann2} which place constraints on consecutive discontinuities of amplitudes \\cite{Bartels:2008ce}. Such relations can be made manifest on appropriately defined infrared finite quantities and then become a powerful constraint in the analytic bootstrap programme \\cite{Caron-Huot:2016owq}.\n\nWe will develop the connection between singularities and cluster algebras further. We emphasise that, although the connection to cluster algebras is phrased in algebraic terms, there is also a very geometric picture to the structure of relations between branch point singularities. The geometry in question is that of cluster polytopes and in particular the intricate structure of their boundaries, which captures the possible nested sequences of cluster subalgebras. The picture which emerges is different from, but shares many features with, the positive geometry arising in the description of integrands in \\cite{ArkaniHamed:2012nw,Arkani-Hamed:2013jha}.\n\nSince the monodromies of analytic functions in general and amplitudes in particular are typically non-abelian in nature, the cluster adjacency controlling their appearance has a non-abelian character; the order in which $\\mathcal{A}$-coordinates appear in the symbol is important. Here we also develop an abelian version of adjacency which controls the poles of individual terms in tree-level amplitudes. We find that precisely the same notion of adjacency holds for individual BCFW terms for NMHV amplitudes and beyond. Since poles multiply in a commutative fashion the adjacency constraints apply to all poles in a given term.\n\nWhen considering NMHV loop amplitudes we have expressions which simultaneously exhibit non-trivial poles and branch cuts. We find that the cluster structure also imposes relations between the two. Specifically we find that the derivatives of  individual terms in NMHV loop amplitudes are constrained in such a way that they are compatible with the poles of the multiplying rational function. The cluster adjacency we find actually comprises a subset of the constraints which follow from dual superconformal symmetry \\cite{Drummond:2008vq}. At loop level these constraints are expressed through the $\\bar{Q}$ equation of \\cite{CaronHuot:2011kk,Bullimore:2011kg}. So the cluster adjacency structure simultaneously implies both branch cut relations, e.g. the Steinmann relations, and derivative relations such as those following from dual superconformal symmetry .\n\n\n\n\n\n\n\n\\section{Amplitudes in planar $\\mathcal{N}=4$ super Yang-Mills theory}\nHere we recall a few basic properties of scattering amplitudes which are necessary for the discussion of singularities and the link to cluster algebras.\n\n\\subsection{Kinematics and symmetries}\n\nThe $\\mathcal{N}=4$ super Yang-Mills on-shell multiplet may be organised into an on-shell superfield $\\Phi$, a function of an on-shell momentum $p^{\\alpha \\dot{\\alpha}} = \\lambda^\\alpha \\tilde{\\lambda}^{\\dot{\\alpha}}$ and a Grassmann variable $\\eta^A$ transforming in the $su(4)$ fundamental representation,\n\\be\n\\Phi=G^+ +\\eta^A\\Gamma_A+\\tfrac{1}{2!}\\eta^A\\eta^B S_{AB}+\\tfrac{1}{3!}\\eta^A\\eta^B\\eta^C\\epsilon_{ABCD}\\bar \\Gamma^D+\\tfrac{1}{4!}\\eta^A\\eta^B\\eta^C\\eta^D\\epsilon_{ABCD}G^-\\,,\n\\ee\nThe colour-ordered partial amplitudes in planar $\\mathcal{N}=4$ super Yang-Mills exhibit dual superconformal symmetry \\cite{Drummond:2008vq} which motivates the introduction of dual variables as follows,\n\\be\np_i^{\\alpha \\dot\\alpha} =\n\\lambda_i^\\alpha \\tilde{\\lambda}_i^{\\dot\\alpha}\n= x_{i+1}^{\\alpha \\dot\\alpha} - x_{i}^{\\alpha \\dot\\alpha}\\,, \\qquad\nq_i^{\\alpha A} = \\lambda_i^{\\alpha} \\eta_i^A\n= \\theta_{i+1}^{\\alpha A} - \\theta_{i}^{\\alpha A}\\,.\n\\label{xthetadef}\n\\ee\nThe dual symmetries act as superconformal transformations in the $(x,\\theta)$ space. The fact that the momenta are null means that the geometry in the dual space is associated with null lines, for which Penrose's (super)twistor variables are most appropriate \\cite{Hodges:2009hk},\n\\be\n\\mathcal{Z}_i = (Z_i \\, | \\, \\chi_i)\\,, \\qquad\nZ_i^{\\alpha,\\dot\\alpha} =\n(\\lambda_i^\\alpha , x_i^{\\beta \\dot\\alpha}\\lambda_{i\\beta})\\,,\n\\qquad\n\\chi_i^A= \\theta_i^{\\alpha A}\\lambda_{i \\alpha} \\,.\n\\ee\nHere the $Z_i$ variables correspond to points in $\\mathbb{P}^3$.\n\nWhen considering amplitudes we should take care of the structure of infrared divergences and the associated dual conformal anomaly \\cite{Drummond:2007cf,Drummond:2007au}. For our considerations here the appropriate way to do this will be to extract from the amplitude the so-called `BDS-like' form of the MHV superamplitude (denoted $\\tilde{A}_n$),\n\\begin{equation}\n\\label{}\nA_{n} = \\tilde{A}_{n} \\mathcal{E}_n\\,.\n\\end{equation}\nThe precise form of $\\tilde{A}_n$ can be found in \\cite{Alday:2009yn} and is not of great relevance here. The important point is that the remaining factor $\\mathcal{E}_n$ is dual conformally invariant and comprises all of the non-trivial information about the scattering amplitudes, once the dual conformal Ward identity of \\cite{Drummond:2007cf,Drummond:2007au} is taken into account. The function $\\mathcal{E}_n$ can be written purely in terms of the supertwistors $\\mathcal{Z}_i$ and has an expansion in Grassmann degree which encompasses the decomposition of different amplitudes into MHV, NMHV and so on,\n\\be\n\\label{MHVexp}\n\\mathcal{E}_n  = \\mathcal{E}_{n,{\\rm MHV}} + \\mathcal{E}_{n,{\\rm NMHV}} + \\ldots\\,.\n\\ee\n\nThe MHV term in (\\ref{MHVexp}) is of degree zero in the Grassmann $\\chi_i$ variables and hence is just a function of the $Z_i$. Dual conformal symmetry implies it is a function of the four-brackets $\\langle ijkl \\rangle$. It is homogeneous of degree zero in each $Z_i$ and so is a function on the space ${\\rm Conf}_n(\\mathbb{P}^3)$ (the configuration space of $n$ points in $\\mathbb{P}^3$).\n\nThe NMHV term in (\\ref{MHVexp}) is of Grassmann degree four and can be written in terms of the Yangian invariants (called \\emph{R-invariants}),\n\\be\n[ijklm] = \\frac{(\\langle\\langle ijklm \\rangle\\rangle}{\\langle ijkl \\rangle \\langle jklm \\rangle \\langle klmi \\rangle \\langle lmij \\rangle \\langle mijk \\rangle}\\,,\n\\ee\nmultiplied by dual conformally invariant functions $E_{ijklm}$ on ${\\rm Conf}_n(\\mathbb{P}^3)$,\n\\be\n\\mathcal{E}_{n,{\\rm NMHV}} = \\sum [ijklm] E_{ijklm}(Z_1,\\ldots,Z_n)\\,,\n\\ee\nwhere $\\langle \\langle i j k l m \\rangle \\rangle = (\\chi_i \\langle j k l m \\rangle + \\text{cyclic})^4$. In what follows the functions $\\mathcal{E}_{n,{\\rm MHV}}$ and $\\mathcal{E}_{n,{\\rm NMHV}}$ (and hence the functions $E_{ijklm}$) admit perturbative expansions of the form\n\\be\\label{eq:gLoopExpansion}\nF=\\sum_{L=0}^\\infty g^{2L} F^{(L)}\\,.\n\\ee\nFor the hexagon and heptagon amplitudes that we focus on here we need only consider MHV and NMHV terms in the expansion (\\ref{MHVexp}) since other amplitudes are obtained by parity conjugation of these ones. \n\n\\subsection{Analytic structure in perturbation theory}\n\nIn perturbation theory the functions appearing in hexagon and heptagon amplitudes are (according to all current evidence) polylogarithms of degree $2L$ where $L$ is the loop order. Polylogarithms are a class of iterated integrals over logarithmic singularities. Here we will define them in a recursive fashion. We declare that polylogarithms come with a grading and that a polylogarithm $f^{(k)}$ of degree (or \\emph{weight}) $k$ obeys\n\\be\n\\label{dlogpolys}\nd f^{(k)} = \\sum_{a \\in \\mathcal{A}} f_{[a]}^{(k-1)} d \\log a\\,,\n\\ee\nwhere the $a$ are some rational (or algebraic) functions of some number of variables (called \\emph{letters}) and the sum runs over a finite set $\\mathcal{A}$ of such functions (an \\emph{alphabet}). The space of functions of degree one is spanned by the set of logarithms of the letters $a$ themselves. The choice of the set $\\mathcal{A}$ then determines a class of polylogarithmic functions recursively in the degree. For example, in the case of functions of a single variable $x$, the choice $\\mathcal{A} = \\{ x, 1-x \\}$ yields the class of harmonic polylogarithms \\cite{Remiddi:1999ew} with indices $0$ or $1$. In particular this example includes the classical polylogarithms ${\\rm Li}_n(x)$.\n\nThe formula (\\ref{dlogpolys}) encodes the $(k-1,1)$ part of the coproduct of the function $f^{(k)}$. We write this as\n\\be\nf^{(k-1,1)}  = \\sum_{a \\in \\mathcal{A}} [f_{[a]}^{(k-1)} \\otimes a]\\,,\n\\ee\nwhere by convention we just record the argument of the $d \\log$ in the second argument of the tensor product. The arguments of the $(k-1,1)$ coproduct must obey the integrability relation\n\\be\n\\label{intcond}\n\\sum_{a \\in \\mathcal{A}} d f_{[a]}^{(k-1)} \\wedge d \\log a = 0\\,,\n\\ee\nwhich follows from $d^2 f^{(k)}=0$.\n\nIf we continue applying the definition of the $(n,1)$ coproduct iteratively to each of the functions $f^{(k-1)}_{[a]}$ all the way down to weight zero we obtain the \\emph{symbol}, an element of the $k$-fold tensor product of the space of one-forms spanned by the $d \\log a$ for $a \\in \\mathcal{A}$ (or more compactly a \\emph{word} in the alphabet $\\mathcal{A}$),\n\\be\n\\label{fksymbol}\nS[f^{(k)}] = f^{(1,\\dots,1)} = \\sum_{(a_1,\\ldots,a_k)} \\!\\! c_{a_1,\\ldots,a_k} \\, [a_1 \\otimes a_2 \\otimes \\ldots \\otimes a_k]\\,, \\quad c_{a_1,\\ldots,a_k} \\in \\mathbb{Q}\\,, \\quad a_i \\in \\mathcal{A}\\,.\n\\ee\nNote that by common convention we write the letters $a$ rather than $d \\log a$ in the arguments of the tensor product. This leads to the property that symbols with products of functions in their arguments decompose as follows,\n\\be\n\\label{symmult}\n[a \\otimes b\\, b ' \\otimes c] = [a \\otimes b \\otimes c] + [a \\otimes b' \\otimes c]\\,,\n\\ee\nand similarly that symbols with powers of functions in their arguments obey\n\\be\n\\label{sympower}\n[a \\otimes b^{\\, p} \\otimes c] = p\\, [a \\otimes b \\otimes c]\\, \\qquad p \\in \\mathbb{Q}\\,.\n\\ee\nIn the following we will discuss examples where the alphabet $\\mathcal{A}$ is given by the set of $\\mathcal{A}$-coordinates associated to a cluster algebra.\n\nThe symbol $S[f^{(k)}]$ displays both the branch cut structure and the differential structure of the function $f^{(k)}$. From the definition of the symbol (\\ref{fksymbol}) and the behaviour of polylogarithms under derivative action, (\\ref{dlogpolys}) we see derivatives act on the symbol by action on the rightmost element of the tensor product,\n\\be\nd \\, [a_1 \\otimes \\ldots \\otimes a_k] = [a_1 \\otimes \\ldots \\otimes a_{k-1}]\\, d \\log a_k\\,.\n\\ee\n\n\nThe symbol (\\ref{fksymbol}) obeys integrability relations,\n\\be\n\\sum_{\\vec{a}} c_{\\vec{a}}\\, [a_1 \\otimes \\ldots \\otimes a_{i-1} \\otimes a_{i+2} \\otimes \\ldots \\otimes a_k]\\, (d \\log a_{i} \\wedge d \\log a_{i+1}) = 0\\,, \\quad i=1,\\ldots,k-1\n\\ee\nwhich follow from the fact that $d^2 f  = 0$ for all the functions of all weights and encode the commutativity of partial derivatives.\n\nSimilarly a logarithmic branch cut discontinuity around a singularity at $a=0$ is obtained from terms beginning with the letters $a$, assuming the alphabet is chosen so that no other letter vanishes at $a=0$,\n\\be\n{\\rm disc}_{a=0} [a_1 \\otimes a_2 \\otimes \\ldots \\otimes a_k] = (2 \\pi i)[ a_2 \\otimes  \\ldots \\otimes a_k]\\,.\n\\ee\n\nThe symbol is an efficient tool for simplifying polylogarithmic expressions, as demonstrated in the derivation of the simple formula of \\cite{Goncharov:2010jf} for the two-loop MHV hexagon amplitude \\cite{DelDuca:2009au}.\nA first step in the bootstrap calculations of \\cite{Dixon:2011pw,Dixon:2011nj,Dixon:2013eka,Dixon:2014voa,Dixon:2014iba,Drummond:2014ffa,Dixon:2015iva,Caron-Huot:2016owq,Dixon:2016nkn} is to build integrable words in a given alphabet. We quickly review here the method described in \\cite{Dixon:2016nkn} for performing this task. The construction of integrable words can be done iteratively in the weight. We suppose that we have a basis $\\{ f^{(k)}_i \\}$ of integrable words up to weight $k$. This means that we know how to decompose integrable words of weight $k$ into their $(k-1,1)$ coproducts\n\\be\nf^{(k)}_i = \\sum_{a,j} M^{(k)}_{ija} [f^{(k-1)}_j \\otimes a]\\,.\n\\ee\n\nNow we would like to construct integrable words of weight $(k+1)$. We build an ansatz for the $(k,1)$ coproduct with constants $c_{ai}$,\n\\be\nf^{(k,1)} = \\sum_{a,i} c_{ai} [f^{(k)}_i \\otimes a]\\,.\n\\ee\nThe constraints we have to solve come from the integrability condition (\\ref{intcond}),\n\\be\n\\label{bootstrapint}\n\\sum_{a,i} c_{ai} df^{(k)} \\wedge d\\log a  = \\sum_{a,i} c_{ai} \\sum_{b,j} M^{(k)}_{ijb} f^{(k-1)}_j d \\log b \\wedge d \\log a = 0\\,.\n\\ee\nwhere the first equality expresses $df^{(k)}$ using (\\ref{dlogpolys}).\n\nThe two-forms $d\\log a \\wedge d\\log b$ are not generally all linearly independent. They satisfy linear relations known as Arnold relations which essentially come from partial fraction identities. We suppose that $\\{ \\omega^{(2)}_m \\}$ form a basis for the space of independent two forms. Then there exists a tensor $Y$ which expresses each two-form $d\\log a \\wedge d \\log b$ in terms of the independent basis\n\\be\nd \\log a \\wedge d \\log b = \\sum_m Y_{ab,m}\\, \\omega^{(2)}_m\\,.\n\\ee\nIt follows that the condition (\\ref{bootstrapint}) becomes\n\\be\n\\sum_{a,i} c_{ai} \\sum_{b,j} M^{(k)}_{ijb} f^{(k-1)}_j Y_{ab,m} \\, \\omega^{(2)}_m = 0\\,.\n\\ee\nSince the $\\omega^{(2)}_m$ form a basis for the independent two-forms and the $f^{(k-1)}_j$ form a basis for the integrable words of weight $(k-1)$ the condition becomes\n\\be\n\\sum_{a,i} c_{ai} \\sum_{b} M^{(k)}_{ijb} Y_{ab,m} = 0\\,.\n\\ee\nIn other words we need to compute the kernel of the matrix\n\\be\n\\label{linalg}\n\\mathcal{M}_{AB} = \\sum_b M^{(k)}_{ijb} Y_{ab,m}\\,,\\qquad A=(jm)\\,,\\, B=(ai)\\,,\n\\ee\nwhere we grouped indices into multi-indices $A,B$. \n\nTo obtain a solution to (\\ref{linalg}) is a linear algebra problem that can be helpfully addressed with available packages. The package SpaSM \\cite{spasm} for sparse modular linear algebra operations is particularly helpful as the matrices involved are typically sparse and all quantities involved can be chosen to be integer-valued. \nHowever it is solved, one obtains a basis for the kernel of $\\mathcal{M}$, i.e. a set of linearly independent null vectors $\\{ v_{A, l} \\}$ where $l=1,\\ldots,{\\rm dim} (\\ker \\mathcal{M})$. Expanding the multi-index $A=(ai)$ we obtain the desired basis of weight $(k+1)$ words,\n\\be\nf^{(k+1)}_l = \\sum_{a,i} M^{(k+1)}_{lia} [f^{(k)}_i \\otimes a]\\,, \\qquad M^{(k+1)}_{lia} = v_{ai,l}\\,.\n\\ee\n\nThe above procedure has been used extensively in several works as a first step in the analytic bootstrap programme for amplitudes.\n\n\n\n\\section{Cluster Algebras and Grassmannians}\n\n\\sloppy In \\cite{Golden:2013xva} the important observation was made that the symbols of the two-loop MHV remainder functions constructed in \\cite{CaronHuot:2011ky} were written in terms of alphabets that exclusively contained $\\mathcal{A}$-coordinates of cluster algebras associated to Grassmannians ${\\rm Gr}(4,n)$, or more precisely, the $(3n-15)$-dimensional spaces $\\allowbreak{\\rm Conf}_n(\\mathbb{P}^3) \\allowbreak= {\\rm Gr}(4,n)\/(\\mathbb{C}^*)^{n-1}$. Beyond two loops, great progress has been made in understanding the hexagon ($n=6$) and heptagon ($n=7$) amplitudes via the analytic bootstrap programme. All current evidence is compatible with the hypothesis that the hexagon and heptagon amplitudes are polylogarithmic at all orders in perturbation theory and moreover that their symbol alphabets are given by the set of $\\mathcal{A}$-coordinates for the cases ${\\rm Conf}_6(\\mathbb{P}^3)$ and ${\\rm Conf}_7(\\mathbb{P}^3)$ respectively. The associated cluster algebras are isomorphic to the ones based on $A_3$ and $E_6$ respectively. Here we will review some of the important aspects of cluster algebras. Many of the points we recall here are covered already in \\cite{Golden:2013xva} but we review them as we will need many of the ideas to explain the notion of adjacency for the cluster polylogarithms appearing in the expressions for scattering amplitudes.\n\nCluster algebras are commutative associative algebras with generators referred to as cluster coordinates which arise in families called clusters. They can be specified by giving an initial cluster with a set of $\\mathcal{A}$-coordinates together with a mutation rule which allows the generation of further clusters and cluster coordinates. To each cluster can be associated a quiver diagram with $\\mathcal{A}$-coordinates associated to the nodes. Such a quiver is described by the adjacency matrix $b_{ij}$ defined via\n\\be\nb_{ij} = (\\text{no. of arrows } i \\rightarrow j) - (\\text{no. of arrows } j \\rightarrow i)\\,.\n\\ee\nThe adjacency matrix specifies how the cluster changes under a mutation. If one performs a mutation on a node labelled by $\\mathcal{A}$-coordinate $a_k$ then the adjacency matrix of the new cluster is given by \n\\begin{equation}\n\\label{mutationb}\n  b'_{ij} =\n  \\begin{cases}\n    -b_{ij}                  &   k \\in \\{i,j\\}\\,,\\\\\n    b_{ij}                   &   b_{ik}b_{kj} \\leq 0\\,,\\\\\n    b_{ij}  + b_{ik}b_{kj}    &   b_{ik}, b_{kj} > 0\\,,\\\\\n    b_{ij}  - b_{ik}b_{kj}    &   b_{ik}, b_{kj} < 0\\,.\\\\\n  \\end{cases}\n\\end{equation}\nand the $\\mathcal{A}$-coordinate $a_k$ associated to that node is replaced by\n\\be\n\\label{mutationa}\na_k' = \\frac{1}{a_k}\\biggl[\\prod_{i|b_{ik}>0} a_i^{b_{ik}} +\\prod_{i|b_{ik}<0} a_i^{-b_{ik}} \\biggr]\\,.\n\\ee\n\n \n \nFor the set of cluster algebras associated to ${\\rm Conf}_n(\\mathbb{P}^3)$ we take the initial cluster depicted in Fig. \\ref{Gr4ninitial}.\n\\begin{figure}\n\\begin{center}\n{\\footnotesize\n\\begin{tikzpicture}\n\\pgfmathsetmacro{\\nw}{1.3}\n\\pgfmathsetmacro{\\vvwnw}{2.5}\n\\pgfmathsetmacro{\\vvvwnw}{2.85}\n\\pgfmathsetmacro{\\nh}{0.6}\n\\pgfmathsetmacro{\\aa}{0.6}\n\\pgfmathsetmacro{\\ep}{0.1}\n\\node at (-0.5*\\nw -\\aa,\\aa+0.5*\\nh) {$\\langle 1\\,2\\,3\\,4 \\rangle$};\n\\draw[] (-\\aa,\\aa) -- (-\\aa -\\nw,\\aa) -- (-\\aa -\\nw, \\aa+\\nh) -- (-\\aa,\\aa+\\nh) -- cycle;\n\\node at (0.5*\\nw +0*\\aa,-0*\\aa-0.5*\\nh) {$\\langle 1\\,2\\,3\\,5 \\rangle$};\n\\node at (0.5*\\nw +0*\\aa,-1*\\aa-1.5*\\nh) {$\\langle 1\\,2\\,4\\,5 \\rangle$};\n\\node at (0.5*\\nw +0*\\aa,-2*\\aa-2.5*\\nh) {$\\langle 1\\,3\\,4\\,5 \\rangle$};\n\\node at (0.5*\\nw +0*\\aa,-3*\\aa-3.5*\\nh) {$\\langle 2\\,3\\,4\\,5 \\rangle$};\n\\draw[] (0,-3*\\aa-3*\\nh) -- (0,-3*\\aa-4*\\nh) -- (\\nw,-3*\\aa-4*\\nh) -- (\\nw,-3*\\aa-3*\\nh) -- cycle;\n\\node at (1.5*\\nw +1*\\aa,-0*\\aa-0.5*\\nh) {$\\langle 1\\,2\\,3\\,6 \\rangle$};\n\\node at (1.5*\\nw +1*\\aa,-1*\\aa-1.5*\\nh) {$\\langle 1\\,2\\,5\\,6 \\rangle$};\n\\node at (1.5*\\nw +1*\\aa,-2*\\aa-2.5*\\nh) {$\\langle 1\\,4\\,5\\,6 \\rangle$};\n\\node at (1.5*\\nw +1*\\aa,-3*\\aa-3.5*\\nh) {$\\langle 3\\,4\\,5\\,6 \\rangle$};\n\\draw[] (\\nw+\\aa,-3*\\aa-3*\\nh) -- (\\nw+\\aa,-3*\\aa-4*\\nh) -- (2*\\nw+\\aa,-3*\\aa-4*\\nh) -- (2*\\nw+\\aa,-3*\\aa-3*\\nh) -- cycle;\n\\node at (3*\\nw + 2*\\aa+0.5*\\vvvwnw,-0*\\aa-0.5*\\nh) {$\\langle 1\\,2\\,3\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 1 \\rangle$};\n\\node at (3*\\nw + 2*\\aa+0.5*\\vvvwnw,-1*\\aa-1.5*\\nh) {$\\langle 1\\,2\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 2\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 1 \\rangle$};\n\\node at (3*\\nw + 2*\\aa+0.5*\\vvvwnw,-2*\\aa-2.5*\\nh) {$\\langle 1\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 3\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 2\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 1 \\rangle$};\n\\node at (3*\\nw + 2*\\aa+0.5*\\vvvwnw,-3*\\aa-3.5*\\nh) {$\\langle n \\scalebox{0.65}[1.0]{\\( - \\)} 4\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 3\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 2\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 1 \\rangle$};\n\\draw[] (3*\\nw+2*\\aa,-3*\\aa-3*\\nh) -- (3*\\nw+2*\\aa,-3*\\aa-4*\\nh) -- (3*\\nw+2*\\aa+\\vvvwnw,-3*\\aa-4*\\nh) -- (3*\\nw+2*\\aa+\\vvvwnw,-3*\\aa-3*\\nh) -- cycle;\n\\node at (3*\\nw + 3*\\aa+1*\\vvvwnw+0.5*\\vvwnw,-0*\\aa-0.5*\\nh) {$\\langle 1\\,2\\,3\\,n \\rangle$};\n\\draw[] (3*\\nw+3*\\aa+1*\\vvvwnw,-0*\\aa-0*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw,-0*\\aa-1*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw+1*\\vvwnw,-0*\\aa-1*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw+1*\\vvwnw,-0*\\aa-0*\\nh) -- cycle;\n\\node at (3*\\nw + 3*\\aa+1*\\vvvwnw+0.5*\\vvwnw,-1*\\aa-1.5*\\nh) {$\\langle 1\\,2\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 1\\,n \\rangle$};\n\\draw[] (3*\\nw+3*\\aa+1*\\vvvwnw,-1*\\aa-1*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw,-1*\\aa-2*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw+1*\\vvwnw,-1*\\aa-2*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw+1*\\vvwnw,-1*\\aa-1*\\nh) -- cycle;\n\\node at (3*\\nw + 3*\\aa+1*\\vvvwnw+0.5*\\vvwnw,-2*\\aa-2.5*\\nh) {$\\langle 1\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 2\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 1\\,n \\rangle$};\n\\draw[] (3*\\nw+3*\\aa+1*\\vvvwnw,-2*\\aa-2*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw,-2*\\aa-3*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw+1*\\vvwnw,-2*\\aa-3*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw+1*\\vvwnw,-2*\\aa-2*\\nh) -- cycle;\n\\draw[] (3*\\nw+3*\\aa+1*\\vvvwnw,-3*\\aa-3*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw,-3*\\aa-4*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw+1*\\vvwnw,-3*\\aa-4*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw+1*\\vvwnw,-3*\\aa-3*\\nh) -- cycle;\n\\node at (3*\\nw + 3*\\aa+1*\\vvvwnw+0.5*\\vvwnw,-3*\\aa-3.5*\\nh) {$\\langle n \\scalebox{0.65}[1.0]{\\( - \\)} 3\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 2\\,n \\scalebox{0.65}[1.0]{\\( - \\)} 1\\,n \\rangle$};\n\\node at (2.25*\\nw + 2*\\aa,-0.5*\\nh) {$\\ldots$};\n\\node at (2.25*\\nw + 2*\\aa,-1.5*\\nh-\\aa) {$\\ldots$};\n\\node at (2.25*\\nw + 2*\\aa,-2.5*\\nh-2*\\aa) {$\\ldots$};\n\\node at (2.25*\\nw + 2*\\aa,-3.5*\\nh-3*\\aa) {$\\ldots$};\n\\draw[->] (-\\aa+0*\\ep,\\aa-\\ep) -- (0-0*\\ep,0+\\ep);\n\\draw[->] (0.5*\\nw,-\\nh-\\ep) -- (0.5*\\nw,-\\nh-\\aa+\\ep);\n\\draw[->] (0.5*\\nw,-2*\\nh-\\aa-\\ep) -- (0.5*\\nw,-2*\\nh-2*\\aa+\\ep);\n\\draw[->] (0.5*\\nw,-3*\\nh-2*\\aa-\\ep) -- (0.5*\\nw,-3*\\nh-3*\\aa+\\ep);\n\\draw[->] (1*\\nw+\\ep,-0.5*\\nh) -- (1*\\nw+\\aa-\\ep,-0.5*\\nh);\n\\draw[->] (1*\\nw+\\ep,-1.5*\\nh-\\aa) -- (1*\\nw+\\aa-\\ep,-1.5*\\nh-\\aa);\n\\draw[->] (1*\\nw+\\ep,-2.5*\\nh-2*\\aa) -- (1*\\nw+\\aa-\\ep,-2.5*\\nh-2*\\aa);\n\\draw[->] (1*\\nw+\\aa-0*\\ep,-\\nh-\\aa+\\ep) -- (1*\\nw+0*\\ep,-\\nh-\\ep);\n\\draw[->] (1*\\nw+\\aa-0*\\ep,-2*\\nh-2*\\aa+\\ep) -- (1*\\nw+0*\\ep,-2*\\nh-1*\\aa-\\ep);\n\\draw[->] (1*\\nw+\\aa-0*\\ep,-3*\\nh-3*\\aa+\\ep) -- (1*\\nw+0*\\ep,-3*\\nh-2*\\aa-\\ep);\n\\draw[->] (1.5*\\nw+1*\\aa,-\\nh-\\ep) -- (1.5*\\nw+1*\\aa,-\\nh-\\aa+\\ep);\n\\draw[->] (1.5*\\nw+1*\\aa,-2*\\nh-1*\\aa-\\ep) -- (1.5*\\nw+1*\\aa,-2*\\nh-2*\\aa+\\ep);\n\\draw[->] (1.5*\\nw+1*\\aa,-3*\\nh-2*\\aa-\\ep) -- (1.5*\\nw+1*\\aa,-3*\\nh-3*\\aa+\\ep);\n\\draw[->] (3*\\nw+0.5*\\vvvwnw+2*\\aa,-\\nh-\\ep) -- (3*\\nw+0.5*\\vvvwnw+2*\\aa,-\\nh-\\aa+\\ep);\n\\draw[->] (3*\\nw+0.5*\\vvvwnw+2*\\aa,-2*\\nh-\\aa-\\ep) -- (3*\\nw+0.5*\\vvvwnw+2*\\aa,-2*\\nh-2*\\aa+\\ep);\n\\draw[->] (3*\\nw+0.5*\\vvvwnw+2*\\aa,-3*\\nh-2*\\aa-\\ep) -- (3*\\nw+0.5*\\vvvwnw+2*\\aa,-3*\\nh-3*\\aa+\\ep);\n\\draw[->] (3*\\nw+2*\\aa+1*\\vvvwnw+\\ep,-0.5*\\nh) -- (3*\\nw+3*\\aa+1*\\vvvwnw-\\ep,-0.5*\\nh);\n\\draw[->] (3*\\nw+2*\\aa+1*\\vvvwnw+\\ep,-1.5*\\nh-1*\\aa) -- (3*\\nw+3*\\aa+1*\\vvvwnw-\\ep,-1.5*\\nh-1*\\aa);\n\\draw[->] (3*\\nw+2*\\aa+1*\\vvvwnw+\\ep,-2.5*\\nh-2*\\aa) -- (3*\\nw+3*\\aa+1*\\vvvwnw-\\ep,-2.5*\\nh-2*\\aa);\n\\draw[->] (3*\\nw+3*\\aa+1*\\vvvwnw-0*\\ep,-\\nh-\\aa+\\ep) -- (3*\\nw+2*\\aa+1*\\vvvwnw+0*\\ep,-\\nh-\\ep);\n\\draw[->] (3*\\nw+3*\\aa+1*\\vvvwnw-0*\\ep,-2*\\nh-2*\\aa+\\ep) -- (3*\\nw+2*\\aa+1*\\vvvwnw+0*\\ep,-2*\\nh-1*\\aa-\\ep);\n\\draw[->] (3*\\nw+3*\\aa+1*\\vvvwnw-0*\\ep,-3*\\nh-3*\\aa+\\ep) -- (3*\\nw+2*\\aa+1*\\vvvwnw+0*\\ep,-3*\\nh-2*\\aa-\\ep);\n\\end{tikzpicture}\n}\n\\end{center}\n\\caption{The initial cluster of the Grassmannian series ${\\rm Gr}(4,n)$.}\n\\label{Gr4ninitial}\n\\end{figure}\nThe boxed nodes are referred to as \\emph{frozen} nodes and the remainder are \\emph{unfrozen}. Other clusters (and hence other $\\mathcal{A}$-coordinates) are obtained by mutating on the unfrozen nodes according to the above rules. In the cases of $n=6,7$ the number of distinct clusters obtained is finite. For $n=6$ all $\\mathcal{A}$-coordinates are Pl\\\"ucker coordinates of the form $\\langle ijkl \\rangle$ while $n=7$ some $\\mathcal{A}$-coordinates are quadratic in Pl\\\"uckers.\n\nIf we pick a particular $\\mathcal{A}$-coordinate $a$ and look at all clusters containing $a$ we obtain a cluster subalgebra. Such clusters may be generated by starting in one cluster containing $a$ and performing all possible combinations of mutations on the other nodes. In this way, to each $\\mathcal{A}$-coordinate we associate a codimension-one subalgebra. Similarly we may pick a pair of coordinates $\\{ a, b \\}$ and, as long as there is at least one cluster where they both appear, associate to them a codimension-two subalgebra by performing all possible mutations on the other nodes. If there is no cluster where $a$ and $b$ appear together then there is no such subalgebra. The fact that some pairs can be found together (we call them `admissible' or `adjacent') while other pairs cannot is at the heart of the cluster adjacency property describing the behaviour of singularities of scattering amplitudes. Note that frozen nodes are present in every cluster and hence are always admissible with any other $\\mathcal{A}$-coordinate.\n\nWe can continue further and associate codimension-three subalgebras with admissible triplets $\\{a,b,c\\}$ where $a$, $b$ and $c$ can all be found together in some cluster and so on. Finally when we have fixed an admissible set of $(3n-15)$ $\\mathcal{A}$-coordinates we uniquely specify a cluster which we could alternatively describe as a dimension-zero subalgebra.\n\n\n\n\nNote that while $\\mathcal{A}$-coordinates are called `coordinates' they are not strictly coordinates on ${\\rm Conf}_n(\\mathbb{P}^3)$ because they are not homogeneous under rescalings of the twistors. A natural set of homogeneous coordinates for ${{\\rm Conf}_n(\\mathbb{P}^3)}$  are the cluster $\\mathcal{X}$-coordinates. They are defined with respect to a given cluster for each unfrozen node $j$ and are related to the $\\mathcal{A}$-coordinates and the adjacency matrix of the cluster via\n\\be\nx_j = \\prod_i a_i^{b_{ij}}\\,,\n\\ee\nwhere the product runs over all nodes (frozen and unfrozen) labelled by $i$.\nUnder mutation on an unfrozen node $k$ the $\\mathcal{X}$-coordinates change according to \n\\be\nx_i' = \\begin{cases}\n    1\/x_i              &   k = i\\,,\\\\\n    x_i\\bigl(1+x_k^{{\\rm sgn}(b_{ik})}\\bigr)^{b_{ik}} & k \\neq i\\,.\n  \\end{cases}\n\\ee\nNote that if node $i$ is not connected to node $k$ then $b_{ik}=0$ and $x_i'=x_i$.\n\nThe adjacency matrix $b_{ij}$ actually defines a Poisson structure on the space ${\\rm Conf}_n(\\mathbb{P})$ via the formula\n\\be\n\\label{poisson}\n\\{ x_i , x_j \\} = b_{ij} x_i x_j.\n\\ee\nThe choice of cluster is irrelevant since the formula (\\ref{poisson}) is preserved under mutation. Note that only the restriction of the adjacency matrix to the unfrozen nodes actually appears in (\\ref{poisson}). We recall that a Poisson structure can be described in terms of a bivector $b$ such that $b(d f , d g) = \\{f,g\\}$. The adjacency matrix of a cluster then gives the components of the Poisson bivector in the coordinate system given by the (logarithms of the) cluster $\\mathcal{X}$-coordinates for that cluster. \n\nIf we restrict attention to the real case then the condition that all $\\mathcal{X}$-coordinates obey $0<x<\\infty$ defines a region inside ${\\rm Conf}_n(\\mathbb{RP}^3)$. The region should be visualised as a polytope with a boundary that is made of facets corresponding to codimension-one sub-algebras of the original cluster algebra (which, as we described before are associated to individual $\\mathcal{A}$-coordinates). Each facet has boundaries corresponding to codimension-two subalgebras (associated to admissible pairs of $\\mathcal{A}$-coordinates) and so on. If we continue all the way down we arrive at dimension-zero subalgebras given by the clusters themselves and corresponding to corners of the polytope in the sense that the corner is the origin in the associated set of $\\mathcal{X}$-coordinates.\n\nThe cluster $\\mathcal{X}$-coordinates are edge coordinates in that they can be associated to the one-dimensional edges (axes) which meet at the vertex corresponding to the cluster. On each edge the associated $\\mathcal{X}$-coordinate runs over $0<x<\\infty$, in correspondence with the fact that the $\\mathcal{X}$-coordinate associated to a given edge inverts under the mutation along that edge.\n\n\n\\subsection{Hexagons and the $A_3$ associahedron}\nFor $\\mathrm{Conf}_6(\\mathbb{P}^3)$, the initial cluster is represented by the quiver diagram given in Fig. \\ref{hexinitial} with Pl\\\"ucker coordinates at each of the nodes. The unfrozen $\\mathcal{A}$-coordinates of this cluster are\n\\be\na_1 = \\langle 1235 \\rangle\\,, \\qquad a_2 = \\langle 1245 \\rangle\\,, \\qquad a_3 = \\langle 1345 \\rangle\\,.\n\\ee\n\\begin{figure}\n{\\footnotesize\n\\begin{center}\n\\makeatletter  \n\\newcommand{\\phantombox}[1]{%\n  \\setbox0=\\hbox{#1}%\n \n  \\begin{tcolorbox}[colframe=white,colback=white,boxrule=0.4pt,\n    left=2pt,right=2pt,top=3pt,bottom=3pt,boxsep=0pt,width=1.2cm, valign = center,  halign=center, sharp corners = all]\n    #1\n  \\end{tcolorbox}\n}\n\\newcommand{\\frozenbox}[1]{%\n  \\setbox0=\\hbox{#1}%\n  \\begin{tcolorbox}[colframe=black,colback=white,boxrule=0.5pt,\n      left=2pt,right=2pt,top=2pt,bottom=2pt,boxsep=0pt,width=1.2cm, halign=center, sharp corners = all]\n    #1\n  \\end{tcolorbox}\n}\n\\makeatother\n\\begin{tikzpicture}%\n    [\n    unfrozen\/.style={},\n    frozen\/.style={inner sep=1.2mm,outer sep=0mm,yshift=0},\n    node distance = 0.4cm\n    ]\n    \\node[frozen]        (f0) at (0,5) {$\\frozenbox{$\\langle 1234 \\rangle$}$};\n    \\node[frozen, below right = of f0]        (t1)  {$\\phantombox{$\\langle1235 \\rangle$}$};\n    \\node[frozen, right = of t1]        (t2)  {$\\frozenbox{$\\langle 1236 \\rangle$}$};\n    %\n    \\node[frozen, below = of t1]        (m1)  {$\\phantombox{$\\langle 1245 \\rangle$}$};\n    \\node[frozen, right = of m1]        (m2)  {$\\frozenbox{$\\langle 1256 \\rangle$}$};\n    %\n    \\node[frozen, below = of m1]        (b1)  {$\\phantombox{$\\langle1345 \\rangle$}$};\n    \\node[frozen, right = of b1]        (b2)  {$\\frozenbox{$\\langle1456 \\rangle$}$};\n    %\n   \n   \n   \n    %\n    \\node[frozen, below = of b1]        (f4)  {$\\frozenbox{$\\langle 2345 \\rangle$}$};\n    \\node[frozen, right = of f4]        (f5)  {$\\frozenbox{$\\langle 3456 \\rangle$}$};\n   \n    \\draw[->] (f0) -- (t1);\n    \\draw[->] (t1) -- (t2);    \\draw[->] (t1) -- (m1) ;  \n   \n    \\draw[->] (m1) -- (m2);  \\draw[->] (m1) -- (b1) ; \\draw[->] (m2) -- (t1);     \n   \n   \n   \n    \\draw[->] (b1) -- (b2);  \n    \\draw[->] (b1) -- (f4) ; \n    \\draw[->] (b2) -- (m1);     \n   \n   \n   \n   \n    \\draw[->] (f5) -- (b1);\n\\end{tikzpicture}\n\\end{center}}\n\\caption{The quiver diagram for the initial cluster for the algebra associated to ${\\rm Conf}_6(\\mathbb{P}^3)$.}\n\\label{hexinitial}\n\\end{figure}\n\nBy repeated mutation of the above data according to (\\ref{mutationb}) and (\\ref{mutationa}) one obtains 14 distinct clusters arranged in the topology of the Stasheff polytope or associahedron illustrated in Fig. \\ref{Stasheff}. In total nine distinct unfrozen $\\mathcal{A}$-coordinates are obtained, corresponding to the nine faces of the polytope, in addition to the six frozen ones present in every cluster. Three are square faces and six are pentagonal. Each cluster corresponds to a vertex, with the unfrozen $\\mathcal{A}$-coordinates of the cluster corresponding to the faces of the polytope which meet at the vertex. The frozen $\\mathcal{A}$-coordinates $\\langle i\\, i+1\\,i+2\\,i+3 \\rangle$, being present in every cluster, are not shown in Fig. \\ref{Stasheff}. The initial cluster drawn in Fig. \\ref{hexinitial} corresponds to the cluster in the top left of Fig. \\ref{Stasheff}. The edges between clusters correspond to mutation operations.\n\n\\begin{figure}\n{\\footnotesize\n\\begin{center}\n\\begin{tikzpicture}[scale=0.46]\n \\draw[join=bevel,thick,gray,dashed]  (5,-2.75) -- (5,-5.25);\n \\draw[join=bevel,->,gray,dashed] (1.25,3.75) -- (2.5,2.5);\n\\node[gray] at (0.25,4.25){$\\langle 1345 \\rangle$};\n\\draw[join=bevel,->,gray,dashed] (8.75,3.75) -- (7.5,2.5);\n\\node[gray] at (9.75,4.25){$\\langle 1356 \\rangle$};\n\\draw[join=bevel,->,gray,dashed] (8.75,-4.25) -- (7.5,-3);\n\\node[gray] at (9.75,-4.75){$\\langle 2346 \\rangle$};\n\\draw[join=bevel,->,gray,dashed] (1.25,-4.25) -- (2.5,-3);\n\\node[gray] at (0.25,-4.75){$\\langle 1246 \\rangle$};\n \\draw[join=bevel,thick,gray,dashed] (0,0) -- (2.75,0.25) ;\n  \\draw[join=bevel,thick,gray,dashed]  (7.25,0.25) -- (10,0);\n  \\draw[join=bevel,thick,gray,dashed]  (2.75,0.25) -- (5,3.25) -- (7.25,0.25) -- (5,-2.75) -- cycle;\n  \\draw[join=bevel,thick,gray,dashed]  (5,3.25) -- (5,5);\n  \\draw[join=bevel,thick,fill=none] (0,0) -- (1,3) -- (2,-0.25) -- (1,-3.5)  -- cycle;\n  \\draw[join=bevel,thick,fill=none] (2,-0.25) -- (8,-0.25);\n  \\draw[join=bevel,thick,fill=none] (10,0) -- (9,3) -- (8,-0.25) -- (9,-3.5)  -- cycle;\n  \\draw[join=bevel,thick,fill=none] (1,3) -- (5,5) -- (9,3);\n  \\draw[join=bevel,thick,fill=none] (1,-3.5) -- (5,-5.25) -- (9,-3.5);\n    \\draw[thick,fill=none] (0,0) -- (1,3) -- (5,5) -- (9,3) -- (10,0) -- (9,-3.5) -- (5,-5.25) -- (1,-3.5) -- cycle;\n\\draw[join=bevel,->] (-0.75,-0.125) -- (1,-0.125);\n\\node at (-2,-0.125){$\\langle 1245 \\rangle$};\n\\draw[join=bevel,->] (10.75,-0.125) -- (9,-0.125);\n\\node at (12,-0.125){$\\langle 2356 \\rangle$};\n\\draw[join=bevel,->] (5,5.75) arc (-200:-160:5.75);\n\\node at (5,6.5){$\\langle 1235 \\rangle$};\n\\draw[join=bevel,->] (5,-6) arc (-20:20:6.25);\n\\node at (5,6.5){$\\langle 1235 \\rangle$};\n\\node at (5,-6.75){$\\langle 2456 \\rangle$};\n\\node[gray] at (5,0.25){$\\langle 1346 \\rangle$};\n\\draw[join=bevel,thick,->] (1,3) -- (1.5,1.375);\n\\draw[join=bevel,thick,->] (2,-0.25) -- (5,-0.25);\n\\draw[join=bevel,thick,->] (8,-0.25) -- (8.5,-1.875);\n \\end{tikzpicture}\n \\end{center}\n }\n \\caption{The $A_3$ Stasheff polytope with six pentagonal faces and three square faces, each labelled with the corresponding $\\mathcal{A}$-coordinate. The initial cluster corresponds to the vertex at the top left corner at the intersection of the faces labelled by $\\langle 1235 \\rangle$, $\\langle 1245 \\rangle$, $\\langle 1345\\rangle$. The three-step path leads from the initial cluster to one obtained by a cyclic rotation by one unit.}\n \\label{Stasheff}\n \\end{figure}\n\n\n\nFig. \\ref{Stasheff} also makes manifest the discrete symmetries of the ${\\rm Conf}_6(\\mathbb{P}^3)$ cluster algebra. A cyclic rotation of the initial cluster can be generated by a threefold sequence of mutations, as indicated by the arrows. This corresponds to mutating on the three unfrozen nodes in Fig. \\ref{hexinitial} in turn, starting at the bottom and moving to the top. A threefold cyclic rotation corresponds to a reflection in the equatorial plane of Fig. \\ref{Stasheff} and also corresponds to the parity transformation $Z_i \\mapsto Z_{i-1} \\wedge Z_i \\wedge Z_{i+1}$ when applied to homogeneous quantities. Finally, the reflection $Z_i \\mapsto Z_{7-i}$ corresponds to a left-right reflection of Fig. \\ref{Stasheff} together with a reflection in the equatorial plane.\n\nThe space ${\\rm Conf}_6(\\mathbb{P}^3)$ can be identified with the space ${\\rm Conf}_6(\\mathbb{P}^1) \\cong \\mathcal{M}_{0,6}$, that is the moduli space of six points on the Riemann sphere modulo $sl_2$ transformations. At the level of Pl\\\"ucker coordinates this can be achieved by identifying an ordered four-bracket $\\langle ijkl \\rangle$ (such that $i<j<k<l$) with an ordered two-bracket $(mn)$ (with  $m<n$) made of the absent labels from the set $\\{1, \\ldots , 6\\}$. In other words we make the identifications $\\langle 1345 \\rangle = (26)$, $\\langle 1245 \\rangle = (36)$, $\\langle 1235 \\rangle = (46)$ and so on. In this way the nodes of each quiver diagram can be identified with chords of a hexagon. The edges of the hexagon correspond to adjacent two-brackets, e.g. $(12) = \\langle 3456 \\rangle$. Such an identification is described in Fig. \\ref{hexagonchords}. With the triangulation labelling of clusters to hand we may illustrate all triangulations on the Stasheff polytope, as shown in Fig. \\ref{Stasheff_triang}. \n\n\\begin{figure}\n{\\footnotesize\n\\begin{center}\n\\begin{tikzpicture}[scale=0.8]\n   \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at (60+60*\\i:1.6);\n   }\n   \\draw[thick]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw[thick] (a6) -- (a2);\n   \\draw[thick] (a6) -- (a3);\n   \\draw[thick] (a6) -- (a4);\n    \\foreach \\i in {1,...,6}\n   {\n     \\node at (60+60*\\i:1.9) {$\\i$};\n   }\n\\end{tikzpicture}\n\\end{center}\n}\n\\caption{The two-brackets $(ij)$ can be identified with chords on a hexagon between the vertices $i$ and $j$. A triangulation of the hexagon then corresponds to a cluster of the $A_3$ or ${\\rm Conf}_6(\\mathbb{P}^3)$ polytope. Above is shown the triangulation corresponding to the initial cluster of Fig. \\ref{hexinitial} comprised of the chords $(26) = \\langle 1345 \\rangle$, $(36) = \\langle 1245 \\rangle$ and $(46) = \\langle 1235 \\rangle$ together with the six edges which correspond to the frozen nodes.}\n\\label{hexagonchords}\n\\end{figure}\n\n\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.54]\n  \\draw[thick,fill=none] (0,0) -- (1,3) -- (2,-0.25) -- (1,-3.5)  -- cycle;\n  \\draw[thick,fill=none] (2,-0.25) -- (8,-0.25);\n  \\draw[thick,fill=none] (10,0) -- (9,3) -- (8,-0.25) -- (9,-3.5)  -- cycle;\n  \\draw[thick,fill=none] (1,3) -- (5,5) -- (9,3);\n  \\draw[thick,fill=none] (1,-3.5) -- (5,-5.25) -- (9,-3.5);\n  \\draw[thick,gray,dashed] (0,0) -- (2.75,0.25) ;\n  \\draw[thick,gray,dashed]  (7.25,0.25) -- (10,0);\n  \\draw[thick,gray,dashed]  (2.75,0.25) -- (5,3.25) -- (7.25,0.25) -- (5,-2.75) -- cycle;\n  \\draw[thick,gray,dashed]  (5,3.25) -- (5,5);\n  \\draw[thick,gray,dashed]  (5,-2.75) -- (5,-5.25);\n\n\\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(0,0)$);\n   }\n   \\draw[fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw (a6) -- (a2);\n   \\draw (a6) -- (a3);\n   \\draw (a3) -- (a5);\n\n\n \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(1,3)$);\n   }\n   \\draw[fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw (a6) -- (a2);\n   \\draw (a6) -- (a3);\n   \\draw (a6) -- (a4);\n\n \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(2,-0.25)$);\n   }\n   \\draw[fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw (a1) -- (a3);\n   \\draw (a6) -- (a3);\n   \\draw (a6) -- (a4);\n\n \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(1,-3.5)$);\n   }\n   \\draw[fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw (a1) -- (a3);\n   \\draw (a6) -- (a3);\n   \\draw (a3) -- (a5);\n\n \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(5,5)$);\n   }\n   \\draw[fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw (a2) -- (a4);\n   \\draw (a4) -- (a6);\n   \\draw (a6) -- (a2);\n\n\\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(5,-5.25)$);\n   }\n   \\draw[fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw (a1) -- (a3);\n   \\draw (a3) -- (a5);\n   \\draw (a5) -- (a1);\n\n\\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(8,-0.25)$);\n   }\n   \\draw[fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw (a1) -- (a3);\n   \\draw (a1) -- (a4);\n   \\draw (a4) -- (a6);\n\n\\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(9,3)$);\n   }\n   \\draw[fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw (a1) -- (a4);\n   \\draw (a2) -- (a4);\n   \\draw (a4) -- (a6);\n   \n\\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(10,0)$);\n   }\n   \\draw[fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw (a1) -- (a4);\n   \\draw (a1) -- (a5);\n   \\draw (a2) -- (a4);\n   \n   \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(9,-3.5)$);\n   }\n   \\draw[fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw (a1) -- (a3);\n   \\draw (a1) -- (a4);\n   \\draw (a1) -- (a5);\n\n   \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(2.75,0.25)$);\n   }\n   \\draw[gray,fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw[gray] (a2) -- (a5);\n   \\draw[gray] (a2) -- (a6);\n   \\draw[gray] (a3) -- (a5);\n\n   \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(5,3.25)$);\n   }\n   \\draw[gray,fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw[gray] (a2) -- (a4);\n   \\draw[gray] (a2) -- (a5);\n   \\draw[gray] (a2) -- (a6);\n   \n      \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(7.25,0.25)$);\n   }\n   \\draw[gray,fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw[gray] (a1) -- (a5);\n   \\draw[gray] (a2) -- (a5);\n   \\draw[gray] (a2) -- (a4);\n   \n      \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at ($(60+60*\\i:0.32)+(5,-2.75)$);\n   }\n   \\draw[gray,fill=white]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw[gray] (a1) -- (a5);\n   \\draw[gray] (a2) -- (a5);\n   \\draw[gray] (a3) -- (a5);   \n\n \\end{tikzpicture}\n \\end{center}\n \\caption{The Stasheff polytope for ${\\rm Conf}_6(\\mathbb{P}^3) \\cong \\mathcal{M}_{0,6}$ with the clusters labelled by the different triangulations of a hexagon.}\n \\label{Stasheff_triang}\n \\end{figure}\n\nIt is important to stress again that Fig. \\ref{Stasheff} is not just a pictorial representation of a set of topological relations between clusters. If we restrict attention to the case of real twistors then ${\\rm Conf}_6(\\mathbb{RP}^3) \\cong \\mathcal{M}_{0,6}(\\mathbb{R})$ is a three-dimensional space. The interior of the polytope in Fig. \\ref{Stasheff} is precisely the region inside ${\\rm Conf}_6(\\mathbb{RP}^3)$ where all the cluster $\\mathcal{X}$-coordinates obey $0<x<\\infty$. Each corner is the origin in the set of $\\mathcal{X}$-coordinates defined by the corresponding cluster. As an example, the $\\mathcal{X}$-coordinates of the initial cluster for the ${\\rm Conf}_6(\\mathbb{P}^3)$ polytope are\n\\begin{align}\nx_1 &= \\frac{\\langle 1234 \\rangle \\langle 1256 \\rangle}{\\langle 1236 \\rangle \\langle 1245 \\rangle} = \\frac{(56)(34)}{(45)(36)}\\,, \\notag \\\\\nx_2 &= \\frac{\\langle 1235 \\rangle \\langle 1456 \\rangle}{\\langle 1256 \\rangle \\langle 1345 \\rangle} = \\frac{(46)(23)}{(34)(26)}\\,, \\notag \\\\\nx_3 &= \\frac{\\langle 1245 \\rangle \\langle 3456 \\rangle}{\\langle 2345 \\rangle \\langle 1456 \\rangle} = \\frac{(36)(12)}{(16)(23)}\\,.\n\\end{align}\nThe vertex corresponding to the initial cluster is the origin $x_1=x_2=x_3=0$ in this coordinate system. The cluster coordinates run from $0$ to $\\infty$ along the three one-dimensional edges which meet at the vertex. This is in accord with the fact that under a mutation (which corresponds to moving along an edge to an adjacent vertex) the associated $\\mathcal{X}$-coordinate inverts.\n\nThe adjacency matrix for the unfrozen nodes of the initial cluster is \n\\be\n\\label{A3bmatrix}\nb = \n\\left(\n\\begin{tabular}{ccc}\n0 & 1 & 0 \\\\\n-1 & 0 & 1 \\\\\n0 &-1&0\n\\end{tabular}\n\\right)\\,.\n\\ee\nWe recall that the $\\mathcal{X}$-coordinates are (log) canonical coordinates for the Poisson bracket. The adjacency matrix $b$ in (\\ref{A3bmatrix}) is singular and has rank two. This means that there is a coordinate $\\Delta$ which Poisson commutes with every function $\\{ \\Delta , f \\} = 0$. It is the product of two of the $x_i$ above,\n\\be\n\\label{Delta}\n\\Delta = x_1 x_3 = \\frac{(12)(34)(56)}{(16)(23)(45)} = \\frac{\\langle 1234 \\rangle \\langle 1256 \\rangle \\langle 3456 \\rangle}{\\langle 1236 \\rangle \\langle 1456 \\rangle \\langle 2345 \\rangle}\\,.\n\\ee\nEquivalently, there is a canonical (up to a constant rescaling) one-form $d \\log \\Delta$ which is null under the action of the Poisson bivector,\n\\be\nb(d\\log \\Delta , \\cdot) = 0\\,.\n\\ee\nNote that $\\Delta$ is built purely from frozen $\\mathcal{A}$-coordinates. \n\nAnother natural set of coordinates are \\emph{dihedral} coordinates \\cite{Brown:2009qja} which can be defined (here with respect to the trivial ordering $\\{1,\\ldots,n\\}$) for all the moduli spaces $\\mathcal{M}_{0,n}$ (or $A_{n-3}$ cluster algebras) via\n\\be\nu_{ij} = \\frac{(i\\,j+1)(i+1\\,j)}{(i\\,j)(i+1\\,j+1)}\\,.\n\\ee\nWe require that the labels $i$ and $j$ are separated by at least two (as for unfrozen two-brackets). For the case $n=6$ this implies that there are nine such dihedral coordinates, each labelled by the chords of the hexagon. \n\n\n\nThe interior of the Stasheff polytope is the region where all nine $u_{ij}$ obey $0< u_{ij} < 1$. A face of the polytope is the locus defined by $u_{ij}=0$ where $(ij)$ is the chord associated to that face. When a particular $u_{ij}=0$ then all the $u_{kl}$ such that the chord $(kl)$ intersects the chord $(ij)$ take the value $1$. The vertices of the polytope are then the origin in the coordinate system defined by taking the dihedral coordinates associated to the triangulation of the corresponding cluster. For example, the initial cluster is associated to the origin in the coordinates $\\{u_{26},u_{36},u_{46}\\}$ and the equations $u_{26}=0$, $u_{36}=0$ and $u_{46}=0$ define the faces labelled by $\\langle 1345 \\rangle = (26)$, $\\langle 1245 \\rangle = (36)$ and $\\langle 1235 \\rangle = (46)$ respectively in Fig. \\ref{Stasheff}. These dihedral coordinates are related to the $\\mathcal{X}$-coordinates above via\n\\begin{align}\nu_{26} = \\frac{x_3}{1+x_3}\\,, \\qquad\nu_{36} = \\frac{x_2(1+x_3)}{1+x_2+x_2 x_3}\\,, \\qquad\nu_{46} = \\frac{x_1(1+x_2+x_2 x_3)}{1+x_1+x_1x_2+x_1x_2x_3} \\,. \n\\end{align}\nFrom the above relations it is clear that the three faces meeting at the vertex are equivalently defined either by vanishing of dihedral coordinates or by vanishing of cluster $\\mathcal{X}$-coordinates.\n\nThe dihedral coordinates form a complete set of nine multiplicatively independent homogeneous combinations of the $\\mathcal{A}$-coordinates. They can therefore be taken as an alphabet for the construction of polylogarithms on ${\\rm Conf}_6(\\mathbb{P}^3) = \\mathcal{M}_{0,6}$. They are related to the nine letters taken in e.g. \\cite{Dixon:2011nj} for the construction of hexagon functions as follows,\n\\begin{eqnarray}\n\\label{hexusual}\nu = u_{26} u_{35} u_{25} u_{36}\\,, \\qquad & 1-u = u_{14}\\,, \\qquad &y_u = \\frac{u_{35}}{u_{26}}\\,,\\notag \\\\\nv = u_{13} u_{46} u_{36} u_{14}\\,, \\qquad & 1-v = u_{25}\\,, \\qquad &y_v = \\frac{u_{13}}{u_{46}}\\,,\\notag \\\\\nw = u_{24} u_{15} u_{14} u_{25}\\,, \\qquad & 1-w = u_{36}\\,, \\qquad &y_w = \\frac{u_{15}}{u_{24}}\\,.\n\\end{eqnarray}\nThe null Poisson coordinate $\\Delta$ is given by\n\\be\n\\label{Delta}\n\\Delta = \\frac{u_{24} u_{26} u_{46}}{u_{13} u_{15} u_{35}} = \\frac{1}{y_u y_v y_w}\\,.\n\\ee\n\nThere are two distinct types of codimension-one subalgebras in the $A_3$ polytope. Each pentagonal face of Fig. \\ref{Stasheff} corresponds to an $A_2$ subalgebra. For example, freezing the node labelled by $\\langle 1235 \\rangle = (46)$ in the initial cluster, and mutating the other nodes generates the pentagon of clusters around the edge of the corresponding face of the polytope. The condition $u_{46}=0$ corresponds to restricting to the pentagonal boundary. Physically, taking the limit $u_{46} \\rightarrow 0$ corresponds to taking the double scaling limit where $v\\rightarrow 0$ on the branch where $y_v \\rightarrow \\infty$. Its parity conjugate version is the limit $u_{13} \\rightarrow 0$ which corresponds to $v \\rightarrow 0$ on the branch where $y_v \\rightarrow 0$. These double-scaling limits are highlighted in red in Fig. \\ref{Stasheffhighlights}.\n\n\\begin{figure}\n{\\footnotesize\n\\begin{center}\n\\begin{tikzpicture}[scale=0.4]\n  \\draw[join=bevel,thick,fill=blue!20] (0,0) -- (1,3) -- (2,-0.25) -- (1,-3.5)  -- cycle;\n  \\draw[join=bevel,thick,fill=red!10] (2,-0.25) -- (8,-0.25) -- (9,3) -- (5,5) -- (1,3) -- cycle ;\n  \\draw[join=bevel,thick,fill=blue!20] (10,0) -- (9,3) -- (8,-0.25) -- (9,-3.5)  -- cycle;\n  \\draw[thick,fill=none] (1,3) -- (5,5) -- (9,3);\n  \\draw[join=bevel,thick,fill=red!20] (1,-3.5) -- (5,-5.25) -- (9,-3.5) -- (8,-0.25) --  (2,-0.25) -- cycle;\n   \\draw[thick,gray,dashed] (0,0) -- (2.75,0.25) ;\n  \\draw[thick,gray,dashed]  (7.25,0.25) -- (10,0);\n  \\draw[thick,gray,dashed]  (2.75,0.25) -- (5,3.25) -- (7.25,0.25) -- (5,-2.75) -- cycle;\n  \\draw[thick,gray,dashed]  (5,3.25) -- (5,5);\n  \\draw[thick,gray,dashed]  (5,-2.75) -- (5,-5.25);\n   \\draw[join=bevel,thick,fill=none] (0,0) -- (1,3) -- (2,-0.25) -- (1,-3.5)  -- cycle;\n    \\draw[join=bevel,thick,fill=none] (2,-0.25) -- (8,-0.25) -- (9,3) -- (5,5) -- (1,3) -- cycle ;\n  \\draw[join=bevel,thick,fill=none] (10,0) -- (9,3) -- (8,-0.25) -- (9,-3.5)  -- cycle;\n   \\draw[join=bevel,thick,fill=none] (1,-3.5) -- (5,-5.25) -- (9,-3.5) -- (8,-0.25) --  (2,-0.25) -- cycle;\n     \\draw[thick,fill=none] (0,0) -- (1,3) -- (5,5) -- (9,3) -- (10,0) -- (9,-3.5) -- (5,-5.25) -- (1,-3.5) -- cycle;\n\\draw[->] (-0.75,-0.125) -- (1,-0.125);\n\\node at (-2,-0.125){$u_{36}$};\n\\draw[->] (10.75,-0.125) -- (9,-0.125);\n\\node at (12,-0.125){$u_{14}$};\n\\draw[->] (5,5.75) arc (-200:-160:5.75);\n\\node at (5,6.5){$u_{46}$};\n\\draw[->] (5,-6) arc (-20:20:6.25);\n\\node at (5,-6.75){$u_{13} $};\n \\end{tikzpicture}\n \\end{center}\n }\n \\caption{The $A_3$ polytope with four faces labelled by their dihedral coordinates. The double scaling limits $u_{46}\\rightarrow0$ and its parity conjugate version $u_{13} \\rightarrow 0$ are are the highlighted red pentagons. The soft limits $u_{36} \\rightarrow 0$ and $u_{14} \\rightarrow 0$ are the blue squares. The line joining the two squares corresponds to the collinear limit $u_{13} = u_{46} = 0$.}\n \\label{Stasheffhighlights}\n \\end{figure}\n\nThe other type of codimension-one subalgebra is $A_1 \\times A_1$, corresponding to a square face, as can be obtained from freezing the node $\\langle 1245\\rangle = (36)$ in the initial cluster and mutating the others. The condition $u_{36}=0$ defines this face and taking the limit $u_{36} \\rightarrow 0 $ corresponds to taking the soft limit where $u \\rightarrow 0$, $v\\rightarrow 0$, $w \\rightarrow 1$. Note that this limit is a limit to a codimension one (i.e. dimension two) subspace. This is important because, although the soft limit itself (of the remainder function) is independent of the location approached on the face, after analytic continuation the same limit corresponds to a Regge limit which is not independent of where on the face is being approached. The remaining transverse kinematic dependence of the amplitude in the Regge limit is precisely parametrised by the two-dimensional square face. The limit $u_{36} \\rightarrow 0$ and a cyclically rotated one $u_{14} \\rightarrow 0$ are highlighted as blue squares in Fig. \\ref{Stasheffhighlights}.\n\nAdmissible pairs of unfrozen nodes are pairs of faces which intersect on the boundary, e.g. the pair $\\{\\langle 1235 \\rangle , \\langle 2456 \\rangle\\} = \\{(46),(13)\\}$ is admissible and intersects in a codimension-two (i.e. dimension-one) $A_1$ subalgebra corresponding to the shared edge of those two faces. The edge in question is defined by $u_{46} = u_{13} = 0$ and corresponds to taking the collinear limit of the hexagon amplitudes. Note that the collinear limit indeed interpolates between two soft limits corresponding to the square faces labelled by $(36)$ and $(14)$.\n\nThe pair $\\{\\langle 1245 \\rangle  , \\langle 2356 \\rangle \\} = \\{(36),(14)\\}$ on the other hand is not admissible as the corresponding faces do not intersect on the boundary of Fig. \\ref{Stasheff}. The absence of such an intersection is directly related to the Steinmann relations obeyed by scattering amplitudes, or even more basically, to the absence of overlapping factorisation poles in tree-level amplitudes. In general we can describe admissible pairs as non-intersecting chords $(ij)$ of the polygon while intersecting chords give non-admissible pairs. Frozen $\\mathcal{A}$-coordinates correspond to the edges of the polygon and therefore do not intersect any chord and hence are admissible with every other $\\mathcal{A}$-coordinate.\n\nFinally, admissible triples correspond to corners of Fig. \\ref{Stasheff}, i.e. to clusters themselves. They are codimension-three or dimension-zero subalgebras and as an example we could take the triplet $\\{ \\langle 1235 \\rangle , \\langle 1245 \\rangle , \\langle 1345 \\rangle\\}$ which defines the initial cluster.\n\nThe full space ${\\rm Conf}_6(\\mathbb{RP}^3) \\cong \\mathcal{M}_{0,6}(\\mathbb{R})$ is tiled by 60 regions identical to the Stasheff polytope of Fig. \\ref{Stasheff}. In general \\cite{Brown:2009qja}, the moduli spaces $\\mathcal{M}_{0,n}(\\mathbb{R})$ are tiled by $n!\/(2n)$ regions which are $(n-3)$-dimensional polytopes, each corresponding to a choice of dihedral structure (i.e. an ordering modulo cyclic  transformations and reflections) on the $n$ points in $\\mathbb{RP}^1$. \n\nEach vertex of the polytope provides a natural base point for the contour of integration over which a symbol made of homogeneous combinations of the $\\mathcal{A}$-coordinates can be iteratively integrated to produce a polylogarithmic function \\cite{Brown:2009qja}. \n\n\n\\subsection{Heptagons and the $E_6$ polytope}\n\nFor $\\mathrm{Gr}(4,7)$, the initial cluster is represented by the quiver diagram of Fig. \\ref{heptinitial}. Each cluster contains six unfrozen nodes as well as the seven frozen ones labelled by the adjacent four-brackets $\\langle i\\, i+1\\,i+2\\,i+3\\rangle$. Repeated mutation generates a total of 833 distinct clusters containing a total of 42 distinct unfrozen $\\mathcal{A}$-coordinates in addition to the 7 frozen ones. \n\n\\begin{figure}\n{\\footnotesize\n\\begin{center}\n\\makeatletter  \n\\newcommand{\\phantombox}[1]{%\n  \\setbox0=\\hbox{#1}%\n \n  \\begin{tcolorbox}[colframe=white,colback=white,boxrule=0.4pt,\n    left=2pt,right=2pt,top=3pt,bottom=3pt,boxsep=0pt,width=1.2cm, valign = center,  halign=center, sharp corners = all]\n    #1\n  \\end{tcolorbox}\n}\n\\newcommand{\\frozenbox}[1]{%\n  \\setbox0=\\hbox{#1}%\n  \\begin{tcolorbox}[colframe=black,colback=white,boxrule=0.5pt,\n      left=2pt,right=2pt,top=2pt,bottom=2pt,boxsep=0pt,width=1.2cm, halign=center, sharp corners = all]\n    #1\n  \\end{tcolorbox}\n}\n\\makeatother\n\\begin{tikzpicture}%\n    [\n    unfrozen\/.style={},\n    frozen\/.style={inner sep=1.2mm,outer sep=0mm,yshift=0},\n    node distance = 0.5cm\n    ]\n    \\node[frozen]        (f0) at (0,5) {$\\frozenbox{$\\langle 1234 \\rangle$}$};\n    \\node[frozen, below right = of f0]        (t1)  {$\\phantombox{$\\langle1235 \\rangle$}$};\n    \\node[frozen, right = of t1]        (t2)  {$\\phantombox{$\\langle1236 \\rangle$}$};\n    %\n    \\node[frozen, below = of t1]        (m1)  {$\\phantombox{$\\langle1245 \\rangle$}$};\n    \\node[frozen, below = of t2]        (m2)  {$\\phantombox{$\\langle1256 \\rangle$}$};\n    %\n    \\node[frozen, below = of m1]        (b1)  {$\\phantombox{$\\langle1345 \\rangle$}$};\n    \\node[frozen, below = of m2]        (b2)  {$\\phantombox{$\\langle1456 \\rangle$}$};\n    %\n    \\node[frozen, right = of t2]        (f1)  {$\\frozenbox{$\\langle1237 \\rangle$}$};\n    \\node[frozen, right = of m2]        (f2)  {$\\frozenbox{$\\langle1267 \\rangle$}$};\n    \\node[frozen, right = of b2]        (f3)  {$\\frozenbox{$\\langle1567 \\rangle$}$};\n    %\n    \\node[frozen, below = of b1]        (f4)  {$\\frozenbox{$\\langle2345 \\rangle$}$};\n    \\node[frozen, below = of b2]        (f5)  {$\\frozenbox{$\\langle3456 \\rangle$}$};\n    \\node[frozen, right= of f5]         (f6)  {$\\frozenbox{$\\langle4567 \\rangle$}$};\n    \n    \\draw[->] (f0) -- (t1);\n    \\draw[->] (t1) -- (t2);    \\draw[->] (t1) -- (m1) ;    \\draw[->] (t2) -- (f1) ;  \\draw[->] (t2) -- (m2) ;\n    \\draw[->] (m1) -- (m2);  \\draw[->] (m1) -- (b1) ; \\draw[->] (m2) -- (t1);      \\draw[->] (m2) -- (f2) ;  \\draw[->] (m2) -- (b2) ; \\draw[->] (f2) -- (t2);\n    \\draw[->] (b1) -- (b2);  \\draw[->] (b1) -- (f4) ; \\draw[->] (b2) -- (m1);      \\draw[->] (b2) -- (f3) ;  \\draw[->] (b2) -- (f5) ; \\draw[->] (f3) -- (m2);\n    \\draw[->] (f6) -- (b2);\n    \\draw[->] (f5) -- (b1);\n\\end{tikzpicture}\n\\end{center}}\n\\caption{The initial cluster of the ${\\rm Conf}_7(\\mathbb{P}^3)$ cluster algebra, relevant for heptagon amplitudes.}\n\\label{heptinitial}\n\\end{figure}\n\n\nA useful feature of cases of ${\\rm Gr}(k,n)$ where the pair $(k,n)$ is coprime (such as the heptagon case) is that one may use the frozen $\\mathcal{A}$-coordinates to render the unfrozen ones homogeneous \\cite{Drummond:2014ffa}. In this way one can make a natural set of 42 homogeneous letters labelled in one-to-one correspondence with the 42 unfrozen $\\mathcal{A}$-coordinates. They are given by the following six quantities together with their cyclic rotations,\n\\begin{equation}\n\\begin{aligned}[b]\n  a_{11} &= \\frac{\\langle 1234\\rangle\\langle1567\\rangle\\langle2367\\rangle}{\\langle1237\\rangle\\langle1267\\rangle\\langle3456\\rangle}\\\\\n  a_{31} &= \\frac{\\langle1567\\rangle\\langle2347\\rangle}{\\langle1237\\rangle\\langle4567\\rangle}\\\\\n  a_{51} &= \\frac{\\langle1(23)(45)(67)\\rangle}{\\langle1234\\rangle\\langle1567\\rangle}\n\\end{aligned}\n\\,\\,\\,\n\\begin{aligned}[b]\n  a_{21} &= \\frac{\\langle1234\\rangle\\langle2567\\rangle}{\\langle1267\\rangle\\langle2345\\rangle}\\\\\n  a_{41} &= \\frac{\\langle2457\\rangle\\langle3456\\rangle}{\\langle2345\\rangle\\langle4567\\rangle}  \\\\\n  a_{61} &= \\frac{\\langle1(34)(56)(72)\\rangle}{\\langle1234\\rangle\\langle1567\\rangle}\\,,\n\\end{aligned}\\,\n\\label{heptletters}\n\\end{equation}\nHere we use the notation \n\\be\n\\langle 1 (23) (45) (67) \\rangle = \\langle 1234 \\rangle \\langle 5671 \\rangle - \\langle 1235 \\rangle \\langle 4671\\rangle\\,.\n\\ee\nBy labelling the nodes of the quiver diagram with the homogenised $\\mathcal{A}$-coordinates, the initial cluster can be illustrated as in Fig. \\ref{heptinitialhom}.\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}%\n    [\n    unfrozen\/.style={},\n    frozen\/.style={inner sep=1.5mm,outer sep=0mm,yshift=0},\n    node distance = 0.6cm\n    ]\n    \\node[frozen]  (t1)  at (1,4) {$a_{24}$};\n    \\node[frozen, right = of t1]        (t2)  {$a_{37}$};\n    %\n    \\node[frozen, below = of t1]        (m1)  {$a_{13}$};\n    \\node[frozen, below = of t2]        (m2)  {$a_{17}$};\n    %\n    \\node[frozen, below = of m1]        (b1)  {$a_{32}$};\n    \\node[frozen, below = of m2]        (b2)  {$a_{27}$};\n    \n    \\draw[->] (t1) -- (t2); \\draw[->] (t1) -- (m1); \\draw[->] (t2) -- (m2);\n    \\draw[->] (m1) -- (m2); \\draw[->] (m1) -- (b1); \\draw[->] (m2) -- (t1); \\draw[->] (m2) -- (b2) ;\n    \\draw[->] (b1) -- (b2); \\draw[->] (b2) -- (m1);      \n\n  \\end{tikzpicture}\\,.\n\\end{center}\n\\caption{The initial cluster for ${\\rm Conf}_7(\\mathbb{P}^3)$ labelled by homogenised $\\mathcal{A}$-coordinates.}\n\\label{heptinitialhom}\n\\end{figure}\n\n\n\nJust as in the hexagon case we should try to visualise the 833 clusters being connected together in a polytope (the $E_6$ polytope). The polytope is a six-dimensional space with 42 codimension one (i.e dimension five) boundary faces, corresponding to the 42 unfrozen $\\mathcal{A}$-coordinates. Considering the dimension and the number of vertices it is not as visually instructive to plot the full polytope as a graph. Nevertheless similar general features are present as in the hexagon case. \n\nTo illustrate the structure of possible subalgebras it is helpful to bring the initial cluster to a cluster with the topology of an $E_6$ Dynkin diagram by a sequence of mutations as shown in Fig. \\ref{howtonshep}.\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.8]\n\n\n\\node[] (a51) at (-2, 0) {$a_{51}$};\n\\node[] (a24) at (-4, 0)   {$a_{24}$};          \n\\node[] (a62) at (0, 0)    {$a_{62}$};         \n\\node[] (a41) at (2, 0)  {$a_{41}$};         \n\\node[] (a33) at (4, 0)    {$a_{33}$};          \n\\node[] (a13) at (0, 2)    {$a_{13}$};          \n\n\\draw[norm] (a33) -- (a41);\n\\draw[norm] (a41) -- (a62) ;\n\\draw[norm] (a62) -- (a13) ;\n\\draw[norm] (a24) -- (a51) ;\n\\draw[norm] (a51) -- (a62) ;\n\n\n\n\\end{tikzpicture}\n\n\n\\end{center}\n\\caption{The initial cluster of ${\\rm Conf}_7(\\mathbb{P}^3)$ does not have the\n  topology of an $E_6$ Dynkin diagram but it is possible to mutate it\n  to one which does. This cluster contains homogenised $\\mathcal{A}$-coordinates of all six types given in  (\\ref{heptletters}). \n \n  }\n\\label{howtonshep}\n\\end{figure}\nA helpful feature of the $E_6$-shaped cluster is that its homogenised $\\mathcal{A}$-coordinates contain one representative of each of the six cyclically related classes given in eq. (\\ref{heptletters}). The codimension-one subalgebras obtained by freezing any given letter are then obvious. Freezing $a_{13}$ and mutating on the other nodes generates an $A_5$ subalgebra. Freezing $a_{25}$ or $a_{33}$ will generate a $D_5$ subalgebra. Freezing $a_{41}$ or $a_{51}$ generates an $A_4 \\times A_1$ subalgebra. Finally freezing $a_{62}$ generates an $A_2 \\times A_2 \\times A_1$ subalgebra. The $E_6$-shaped cluster is special in this regard. For example, the initial cluster contains only $a_{1i}$, $a_{2i}$ and $a_{3i}$ types of coordinates and therefore is at the intersection only of $D_5$ and $A_5$ type subalgebras.\n\nAdmissible pairs in the $E_6$ case correspond to codimension two subalgebras, i.e. dimension four subalgebras. For example the admissible pair $\\{ a_{13}, a_{62} \\}$ corresponds to an $A_2 \\times A_2$ subalgebra while the pair $\\{ a_{51} , a_{41} \\}$ corresponds to an $A_2 \\times A_1 \\times A_1$ subalgebra. Admissible triplets correspond to dimension three subalgebras and so on.\n\nEach cluster (or dimension zero subalgbera) corresponds to a vertex on the boundary of the $E_6$ polytope and the six associated cluster $\\mathcal{X}$-coordinates define a local coordinate system such that the vertex is the origin. Once again the $\\mathcal{X}$-coordinates can be associated to the one-dimensional edges of the polytope and the interior of the polytope is the region where all $\\mathcal{X}$-coordinates obey $0<x<\\infty$. The six $\\mathcal{X}$-coordinates for the $E_6$-shaped cluster are shown in Fig. \\ref{E6xcoords}. The five-dimensional face corresponding to the $A_5$ subalgebra is the boundary component defined by $x_6 =0$ with all other $x_i$ obeying $0<x_i<\\infty$. The condition $x_4 = 0$ defines a face corresponding to an $A_4 \\times A_1$ subalgebra and so on.\n\nAs in the $A_3$ case we may define another set of coordinates $u_{ij}$ such that the $u_{ij}=0$ defines the codimension one face labelled by $a_{ij}$. In terms of the cluster $\\mathcal{X}$-coordinates of the $E_6$ shaped cluster we have the following six face coordinates\\footnote{Such variables have already been derived by Arkani-Hamed and collaborators \\cite{Nimaslides} for finite cluster algebras from a different perspective. Here we obtain them from the cluster $\\mathcal{X}$-coordinates. We would like to thank Nima Arkani-Hamed for discussions of this point.},\n\\begin{align}\n\\label{usfromxs}\nu_{13} &= \\frac{x_1}{1+x_1}         &  u_{62} &= \\frac{x_6(1+x_1)}{1+x_6+x_1 x_6}             \\\\\nu_{51} &= \\frac{x_5(1+x_6+x_1x_6)}{1+x_5+x_5x_6+x_1x_5x_6}         &   u_{41} &=\\frac{x_4(1+x_6+x_1x_6)}{1+x_4+x_4x_6+x_1x_4x_6}  \\notag \\\\\nu_{24} &= \\frac{x_2(1+x_5+x_5x_6+x_1x_5x_6)}{1+x_2+x_2x_5+x_2x_5x_6+x_1x_2x_5x_6}   &  u_{33}&= \\frac{x_3(1+x_4+x_4x_6+x_1x_4x_6)}{1+x_3+x_3x_4+x_3x_4x_6+x_1x_3x_4x_6} \\,.\\notag\n\\end{align}\nAgain the origin in the cluster $\\mathcal{X}$-coordinates coincides with the origin in the face coordinates.\nIn terms of the homogenised $\\mathcal{A}$-coordinates we have\n\\begin{align}\nu_{13} &= \\frac{a_{62}}{a_{11}a_{13}}         &  u_{62} &= \\frac{a_{11}a_{41}a_{51}}{a_{62}a_{67}}         \\notag    \\\\\nu_{51} &= \\frac{a_{24}a_{67}}{a_{46}a_{51}}         &   u_{41} &=\\frac{a_{33}a_{67}}{a_{41}a_{56}}  \\notag \\\\\nu_{24} &= \\frac{a_{46}}{a_{24}a_{31}}   &  u_{33}&= \\frac{a_{56}}{a_{22}a_{33}}\\,.\n\\label{usfromas}\n\\end{align}\nAgain we clearly have $0<u_{ij}<1$ in the interior of the polytope from (\\ref{usfromxs}).\nFrom the equations (\\ref{usfromas}) and cyclically related equations one can define a complete set of 42 homogeneous coordinates $u_{ij}$ which makes an alternative multiplicatively independent set to the $a_{ij}$. The variables $u_{ij}$ have the property that $u_{ij}=0$ implies $u_{kl}=1$ if the face labelled by $a_{kl}$ is not adjacent to the face labelled by $a_{ij}$. In other words, setting one $u_{ij}$ to zero for a given face means that all the $u_{kl}$ corresponding to non-adjacent faces go to 1.\n\nJust as in the $A_3$ case there are specific sequences of mutations which generate a cyclic transformation of the $\\mathcal{A}$-coordinates in a given cluster. Rather than describe it here for $E_6$ we give a general discussion for ${\\rm Conf}_n(\\mathbb{P}^{k-1})$ in the next section.\n\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.7]\n\n\\node[] (x51) at (-2, 0) {$x_5$};\n\\node[] (x24) at (-4, 0)   {$x_2$};          \n\\node[] (x62) at (0, 0)    {$x_6$};         \n\\node[] (x41) at (2, 0)  {$x_4$};         \n\\node[] (x33) at (4, 0)    {$x_3$};          \n\\node[] (x13) at (0, 2)    {{$x_1$}};          \n\\node[] (a51) at (9, 0) {$\\frac{a_{24}}{a_{62}} $};\n\\node[] (a24) at (7, 0)   {$\\frac{1}{a_{51}}$};          \n\\node[] (a62) at (11, 0)    {$\\frac{a_{41} a_{51}}{a_{13}}$};         \n\\node[] (a41) at (13, 0)  {$\\frac{a_{33}}{a_{62}}$};         \n\\node[] (a33) at (15, 0)    {$\\frac{1}{a_{41}}$};          \n\\node[] (a13) at (11, 2)    {{\\footnotesize $a_{62}$}};          \n\\node[] (eq) at (5.5, 1)    {{$=$}};\n\\draw[norm] (a33) -- (a41);\n\\draw[norm] (a41) -- (a62) ;\n\\draw[norm] (a62) -- (a13) ;\n\\draw[norm] (a24) -- (a51) ;\n\\draw[norm] (a51) -- (a62) ;\n\\draw[norm] (x33) -- (x41);\n\\draw[norm] (x41) -- (x62) ;\n\\draw[norm] (x62) -- (x13) ;\n\\draw[norm] (x24) -- (x51) ;\n\\draw[norm] (x51) -- (x62) ;\n\n\n\\end{tikzpicture}\n\n\n\\end{center}\n\\caption{The $E_6$-shaped cluster with $\\mathcal{X}$-coordinates shown at each of the nodes.}\n\\label{E6xcoords}\n\\end{figure}\n\n\n\n\n\\subsection{General cyclic mutations for $n>7$}\n\\label{gencyc}\n\nFor $n>7$, the ${\\rm Conf}_n(\\mathbb{P}^3)$ cluster algebra is infinite. We can still define a positive region where all $\\mathcal{X}$-coordinates are positive but the structure of its boundary is much less clear. We can still, however, understand certain finite aspects of these infinite algebras. For instance we can mutate from the initial cluster in Fig. \\ref{Gr4ninitial} to another one in which all the $\\mathcal{A}$-coordinate labels have been rotated by one unit. We do this by mutating in a manner that mirrors building Young tableaux, instead building from the bottom-left to the top-right (as opposed from top-left to bottom-right) as demonstrated in Fig. \\ref{fig:YoungTableaux}.\n\\begin{figure}\n\\begin{center}\n\\begin{ytableau}\n\\none[\\cdot] & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\none[\\cdot] & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\cdot & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\end{ytableau}\n\\qquad\n\\begin{ytableau}\n\\none[\\cdot] & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\none[\\cdot] & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\cdot & \\cdot & \\none[\\cdot] \\\\\n\\end{ytableau}\n\\qquad\n\\begin{ytableau}\n\\none[\\cdot] & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\cdot & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\cdot & \\cdot & \\none[\\cdot] \\\\\n\\end{ytableau}\n\\qquad\n\\begin{ytableau}\n\\none[\\cdot] & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\cdot & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\cdot & \\cdot & \\cdot \\\\\n\\end{ytableau}\n\\qquad\n\\begin{ytableau}\n\\cdot & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\cdot & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\cdot & \\cdot & \\cdot \\\\\n\\end{ytableau}\n\\\\ \\vspace{3mm}\n\\begin{ytableau}\n\\cdot & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\cdot & \\cdot & \\none[\\cdot] \\\\\n\\cdot & \\cdot & \\cdot \\\\\n\\end{ytableau}\n\\qquad\n\\begin{ytableau}\n\\cdot & \\none[\\cdot] & \\none[\\cdot] \\\\\n\\cdot & \\cdot & \\cdot \\\\\n\\cdot & \\cdot & \\cdot \\\\\n\\end{ytableau}\n\\qquad\n\\begin{ytableau}\n\\cdot & \\cdot & \\none[\\cdot] \\\\\n\\cdot & \\cdot & \\cdot \\\\\n\\cdot & \\cdot & \\cdot \\\\\n\\end{ytableau}\n\\qquad\n\\begin{ytableau}\n\\cdot & \\cdot & \\cdot \\\\\n\\cdot & \\cdot & \\cdot \\\\\n\\cdot & \\cdot & \\cdot \\\\\n\\end{ytableau}\n\\end{center}\n\\caption{A series of mutations which result in a rotation of the Gr$(4,8)$ initial cluster by one unit. The dots represent unfrozen nodes (arrows have been removed for clarity) and the squares represent the mutated nodes. Note there are no gaps between mutated nodes and we always mutate from the bottom up and from left to right. }\n\\label{fig:YoungTableaux}\n\\end{figure}\n\nWe can use this method to rotate initial-type sub-algebras within a cluster in order to search for clusters with specific Pl\\\"uckers. In fact we will use this method later to prove that all R-invariants are cluster adjacent. An example is given in Fig. \\ref{cycexample}.\n\\begin{figure}\n\t\\begin{subfigure}{0.4\\textwidth}\n\t\t\\begin{tikzpicture}[scale=0.75]\n\n\t\n\t\t\\node[frozen] (123) at (-2, 1.25) {$\\ab{123}$};\n\t\t\\node (124) at (0, 0) {$\\ab{124}$};\n\t\t\\node (125) at (2, 0) {$\\ab{125}$};\n\t\t\\node (126) at (4,0) {$\\ab{126}$};\n\t\t\\node[frozen] (127) at (6,0) {$\\ab{127}$};\n\t\t\\node (134) at (0,-1.25) {$\\ab{134}$};\n\t\t\\node (145) at (2,-1.25) {$\\ab{145}$};\n\t\t\\node (156) at (4, -1.25) {$\\ab{156}$};\n\t\t\\node[frozen] (167) at (6,-1.25) {$\\ab{167}$};\n\t\t\\node[frozen] (234) at (0,-2.5) {$\\ab{234}$};\n\t\t\\node[frozen] (345) at (2,-2.5) {$\\ab{345}$};\n\t\t\\node[frozen] (456) at (4, -2.5) {$\\ab{456}$};\n\t\t\\node[frozen] (567) at (6, -2.5) {$\\ab{567}$};\n\n\t\n\t\t\\draw[->,shorten <=4pt, shorten >=3pt] (123.south east) -- (124.north \t\twest);\n\t\t\\draw[norm] (124) -- (125);\n\t\t\\draw[norm] (125) -- (126);\n\t\t\\draw[norm] (126) -- (127);\n\t\t\\draw[norm] (124) -- (134);\n\t\t\\draw[norm] (125) -- (145);\n\t\t\\draw[norm] (126) -- (156);\n\t\t\\draw[norm] (134) -- (145);\n\t\t\\draw[norm] (145) -- (156);\n\t\t\\draw[norm] (156) -- (167);\n\t\t\\draw[norm] (134) -- (234);\n\t\t\\draw[norm] (145) -- (345);\n\t\t\\draw[norm] (156) -- (456);\n\t\t\\draw[diag] (145.north west) -- (124.south east);\n\t\t\\draw[diag] (156.north west) -- (125.south east);\n\t\t\\draw[diag] (167.north west) -- (126.south east);\n\t\t\\draw[diag] (345.north west) -- (134.south east);\n\t\t\\draw[diag] (456.north west) -- (145.south east);\n\t\t\\draw[diag] (567.north west) -- (156.south east);\n\n\t\t\\end{tikzpicture}\n\t\\end{subfigure}\n\\hspace*{1.5cm}\n\\begin{subfigure}{0.4\\textwidth}\n\t\t\\begin{tikzpicture}[scale=0.75]\n\n\t\n\t\t\\node[frozen] (234) at (-2, 1.25) {$\\ab{234}$};\n\t\t\\node[frozen] (123) at (2, 1.25) {$\\ab{123}$};\n\t\t\\node (235) at (0, 0) {$\\ab{235}$};\n\t\t\\node (236) at (2, 0) {$\\ab{236}$};\n\t\t\\node (126) at (4,0) {$\\ab{126}$};\n\t\t\\node[frozen] (127) at (6,0) {$\\ab{127}$};\n\t\t\\node (245) at (0,-1.25) {$\\ab{245}$};\n\t\t\\node (256) at (2,-1.25) {$\\ab{256}$};\n\t\t\\node (156) at (4, -1.25) {$\\ab{156}$};\n\t\t\\node[frozen] (167) at (6,-1.25) {$\\ab{167}$};\n\t\t\\node[frozen] (345) at (0,-2.5) {$\\ab{345}$};\n\t\t\\node[frozen] (456) at (2,-2.5) {$\\ab{456}$};\n\t\t\\node[frozen] (567) at (6, -2.5) {$\\ab{567}$};\n\n\t\n\t\t\\begin{scope}[blend mode=overlay,overlay]\n          \\node[rectangle,fill=green!20, rounded corners, fit=(235)(236)(245)(256), inner sep=0.3pt] {};\n        \\end{scope}\n\t\t\\node (235) at (0, 0) {$\\ab{235}$};\n\t\t\\node (236) at (2, 0) {$\\ab{236}$};\n\t\t\\node (126) at (4,0) {$\\ab{126}$};\n\t\t\\node[frozen] (127) at (6,0) {$\\ab{127}$};\n\t\t\\node (245) at (0,-1.25) {$\\ab{245}$};\n\t\t\\node (256) at (2,-1.25) {$\\ab{256}$};\n\t\t\\node (156) at (4, -1.25) {$\\ab{156}$};\n\n\t\n\t\t\\draw[->,shorten <=4pt, shorten >=3pt] (234.south east) -- (235.north west);\n\t\t\\draw[norm] (235) -- (236);\n\t\t\\draw[norm] (126) -- (236);\n\t\t\\draw[norm] (126) -- (127);\n\t\t\\draw[norm] (235) -- (245);\n\t\t\\draw[norm] (236) -- (256);\n\t\t\\draw[norm] (236) -- (123);\n\t\t\\draw[norm] (126) -- (156);\n\t\t\\draw[norm] (245) -- (256);\n\t\t\\draw[norm] (156) -- (256);\n\t\t\\draw[norm] (156) -- (167);\n\t\t\\draw[norm] (245) -- (345);\n\t\t\\draw[norm] (256) -- (456);\n\t\t\\draw[diag] (123.south east) -- (126.north west);\n\t\t\\draw[diag] (256.north west) -- (235.south east);\n\t\t\\draw[diag] (167.north west) -- (126.south east);\n\t\t\\draw[diag] (456.north west) -- (245.south east);\n\t\t\\draw[diag] (567.north west) -- (156.south east);\n\t\t\\draw[diag] (256.north east) -- (126.south west);\n\n\t\t\\end{tikzpicture}\n\t\\end{subfigure}\n\\caption{The ${\\rm Conf}_7(\\mathbb{P}^2)$ initial cluster (left) and the cluster resulting from a cyclic mutation of a ${\\rm Conf}_6(\\mathbb{P}^2)$ subalgebra, highlighted in green (right). ${\\rm Conf}_7(\\mathbb{P}^2) \\sim {\\rm Conf}_7(\\mathbb{P}^3)$ but we have given this example to demonstrate this procedure is valid for ${\\rm Conf}_n(\\mathbb{P}^{k-1}) \\text{ } \\forall \\text{ } k,n$.}\n\\label{cycexample}\n\\end{figure}\nAs we can see, the ${\\rm Conf}_6(\\mathbb{P}^2)$ sub-topology remains unchanged but the labels have all been rotated by one unit. The other nodes have rearranged themselves such that the frozen nodes connected to the sub-algebra have shifted round the cluster. Mutating on $\\ab{156}$ followed by $\\ab{126}$ will result in the same topology as the left cluster but with each label rotated by one unit. We can repeat this process any number of times to achieve the desired number of rotations.\n\n\\section{Cluster adjacent polylogarithms}\n\nIn \\cite{Drummond:2017ssj} the notion of cluster adjacency for symbols\nand polylogarithms was introduced. It extends the role\nof cluster algebras in describing the analytic structure of the\nscattering amplitudes, at least in the hexagon and heptagon cases for planar $\\mathcal{N}=4$ super Yang-Mills theory. The\nstructure of the cluster algebra restricts the way given\n$\\mathcal{A}$-coordinates may appear next to each other in the\nsymbol of appropriately defined IR finite quantities. In particular,\n\\emph{for two $\\mathcal{A}$-coordinates to appear next\nto each other in the symbol they must appear together in some cluster}.\nIn other words they must either be a repeat of the same $\\mathcal{A}$-coordinate or be an admissible pair.\n\n\nThe property of cluster adjacency is closely related to the Steinmann relations whose role in constraining the analytic structure of scattering amplitudes was stressed in \\cite{Bartels:2008ce}. In \\cite{Caron-Huot:2016owq} it was realised that the Steinmann relations were employed to greatly increase the power of the hexagon bootstrap programme and in \\cite{Dixon:2016nkn} the same conditions were extended to the heptagon case. In fact the Steinmann conditions can be extended to hold on all adjacent pairs in the symbol \\cite{DP,Yorgosslides}, not only in the first two entries. The cluster adjacency property outlined above implies the Steinmann conditions, including the extended ones. In the hexagon (or $A_3$) case this is simply the statement that the square faces of the associahedron in Fig. \\ref{Stasheff} are not adjacent to each other. In the heptagon ($E_6$) case it follows from the fact that the face labelled by $a_{11}$ only intersects those labelled by $a_{14}$ and $a_{15}$ but not those labelled by the other $a_{1i}$. What is less obvious but nevertheless appears to hold for the hexagon and heptagon symbols is that the extended Steinmann relations \\emph{together with the physical initial entry conditions} actually imply cluster adjacency.\n\nNote that the property of cluster adjacency is described in terms of the inhomogeneous $\\mathcal{A}$-coordinates. The polylogarithms describing the known dual conformal invariant amplitudes are functions on the space ${\\rm Conf}_n(\\mathbb{P}^3)$ and their symbols are normally described in terms of homogeneous multiplicative combinations of $\\mathcal{A}$-coordinates. Such combinations can be expanded out into non-manifestly homogeneous combinations by the identities (\\ref{symmult}) and (\\ref{sympower}). The resulting expressions are the ones which obey the adjacency criterion. \n\nIn the heptagon case we may take the homogenised $\\mathcal{A}$-coordinates (\\ref{heptletters}) as our symbol alphabet and the statement of adjacency becomes very direct. In the hexagon case this is not possible, essentially due to the existence of the purely frozen homogeneous combination $\\Delta$ defined eq. (\\ref{Delta}).\n\nIn general, beyond the hexagon and heptagon amplitudes we discuss here, we expect a number of new features whose interplay with cluster adjacency is not yet clear. Firstly there will exist algebraic symbol letters with square roots which are not immediately related to $\\mathcal{A}$-coordinates which are all polynomials in the Pl\\\"ucker coordinates. These already appear in the N${}^2$MHV octagon at one loop in the four-mass box contributions. Moreover at high enough multiplicity and loop order there will appear non-polylogarithmic functions, e.g. in the ten-point N${}^3$MHV amplitude at two loops \\cite{CaronHuot:2012ab}. Nevertheless we believe that some suitably extended notion of cluster adjacency will also hold beyond the hexagon and heptagon amplitudes.\n\n\\subsection{Neighbour sets}\n\\label{sec:neighbour-sets}\n\nWe define the \\emph{neighbour set} $\\ns{a}$ of a given ${\\cal A}$-coordinate $a$ as the set of $\\mathcal{A}$-coordinates $b$ such that $\\{a,b\\}$ form an admissible pair together with $a$ itself. This set automatically includes all the frozen $\\mathcal{A}$-coordinates. In terms of the polytope the unfrozen nodes in the neighbour set correspond to all faces that share a codimension-two boundary with the face labelled by $a$ (i.e. are adjacent to $a$) together with the face labelled by $a$ itself. One way of systematically\nconstructing neighbour sets is to go to a convenient cluster and freeze the ${\\cal A}$-coordinate whose neighbour set is being considered. The neighbour set then consists of all unfrozen ${\\cal A}$-coordinates generated in this codimension-one subalgebra, the frozen\ncoordinates and the coordinate $a$ itself. This is demonstrated in Figure\n\\ref{howtonshex}. Note that the notion of a neighbour set depends on\nthe cluster algebra in question, as well as the choice of $\\mathcal{A}$-coordinate $a$. \n\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}\n\n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node[blue] (1235) at (0, 0) {$\\ab{1235}$};\n\\node[frozenblue] (1236) at (2.25, 0) {$\\ab{1236}$};\n\\node (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node[frozenblue] (1256) at (2.25,-1.25) {$\\ab{1256}$};\n\\node (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node[frozenblue] (1456) at (2.25,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (0,-3.75) {$\\ab{2345}$};\n\\node[frozenblue] (3456) at (2.25,-3.75) {$\\ab{3456}$};\n\\begin{scope}[blend mode=overlay,overlay]\n          \\node[rectangle, rounded corners, fit=(1245)(1345),fill=red!20, inner sep=0pt] {};\n        \\end{scope}\n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node[blue] (1235) at (0, 0) {$\\ab{1235}$};\n\\node[frozenblue] (1236) at (2.25, 0) {$\\ab{1236}$};\n\\node (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node[frozenblue] (1256) at (2.25,-1.25) {$\\ab{1256}$};\n\\node (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node[frozenblue] (1456) at (2.25,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (0,-3.75) {$\\ab{2345}$};\n\\node[frozenblue] (3456) at (2.25,-3.75) {$\\ab{3456}$};\n\n\n\\draw[->,shorten <=4pt, shorten >=3pt] (1234.south east) -- (1235.north west);\n\\draw[norm] (1235) -- (1236);\n\\draw[norm] (1245) -- (1256);\n\\draw[norm] (1345) -- (1456);\n\\draw[norm] (1235) -- (1245);\n\\draw[norm] (1245) -- (1345);\n\\draw[norm] (1345) -- (2345);\n\\draw[diag] (1256.north west) -- (1235.south east);\n\\draw[diag] (1456.north west) -- (1245.south east);\n\\draw[diag] (3456.north west) -- (1345.south east);\n\n\\end{tikzpicture}\n\\qquad\\qquad\n\\begin{tikzpicture}\n\n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node[] (1235) at (0, 0) {$\\ab{1235}$};\n\\node[frozenblue] (1236) at (2.25, 0) {$\\ab{1236}$};\n\\node[blue] (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node[frozenblue] (1256) at (2.25,-1.25) {$\\ab{1256}$};\n\\node (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node[frozenblue] (1456) at (2.25,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (0,-3.75) {$\\ab{2345}$};\n\\node[frozenblue] (3456) at (2.25,-3.75) {$\\ab{3456}$};\n\n\\begin{scope}[blend mode=overlay,overlay]\n          \\node[rectangle, rounded corners, fit=(1235),fill=red!20, inner sep=0pt] {};\n        \\end{scope}\n\\begin{scope}[blend mode=overlay,overlay]\n          \\node[rectangle, rounded corners, fit=(1345),fill=red!20, inner sep=0pt] {};\n        \\end{scope}\n\n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node[] (1235) at (0, 0) {$\\ab{1235}$};\n\\node[frozenblue] (1236) at (2.25, 0) {$\\ab{1236}$};\n\\node[blue] (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node[frozenblue] (1256) at (2.25,-1.25) {$\\ab{1256}$};\n\\node (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node[frozenblue] (1456) at (2.25,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (0,-3.75) {$\\ab{2345}$};\n\\node[frozenblue] (3456) at (2.25,-3.75) {$\\ab{3456}$};\n\n\\draw[->,shorten <=4pt, shorten >=3pt] (1234.south east) -- (1235.north west);\n\\draw[norm] (1235) -- (1236);\n\\draw[norm] (1245) -- (1256);\n\\draw[norm] (1345) -- (1456);\n\\draw[norm] (1235) -- (1245);\n\\draw[norm] (1245) -- (1345);\n\\draw[norm] (1345) -- (2345);\n\\draw[diag] (1256.north west) -- (1235.south east);\n\\draw[diag] (1456.north west) -- (1245.south east);\n\\draw[diag] (3456.north west) -- (1345.south east);\n\n\\end{tikzpicture}\n\n\n\\end{center}\n\\caption{The initial cluster of ${\\rm Conf}_6(\\mathbb{P}^3)$ has the topology of\n  an $A_3$ Dynkin diagram. Freezing $\\ab{1235}=(46)$ results in a\n  $A_2$ subalgebra whereas freezing $\\ab{1245}=(36)$ results in a\n  $A_1 \\times A_1$ subalgebra. These subalgebras generate the letters\n  in $\\ns{\\ab{1235}}$ and $\\ns{\\ab{1245}}$, respectively.}\n\\label{howtonshex}\n\\end{figure}\n\nThrough this procedure we find the following neighbour sets for the unfrozen \nhexagon $\\mathcal{A}$-coordinates:\n\\begin{equation}\n  \\begin{aligned}[t]\n    \\ns{\\ab{1235}} &= \\{\\ab{1235}, \\ab{2456}, \\ab{2356}, \\ab{1356}, \\ab{1345}, \\ab{1245}, \\,\\text{\\& frozen coordinates.}\\}\\\\\n    \\ns{\\ab{1245}} &= \\{\\ab{1245}, \\ab{2456}, \\ab{1345}, \\ab{1246},  \\ab{1235},\\, \\text{\\& frozen coordinates.}\\}\\,.\\\\\n\\end{aligned}\n\\end{equation}\nAs stated above, apart from $a$ itself, the unfrozen elements of the neighbour set of $a$ are associated with the faces of the Stasheff\npolytope which neighbour the face associated with $a$. The edges\nwhere these faces intersect correspond to the remaining $A_1$ algebra\nin a cluster containing the two letters associated with the two\nfaces, cf. Figure~\\ref{Stasheff}.\n\nAn equivalent way to state the neighbouring principle for the $A_3$ case (and more generally for the $A_n$ case) is that $\\mathcal{A}$-coordinates corresponding to chords on the hexagon which cross are non-neighbouring, i.e. are forbidden to appear next to each other in the symbol. Examples are shown in Fig. \\ref{hexagonchordscrossing}.\n\\begin{figure}\n{\\footnotesize\n\\begin{center}\n\\begin{tikzpicture}[scale=0.8]\n   \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at (60+60*\\i:1.6);\n   }\n   \\draw[thick]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw[] (a6) -- (a2);\n   \\draw[] (a1) -- (a5);\n    \\foreach \\i in {1,...,6}\n   {\n     \\node at (60+60*\\i:1.9) {$\\i$};\n   }\n  \\end{tikzpicture}\n  \\qquad\n  \\begin{tikzpicture}[scale=0.8]\n   \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at (60+60*\\i:1.6);\n   }\n   \\draw[thick]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw[] (a6) -- (a2);\n   \\draw[] (a1) -- (a4);\n    \\foreach \\i in {1,...,6}\n   {\n     \\node at (60+60*\\i:1.9) {$\\i$};\n   }\n  \\end{tikzpicture}\n  \\qquad\n  \\begin{tikzpicture}[scale=0.8]\n   \\foreach \\i in {1,...,6}\n   {\n     \\coordinate (a\\i) at (60+60*\\i:1.6);\n   }\n   \\draw[thick]\n     (a1)\n     \\foreach \\i in {2,...,6}\n   {\n     -- (a\\i)\n   } -- cycle;\n\n   \\draw[] (a6) -- (a3);\n   \\draw[] (a1) -- (a4);\n    \\foreach \\i in {1,...,6}\n   {\n     \\node at (60+60*\\i:1.9) {$\\i$};\n   }\n  \\end{tikzpicture}\n  \n\\end{center}\n}\n\\caption{Forbidden pairs correspond to crossing chords of the hexagon.}\n\\label{hexagonchordscrossing}\n\\end{figure}\n\nThere are 12 coordinates in the\nneighbour set of the ${\\cal A}$-coordinate $\\langle 2456 \\rangle = (13)$ including itself\nand the 6 frozen coordinates. When writing down homogeneous functions,\nit convenient to work with a homogeneous alphabet and there are 6\nhomogeneous combinations that can be constructed using the allowed\nneighbours of $\\langle 1235 \\rangle = (46)$. Such a \\emph{homogeneous neighbour set} can be chosen as the five $\\mathcal{X}$-coordinates associated to the edges of the pentagonal face labelled by $(46)$ together with $\\Delta$ from eq. (\\ref{Delta}) as follows:\n\\begin{equation}\n  \\begin{minipage}[c]{2cm}\n    \\small\n    ${\\rm  hns}[(46)] = $\n  \\end{minipage}\\left\\{\n  \\frac{(13)(46)}{(16)(34)},\n  \\frac{(24)(16)}{(12)(46)},\n  \\frac{(36)(12)}{(23)(16)},\n  \\frac{(14)(23)}{(12)(34)},\n  \\frac{(26)(34)}{(23)(46)},\n  \\frac{(12)(34)(56)}{(23)(45)(16)}\\right\\}\\,.\n\\end{equation}\nSimilarly, there are five homogeneous combinations that are\nmade out of the 11 allowed neighbours of $\\langle 1245 \\rangle = (36)$. They may be taken as the two $\\mathcal{X}$-coordinates associated to the square (opposite edges on a square have the same $\\mathcal{X}$-coordinate) as well as any three of the four $\\mathcal{X}$-coordinates which are associated to the edges which lead away from the square face. A choice is as follows:\n\\begin{equation}\n  \\begin{minipage}[c]{2cm}\n    \\small\n     ${\\rm  hns}[(36)] =$\n  \\end{minipage}\n  \\left\\{\n  \\frac{(14)(23)}{(12)(34)},\n  \\frac{(14)(56)}{(16)(45)},\n  \\frac{(13)(24)}{(12)(34)},\n  \\frac{(15)(46)}{(16)(45)},\n  \\frac{(13)(45)}{(34)(15)}\\right\\}\\,.\\hfill\n\\end{equation}\n\nFor the cases of the cluster algebras associated to ${\\rm Conf}_n(\\mathbb{P}^{k-1})$ with $(k,n)$ coprime, one has the advantage of using frozen\ncoordinates to homogenise all remaining letters to construct a\nhomogeneous alphabet. Since frozen coordinates appear in every cluster\nby definition, they cannot spoil cluster adjacency. Hence for $(k,n)$ coprime,\nit is possible to talk about the cluster adjacency directly in terms of homogeneous\nletters such as those in equation (\\ref{heptletters}) for seven-particle\nscattering and ignore the frozen coordinates altogether.\n\n\nThe heptagon alphabet (\\ref{heptletters}) consists of 42 letters $a_{ij}$\ngrouped into six types. The neighbour sets of\nthese letters can be worked out in the same way as in the hexagon\ncase, for example starting with the $E_6$-shaped cluster in Fig.\n\\ref{howtonshep}, freezing the letter one is interested in and performing all possible mutations on the others. One finds\nthe following homogeneous neighbour sets for the letters $a_{11}$, $a_{21}$,\n$a_{41}$ and $a_{61}$:\n\\begin{equation}\n\\label{heptns}\n  \\begin{aligned}[t]\n    \\hns{a_{11}} =\\{ &a_{11}, a_{14}, a_{15}, a_{21}, a_{22}, a_{24}, a_{25}, a_{26}, a_{31}, a_{33}, a_{34}, a_{35}, a_{37}, a_{41},a_{43}, a_{46}, a_{51},\\\\\n     \\qquad        &a_{53}, a_{56}, a_{62}, a_{67}\\}\\\\\n    \\hns{a_{21}} =\\{&a_{11}, a_{13}, a_{14}, a_{15}, a_{17}, a_{21}, a_{23}, a_{24}, a_{25}, a_{26},a_{31}, a_{33}, a_{34}, a_{36},a_{37}, a_{41}, a_{43},\\\\\n    \\qquad        &a_{45}, a_{46}, a_{52}, a_{53}, a_{55}, a_{57}, a_{62}, a_{64}, a_{66}\\}\\\\\n    \\hns{a_{41}} =\\{&a_{11}, a_{13}, a_{16}, a_{21}, a_{23}, a_{24}, a_{26}, a_{31}, a_{33}, a_{35}, a_{36}, a_{41}, a_{43}, a_{46},a_{51}, a_{62}, a_{67}\\}\\\\\n    \\hns{a_{61}} =\\{&a_{12}, a_{17}, a_{23}, a_{25}, a_{27}, a_{32}, a_{34}, a_{36}, a_{42}, a_{47}, a_{52}, a_{57}, a_{61}\\}\\,.\n\\end{aligned}\n\\end{equation}\nAll other homogeneous neighbour sets for $\\text{Conf}_7(\\mathbb{P}^3)$ can be obtained as cyclic\nrotations, reflections or parity conjugates of these.\n\n\\subsection{Definition of cluster adjacent polylogarithms}\n\\label{CApolysdef}\n\nWe recall a polylogarithm of weight $k$ obeys\n\\be\nd f^{(k)} = \\sum_{a \\in \\mathcal{A}} f_{[a]}^{(k-1)} d \\log a\\,,\n\\ee\nwhere for us $\\mathcal{A}$ is the set of all $\\mathcal{A}$-coordinates of our cluster algebra. A cluster adjacent polylogarithm is one where the $f_{[a]}^{(k-1)}$ above additionally obey\n\\be\n\\label{adjacentdef}\nd f_{[a]}^{(k-1)}  = \\sum_{b \\in \\ns{a}} f^{(k-2)}_{[b],a} d \\log b\\,,\n\\ee\nwhere the sum is only over $b$ in the neighbour set of $a$. We also insist that the $f_{[a]}^{(k-1)}$ are themselves cluster adjacent polylogarithms in the same sense, i.e.\n\\be\n\\label{adjacency2nd}\nd f_{[b],a}^{(k-2)} = \\sum_{c \\in \\ns{b}} f_{[c],ba}^{(k-3)} d \\log c\\,,\n\\ee\nand so on all the way down to weight zero. It follows from the above that all adjacent pairs in the symbol of a cluster adjacent polylogarithm  $[\\ldots \\otimes a \\otimes b \\otimes \\ldots]$ are such that $a \\in \\ns{b}$ or equivalently $b \\in \\ns{a}$. \n\nNote that the above discussion is phrased in terms of the inhomogeneous $\\mathcal{A}$-coordinates, even though we are always interested in homogeneous functions $f^{(k)}$. This simply means that all the $d f^{(k)}$ above can be rewritten purely in terms of homogeneous combinations of $\\mathcal{A}$-coordinates and the sum in (\\ref{adjacentdef}) could be taken over the homogeneous neighbour set of $a$. In general, not all the cluster adjacency properties will be manifest in such a homogeneous representation, as happens in the hexagon case. In particular if we choose to write take sum in (\\ref{adjacentdef}) over the homogeneous neighbour set of $a$, then each homogeneous $b$ should be expanded in terms of the inhomogeneous $\\mathcal{A}$-coordinates in order to then reveal the cluster adjacent nature of the expression (\\ref{adjacency2nd}).\n\nIn the heptagon case one can phrase the whole discussion in terms of the homogenised unfrozen coordinates and the sum in (\\ref{adjacentdef}) can be taken over the homogeneous neighbour sets given in (\\ref{heptns}). Since the frozen factors play no role in cluster adjacency this property can be made manifest at the same time as homogeneity.\n\n\n\n\n\\subsection{Neighbour-set functions}\n\\label{sec:neisets}\n\nWhen constructing integrable cluster-adjacent functions, it is natural\nto introduce the concept of \\emph{neighbour-set functions}. They are\ndefined as polylogarithms which satisfy\n\\be\nd f^{(k)} = \\sum_{b \\in \\ns{a}} f_{[b]}^{(k-1)} d \\log b\n\\ee\nfor a given choice of $\\mathcal{A}$-coordinate $a$. The final entries of the symbols of such functions are selected only from the\nneighbour set of a given $\\mathcal{A}$-coordinate.  As can be seen from (\\ref{adjacentdef}) above, any cluster adjacent weight-$k$\nfunction only requires neighbour set functions in its $(k-1,1)$ coproduct. \nHence, when constructing cluster adjacent functions of weight $k$ one can use a reduced ansatz for the $(k-1,1)$ coproduct\n\\begin{equation}\n\\label{CAans}\n  f^{(k-1,1)}\n  =\n  \\sum_{a \\in {\\cal A}}\n  \\sum_{i=1}^{d_{[a]}^{(k-1)}}\n  \\, c_{ai}\\,\\bigl[f^{(k-1)}_{[a], i} \\otimes a\\bigr] \\, ,\n\\end{equation}\nwhere $f^{(k-1)}_{[a], i}$ are elements of a basis for \nthe space of homogeneous weight-($k-1$) functions whose final entries are in the neighbour-set of $a$ and\n$d_{[a]}^{(k-1)}$ is the dimension of this\nspace. If the $\\mathcal{A}$-coordinates $a$ in (\\ref{CAans}) above cannot be chosen as unfrozen ones homogenised purely in terms of frozen ones, then the coefficients $c_{ai}$ are assumed to be constrained to ensure homogeneity of the resulting expression. Eliminating any cluster-adjacency violation in the ansatz reduces the size of the resulting linear algebra problem. The notion of a neighbour set function is compatible with any possible choices of constraints in the initial entries, for example when constructing hexagon symbols to describe six-point amplitudes in planar $\\mathcal{N}=4$ super Yang-Mills theory.\n\n\n\nWe now illustrate neighbour set functions for $\\text{Conf}_6(\\mathbb{P}^3)$. In this case, there are two types of unfrozen\n${\\cal A}$-coordinates with neighbour set functions:\n$(13)$ \\& cyclic and $(14)$ \\& cyclic. \nThe neighbour-set functions for the hexagon are then defined as\nhomogeneous, cluster-adjacent functions that obey the initial entry\ncondition, i.e. begin with the three-cross ratios of the hexagon ($u$, $v$ or $w$ from eq. \\ref{hexusual}), and\nend with aforementioned homogeneous combinations that are\ncluster-adjacent to $(13)$ or $(14)$. The dimensions of such spaces\nfor a few weights are compared to the full space of\ncluster-adjacent hexagon symbols is given in Table \\ref{neidimshex}.\n{\n  \\renewcommand{\\arraystretch}{1.2}\n\\begin{table}\n  \\centering\n  {\\small\n    \\begin{tabular}{@{}lllllllllllllll@{}}\n      \\toprule\n    Weight&2&3&4&5&6&7&8&9&10&11&12&13&14\\\\\n    \\midrule\n    ${\\rm hns}[(13)]$&3&6&11&21&39&73&132&237&415&717&1216&2036&3358\\\\\n\n    ${\\rm hns}[(14)]$&3&5&10&19&36&66&120&213&374&644&1096&1835&3041\\\\\n\n      Full $A_3$&6&13&26&51&98&184&340&613&1085&1887&3224&5431&9014\\\\\n      \\bottomrule\n  \\end{tabular}\n  }\n  \\caption{Dimensions of the spaces of integrable words in the\n    hexagon alphabet with hexagon initial entries $\\{u,v,w\\}$ only and final entries drawn from the neighbour sets ${\\rm hns}[(13)]$, ${\\rm hns}[(14)]$ or from the full nine-letter $A_3$ alphabet.}\n  \\label{neidimshex}\n\\end{table}\n}\n\nWe have also computed the neighbour-set functions of the heptagon\nletters up to weight seven. The dimensions of the neighbour-set\nfunction spaces depend on the letter and they are summarised in Table\n\\ref{neisets}. For weights 2-7 we find the span of all $a_{2i}$ and\n$a_{3i}$ neighbour-set function spaces covers the entire\ncluster-adjacent function space of the corresponding weight.\n\n\n{\n  \\renewcommand{\\arraystretch}{1.2}\n\\begin{table}\n  \\centering\n  \\begin{tabular}{lllllll}\n    \\toprule\n    Weight&2&3&4&5&6&7\\\\\n    \\midrule\n    ${\\rm hns}[a_{1i}]$&10&29&83&229&612&1577\\\\\n    ${\\rm hns}[a_{2i}]$ &15&43&117&311&804&2025\\\\\n    ${\\rm hns}[a_{4i}]$ &6&14&34&87&224&570\\\\\n    ${\\rm hns}[a_{6i}]$&4&11&29&76&193&476\\\\\n    Full $E_6$&28&97&308&911&2555&6826\\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption{Dimensions of the neighbour-set function spaces of the\n    heptagon alphabet with initial entries $a_{1i}$ and the dimensions\n    of the full cluster-adjacent heptagon functions}\n  \\label{neisets}\n\\end{table}\n}\n\n\\subsection{Integrability}\n\nIt is interesting to investigate in low weights the spaces of cluster adjacent functions without any initial entry condition. At weight two we may split the space of integrable words into those which are symmetric in the two entries of the symbol and those which are antisymmetric. The symmetric ones are trivially integrable: any word of the form $[a\\otimes b]+[b\\otimes a]$ is the symbol of $\\log a  \\, \\log b$. Adjacency however constrains the possible choices of $a$ and $b$ - they must come from a common cluster, i.e. they must not correspond to distant faces on the polytope. The antisymmetric words on the other hand are not trivially integrable. However, they do automatically obey the adjacency condition, in the sense that all antisymmetric integrable weight two words are cluster adjacent, even if that condition was not imposed in constructing them. Actually they obey a stronger condition, namely that the $\\mathcal{A}$-coordinates appearing in the two slots can be found in some cluster together where they are connected by an arrow.\n\nWhen we investigate weight three words we find that the associated triplets of $\\mathcal{A}$-coordinates are of two possible types. Each term $[a\\otimes b \\otimes c]$ is either of the form where $a$, $b$ and $c$ can all be found together in the same cluster or we have $c=a'$ where $a'$ is the result of mutating on $a$ in some cluster. In fact there is an even stronger condition in this latter case: if we find triplets of the form $[a\\otimes b \\otimes a']$ then they can always be combined so that the intermediate letter becomes the $\\mathcal{X}$-coordinate associated with the mutation pair $(a,a')$. Recall that $\\mathcal{X}$-coordinates are associated to one-dimensional edges of the polytope which are also associated to mutations. Moreover if there is more than one edge between the two faces labelled by $a$ and $a'$ those edges are associated to the same $\\mathcal{X}$-coordinate. In other words $\\mathcal{X}$-coordinates are associated to mutation pairs of $\\mathcal{A}$-coordinates, hence we may denote them by $x(a,a')$.  So we have triplets of the form $[a\\otimes x(a,a') \\otimes a']$ or triplets $[a\\otimes b \\otimes c]$ where all three letters can be found together in some cluster.\n\n\n\n\\subsection{Cluster adjacency in hexagon and heptagon loop amplitudes}\n\nWe have confirmed that all the currently available results for hexagon and heptagon functions appearing in the loop expansion of MHV and NMHV amplitudes are cluster adjacent polylogarithms. That is, the functions $\\mathcal{E}^{{\\rm MHV}, (L)}$ and $E_{ijklm}^{(L)}$ are weight $2L$ polylogarithms whose symbols obey the cluster adjacency conditions and whose initial entries are constrained to be compatible with the physical branch cut conditions. In the hexagon case this means the initial entries are drawn from the set $\\{u,v,w\\}$ from (\\ref{hexusual}) and in the heptagon case that they are of the form $a_{1i}$ from the heptagon alphabet given in (\\ref{heptletters}).\n\nIn the MHV case the $(2L-1,1)$ coproduct of the polylogarithmic functions which appear is constrained in the the final entries are drawn only from $\\mathcal{A}$-coordinates of the form $\\langle i \\,j-1\\,j\\,j+1\\rangle$. This behaviour follows from an analysis of the $\\bar{Q}$-equation of \\cite{CaronHuot:2011kk,Bullimore:2011kg}. This has the consequence that the $(n-1,1)$ coproduct of the MHV amplitudes is heavily constrained,\n\\be\n\\mathcal{E}^{(2L-1,1)} = \\sum_{i,j} [\\mathcal{E}_{ij} \\otimes \\langle i\\, j-1\\, j\\, j+1 \\rangle]\\,,\n\\ee\nwhere $\\mathcal{E}_{ij}$ is a neighbour set function of the $\\mathcal{A}$-coordinate $\\langle i\\, j-1\\, j\\, j+1\\rangle$, i.e. it is a weight $(2L-1)$ polylogarithm whose symbol's final entries are drawn from the neighbour set of $\\langle i \\, j-1 \\,j \\, j+1\\rangle$.\n\nIn the NMHV case there is an interplay between the R-invariants and the final entries of the symbols of the polylogarithms which appear. We will address this point in greater detail in Sect. \\ref{NMHVloops}.\n\n\\section{Cluster adjacency of tree-level BCFW recursion}\n\nIt is clear from the above discussion that cluster adjacency of polylogarithms or symbols has a non-abelian character. Two $\\mathcal{A}$-coordinates $a$ and $a'$ which cannot appear next to each other are allowed to appear in the same word if they are appropriately separated by intermediate $\\mathcal{A}$-coordinates. For example, if they are separated by one step only the $\\mathcal{X}$-coordinate associated to the relevant mutation appears between them, as discussed above. This non-abelian behaviour is due to the fact that the symbol comes with an ordering which ultimately reflects the fact that monodromies of the associated iterated integrals do not commute with each other.\n\nHowever we now discuss a setting where an abelian form of cluster adjacency holds. It is in the context of the poles of rational functions contributing to tree-level amplitudes. Here we will restrict our discussion to the cluster adjacency properties of BCFW tree-amplitudes for NMHV and N$^2$MHV helicity configurations. The superconformal and dual superconformal symmetries are known to combine into a Yangian structure \\cite{Drummond:2009fd}. BCFW expansions for tree amplitudes are solved in terms of Yangian invariants. These quantities can be found as residues in the Grassmannian integral of \\cite{ArkaniHamed:2009dn,Mason:2009qx}.\n\nThe pattern we find can be stated as follows: \\emph{every Yangian invariant in the BCFW expansion of tree amplitudes has poles given by $\\mathcal{A}$-coordinates which can be found together in a common cluster.}\n\nExpressions for BCFW expansions may be generated directly in momentum twistor variables using the {\\tt bcfw.m} package provided in \\cite{Bourjaily:2010wh}. We give explicit examples showing all BCFW terms obey the cluster adjacency property up to eight points. \nAs well as providing another example in which the cluster algebra structure plays a role in controlling the singularities of amplitudes, the discussion of R-invariants will be relevant later when we consider NMHV loop amplitudes. \n\n\n\n\\subsection{NMHV}\n\\label{NMHVBCFW}\nThe BCFW expansion of the $n$-point NMHV tree amplitude of $\\mathcal{N}=4$ SYM (divided by the MHV tree) is given by\n\\begin{equation}\nA_{n,1}^{\\text{tree}} = \\sum_{1<i<j<n} [1 i i+1 j j+1]\n\\label{BCFWtree}\n\\end{equation}\nwhere we remind that the R-invariant $[ijklm]$ is given by\n\\begin{equation}\n[i j k l m] = \\frac{\\langle \\langle i j k l m \\rangle \\rangle }{\\ab{ijkl}\\ab{jklm}\\ab{klmi}\\ab{lmij}\\ab{mijk}}\\,.\n\\end{equation}\nHere the denominator is a product of Pl\\\"ucker coordinates which are examples of $\\mathcal{A}$-coordinates of the cluster algebra associated to ${\\rm Conf}_n(\\mathbb{P}^3)$. The numerator is a polynomial in momentum twistors and the Grassmann parameters $\\chi_i$ encoding the supermultiplet structure, $\\langle \\langle i j k l m \\rangle \\rangle = (\\chi_i \\langle j k l m \\rangle + \\text{cyclic})^4$.\n\nThe R-invariants are not all independent; there are ${n-1 \\choose 4}$ linearly independent ones due to identities of the form \n\\begin{equation}\\label{eq:Ridentity}\n[abcde] - [bcdef] + [cdefa] - [defab] + [efabc] - [fabcd] = 0\\,.\n\\end{equation}\n\nWe will now show that the $\\mathcal{A}$-coordinates which describe the poles of R-invariants obey an abelian form of cluster adjacency: it is always possible to find a cluster where all the poles of an R-invariant appear together. Since the poles multiply in a commutative fashion there is no ordering to them and it is natural therefore that adjacency simply requires them all to appear together in some cluster.\n\n\\subsubsection*{Five points}\nFive-points is a trivial example as there is just one R-invariant and hence the amplitude is simply\n\\begin{equation}\\label{eq:5ptAmp}\n\\mathcal{A}_{5,1} = [12345],\n\\end{equation}\nalso ${\\rm Conf}_5(\\mathbb{P}^3)$ contains just one cluster containing all frozen nodes.\n\\begin{figure}\n\t\\begin{center}\n\t\t\\begin{tikzpicture}[scale=0.75]\n\t\t{\\footnotesize\n\t\t\\node[frozenblue] (1234) at (0,0) {$\\ab{1234}$};\n\t\t\\node[frozenblue] (1235) at (2.5,0) {$\\ab{1235}$};\n\t\t\\node[frozenblue] (1245) at (5,0) {$\\ab{1245}$};\n\t\t\\node[frozenblue] (1345) at (7.5,0) {$\\ab{1345}$};\n\t\t\\node[frozenblue] (2345) at (10,0) {$\\ab{2345}$};\n\n\t\t\\draw[norm] (1234) edge (1235) (1235) edge (1245) (1245) edge (1345) (1345) edge (2345);\n\t\t}\n\t\t\\end{tikzpicture}\n\t\\end{center}\n\t\\caption{The single ${\\rm Conf}_5(\\mathbb{P}^3) \\sim A_0$ cluster. All nodes are frozen.}\n\t\\label{5point}\n\\end{figure}\n\\noindent The nodes in Fig. \\ref{5point} are coloured blue to indicate that they are present as poles in the R-invariant $[12345]$.\nThe basic R-invariant  \\eqref{eq:5ptAmp} and its associated cluster will be the starting point for analysing all other NMHV R-invariants.\n\n\\subsubsection*{Six points}\nAt six-points there is only one type of R-invariant, $[12345]$ and its cyclic rotations, which make up the six-point, NMHV, tree given as\n\\begin{equation}\\label{eq:6ptAmp1}\n\\mathcal{A}_{6,1}= [12345] + [12356] + [13456] = [12346] + [12456] + [23456].\n\\end{equation}\nSince every R-invariant at six points is a rotation of \\eqref{eq:5ptAmp} in ${\\rm Conf}_6(\\mathbb{P}^3)$ we can identify each one with a single cluster in the polytope, one of which is\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.75]\n{\\footnotesize\n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node[blue] (1235) at (0, 0) {$\\ab{1235}$};\n\\node[frozen] (1236) at (2.25, 0) {$\\ab{1236}$};\n\\node[blue] (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node[frozen] (1256) at (2.25,-1.25) {$\\ab{1256}$};\n\\node[blue] (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node[frozen] (1456) at (2.25,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (0,-3.75) {$\\ab{2345}$};\n\\node[frozen] (3456) at (2.25,-3.75) {$\\ab{3456}$};\n\n\\draw[->,shorten <=4pt, shorten >=3pt] (1234.south east) -- (1235.north west);\n\\draw[norm] (1235) -- (1236);\n\\draw[norm] (1245) -- (1256);\n\\draw[norm] (1345) -- (1456);\n\\draw[norm] (1235) -- (1245);\n\\draw[norm] (1245) -- (1345);\n\\draw[norm] (1345) -- (2345);\n\\draw[diag] (1256.north west) -- (1235.south east);\n\\draw[diag] (1456.north west) -- (1245.south east);\n\\draw[diag] (3456.north west) -- (1345.south east);\n}\n\\end{tikzpicture}\n\\end{center}\n\\caption{The cluster containing the poles of $[12345]$ in ${\\rm Conf}_6(\\mathbb{P}^3)$.}\n\\end{figure}\nOne would obtain the other five R-invariants and their associated clusters through cyclic rotations of this cluster. This can be achieved by applying the sequence of mutations illustrated in Fig. \\ref{Stasheff} which generates a cyclic rotation. The clusters associated to the R-invariants are the six associated to the top and bottom corners of the square faces in Fig. \\ref{Stasheff}.\n\nNote that while the full tree amplitude (\\ref{eq:6ptAmp1}) only contains physical poles of the form $\\langle 1245 \\rangle \\sim 1\/x_{25}^2 = 1\/(p_2 + p_3 + p_4)^2$ and rotations, the adjacency property holds term by term in the BCFW expansion. Hence it also constrains the way in which the spurious poles at $\\langle 1235 \\rangle = 0$ and its cyclic rotations may appear. A consequence of the adjacency property is the well-known fact that the tree amplitude cannot have simultaneous poles in two different factorisation channels. For example, there is no term with both $\\langle 1245 \\rangle$ and $\\langle 2356 \\rangle$ in the denominator. This statement is the analogue of the fact that the Steinmann relations follow from cluster adjacency in the loop amplitudes.\n\n\\subsubsection*{Seven points and beyond}\nAt seven points there are three types of R-invariant, \n\\begin{equation}\n\\label{Rinvs7pts}\n[12345] \\text{ \\& cyclic,} \\quad [12346]  \\text{ \\& cyclic,} \\quad [12356] \\text{ \\& cyclic.}\n\\end{equation}\nThe tree amplitude takes the form\n\\begin{equation}\n\\label{A71}\n\\mathcal{A}_{7,1} = [12345] + [12356] + [12367] + [13456] + [13467] + [14567]\\,. \n\\end{equation}\nAs with \\eqref{eq:6ptAmp1}, the BCFW representation of this amplitude is not unique due to the identity among the R-invariants \\eqref{eq:Ridentity}.\nAt seven points multiple clusters contain the poles of a given R-invariant and hence R-invariants are associated to sub-algebras in the full ${\\rm Conf}_7(\\mathbb{P}^3)$ cluster algebra. For example, the initial cluster in Fig. \\ref{heptinitial} contains all the poles of $[12345]$. It also contains three more unfrozen nodes in the second column. Performing all possible mutations in the second column generates an entire $A_3$ subalgebra, all of whose clusters contain the poles of $[12345]$. This is illustrated in Fig. \\ref{fig:R67cluster}. The other two types of R-invariants in (\\ref{Rinvs7pts}) appear respectively in $A_2$ and $A_1$ subalgebras.\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.75]\n{\\footnotesize\n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node[blue] (1235) at (0, 0) {$\\ab{1235}$};\n\\node (1236) at (2.25, 0) {$\\ab{1236}$};\n\\node[blue] (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node (1256) at (2.25,-1.25) {$\\ab{1256}$};\n\\node[blue] (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node (1456) at (2.25,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (0,-3.75) {$\\ab{2345}$};\n\\node[frozen] (3456) at (2.25,-3.75) {$\\ab{3456}$};\n\\node[frozen] (1237) at (4.5,0) {$\\ab{1237}$};\n\\node[frozen] (1267) at (4.5,-1.25) {$\\ab{1267}$};\n\\node[frozen] (1567) at (4.5,-2.5) {$\\ab{1567}$};\n\\node[frozen] (4567) at (4.5,-3.75) {$\\ab{4567}$};\n\n\\begin{scope}[blend mode=overlay,overlay]\n          \\node[rectangle, rounded corners, fit=(1236)(1256)(1456),fill=red!20, inner sep=0pt] {};\n        \\end{scope}\n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node[blue] (1235) at (0, 0) {$\\ab{1235}$};\n\\node (1236) at (2.25, 0) {$\\ab{1236}$};\n\\node[blue] (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node (1256) at (2.25,-1.25) {$\\ab{1256}$};\n\\node[blue] (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node (1456) at (2.25,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (0,-3.75) {$\\ab{2345}$};\n\\node[frozen] (3456) at (2.25,-3.75) {$\\ab{3456}$};\n\\node[frozen] (1237) at (4.5,0) {$\\ab{1237}$};\n\\node[frozen] (1267) at (4.5,-1.25) {$\\ab{1267}$};\n\\node[frozen] (1567) at (4.5,-2.5) {$\\ab{1567}$};\n\\node[frozen] (4567) at (4.5,-3.75) {$\\ab{4567}$};\n\n        \n\\draw[norm] (1235) -- (1236);\n\\draw[norm] (1236) -- (1237);\n\\draw[norm] (1245) -- (1256);\n\\draw[norm] (1256) -- (1267);\n\\draw[norm] (1345) -- (1456);\n\\draw[norm] (1456) -- (1567);\n\n\\draw[norm] (1235) -- (1245);\n\\draw[norm] (1236) -- (1256);\n\\draw[norm] (1245) -- (1345);\n\\draw[norm] (1256) -- (1456);\n\\draw[norm] (1345) -- (2345);\n\\draw[norm] (1456) -- (3456);\n\n\\draw[->,shorten <=4pt, shorten >=3pt] (1234.south east) -- (1235.north west);\n\\draw[diag] (1256.north west) -- (1235.south east);\n\\draw[diag] (1267.north west) -- (1236.south east);\n\\draw[diag] (1456.north west) -- (1245.south east);\n\\draw[diag] (1567.north west) -- (1256.south east);\n\\draw[diag] (3456.north west) -- (1345.south east);\n\\draw[diag] (4567.north west) -- (1456.south east);\n}\n\\end{tikzpicture}\n\\end{center}\n\\caption{A cluster containing the poles of $[12345]$ in ${\\rm Conf}_7(\\mathbb{P}^3)$. The unfrozen nodes highlighted in red generate an $A_3$ subalgebra by repeated mutation.}\n\\label{fig:R67cluster}\n\\end{figure}\n\n\nOne form of the eight-point NMHV tree amplitude is given by\n\\begin{equation}\n\\begin{aligned}\n\\mathcal{A}_{8,1} = &[12345] + [12356] + [12367] + [12378] + [13456] \\\\ \n\t\t\t\t\t+ &[13467] + [13478] + [14567] + [14578] + [15678].\n\\end{aligned}\n\\end{equation}\nAs we can see, more types of R-invariants begin to appear at eight points so we have presented their subalgebras in Table \\ref{table:Rinvs} below along with their subalgebras at lower points.\n\\begin{table}\n  \\renewcommand{\\arraystretch}{1.3}\n\\centering\n\t\\begin{tabular}{@{} c  c  c  c  c @{}}\n          \\toprule\n\t$n$ & 5 & 6 & 7 & 8 \\\\ \\hline\n\t$[12345]$ & $A_0$ & $A_0$ & $A_3$ & $E_6$ \\\\ \n\t$[12356]$ & $\\--$ & $A_0$ & $A_1$ & $A_4$ \\\\ \n\t$[12346]$ & $\\--$ & $A_0$ & $A_2$ & $A_5$ \\\\ \n\t$[13467]$ & $\\--$ & $\\--$ & $A_1$ & $A_2 \\times A_1 \\times A_1$ \\\\ \n          $[12357]$ & $\\--$ & $\\--$ & $A_2$ & $A_4$ \\\\ \n          \\bottomrule\n\t\\end{tabular}\n\t\\caption{Various R-invariants and their subalgebras in ${\\rm Conf}_n(\\mathbb{P}^3)$ at different multiplicities $n$.}\n\t\\label{table:Rinvs}\n\\end{table}\n\\noindent The notation $A_0$ in Table \\ref{table:Rinvs} indicates that a single cluster is associated to that R-invariant.\nThe last R-invariant $[12357]$ does not appear in the BCFW expansion of any tree in formula (\\ref{BCFWtree}) amplitude but we can nevertheless associate a sub-algebra to this Yangian invariant object.\n\nAs described in Sect. \\ref{gencyc} above, one can rotate the nodes in an initial-type cluster by mutating up all consecutive columns. Using this we can show that one can obtain any R-invariant by starting with the initial cluster, which we associate to $[12345]$, and mutating in different ${\\rm Conf}_n(\\mathbb{P}^3)$ sub-algebras. We illustrate this procedure with the following eight-point example: we will find a cluster in ${\\rm Conf}_8(\\mathbb{P}^3)$ which contains the poles of $[13467]$. \n\nStarting from $[12345]$, the sequence of rotations to get $[13467]$ is\n\\begin{equation}\n[12345] \\xrightarrow{+4} [12356] \\xrightarrow{+5} [13467]\n\\end{equation}\nwhere the rotations are in ${\\rm Conf}_6(\\mathbb{P}^3)$ and ${\\rm Conf}_7(\\mathbb{P}^3)$ respectively. To find a cluster in ${\\rm Conf}_8(\\mathbb{P}^3)$ with all the $\\mathcal{A}$-coordinates we need we start from the initial cluster (shown in Fig. \\ref{fig:8ptinitcluster}) and mutate in the ${\\rm Conf}_7(\\mathbb{P}^3)$ subalgebra (the first two columns) such that its nodes rotate by five to arrive at the cluster shown in Fig. \\ref{intermediate}. Then we mutate in the ${\\rm Conf}_6(\\mathbb{P}^3)$ subalgebra (the first column only) such that its nodes rotate by four.\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=0.75]\n{\\footnotesize\n\\node[frozen] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node (1235) at (0, 0) {$\\ab{1235}$};\n\\node (1236) at (2.25, 0) {$\\ab{1236}$};\n\\node (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node (1256) at (2.25,-1.25) {$\\ab{1256}$};\n\\node (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node (1456) at (2.25,-2.5) {$\\ab{1456}$};\n\\node[frozen] (2345) at (0,-3.75) {$\\ab{2345}$};\n\\node[frozen] (3456) at (2.25,-3.75) {$\\ab{3456}$};\n\\node (1237) at (4.5,0) {$\\ab{1237}$};\n\\node (1267) at (4.5,-1.25) {$\\ab{1267}$};\n\\node (1567) at (4.5,-2.5) {$\\ab{1567}$};\n\\node[frozen] (1238) at (6.75,0) {$\\ab{1238}$};\n\\node[frozen] (1278) at (6.75,-1.25) {$\\ab{1278}$};\n\\node[frozen] (1678) at (6.75,-2.5) {$\\ab{1678}$};\n\\node[frozen] (5678) at (6.75,-3.75) {$\\ab{5678}$};\n\\node[frozen] (4567) at (4.5,-3.75) {$\\ab{4567}$};\n        \n\\draw[norm] (1235) -- (1236);\n\\draw[norm] (1236) -- (1237);\n\\draw[norm] (1245) -- (1256);\n\\draw[norm] (1256) -- (1267);\n\\draw[norm] (1345) -- (1456);\n\\draw[norm] (1456) -- (1567);\n\\draw[norm] (1237) -- (1238);\n\\draw[norm] (1267) -- (1278);\n\\draw[norm] (1567) -- (1678);\n\n\\draw[norm] (1235) -- (1245);\n\\draw[norm] (1236) -- (1256);\n\\draw[norm] (1245) -- (1345);\n\\draw[norm] (1256) -- (1456);\n\\draw[norm] (1345) -- (2345);\n\\draw[norm] (1456) -- (3456);\n\\draw[norm] (1237) -- (1267);\n\\draw[norm] (1267) -- (1567);\n\\draw[norm] (1567) -- (4567);\n\n\\draw[->,shorten <=4pt, shorten >=3pt] (1234.south east) -- (1235.north west);\n\\draw[diag] (1256.north west) -- (1235.south east);\n\\draw[diag] (1267.north west) -- (1236.south east);\n\\draw[diag] (1456.north west) -- (1245.south east);\n\\draw[diag] (1567.north west) -- (1256.south east);\n\\draw[diag] (3456.north west) -- (1345.south east);\n\\draw[diag] (4567.north west) -- (1456.south east);\n\\draw[diag] (1278.north west) -- (1237.south east);\n\\draw[diag] (1678.north west) -- (1267.south east);\n\\draw[diag] (5678.north west) -- (1567.south east);\n}\n\\end{tikzpicture}\n\n\n\\caption{A cluster containing the poles of $[12345]$ in ${\\rm Conf}_8(\\mathbb{P}^3)$.}\n\\label{fig:8ptinitcluster}\n\\end{figure}\nBeginning with the ${\\rm Conf}_8(\\mathbb{P}^3)$ initial cluster we employ our mutation prescription by mutating up the first column, followed by the second column, repeating this another four times which results in the cluster shown in Fig. \\ref{intermediate} where the unchanged topology of the ${\\rm Conf}_7(\\mathbb{P}^3)$ subalgebra is given in green.\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=0.75]\n{\\footnotesize\n\\node (1236) at (0,0) {$\\ab{1236}$};\n\\node (2367) at (0,1.25) {$\\ab{2367}$};\n\\node (1367) at (0,2.5) {$\\ab{1367}$};\n\\node (2346) at (2.25,0) {$\\ab{2346}$};\n\\node (3467) at (2.25,1.25) {$\\ab{3467}$};\n\\node (1467) at (2.25,2.5) {$\\ab{1467}$};\n\\node[frozen] (1678) at (0,3.75) {$\\ab{1678}$};\n\\node (1567) at (2.25,3.75) {$\\ab{1567}$};\n\\node (1267) at (-2.25,2.5) {$\\ab{1267}$};\n\\node (1237) at (-2.25,1.25) {$\\ab{1237}$};\n\\node[frozen] (1234) at (2.25,-1.25) {$\\ab{1234}$};\n\\node[frozen] (2345) at (4.5,-1.25) {$\\ab{2345}$};\n\\node[frozen] (1278) at (-4.5,2.5) {$\\ab{1278}$};\n\\node[frozen] (1238) at (-4.5,1.25) {$\\ab{1238}$};\n\\node[frozen] (3456) at (4.5,0) {$\\ab{3456}$};\n\\node[frozen] (4567) at (4.5,2.5) {$\\ab{4567}$};\n\\node[frozen] (5678) at (4.5,3.75) {$\\ab{5678}$};\n\n\\begin{scope}[blend mode=overlay,overlay]\n          \\node[rectangle,fill=green!20, rounded corners, fit=(1367)(1467)(2367)(3467)(1236)(2346), inner sep=0.3pt] {};\n        \\end{scope}\n\\node (1236) at (0,0) {$\\ab{1236}$};\n\\node (2367) at (0,1.25) {$\\ab{2367}$};\n\\node (1367) at (0,2.5) {$\\ab{1367}$};\n\\node (2346) at (2.25,0) {$\\ab{2346}$};\n\\node (3467) at (2.25,1.25) {$\\ab{3467}$};\n\\node (1467) at (2.25,2.5) {$\\ab{1467}$};\n\\node[frozen] (1678) at (0,3.75) {$\\ab{1678}$};\n\\node (1567) at (2.25,3.75) {$\\ab{1567}$};\n\\node (1267) at (-2.25,2.5) {$\\ab{1267}$};\n\\node (1237) at (-2.25,1.25) {$\\ab{1237}$};\n\\node[frozen] (1234) at (2.25,-1.25) {$\\ab{1234}$};\n\\node[frozen] (2345) at (4.5,-1.25) {$\\ab{2345}$};\n\\node[frozen] (1278) at (-4.5,2.5) {$\\ab{1278}$};\n\\node[frozen] (1238) at (-4.5,1.25) {$\\ab{1238}$};\n\\node[frozen] (3456) at (4.5,0) {$\\ab{3456}$};\n\\node[frozen] (4567) at (4.5,2.5) {$\\ab{4567}$};\n\\node[frozen] (5678) at (4.5,3.75) {$\\ab{5678}$};\n\n\\draw[norm] (1236) -- (2346);\n\\draw[norm] (2367) -- (3467);\n\\draw[norm] (1367) -- (1467);\n\\draw[norm] (4567) -- (1467);\n\\draw[norm] (2346) -- (3456);\n\\draw[norm] (1567) -- (1678);\n\\draw[norm] (5678) -- (1567);\n\\draw[norm] (1267) -- (1367);\n\\draw[norm] (1267) -- (1278);\n\\draw[norm] (1237) -- (1238);\n\n\\draw[norm] (1367) -- (2367);\n\\draw[norm] (2367) -- (1236);\n\\draw[norm] (1467) -- (3467);\n\\draw[norm] (3467) -- (2346);\n\\draw[norm] (2346) -- (1234);\n\\draw[norm] (1237) -- (1267);\n\\draw[norm] (1467) -- (1567);\n\n\\draw[diag] (2346.north west) -- (2367.south east);\n\\draw[diag] (3467.north west) -- (1367.south east);\n\\draw[diag] (1234.north west) -- (1236.south east);\n\\draw[diag] (2345.north west) -- (2346.south east);\n\\draw[diag] (3456.north west) -- (3467.south east);\n\\draw[diag] (1236.north west) -- (1237.south east);\n\\draw[diag] (3467.north east) -- (4567.south west);\n\\draw[diag] (1278.south east) -- (1237.north west);\n\\draw[diag] (1567.south east) -- (4567.north west);\n\\draw[diag] (1678.south west) -- (1267.north east);\n}\n\\end{tikzpicture}\n\\caption{The cluster obtained after five cyclic mutations of Fig. \\ref{fig:8ptinitcluster} in the first two columns.}\n\\label{intermediate}\n\\end{figure}\n We now mutate up the first column in the green section four times, resulting in the final cluster shown in Fig. \\ref{final} where the poles of $[13467]$ are in blue and the $A_2 \\times A_1 \\times A_1$ subalgebra is in red in agreement with Table \\ref{table:Rinvs}.\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=0.75]\n{\\footnotesize\n\\node[blue] (1346) at (0,0) {$\\ab{1346}$};\n\\node[blue] (1347) at (0,1.25) {$\\ab{1347}$};\n\\node[blue] (1367) at (0,2.5) {$\\ab{1367}$};\n\\node[red] (2346) at (2.25,-1.25) {$\\ab{2346}$};\n\\node[blue] (3467) at (2.25,1.25) {$\\ab{3467}$};\n\\node[blue] (1467) at (2.25,2.5) {$\\ab{1467}$};\n\\node[frozen] (1678) at (0,3.75) {$\\ab{1678}$};\n\\node[red] (1567) at (2.25,3.75) {$\\ab{1567}$};\n\\node (1267) at (-2.25,2.5) {$\\ab{1267}$};\n\\node (1237) at (-2.25,1.25) {$\\ab{1237}$};\n\\node[frozen] (1234) at (-2.25,-1.25) {$\\ab{1234}$};\n\\node[frozen] (2345) at (4.5,-1.25) {$\\ab{2345}$};\n\\node[frozen] (1278) at (-4.5,2.5) {$\\ab{1278}$};\n\\node[frozen] (1238) at (-4.5,1.25) {$\\ab{1238}$};\n\\node[frozen] (3456) at (4.5,0) {$\\ab{3456}$};\n\\node[frozen] (4567) at (4.5,2.5) {$\\ab{4567}$};\n\\node[frozen] (5678) at (4.5,3.75) {$\\ab{5678}$};\n\n\\begin{scope}[blend mode=overlay,overlay]\n          \\node[rectangle, rounded corners, fit=(1267)(1237),fill=red!20, inner sep=0pt] {};\n        \\end{scope}\n\\node[blue] (1346) at (0,0) {$\\ab{1346}$};\n\\node[blue] (1347) at (0,1.25) {$\\ab{1347}$};\n\\node[blue] (1367) at (0,2.5) {$\\ab{1367}$};\n\\node[red] (2346) at (2.25,-1.25) {$\\ab{2346}$};\n\\node[blue] (3467) at (2.25,1.25) {$\\ab{3467}$};\n\\node[blue] (1467) at (2.25,2.5) {$\\ab{1467}$};\n\\node[frozen] (1678) at (0,3.75) {$\\ab{1678}$};\n\\node[red] (1567) at (2.25,3.75) {$\\ab{1567}$};\n\\node (1267) at (-2.25,2.5) {$\\ab{1267}$};\n\\node (1237) at (-2.25,1.25) {$\\ab{1237}$};\n\\node[frozen] (1234) at (-2.25,-1.25) {$\\ab{1234}$};\n\\node[frozen] (2345) at (4.5,-1.25) {$\\ab{2345}$};\n\\node[frozen] (1278) at (-4.5,2.5) {$\\ab{1278}$};\n\\node[frozen] (1238) at (-4.5,1.25) {$\\ab{1238}$};\n\\node[frozen] (3456) at (4.5,0) {$\\ab{3456}$};\n\\node[frozen] (4567) at (4.5,2.5) {$\\ab{4567}$};\n\\node[frozen] (5678) at (4.5,3.75) {$\\ab{5678}$};\n\n\\draw[norm] (1234) -- (1237);\n\\draw[norm] (2346) -- (1234);\n\\draw[norm] (1237) -- (1267);\n\\draw[norm] (1237) -- (1347);\n\\draw[norm] (1237) -- (1267);\n\\draw[norm] (1237) -- (1238);\n\\draw[norm] (1267) -- (1278);\n\\draw[norm] (1267) -- (1367);\n\\draw[norm] (1367) -- (1467);\n\\draw[norm] (1467) -- (1567);\n\\draw[norm] (1467) -- (3467);\n\\draw[norm] (1567) -- (1678);\n\\draw[norm] (5678) -- (1567);\n\\draw[norm] (2345) -- (2346);\n\\draw[norm] (1346) -- (1347);\n\\draw[norm] (1347) -- (1367);\n\\draw[norm] (4567) -- (1467);\n\n\\draw[->,shorten >=2pt] (1347.south west) -- (1234.35);\n\\draw[->,shorten <=3pt] (1234.north east) -- (1346.south west);\n\\draw[diag] (1346.south east) -- (2346.north west);\n\\draw[diag] (2346.north east) -- (3456.south west);\n\\draw[diag] (3456.north west) -- (3467.south east);\n\\draw[diag] (3467.south west) -- (1346.north east);\n\\draw[diag] (3467.north east) -- (4567.south west);\n\\draw[diag] (1678.south west) -- (1267.north east);\n\\draw[diag] (1278.south east) -- (1237.north west);\n\\draw[diag] (1367.south west) -- (1237.north east);\n\\draw[diag] (1567.south east) -- (4567.north west);\n}\n\\end{tikzpicture}\n\\caption{A cluster containing the poles of the R-invariant $[13467]$.}\n\\label{final}\n\\end{figure}\nUsing this procedure one can locate a cluster which contains the poles of any R-invariant for an arbitrary number of points.\n\n\\subsection{Beyond NMHV}\nBeyond NMHV, terms in BCFW tree amplitudes are more complicated than simple R-invariants so it is less obvious that one could associate subalgebras of ${\\rm Conf}_n(\\mathbb{P}^3)$ cluster algebras to individual terms. We show, up to eight points, that one can do this in much the same way as for NMHV.\n\n\\subsubsection*{Six points}\nAt six points the N$^2$MHV amplitude is equivalent to the $\\overline{\\text{MHV}}$ amplitude. It is given by\n\\begin{equation} \\label{eq:6ptAmp}\n\\mathcal{A}_{6,2} = \\frac{\\dab{123456}}{\\ab{1234}\\ab{1236}\\ab{1256}\\ab{1456}\\ab{2345}\\ab{3456}}\n\\end{equation}\nwhere\n\\begin{equation}\n\\dab{ijklmn} = \\frac{\\dab{ijkmn} \\dab{jklmn}}{\\ab{jkmn}^4}\n\\end{equation}\nis cyclically invariant and polynomial although not manifestly so in this form.\n\n\nIdentifying a cluster with \\eqref{eq:6ptAmp} is trivial since every pole is an adjacent bracket and hence appears in every cluster in ${\\rm Conf}_6(\\mathbb{P}^3)$ i.e. one can associate this amplitude with the entire $A_3$ cluster algebra.\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.75]\n{\\footnotesize\n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node (1235) at (0, 0) {$\\ab{1235}$};\n\\node[frozenblue] (1236) at (2.25, 0) {$\\ab{1236}$};\n\\node (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node[frozenblue] (1256) at (2.25,-1.25) {$\\ab{1256}$};\n\\node (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node[frozenblue] (1456) at (2.25,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (0,-3.75) {$\\ab{2345}$};\n\\node[frozenblue] (3456) at (2.25,-3.75) {$\\ab{3456}$};\n\n\\begin{scope}[blend mode=overlay,overlay]\n          \\node[rectangle, rounded corners, fit=(1235)(1245)(1345),fill=red!20, inner sep=0pt] {};\n        \\end{scope}\n\n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node (1235) at (0, 0) {$\\ab{1235}$};\n\\node[frozenblue] (1236) at (2.25, 0) {$\\ab{1236}$};\n\\node (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node[frozenblue] (1256) at (2.25,-1.25) {$\\ab{1256}$};\n\\node (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node[frozenblue] (1456) at (2.25,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (0,-3.75) {$\\ab{2345}$};\n\\node[frozenblue] (3456) at (2.25,-3.75) {$\\ab{3456}$};\n\n        \n\\draw[->,shorten <=4pt, shorten >=3pt] (1234.south east) -- (1235.north west);\n\\draw[norm] (1235) -- (1236);\n\\draw[norm] (1245) -- (1256);\n\\draw[norm] (1345) -- (1456);\n\\draw[norm] (1235) -- (1245);\n\\draw[norm] (1245) -- (1345);\n\\draw[norm] (1345) -- (2345);\n\\draw[diag] (1256.north west) -- (1235.south east);\n\\draw[diag] (1456.north west) -- (1245.south east);\n\\draw[diag] (3456.north west) -- (1345.south east);\n}\n\\end{tikzpicture}\n\\end{center}\n\\caption{A cluster in $A_3$ corresponding to the six-point N$^{2}$MHV amplitude.}\n\\end{figure}\nThe blue nodes correspond to poles in the amplitude and the nodes highlighted in red correspond to the full $A_3$ algebra in which the amplitude lives.\n\\subsubsection*{Seven points}\nThe seven-point, N$^2$MHV, tree-amplitude is equivalent to the $\\overline{\\text{NMHV}}$ amplitude\n\\begin{equation} \\label{eq:7ptAmp}\n\\begin{aligned}\n\\mathcal{A}_{7,2} &= \\mathcal{A}_{6,2} \\\\ &+ \\frac{\\dab{134567}}{\\ab{1345}\\ab{1347}\\ab{1367}\\ab{1567}\\ab{3456}\\ab{4567}} \\\\ \n&+ \\frac{\\dab{123467}}{\\ab{1234}\\ab{1237}\\ab{1267}\\ab{1467}\\ab{2346}\\ab{3467}} \\\\ \n&+ \\frac{\\dab{12345} \\dab{14567}}{\\ab{1234}\\ab{1245}\\ab{1345}\\ab{1456}\\ab{1457}\\ab{1567}\\ab{2345}\\ab{4567}\\langle 1 (23)(45)(67) \\rangle} \\\\ \n&+ \\frac{\\dab{12367} \\dab{23456}}{\\ab{1236}\\ab{1237}\\ab{1267}\\ab{2345}\\ab{2346}\\ab{2356}\\ab{2367}\\ab{3456}\\langle 6 (23)(45)(17) \\rangle} \\\\ \n&+ \\frac{\\dab{12367} \\dab{14567}}{\\ab{1237}\\ab{1267}\\ab{1367}\\ab{1467}\\ab{1567}\\ab{4567}\\langle 1 (23)(45)(67) \\rangle \\langle 6 (23)(45)(17) \\rangle}.\n\\end{aligned}\n\\end{equation}\nThe first term is equal to the expression (\\ref{eq:6ptAmp}) for the six-point amplitude. It is now in ${\\rm Conf}_7(\\mathbb{P}^3) \\sim E_6$ therefore some of the poles are now unfrozen and the $A_3$ algebra is now a subalgebra of the full $E_6$ algebra, as shown in Fig. \\ref{fig:7ptInitialCluster}.\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.75]\n{\\footnotesize\n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node (1235) at (0, 0) {$\\ab{1235}$};\n\\node[blue] (1236) at (2.25, 0) {$\\ab{1236}$};\n\\node (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node[blue] (1256) at (2.25,-1.25) {$\\ab{1256}$};\n\\node (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node[blue] (1456) at (2.25,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (0,-3.75) {$\\ab{2345}$};\n\\node[frozenblue] (3456) at (2.25,-3.75) {$\\ab{3456}$};\n\\node[frozen] (1237) at (4.5,0) {$\\ab{1237}$};\n\\node[frozen] (1267) at (4.5,-1.25) {$\\ab{1267}$};\n\\node[frozen] (1567) at (4.5,-2.5) {$\\ab{1567}$};\n\\node[frozen] (4567) at (4.5,-3.75) {$\\ab{4567}$};\n\n\\begin{scope}[blend mode=overlay,overlay]\n          \\node[rectangle, rounded corners, fit=(1235)(1245)(1345),fill=red!20, inner sep=0pt] {};\n        \\end{scope}\n\n       \n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node (1235) at (0, 0) {$\\ab{1235}$};\n\\node[blue] (1236) at (2.25, 0) {$\\ab{1236}$};\n\\node (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node[blue] (1256) at (2.25,-1.25) {$\\ab{1256}$};\n\\node (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node[blue] (1456) at (2.25,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (0,-3.75) {$\\ab{2345}$};\n\\node[frozenblue] (3456) at (2.25,-3.75) {$\\ab{3456}$};\n\\node[frozen] (1237) at (4.5,0) {$\\ab{1237}$};\n\\node[frozen] (1267) at (4.5,-1.25) {$\\ab{1267}$};\n\\node[frozen] (1567) at (4.5,-2.5) {$\\ab{1567}$};\n\\node[frozen] (4567) at (4.5,-3.75) {$\\ab{4567}$};\n\n\\draw[norm] (1235) -- (1236);\n\\draw[norm] (1236) -- (1237);\n\\draw[norm] (1245) -- (1256);\n\\draw[norm] (1256) -- (1267);\n\\draw[norm] (1345) -- (1456);\n\\draw[norm] (1456) -- (1567);\n\n\\draw[norm] (1235) -- (1245);\n\\draw[norm] (1236) -- (1256);\n\\draw[norm] (1245) -- (1345);\n\\draw[norm] (1256) -- (1456);\n\\draw[norm] (1345) -- (2345);\n\\draw[norm] (1456) -- (3456);\n\n\\draw[->,shorten <=4pt, shorten >=3pt] (1234.south east) -- (1235.north west);\n\\draw[diag] (1256.north west) -- (1235.south east);\n\\draw[diag] (1267.north west) -- (1236.south east);\n\\draw[diag] (1456.north west) -- (1245.south east);\n\\draw[diag] (1567.north west) -- (1256.south east);\n\\draw[diag] (3456.north west) -- (1345.south east);\n\\draw[diag] (4567.north west) -- (1456.south east);\n}\n\\end{tikzpicture}\n\\end{center}\n\n\\caption{A cluster containing the poles of $\\mathcal{A}_{6,2}$ in ${\\rm Conf}_7(\\mathbb{P}^3)$.}\n\\label{fig:7ptInitialCluster}\n\\end{figure}\nAs before, the blue nodes correspond to poles in the term while the nodes highlighted in red correspond to an $A_3$ subalgebra inside the full $E_6$ algebra in which all the poles of \\eqref{eq:6ptAmp} can be found. The second and third terms of \\eqref{eq:7ptAmp} can be obtained by rotating the momentum twistors in \\eqref{eq:6ptAmp} by two and five units respectively and hence one can obtain clusters containing their poles by rotating Fig. \\ref{fig:7ptInitialCluster} by the same amounts. We can associate the fourth term of \\eqref{eq:7ptAmp} with an $A_1$ subalgebra as shown in Fig. \\ref{fig:7ptClusterTerm4}. \n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.75]\n{\\footnotesize\n\\node[frozen] (3456) at (0,0) {$\\ab{3456}$};\n\\node[frozenblue] (4567) at (-3,0) {$\\ab{4567}$};\n\\node[frozenblue] (2345) at (3,0) {$\\ab{2345}$};\n\\node[blue] (1456) at (-1.5,1.25) {$\\ab{1456}$};\n\\node[blue] (1345) at (1.5,1.25) {$\\ab{1345}$};\n\\node[blue] (1457) at (-1.5,2.5) {$\\ab{1457}$};\n\\node[blue] (1245) at (1.5,2.5) {$\\ab{1245}$};\n\\node[frozen] (1237) at (-3,3.75) {$\\ab{1237}$};\n\\node[frozen] (1267) at (3,3.75) {$\\ab{1267}$};\n\\node[blue] (s1) at (0,3.75) {$\\langle 1 (23) (45) (67) \\rangle$};\n\\node[red] (1467) at (0, 5) {$\\ab{1467}$};\n\\node[frozenblue] (1234) at (-3,5) {$\\ab{1234}$};\n\\node[frozenblue] (1567) at (3,5) {$\\ab{1567}$};\n\n\\draw[norm] (1345) -- (1456);\n\\draw[norm] (s1) -- (1237);\n\\draw[norm] (1267) -- (s1);\n\\draw[norm] (1467) -- (1234);\n\\draw[norm] (1467) -- (1567);\n\n\\draw[norm] (s1) -- (1467);\n\\draw[norm] (1456) -- (1457);\n\\draw[norm] (1245) -- (1345);\n\n\\draw[diag] (4567.north) -- (1456.south west);\n\\draw[diag] (1456.south east) -- (3456.north);\n\\draw[diag] (3456.north) -- (1345.south west);\n\\draw[diag] (1345.south east) -- (2345.north);\n\\draw[diag] (1457.north east) -- (s1.south);\n\\draw[diag] (s1.south) -- (1245.north west);\n\\draw[diag] (1237.south) -- (1457.north west);\n\\draw[diag] (1245.north east) -- (1267.south);\n\\draw[diag] (1234.south east) -- (s1.north west);\n}\n\\end{tikzpicture}\n\\end{center}\n\\caption{A cluster corresponding to the $4^{\\text{th}}$ term in $\\mathcal{A}_{7,2}$.}\n\\label{fig:7ptClusterTerm4}\n\\end{figure}\n\\noindent One can obtain the fifth term by rotating the fourth term by five units hence it also lives in an $A_1$ subalgebra found by rotating Fig. \\ref{fig:7ptClusterTerm4} by five units.\nFinally, the sixth term can be associated to an $A_2$ subalgebra as illustrated in Fig. \\ref{n2mhvterm6}.\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.75]\n{\\footnotesize\n\\node[red] (2367) at (0,0) {$\\ab{2367}$};\n\\node[frozenblue] (1237) at (-3,0) {$\\ab{1237}$};\n\\node[frozenblue] (4567) at (3,0) {$\\ab{4567}$};\n\\node[red] (s3) at (0,1.25) {$\\langle 7 (23) (45) (16) \\rangle $};\n\\node[frozenblue] (1567) at (-6,1.25) {$\\ab{1567}$};\n\\node[frozenblue] (1267) at (6,1.25) {$\\ab{1267}$};\n\\node[blue] (1467) at (-6, 2.5) {$\\ab{1467}$};\n\\node[blue] (s1) at (-2.5, 2.5) {$\\langle 1 (23) (45) (67) \\rangle $};\n\\node[blue] (s2) at (2.5, 2.5) {$\\langle 6 (23) (45) (17) \\rangle $};\n\\node[blue] (1367) at (6, 2.5) {$\\ab{1367}$};\n\\node[frozen] (1234) at (-4.65,3.75) {$\\ab{1234}$};\n\\node[frozen] (2345) at (0,3.75) {$\\ab{2345}$};\n\\node[frozen] (3456) at (4.65,3.75) {$\\ab{3456}$};\n\n\\draw[norm] (s1) -- (1467);\n\\draw[norm] (1367) -- (s2);\n\\draw[norm] (2367) -- (1237);\n\\draw[norm] (2367) -- (4567);\n\n\\draw[norm] (1467) -- (1567);\n\\draw[norm] (1367) -- (1267);\n\\draw[norm] (s3) -- (2367);\n\\draw[norm] (s1.210) -- (1237);\n\\draw[norm] (4567) -- (s2.330);\n\n\\draw[->, shorten <=3pt] (1237.north east) -- (s3.south west);\n\\draw[->, shorten >=3pt] (s3.north west) -- (s1.south);\n\\draw[->, shorten <=3pt] (s2.south) -- (s3.north east);\n\\draw[diag] (1467.north) -- (1234.south west);\n\\draw[diag] (1234.south east) -- (s1.160);\n\\draw[diag] (s1.20) -- (2345.south west);\n\\draw[diag] (2345.south east) -- (s2.160);\n\\draw[diag] (s2.20) -- (3456.south west);\n\\draw[diag] (3456.south east) -- (1367.north);\n\n\\begin{scope}[on background layer]\n\\draw[fill=red!20, thin, red!20] (s3.240) rectangle (2367.60);\n\\end{scope}\n}\n\\end{tikzpicture}\n\\end{center}\n\\caption{A cluster corresponding to the $6^{\\text{th}}$ term in $\\mathcal{A}_{7,2}$.}\n\\label{n2mhvterm6}\n\\end{figure}\n\n\\subsubsection*{Eight points}\nThe eight-point N$^{2}$MHV amplitude is the first true N$^{2}$MHV amplitude in that it is not equivalent to the parity conjugate of another N$^{k<2}$MHV amplitude. Explicitly it is given by\n\n{\\footnotesize\n\\begin{align}\n\\label{eq:8ptAmp}\n\\!\\!\\!\\mathcal{A}_{8,2} &=  \\mathcal{A}_{7,2} \\notag \\\\\n&+ \\frac{\\dab{123478}}{\\ab{1234}\\ab{1238}\\ab{1278}\\ab{1478}\\ab{2347}\\ab{3478}} \\notag \\\\\n&+ \\frac{\\dab{134578}}{\\ab{1345}\\ab{1348}\\ab{1378}\\ab{1578}\\ab{3457}\\ab{4578}} \\notag \\\\\n&+ \\frac{\\dab{145678}}{\\ab{1456}\\ab{1458}\\ab{1478}\\ab{1678}\\ab{4567}\\ab{5678}} \\notag \\\\\n&+ \\frac{\\dab{12345} \\dab{15678}}{\\ab{1234}\\ab{1235}\\ab{1245}\\ab{1345}\\ab{1567}\\ab{1568}\\ab{1578}\\ab{1678}\\ab{2345}\\ab{5678}} \\notag \\\\\n&- \\frac{\\dab{12378} \\dab{23456}}{\\ab{1237}\\ab{1238}\\ab{1278}\\ab{2345}\\ab{2346}\\ab{2356}\\ab{2378}\\ab{3456}\\langle 2 3 \\bar{5} \\cap \\bar{8} \\rangle } \\notag \\\\\n&+ \\frac{\\dab{12345} \\dab{14578}}{\\ab{1234}\\ab{1245}\\ab{1345}\\ab{1457}\\ab{1458}\\ab{1578}\\ab{2345}\\ab{4578}\\quadd{1}{23}{45}{78}} \\notag \\\\\n&+ \\frac{\\dab{12356} \\dab{15678}}{\\ab{1235}\\ab{1256}\\ab{1356}\\ab{1567}\\ab{1568}\\ab{1678}\\ab{2356}\\ab{5678}\\quadd{1}{23}{56}{78}} \\notag \\\\\n&+ \\frac{\\dab{13456} \\dab{15678}}{\\ab{1345}\\ab{1356}\\ab{1456}\\ab{1567}\\ab{1568}\\ab{1678}\\ab{3456}\\ab{5678}\\quadd{1}{34}{56}{78}} \\notag \\\\\n&+ \\frac{\\dab{12378} \\dab{23467}}{\\ab{1237}\\ab{1238}\\ab{1278}\\ab{2346}\\ab{2347}\\ab{2367}\\ab{2378}\\ab{3467}\\quadd{7}{23}{46}{18}} \\notag \\\\\n&+ \\frac{\\dab{13478} \\dab{34567}}{\\ab{1347}\\ab{1348}\\ab{1378}\\ab{3456}\\ab{3457}\\ab{3467}\\ab{3478}\\ab{4567}\\quadd{7}{34}{56}{18}} \\notag \\\\\n&+ \\frac{\\dab{12378} \\dab{14578}}{\\ab{1238}\\ab{1278}\\ab{1378}\\ab{1478}\\ab{1578}\\ab{4578}\\quadd{1}{23}{45}{78}\\quadd{7}{23}{45}{18}} \\notag \\\\\n&+ \\frac{\\dab{12378} \\dab{15678}}{\\ab{1238}\\ab{1278}\\ab{1378}\\ab{1578}\\ab{1678}\\ab{5678}\\quadd{1}{23}{56}{78}\\quadd{7}{23}{56}{18}} \\notag \\\\\n&+ \\frac{\\dab{13478} \\dab{15678}}{\\ab{1348}\\ab{1378}\\ab{1478}\\ab{1578}\\ab{1678}\\ab{5678}\\quadd{1}{34}{56}{78}\\quadd{7}{34}{56}{18}} \\notag \\\\\n&+ \\frac{\\dab{12378} \\Delta}{\\ab{1237}\\ab{1238}\\ab{1378}\\ab{2378}\\ab{4567}\\langle 2 3 \\bar{5} \\cap \\bar{8} \\rangle \\quadd{7}{23}{45}{18}\\quadd{7}{23}{46}{18}\\quadd{7}{23}{56}{18}}\n\\end{align}\n}\nwhere in the last term we have the quantity $\\Delta^{0|4} = \\delta^{0|4}(\\chi_2 \\ab{1378}\\ab{4567} - \\chi_3 \\ab{1278}\\ab{4567} - \\chi_4 \\quadd{7}{23}{56}{18} + \\chi_5 \\quadd{7}{23}{46}{18} - \\chi_6 \\quadd{7}{23}{45}{18} - \\chi_7 \\langle 2 3 \\bar{5} \\cap \\bar{8} \\rangle)$.\n\nAt eight points, ${\\rm Conf}_8(\\mathbb{P}^3)$ is an infinite cluster algebra, however we can still associate finite subalgebras to each of the 20 terms in the amplitude. These subalgebras are displayed in Table \\ref{n2mhv-tab} where terms 1-6 are those in \\eqref{eq:7ptAmp}.\n\\begin{table}\n\\centering\n\t{\n\t\\begin{tabular}{@{} c  c  c  c  c  c  c  c @{}}\n          \\toprule\n\tTerm & Sub-Algebra & Term & Sub-Algebra & Term & Sub-Algebra & Term & Sub-Algebra \\\\ \\midrule\n\t$1$ & $A_3 \\times A_3$ & $6$ & $A_2$ & $11$ & $A_3 \\times A_1$ & $16$ & $A_1 \\times A_1$ \\\\ \n\t$2$ & $A_3 \\times A_2$ & $7$ & $A_3 \\times A_3$ & $12$ & $A_1 \\times A_1$ & $17$ & $A_2 \\times A_1$ \\\\ \n\t$3$ & $A_3 \\times A_1$ & $8$ & $A_3$ & $13$ & $A_1 \\times A_1$ & $18$ & $A_3 \\times A_2$ \\\\ \n\t$4$ & $A_2 \\times A_1$ & $9$ & $A_3 \\times A_3$ & $14$ & $A_2 \\times A_1$ & $19$ & $A_2 \\times A_1$ \\\\ \n          $5$ & $A_1 \\times A_1$ & $10$ & $A_3\u00a3$ & $15$ & $A_1 \\times A_1$ & $20$ & $A_2$ \\\\\n          \\bottomrule\n\t\\end{tabular}\n\t}\n\t\\caption{Subalgebras associated to terms in $\\mathcal{A}_{8,2}$.}\n\t\\label{n2mhv-tab}\n\\end{table}\nAlthough the subalgebras shown in Table \\ref{n2mhv-tab} are all finite, at higher points they may become infinite. For example, the subalgebra associated to \\eqref{eq:6ptAmp} at ten points will be $A_3 \\times {\\rm Conf}_8(\\mathbb{P}^3)$ which is infinite as ${\\rm Conf}_8(\\mathbb{P}^3)$ is infinite. \n\nThe tenth term is a new type of term of the form\n\\begin{equation}\n[12345][56781]\\,,\n\\end{equation}\nto which we can associate an $A_3$ subalgebra, a cluster belonging to which takes the form shown in Fig. \\ref{fig:RProductCluster} below. The left and right columns of blue nodes in Fig. \\ref{fig:RProductCluster} correspond to the poles of $[12345]$ and $[56781]$ respectively while the red column signifies the $A_3$ subalgebra to which we associate this term.\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}[scale=0.75]\n{\\footnotesize\n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node[frozen] (1238) at (1.25,1.25) {$\\ab{1238}$};\n\\node[frozen] (1278) at (3.75,1.25) {$\\ab{1278}$};\n\\node[frozenblue] (1678) at (7, 1.25) {$\\ab{1678}$};\n\\node[blue] (1235) at (0, 0) {$\\ab{1235}$};\n\\node (1258) at (2.5, 0) {$\\ab{1258}$};\n\\node[blue] (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node (1256) at (2.5,-1.25) {$\\ab{1256}$};\n\\node[blue] (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node (1456) at (2.5,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (-2,-3.75) {$\\ab{2345}$};\n\\node[frozen] (3456) at (1.25,-3.75) {$\\ab{3456}$};\n\\node[blue] (1578) at (5,0) {$\\ab{1578}$};\n\\node[blue] (1568) at (5,-1.25) {$\\ab{1568}$};\n\\node[blue] (1567) at (5,-2.5) {$\\ab{1567}$};\n\\node[frozen] (4567) at (3.75,-3.75) {$\\ab{4567}$};\n\\node[frozenblue] (5678) at (7, -3.75) {$\\ab{5678}$};\n\n\\begin{scope}[blend mode=overlay,overlay]\n          \\node[rectangle, rounded corners, fit=(1258)(1256)(1456),fill=red!20, inner sep=0pt] {};\n        \\end{scope}\n\\node[frozenblue] (1234) at (-2, 1.25) {$\\ab{1234}$};\n\\node[frozen] (1238) at (1.25,1.25) {$\\ab{1238}$};\n\\node[frozen] (1278) at (3.75,1.25) {$\\ab{1278}$};\n\\node[frozenblue] (1678) at (7, 1.25) {$\\ab{1678}$};\n\\node[blue] (1235) at (0, 0) {$\\ab{1235}$};\n\\node (1258) at (2.5, 0) {$\\ab{1258}$};\n\\node[blue] (1245) at (0,-1.25) {$\\ab{1245}$};\n\\node (1256) at (2.5,-1.25) {$\\ab{1256}$};\n\\node[blue] (1345) at (0,-2.5) {$\\ab{1345}$};\n\\node (1456) at (2.5,-2.5) {$\\ab{1456}$};\n\\node[frozenblue] (2345) at (-2,-3.75) {$\\ab{2345}$};\n\\node[frozen] (3456) at (1.25,-3.75) {$\\ab{3456}$};\n\\node[blue] (1578) at (5,0) {$\\ab{1578}$};\n\\node[blue] (1568) at (5,-1.25) {$\\ab{1568}$};\n\\node[blue] (1567) at (5,-2.5) {$\\ab{1567}$};\n\\node[frozen] (4567) at (3.75,-3.75) {$\\ab{4567}$};\n\\node[frozenblue] (5678) at (7, -3.75) {$\\ab{5678}$};\n        \n\\draw[norm] (1578) -- (1258);\n\\draw[norm] (1245) -- (1256);\n\\draw[norm] (1568) -- (1256);\n\\draw[norm] (1345) -- (1456);\n\\draw[norm] (1456) -- (1567);\n\\draw[norm] (1258) -- (1235);\n\n\\draw[norm] (1235) -- (1245);\n\\draw[norm] (1256) -- (1258);\n\\draw[norm] (1245) -- (1345);\n\\draw[norm] (1256) -- (1456);\n\\draw[norm] (1568) -- (1578);\n\\draw[norm] (1567) -- (1568);\n\n\\draw[->,shorten <=4pt, shorten >=3pt] (1234.south east) -- (1235.north west);\n\\draw[->,shorten <=4pt, shorten >=3pt] (1345.south west) -- (2345.north east);\n\\draw[->,shorten <=4pt, shorten >=3pt] (1578.north east) -- (1678.south west);\n\\draw[->,shorten <=4pt, shorten >=3pt] (5678.north west) -- (1567.south east);\n\\draw[diag] (1258.south east) -- (1568.north west);\n\\draw[diag] (1456.north west) -- (1245.south east);\n\\draw[norm] (3456) -- (1345);\n\\draw[norm] (4567) -- (1456);\n\\draw[norm] (1456) -- (3456);\n\\draw[norm] (1567) -- (4567);\n\\draw[norm] (1235) -- (1238);\n\\draw[norm] (1238) -- (1258);\n\\draw[norm] (1258) -- (1278);\n\\draw[norm] (1278) -- (1578);\n}\n\\end{tikzpicture}\n\\caption{A cluster containing the poles of $[12345][56781]$ in ${\\rm Conf}_8(\\mathbb{P}^3)$.}\n\\label{fig:RProductCluster}\n\\end{figure}\n\n\n\\subsection{Discussion}\n\nWe have shown that all NMHV R-invariants obey the cluster adjacency property in that their poles can all be found together in some cluster. We have also shown that the BCFW terms in the expansion of N$^2$MHV trees also obey cluster adjacency for six, seven, and eight points. To each term is associated some subalgebra in the full polytope where every cluster contains all of the poles. Similar structures have emerged in the study of the Grassmannian integrals of \\cite{ArkaniHamed:2009dn,Mason:2009qx} and on-shell diagrams \\cite{ArkaniHamed:2012nw}. The difference here is that the properties we observe between poles (both physical and spurious) are phrased in the same language that we have found relates the branch cuts (symbol entries) of the integrated amplitudes.\n\nThe results for tree-level NMHV and N$^2$MHV are highly suggestive that there should exist a general relation between the singularities of the Yangian invariant leading singularities and the cluster algebras associated to ${\\rm Conf}_n(\\mathbb{P}^3)$.  A natural question is whether an extension of the notion of cluster adjacency holds for all Yangian invariants. This would lead us to consider quantities which go beyond $\\mathcal{A}$-coordinates for ${\\rm Conf}_n(\\mathbb{P}^3)$ such as the four-mass box leading singularity which exhibits square root branch cuts in momentum twistor variables. Studying such quantities should lead to insight on what cluster adjacency has to say beyond rational $\\mathcal{A}$-coordinates and should have implications for understanding the boundary structure of higher polytopes and the type of transcendental functions which appear beyond seven-point amplitudes.\n\n\n\n\nCertain operations can also be performed on Yangian invariants \\cite{ArkaniHamed:2010kv}, e.g. the `fusing' of two Yangian invariants is also a Yangian invariant. Could one find a cluster interpretation of such an operation? The cluster shown in Fig. \\ref{fig:RProductCluster} contains the poles of the product of two Yangian invariants and could also be indicative of the amalgamation procedure \\cite{ArkaniHamed:2012nw} whereby two clusters can be joined together to produce a cluster in a larger algebra.\n\n\\section{NMHV loop amplitudes}\n\\label{NMHVloops}\n\nNow we are in a position to relate the cluster adjacency properties described in the two previous sections. The first amplitudes which exhibit both poles and cuts non-trivially are the NMHV loop amplitudes. \n\n\n\n\n\\subsection{Hexagons}\n\nThe BDS-like subtracted NMHV hexagon is often written in terms of a parity even function $E(u,v,w)=E(Z_1,\\ldots,Z_6)$ and a parity odd function\\footnote{Sometimes $\\tilde{E}(y_u,y_v,y_w)$ denoted simply as $\\tilde{E}(u,v,w)$, in which case one should in addition take care to remember its odd parity.} $\\tilde{E}(y_u,y_v,y_w)=\\tilde{E}(Z_1,\\ldots,Z_6)$, where we have drawn attention to their dependence on the twistor variables. Here we will adopt a shorthand notation which makes reference to the which of the cyclically ordered twistors $Z_i$ sits in the first argument,\n\\begin{equation}\n\\begin{aligned}\nE_1 &= E(u,v,w)\\,, \\quad &E_2 &= E(v,w,u)\\,,\\quad &E_3 &= E(w,u,v)\\,, \\\\\n\\tilde{E}_1 &= E(y_u,y_v,y_w)\\,, \\quad &\\tilde{E}_2 &= -\\tilde{E}(y_v,y_w,y_u)\\,,\\quad &\\tilde{E}_3 &= E(y_w,y_u,y_v)\\,.\n\\end{aligned}\n\\end{equation}\nThe parity properties of $E$ and $\\tilde{E}$ imply\n\\be\nE_4 = E_1\\,, \\qquad \\tilde{E}_4 = - \\tilde{E}_1\\,.\n\\ee\nWith this notation the hexagon NMHV amplitude takes the form\n\\begin{align}\n\\label{NMHV6pt}\n\\mathcal{E}_{6,{\\rm NMHV}}   = &E_{1}[(1)+(4)] + E_{2}[(2)+(5)]+E_{3}[(3)+(6)] \\notag\\\\\n+ &\\tilde{E}_{1}[(1)-(4)] + \\tilde{E}_{2}[(2)-(5)] + \\tilde{E}_{3}[(3)-(6)] \\,.\n\\end{align}\nHere we have adopted a common shorthand notation for the R-invariants: we write $(1) = [23456]$ and cyclically related formulae. The function $\\tilde{E}$ is taken to obey\n\\begin{equation}\n\\tilde{E}_{1} - \\tilde{E}_{2} + \\tilde{E}_{3}=0\\,.\n\\end{equation}\nWe may equivalently write $\\mathcal{E}_6^{\\rm NMHV}$ as follows,\n\\begin{equation}\n\\label{NMHV6}\n\\mathcal{E}_{6,{\\rm NMHV}} = (1) F_{1} + \\text{ cyc.} \\qquad F_1=E_1+\\tilde{E}_1\\,.\n\\end{equation}\nIn (\\ref{NMHV6}) the notation `cyc' refers to all cyclic rotations of the momentum twistors. At $L$ loops the functions $E$ and $\\tilde{E}$ are weight $2L$ polylogarithms.\n\nTo discuss the cluster adjacency properties of the hexagon NMHV amplitudes we should consider the $(2L-1,1)$ coproduct of $\\mathcal{E}_{6,{\\rm NMHV}}$,\n\\be\n\\label{dFNMHV}\n\\mathcal{E}_{6,{\\rm NMHV}}^{(2L-1,1)} = (1) \\sum_{i<j<k<l} [F_{1}^{\\langle ijkl \\rangle} \\otimes \\langle ijkl \\rangle]  + \\text{ cyc.}\n\\ee\nCluster adjacency manifest itself in two ways in the above expression. Firstly the $F^{\\langle ijkl \\rangle}$ are neighbour set functions for $\\langle ijkl \\rangle$. This is the statement that $F$ and hence $E$ and $\\tilde{E}$ cluster adjacent polylogarithms in the sense described in Sect. \\ref{CApolysdef}. Secondly we find that the different functions $F_1^{\\langle ijkl \\rangle}$ appearing in (\\ref{dFNMHV}) are constrained by the fact that $F_1$ appears with the R-invariant $(1)$ in (\\ref{NMHV6}).\n\nIn order to reveal the additional constraints that cluster adjacency places on the form of $F$ we exploit the fact that the R-invariants obey the identity\n\\begin{equation}\n(1)-(2)+(3)-(4)+(5)-(6)=0\\,.\n\\end{equation}\nThis allows us to modify the presentation of $\\mathcal{E}_{6,{\\rm NMHV}}^{(2L-1,1)}$ by adding to it a vanishing term of the form\n\\begin{equation}\n\\label{addzero}\n[(1)-(2)+(3)-(4)+(5)-(6)] Z_{1} \\,,\n\\end{equation}\nwhere $Z$ is given by\n\\be\nZ_{1} = \\sum_{i<j<k<l} [Z^{\\langle ijkl \\rangle}_{1}\\otimes \\langle ijkl \\rangle]\\,.\n\\ee\nHere (by cyclically symmetrising (\\ref{addzero}) if necessary) we can require that $Z$ is anti-cyclic,\n\\begin{equation}\n\\label{cycZ}\nZ_{2}= - Z_{1}\\,.\n\\end{equation}\nThis means that the presentation of $\\mathcal{E}_{6,{\\rm NMHV}}^{(2L-1,1)}$ is still manifestly cyclic,\n\\begin{equation}\n\\label{E6copmod}\n\\mathcal{E}_{6,{\\rm NMHV}}^{(2L-1,1)} = (1) \\sum_{i<j<k<l} [(F_{1}^{\\langle ijkl \\rangle} + Z_1^{\\langle ijkl \\rangle}) \\otimes \\langle ijkl \\rangle]  + \\text{ cyc.}\n\\end{equation}\n\n\nWe find the following additional cluster adjacency property of all hexagon NMHV loop amplitudes:\n\\emph{there exists a $Z$ such that the only $\\mathcal{A}$-coordinates $\\langle ijkl \\rangle$ appearing in (\\ref{E6copmod}) are in the neighbour set of every $\\mathcal{A}$-coordinate in the denominator of the R-invariant $(1)$}. \n\nAs we have discussed in Sect. \\ref{NMHVBCFW}, the R-invariant $(1)=[23456]$ is associated to a single cluster in ${\\rm Conf}_6(\\mathbb{P}^3)$ (in fact it is the one whose triangulation involves all the chords of the form $(1i)$). It follows that the only unfrozen $\\mathcal{A}$-coordinates allowed in the final entries are the ones of that cluster, namely $\\langle 2346 \\rangle = (15)$, $\\langle 2356 \\rangle = (14)$ and $\\langle 2456\\rangle=(13)$.  The following unfrozen $\\mathcal{A}$-coordinates,\n\\begin{equation}\n\\label{absentAs}\n\\{\\langle 1235 \\rangle,\\langle 1245 \\rangle,\\langle 1246 \\rangle,\\langle 1345 \\rangle,\\langle 1346 \\rangle,\\langle 1356 \\rangle\\}\\,,\n\\end{equation}\nare therefore forbidden in the sum in (\\ref{E6copmod}) above.\n\nNote that since $Z$ is multiplied by zero in (\\ref{addzero}) we do not need to require that it is integrable, nor even that it is homogeneous.  Nevertheless, the fact that it exists and obeys (\\ref{cycZ}) has the following implications for the final entries (or $(n-1,1)$ coproduct) of $F$,\n\\begin{align}\nF_{1}^{\\langle 1235 \\rangle}  &= - Z_{1}^{\\langle 1235 \\rangle}\\,,\\notag\\\\\nF_{1}^{\\langle 1246 \\rangle}  &= - Z_{1}^{\\langle 1246 \\rangle}\\,,\\notag\\\\\nF_{1}^{\\langle 1345 \\rangle}  &= - Z_{1}^{\\langle 1345 \\rangle}\\,,\\notag\\\\\nF_{1}^{\\langle 1356 \\rangle}  &= - Z_{1}^{\\langle 1356 \\rangle}\\,,\\notag\\\\\nF_{1}^{\\langle 1245 \\rangle}  &= - Z_{1}^{\\langle 1245 \\rangle}\\,, \\notag\\\\\nF_{1}^{\\langle 1346 \\rangle}  &= - Z_{1}^{\\langle 1346 \\rangle}\\,.\n\\end{align}\nThe anti-cyclicity of $Z$ implies\\footnote{We remind the reader that the subscripts refer to the arguments of functions. For example, $Z_6^{\\langle 1235 \\rangle}$ means $Z_1^{\\langle 1235 \\rangle}|_{Z_i \\rightarrow Z_{i-1}}$ and not the $\\langle1235 \\rangle$ coproduct element of $Z_6$.}\n\\begin{align}\nZ^{\\langle 1246 \\rangle}_{1} &= - Z^{\\langle 1235 \\rangle}_{6}\\,,\\notag\\\\\nZ^{\\langle 1345 \\rangle}_{1} &= + Z^{\\langle 1235 \\rangle}_{3}\\,,\\notag\\\\\nZ^{\\langle 1356 \\rangle}_{1} &= + Z^{\\langle 1235 \\rangle}_{5}\\,,\\notag \\\\\nZ^{\\langle 1346 \\rangle}_{1} &= + Z^{\\langle 1245 \\rangle}_{3}\\,.\n\\end{align}\nCombining the above two sets of relations we deduce that adjacency implies the following relations among the coproducts of $F$\\,,\n\\begin{align}\nF_{1}^{\\langle 1246 \\rangle}  &= -F_{6}^{\\langle 1235 \\rangle} \\,,\\notag\\\\\nF_{1}^{\\langle 1345 \\rangle}  &=+F_{3}^{\\langle 1235 \\rangle} \\,, \\notag\\\\\nF_{1}^{\\langle 1356 \\rangle}  &=+ F_{5}^{\\langle 1235 \\rangle} \\,, \\notag\\\\\nF_{1}^{\\langle 1346 \\rangle}  &=+ F_{3}^{\\langle 1245 \\rangle}  \\,.\n\\label{Fcoproductrels}\n\\end{align}\nThe equations (\\ref{Fcoproductrels}) are the consequences of cluster adjacency between the final entries of the coproduct of $F=E+\\tilde{E}$ and the R-invariants.\n\nAs discussed in \\cite{Dixon:2015iva}, similar coproduct relations follow from the $\\bar{Q}$-equation of \\cite{CaronHuot:2011kk,Bullimore:2011kg}. We may ask how the $\\bar{Q}$ conditions are related to the adjacency ones. To do this it is simplest to count how many homogeneous (final entry)$\\otimes$(R-invariant) combinations are allowed by cluster adjacency. To do this one may choose five independent R-invariants, say $(1),(2),(3),(4),(5)$, and nine $d \\log$'s of multiplicatively independent homogeneous letters and make an arbitrary linear combination of all 45 possible products. We expand the resulting expression into the $d \\log \\langle ijkl \\rangle$ and eliminate all pairs $(m)\\,d\\log \\langle ijkl \\rangle $ which obey adjacency (taking care to remember that some $\\mathcal{A}$-coordinates are compatible with the R-invariant $(6) = (1)-(2)+(3)-(4)+(5)$) and require the resulting combination to vanish. This yields 27 conditions, leaving 18 linearly independent homogeneous (final entry)$\\otimes$(R-invariant) combinations. This is exactly the same number of linearly independent combinations which are compatible with the $\\bar{Q}$ final entry conditions described in \\cite{Dixon:2015iva}. \n\nWe conclude that for the NMHV hexagon, the cluster adjacency property is equivalent to the $\\bar{Q}$ final entry conditions. One should nevertheless stress that the $\\bar{Q}$ equation itself is stronger than just the final entry conditions as it expresses the $(2L-1,1)$ coproduct entries in terms of and integral over a limit of certain heptagon amplitudes.\nWe find it remarkable that cluster adjacency property in its various forms encompasses both the (extended) Steinmann conditions as well as some of the implications of dual superconformal symmetry.\n\n\n\\subsection{Heptagons}\n\nIn the case of heptagons it is possible to write down 21 R-invariants,\n\\begin{equation}\n  [34567]=(12), \\,\\,\\,\n  [24567]=(13), \\,\\,\\,\n  [23567]=(14)\\,\\,\\,\\text{\\& cyclic}\\,.\n\\end{equation}\nThey satisfy seven six-term identities of the form\n\\begin{equation}\n\\label{eq:sixterm}\n(12) - (13) + (14) - (15) + (16) - (17) = 0 \\quad \\text{\\& cyclic} \\,.\n\\end{equation}\nOnly six of these identities are linearly independent and the number\nof independent R-invariants is therefore 15. In a canonical basis (as\nused in \\cite{Dixon:2016nkn,CaronHuot:2011kk}) which comprises the\ntree amplitude and 14 other R-invariants, the BDS-like-normalised\namplitude is expressed as follows:\n\\begin{equation}\n  \\label{eq:hepnonred}\n \\mathcal{E}_{7,{\\rm NHMV}}\n  =\n  \\mathcal{A}_{7,1} E_0\n  +\\bigl[(12)\\, E_{12}+\\text{cyclic} + (14)\\, E_{14}+\\text{cyclic}\\bigr]\\, ,\n\\end{equation}\nwhere $\\mathcal{A}_{7,1}$ is equal to the NMHV tree amplitude, given in (\\ref{A71}).\n\nThe property of cluster adjacency again manifests itself in the heptagon NMHV amplitudes. It is possible to find a representation of the $(2L-1,1)$ coproduct of the form\n\\be\n\\label{heptcoproduct}\n\\mathcal{E}_{7,{\\rm NMHV}}^{(2L-1,1)} = \\sum_{a\\in\\mathcal{A}} \\bigl[[(12) e_{12}^{a} + (13)e_{13}^{a} + (14)e_{14}^{a}]\\otimes a \\bigr] + \\text{ cyc.}\n\\ee\nHere the sum is over the heptagon alphabet (\\ref{heptletters}). As in the hexagon case, adjacency manifests itself in two ways in (\\ref{heptcoproduct}). Firstly each of the $e_{ij}^a$ is a weight $(2L-1)$ heptagon neighbour set function for the letter $a$. This implies that the functions $E_0$ and $E_{ij}$ in (\\ref{eq:hepnonred}) are cluster adjacent polylogarithms. Secondly, only some of the $e_{ij}^a$ are non-zero: the ones where the letter $a$ is cluster adjacent to all of the poles of the R-invariant $(ij)$. For example, the\nR-invariant $(12)$ contains three poles that are non-frozen cluster\n${\\cal A}$ coordinates, namely $\\langle 3567 \\rangle \\sim a_{34}$,\n$ \\langle 3467 \\rangle \\sim a_{15}$, and $\\langle 3457 \\rangle \\sim a_{26}$:\n\\begin{equation}\n  (12) =\n  \\frac{\\bigl(\\langle 3456 \\rangle \\chi_7 + \\text{cyclic}\\bigr)^4}\n  {\\langle 4567 \\rangle \\langle 3567 \\rangle \\langle 3467 \\rangle \\langle 3457 \\rangle \\langle 3456 \\rangle}\\,.\n\\end{equation}\nThe intersection of the homogeneous neighbour sets of these coordinates defines the neighbour set of the R-invariant $(12)$, and similarly for the other R-invariants:\n\\begin{equation}\n  \\begin{aligned}[t]\n    \\hns{(12)} &= \\hns{a_{34}} \\, \\cap \\hns{a_{15}}\\, \\cap \\hns{a_{26}}\\\\\n    &= \\{a_{11}, a_{12}, a_{15}, a_{21},a_{22},a_{26},a_{31}, a_{32}, a_{34}, a_{53}, a_{55}, a_{57}\\}\\,,\\\\\n     \\hns{(13)}&= \\hns{a_{21}} \\, \\cap \\hns{a_{33}}\\, \\cap \\hns{a_{41}}\\, \\cap \\hns{a_{43}}\\\\\n     &= \\{a_{11}, a_{13}, a_{21}, a_{23},a_{31},a_{33},a_{41}, a_{43}, a_{62}\\}\\,,\\\\\n     \\hns{(14)}&= \\hns{a_{11}} \\, \\cap \\hns{a_{14}}\\, \\cap \\hns{a_{21}}\\, \\cap \\hns{a_{34}}\\, \\cap \\hns{a_{46}}\\\\\n     &= \\{a_{11}, a_{14}, a_{21}, a_{24},a_{31},a_{34},a_{46}\\}\\,.\n  \\end{aligned}\n  \\label{heptNMHVns}\n\\end{equation}\nOnly the (final entry)$\\otimes$(R-invariant) combinations compatible with the above and their cyclic rotations are allowed by cluster adjacency.\n\nNote that the representation (\\ref{heptcoproduct}) employs the full redundant set of R-invariants. Upon elimination of the redundant R-invariants, the coproducts of the functions $E_0$ and $E_{ij}$ in (\\ref{eq:hepnonred}) above are seen to be related to the quantities $e_{ij}$ via\n\\be\nE_0^a = \\sum_i e^a_{i,i+2}\\,, \\qquad E^a_{12} = e^a_{12} -e^a_{16} - e^a_{24} - e^a_{46}\\,, \\qquad E^a_{14} = e^a_{14} - e^a_{16} - e^a_{46}\\,.\n\\ee\nAs in the hexagon case, we do not require that the combinations $\\sum_a [e^a_{ij} \\otimes a]$ are integrable; only $\\sum_a [E_0^a \\otimes a]$ and $\\sum_a [E_{ij}^a \\otimes a]$ are integrable. Nevertheless, just as in the hexagon case, the existence and adjacency properties of the $e^a_{ij}$ imply relations on the coproducts of the functions $E_0$ and $E_{ij}$.\n\n\n\n\nOut of the $7\\times(7+9+12) = 196$ cluster adjacent (final entry)$\\otimes$(R-invariant) combinations allowed by (\\ref{heptNMHVns}). The following linear combinations of cluster adjacent (final entry)$\\otimes$(R-invariant) products vanish due to identities,\n\\begin{align}\n[(12) - (13) + (14) - (15) + (16) - (17)]\\otimes\\{a_{11},a_{21},a_{31}\\}\n\\end{align}\nas do their cyclic rotations. This allows us to eliminate 21 such combinations leaving 175 independent cluster adjacent combinations.\n\n\nThe 175 combinations form a \nlarger set than the more restricted set of 147 NMHV (final entry)$\\otimes$(R-invariant) combinations derived by Caron-Huot which are\ncompatible with the $\\bar Q$ equation. These 147 combinations are listed in \\cite{Dixon:2016nkn}. Using the\nidentities (\\ref{eq:sixterm}), these NMHV final entries can be\nrewritten in the following manifestly cluster-adjacent way in which\nthe final entries of the function multiplying the R-invariant $(ij)$\nare in the set $\\hnsqbar{(ij)}$ where:\n\\label{sec:clust-adjc-nmhv}\n    \\label{eq:qbarca}\n\\begin{align}\n  \\hnsqbar{(12)} &= \\{ a_{15}, a_{21}, a_{26}, a_{32}, a_{34}, a_{53}, a_{57}\\}\n                  \\subset \\hns{(12)}\\notag \\\\\n  \\hnsqbar{(13)} &= \\{ a_{21}, a_{23}, a_{31}, a_{33}, a_{41}, a_{43}, a_{62}\\}\n                  \\subset \\hns{(13)\\notag }\\\\\n  \\hnsqbar{(14)} &= \\{ a_{11}, a_{14}, a_{21}, a_{24}, a_{31}, a_{34}, a_{46}\\}\n                  \\subset \\hns{(14)}\\qquad  \\text{\\& cyclic}\\,.\n\\end{align}\nThe above set of $7\\times3\\times7 = 147$ (final entry)$\\otimes$(R-invariant) pairs are equivalent up to using identities to the set presented in \\cite{Dixon:2016nkn}. In contrast to the form presented in \\cite{Dixon:2016nkn}, the\n$\\bar Q$-compatible final entries are monomials in the letters, which\nmakes it trivial to verify cluster adjacency properties. Note that\nthe list of (final entry)$\\otimes$(R-invariant) pairs (\\ref{eq:qbarca}) is not\nunique since it is possible to trade some combinations with others\nusing the six-term identities (\\ref{eq:sixterm}).\n\n\nIn \\cite{NMHV4} we will make use of the above cluster adjacent form for the NMHV heptagon amplitude to allow for an efficient implementation of the bootstrap programme at four loops.\n\n\n\n\n  \\section{Conclusions}\n\nWe have explored and extended the role of cluster algebras and their relation to the appearance of singularities in scattering amplitudes in $\\mathcal{N}=4$ super Yang-Mills theory. The picture which emerges is very geometric in nature, the boundary structure of the cluster polytope controls the way in which both poles and branch cuts appear. Codimension-one faces of the cluster polytope correspond to unfrozen $\\mathcal{A}$-coordinates which appear in the symbol alphabet. The branch cuts exhibit a non-abelian structure, with sequential cuts corresponding to faces which do not touch being forbidden. Poles in BCFW terms for tree amplitudes (and more conjecturally Yangian invariants) exhibit an abelianised version of adjacency; they all correspond to $\\mathcal{A}$-coordinates from the same cluster. The same adjacency structure also relates the poles of R-invariants and the final entries (i.e. derivatives) of the polylogarithms which appear in the NMHV amplitudes.\n\nThe structures we have uncovered naturally lead to many further questions.\n\\begin{itemize}\n\\item Can we use adjacency to construct integrable words without\n  having to apply the bootstrap techniques? This question is even of\n  interest if we do not insist on the physical initial entry\n  conditions and indeed one can ask it for all finite cluster\n  algebras, not just the cases of physical interest described here. A\n  hint that this might be possible comes from the observation that\n  mutation pairs $\\{a,a'\\}$ appearing in a triple always appear in the\n  form $[a \\otimes x(a,a') \\otimes a']$ where $x(a,a')$ is the\n  $\\mathcal{X}$-coordinate associated to any mutation which takes $a$\n  to $a'$.\n\n\\item Can we extend our results to general N${}^k$MHV BCFW terms or\n  more generally Yangian invariants? Going beyond BCFW terms will lead\n  to expressions which involve quantities more complicated that\n  $\\mathcal{A}$-coordinates. Perhaps we will learn something about how\n  such singularities interact with the known ones and how they relate\n  to adjacency.\n\n\\item When considering loop amplitudes, to what extent does the\n  structure seen here extend to the octagon and beyond? There are\n  several issues at stake here. Firstly, we would ideally like to\n  define a Steinmann IR finite quantity for all $n$, while the\n  BDS-like subtraction only exists for $n\\neq0 \\text{ mod } 4$. We\n  also know that at eight points and beyond we will have to deal with\n  letters which involve square roots and are not just rational\n  functions of Pl\\\"ucker coordinates. It is important to understand\n  what role adjacency plays when these are included - it will\n  necessarily go beyond the definition of adjacency in terms of\n  $\\mathcal{A}$-coordinates we have used here. This question is\n  connected to the question of whether we can understand the boundary\n  structure of the cluster polytope for ${\\rm Conf}_n(\\mathcal{P}^3)$\n  for $n\\geq8$. Further, there will be the question of whether\n  adjacency can be extended beyond the polylogarithmic case to include\n  the elliptic functions appearing in e.g. the ten-point two-loop\n  N${}^3$MHV amplitude.\n\n\\item To what extent do adjacency constraints arise beyond planar\n  $\\mathcal{N}=4$ amplitudes? For sufficiently many external legs\n  there will always be Steinmann constraints on scattering\n  amplitudes. A natural question is whether these extend to further\n  constraints between pairs of singularities which are not both simple\n  unitarity cuts of amplitudes. The geometrical picture of the\n  relations between singularities described here suggests that it is\n  important to understand the relevant geometry and its boundary\n  structure in the more general setting. This geometry is necessarily\n  more complicated in the general case of massless scattering where\n  dual conformal symmetry is broken.\n\n\\end{itemize}\n\nIt will be fascinating to explore the above questions. Ultimately we\nmight hope to be able to give a simple geometric or algebraic\nconstruction of physical scattering amplitudes.\n\n\\section*{Acknowledgments}\nJMD, JAF and \\\"OCG are supported by ERC consolidator grant 648630 IQFT.\n\n  \\bibliographystyle{JHEP}\n  \n  ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nThe upcoming launch of the James Webb Space Telescope (JWST) promises to usher in a new astrochemical ``ice age'' by allowing for the collection of an unprecedented amount of data regarding species frozen onto dust grains in the interstellar medium (ISM). Astrochemical models will surely figure prominently in both interpreting these data as well as in informing future observing proposals. This opportunity, though, calls for models that are equal to the task and, unfortunately, there remain significant uncertainties regarding the chemistry of dust grains and dust grain ice-mantles \\citep{cuppen_grain_2017} - a situation that has given it the dubious reputation of ``the last refuge of the scoundrel'' \\citep{charnley_molecular_1992}.  \n\nCurrently, it is a common practice in modern astrochemical codes to distinguish between species in the top few monolayers of dust grain ice-mantles (the selvedge), and those underneath in the interior (the bulk), though there exists considerable disagreement as to whether, or to what degree, such bulk species are chemically active \\citep{cuppen_grain_2017}. The hypothesis that chemistry within the ice-mantle is less significant than in the selvedge rests on the following three main lines of reasoning: \n\n\n\\begin{enumerate}[label=(\\alph*)]\n\\item  species in the bulk are more tightly bound,\n\\item  a richer chemistry is possible in the selvedge, given the constant exchange of species with the surrounding gas, and\n\\item  photo-processes, which further drive chemical reactions, are more efficient in the topmost monolayers. \n\\end{enumerate}\n\n\\noindent\nAn implicit assumption behind (a) is that reactions in the bulk occur largely via thermal diffusion, as on surfaces, and thus that reaction rates will be low or negligible for species with high binding energies in cold environments. However, some previous experimental studies strongly suggest that reactions in the bulk can quickly occur among neighboring species, without the need of diffusion \\citep{baragiola_solid-state_1999,bennett_laboratory_2005,abplanalp_study_2016,ghesquiere_reactivity_2018}. \n\nSimilarly, just as selvedge chemistry is stimulated via photo-processes and interactions with the gas - as noted in (b) and (c) - so too have experiments shown that bombardment by energetic ions, which penetrate solids much more efficiently than photons \\citep{gerakines_energetic_2001,spinks_introduction_1990}, causes physicochemical changes that drive a rich chemistry within the ice at low temperatures. This \\textit{radiation chemistry} is quite relevant to the ISM, since ices there are continually bombarded by energetic particles of one kind or another, though typically in the form of cosmic rays \\citep{indriolo_cosmic-ray_2013,rothard_modification_2017}, stellar winds \\citep{madey_far-out_2002,hudson_radiation_2001}, or radionuclide emission \\citep{cleeves_radionuclide_2013}. \n\nThus, the processing of dust grain ice-mantles by ionizing radiation is a very real phenomenon and the fact that such interactions can lead to the formation of astrochemically interesting, even prebiotic, species makes radiation chemistry very promising from a modeling standpoint \\citep{hudson_amino_2008,holtom_combined_2005,kaiser_theoretical_1997,abplanalp_study_2016}. Until recently, though, the variety and complexity of the underlying microscopic processes have hindered attempts to add radiation-chemical reactions to astrochemical codes \\citep{abplanalp_study_2016,shingledecker_cosmic-ray-driven_2018}. \n\nThese processes result from collisions between an incoming particle (the primary ion), and species in the target material. Following the formalism of \\citet{bohr_theory_1913}, it is customary to divide such collisions into two categories, namely, those in which energy is transferred to target nuclei (elastic) or electrons (inelastic). One can therefore express the energy loss per unit path length of the primary ion - called the stopping power - as follows:\n\n\\begin{equation}\n  \\frac{\\mathrm{d}E}{\\mathrm{d}t} \\approx n (S_\\mathrm{n} + S_\\mathrm{e})\n\\end{equation}\n\n\\noindent\nwhere $n$ is the density of the target  (cm$^{-3}$) and $S_\\mathrm{n}$ and $S_\\mathrm{e}$ are, respectively, the so-called nuclear\/elastic and electronic\/inelastic stopping cross sections  (eV-cm$^{2}$).  Inelastic collisions, in turn, can further be divided into those in which target species are ionized or excited. Such ionizations result in the formation of so-called ``secondary electrons'' which typically have energies under 50 eV and can themselves ionize or excite target species, thereby further propagating the physicochemical effects initiated by the primary ion \\citep{spinks_introduction_1990,johnson_energetic_1990}.\n\nRecently in \\citet{shingledecker_general_2018} (hereafter, SH), we introduced a method for treating radiation-chemical processes that is simple enough to include in existing astrochemical models, while simultaneously preserving salient features inferred from experimental data. This method was later added to the astrochemical-type Nautilus v1.1 code \\citep{ruaud_gas_2016}, and used by us to determine what, if any, effects these new processes would have on cold core models \\citep{shingledecker_cosmic-ray-driven_2018}. Preliminary results from that study indicated that cosmic ray induced reactions could indeed result in enhanced abundances for a number of species, including methyl formate and ethanol, and allowed us to include novel reactions such as  insertions \\citep{bergner_methanol_2017,bergantini_mechanistical_2018}. However, given the uncertainties sometimes involved in comparing the results of astrochemical models with observations of interstellar environments - particularly in cases where grain-surface chemistry is involved - questions lingered regarding how well astrochemical models were actually simulating irradiation chemistry. \n\n\nTo that end, we have carried out simulations of two different ice irradiation experiments using an existing astrochemical-type model \\citep{vasyunin_formation_2017}, modified to include the SH method for treating radiation chemistry. Specifically, we have modeled the bombardment of a 5 K pure \\ce{O2} ice by 100 keV protons as described in \\citet{baragiola_solid-state_1999} as well as that of a pure water ice at both 16 K and 77 K by 200 keV protons, as reported in \\citet{gomis_hydrogen_2004-1}. Doing so has not only afforded us the opportunity to compare how well our simulations replicate the experimental data, but also to directly gauge how well our models perform, more generally, with handling reactions in the bulk ice over a range of temperatures. By thus testing our approaches to ice chemistry in well constrained experimental systems, confidence can be increased in the results of models of much less well constrained interstellar environments. \n\nThe rest of the paper is as follows: in \\S \\ref{sec:model} we discuss details of both the model and chemical network used in this work. In \\S \\ref{sec:results} our major results are described and their significance is discussed. Finally, our conclusions and suggestions for future work are presented in \\S \\ref{sec:conclusions}. \n\n\n\n\n\\section{Model and Network} \\label{sec:model}\n\nThe basis of this work is the \\texttt{MONACO} code, described in \\citet{vasyunin_formation_2017}, which uses a rate-equation approach for reactions in the gas phase, and the so-called ``modified rate-equation'' method  for reactions on and in dust grain ice mantles \\citep{garrod_new_2008,garrod_new_2009}. As in \\citet{vasyunin_unified_2013} we assume the selvedge,  a term for the altered layers near the top of the mantle,  is comprised of the top four monolayers of the ice \\citep{vasyunin_unified_2013}.  \n\nWe have modified the code to account for irradiation-chemical reactions on and in ice using the SH method, the foundation of which is the assumption that one of the following outcomes is possible upon collision between an energetic particle, the primary ion or a secondary electron, and some target species, $A$:\n\n\\begin{equation}\nA  \\leadsto A^+ + e^- \n\\tag{P1}\n\\label{p1}\n\\end{equation}\n\n\\begin{equation}\nA  \\leadsto A^+ + e^- \\rightarrow A^* \\rightarrow B^* + C^* \n\\tag{P2}\n\\label{p2}\n\\end{equation}\n\n\\begin{equation}\nA  \\leadsto A^* \\rightarrow B + C \n\\tag{P3}\n\\label{p3}\n\\end{equation}\n\n\\begin{equation}\nA  \\leadsto A^* \n\\tag{P4}\n\\label{p4}\n\\end{equation}\n\n\\noindent\nwhere the curly arrow  indicates a collision. In processes (P1) and (P2), $A$ is ionized, though with fast charge recombination leading to the formation of suprathermal dissociation products, designated with an asterisk, assumed to occur in \\eqref{p2}. Processes (P3) and (P4) represent collisions in which a suprathermal product is electronically excited directly into some higher state which, in the case of (P3), is assumed to lead to the formation of thermal products. The efficiencies of (P1) - (P4) are characterized by radiochemical yields, called $G$-values, which give the number of molecules created (or destroyed) per 100 eV transferred to the system from bombarding particles \\citep{dewhurst_general_1952,spinks_introduction_1990}. A full list of the radiolysis processes used here is given in Table \\ref{tab:gvalues} of Appendix \\ref{sec:radiolysis}. The yields given there were arrived at through a two-step process where first, initial values were estimated usnig the SH method and subsequently adjusted to improve agreement with the experimental data.\nIt should be noted that, following \\citet{shingledecker_cosmic-ray-driven_2018}, we assume the rate of process \\eqref{p1} is zero, i.e. that all charged species produced in the bulk via ionization are quickly neutralized. \n\nFirst-order rate coefficients for each radiolysis process are calculated using\n\n\\begin{equation}\n\tk_\\mathrm{rad} = \\left(\\frac{G}{100\\;\\mathrm{eV}}\\right) S_\\mathrm{e}  \\phi,\n   \\label{krad}\n\\end{equation} \n\n\\noindent\nwith $\\phi$ being the radiation flux and $S_\\mathrm{e}$ the previously described inelastic\/electronic stopping cross section. In this work, we assume the flux to be $1.0\\times10^{11}$ \\ce{H+} $\\mathrm{cm}^{-2}$ $\\mathrm{s^{-1}}$, and use the stopping cross sections given in Table \\ref{tab:experiment}.  \n\nWe assume that, in the bulk, suprathermal species produced via radiolysis rapidly undergo either (a) reaction with a neighbor, or (b) are quenched by the surrounding material. Rate coefficients for the reaction between $A$ and $B$ - where either one could be suprathermal - are calculated using the expression\n\n\\begin{equation}\n  k_\\mathrm{fast} = \n  f_\\mathrm{br}\n  \\left[\n  \\frac{\n  \\nu_0^A + \n  \\nu_0^B\n  }\n  {\n  {N_\\mathrm{bulk}}\n  }\n  \\right]\n  \\mathrm{exp}\n  \\left(-\n  \\frac \n  {E_\\mathrm{act}^{AB}}\n  {T_\\mathrm{ice}}\n  \\right),\n  \\label{ksup}\n\\end{equation}\n\n\\noindent\nwhere  $f_\\mathrm{br}$ is the branching fraction, $N_\\mathrm{bulk}$ is the total number of bulk species in the simulated ice,  \nand $E_\\mathrm{act}^{AB}$ is the activation energy for reaction. Absent other data, we assume here that suprathermal species react barrierlessly. The characteristic vibrational frequency, $\\nu^A_0$, is given by\n\n\\begin{equation}\n  \\nu_{0}^{A} = \\sqrt{\n  \\frac\n  {2n_\\mathrm{s}E^A_\\mathrm{D}}\n  {\\pi^2m_A}\n  },\n  \\label{trialnu}\n\\end{equation}\n\n\\noindent\nwith  $E^A_\\mathrm{D}$ and $m_A$ being the desorption energy  and mass of $A$, respectively, and $n_\\mathrm{s}$ the phsysisorption site density - here $6.55\\times10^{14}$ cm$^{-2}$ \\citep{vasyunin_formation_2017}. This characteristic vibrational frequency is also used as the first-order rate coefficient for the quenching of the suprathermal species by the ice. It should be noted that, particularly in water ice, these electronic excitations, also called excitons, may diffuse from the interior of the ice to the selvedge, where they can drive the desorption of species into the gas \\citep{thrower_highly_2011,marchione_efficient_2016}. Thus, our assumption that these suprathermal species are rapidly quenched should be seen as a first-order approximation of their real behavior.\n\nAs an example of how the above equations are utilized in our code, consider the time-dependent abundance of $B^*$. Assuming $B^*$ is produced via the radiolysis of $A$ as in \\eqref{p2}, and destroyed both via quenching and reaction with some other bulk species, $X$, the rate of change in the number of $B^*$ as a function of time can be described by\n\n\\begin{equation}\n    \\frac{d\\,N_{B^*}}{dt} = k_\\mathrm{rad}^\\mathrm{\\ref{p2}}N_A - \\nu_0^B N_{B^*} - k_\\mathrm{fast}N_X N_{B^*},\n\\end{equation}\n\n\\noindent\nwhere the reaction between $B^*$ and $X$ is, as with all suprathermal species here, assumed to occur barrierlessly.\n\n\\begin{deluxetable*}{lcc}\n\\tablecaption{Physical parameters used in simulations of experiments. \\label{tab:experiment}}\n\\tablehead{\n\\colhead{Parameter} & \\colhead{Value} & \\colhead{Source}  \n}\n\\startdata\n\\multicolumn{3}{c}{\\ce{O2}} \\\\ \\\\\nTemperature     & 5 K   & \\citet{baragiola_solid-state_1999} \\\\\nDensity         & $5.7\\times10^{22}$ cm$^{-3}$ & \\citet{horl_structure_1982} \\\\\n\\ce{H+} Energy & 100 keV &  \\citet{baragiola_solid-state_1999} \\\\\n$S_\\mathrm{e}$  &  $4.6\\times10^{-14}$ eV cm$^2$ & \\citet{baragiola_solid-state_1999} \\\\ \\hline\n\\multicolumn{3}{c}{\\ce{H2O}} \\\\ \\\\\nTemperature  & 16 K, 77 K & \\citet{gomis_hydrogen_2004} \\\\\nDensity      & $3.1\\times10^{22}$ cm$^{-3}$ & \\citet{martonak_evolution_2005} \\\\\n\\ce{H+} Energy & 200 keV & \\citet{gomis_hydrogen_2004} \\\\\n$S_\\mathrm{e}$  &  $3.3\\times10^{-14}$ eV cm$^2$ & \\citet{uehara_calculations_2000} \\\\ \\hline\n\\enddata\n\\end{deluxetable*}\n\nIn order to gauge how well astrochemical models are able to reproduce bulk chemistry, we have run the following three sets of simulations for each experiment considered here using the parameters listed in Table \\ref{tab:experiment}:\n\n\\begin{enumerate}[label=(\\alph*)]\n    \\item a set of models in which thermal radicals and atomic oxygen were assumed to react non-diffusively with neighboring species in the ice with rate coefficients calculated using Eq. \\eqref{ksup},\n    \\item a set of models in which all bulk rate coefficients were calculated using the standard diffusive formula \\citep{hasegawa_models_1992} along with diffusion barriers obtained from the desorption energies listed in Table \\ref{tab:desorption} and no tunneling under diffusion barriers for any species, and \n    \\item a set of models similar to (b) but with tunneling through diffusion barriers allowed for H, \\ce{H2}, and O.\n\\end{enumerate}\n\n\\noindent\n\nFor the third set of simulations, we treat tunneling under diffusion barriers by H and \\ce{H2} using the standard formalism in \\citet{hasegawa_models_1992} with a barrier width of 1.0 \\AA. For atomic oxygen, following results from \\citet{minissale_quantum_2013}, we use a desorption energy of 1040 K and a barrier width of 0.7 \\AA. In all simulations, H and \\ce{H2} were assumed to tunnel through reaction barriers \\citep{tielens_model_1982}. The competition mechanism for systems with chemical activation energies \\citep{chang_gas-grain_2007,herbst_chemistry_2008} is used here in all models only for reactions on the surface.\n\n\n\\begin{deluxetable*}{lcc}\n\\tablecaption{Desorption energies for relevant species. \\label{tab:desorption}}\n\\tablehead{\n\\colhead{Species} & \\colhead{$E_\\mathrm{D}$ (K)} & \\colhead{Source} \n}\n\\startdata\n\\ce{O}  & 1\\,660\\tablenotemark{a} & \\citet{he_new_2015} \\\\\n        &  1\\,040\\tablenotemark{b} & \\citet{minissale_quantum_2013} \\\\\n\\ce{O2}  & 930 & \\citet{jing_formation_2012} \\\\\n\\ce{O3}  & 1\\,833  & \\citet{jing_formation_2012} \\\\\n\\ce{H} & 450  & \\citet{garrod_formation_2006} \\\\\\citep{thrower_highly_2011,marchione_efficient_2016}\n\\ce{H2} & 430  & \\citet{garrod_formation_2006} \\\\\n\\ce{OH} & 2\\,850  & \\citet{garrod_formation_2006} \\\\\n\\ce{H2O} & 5\\,700  & \\citet{garrod_formation_2006} \\\\\n\\ce{HO2} & 4\\,510  & $E_\\mathrm{D}^\\mathrm{O} + E_\\mathrm{D}^\\mathrm{OH}$ \\\\\n\\ce{H2O2} & 5\\,700  & $E_\\mathrm{D}^\\mathrm{OH} + E_\\mathrm{D}^\\mathrm{OH}$ \\\\\n\\enddata\n\\tablenotetext{a}{Used in models (a) and (b)}\n\\tablenotetext{b}{Used in model (c)}\n\\tablecomments{Diffusion barriers in the selvedge were assumed to be $0.5\\times E_\\mathrm{D}$, while those in the bulk to be $0.7\\times E_\\mathrm{D}$.}\n\\end{deluxetable*}\n\nFor the chemical network, we used the reactions listed in Table \\ref{tab:h2onetwork} of Appendix \\ref{sec:network}, as well as the neutral-neutral oxygen reactions listed in Table 7 of \\citet{shingledecker_new_2017}. In cases where we have been unable to find values for activation energies, we have assumed a barrier of $10^4$ K, except for radical-radical reactions, which were assumed to be barrierless. For every thermal reaction of the type given in Table \\ref{tab:h2onetwork}, we included variants involving one suprathermal reactant. To illustrate this point, for every reaction of the form $\\mathrm{A + B \\rightarrow} \\; products$, we include the following suprathermal variants\n\n\\begin{equation}\n    \\mathrm{A^* + B \\rightarrow} \\; products\n    \\tag{R1}\n    \\label{r1}\n\\end{equation}\n\n\\begin{equation}\n    \\mathrm{A + B^* \\rightarrow} \\; products.\n    \\tag{R2}\n    \\label{r2}\n\\end{equation}\n\n\\noindent\nA number of reactions listed in Table \\ref{tab:h2onetwork} have barriers that are quite high for astrochemical networks,\nbut are more common in the combustion-chemical literature from which most are taken. Even though the rates of such reactions involving thermal species will indeed be negligible in our models, the variants involving suprathermal species should be efficient even at the temperatures considered here. For example, even a $40\\,000$ K barrier corresponds to only $\\sim3.4$ eV, a realistic energy for an electronically excited suprathermal species. \n\n\\section{Results and Discussion} \\label{sec:results}\n\nIn order to examine how well astrochemical models using the SH method are capable of reproducing experimental data, as well as to determine the accuracy of bulk chemistry in such codes, we have simulated ice-irradiation systems described in two previous radiation-chemical studies discussed above - the results of which are described below.\n\n\\subsection{H$^+$ Bombardment of Pure O$_2$ Ice}\n\n\\begin{figure*}[ht!]\n\\gridline{\n          \\fig{f1a.eps}{0.45\\textwidth}{(a)}\n         }\n\\gridline{\n          \\fig{f1b.eps}{0.45\\textwidth}{(b)}\n          \\fig{f1c.eps}{0.45\\textwidth}{(c)}\n         }\n\\caption{Results from simulations of an irradiated pure O$_2$ ice using (a) fast bulk reaction of O, (b) standard bulk rate coefficients based on thermal hopping, and (c) O atom tunneling. \n\\label{fig:o2ice}}\n\\end{figure*}\n\nIn this system, the only radical used is atomic oxygen, which can be thermal or suprathermal. We first simulated the irradiation of a pure \\ce{O2} ice at 5 K by 100 keV \\ce{H+}, following the study by \\citet{baragiola_solid-state_1999}, which has previously been successfully modeled by us using the much more detailed Monte Carlo code, \\texttt{CIRIS} \\citep{shingledecker_new_2017}. An irradiated \\ce{O2} ice is, in many respects, an ideal system for testing simulations of radiation chemistry, given the limited number of possible neutral species, i.e. O, \\ce{O2}, and \\ce{O3}. However, in the Monte Carlo code even this seemingly simple system required a network of $\\sim50$ reactions or processes and involved a level of detail beyond the practical capabilities of rate equations-based astrochemical codes, such as explicitly calculating the tracks of incident ions and secondary electrons or following each ice species on a hop-by-hop basis.  \n\nDespite these limitations though,  as shown in Fig. \\ref{fig:o2ice}a, even our simplified approach is capable of reasonably reproducing the concentration of  \\ce{O3} as a function of fluence. Here, fluence - the product of the \\ce{H+} flux and irradiation time - represents the number of ions that have bombarded the ice per cm$^2$. As can be seen in Fig. \\ref{fig:o2ice}, the best results were obtained in model (a), where atomic oxygen is assumed to react non-diffusively in the bulk, yielding excellent agreement between calculated and experimental abundances in the steady stage regime. An analogous assumption was made in our previous Monte Carlo simulations using \\texttt{CIRIS}, with similarly good results  \\citep{shingledecker_new_2017}. The slight under-prediction of \\ce{O3} at fluences under $\\sim10^{13}$ \\ce{H+} cm$^{-1}$ is likely due to additional processes - such as dissociative electron attachment (DEA) - which are also driven by energetic ion bombardment but not currently considered in this simplified treatment of radiolysis \\citep{arumainayagam_low-energy_2010}. Also noticeable in Fig. \\ref{fig:o2ice}a is the fact that atomic oxygen abundances remained low throughout the simulation, despite its continuous production from the dissociation of \\ce{O2} and, to a lesser degree, \\ce{O3}. \n\nComparison with Fig. \\ref{fig:o2ice}b reveals that model (b), using standard Langmuir-Hinshelwood rate coefficients, is the least accurate. There, though the modeled \\ce{O3} abundance is only a factor of a few lower than the experimental values, an unrealistically large abundance of atomic oxygen is predicted, contrary to what has been suggested by previous studies  \\citep{gerakines_ultraviolet_1996,baragiola_solid-state_1999,bennett_laboratory_2005}. This buildup is caused by the very slow rates of thermal diffusion reactions at 5 K, given the high atomic oxygen desorption energy of 1660 K \\citep{he_new_2015}. \n\nIn order to determine the possible effects of quantum tunneling by atomic oxygen through diffusion barriers, as suggested by \\citet{minissale_quantum_2013}, we ran further simulations using their best-fit results of a barrier width of $0.7$ \\AA $\\;$ and a desorption energy of 1040 K (assuming $E_\\mathrm{b} = 0.5 E_\\mathrm{D}$). Tunneling for H and \\ce{H2} is treated using the method of \\citet{hasegawa_models_1992}, in which a barrier width of 1 \\AA$\\;$ is assumed.\nComparison of the results of this model, shown in Fig. \\ref{fig:o2ice}c, with \\ref{fig:o2ice}b reveals that though agreement between the calculated and experimental ozone abundances is improved, the abundance of atomic oxygen remains too high. \n\n\nThe results of these three models demonstrate that best agreement with experimental data - both the measured \\ce{O3} abundance as well as the inferred low O abundance - is obtained when atomic oxygen is assumed to react quickly in the bulk. We note, however, that the predicted atomic oxygen abundances in model (a) may in fact be somewhat too low, based on the observation by \\citet{bennett_laboratory_2005} of an IR feature associated with an [\\ce{O3}...O] complex, though no estimate of the atomic oxygen abundance (or an upper limit) was derived. In such irradiated ices, it is likely that some fraction of the oxygen atoms cannot react quickly, either due to trapping by \\ce{O3} or steric effects, in which case their abundance would indeed be greater than that predicted in our models. These effects could be accounted for in the method used here by decreasing the value of the pre-exponential frequency, $\\nu_0$, thereby increasing the abundance of O. Here, we use the characteristic vibrational frequency given in Eq. \\eqref{trialnu} (which typically has a value on the order of $10^{12}$ s$^{-1}$) as a rough approximation, though in fact, the value of this parameter can be as low as $10^{-3}$ s$^{-1}$ \\citep{theule_thermal_2013}, and rate coefficients obtained using \\eqref{trialnu} should probably be interpreted as upper limits.\n\n\n\\subsection{H$^+$ Bombardment of Pure H$_2$O Ice}\n\nWe next simulated the irradiation of a pure \\ce{H2O} ice by 200 keV \\ce{H+} at both 16 and 77 K \nusing the three different bulk chemistry schemes described in Sec. \\ref{sec:model}. Here, in addition to atomic oxygen, we further assume that all radicals in our network -- H, OH, and \\ce{HO2} -- react quickly in model (a). This system, though substantially more complex than \\ce{O2}, is of greater astronomical interest, given the ubiquity of water ice in planetary bodies \\citep{altwegg_67p\/churyumov-gerasimenko_2015,carlson_europas_2009}, as well as interstellar dust grain ice mantles \\citep{gibb_interstellar_2004}. \n\nShown in Fig. \\ref{fig:relative} are both our calculated \\ce{H2O2} abundances vs proton fluence as well as the approximate steady-state bf experimental values from \\citet{gomis_hydrogen_2004-1} relative to \\ce{H2O} of $\\sim 1.0$\\% at 16 K and $0.25$\\% at 77 K. As with the pure \\ce{O2} ice, model (a) once again provides the best agreement with experimental data and, notably, is the only one of the three to predict a drop in hydrogen peroxide abundance at higher temperatures. \\citet{moore_ir_2000}, who first detected this trend, speculated that it was due to the reaction\n\n\\begin{equation}\n  \\ce{OH + H2O2 -> H2O + HO2}\n  \\tag{R3}\n  \\label{r3}\n\\end{equation}\n\n\\noindent\n\nWe find that, in agreement with \\citet{moore_ir_2000}, reaction \\eqref{r3} is indeed behind the drop in \\ce{H2O2} abundance at $\\sim 80$ K, since, even though it is assumed that radicals such as OH react quickly, \\eqref{r3} only becomes competitive at 77 K due to the 755.3 K barrier \\citep{ginovska_reaction_2007}. In models (b) and (c) the effect of reaction \\eqref{r3} is  reduced by the slow diffusion rates of OH and \\ce{H2O2}. As can be seen in Fig. \\ref{fig:radicals}, without the assumption that radicals react quickly in the bulk, OH, in particular, becomes quite abundant in the ice, even at 77 K.\n\nThe increase in hydrogen peroxide abundance at 77 K in models (b) and (c)  is due to the following reactions:\n\n\\begin{equation}\n  \\ce{OH + OH -> H2O2}\n  \\tag{R4}\n  \\label{r4}\n\\end{equation}\n\\begin{equation}\n  \\ce{H + HO2 -> H2O2},\n  \\tag{R5}\n  \\label{r5}\n\\end{equation}\n\n\\noindent\nwhere here, the higher rate of \\ce{H2O2} formation is due to the increased mobility of the reactants. Enabling quantum tunneling through diffusion barriers, as in model (c), further speeds up the rate of reaction \\eqref{r5}, as well as the formation of the precursor species, \\ce{HO2}, via \n\n\\begin{equation}\n  \\ce{H + O2 -> HO2},\n  \\tag{R6}\n  \\label{r6}\n\\end{equation}\n\n\\noindent\nthereby contributing to the even higher \\ce{H2O2} abundance at 77 K in model (c) compared with model (b).  \n\nWe can gain further insights into how closely our simulations are replicating the experiment by examining $G$(\\ce{H2O2}), the radiolytic hydrogen peroxide formation yield.\nUnfortunately, we cannot simply compare the $G$-values given in Table \\ref{tab:gvalues} with experimentally determined ones directly,\nsince our values are more representative of the immediate creation (or destruction) of target species, i.e. the efficiencies of each of the microscopic radiolytic processes given in \\eqref{p1}-\\eqref{p4}, than the single effective experimental value, which is sensitive to the temperature-dependent chemistry of the system \\citep{spinks_introduction_1990}. However, following the method used in \\citet{moore_ir_2000}, we can estimate what the experimental $G$-value might be from the slope of a linear fit to the abundance curves in Fig. \\ref{fig:doseplot} over the pre-steady-state regime - corresponding to doses of ca. 0-10 eV, where dose is the product of the fluence and $S_\\mathrm{e}$. From this, we calculate the yield of H$_2$O$_2$ at 16 K to be $0.1$ molecules\/100 eV - exactly what was mentioned by \\citet{moore_ir_2000} as the yield in pure \\ce{H2O}.\n\n\\begin{figure*}[p]\n\\gridline{\n          \\fig{f2a.eps}{0.45\\textwidth}{(a)}\n          }\n\\gridline{\n          \\fig{f2b.eps}{0.45\\textwidth}{(b)}\n          \\fig{f2c.eps}{0.45\\textwidth}{(c)}\n          }\n\\caption{\nCalculated abundances of \\ce{H2O2} versus proton fluence from simulations of a pure \\ce{H2O} ice bombarded by 200 keV \\ce{H+} assuming (a) fast bulk reactions of radicals and atomic oxygen, (b) only thermal bulk diffusion, and (c) diffusion barrier tunneling for H, \\ce{H2}, and O. Approximate steady-state hydrogen peroxide abundances from \\citet{gomis_hydrogen_2004-1} at both 16 K and 77 K are represented by the solid, and line-filled boxes, respectively. \n\\label{fig:relative}}\n\\end{figure*}\n\n\\begin{figure*}[p]\n \\gridline{\n          \\fig{f4a.eps}{0.45\\textwidth}{(a)}\n          }         \n \\gridline{\n          \\fig{f4b.eps}{0.45\\textwidth}{(b)}\n          \\fig{f4c.eps}{0.45\\textwidth}{(c)}\n          }         \n\\caption{\nCalculated abundances of \\ce{H}, \\ce{O}, \\ce{OH}, and \\ce{HO2} versus proton fluence at 16 K (solid lines) and 77 K (dashed lines) from simulations of a pure \\ce{H2O} ice bombarded by 200 keV \\ce{H+} assuming (a) fast bulk reactions of radicals and atomic oxygen, (b) only thermal bulk diffusion, and (c) diffusion barrier tunneling for H, \\ce{H2}, and O. \n\\label{fig:radicals}}\n\\end{figure*}\n\n\\begin{figure*}[ht!]\n          \\fig{doseplot.eps}{0.45\\textwidth}{}\n\\caption{\nPercentage \\ce{H2O2} vs. dose for model (a) in which radicals are assumed to react quickly. Following \\citet{moore_ir_2000}, based on the slopes of linear fits to these data, we estimate equivalent measured $G(\\ce{H2O2})$-values of 0.1 and 0.03 molecules\/100 eV for the 16 K and 77 K simulations, respectively. \n\\label{fig:doseplot}}\n\\end{figure*}\n\n\\begin{figure*}[p]\n\\gridline{\n          \\fig{f3a.eps}{0.45\\textwidth}{(a)}\n          }\n\\gridline{\n          \\fig{f3b.eps}{0.45\\textwidth}{(b)}\n          \\fig{f3c.eps}{0.45\\textwidth}{(c)}\n          }\n\\caption{\nCalculated abundances of \\ce{H2}, \\ce{O2}, and \\ce{O3} at 16 K (solid line) and 77 K (dashed line) from simulations of a pure \\ce{H2O} ice bombarded by 200 keV \\ce{H+} assuming (a) fast bulk reactions of radicals and atomic oxygen, (b) only thermal bulk diffusion, and (c) diffusion barrier tunneling for H, \\ce{H2}, and O. \n\\label{fig:others}}\n\\end{figure*}          \n\n\nShown in Fig. \\ref{fig:others} are the abundances of \\ce{H2}, \\ce{O2}, and \\ce{O3} versus fluence. In model (a), the abundance of \\ce{O2} at 16 K is kept low because of destruction via \\eqref{r6} to form \\ce{HO2} but increases at 77 K due, in part, to more efficient formation via \n\n\\begin{equation}\n    \\ce{HO2 + O3 -> O2 + O2 + OH}.\n    \\tag{R7}\n    \\label{r7}\n\\end{equation}\n\n\\noindent\nwhich has a small barrier of 490 K \\citet{burkholder_nasa_2014}. Similarly, the increase in molecular oxygen abundance at 77 K in model (c) is further driven by \n\n\\begin{equation}\n    \\ce{OH + HO2 -> H2O + O2}\n    \\tag{R8}\n    \\label{r8}\n\\end{equation}\n\n\\noindent\nwhere the abundances of OH and \\ce{HO2} are enhanced relative to model (b) because of the effects of* quantum tunneling through diffusion barriers by H, \\ce{H2}, and O - as shown in Fig. \\ref{fig:radicals}. The decreased abundance of these radicals at 77 K in model (b), combined with destruction with atomic oxygen, leads to the drop in [\\ce{O2}] in Fig. \\ref{fig:others}b.\n\nUnfortunately, comparison of our \\ce{O2} results with experimental data, as with \\ce{H2O2}, is complicated by the fact that homonuclear diatomic molecules, lacking permanent dipoles, are IR inactive. Thus, their abundances cannot be measured using standard techniques, such as Fourier Transform Infrared Spectroscoppy (FTIR). Nevertheless, it is well known that \\ce{H2} and \\ce{O2} form during water ice radiolysis based on  analysis of both sputtering products as well as post-irradiation temperature-programmed desorption (TPD) of the ice via mass spectrometry \\citep{johnson_photolysis_1997,teolis_water_2017}. In principle, though, such measurements should be possible using Raman techniques \\citep{rothard_modification_2017} and would be of great value, in part, by enabling us to further refine both our radiochemical yields and chemical network. Interest in constraining \\ce{O2} abundances in irradiated water was recently renewed following its detection in the coma of comet 67P\/C-G by \\citet{bieler_abundant_2015}. As can be seen in Fig. \\ref{fig:others}, the maximum abundance of \\ce{O2} with respect to water achieved here is $\\sim0.1$\\% in model (a) at 77 K, a value which increased only negligibly at still higher temperatures. Thus, our models predict that the radiolysis of pure \\ce{H2O} ice is not the dominant mechanism behind the $\\sim3.8$\n\\% \\ce{O2} abundances relative to water measured by Rosetta \\citep{bieler_abundant_2015}. In that study, moreover, no evidence for ozone was found, though an upper limit of $1\\times10^{-4}$ \\% relative to water was established. Interestingly, only in model (a) are the ozone abundances predicted to remain below this limit, even at 77 K.\n\n\n\\section{Conclusions \\& Outlook} \\label{sec:conclusions}\n\nIn this work, we have simulated the bombardment of pure \\ce{O2} and \\ce{H2O} ices by energetic protons using a general rate-equation-based astrochemical code, modified to include radiation-chemical processes using the SH method. These models were carried out with the \\texttt{MONACO} program \\citep{vasyunin_formation_2017}, and a network consisting of the radiolysis processes listed in Table \\ref{tab:gvalues} of Appendix \\ref{sec:radiolysis} and the reactions noted in Appendix \\ref{sec:network}. As illustrated in Figs. \\ref{fig:o2ice} and \\ref{fig:relative}, we were able to qualitatively reproduce both the abundance of \\ce{O3} in pure \\ce{O2} \\citep{baragiola_solid-state_1999} and \\ce{H2O2} in pure \\ce{H2O} \\citep{gomis_hydrogen_2004-1} utilizing the radiochemical processes given in Table \\ref{tab:gvalues}. Thus validated, these processes, along with the reactions given in Table \\ref{tab:h2onetwork}, can be added to existing chemical networks in order to better account for physicochemical effects caused by cosmic ray bombardment of dust grain ice-mantles.\n\nMoreover, by simulating well-constrained experiments rather than interstellar environments we have been afforded a unique opportunity to compare the accuracy and physical realism of several approaches to modeling bulk chemistry over a variety of temperatures relevant to the ISM. As reported here, we have found that the standard approaches to bulk chemistry based on thermal diffusion or quantum tunneling through diffusion barriers did more poorly at reproducing the experimental data - particularly at low temperatures - than our model in which radicals and atomic oxygen were assumed to react quickly with neighboring species in the ice. This finding is in agreement with recent experiments by \\citet{ghesquiere_reactivity_2018}, who found no evidence for true bulk diffusion. \n\nRegrettably, despite the large body of work in laboratory astrophysics on the irradiation of interstellar ice-analogues, it has not been possible, until recently, to incorporate many of the results of these experiments into astrochemical codes \\citep{shingledecker_cosmic-ray-driven_2018}. However, our work presented here shows that not only can such models simulate radiation-chemical reactions, they might even be fruitfully used as a replacement for the simpler kinetic models sometimes employed (e.g. \\citet{gomis_hydrogen_2004,baragiola_solid-state_1999}) in understanding and analyzing experimental data.\n\nIn summary, this study represents an attempt to shrink the existing gap between experiments and models, an increasingly urgent task in light of the upcoming launch of JWST. However, there is ample opportunity for even further refinements to our approach by, for example, considering the effects of the implantation and subsequent reactions of the bombarding \\ce{H+} ions, of ice heating along the particle track, or of the effects caused by the nuclear\/elastic component of the stopping, which begins to dominate over the electronic\/inelastic component considered here at lower particle energies \\citep{spinks_introduction_1990}. In addition to the synthesis of molecules, charged particle bombardment is well known to drive the non-thermal desorption of even large molecules such as benzene \\citep{thrower_highly_2011,marchione_efficient_2016}. From experiments it is known that, particularly in water ice, excitons migrating to the surface represent one such mechanism that can stimulate this desorption. Given lingering questions about how molecules formed in dust-grain ice mantles are introduced into the surrounding gas in cold environments, future improvements to our approach in this area are warranted. Finally, experiments in which the abundances of multiple species are followed during irradiation would further advance our knowledge of radiation-chemical processes in ices and help to reduce uncertainties in future modeling research. In particular, Raman spectroscopic analysis - where even the behavior of IR-inactive species like \\ce{O2} could be monitored - represents a powerful, yet perhaps underutilised, technique that should be considered in future studies.\n\n\\acknowledgments\n\nCNS gratefully acknowledges the support of the Alexander von Humboldt Foundation. EH acknowledges the support of the National Science Foundation through grant AST-1514844. AV acknowledges the support of the Russian Science Foundation through grant 18-12-00351.\n\n\\software{MONACO \\citep{vasyunin_formation_2017}}\n\n\\FloatBarrier\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}