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+{"text":"\\section*{Introduction}\n\n\n\nLet $X$ denote a compact Riemann surface of genus $g \\geq 2$. By the\nretrosection theorem of Koebe-Courant (e.g., \\cite{Bers}) $X$ can be\nrepresented by a Schottky group $\\Gamma$: we can write $$X=\\Gamma\n\\backslash(\\mathbf{P}^1(\\mathbf{C})-\\Lambda_\\Gamma),$$ where $\\Lambda=\\Lambda_\\Gamma$\nis the set of limit points of $\\Gamma$ and $\\Gamma$ is a free group of\nrank $g$, discrete in $\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$. In complex analysis, it is well\nknown that the dynamics of the action of $\\Gamma$ on the limit set\nendowed with Patterson-Sullivan measure encodes a lot about the\nstructure of the Riemann surface. Our purpose is to show that this\naction can be conveniently encoded by a notion from non-commutative\ngeometry, namely a \\emph{spectral triple} (\\cite{ConnesLMP}) which\nprovides the non-commutative analogue of a Riemannian manifold. As it\nwill turn out, the spectral triple we will consider is\ncommutative. But, as has been observed frequently, ``even for\nclassical spaces, which correspond to commutative algebras, the new\npoint of view [of noncommutative geometry] will give new tools and\nresults.'' (Connes, \\cite{Connesbook}, p.\\ 1).  \nWe show that the isometry class of the boundary action is encoded in\nthe \\emph{zeta function} formalism of the spectral triple.  \n\n\\paragraph{Construction of the spectral triple} At the\ntopological level, we consider the commutative algebra $A=C(\\Lambda)$.\nWe also need the dense subalgebra $A_\\infty=C(\\Lambda,\\mathbf{Z})\\otimes_\\mathbf{Z}\n\\mathbf{C}$ of locally constant functions on $\\Lambda$. \n\nIt might be natural to consider the boundary operator algebra\n$C(\\Lambda) \\rtimes \\Gamma$ instead. However, by the non-amenability\nof the group $\\Gamma$, the hyperfiniteness result of Connes implies\nthat this algebra does not carry any finitely summable spectral triple\n(\\cite{Conneshyper}). As we will indicate at the end of the proof of\nthe main theorem, our construction can be extented to an AF-algebra\nthat is Morita equivalent to a large subalgebra of the boundary\noperator algebra. \n\nTo retrieve the actual conformal structure, we need to make the\noperator algebra act on a Hilbert space in a way compatible with a\nDirac operator. The Hilbert space $H$ is a particular\nGNS-representation of $A$. Its construction depends on chosing a\nstate, and we make this choice in such a way that it encodes the\nmetric action of $\\Gamma$ on $\\Lambda$, expressed via the\nPatterson-Sullivan measure. More specifically, on certain elements of\n$A$ related to words in a presentation of $\\Gamma$, it gives the\nmeasure of the subset of $\\Lambda$ reached from that word in  the\nrepresentation of the limit set of $\\Gamma$ via word group completion\n(Floyd \\cite{Floyd}). Finally, the Dirac operator $D$ is composed from\nprojection operators depending on the word length grading in a\npresentation for $\\Gamma$. Let ${\\mathcal S}_X$ denote the spectral triple so\nconstructed (see Section \\ref{s1} for details). \n\n\\begin{introprop} \\label{tA}\nIf $X$ is a compact Riemann surface of genus at least $2$, ${\\mathcal S}_X$ is a 1-summable spectral triple.\n\\end{introprop}\n\n\\paragraph{Zeta function rigidity} The theory of finitely summable\nspectral triples comes with an elegant framework of zeta functions:\nfor any $a \\in A_\\infty$, one has the spectral zeta function\n$\\zeta_{X,a}(s):=\\mathrm{tr} (a|D|^s)$. \n\n\\begin{introthm}\\label{tB}\nIf $X_1$ and $X_2$ are compact Riemann surfaces of genus at least $2$,\nsuch that  \n$\\zeta_{X_1,a}(s)=\\zeta_{X_2,a}(s)$ for all $a \\in A_\\infty$, then\n$X_1$ and $X_2$ are conformally or anti-conformally equivalent as\nRiemann surfaces. In particular, the spectral triple ${\\mathcal S}_X$ encodes\nthe conformal\/anticonformal isomorphism type of $X$. \\end{introthm}  \n\nIn the theorem, equality of zeta functions should be understood as\nfollows: both the algebras $A_1$ and $A_2$ of $X_1$ and $X_2$,\nrespectively, have a unit. If the zeta functions for this unit are\nequal, we first deduce that the Riemann surfaces have the same\ngenus. This allows us to identify the algebras $A_1$ and $A_2$, along\nwith the corresponding dense subalgebras of locally constant\nfunctions. It then makes sense to interpret the expression\n$\\zeta_{X_1,a}(s)=\\zeta_{X_2,a}(s)$. \n\nAbout the proof: by the analogue of Fenchel-Nielsen theory (cf.\\ Tukia\n\\cite{Tukia2}), the abstract isomorphism of Schottky groups of $X_1$\nand $X_2$ induces a unique homeomorphism of limit sets, equivariant\nwith respect to the group isomorphism (the \\emph{boundary map}). We\nshow that equality of zeta functions implies that this boundary map is\nabsolutely continuous. For this, one has to trace through the\nrepresentation of the limit set in the sense of Floyd (\\cite{Floyd}) to deduce fairly explicit expressions for various zeta functions. We do this by computing traces in an explicit orthonormal basis for $H$.\n\nWe then apply an ergodic rigidity theorem for Schottky uniformization\nthat we deduce from a theorem of Yue (cf.\\ \\cite{Yue}; from the long\nhistory we also mention the names Mostow, Kuusalo, Bowen, Sullivan,\nTukia).  This says that there are only two alternatives for the\nboundary map: either the Patterson-Sullivan measures are mutually\nsingular with respect to the boundary map, or the map extends to a\ncontinuous automorphism of $\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$. Absolute continuity excludes\nthe first case.  \n\nWe will end the paper with a list  of open questions. \n\n\\paragraph{Can you hear the shape of a Riemann surface?} Theorem\n\\ref{tB} fits into the framework of isospectrality questions, as\ncoined by I.M.~Gelfand for compact Riemannian manifolds. Vign\\'eras (\\cite{V1}, \\cite{V2})  and Sunada\n(\\cite{Sunada}) constructed non-isometric surfaces with identical Laplace operator zeta function $\\mathop{\\mathrm{tr}}\\nolimits(\\Delta^s)$ (`isospectral'). The work of Sunada, in particular, transports an idea from algebraic number theory due to Gassmann (\\cite{Gassmann}), where the same phenomenon is visible: there exist non-isomorphic algebraic number fields with identical Dedekind zeta function.\n\n\n For the specific case of Riemann surfaces, Buser (\\cite{Buser}) obtained  a\n(finite) upper bound on the number of Riemann surfaces isospectral with a given Riemann surface, depending only on the genus of that surface, and work of Brooks, Gornet and Gustafson (\\cite{BGG})\nshows this bound is of the correct order of magnitude. \n\nAs for Dirac operators instead of Laplace operators, B\\\"ar has\nconstructed non-isometric space forms with the same Dirac spectrum\n(\\cite{Baer1}).  \n\nFor the case of planar domains, the problem of isospectrality was\ncoined by Bochner, to quote Kac --- quoting Lipman Bers ---  ``can you\nhear the shape of a drum?'' (\\cite{Kac},  solved by Gordon, Webb and\nWolpert \\cite{GWW}). \n\nIn this phrasing, Theorem \\ref{tB} says you can\nhear the complex analytic type of a compact Riemann surface from\nlistening to the noncommutative spectra of its associated spectral\ntriple (that is, to the collection of the associated zeta\nfunctions). A main difference with respect to the classical isospectrality\nquestion is that in this case you do have to listen to $\\mathop{\\mathrm{tr}}\\nolimits(aD^s)$ for a\ndense subset of operators $a \\in A$, and the eigenvalues of $D$\nthemselves are not so interesting. For example, at the unit $1 \\in A$, we find the innocent zeta function (cf.\\ Formula (\\ref{zeta1}))\n$$ \\zeta_{X,1}(s) = 1 +\\frac{2g-2}{2g-1} \\cdot \\frac{(2g)^{3s+1}}{1-(2g-1)^{3s+1}}. $$\n We note that in the completely\ndifferent construction of a  ``conformal'' spectral triple by B\\\"ar,\nthe eigenvalues themselves are uninteresting, too (\\cite{Baer}). \n\nThus, the main point of our discussion is that the ``spectral object'' ${\\mathcal S}_X$ determines the ``conformal object'' $X$, up to complex conjugation.\n\n\n\n\n\\section{A spectral triple associated to a Kleinian Schottky group} \\label{s1}\n\n\n\\begin{se}The aim of this section is to introduce a finitely summable spectral\ntriple ${\\mathcal S}_X:=(A,H,D)$ associated to a Schottky group $\\Gamma$ that\nuniformizes a compact Riemann surface $X$ of genus $g \\geq 2$.  Recall\nthat a spectral triple is a noncommutative analogue of a \nRiemannian spin manifold, \nwhere $A$ is a $C^*$-algebra, $H$ is a Hilbert space on which\n$A$ acts by bounded operators, and $D$ is an unbounded self\nadjoint operator on $H$ with compact resolvent $(D-z)^{-1}$ for\n$z\\notin \\mathbf{R}$, and such that the commutators $[D,a]$ are bounded\noperators for all $a$ in a dense involutive subalgebra $A_\\infty$ of\n$A$. Connes has shown \\cite{ConnesLMP} that if $(A,H,D)$ arises from a\nRiemannian spin manifold, then the distance element is encoded by the\ninverse of the Dirac operator.  \n\\end{se}\n\n\n\\begin{se}Let $\\Gamma$ denote a Schottky group of rank $g \\geq 2$.  As an abstract group, $\\Gamma$ is isomorphic to $F_g$, the free group on $g$ generators. \nWe think of $\\Gamma$ as being specified by an injective group\nhomomorphism $\\rho  :  F_g \\hookrightarrow \\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$. Let\n$\\Lambda=\\Lambda_\\Gamma$ denote the limit set of the action of\n$\\Gamma$ on $\\mathbf{P}^1(\\mathbf{C})$. \n \\end{se}\n\n\\begin{se}[Group completion and limit set] We recall what we need from\nFloyd's relation between the group completion of $F_g$ and the limit\nset $\\Lambda$ of $\\Gamma$ (\\cite{Floyd}).  Let $Y_g$ denote the Cayley\ngraph (with unordered edges) of $F_g$ for a presentation of $F_g$ in a\nfixed alphabet on $g$ letters, and let $\\bar{Y}_g$ denote the\ncompletion of the Cayley graph as a metric space for the following\nmetric. Let $|w|$ denote the reduced word length of a word in the\ngenerators of $F_g$. The edge between two words $w_1$ and $w_2$ is\ngiven length $\\min \\{ |a|^{-2}, |b|^{-2} \\}$ (with $|e|^{-2}:=1$ for\nthe empty word $e$).  \nThe \\emph{group completion} of $F_g$ is by definition the space\n$\\bar{F}_g:=\\bar{Y}_g-Y_g$. It is a compact metric space. A different\n(finite) presentation for $F_g$ leads to a Lipshitz equivalent group\ncompletion. Since $F_g$ has no ``parabolic ends'' in the sense of\nFloyd, we have the following: \n\\end{se}\n\n\\begin{lem}[Floyd, \\cite{Floyd}, p.\\ 213--217] Given a point $x_0 \\in\n\\mathbf{P}^1(\\mathbf{C})$ and an embedding $\\rho  :  F_g \\hookrightarrow \\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$\nas above, the following map is a continuous bijection: \n$$ \\begin{array}{lccc} \\iota_\\rho \\ : \\  & \\bar{F}_g & \\rightarrow &\n\\Lambda \\\\ &  \\lim\\limits_i w_i & \\mapsto & \\lim\\limits_i\n\\rho(w_i)(x_0). \\end{array} \\ \\ \\ \\ \\Box $$  \n\\label{Floyd} \\end{lem}\n\n\\begin{df} Given a reduced word $w$ in the generators of $F_g$, let\n$i(w)$ respectively $t(w)$ denote the initial, respectively terminal\nletter of $w$. \nFor two reduced words $w$ and $v$ (or $v$ a limit of such), we write $$w \\subseteq v \\mbox{ if }(\\exists w_0)(v=w \\cdot w_0) \\mbox{ with }t(w) \\neq i(w_0)^{-1}.$$\n We write $w \\subset v$ if $w \\subseteq v$ and $w \\neq v$. \n\nGiven $\\rho: F_g \\rightarrow \\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$ and a word $w \\in F_g$,\ndefine the subset of $\\Lambda$ of \\emph{ends of $w$ with respect to\n$\\rho$} to be  \n$$ \\overrightarrow{w}_\\rho := \\{ \\iota_\\rho(v) \\ : \\ v \\in \\bar{F}_g \\mbox{ and } w \\subseteq v \\}. $$ \n\\end{df}\n\n\n\\begin{lem}\n$A=C(\\Lambda)$ is the closure of the span of the characteristic\nfunctions $\\chi_{\\overrightarrow{w}_\\rho}$ of the sets\n$\\overrightarrow{w}_\\rho$ for $w \\in F_g$.  \n\\end{lem}\n\n\\begin{proof}\nThis is immediate, since $\\Lambda$ is a totally disconnected compact\nHausdorff space, and the sets $\\overrightarrow{w}$ form a basis of\nclopen sets for its topology. \n\\end{proof}\n\nWe denote by $A_\\infty$ the dense involutive subalgebra of $A$ spanned\nby the characteristic functions $\\chi_{\\overrightarrow{w}_\\rho}$. \n\n\\begin{df} Let $\\mu_\\Lambda$ denote the Patterson-Sullivan measure on\n$\\Lambda$ (cf. \\cite{Patterson}, \\cite{SullivanIHES}). Its main property is scaling by the Hausdorff dimension\n$\\delta_H$ of $\\Lambda$: \\begin{equation*}\\label{PSmeas} \n(\\gamma^* d\\mu)(x)= |\\gamma^\\prime (x)|^{\\delta_H} \\, d\\mu(x), \\ \\ \\\n\\forall \\gamma\\in \\Gamma.\n\\end{equation*} We define a state $\\tau : A_\\infty \\rightarrow \\mathbf{R}$ by \n$$ \\tau(\\chi_{\\overrightarrow{w}_\\rho}) := \\int\\limits_{\\Lambda}\n\\chi_{\\overrightarrow{w}_\\rho} \\mathrm{d}\\mu_\\Lambda =\n\\mu_\\Lambda(\\overrightarrow{w}_\\rho). $$ \nThe above lemma shows that $\\tau$ extends uniquely to a state on\n$A$. We define the Hilbert space $H$ to be the GNS-representation of\n$A$ arising from this state $\\tau$, that \nis, the completion of $A$ with respect to the inner product $\\langle\na | b \\rangle := \\tau(b^*a)$.  \n\\end{df} \n\n\\begin{df} We now take our inspiration from the construction of\nAntonescu and Christensen in \\cite{AntChris}.  \nThe subalgebra $A_\\infty$ of $A=C(\\Lambda)$ is a limit of finite\ndimensional subspaces $A_\\infty =\n\\lim\\limits_{\\xrightarrow[]{}} A_n$ with $A_n$ the span of the\ncharacteristic functions of sets of ends of reduced words of length\n$\\leq n$. This filtration is inherited by  $H$. We denote by $H_n$ the\nterm of the filtration of $H$ corresponding to $A_n$ and we let\n$P_n$ denote the orthogonal projection operator onto $H_n$. \nWe define the Dirac operator to be \n$$ D:= \\lambda_0 P_0 + \\sum_{n \\geq 1} \\lambda_n (P_n-P_{n-1}), $$\nwhere $\\lambda_n = (\\dim A_n)^3$. Note that $Q_n:=P_n-P_{n-1}$ is the\nprojection onto the graded pieces, identified with the orthogonal\ncomplements $H_n \\ominus H_{n-1}$, which correspond to words of exact \nlength $n$. The choice of $\\lambda_n$ arises from the fact that we\nthen arrive at 1-summability (Proposition \\ref{tA}): \n\\end{df}\n\n\\begin{prop}\nThe triple ${\\mathcal S}_{X}=(A,H,D)$ is a 1-summable spectral\ntriple.\n\\end{prop}\n\\begin{proof} The $*$-operation is complex conjugation, and since $D$\nis real, it is self-adjoint. For $a\\in A_{n}$ and for any $m>n$, multiplication by $a$ maps $A_{m-1}$ and $A_m$ into itself. Therefore, $a$ commutes with the projections $P_m$ and $P_{m-1}$ and so $[Q_m,a]=0$. Hence \n$$ [D,a] = \\sum_{i=0}^n \\lambda_i [Q_i,a]$$\nis a finite sum (we set $P_{-1}=0$ for convenience). \nThus, the commutators of $D$ with elements in  \nthe dense subalgebra $A_{\\infty}$ of $A$ are bounded. \n\nMoreover, one has $\\dim A_{n} \\geq\nn+1$, hence the 1-summability (and compact resolvent): \n$$ \\mathop{\\mathrm{tr}}\\nolimits((1+D^2)^{-1\/2})=1 +\\sum_{n=1}^\\infty (1+\\lambda_n^2)^{-1\/2} (\\dim\nH_{n}-\\dim H_{n-1}) $$\n$$ \\leq 1+\\sum_{n=1}^\\infty (1+\\lambda_n^2)^{-1\/2}\\dim  H_{n}\n\\leq 1+\\sum_{n=1}^\\infty (1+\\lambda_n^2)^{-1\/2}\\dim  A_{n} $$\n$$ \\leq 1+\\sum_{n=1}^\\infty (\\dim  A_{n})^{-2} \\leq  \n1+\\sum_{n=1}^\\infty (n+1)^{-2} \\leq  2, $$ where we used $\\lambda_n =\n(\\dim A_n)^3$ in the second-to-last inequality. This proves the\nproposition. \n\\end{proof}\n\n\\begin{rem}\nA recent deep theorem of Rennie and V\\'arilly (\\cite{RV}) allows one\nto decide whether a given spectral triple is associated to an actual\ncommutative Riemannian spin manifold. For the purpose of this paper,\nsince we are mostly interested in the zeta functions, we do not\nconsider any additional structure on the spectral triple. In\nparticular, our Dirac operator is only considered up to sign, since\nthe sign does not play a role in the zeta functions, while for\n\\cite{RV}, the sign provides the essential information on the\n$K$-homology fundamental class. It is possible that our construction\nmay be refined to incorporate the further necessary properties of an\nabelian spectral triple to which the reconstruction theorem can be\napplied. In that case, it seems that the underlying metric geometry\nshould probably relate to the existence of quasi-circles of limit sets\nof Schottky groups as in  \\cite{Bowen} --- see also the next\nremark. \\end{rem} \n\n\\begin{rem}\nNotice that our construction provides a 1-summable spectral triple on\nthe limit set, regardless of the actual value of its Hausdorff\ndimension (which can be greater than one). Thus, the metric dimension\nseen from this construction will be in general different from the\nactual metric dimension of the limit set embedded in $\\mathbf{P}^1(\\mathbf{C})$. The\nexistence in all cases of a 1-summable spectral triple on the limit\nset reflects the fact that topologically $\\Lambda$ is always a Cantor\nset that can be embedded in a topologically 1-dimensional quasi-circle\n(Bowen \\cite{Bowen}). In the metric induced by the embedding in\n$\\mathbf{P}^1(\\mathbf{C})$, the quasi-circle need not be rectifiable (when the\nHausdorff dimension of the limit set exceeds 1), but the existence of\n1-summable spectral triples is compatible with the topological\ndimension being one in all cases.   \n\\end{rem}\n\n\\section{Boundary isometry from the\nspectral zeta function} \\label{boundary}\n\n\nIn this section we study the effect of equality of zeta functions on\nmetric properties of the limit sets. Since we are dealing with two\nRiemann surfaces $X_1, X_2$, we will now sometimes index symbols\n($H,D,\\zeta,\\lambda,\\dots)$ by the index of the corresponding Riemann\nsurface and will do so without further mention. If there is no index,\nwe refer to any of the two Riemann surfaces. \n\nAs was already observed by Connes for the spectral triple associated\nto a usual spin manifold, only the action of $A$ on $H$, or of $D$ on\n$H$, doesn't capture interesting (metrical\/conformal)  information\nabout the space, it is the  interaction of the action of $A$ and $D$\nthat is important (\\cite{Connesbook}, VI.1). For our purposes, this\ninteraction will be encoded in the framework of zeta functions of\nspectral triples. The zeta functions are $\\zeta_a(s):=\\mathop{\\mathrm{tr}}\\nolimits(a|D|^s)$, a\npriori defined for $\\mathrm{Re}(s)$ sufficiently negative, but then\nmeromorphically extended to the whole complex plane with poles at the\ndimension spectrum of the spectral triple (see \\cite{ConnesLMP}). \n\n\\begin{thm}\\label{zetalambda}\nLet $X_1$ and $X_2$ be compact Riemann surfaces of genus at least\n$2$. If $ \\zeta_{a,{X_1}}(s) = \\zeta_{a,X_2}(s)$ for all $a \\in\nA_\\infty$, then $g_1=g_2$ and $$ \\forall \\eta \\in F_g \\ : \\\n\\mu_1(\\overrightarrow{\\eta}_{\\rho_1}) =\n\\mu_2(\\overrightarrow{\\eta}_{\\rho_2}). $$ \\end{thm} \n\n\n\\begin{rem}\nAs was indicated in the introduction, equality of zeta functions\nshould be understood as follows: both the algebras $A_1$ and $A_2$ of\n$X_1$ and $X_2$, respectively, have a unit. If the zeta functions for\nthis unit are equal, then we will conclude from this that the Riemann\nsurfaces have the same genus. Therefore, the algebras $A_1$ and $A_2$\nare isomorphic via the homeomorphism $\\Phi: \\Lambda_1 \\rightarrow\n\\Lambda_2$ induced from the Floyd maps in the triangle \n$$ \\xymatrix{    & & \\Lambda_1 \\ar[dd]^{\\Phi} \\\\ \\bar{F}_g\n\\ar[urr]^{\\iota_{\\rho_1}} \\ar[drr]^{\\iota_{\\rho_2}} & \\\\ & &\n\\Lambda_2}$$ It then makes sense to interpret the expression\n$\\zeta_{X_1,a}(s)=\\zeta_{X_2,a}(s)$ for elements $a\\neq 1$ in\n$A_\\infty$. \n\\end{rem}\n\n\n\n\\begin{proof} We make the convention that all words are reduced.\n\nLet $X$ be a Riemann surface of genus $g \\geq 2$ and ${\\mathcal S}_X$ its\nassociated spectral triple.  Suppose given an element $a=\\chi_U$ in\n$A_\\infty$. We can assume $U=\\overrightarrow{\\eta}$ for a given word\n$\\eta$ of length $|\\eta|=m$, since any  $a$ is a linear combination of such.  \n\nWe now construct an orthogonal basis for $H$.\nFirst, we prove a lemma about ends of words.\n\n\\begin{lem} \\label{lemwords} Let $w_1, w_2$ denote two words. If\n$\\overrightarrow{w_1} \\cap \\overrightarrow{w_2} \\neq \\emptyset$, then\n$\\overrightarrow{w}_1 \\subseteq \\overrightarrow{w}_2$ or conversely\n$\\overrightarrow{w}_2 \\subseteq \\overrightarrow{w}_1$. In particular, if we set $\\max\\{w,v\\}$ to be the largest of the words $w$ and $v$ (if they are comparable in the order $\\subseteq$) and $\\emptyset$ otherwise, we find that $$ \\overrightarrow{w_1} \\cap \\overrightarrow{w_2} = \\overrightarrow{\\max\\{w_1,w_2\\}}$$ with the convention \n$\\overrightarrow{\\emptyset}=\\emptyset$, see Figure \\ref{figure1}. \n\\begin{figure}[h] \\begin{center} \\input{maxlength.pstex_t}  \n\\end{center}\n\\caption{{Illustration of Lemma \\ref{lemwords}.}}\n\\label{figure1}\n\\end{figure}\n \n\\end{lem}\n\\begin{proof}   \n\nIf this were not the case, then there is an end that lands in\nthe nonempty intersection and starts from both segments $w_1$ and\n$w_2$, leading to a loop in the Cayley graph $Y_g$, but $Y_g$ is a\ntree, so this is impossible, see Figure \\ref{figure2} (as usual, we identify the limit set $\\Lambda$ topologically with $\\bar{F}_g$, by Floyd's Lemma).\n\n\\begin{figure}[h] \\begin{center} \\input{forbidden.pstex_t} \n\\end{center}\n\\caption{{ A forbidden situation.}}\n\\label{figure2}\n\\end{figure}\n\n\\end{proof} \n\nIt is then easy to find a basis for the individual spaces $H_n$. \n\n\\begin{lem}\\label{basis}\nThe functions $\\chi_w$ for $|w|=n$ for a linear basis for $H_n$, and $$ \\langle \\chi_w |  \\chi_v \\rangle = \n\\mu(\\overrightarrow{\\max\\{v,w\\}}). $$\n \\end{lem}\n\n\\begin{proof} The characteristic functions $\\chi_{\\overrightarrow{w}}$\nfor $|w|=n$ give a linear basis for $H_n$: they are\nlinearly independent as their supports are disjoint, and they generate the space since for any word $u$ of length $|u|< n$ one\nhas $$\\chi_{\\overrightarrow{u}} =\\sum_{\\substack{|w|=n \\\\u \\subset w}}\n\\chi_{\\overrightarrow{w}}.$$ Indeed, by the previous Lemma, all occuring $\\chi_{\\overrightarrow{w}}$ have disjoint support, and their union $\\bigcup \\overrightarrow{w}$ equals $\\overrightarrow u$.\n\nNow\n$$  \\langle \\chi_{\\overrightarrow{w}_1} |  \\chi_{\\overrightarrow{w}_2}\n\\rangle = \\tau(\\chi_{\\overrightarrow{w}_1}^*\n\\chi_{\\overrightarrow{w}_2}) =\\mu_X (\\overrightarrow{w}_1 \\cap\n\\overrightarrow{w}_2), $$\nand the previous lemma applies. \n\\end{proof}\n\n\\begin{lem} For all $n>0$, we have \n$$\\dim A_n=\\dim H_n = 2g(2g-1)^{n-1}.$$ \n\\end{lem}\n\n\\begin{proof} The space $A_{n}$ is spanned by the linear basis \n$\\chi_{\\overrightarrow{w}_\\rho}$ with $w$ a word of exact length $\nn$, since as in the proof of Lemma \\ref{basis}, all functions corresponding to shorter words are dependent on these functions. An easy count gives the result: we pick the first letter from the alphabet on $g$ letters or its inverses, and consecutive letters with the condition that they differ from the terminal letter of the word already constructed. \n\\end{proof}\nWe now construct a complete orthonormal basis for $H$ inductively, by adding to a basis of $H_{n}$ suitable elements of $H_{n+1}$ in the style of a Gram-Schmidt process. Initially, we set\n$| \\Psi_e \\rangle = \\chi_\\Lambda$ and \\begin{equation} \\label{length1}  |\\Psi_w \\rangle := \\frac{1}{\\sqrt{\\mu_X(\\overrightarrow{w})}}\\,\n\\chi_{\\overrightarrow{w}} \\ \\ \\ \\ (|w|=1) \\end{equation}\nfor $w$ running through a set $S$ of words of length one (viz., letters in the alphabet, and their inverses) unequal to one (arbitrarily chosen) letter. Set $I_1:=S \\cup \\{e\\}$; then $\\{|\\Psi_w\\rangle\\}_{w \\in I_1}$  is an orthonormal basis for $H_1$ by Lemma \\ref{basis}. \n\nNow suppose $$\\{|\\Psi_w\\rangle \\, : \\, w \\in I_n \\}$$ is our inductively constructed basis for $H_n$, where  $I_n$ is an index set. For every word $w$ of length $n$, choose a set $V_w$ of $2g-2$ letters\nfrom the alphabet and its inverses, that are unequal to $t(w)^{-1}$, the inverse of the terminal letter of the fixed $w$, i.e., leave out one arbitrarily chosen letter from the possible extensions of $w$ to an admissible word of length $n+1$. Let $$I_{n+1} = I_n \\cup \\bigcup\\limits_{|w|=n} V_w.$$ Figure \\ref{figure3} has an example in the length $\\leq 3$ words in the Cayley graph $Y_2$ for $g=2$.\n\n\\begin{figure}[h] \\begin{center} \\begin{picture}(120,120)\\includegraphics{basis3.eps}\\end{picture} \\end{center}\n\\begin{center} \\caption{{ Black dots form a possible $I_3 \\subseteq Y_2$}}\\end{center}\n\\label{figure3}\n\\end{figure}\n\n\nWe claim that $\\{\\chi_{\\overrightarrow{w}}\\}_{w \\in I_{n+1}-I_n}$ is a basis for $H_{n+1} \\ominus H_n$. The functions are linearly independent since their supports are disjoint. Hence it suffices to check dimensions. But $$\\dim (H_{n+1} \\ominus H_n) = 2g(2g-1)^{n-1}(2g-2) = |I_{n+1}-I_n|.$$\n\nWe define $\\{|\\Psi_w \\rangle\\}_{w \\in I_{n+1}}$ as the Gram-Schmidt orthonormalisation of $$\\{ |\\Psi_w\\rangle \\}_{w \\in I_n} \\, \\cup \\, \\{ \\chi_{\\overrightarrow{w}} \\}_{w \\in I_{n+1} - I_n}.$$ \nWe recall that this means we choose an enumeration  of the words in $I_{n+1}-I_n:$ $$ I_{n+1}-I_n = \\{w_1,\\dots,w_r\\}$$ and set inductively for $i=1,\\dots,r$\n\\begin{equation} \\label{defPsi}  |\\Psi_{w_i}\\rangle = \\frac{|\\phi_{w_i} \\rangle}{\\| \\phi_{w_i} \\|} \\end{equation}\nwith\n\\begin{equation} \\label{defphi}  |\\phi_{w_i} \\rangle :=\\chi_{\\overrightarrow{w_{i}}} -\\sum_{w \\in I_n \\cup\\{w_1,\\dots,w_{i-1}\\}} \n\\langle \\Psi_w |\\chi_{\\overrightarrow{w_{i}}}\\rangle |\\Psi_w\\rangle. \\end{equation}\nThen indeed,\n$ \\langle \\Psi_v |\\Psi_w\\rangle =\\delta_{v,w}, $\nfor all $|v|,|w|\\leq n+1$.  \n\n\nSet $$I_\\infty = \\bigcup_{n \\geq 0} I_n.$$ We use the complete basis $\\{ |\\Psi_w\\rangle \\}_{w \\in I_\\infty}$ of $H$ to compute the trace of a trace-class\noperator $T$ in the form\n$ \\mathop{\\mathrm{tr}}\\nolimits(T)=\\sum \\langle \\Psi_w | T \\Psi_w\\rangle. $\nWith $T=aD^s$, we have\n$$ \\mathop{\\mathrm{tr}}\\nolimits(aD^s)=1+\\sum_w \\langle \\Psi_w |a \\sum_{n \\geq 1} \\lambda_n^s\n(P_n-P_{n-1}) \\Psi_w \\rangle . $$\nNow the projector $P_n-P_{n-1}$ onto $H_n \\ominus H_{n-1}$ is $\\sum\\limits_{r \\in I_n - I_{n-1}}| \\Psi_r \\rangle \\langle \\Psi_r |$, so we get\n$$ (P_n - P_{n-1}) |\\Psi_w\\rangle =\\sum_{r \\in I_n-I_{n-1}}| \\Psi_r \\rangle \\langle \\Psi_r |\n\\Psi_w \\rangle = \\delta_{|w|,n} \\delta_{r,w} | \\Psi_r \\rangle =  \\delta_{|w|,n} | \\Psi_w \\rangle.$$\nThus, we rewrite the above as\n$$ \\mathop{\\mathrm{tr}}\\nolimits(aD^s)=1+\\sum_{n \\geq 1} \\sum_{w \\in I_n-I_{n-1}} \\lambda_n^s  \\langle \\Psi_w |\na \\Psi_w \\rangle . $$\nIf we denote by \n$$ c_n(a) =\\sum_{w \\in I_n-I_{n-1}} \\langle \\Psi_w |\na  \\Psi_w \\rangle, $$\nwe can write\n$$ \\zeta_{a,X}(s) = \\mathop{\\mathrm{tr}}\\nolimits(a D^s)=1+ \\sum\\limits_{n\\geq 1} \\lambda_{n}^s\\,\n c_{n}(a), \\ \\ \\mathrm{Re}(s) \\ll 0. $$ \n\n\\begin{lem} \\label{genusequal} If $ \\zeta_{a,1}(s) = \\zeta_{a,2}(s) $ for\n$a=1=\\chi_\\Lambda$ the identity of $A_1$, respectively $A_2$, then\n$g_1=g_2$. \\end{lem} \n\n\\begin{proof} We know that $\\lambda_{n}=(\\dim\nA_{n})^3=(2g)^3(2g-1)^{3n-3}. $ By orthnormality, we find that $$c_n(1)= \\sum_{|w| \\in I_n-I_{n-1}} \\langle \\Psi_w | \\Psi_w \\rangle = \\sum_{|w| \\in I_n-I_{n-1}} 1 \\, = \\, 2g(2g-1)^{n-2}(2g-2).$$\nHence we find for $a=1$ that $$ \\zeta_{1}(s) = 1 + \\sum_{n \\geq 1} \\lambda_n^s c_n(1) = 1+ (2g)^{3s+1} \\, \\frac{2g-2}{2g-1} \\, \\sum_{n \\geq 1} (2g-1)^{(3s+1)(n-1)},  $$  and thus \\begin{equation} \\label{zeta1} \\zeta_1(s) = 1 +\\frac{2g-2}{2g-1} \\cdot \\frac{(2g)^{3s+1}}{1-(2g-1)^{3s+1}}. \\end{equation} \nFor $a=1$, the \ncondition $ \\zeta_{a,1}(s) = \\zeta_{a,2}(s) $ is thus equivalent to \n$$\\frac{2g_1-2}{2g_1-1} \\cdot \\frac{2g_2-1}{2g_2-2} \\cdot  \\left( \\frac{g_1}{g_2} \\right)^{3s+1}  = \\frac{1-(2g_1-1)^{3s+1}}{1-(2g_2-1)^{3s+1}} \\mbox{ for } \\mathrm{Re}(s) \\ll 0 $$\nIf we let $s$ tend to $- \\infty$, the right hand side tends to $1$. However, unless $g_1=g_2$, the left hand side tends to zero. This finishes the proof that $g_1=g_2$. \n\\end{proof}\n\nAs mentioned in the remark above, we conclude from this lemma that the\nalgebras $A_1$ and $A_2$ are isomorphic via the induced Floyd\nhomeomorphism $\\Phi: \\Lambda_1 \\rightarrow \\Lambda_2$. \n This makes the condition $ \\zeta_{a,1}(s) = \\zeta_{a,2}(s) $ meaningful.\n \n\\begin{lem}\\label{equal} $c_{n,{1}}(a)=c_{n,{2}}(a)$ for all $a \\in A_\\infty$. \n\\end{lem}  \n\\begin{proof}\nThe equality $\\zeta_{a,1}(s) = \\zeta_{a,2}(s) $ is equivalent to \n $$\\sum_{n \\geq 0} \\left( c_{n,1}(a)-c_{n,2}(a)\\right) \\lambda_n^{s} \\equiv 0$$ \nfor $\\mathrm{Re}(s) \\ll 0$. Here, $\\lambda_n$ is the same for the two Riemann surfaces, since it only depends on their genus and those have just been shown to be equal in Lemma \\ref{genusequal}. Now since all $\\lambda_n$ are \\emph{distinct}\npositive integers, we also have an identically zero Dirichlet series \n$$ \\sum_{N \\geq 0} \\til{c}_N N^s \\equiv 0 \\mbox{ for $\\mathrm{Re}(s) \\ll 0$} $$\nwith $\\til{c}_N = c_{n,1}(a)-c_{n,2}(a)$ if $N=\\lambda_n$ for some $n$, and\n$\\til{c}_N=0$ otherwise. Now clearly $\\til{c}_N=0$ for all $N$, by the identity theorem for Dirichlet series (e.g., \\cite{HW}, 17.1). \n\\end{proof} \n\n\\begin{lem} \\label{algfunc}\nFor $a = \\chi_{\\overrightarrow{\\eta}}$  and $w$ a word of length $n<|\\eta|$, we have that \n$$ \\langle \\Psi_w | a \\Psi_w \\rangle = \\mu(\\overrightarrow{\\eta}) \\cdot \\kappa$$\nwhere $\\kappa$ depends only on measures $\\mu(\\overrightarrow{v})$ of certain words $v$ of length $|v|<|\\eta|$.\n\\end{lem}\n\n\\begin{proof} This holds for $w$ a word of length one, since by definition (\\ref{length1}) and Lemma \\ref{basis}, we have\n$$  \\langle \\Psi_w | a \\Psi_w \\rangle = \\frac{\\mu(\\overrightarrow{w} \\cap \\overrightarrow{\\eta})}{\\mu(\\overrightarrow{w})}  =\\left\\{ \\begin{array}{ll}  \\frac{\\mu(\\overrightarrow{\\eta})}{\\mu(\\overrightarrow{w})} & \\mbox{if } w \\subset \\eta; \\\\ 0 & \\mbox{otherwise.} \\end{array} \\right. $$\nWe then use induction on the word length of $w$. By construction of $\\Psi_w$ (looking at the definitions in {Formul\\ae}  (\\ref{defphi}) and (\\ref{defPsi})) it suffices to prove that \nfor $w,u$ of length $ \\leq n$, we have that $\\langle\\chi_{\\overrightarrow{w}}|a\\chi_{\\overrightarrow{u}}\\rangle$ is of the required form $\\mu(\\overrightarrow{\\eta}) \\cdot \\kappa$\nwhere $\\kappa$ depends only on measures $\\mu(\\overrightarrow{v})$ of certain words $v$ of length $|v|<|\\eta|$: $\\Psi_w$ is a linear combination of such terms. Now\n$$\\langle\\chi_{\\overrightarrow{w}}|a\\chi_{\\overrightarrow{u}}\\rangle = \\mu(\\overrightarrow{w} \\cap \\overrightarrow{\\eta} \\cap \\overrightarrow{u}) =\\left\\{ \\begin{array}{ll}  \\mu(\\overrightarrow{\\eta}) & \\mbox{if } w,u \\subset \\eta \\\\ 0 & \\mbox{otherwise,} \\end{array} \\right. $$\nsince $\\eta$ is longer than $w$ and $u$. This proves the claim.\n\\end{proof} \n\nComputing $c_{m-1}(\\chi_{\\overrightarrow{\\eta}})$ as a linear combination of terms of the form $\\langle \\Psi_w | a \\Psi_w \\rangle,$ we find from Lemma \\ref{algfunc} (now indicating the representation $\\rho$, since we will soon vary it) :\n\\begin{equation} \\label{inductmu} c_{m-1}(\\chi_{\\overrightarrow{\\eta}_\\rho}) =  \\mu(\\overrightarrow{\\eta}_{\\rho})  \\cdot \\kappa, \\end{equation}\nwhere $\\kappa$ only depends on $\\mu(\\overrightarrow{v}_\\rho)$ for $|v|<m$. \n\nWe need one more technical observation, namely that $\\kappa \\neq 0$ in (\\ref{inductmu}) or, what is the same: \n\\begin{lem} $c_{m-1}(a) \\neq 0$ for $a=\\chi_{\\overrightarrow{\\eta}}$ with $|\\eta|=m$. \\end{lem}\n\\begin{proof}\nRecall $c_{m-1}(a) = \\sum\\limits_{w \\in I_{m-1}-I_{m-2}} \\langle \\Psi_w | a \\Psi_w \\rangle.$ The terms $$ \\langle \\Psi_w | a \\Psi_w \\rangle = \\int |\\Psi_w|^2 \\cdot \\chi_{\\overrightarrow{\\eta}} \\, \\, d\\mu \\geq 0$$ are all positive, but some might be zero. It therefore suffices to prove that at least one of them is non-zero, and for this, it suffices to find $w$ such that the support of $\\Psi_w$ intersects $\\overrightarrow{\\eta}$. But if $x \\in \\overrightarrow{\\eta}$, then there is a word $v$ of length $m-1$ such that $x \\in \\overrightarrow{v}$, too, since $$\\bigcup\\limits_{|v|=m-1} \\overrightarrow{v} = \\Lambda.$$ Then $\\chi_{\\overrightarrow{v}}(x)  \\neq 0$, but,  as $\\{\\Psi_w\\}_{w \\in I_{m-1}-I_{m-2}}$ is a basis for $H_{m-1} \\ominus H_{m-2}$, we have $$\\chi_{\\overrightarrow{v}}(x) = \\sum_{w \\in I_{m-1}-I_{m-2}} a_w \\Psi_w(x)$$ for some coefficients $a_w$, hence there exists $w$ of length $m-1$ such that $\\Psi_w(x) \\neq 0$. \n\\end{proof}\n\n\\begin{prop}\nFor all $\\eta \\in F_g$, $\\mu_1(\\overrightarrow{\\eta}_{\\rho_1}) =\n\\mu_2(\\overrightarrow{\\eta}_{\\rho_2})$.  \n\\end{prop}\n\n\\begin{proof} We prove this by induction on the word length of $\\eta \\in F_g$\nIf $|\\eta|=0$, we find $\\overrightarrow{\\eta}_{\\rho_i}=\\Lambda_i$ for $i=1,2$, so the identity holds. If it holds for all words of length $<m$, let $\\eta$ denote a word of length $m$. \n\nRecall that the map $\\Phi^*: A_1 \\rightarrow A_2$ is such that\n$\\Phi^*(\\chi_{\\overrightarrow{w}_{\\rho_1}} )=\n\\chi_{\\overrightarrow{w}_{\\rho_2}}$. \n\nWe apply the expression (\\ref{inductmu}) to both Riemann surfaces, substituting $\\rho=\\rho_1$ and $\\rho=\\rho_2$, respectively. Since $\\kappa \\neq 0$, this is a genuine formula for $\\mu(\\overrightarrow{\\eta}_\\rho)$. \nThe equality $c_{m-1,1}(\\chi_{\\overrightarrow{\\eta}_{{\\rho_1}}})=c_{m-1,2}(\\chi_{\\overrightarrow{\\eta}_{{\\rho_2}}})$ is our assumption, and for the second factor on the right hand side we can inductively assume that the measures on the occuring words  of length $<m$ agree in both representations. Hence we find indeed $\\mu(\\overrightarrow{\\eta_{\\rho_1}})=\\mu(\\overrightarrow{\\eta_{\\rho_2}})$.\n\\end{proof}\nThis proposition finishes the proof of Theorem \\ref{zetalambda}.\n\\end{proof}\n\n\\section{Rigidity from boundary isometry} \n\nWe now prove Theorem \\ref{tB} from the introduction: \n\n\\begin{thm}\nIf $X_1$ and $X_2$ are compact Riemann surfaces of respective genus\n$g_1, g_2 \\geq 2$, such that  \n$\\zeta_{X_1,a}(s)=\\zeta_{X_2,a}(s)$ for all $a \\in A$, then $X_1$ and\n$X_2$ are conformally or anti-conformally equivalent as Riemann\nsurfaces.\\end{thm}  \n\n\\begin{proof} From Theorem \\ref{zetalambda}, we find that $X_1$ and\n$X_2$ have the same genus $g \\geq 2$.  \nWe consider the two Schottky groups $\\Gamma_1$ and $\\Gamma_2$\ncorresponding to $X_1$ and $X_2$, respectively. Let $\\rho_i :  F_g\n\\hookrightarrow \\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$ denote the corresponding embeddings of the\nabstract group $F_g$ (so $\\rho_i(F_g)=\\Gamma_i$), and let\n$\\alpha:=\\rho_2 \\circ \\rho_1^{-1}$ denote the induced group\nisomorphism $\\Gamma_1 \\rightarrow \\Gamma_2$. We consider the map $\\Phi\n: \\Lambda_1 \\rightarrow \\Lambda_2$ as in the diagram of the proof of\n\\ref{zetalambda}: $x\\in \\Lambda_1$ can be written as\n$\\iota_{\\rho_1}(\\lim w_i)$ for some Cauchy sequence $\\lim w_i$ in\n$Y_g$. We then define $\\Phi(x):=\\iota_{\\rho_2}(\\lim w_i)$. As was\nremarked before, by Floyd's Lemma \\ref{Floyd}, $\\Phi$ is a\nhomeomorphism of $\\Lambda_1$ onto $\\Lambda_2$.  \n\n\\begin{lem} $\\Phi$ is $\\alpha$-equivariant. \n\\end{lem}\n\\begin{proof} Given $\\gamma \\in \\Gamma_1$, we can find $g \\in F_g$\nsuch that $\\gamma=\\rho_1(g)$. For $x = \\iota_{\\rho_1}(\\lim w_i)$, we\nfind that $\\gamma \\cdot x = \\iota_{\\rho_1}(\\lim gw_i)$. Hence\n$$\\Phi(\\gamma \\cdot x) = \\iota_{\\rho_2}(\\lim gw_i) = \\iota_{\\rho_2}(g)\n\\cdot \\iota_{\\rho_2}(\\lim w_i) = \\rho_2(\\rho_1^{-1}(\\gamma)) \\cdot\n\\Phi(x) = \\alpha(\\gamma) \\Phi(x).$$ So we do find that $\\Phi$ is\n$\\alpha$-equivariant. \\end{proof}  \n\nThis means that $\\Phi$ is a boundary homeomorphism in the sense of\nFenchel-Nielsen, see Tukia \\cite{Tukia2}, 3C.  \n\nBy  Theorem \\ref{zetalambda}, the equality of zeta functions implies\nthat $\\Phi$ is an isometry. Indeed,  \n$$\\mu_2(\\Phi^*(\\chi_{\\overrightarrow{w}_{\\rho_1}})) =\n\\mu_2(\\chi_{\\Phi(\\overrightarrow{w}_{\\rho_1})})= \n\\mu_2(\\chi_{\\overrightarrow{w}_{\\rho_2}})=\n\\mu_1(\\chi_{\\overrightarrow{w}_{\\rho_1}}),$$\nwhere we use the definition of $\\Phi$ in the second equality and the\nproposition in the third. Thus, since\n$\\{\\overrightarrow{w}_{\\rho_i}\\}$ is a basis for $\\Lambda_i$, we find\nthat $\\mu_2 \\circ \\Phi^* = \\mu_1$.  \n\nNow recall the following ergodic rigidity theorem: \n\n\\begin{lem}[Chengbo Yue] Let $\\Gamma_1$ and $\\Gamma_2$ be\ngeometrically  finite subgroups in two simple, \nconnected and adjoint Lie groups $G_1$ and $G_2$ of real rank one,\nsuch that $\\Gamma_1$ \nis Zariski dense in $G_1$. Let  $\\alpha : \\Gamma_1 \\rightarrow\n\\Gamma_2$ be a type-preserving isomorphism. \nThen there exists a homeomorphism  $\\phi: \\Lambda_{\\Gamma_1}\n\\rightarrow \\Lambda_{\\Gamma_2}$ which is equivariant \nwith respect to  $\\alpha$. If $\\phi$ is absolutely continuous with\nrespect to the Patterson--Sullivan measure, then  $\\alpha$ can be\nextended to a continuous homomorphism $G_1 \\rightarrow G_2$.\\end{lem}\n\n\\begin{proof} This is literally Corollary B from  \\cite{Yue}, apart\nfrom the fact that the extended homomorphism $G_1 \\rightarrow G_2$ can\nbe assumed \\emph{continuous}, but this follows by looking at the\nstatement of Theorem A from which the corollary follows. \\end{proof} \n\nWe want to apply this corollary with $G_1=G_2=\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$ and\n$\\Gamma_i$ our Schottky groups, so let us check the conditions:  \nBoth Schottky groups are geometrically finite subgroups of $\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$;\n$\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})=\\mathop{\\mathrm{PSL}}\\nolimits(2,\\mathbf{C})$ is a simple and connected adjoint\nreal-rank-one Lie group; \nand finally:\n\n\\begin{lem} A noncommutative Schottky group is Zariski dense in $\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$. \n\\end{lem}\n\n\\begin{proof} Since the group operations on such a Schottky group are\ninduced from the algebraic operations on the algebraic group\n$\\mathop{\\mathrm{PGL}}\\nolimits(2)$, the Zariski closure $\\hat{\\Gamma}$ is itself an algebraic\nsubgroup of $\\mathop{\\mathrm{PGL}}\\nolimits(2)$. Assume $\\hat{\\Gamma}$ is a strict algebraic\nsubgroup of $\\mathop{\\mathrm{PGL}}\\nolimits(2)$. Let $\\hat{\\Gamma}_0$ denote its connected\ncomponent of the identity. The group of connected components\n$\\hat{\\Gamma}\/\\hat{\\Gamma}_0$ is finite, and $\\Gamma \\cap\n\\hat{\\Gamma}_0$ \nis a finite index subgroup of the free group $\\Gamma$, hence free of\nthe same rank $g$; and its Zariski closure is connected. It suffices\nthat this group has full Zariski closure, hence we can assume without\nloss of generality that $\\hat{\\Gamma}$ is connected. However, if\n$\\hat{\\Gamma}$ is connected of dimension $\\leq 2$, then it is solvable\n(cf.\\ \\cite{LAG}, IV.11.6), and a solvable group cannot contain a free\ngroup of rank $g \\geq 2$ (since the composition series of\n$\\hat{\\Gamma}$ would descend to one for $\\Gamma$). On the other hand,\nif $\\dim \\hat{\\Gamma} = 3$, then since $\\mathop{\\mathrm{PGL}}\\nolimits(2)$ is connected, we have\n$\\hat{\\Gamma}=\\mathop{\\mathrm{PGL}}\\nolimits(2)$. \\end{proof} \n\nSince $F_g$ has no parabolic points, we know that the equivariant\nboundary homeomorphism $\\Phi$ is unique and type-preserving (Tukia,\n\\cite{Tukia1}, p.\\ 426), hence it coincides with the boundary\nhomeomorphism $\\phi$ in Yue's result. Since all conditions are\nsatisfied, we can apply the result (replacing $\\phi$ by $\\Phi$) to\nboth the isometry $\\Lambda_1 \\rightarrow \\Lambda_2$ and its inverse,\nwe find that $\\alpha$ extends to a continuous group automorphism\n$\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C}) \\rightarrow \\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$. Now recall that the automorphisms\nof $\\mathop{\\mathrm{PGL}}\\nolimits(2,k)$ over a field $k$ have been classified by Schreier and\nvan der Waerden (cf.\\ \\cite{SW}, see also the supplement to\n\\cite{Hua}): the outer automorphisms are induced from field\nautomorphisms of $k$. Now all  continuous field automorphisms of $\\mathbf{C}$\nfix $\\mathbf{R}$. \n\nWe conclude that there is an isomorphism $\\Gamma_1 \\rightarrow\n\\Gamma_2$ of the form $$\\gamma_1 \\rightarrow g \\gamma_1^\\sigma\ng^{-1}$$ for $g \\in \\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$ and $\\sigma \\in \\mathop{\\mathrm{Aut}}\\nolimits(\\mathbf{C}\/\\mathbf{R})$, that is,\n$\\Gamma_1$ and $\\Gamma_2^\\sigma$ are conjugate in $\\mathop{\\mathrm{PGL}}\\nolimits(2,\\mathbf{C})$. Now\nnote that $\\Gamma_2^\\sigma$ uniformizes the curve $X_2^\\sigma$: \n$$ (\\mathbf{P}^1(\\mathbf{C})-\\Lambda_{\\Gamma_2^\\sigma})\/ \\Gamma_2^\\sigma = \\left(\n(\\mathbf{P}^1(\\mathbf{C})-\\Lambda_{\\Gamma_2})\/ \\Gamma_2\\right)^\\sigma = X_2^\\sigma.$$ \n Hence $X_1$ and $X_2^\\sigma$ are isomorphic Riemann surfaces, so\n$X_1$ and $X_2$ are conformally or anti-conformally equivalent, the\nformer case arising when $\\sigma$ is trivial, and the latter case\narising when $\\sigma$ is complex conjugation. \n\\end{proof}\n\n\n\\begin{rem}\nThe statement that two hyperbolic compact Riemann surfaces of the same\ngenus are isomorphic if and only if the boundary map $S^1 \\rightarrow\nS^1$ induced from the isomorphism of their fundamental groups, seen as\nFuchsian groups of the first kind, is absolutely continuous in\nLebesgue measure is originally part of Mostous rigidity theorem (cf.\\\nMostow \\cite{Mostow} 22.14, p.\\ 178). An easy proof for this case is\nin Kuusalo \\cite{Kuusalo}, and  Bowen has given another proof using\nGibbs measures in \\cite{Bowen}. For more general M\\\"obius groups\n(whose limit set is not necessarily the full boundary of the symmetric\nspace, such as Schottky groups, and in higher dimensions), there is\nthe work of Sullivan (\\cite{Sullivan}) and Tukia (\\cite{Tukia1},\n\\cite{Tukia2}). The typical ergodic rigidity theorem in this setting\nis that an absolutely continuous boundary map is identical to the\nrestriction of a M\\\"obius transformation \\emph{on the limit set}. What\nhappens outside the limit set, however, depends on other\nconsiderations (cf. \\cite{Tukia2}, Marden \\cite{Marden}). See, e.g.,\nTukia (\\cite{Tukia1}), Section 4D for a 3-dimensional example of an\nisomorphism of Schottky groups with absolutely continuous boundary\nmap, that does \\emph{not} extend to a M\\\"obius map outside the limit\nset.\\end{rem} \n\n\\begin{rem}\nThe construction cannot be extended to the crossed product boundary\noperator algebra $C(\\Lambda) \\rtimes \\Gamma$, since Connes\nhyperfiniteness result \\cite{Conneshyper} and the hyperbolic growth of\nthe group $\\Gamma$ ($g \\geq 2$) prevent this algebra from carrying\nfinitely summable spectral triples.  \nHowever, $C(\\Lambda) \\rtimes \\Gamma$ can \nbe identified with a Cuntz--Krieger algebra (\\cite{CK}), which has a\nstandard AF-subalgebra $\\til{A}$. This has a maximal abelian\nsubalgebra that can be identified with our algebra $A$ (\\cite{CK},\n2.5). One can construct a conditional expectation $E$ from $\\til{A}$\nto $A$. Thus,  the construction of the spectral triple extends to this\nAF-algebra $\\til{A}$ by using an \\emph{unfaithful} state $\\tau \\circ\nE$ that is zero outside $A$ and equals our state $\\tau$ on $A$. Since\nthis construction, however, is just a ``factorisation'' through the\nabove commutative spectral triple, it doesn't give a lot of new\ninformation. \n\\end{rem}\n\n\n\\section{Remarks and Questions} \n\n\\begin{se} \n An interesting question (e.g., in the light of Arakelov geometry) is\nhow to generalize the result to the case of $p$-adic Mumford curves\n(\\cite{GvdP}, \\cite{Mumford}). See \\cite{CMRV} for some results on\ntrees, inspired by earlier work on non-finitely summable spectral\ntriples for tree actions by Consani and Marcolli\n\\cite{ConsaniMarcolli}. \n\\end{se}\n\\begin{se}  Is there such a theory for Fuchsian uniformisation instead\nof Schottky uniformisation? In the noncompact case? \n\\end{se}\n\\begin{se} There are other ways of looking at Riemann surfaces from\nthe point of view of non-commutative geometry. For example, as a\ncommutative conformal manifold, it carries the non-commutative\ndifferential geometry of ``quantized calculus'', whose Fredholm module\ndetermines the conformal isomorphism type of the surface (Connes,\nDonaldson, Sullivan, N.~Teleman \\cite{Connesbook},\nIV.4.$\\alpha$). Also, Consani and Marcolli (\\cite{CMinf}) have\nconstructed $\\theta$-summable, but non finitely summable spectral\ntriples from the boundary action, where the Hilbert space is a\nsymmetrized version of the $L^2$-space of the boundary that should\nhave Hausdorff dimension $<1$. There is the work of B\\\"ar\n(\\cite{Baer}) mentioned before. All of these constructions are rather\ndifferent from the one in this paper. In particular, our spectral\ntriple is finitely summable, so better suited to tools such as the\nlocal index formula of Connes and Moscovici (\\cite{CoMo}).  \nNevertheless, the question arises whether the Consani-Marcolli\nspectral triple hears the conformal shape of the Riemann surface.  B\\\"ar's spectral tiple does enjoy this property.\n\\end{se}\n\\begin{se}  Does our spectral triple ${\\mathcal S}_X$ carry a real structure?\nDoes it then arise from a commutative spin manifold (\\cite{GVF},\n\\cite{RV})? Can one enhance ${\\mathcal S}_X$ by additional classical structure,\nso this structure determines the conformal structure of $X$ completely\n(not just conformal or anti-conformal)? \nNotice that in our construction we work only with the absolute value\nof the Dirac operator, and we use zeta functions that do not see the\nsign. In order to relate it to spectral triples coming from a\nRiemannian manifold one would need to first enrich it with a sign\n(which gives the fundamental class in $K$-homology) and then with a\ncompatible real structure. It seems that the information on the sign\nwill be needed to reconstruct completely the conformal structure. \n\\end{se}\n\n\\begin{se}  Since special values of the spectral zeta functions are\njust 0-Hochschild homology, it is interesting to compute higher\ncharacteristic classes of the spectral triple and relate them to the\nactual geometry of the original Riemann surface. This is especially\ntempting in arithmetically interesting cases, such as modular curves. \n\\end{se}\n\\begin{se}  Can Theorem \\ref{tB} be extended to an injective functor\nfrom the category of Riemann surfaces with conformal and\nanti-conformal morphisms to a category of spectral triples? \n\\end{se}\n\n\n\n\n\n\n\n\n\\begin{small}\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n{\\bf Introduction:} Being the heaviest particle in the Standard Model (SM), the top quark is\nconsidered to play a special role in the electroweak (EWK) symmetry breaking. Precise\nunderstanding of the top quark production processes is essential for the study of physics\nbeyond the SM since they are usually the most important background of these new physics.\nDue to the short lifetime of top quark, its spin is fully inherited by its decay products.\nThus, top quark can be used as a tool to study the spin information of the related processes.\nSimilarly, we study in this paper that the top quark can also be utilized to study\nthe color properties of the production processes. An efficient treatment is just to\nanalyze the shape of the b jet from the top decay, as the top quark color are fully\ncarried on by its decay product b quark.\n\n\nRecently, the anomaly of top quark forward-backward asymmetry~($A_{FB}$) has been a hot topic.\nIt was observed by the CDF and D0 Collaborations at the Tevatron and the CDF latest result\nstill supports that there is an excess in top $A_{FB}$~\\cite{Aaltonen:2012it}. Disappointingly,\nthe top $A_{FB}$~(or sometimes named as the charge asymmetry) is very difficult to be measured\nprecisely at the LHC due to the large symmetric gluon-gluon fusion component. Two of us\nhave introduced a so called color octet axial-vector-like boson to explain\nthis anomaly~\\cite{Xiao:2010ph}. Such a new particle is also called the light\naxigluon~\\cite{Gross:2012bz}. Recent studies showed the color octet axial-vector-like boson\ncan have the virtue to give the correct differential distributions, such as\n $d\\sigma\/d M_{t\\bar{t}}$, $d A_{FB}\/ d M_{t\\bar{t}}$, and $d A_{FB}\/d Y$, where $\\sigma$ is\nthe production cross section, $M_{t\\bar{t}}$ is the top pair invariant mass, and $Y$ is the\nrapidity of the top quark. However, the other explanations often have difficulties to give\nsuch a good fit~\\cite{Wang:2011hc}. A very important feature of the new boson is that it carries\noctet color and then can not be seen directly due to the color confinement. Interestingly,\nthe color flow in quark pair production does have been studied in Ref.~\\cite{Gallicchio:2010sw}.\nThe mediating particles carrying different colors (color singlet or color octet) can be\ndistinguished successfully. In this paper, we adopt the color flow method in the\ntop $A_{FB}$ analysis, especially in studying the color octet axial-vector-like boson explanation.\n\n\n\n\n{\\bf The Color Octet Axial-Vector-Like Boson:} The color octet axial-vector-like boson\n$Z_C$ is an effective description to explain the top $A_{FB}$ anomaly.\nThe effective lagrangian can be written as\n\\begin{equation}\n\\begin{array}{rl}\n{\\cal L} \\propto &- i g^Q_A \\bar{Q} \\gamma_\\mu \\gamma_5 T_a Q {Z_C}^\\mu_a\\\\\n& - i g^q_A \\bar{q} \\gamma_\\mu \\gamma_5 T_a q {Z_C}^\\mu_a~,\\\\\n\\end{array}\n\\end{equation}\nin which $Q=t, b$ and $q=u, d, c, s$.\nThe squared invariant amplitude with color and spin summed is\n\\begin{equation}\n \\begin{array}{l}\n \\sum\\limits_{\\text{Color, Spin}}|{\\cal M}|^2= \\frac{C_A C_F}{2}\\{4g_s^4 (1+c^2+4m^2)\\\\\\\\\n  +\\frac{8g_s^2 \\hat{s}(\\hat{s}-M_{Z_C}^2)}{(\\hat{s}-M_{Z_C}^2)^2+M_{Z_C}^2\\Gamma_{Z_C}^2}2g_A^q g_A^t c\\\\\\\\\n  +\\frac{4\\hat{s}^2}{(\\hat{s}-M_{Z_C}^2)^2+M_{Z_C}^2\\Gamma_{Z_C}^2}\n (g_A^qg_A^t)^2(1+c^2-4m^2)\\},\n \\end{array}\n\\label{eq.simp_axigluon}\n\\end{equation}\nwhere $C_A=3$~, $C_F=4\/3$, $g_s$ is the strong coupling constant, $m=m_t\/\\sqrt{\\hat s}$, ${\\hat s}$ is the parton level interacting energy, $\\beta=\\sqrt{1-4m^2}$, and $c=\\beta\\cos\\theta$. The decay width $\\Gamma_{Z_C}$ of $Z_C$ is given by\n\\begin{equation}\n\\Gamma_{Z_C}=\\sum\\limits_{i}^{2m_i<M_{Z_C}}\\frac{g_A^i}{4\\pi}\\frac{C_F}{8}\n(1-4m_i^2\/M_{Z_C}^2)^{\\frac{3}{2}}M_{Z_C}\n\\label{ZCwidth}\n\\end{equation}\nIn Eq.~(\\ref{eq.simp_axigluon}), the first term is the contribution from SM gluon mediation,\n the second term is the interference between $q\\bar{q}\\to g \\to t\\bar{t}$ and\n $q\\bar{q}\\to Z_C \\to t\\bar{t}$, and the last term is the self-conjugation of the $Z_C$\nmediating process. Some characters can be described for the $Z_C$ explanation:\n (1) The additional asymmetry comes from the interference between the\ngluon and $Z_C$ processes. This can only change the angular distribution of\nthe produced $t\\bar{t}$ events and has no contribution to\n the total $t\\bar{t}$ cross section; (2) The third term of $Z_C$ self-conjugation can\ncontribute to the $t\\bar{t}$ total cross section. However, this contribution can be\nsuppressed by adopting proper couplings;  (3) The Breit-Wigner propagator of $Z_C$\nshows that $Z_C$ can only impact on its near mass peak region. Actually,\nour original idea was to introduce a near top pair production threshold\nparticle to replace the usual heavy ($>1$~TeV) axigluon.\n\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.23\\textwidth]\n{plot_afb_mass.eps}\n\\includegraphics[width=0.23\\textwidth]\n{plot_afb_dy.eps}\n\\includegraphics[width=0.23\\textwidth]\n{plot_xsec_mass.eps}\n\\end{center}\n\\caption{\\label{Zc-differential-distribution}The differential distributions with best fit $Z_C$ parameters. The experimental data comes from~\\cite{Aaltonen:2012it, Aaltonen:2009iz}.}\n\\end{figure}\n\n\n\n\nFig.~\\ref{Zc-differential-distribution} shows the best fit results by $Z_C$. The experimental data of $A_{FB}$ versus $M_{t\\bar{t}}$ and $\\Delta Y$, together with the differential distributions of the cross section $d\\sigma\/d M_{t\\bar{t}}$ are constructed in the $\\chi^2$ global fit. The free parameters to fit can be taken as $g_A^q$, $g_A^Q$ and $M_{Z_C}$, or equally as $g_A^q g_A^Q$, $\\Gamma_{Z_C}$ and $M_{Z_C}$. Table~\\ref{fitted-parameter} shows the consistent results in the two parametrization scenarios.\n\n\n\\begin{table}[htb]\n\\caption{\\label{fitted-parameter}The best fit parameters in two free parameter selection scenarios. The unit of the mass and width is GeV. }\n\\begin{tabular}{ccc|ccc}\n\\hline\\hline\n$g_A^q$ & $g_A^t$ & $M_{Z_C}$ & $g_A^q g_A^t$&$\\Gamma_{Z_C}$ &$M_{Z_C}$ \\\\\n\\hline\n 0.034&9.5 & 340 &0.32 &400&340 \\\\\n\\hline\\hline\n\\end{tabular}\n\\end{table}\nIt can be seen that $Z_C$ has tiny couplings to the light quarks and has very strong couplings to the heavy quarks. People may argue that $g_A^Q$ is so large that it may destroy the perturbation of the theory. Our statement is that the color octet axial-vector-like particle $Z_C$ is just an effective explanation of the top $A_{FB}$ anomaly. It is not necessary to be taken as a fundamental particle.\n\nActually, the forward-backward asymmetry is just a variate to exhibit the angular distribution of the final produced top quarks. For large enough luminosity, we can study the angular distribution directly and more information can be reserved by avoiding the half plane angular integration. Recently, by analyzing the $9.4~\\mbox{fb}^{-1}$ data at the Tevatron, the CDF Collaboration gave the polar angular distribution of the top quark in the $t\\bar{t}$ rest frame in Ref.~\\cite{angular-distribution}.\nFig.~\\ref{Zc-angular-distribution} shows our result by taking the above best fit parameters. The introduction of an effective color octet axial-vector-like particle can give a pretty good agreement with the experimental data.\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.3\\textwidth]\n{plot_TopCosTheta.eps}\n\\end{center}\n\\caption{\\label{Zc-angular-distribution}The normalized angular distributions of the top quark in $t\\bar{t}$ rest frame at the Tevatron. The experimental data and SM QCD NLO +EWK predicted values are taken from Ref.~\\cite{angular-distribution}. }\n\\end{figure}\n\n{\\bf The Color Flow Method Analysis:} In this paper, we take the method in Ref.~\\cite{Gallicchio:2010sw} to analyze the octet color of the $Z_C$. The color flow method is invented to distinguish the color of the mediate particle in quark pair production process. If we draw the transverse momentum $P_T$ of the particles of the quark pair jets in the rapidity-azimuthal angular $y-\\phi$ plane, the different color of the mediate particle will lead to different shapes of the jets. For a color singlet\nmediating particle, such as $\\gamma\/Z$, the two jet shape seems to attract to each other, while for\na color octet mediating particle, such as gluon, the two jet shape seems to repel to each other. Such a difference is caused by the different color charge flow in the Feynman diagrams. A vivid example can be found as Fig.~2 in Ref.~\\cite{Gallicchio:2010sw}.\n\nTo distinguish the color flow, a measurable variate named as the Pull\nis defined as~\\cite{Gallicchio:2010sw}\n\\begin{equation}\n\\vec{t}=\\sum\\limits_{i\\in {\\rm jet}} \\frac{P_T^i|r_i|}{P^{\\rm jet}_T}\\vec{r}_i,\n\\label{Pull}\n\\end{equation}\nwhere $\\vec{r}_i=(\\delta y_i,\\delta \\phi_i)=\\vec{c}_i-\\vec{J}$, in which\n $\\vec{J}=(y_J,\\phi_J)$ is the location of the jet and $\\vec{c}_i$ is\nthe position of a cell or particle with transverse momentum $P_T^i$.\nWe would like to emphasize that $\\vec{t}$ is a two dimensional vector\nand it can be written as $\\vec{t}=(t_y, t_\\phi)=(|\\vec{t}|\\cos \\theta_t|,|\\vec{t}|\\sin\\theta_t)$. $\\theta_t$ is the polar angle determined by the two components of $\\vec{t}$.\n\nWe adopt the Madgraph~\\cite{Alwall:2011uj} to make the Monte-Carlo (MC) simulation. The top pair can be produced through the SM processes and the $q{\\bar q}\\to Z_C \\to t\\bar{t}$ process. We use\nPythia~\\cite{Sjostrand:2006za} to simulate the top decay and the sequent hadronization to form the jets. Some special characters in the simulation are described as follows\n\\begin{itemize}\n\\item Bottom jet Pull is calculated to replace the ``top jets''. As top quark decays to bottom and the $W$ boson. Its color is fully carried by the bottom quark, and then the selection of bottom jet Pull can avoid the dilution caused by $W$ boson from the top quark.\n\\item When calculating the b jet Pull, a core $\\vec{J}=(y_J,\\phi_J)$ is needed and it is taken as the b quark from the top decay in the Monte-Carlo sample events. Such a b quark is actually invisible in the real data sample.\n\n\\item For the hadronization of the bottom quark from the top decay in the MC simulated events, the b quark firstly forms a string\/cluster together with a light quark. The light quark can be from the hadron beam in the color octet mediating case or from the vacuum in the color singlet case. The string\/cluster then decays into a B meson together with some light quarks which transforms to light hadrons. These light hadrons can generally be divided into two categories. The first category belongs to the beam remnant hadrons, which have large rapidity $y$ and small $P_T$. The second category is the other light hadrons which are likely to locate near to the B meson. The summed momentum of the B meson nearby hadrons and the B meson decay products are approximately equal to the momentum of the b quark from the top decay; the summed momentum of the beam line nearby hadrons is approximately equal to the initial momentum of the light quark which forms a string\/cluster together with the b quark.\n\n\\item In calculating the Pull, we adopt the b quark from top decay as the core, and all the string\/cluster decay products are calculated in the summation in Eq.~(\\ref{Pull}). Although the first category jets actually can not be recorded in the real experiments, their contributions to the Pull are small enough to be neglected for their small $P_T$. So the involving of all the decay products from the string\/cluster can still be a good approximation.\n\n\\item We find that the Pull diagrams are very sensitive to the selection of the core. For example, if we take the B meson to be the core and sum the B meson decay products in the Pull,  the color singlet or octet cases can not be distinguished. This shows that the distortion of the jet shape happens just in time of the B meson production, rather than in its decay. Once the colorless B meson is formed, the color flow is stopped. So its decay is independent of the previous color information.\n\n\\item Based on the above analysis, we can infer that the main contribution to the different Pull really comes from the light hadrons near the B meson, which maybe tagged as a part of the b jet. Experimentally, the b tagging usually requires seeing a secondary vertex from the B meson decay. Inevitably, the light quark jets will fall in the B meson cone,  are irreducible from the B meson decay products, and form a component of the b jet. Such quarks and their decay products carry the color information from the b quark, inherit from the top quark.\n\n\\end{itemize}\n\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]\n{Tevatron_b_kernel_string.eps}\n\\end{center}\n\\caption{\\label{Tevatron_b_kernel_string}The two dimensional diagrams of the Pull $\\vec{t}$ for the different color mediate particles~(upper diagrams), and the histograms of the polar angles $\\theta_t$~(lower diagrams) at the Tevatron. To make the peaks just locate in the central region of the diagrams, we take the $\\bar{p} p$ collide mode in the simulation. The convolution of the parton distribution function is included. }\n\\end{figure*}\n\n\\begin{figure}[htbp]\n\\begin{center}\n\\includegraphics[width=0.23\\textwidth]\n{Tevatron_b_kernel_string_pp_to_tt.eps}\n\\includegraphics[width=0.23\\textwidth]\n{LHC_b_kernel_string_pp_to_tt.eps}\n\\end{center}\n\\caption{\\label{collider_Pull_theta_t}The distribution of polar angle $\\theta_t$ of the Pull vector $\\vec{t}$ at the Tevatron and LHC. $\\epsilon$ is the overall event selection efficiency and ${\\cal L}$ is the integrated luminosity.}\n\\end{figure}\n\nFig.~\\ref{Tevatron_b_kernel_string} shows the Pull vector of the b jet and the histogram of $\\theta_t$ for different processes at the Tevatron. The upper plots are the two dimensional points of the Pull, and the lower plots are the histograms of the polar angle $\\theta_t$ of the Pull. It can be seen that the color octet~($g$ and $Z_C$) and singlet~(SM gauge boson $Z$) mediating particles can be distinguished clearly. For the color octet mediating particles, the b jet shape tends to repel to each other, or in other words, the color flow of the jets are connected to the remnants in the beam direction. Thus, the component of $\\vec{t}$ in the rapidity $y$ direction $t_y$ is larger than the component in the azimuthal angle $\\phi$ direction $t_\\phi$. Consequently, the polar angle $\\theta_t$ of $\\vec{t}$ accumulates more in the $\\pi$ or 2$\\pi$ region. For the top quark produced from the $q\\bar{q}$ initial state, it is more probably be attracted by the remnant beam where the $q$ comes from. That is to say, the b quark from the top decay is likely to form a string\/cluster with a quark from the proton beam. This explains why $\\theta_t$ accumulate more at $\\theta_t=\\pi$ than $\\theta_t=2\\pi$ in $q\\bar{q}\\to g\/Z_C\\to t\\bar{t}$ process. While for the $g g$ initial state processes, the b quark from top decay has equal probability to form a string\/cluster with a quark either from the proton, or from the antiproton. Thus, the peaks of the $\\theta_t$ at $\\pi$ or $2\\pi$ for the $g g$ initial state processes are nearly the same. For the gauge boson $Z$ mediating process, the peaks of $\\theta_t$ locates at $\\pi\/2$ and $3\\pi\/2$. This can be explained as follows: because the top quark is relatively heavy, the longitudinal boost of the top pair is small and then the top quark and the anti-top quark are likely to fly back to back. The b quark pair from the top decay are also nearly back to back\nand their rapidities are small. The $\\phi$ component of Pull $\\vec{t}$ can be significantly larger than the rapidity component $t_y$. Thus, the polar angle $\\theta_t$ accumulates more at $\\pi\/2$ and $3\\pi\/2$ region.\n\nFig.~\\ref{collider_Pull_theta_t} shows the distribution histograms of the $\\theta_t$ at the hadron colliders. For the Tevatron, it is a summation of the sub-processes as shown in Figure~\\ref{Tevatron_b_kernel_string}. As we introduce a color octet particle, its peak also locates at the $\\theta_t=\\pi$ region. However, the peak becomes much wider than the SM case. By measuring the $\\theta_t$ distributions at the hadron collider, the different shapes are distinguishable. This is especially useful at the LHC, in which the top asymmetry is too small to be measured precisely. Because the\nLHC is a proton-proton collider without preferred initial direction, the peaks of $\\theta_t$ at $\\pi$ and 2$\\pi$ have the same altitude.\nTo distinguish the two histograms in Fig.~\\ref{collider_Pull_theta_t}, it is necessary to require $sig=|N_1-N_2|\/(\\sqrt{N_1}+\\sqrt{N_2})>1$, where $N_1$ and $N_2$ are the event numbers in each bin\n for the two histograms. $sig$ gets its largest value at the $\\theta_t$ peak region and the least effective luminosity can be estimated as $\\epsilon{\\cal L}=0.008\/0.00002~\\mbox{fb}^{-1}$ with the selection efficiency $\\epsilon$ being about 0.001\/0.00001 for the Tevatron\/LHC~{\\cite{CDF-dilepton-efficiency,Aad:2011yb}}.\n\n\n\n\n\n{\\bf Conclusion:} In this paper, we study the color properties in the top quark pair production process. Firstly, we show that a color octet axial-vector like new particle with different coupling strengths to heavy and light quarks can provide an excellent effective explanation of\nthe top $A_{FB}$ anomaly discovered at the Tevatron. The color octet property of this new particle makes it very suitable to be studied by adopting the color-flow method. By defining the variate Pull vector, the jet substructure of the b quark from the top decay can be used to exploit the color of the mediating particle in the top pair production. The polar angle of the Pull vector can be measured precisely to distinguish from the SM predictions. This can be a first indirect cross check of the Tevatron top $A_{FB}$ excess at the LHC.\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\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n{\\bf Acknowledgements:} This research is supported in part\nby the Natural Science Foundation of China\nunder grant numbers 11075003, 10821504, 11075194, 11135003,\nand 11275246, and\nby the Postdoctoral Science Foundation of China under grant\nnumber 2012M510564 and 2012M520098.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHistorically, the first planetary system to be resolved around a main-sequence star was the debris disc imaged by \\citet{Smith84} around $\\beta$~Pic, a bright ($V=3.5$) and nearby ($d=19.3$~pc) A5V star. Since then, tremendous instrumental development has allowed the $\\beta$~Pic planetary system to be studied with constantly improving angular resolution and dynamic range. This effort culminated with the discovery of a giant planet orbiting at about 8--9~AU from its host star \\citep{Lagrange10}. This planet is now widely recognised to be responsible for the warp detected in the inner debris disc up to about 85~AU from the star, as first proposed by \\citet{Mouillet97}. The origin of the dynamical excitation in the system, however, remains unclear. No other massive companion seems to be present at larger orbital distance, with upper limits of a few Jupiter masses \\citep{Boccaletti09}. Additional planetary-mass objects were also searched for closer to the star using various observing techniques. So far, coronagraphic observations provided upper limits significantly higher than $10 M_{\\rm Jup}$ in the $0\\farcs2$ to $0\\farcs3$ region \\citep{Boccaletti09,Quanz10}, while interferometric observations ruled out the presence of objects with masses larger than about $50 M_{\\rm Jup}$ within $0\\farcs1$ \\citep{Absil10}. Complementary information was also obtained with radial velocity measurements of the host star, providing detection limits for orbital periods up to 1000 days, that is, a maximum angular separation of about $0\\farcs12$, with values in the planetary mass range for periods of up to 500 days \\citep{Lagrange12}.\n\nIn this Letter, we aim to improve upon these previous studies and provide the best detection limits achieved so far at angular separations ranging from $0\\farcs1$ to $0\\farcs4$, using the new $L'$-band Annular Groove Phase Mask \\citep[AGPM,][]{Mawet05} vector vortex coronagraph on VLT\/NACO \\citep{Mawet13}. The AGPM provides a compelling combination of an exquisite inner working angle (IWA), down to the diffraction limit of the telescope ($\\sim 0\\farcs1$ at $L'$ band), with an operating wavelength that is often regarded as a sweet spot for exoplanet imaging.\n\n\n\\section{Observations and data reduction}\n\n\\begin{figure*}[!t]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=7.5cm]{betpicb_frame_v5.png} & \\includegraphics[width=7.5cm]{betpicb_final_v2.png}\n\\end{tabular}\n\\end{center}\n\\caption{\\textit{Left.} Illustration of an individual image of the cleaned cube (log scale), showing that the companion can be readily identified without further image processing (cf.\\ green circle). The region used for frame selection is located between the two brown circles. \\textit{Right.} Final reduced image obtained as the median of the de-rotated sPCA-processed cleaned cube (linear scale). North is up and east to the left in both images.}\n\\label{fig:image}\n\\end{figure*}\n\nOn 31 January 2013, $\\beta$ Pic was observed for about 3.5 hours at $L'$ band within the science verification observing run of NACO's new AGPM coronagraphic mode. The observations were obtained under fair seeing (${\\rm FWHM} \\sim 1\\farcs0$), but the turbulence was fast ($\\tau_0\\simeq2$~msec), resulting in frequent openings of the AO loop despite the use of the infrared wave-front sensor. The Strehl ratio at $L'$ band typically ranged between 70\\% and 75\\% during the observations. Despite the mediocre observing conditions, $\\beta$~Pic~b could be directly identified on the NACO real time display, thanks to the peak starlight extinction of about 50:1 provided by the AGPM (see Fig.~\\ref{fig:image}, left). Individual observing blocks (OBs) consisted of 200 successive frames of 0.2 sec each, with the detector windowed to a $768\\times768$ pixels region. Sequences of 10 or 20 OBs were obtained, each of them followed by three OBs of 50 frames on a nearby sky region and by re-centring the star under the AGPM mask to achieve maximal extinction. A total of 190 OBs were obtained on $\\beta$~Pic, resulting in an on-source integration time of 114 min, out of an overall observing time of 204 min. Observations were taken in pupil-tracking mode, with a parallactic angle ranging from $-15\\degr$ to $68\\degr$. The undersized Lyot stop APO165 was used as in \\citet{Mawet13}. Before and after the coronagraphic observations, we measured the non-coronagraphic point spread function (PSF) through the AGPM and the Lyot stop by placing the star away from the vortex centre. These observations also serve as photometric reference in our data analysis, where we take the airmass variation into account to properly normalise individual frames.\n\nAfter applying basic cosmetic treatment to individual frames (flat-fielding, bad pixel\/cosmic-ray removal), we started the data processing by performing frame selection to remove the frames affected by strong AO loop openings. We found that a convenient selection criterion is the standard deviation of the pixel intensity in the $1-4 \\lambda\/D$ region ($0\\farcs1 - 0\\farcs4$), which is most affected by residual starlight but does not contain the signal of $\\beta$~Pic~b (located beyond $0\\farcs4$ during our observations). To enhance the contribution of additional residual starlight in bad frames with respect to good ones, we performed a principal component analysis (PCA) of the whole image cube, and subtracted from each individual image its projection onto the first principal component (PC) before computing the standard deviation of the pixel intensity. A histogram of all measured standard deviations was built, and the threshold for frame selection was set at a $1\\sigma$ level above the median. The fraction of frames rejected by this selection process was about $15\\%$. We found this image-processing step to be critical in reaching the best possible image quality.\n\nThe second image-processing step consisted of accurately recentring the frames. We performed a (negative) Gaussian fit to the PSF centre, which resembles a dark hole surrounded by a bright doughnut (see Fig.~\\ref{fig:image}, left). This shape is mainly due to the combination of residual tip-tilt with the off-axis transmission profile of the AGPM. By recentring each image using the fitted Gaussian, we made sure that the centre of AGPM is placed at the exact same position in all individual frames, with a typical accuracy of 0.005 pixel (i.e., about 0.1~mas). We emphasize that this centring accuracy pertains to the position of the AGPM, not of the star itself. We then subtracted from each frame the estimated contribution of the sky based on the median of neighbouring sky observations. A new image cube was then created by averaging 40 successive frames (i.e., 8~sec of effective integration time), resulting in 612 individual images in the cleaned, recentred cube.\n\nFinally, we used our implementation of the KLIP algorithm \\citep{Soummer12} to produce a final image of the $\\beta$~Pic system based on the cleaned cube, taking advantage of angular differential imaging (ADI). In the KLIP algorithm, the whole ADI image sequence is used as a PSF library, to which a PCA treatment is applied. In the presence of an off-axis companion, the PC computed on the ADI cube are expected to contain (part of) the companion signal. To prevent the planet from being partly removed from the individual images when subtracting its projection onto the first $K_{\\rm klip}$ PC, we decided to implement a ``smart'' version of the PCA (or \\emph{sPCA}), where the image library is built only from images where the off-axis companion has rotated by $1\\lambda\/D$ or more with respect to the image under consideration. In this way, the PC will not contain any (or only a very small amount of) signal from the companion at its current position. At the angular separation of $\\beta$~Pic~b ($\\sim 0\\farcs45$), this translates into rejecting from the PSF library all images that have parallactic angles within about $15\\degr$ from the image under consideration. We performed the sPCA in a region of about $3\\arcsec$ in radius around the star, after masking out the region located within the IWA of the AGPM \\citep[$\\sim \\lambda\/D$,][]{Delacroix13}. The final image, obtained as the median of the de-rotated cube after sPCA processing, is shown in Fig.~\\ref{fig:image}. The number of PC kept in this analysis is $K_{\\rm klip}=30$ (out of 612), which is a good compromise to prominently reveal the planetary companion. The black spots on either sides of the planet are artefacts related to the rotation of the planet around the optical axis in the image sequence taken in pupil-tracking mode.\n\n\n\n\\section{Analysis of $\\beta$ Pic b}\n\n\t\\subsection{Photometry}\n\nEven when using sPCA, part of the planetary signal is self-subtracted during the stellar halo removal process. To retrieve the photometric information without bias, we used the negative fake companion technique \\citep{Lagrange10,Bonnefoy11}. The method proceeds as follows: (i) estimate the (biased) position and flux of the companion from the first reduced image; (ii) use the measured off-axis PSF as a template to remove this first estimate from the cleaned data cube before applying PCA; and (iii) iterate on the position ($x$, $y$) and flux until a well-chosen figure of merit -- here, the weighted sum of the squared pixel intensity in a pie chart aperture centred on the first estimate of the companion position, $2.44\\lambda\/D$ in radius and $6 \\times 1.22\\lambda\/D$ in azimuth -- is minimized. The minimization was performed with the simplex-amoeba optimization. An exploration of the figure of merit (equivalent to a $\\chi^2$) around the best-fit position is used to evaluate the statistical error bar on the photometry of the planet ($0.15$~mag at $1\\sigma$ assuming Gaussian noise). Adding the contribution of photometric variations ($0.05$~mag for a photometric night) to the error bar, and correcting the companion photometry for the off-axis transmission profile of the AGPM \\citep{Delacroix13}, we obtain a final contrast $\\Delta L' = 8.01 \\pm 0.16$~mag between the planet and the star. This result agrees within error bars with the contrast ($7.8 \\pm 0.3$~mag) found by \\citet{Lagrange10} in the same photometric band. The final photometry of $\\beta$~Pic~b is given in Table~\\ref{tab:planet}. With an absolute magnitude $M_{L'}=10.02\\pm0.16$, and assuming an age of $12^{+8}_{-4}$~Myr for the system as in \\citet{Bonnefoy13}, the estimated mass of $\\beta$~Pic~b amounts to $8.0^{+3.2}_{-2.1} M_{\\rm Jup}$, based on the BT-Settl models of \\citet{Allard11}. The recent revision of the age \\citep[$21\\pm4$~Myr,][]{Binks13} would lead to a mass of $10.6^{+1.2}_{-1.8} M_{\\rm Jup}$.\n\n\\begin{table}\n\\caption{Measured properties of the $\\beta$~Pic system}\n\\label{tab:planet}\n\\centering\n\\begin{tabular}{c c c}\n\\hline\\hline\nParameter & $\\beta$ Pic A & $\\beta$ Pic b \\\\\n\\hline\n$d$ (pc) & $19.44 \\pm 0.05$ & --- \\\\\n$L'$ (mag) & $3.454\\pm0.003$ & $11.46\\pm0.16$ \\\\\n$M_{L'}$ (mag) & $2.011\\pm0.006$ & $10.02 \\pm 0.16$ \\\\\nSeparation (mas) & --- & $452 \\pm 10$ \\\\\nPA (deg) & --- & $211.2 \\pm 1.3$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\t\\subsection{Astrometry}\n\nThe astrometry of the companion was evaluated using the negative fake companion technique, following the same procedure as for the photometry. The main limitation in the astrometric accuracy in high-contrast imaging generally comes from the imperfect knowledge of the star position in the images (either because it is masked by a coronagraph, or because the star is saturated in the images). In our case, the recentring procedure allowed us to make sure that the AGPM position is the same in all images with an accuracy of about 0.1~mas. This does not mean that the position of the star in the image is known with such an accuracy, however, as it is generally not perfectly centred on the AGPM. We can evaluate the misalignment between the star and the AGPM centre thanks to the instantaneous starlight rejection rate. Assuming that the residual starlight is entirely due to misalignment, we derive a mean upper limit of about 8.5~mas on the error on the position of the star with respect to the AGPM centre. This error is folded into our astrometric error budget. Although this error bar is not drastically improved with respect to saturated $L'$-band data sets \\citep{Chauvin12}, the more robust knowledge of the stellar position is still a clear advantage of the AGPM observing mode.\n\nThe other contributors to the astrometric errors are the statistical error on the companion position and the calibration errors. The statistical error bar is estimated by exploring the figure of merit around the best-fit solution obtained with the negative fake companion technique, yielding a $1\\sigma$ error of 4.5~mas. The calibration errors, related to the platescale, true north orientation and uncertainty on the NACO rotator offset, are computed as in \\citet{Chauvin12}, using an estimate platescale of $27.10 \\pm 0.04$~mas, a true north of $-0\\fdg45 \\pm 0\\fdg09$, and a rotator offset of $104\\fdg84 \\pm 0\\fdg01$. They are negligible in the final error budget. The final astrometry with error bars is given in Table~\\ref{tab:planet}. A comparison with the orbital position predictions of \\citet{Chauvin12} suggests that the planet is now starting to recess.\n\n\n\n\\section{Search for additional, inner companions}\n\nIn this section, we exploit the exquisite IWA of the AGPM to provide new constraints on the presence of planetary companions down to an angular distance of $1\\lambda\/D$ ($0\\farcs1$) at L' band. At the distance of $\\beta$~Pic, this represents a linear separation of about 2~AU. To derive unbiased contrast curves, we removed the contribution of $\\beta$~Pic~b from all individual images in the cube, using the best-fit photometry and astrometry found in the previous section and the off-axis PSF obtained during our observations. The noise level can then be computed as the standard deviation of the pixel intensity in concentric annuli, or equivalently as the azimuthal median of the noise computed locally in small square boxes, even at the location of the (removed) planet. However, the final contrast curve cannot be trusted within a circular region about $1.22 \\lambda\/D$ ($0\\farcs13$) in radius around the position of $\\beta$~Pic~b, because the presence of an additional companion within this zone would result in a partial PSF overlap and in an improper subtraction of $\\beta$~Pic~b from the image cube.\n\nWe implemented an annulus-wise version of the sPCA method, where the exclusion criterion on the parallactic angles to be included in the PSF library is computed separately in thin annuli $2\\lambda\/D$ in width. To assess the amount of self-subtraction of potential companions at any given angular separation from the star, we introduced fake companions in our cube, separated by a few $\\lambda\/D$ from each other and placed on three radial branches separated by $120\\degr$ in azimuth. The fake companions were injected at $20\\sigma$ above the noise computed after a first pass of the sPCA algorithm on the cube without fake companions. By measuring the photometry of the fake companions in the final reduced image after a second pass of the sPCA algorithm and comparing it to their input flux, we inferred the attenuation of the sPCA algorithm in ADI mode. As expected, the self-subtraction strongly depends on the width of the exclusion zone in terms of parallactic angle, but the final contrast curve is only weakly affected by this parameter. Taking self-subtraction into account, small exclusion zones provide slightly better detection limits at very small angular separations, because the frames that are more correlated (i.e., closer in time) to the current frame are then kept in the library. An exclusion zone of only $0.1\\lambda\/D$ is used in the following. With this criterion, and taking $K_{\\rm klip}=20$ in each ring, the companion self-subtraction is significant in the innermost parts of the search region, with only about 40\\% (resp.\\ 20\\%) of the signal making it through at $0\\farcs5$ (resp.\\ $0\\farcs25$).\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=9cm]{contrast_box.pdf}\n\\end{center}\n\\caption{$5\\sigma$ detectability limits in terms of contrast for point-like companions around $\\beta$~Pic (solid line). The dotted line shows the contrast curve that would be derived if self-subtraction encountered in the ADI-PCA data processing was not taken into account. The grey-shaded region recalls that the detection limits are not valid in a small region of $\\sim 0\\farcs13$ in radius around $\\beta$~Pic~b.}\n\\label{fig:contrast}\n\\end{figure}\n\nThe final contrast curves, computed as five times the standard deviation of the pixel intensities in $\\lambda\/D$-wide annuli after applying a median filter on a $\\lambda\/D$-wide moving box, are displayed in Fig.~\\ref{fig:contrast} before and after taking into account companion self-subtraction. Owing to the small number of independent resolution elements in $\\lambda\/D$-wide annuli at short angular separations, the confidence level ($1-3\\times 10^{-7}$) associated to a $5\\sigma$ detection limit for pure Gaussian noise is not preserved, as discussed in Mawet et al.\\ (in prep). Reducing the confidence level (i.e., increasing the false alarm probability) at small angles is, however, not a severe limitation to the validity of the detection limits, not only because no candidate companion is found in our case, but also because one can readily distinguish false positives (bright speckles) from true companions at a few $\\lambda\/D$ with follow-up observations (background point-like sources being very unlikely in such a small search region). \n\nThe detection limits presented in Fig.~\\ref{fig:contrast} are based on the debatable assumption of Gaussian noise. To address this point, which was shown by \\citet{Marois08} to be a potential source of bias on the sensitivity limits, we constructed the histogram of the pixel intensities in the final image (the median of the de-rotated individual sPCA-processed images in the cleaned cube), using annuli $\\lambda\/D$ in width. Using the Shapiro-Wilk test \\citep{Shapiro65}, we verified that for most annuli within the search region, the statistics of the pixel intensity can be considered as Gaussian, with p-values typically ranging from 10\\% to 90\\%.\n\nTo convert our detection limits into sensitivity limits in terms of planetary mass, we used the BT-Settl models of \\citet{Allard11}. The result is given as the solid curve in Fig.~\\ref{fig:detlim}, where the sensitivity limits of previous high-contrast imaging studies have been plotted for comparison using the same BT-Settl models. This figure illustrates the high gain in sensitivity at short angular separation enabled by the $L'$-band AGPM, which allows planets more massive than $5M_{\\rm Jup}$ to be ruled out down to about $0\\farcs2$ from $\\beta$~Pic. Detection limits within the planetary-mass regime are derived down to $0\\farcs1$, although the strong self-subtraction, the bright speckles, and the small number of independent resolution elements urge us to take the detection limit with caution in the $0\\farcs1 - 0\\farcs2$ region. The sensitivity to companions located farther away than $\\beta$~Pic~b is also excellent, down to about $1M_{\\rm Jup}$ beyond $1\\farcs5$. With 768 pixels in our images, the outer working angle is about $10\\arcsec$.\n\n\\begin{figure}[!t]\n\\begin{center}\n\\includegraphics[width=9cm]{betPic_sensitivity_box.pdf}\n\\end{center}\n\\caption{Sensitivity limits in terms of mass derived from our data set (solid line), compared with the results of \\citet[dashed and dashed-dotted]{Boccaletti09} and \\citet[dotted]{Quanz10} using the same evolutionary models (see text). This figure cannot serve as a direct comparison between the sensitivity of these NACO observing modes, as ADI was not used in the three previously published data sets.}\n\\label{fig:detlim}\n\\end{figure}\n\n\n\n\\section{Concluding remarks}\n\nWe have presented the results of our search for additional planetary companions around $\\beta$~Pic. These results nicely complement those obtained at shorter orbital periods  using radial velocity measurements \\citep{Lagrange12}. Combined, these two studies can now exclude the presence of giant planets with masses similar to that of $\\beta$~Pic~b at most orbital distances (only in the 1--2~AU region are the detection limits still in the 10--20\\,$M_{\\rm Jup}$ range). The fact that we do not detect any additional, massive companion around $\\beta$~Pic agrees with the conclusion that $\\beta$~Pic~b is the sole body responsible for the warp in the inner debris disc, as proposed by \\citet{Lagrange12}. It suggests that the dynamical excitation in the $\\beta$~Pic inner system is most probably not directly related to gravitational interactions with other massive bodies ($>5M_{\\rm Jup}$) located in the inner system.\n\nThe excellent sensitivity limits that we obtained down to very short angular separations from $\\beta$~Pic illustrates the potential of the $L'$-band AGPM coronagraph recently installed on NACO, despite the mediocre seeing conditions. We expect that the NACO\/AGPM sensitivity can be further improved under better adaptive optics (AO) correction, with an improved calibration of static instrumental aberrations, and with an optimal Lyot stop. The AGPM mode on NACO would be very well suited to follow-up and characterise new companions found by upcoming near-infrared imagers aided by extreme AO systems. Although our observing sequence on $\\beta$~Pic spans a wide range of parallactic angles ($83\\degr$), we note that the sensitivity limit close to the IWA is still largely affected by the limited displacement of any candidate companion between individual images obtained in pupil tracking mode. We suggest that reference-star differential imaging, an observing strategy mostly dropped in favour of ADI during the past few years, may be the most appropriate way to unleash the full potential of phase masks such as the AGPM in terms of IWA, when operating under good and stable AO correction.\n\n\n\n\\begin{acknowledgements}\nThe research leading to these results has received funding from the European Research Council\nunder the FP7 through ERC Starting Grant Agreement no.\\ 337569, and from the Communaut\\'e fran\\c caise de Belgique -- Actions de recherche concert\\'ees -- Acad\\'emie universitaire Wallonie-Europe.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAccurately describing the behavior of interacting enzymes, proteins, and genes\nrequires spatially extended stochastic models.  However, such models are\ndifficult to implement and fit to data. Hence tractable reduced models are frequently used instead. \nIn many popular models of biological networks, a single ODE is used \nto describe each node, and sigmoidal functions to describe interactions between them.  Even such simplified ODEs are typically intractable, as the number of parameters and\n the potential dynamical complexity make it difficult to analyze the behavior of the system using purely  numerical methods.\n   Reduced models that capture the overall dynamics, or allow approximate solutions\ncan be of great help in this situation [\\cite{Verhulst_GSPT,Hek2010}].\n\nAnalytical treatments are possible in certain limits.  The\napproaches that have been developed  to analyze models of gene interaction\nnetworks can be broadly classified into three\ncategories~[\\cite{PolynikisHoganBernardo2009}]:  \\emph{Quasi Steady State Approximations} (QSSA), \\emph{Piecewise Linear Approximations} (PLA), and\n \\emph{discretization of continuous time\nODEs}. In particular, in certain limits interactions between network elements become\nswitch--like~[\\cite{kauffman69,Snoussi1989,Mochizuki2005, Alon2006,mendoza2006, DavitichBornholdt2008, Wittmann2009, Franke2010, Veliz-CubaArthurHochstetlerKlompsKorpi2012}].   For instance,\nthe Hill function,   $f(x) = x^n \/ (x^n + J^n )$,  approaches the Heaviside function, $H(x - J)$,\nin the limit of large $n$.\nIn this limit the domain on which the network is modeled\nis naturally split into subdomains: The threshold,\ncorresponding to the parameter $J$ in the Hill function, divides the domain into two subdomains within which\nthe Heaviside function is constant.  Within each subdomain a node is either\nfully expressed, or\nnot expressed at all.  When $n$ is large, the Hill function, $f(x)$, is approximately constant in each\nof the subdomains, and boundary layers occur when $x$ is\nclose to the threshold, $x \\approx J$~[\\cite{IroniPanzeriPlahteSimoncini2011}]. To simplify the system further, we can map values of $x$ below the threshold to 0, and the values above the threshold to 1 to obtain a Boolean network (BN); that is, a map\n\\[h=(h_1,\\ldots,h_N):\\{0,1\\}^N\\rightarrow \\{0,1\\}^N,\\]\nwhere each function $h_i$ describes how variable $i$ qualitatively depends on the other variables\n [\\cite{GlassKauffman1973,Snoussi1989,thomasbook,edwards2000,edwards2001}].  Such reduced systems are simpler to analyze, and share the  dynamical properties of the original system, if the reduction is done properly.\n\n\nThe reduced models obtained in the limit of a large Hill coefficient, $n,$ have a long and rich history. Piecewise linear\nfunctions of the form proposed in~[\\cite{GlassKauffman1973}] have been shown to\nbe well suited for the modeling of genetic regulatory networks, and can sometimes be justified rigorously [\\cite{de04}].  In particular,  singular perturbation\ntheory  can\nbe used to obtain reduced equations within each subdomain and the boundary\nlayers,\nand global approximations   within the entire\ndomain~[\\cite{IroniPanzeriPlahteSimoncini2011}]. On the other hand, although BNs have been used to model the dynamics of different biological systems, their relation to more complete models\nwas mostly demonstrated with case studies, heuristically or only for steady states [\\cite{GlassKauffman1973,Snoussi1989, thomasbook, albert2003, mendoza2006, DavitichBornholdt2008, p53, p53ode, Wittmann2009, Franke2010, Veliz-CubaArthurHochstetlerKlompsKorpi2012}].\n\nHere we again start\nwith the Hill function, $ x^n \/ (x^n + J^n )$, but instead of assuming that $n$ is large, we assume that $J$ is small. This case\n has a simple physical\ninterpretation: Consider the Hill function that occurs in the Michaelis-Menten\nscheme, which models the catalysis of the inactive form of some\nprotein to its active form in the presence of an enzyme. When $J$ is small\nthe total enzyme concentration is much smaller than the total protein\nconcentration.\nAlthough the subsequent results hold for any fixed $n$, for simplicity we\nassume $n = 1$. \n\n\n\nMore precisely, we consider a model biological \nnetwork where the activity at each of $N$ nodes is described by $u_i \\in [0,1]$, and evolves according to\n\\begin{align}\\label{eqn:ProblemEquation}\n\t\\frac{du_i}{dt} = A_i\\frac{1-u_i}{J_{i}^A+1-u_i}-I_i\\frac{u_i}{J_{i}^I+u_i},\n\\end{align}\nwhere $J_i^A, J_i^I>0$, and the functions $A_i=A_i(u)$, $I_i=I_i(u)$ are affine functions.\n\n\nThis type of equations have been used successfully in many models [\\cite{GoldbeterKoshland1981, Goldbeter1991, NovakPatakiCilibertoTyson2001, de02, NovakPatakiCilibertoTyson2003, IshiiSugaHagiyaWatanabeMoriYoshinoTomita2007, CilibertoFabrizioTyson2007, DavitichBornholdt2008, vanZwietenRoodaArmbrusterNagy2011}]. Here  $A_i$ and $I_i$  describe how the other variables affect $u_i$  and can represent activation\/phosphorylation\/ production and inhibition\/dephosphorylation\/decay, respectively. The variables $u_i$ can represent species such as protein concentrations, the active form of enzymes, or activation level of genes.  A simple example is provided by a  protein that can exist in an unmodified form, $W,$ and a modified form, $W^*,$ (e.g. proteases, and Cdc2, Cdc25, Wee1, and Mik1 kinases [\\cite{Goldbeter1991, Novak1998, NovakPatakiCilibertoTyson2001}]) where the conversion between the two forms is catalyzed by two enzymes, $E_1$ and $E_2$ [\\cite{GoldbeterKoshland1981, Goldbeter1991,Novak1998, NovakPatakiCilibertoTyson2001}] (See Appendix for details). \nHowever, note that the models of chemical reactions we consider can be rigorously derived from the Chemical Master Equation only in the case of a single reaction~[\\cite{KumarJosic2011}]. The models of networks of chemical reactions that we take as the starting point of \nour reduction should therefore be regarded as phenomenological. \n\nIt is easy to show that the region $0\\leq u_i\\leq 1, 1 \\leq i \\leq N$ is invariant so that Eq.~\\eqref{eqn:ProblemEquation} is a system of equations on $[0,1]^N$. Equations involving this special class of Hill functions are generally referred to as\nMichaelis-Menten type equations, and $J$  the Michaelis-Menten \nconstant~[\\cite{MichaelisMenten1913, GoldbeterKoshland1981, Goldbeter1991, NovakTyson1993, NovakPatakiCilibertoTyson2001, NovakPatakiCilibertoTyson2003, CilibertoFabrizioTyson2007, DavitichBornholdt2008, ChaoTang2009}]. \n\nThe  constants $J$ are frequently very small in practice [\\cite{NovakPatakiCilibertoTyson2001,DavitichBornholdt2008}], which motivates examining Eq.~\\eqref{eqn:ProblemEquation} \nwhen  $0 < J \\ll 0$. In this case, we discuss a two step reduction of the model \n$$\n\\text{full, nonlinear model $\\longrightarrow$ piecewise linear model  (PL) $\\longrightarrow$ Boolean Network (BN)}.\n$$ \nWe first illustrate this reduction using two standard examples, and then provide a general mathematical justification. We note  that the reduction obtained in the first step (see Eq.~\\eqref{eqn:mainReduced_RSTa}) is actually (algebraic) piecewise affine.\nHowever, it is customary  to refer to the equation and the associated model as \\emph{piecewise linear} [\\cite{GlassKauffman1973,Snoussi1989, thomasbook,edwards2000,de02}], and we follow this convention.\n\nThe main idea behind the piecewise linear (PL) reduction is simple:\nIf $J \\ll x$ then the Hill functions, $f(x) = x \/ (x + J ) \\approx 1$. However, when $x$ and $J$ are comparable, $x \\sim J,$ this is no longer true.  In this boundary layer, we  rescale variables by introducing $\\tilde{x} := x\/J$. A similar argument works for the function $(1-x) \/ (J+1-x )$ (see Appendix). We show that using this observation, the domain $[0,1]^N$ naturally decomposes  into a nested sequence of hypercubes.  The dynamics on each hypercube in the sequence is described by a solvable differential-algebraic system of equations.  The PL reduction therefore gives an \\emph{analytically tractable} approximate\nsolution to the original system.\n\nIn the next step of the reduction we obtain a Boolean Network (BN): The PL  approximation is used to divide $[0,1]^N$ into chambers.   Within nearly all of a chamber the rate of change of each element of the network is constant when $J \\ll 1$.   We use these chambers to define a BN. A similar approach was recently used to motivate a Boolean reduction of a model protein interaction\nnetwork~[\\cite{DavitichBornholdt2008}].\n\n\nThe mathematical justification also follows two steps.  We use Geometric Singular Perturbation Theory (GSPT) in Section \\ref{sec:math_PL} to justify the PL approximation. \nThe justification of the BN reduction is given in Section \\ref{sec:math_BN}.\nWe show that  there is a one-to-one correspondence between steady states (equilibrium solutions) of the BN and the full and PL system near the vertices of $[0,1]^N$. Futhermore, we show that this  one-to-one correspondence between steady states is actually global (up to a set of small measure in $[0,1]^N$). BNs have been used to study oscillatory behavior [\\cite{Li_cc_2004,p53}], and we prove in Section \\ref{sec:math_BN_trajectories} that under some conditions oscillations in a BN correspond to oscillations in the full system.\n\n\n\\section{Example problems}\\label{ExampleProblems}\n\nWe start by demonstrating the main idea of our approach using networks of\ntwo and three mutually repressing nodes. These nodes\ncan represent genes that mutually inhibit each other's\nproduction~[\\cite{GardnerCantorCollins2000,ElowitzLeibler2000}].  \nHowever, \nthe theory we develop applies whenever the heuristic model given in Eq.~\\eqref{eqn:ProblemEquation} is applicable.\nWe accompany these examples with a \nheuristic explanation of the different steps in the\nreduction. \n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale = .35]{toggleSwitch_Represscilator_4}\n\\parbox{.9\\textwidth}{\n\\caption{\\footnotesize {(a) Nodes $u_1$, $u_2$ inhibiting each others activity resulting in a switch. The node which starts out stronger suppresses the activity of the other. (b) Nodes $u_1$, $u_2$, and $u_3$ suppress each other in a cyclic fashion. Under certain conditions, this can lead to oscillations.}%\n}\\label{fig:toggle_repressillator}}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{A network of two mutually inhibiting elements}\n\nWe start with the common \\emph{toggle switch} motif, \\emph{i.e} a network of\ntwo mutually repressing elements (see Fig.~\\ref{fig:toggle_repressillator}a) ~[\\cite{NovakPatakiCilibertoTyson2003,GardnerCantorCollins2000}]. Let\n$(u_1,u_2) \\in [0,1]^2$ represent the normalized levels of activity at the two nodes. \nTherefore, when $u_i = 1$ the $i^{\\text{th}}$\nnetwork element is maximally active (expressed). The activity of the two nodes in the system \ncan be modeled by\n\\begin{align}\\label{EquationForToggleSwitch}\n\\begin{split}\n\\frac{du_1}{dt} &= 0.5\\frac{1-u_1}{J+1-u_1}-u_2\\frac{u_1}{J+u_1},  \\\\\n\\frac{du_2}{dt} &= 0.5\\frac{1-u_2}{J+1-u_2}-u_1\\frac{u_2}{J +u_2},\n\\end{split}\n\\end{align}\nwhere $J$ is some positive constant. The structure of\nEq.~\\eqref{EquationForToggleSwitch} implies that the cube $[0,1]^2 = \\{(u_1, u_2)\n\\,|\\, 0 \\le u_1, u_2 \\le 1 \\}$ is invariant (see\nProposition~\\ref{prop:invariantCube}).\n\n\\subsubsection{Piecewise linear approximation}\n\nIn the limit of small $J$, Eq.~\\eqref{EquationForToggleSwitch}  can be\napproximated by a piecewise linear differential equation:\nIf $u_i$ is not too close to zero the expression $u_i\/(J+u_i)$ is approximately unity.\n  More precisely, we fix a small $\\delta > 0 $, which will be chosen to depend on\n$J$.  When $u_i > \\delta$ and $J$ is small then $u_i\/(J+u_i) \\approx1$. Similarly, when \n$u_i > 1-\\delta$ then  $(1-u_i)\/(J+1-u_i) \\approx 1$.\n\n\nWith this convention in mind we  break the cube $[0,1]^2$ into several\nsubdomains, and define\na different reduction of Eq.~\\eqref{EquationForToggleSwitch} within each. Let $\\mathcal{R}_S^T$ to denote the region where $S$ is the set of variables that\nare close to 0, and $T$ is the set of variables close to 1 (See Table~\\ref{table:2d} and Eq.~\\eqref{Rdef}). Also, we omit the curly brackets and commas in $\\mathcal{R}_S^T$ (e.g.  $\\mathcal{R}^{\\{\\}}_{\\{\\}}=\\mathcal{R}$ and $\\mathcal{R}_{\\{1\\}}^{\\{\\}}=\\mathcal{R}_1$) (see Fig.~\\ref{fig:subdomains}). \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale = .65]{subdomains}\n\\parbox{.9\\textwidth}{\n\\caption{\\footnotesize\nSubdomains $\\mathcal{R}_S^T$ for the unit square $[0,1]^2$. \n\\label{fig:subdomains}}\n}\n\\end{center}\n\\end{figure}\n\nWe first reduce Eq.~\\eqref{EquationForToggleSwitch} on each of the subdomains.\nThe interior of the domain $[0,1]^2$ consist of points where neither coordinate is close to 0 nor 1, and\n is defined by\n\\begin{align}\\label{R00}\n\\mathcal{R}:= \\{(u_1,u_2) \\in [0,1]^2 \\,|\\, \\delta \\le u_1 \\le 1-\\delta \\text{ and }\n\\delta \\le u_2 \\le 1-\\delta\\}.\n\\end{align}\n Eq.~\\eqref{EquationForToggleSwitch}, restricted to $\\mathcal{R}$ is\napproximated by the linear differential equation\n\\begin{align}\\label{interiorLinear2d}\n\\frac{du_1}{dt} = 0.5-u_2, \\quad\n\\frac{du_2}{dt} = 0.5-u_1.\n\\end{align}\n \nOn the other hand, if one of the coordinates is near the boundary, while the\nother is in the interior, the approximation is different. For instance, the\nregion\n\\begin{align}\\label{R20}\n\\mathcal{R}_2:= \\{(u_1,u_2) \\in [0,1]^2 \\,|\\,  u_2 < \\delta \\text{ and } \\delta \\le\nu_1 \\le 1-\\delta\\},\n\\end{align}\nforms a boundary layer where $u_2$ is of the same order as $J$. \n \nThe term $u_2\/(J + u_2)$ cannot be approximated by unity.  Instead the approximation takes the form\n\\begin{subequations}\\label{edgeLinear2d}\n\\begin{align}\n\\frac{du_1}{dt} &= 0.5-u_2, \\label{edgeLinear2da} \\\\\n\\frac{du_2}{dt} &= 0.5-u_1 \\frac{u_2}{J + u_2}. \\label{edgeLinear2db}\n\\end{align}\n\\end{subequations}\nThis equation can be simplified further.  Since $\\mathcal{R}_2$ is invariant (for $u_1>.5$), $\\frac{d u_2}{dt}$ must be small inside the boundary layer\n$\\mathcal{R}_2$ (see Fig.~\\ref{fig:twoDimension_nullclines}).  We therefore use the approximations $u_2 \\approx 0$ in \nEq.~\\eqref{edgeLinear2da} and $\\frac{d u_2}{dt}\n\\approx 0$ in Eq.~\\eqref{edgeLinear2db} to obtain\n\\begin{subequations}\\label{smallu}\n\\begin{align}\n\\frac{du_1}{dt} &= 0.5, \\label{smallua}\\\\\n0 &= 0.5-u_1 \\frac{u_2}{J + u_2}.\\label{smallub}\n\\end{align}\n\\end{subequations}\nNote that Eq.~\\eqref{smallua} is linear  and decoupled from Eq.~\\eqref{smallub},\nwhile Eq.~\\eqref{smallub} is an algebraic system which can be solved to obtain\n$u_2 \\approx J\/(2u_1 - 1)$.\nWithin $\\mathcal{R}_2$ we thus obtain the approximation \n\\begin{subequations}\\label{smallu2}\n\\begin{align}\nu_1(t) & = 0.5 t  + u_1(0)  \\label{E:smalleqa}\\\\\nu_2(t) & = \\frac{J}{ t + 2 u_1(0) - 1}  \\label{E:smalleqb}\n\\end{align}\n\\end{subequations}\n\nWe only have the freedom of specifying\nthe initial condition $u_1(0)$, since  $u_2(0)$ is determined by the solution\nof the algebraic equation~\\eqref{smallub}.  As we explain below, this algebraic\nequation defines a slow manifold within the subdomain $\\mathcal{R}_2$.  The\nreduction assumes that solutions are instantaneously attracted to this manifold. \n\nTable~\\ref{table:2d} shows how this approach can be extended to all of $[0,1]^2$. \nThere are  9  subdomains of the cube, one corresponding to the interior and four each to the \nedges and vertices.  On the latter eight subdomains, one or both variables are close to either 0 or 1.\nFollowing the preceding arguments, variable(s)\nclose to 0 or 1 can be described by an algebraic equation. The resulting algebraic-differential systems are given in the last column of\nTable~\\ref{table:2d}. \nFurthermore, by using the approximations $u_i(t)\\approx 0$ for $i\\in S$ and $u_i(t)\\approx 1$ for $i\\in T$, we obtain a simple approximation of the dynamics in each subdomain which is $0$-th order in $J$. For example, in $\\mathcal{R}_2$, we obtain the approximation $u_1(t) \\approx 0.5t + u_1(0), u_2(t) \\approx 0$.\n\n\nEach approximate solution can potentially exit the subdomain within\nwhich it is defined if at some time $u_i\\approx 0$ or $u_i\\approx 1$ and the $i$-th coordinate of the vector field is positive or negative, respectively. This can happen when the sign of some entry of the vector field changes; that is, solutions can exit subdomains when they reach a nullcline. Also, solutions can leave the subdomain if they started on the other side of the nullcline to begin with.\nThe global approximate solution of Eq.~\\eqref{EquationForToggleSwitch} is \nobtained by using the exit point from one subdomain as the initial condition for\nthe approximation in the next.  In\nsubdomains other than $\\mathcal{R}$ some of the initial conditions will be\nprescribed by the algebraic part of the reduced system.  The global\napproximation may therefore be discontinuous, as solutions entering a new\nsubdomain are assumed to instantaneously jump to the slow manifold defined by\nthe algebraic part of the reduced system.\n Fig.~\\ref{fig:twoDimension} shows that when $J$ is small, this approach\nprovides a good approximation. \n\n\n\\begin{table}[t] \n\\[\n\\begin{array}{c|c|c|rcl}\n\\hline\n\\text{Name of the subdomain}\t\t&  u_1 \t& \tu_2\t&\n\\multicolumn{3}{c}{\\text{Approximating system}}  \\\\\n\\hline\n\\multirow{2}{*}{$\\mathcal{R}$}\t&  \\multirow{2}{*}{$ \\delta \\le u_1 \\le 1-\\delta\n$}\t&  \\multirow{2}{*}{ $ \\delta \\le u_2 \\le 1-\\delta $}\t&   \t u_1'\t&=&\n0.5-u_2,\t\t\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n  \t\t &    u_2'\t\t&=& 0.5-u_1\t\t\t\\\\\n\\hline\n\\multirow{2}{*}{$\\mathcal{R}^1$}\t& \\multirow{2}{*}{$ u_1 > 1-\\delta $} \t\n& \\multirow{2}{*}{$ \\delta \\le u_2 \\le 1-\\delta $}\t\t&  \t           \t\n  0\t&=&  \\displaystyle 0.5\\frac{1-u_1}{J+1-u_1}-u_2,\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t\t &  \t u_2'  &=& -0.5\t\t\t\t       \\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}^2$}     & \\multirow{2}{*}{$ \\delta \\le u_1 \\le 1-\\delta\n$}\t\t& \\multirow{2}{*}{$ u_2 > 1-\\delta $}\t&  u_1'\t&=&\t -0.5,\t\n\t\t\t\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t  \t &                         0\t&=&\t\\displaystyle\n0.5\\frac{1-u_2}{J+1-u_2}-u_1\t\\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}_1 $}\t& \\multirow{2}{*}{$ u_1 < \\delta $} \n& \\multirow{2}{*}{$ \\delta \\le u_2 \\le 1-\\delta $}    \t& \t\t\t\n0 \t &=& \\displaystyle 0.5-u_2\\frac{u_1}{J+u_1},  \\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t   \t & \t\tu_2'&=& 0.5  \\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}_2\t$}\t& \\multirow{2}{*}{$ \\delta \\le u_1\n\\le 1-\\delta $}\t\t&   \\multirow{2}{*}{$u_2 <  \\delta $}\t& \tu_1'\t\n&=& 0.5 ,  \\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t\t &\t\t\t0\t\t&=& \\displaystyle 0.5\n-u_1\\frac{u_2}{J +u_2} \\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}^{12}$}& \\multirow{2}{*}{$ u_1 > 1-\\delta$} \t& \n\\multirow{2}{*}{$u_2 > 1-\\delta$} \t&\t0\t&=& \\displaystyle\n0.5\\frac{1-u_1}{J+1-u_1}-1,\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t\t &\t0\t&=& \\displaystyle 0.5\\frac{1-u_2}{J+1-u_2}-1\n\\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}_{12}$}&  \\multirow{2}{*}{$u_1 < \\delta$}\t & \n\\multirow{2}{*}{$u_2 < \\delta$}\t&\t0\t&=& \\displaystyle 0.5- J\n\\frac{u_1}{J+u_1},\t\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t\t &\t0\t&=& \\displaystyle 0.5- J\\frac{u_2}{J +u_2}\t\n\\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}_2^1\t$}\t& \\multirow{2}{*}{$ u_1 > 1-\\delta$}\n\t& \\multirow{2}{*}{$u_2 < \\delta $}\t\t&\t0\t&=&\n\\displaystyle  0.5\\frac{1-u_1}{J+1-u_1},\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t\t &\t0\t&=& \\displaystyle 0.5- \\frac{u_2}{J +u_2}\t\n\\\\\n\\hline\n\\multirow{2}{*}{$ \\mathcal{R}_1^2\t$}\t& \\multirow{2}{*}{$ u_1 < \\delta $}\n&\\multirow{2}{*}{$  u_2 > 1-\\delta$}\t\t&\t0\t&=&\n\\displaystyle 0.5-\\frac{u_1}{J+u_1},\t\t\\\\\n\t\t\t\t\t\t\t\t&\t\t\n\t\t\t\t\t&\t\t\t\t\t\n\t\t &\t0\t&=& \\displaystyle 0.5\\frac{1-u_2}{J+1-u_2}\n\\\\\n\\hline\n\\end{array}\n\\]\n\\begin{center}\n\\parbox{.8\\textwidth}{\n\\caption{\\footnotesize  List of differential--algebraic systems that approximate\nEq.~\\eqref{EquationForToggleSwitch} in different parts of the domain. The\nsubdomains are named so that the superscript (subscript)  lists the coordinates\nthat are close to $1$ (close to 0), with 0 denoting the empty set.  For example,\n$\\mathcal{R}_1^2$ denotes that subdomain with $u_1 \\approx 1$ and $u_2 \\approx\n0$, and $\\mathcal{R}^2$ the subdomain where $u_2$ is near $1$, but $u_1$ is\naway from the boundary. The middle column define the subdomain explicitly.   \nThe right column gives the differential-algebraic system that approximates\nEq.~\\eqref{EquationForToggleSwitch}  within the given subdomain. }\n\\label{table:2d}\n}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale = .45]{twoVariable4}\n\\parbox{.9\\textwidth}{\n\\caption{\\footnotesize\nComparison of the numerical solution of Eq.~\\eqref{EquationForToggleSwitch}\n(dashed black) and the solution of the approximate system as listed in\nTable~\\ref{table:2d} (colors) for two different values of $J$. We used \n$J = 10^{-2}$ in (a); and $J =10^{-4} $ in (b).  Different colors are used for \nthe solution of the reduced system in different subdomains.  Solution of the linear approximation started in the subdomain\n$\\mathcal{R}$ (Initial value: $u_1 = 0.6, u_2 = 0.4$), and as soon as $u_2$\ndecreased below $\\delta = 0.01$, we assumed that the solution entered subdomain  $\\mathcal{R}_2$. \nThe approximate solution is discontinuous since when $u_2 = \\delta$, the solution jumped\n(see inset) to the manifold, described by the algebraic part of the linear\ndifferential algebraic system prevalent in the subdomain $\\mathcal{R}_2$, Eq.~\\eqref{smallub}.\nThe solution finally stopped in the  subdomain $\\mathcal{R}_2^1$.  }\n\\label{fig:twoDimension}\n}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\subsubsection{Boolean approximation}\n\nWe now derive a Boolean approximation, $h=(h_1,h_2):\\{0,1\\}^2\\rightarrow\\{0,1\\}^2$, that captures certain qualitative features of  Eq.~\\eqref{EquationForToggleSwitch}. The idea is to project small values of $u_i$ to 0 and  large values of $u_i$ to 1, and map  the value of the $i$-th variable into 0 and 1 depending on whether $u_i$ is decreasing or increasing, respectively. We will show that the resulting BN can be used directly to detect steady states in the corner subdomains. \n\nNote that for a BN time is discrete; a time step in the Boolean approximation can be interpreted as the time it takes the original system to transition between chambers. Different transitions in the Boolean network may have different duration in the original system; so the time steps in the BN are only used to keep track of the sequence of events, but not their duration.\n\nThe reduction described in the previous section gives a linear ODE  in the interior region $\\mathcal{R}$ (Eq.~\\eqref{interiorLinear2d}), where $\\mathcal{R}$ approaches $[0,1]^2$ as $J\\rightarrow 0$. The approximating  linear system therefore  provides  significant information about the behavior of the full, nonlinear system for $J$ small.\n\nWe first examine the nullclines. In Fig.~\\ref{fig:twoDimension_nullclines} we can see that as $J$ decreases, in the interior of $[0,1]^2$ the nullclines of Eq.~\\eqref{EquationForToggleSwitch} approach the nullclines of Eq.~\\eqref{interiorLinear2d} given by $u_2=.5$ and $u_1=.5$ restricted to $[0,1]^2$.  These lines divide the domain into four chambers, which we denote\n\\begin{equation*}\n\\mathcal{C}_{12}:=[0,0.5)\\times[0,0.5),\\quad\n\\mathcal{C}_{1}^2:=[0,0.5)\\times(0.5,1],\\quad\n\\mathcal{C}_{2}^1 :=(0.5,1]\\times[0,0.5),\\quad\n\\mathcal{C}^{12}:=(0.5,1]\\times(0.5,1].\n\\end{equation*}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale = .55]{twoDimension_nullclines}\\\\\n\\includegraphics[scale = .55]{twoDimension_nullclinesvsmanifold}\n\\parbox{.9\\textwidth}{\n\\caption{\\footnotesize\nBehavior of nullclines as $J$ decreases. Top: Nullclines of Eq.~\\eqref{EquationForToggleSwitch} for $J=10^{-2}$ (left) and $J=10^{-4}$ (right). Bottom: Nullcline $\\frac{du_2}{dt}=0$ of Eq.~\\eqref{EquationForToggleSwitch} (black curve) and the manifold defined by Eq.~\\eqref{smallub} (red) for $J=10^{-2}$ (left), and $J=10^{-4}$ (right). \n\\label{fig:twoDimension_nullclines}}\n}\n\\end{center}\n\\end{figure}\n\n\n\nOn the other hand, the part of the nullclines inside the boundary subdomains are approximately the slow manifolds defined by equivalents of Eq.~\\eqref{E:smalleqb}. Here the slow manifolds converge to the nullclines  as $J \\rightarrow 0$  (See Fig.~\\ref{fig:twoDimension_nullclines}).\n\nAs a shorthand, we define the ``sign'' of a vector $v=(v_1\\ldots,v_N)$ as the vector composed by the signs of its components, $sign(v):=(sign(v_i),\\ldots,sign(v_N))$.\nNote that although the sign of the vector $(.5-u_2,.5-u_1)$ is constant in each chamber, the sign of the vector field of Eq.~\\eqref{EquationForToggleSwitch} may differ. For example, in chamber $\\mathcal{C}_2^1$, the sign of the vector field can take all possible values. However, this difference is small when  $J$ is small, because the regions between the nullclines approach the actual chambers (Fig.~\\ref{fig:twoDimension_nullclines}).  \n\nWe consider Eq.~\\eqref{EquationForToggleSwitch} in each chamber, starting with the first coordinate, $u_1(t)$. \nFor any solution with initial condition in $\\mathcal{C}_{12}$, the sign of $u'_1(0)$ is positive and $u_1(t)$ increases within the chamber.  \nWe use this observation to define $h_1(\\mathcal{C}_{12})=1$.  The formal definition of this function will be given below -- \nintuitively $h_i(\\cdot)$ maps a chamber to 1 if $u_i$ is increasing within the chamber, and to 0 otherwise. \nSimilarly, since $u_1(t)$ initially increases within $\\mathcal{C}_2^1$, we let $h_1(\\mathcal{C}_2^1)=1$. Similarly we  set $h_1(\\mathcal{C}^{12})=0,$ $h_1(\\mathcal{C}_1^2)=0,$ $h_2(\\mathcal{C}_{12})=1$, $h_2(\\mathcal{C}_1^2)=1$, $h_2(\\mathcal{C}_2^1)=0$,  and $h_2(\\mathcal{C}^{12})=0$.  The $i$-th variable is  ``discretized,'' \\emph{i.e.} mapped to 0 and 1 depending on whether $u_i$ is decreasing or increasing, respectively. \n\nMore formally, consider the set $\\{0,1\\}^2$, with each element identified with a chamber (e.g., the element $(0,1)$ represents the chamber $\\mathcal{C}_1^2$). Then $h_1$ and $h_2$ are defined as Boolean functions from $\\{0,1\\}^2$ to $\\{0,1\\}$  by setting $h_1(0,0)=0, h_1(0,1)=0$, $h_1(1,0)=1$, $h_1(1,1)=0$, and $h_2(0,0)=0, h_2(0,1)=1$, $h_2(1,0)=0$, $h_2(1,1)=0$. These two Boolean functions define a BN, $h=(h_1,h_2):\\{0,1\\}^2\\rightarrow\\{0,1\\}^2$.  The functions also define a dynamical system, $x(t+1)=h(x(t)), x \\in \\{0,1\\}^2$. However, other update schedules   can be used [\\cite{AracenaGolesMoreiraSalinas2009}].\n\n\n\n\n\nThe BN reduction can be obtained easily  from the sign of $(.5-u_2,.5-u_1)$ at the vertices of $[0,1]^2$,  since the sign of the vector field is constant within a chamber. To do so we use the the Heaviside  function, $H$, defined by $H(y)=0$ if $y<0$, $H(y)=1$ if $y>0$, and $H(0)=\\frac{1}{2}$. For example, in $\\mathcal{C}_{12}$, both entries increase. We  can see this by evaluating  $H(.5-u_2)=H(.5-u_1)=1$ for $u=(0,0)$.  Using the same argument in each chamber, we obtain the BN \n\\begin{equation}\\label{EquationForToggleSwitchBN}\nh(x)=H(.5-x_2,.5-x_1),\n\\end{equation}\nwhere we used the convention that $H$ acts entrywise on each component in the argument.\n\n\\subsubsection{Steady states of the BN and the PL approximation}\n\nWhile the BN gives information about which variables increase and decrease within a chamber, it is not yet\nclear how or if the dynamics of the BN in Eq.~\\eqref{EquationForToggleSwitchBN}, and the PL approximation in Table \\ref{table:2d} are related. \n\nWe next show that the steady states of the PL approximation near the vertices can be determined by the steady states of the BN.  The reduced equations in the corner subdomains  $\\mathcal{R}_{12},\\mathcal{R}^{12}, \\mathcal{R}_1^2,$ and $\\mathcal{R}_2^1$ are purely algebraic. When $J$ is small, some of these equations have a solution in $[0,1]^2$, indicating a stable fixed point near the corresponding corner (in this case $\\mathcal{R}_1^2$ and $\\mathcal{R}_2^1$). Others will not have a solution in $[0,1]^2$, indicating that approximate solutions do not enter the corresponding subdomain (here $\\mathcal{R}_{12}$ and $\\mathcal{R}^{12}$). To make the relationship between steady states less dependent on the actual parameters, consider the system\n\\begin{align}\n\\begin{split}\\notag\n\\frac{du_1}{dt} &= b_1^+\\frac{1-u_1}{J+1-u_1}-(u_2+b_1^-)\\frac{u_1}{J+u_1},  \\\\\n\\frac{du_2}{dt} &= b_2^+\\frac{1-u_2}{J+1-u_2}-(u_1+b_2^-)\\frac{u_2}{J +u_2},\n\\end{split}\n\\end{align}\nwhere $x^+=\\max{\\{x,0\\}}$ and $x^-=\\max\\{-x,0\\}$. In the previous example $b_1=b_2=.5$, $b_1^+=b_2^+=.5$ and $b_1^-=b_2^-=0$.  \n\nNow, at the corner subdomain $\\mathcal{R}_{12}$ we have the approximate equations,\n\\[\n0= b_1^+ -b_1^-\\frac{u_1}{J+u_1},  0= b_2^+ -b_2^-\\frac{u_2}{J +u_2},\\]\nor equivalently,\n\\[u_1 =\\frac{ -b_1^+ J}{b_1}, u_2 =\\frac{ -b_2^+ J}{b_2}.\\]\n\nThese equations have a solution in $\\mathcal{R}_{12}$ if and only if $b_1<0$ and $b_2<0$, or equivalently, if and only if $H(b_1,b_2)=(0,0)$. A similar analysis  leads to the following conditions for the existence of fixed points in each corner subdomain \n\\begin{align}\n\\begin{split}\\notag\n&\\textrm{On $\\mathcal{R}_{12}$ :}\\ \\ H(b_1,b_2)=(0,0),\\qquad\n\\textrm{On $\\mathcal{R}^{12}$ :}\\ \\ H(b_1-1,b_2-1)=(1,1),\\\\\n&\\textrm{On $\\mathcal{R}^2_1$ :}\\ \\  H(b_1-1,b_2)=(0,1),\\qquad\n\\textrm{On $\\mathcal{R}^1_2$ :}\\ \\  H(b_1,b_2-1)=(1,0).\n\\end{split}\n\\end{align}\nMore compactly, the condition is $H(b_1-x_2,b_2-x_1)=(x_1,x_2)$, where $x=(x_1,x_2)$ is the corner of interest. \nHence, the BN can also be used directly to detect which corner subdomains contain steady states. \n\nThe relationship between steady states in the full system at the corner subdomains and the steady states of the BN is straightforward. However, since there are many update schemes for BNs, the relationship between trajectories is more subtle. For example, using  synchronous update we obtain the transition $(0,0)\\rightarrow (1,1)$ which is not compatible with the solutions of Eq.~\\eqref{EquationForToggleSwitch} (See Fig.~\\ref{fig:twoDimension_ODEandBN}). On the other hand, using  asynchronous update we obtain the transitions $(0,0)\\rightarrow (1,0)$ and $(0,0)\\rightarrow (0,1),$ which are more representative of the solutions of Eq.~\\eqref{EquationForToggleSwitch}. Thus, we will focus on transitions that are independent of the update scheme, that is, transitions where only one entry changes. \n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale = .6]{twoDimension_ODEandBN}\n\\parbox{.9\\textwidth}{\n\\caption{\\footnotesize\nLeft: Solutions of Eq.~\\eqref{EquationForToggleSwitch} for $J=10^{-4}$. When a solution is close to the boundary regions of $\\mathcal{C}_1^2$ and $\\mathcal{C}_2^1$, they enter the invariant region as shown in Fig.~\\ref{fig:twoDimension}b. Right: Graphical representation of the Boolean transitions ($00\\rightarrow 11$, $11\\rightarrow 00$, $01\\rightarrow 01$, $10\\rightarrow 10$).  \n\\label{fig:twoDimension_ODEandBN}\n}} \n\\end{center}\n\\end{figure}\n\n\n\n\\subsection{A network of three mutually inhibiting elements}\\label{sec:3nodes}\n\nThe same reduction can be applied to systems of arbitrary dimension.  As an\nexample consider\nthe\n\\emph{repressilator}~[\\cite{NovakPatakiCilibertoTyson2003,ElowitzLeibler2000}] (see Fig.~\\ref{fig:toggle_repressillator}a)\ndescribed by\n\\begin{eqnarray}\\label{threeNodeMM}\n \\frac{du_1}{dt} &=& 0.6\\frac{1- u_1}{J+1-u_1} -  u_3\\frac{ u_1}{J+u_1}, \\notag\n\\\\\n \\frac{du_2}{dt} &=& 0.4\\frac{1- u_2}{J+1-u_2} -  u_1\\frac{ u_2}{J+u_2}, \\\\\n \\frac{du_3}{dt} &=& 0.3\\frac{1- u_3}{J+1-u_3} -  u_2\\frac{ u_3}{J+u_3}. \\notag\n\\end{eqnarray}\nThe cyclic repression of the three elements in this network leads to oscillatory\nsolutions over a large range of values of $J$.  The domain of this system,\n$[0,1]^3$, can\nbe divided into 27 subdomains corresponding to 1 interior,\n6 faces, 12 edges, and 8 vertices.  \n\nWe can again  approximate Eq.~\\eqref{threeNodeMM} with solvable\ndifferential--algebraic equation within each  subdomain, to obtain a global\napproximate solution (See  Fig.~\\ref{fig:threeDimension}). Note that both the numerically obtained solution to  Eq.~\\eqref{threeNodeMM} and the solution to the piecewise linear equation exhibit oscillations, and that the approximation is discontinuous. Again, in the limit $J \\rightarrow 0$ we obtain a continuous 0-th order approximation.\n\nIn this singular limit, solutions can exit a subdomain when they reach a nullcline of the linear system. For example, when $u_2$ is close to 0 and a solution transitions from $u_1>.4$ to $u_1<.4$, the sign of the second entry of $(0.6-u_3,0.4-u_1,0.3-u_2)$ changes from negative to negative; so the second entry of the solution starts increasing (see Fig.~\\ref{fig:threeDimension}, panel (e)).  Solutions therefore leave the subdomain on which $u_2\\sim  J$ is small and enter the subdomain where $u_2\\gg J$. Similarly when $u_1$ is close to 1 solutions transitions from $u_3<.6$ to $u_3>.6$, and the sign of the first entry of $(0.6-u_3,0.4-u_1,0.3-u_2)$ changes from positive to negative.  Hence the first entry of the solution starts decreasing (see Fig.~\\ref{fig:threeDimension}, panel (f)), and solutions leaves the subdomain where $1-u_1\\sim  J$  and enter another where $1-u_1\\gg J$.\n\nThe BN corresponding to Eq.~\\eqref{threeNodeMM},  $h=(h_1,h_2,h_3):\\{0,1\\}^3\\rightarrow\\{0,1\\}^3,$ is given by $h(x)=H(0.6-x_3,0.4-x_1,0.3-x_2)$, where $H$ is the Heaviside function acting entry wise on the arguments.\nEq.~\\eqref{threeNodeMM} does not have steady states at the corner subdomains, and neither does the corresponding  BN. A subset of states belong to a periodic orbit of the BN: \n$$ (0,1,1) \\rightarrow (0,1,0)\\rightarrow(1,1,0)\\rightarrow(1,0,0)\\rightarrow(1,0,1)\\rightarrow(0,0,1).\n$$  \nNote that subsequent states in this orbit differ in a single entry. Thus, the transitions between the states have an unambiguous interpretation in the original system: The BN predicts that if the initial condition is in chamber $\\mathcal{C}_{12}^3$, then solutions of  Eq.~\\eqref{threeNodeMM}  will go to chamber $\\mathcal{C}_{1}^{23}$, then to $\\mathcal{C}_{13}^2$, and so on. Indeed, solutions of  Eq.~\\eqref{threeNodeMM}, are attracted to a periodic orbit that transitions between the chambers in this order.  The remaining two states form a period two orbit under synchronous update, $(1,1,1) \\leftrightarrow (0,0,0)$.   Here the BN does not give precise information about the dynamics of the original system. We will show that under certain conditions, orbits of the BN where only entry changes \nat each timestep, correspond to oscillations in the original system.\n\n\n\\begin{figure}[h]\n\\centering\n \\includegraphics[width = .95\\textwidth]{3D_timeseries_small}\n\\parbox{.9\\textwidth}{\n\\caption{\\footnotesize\nComparison of the numerical solution of Eq.~\\eqref{threeNodeMM} (dashed black)\nand the solution of the approximate piecewise linear system (colors) for\ntwo different sets of $J$ and $\\delta$. For (a)-(c) $J = 10^{-2}, \\delta = 0.06$; for\n(e)-(g)  $J = 10^{-4}, \\delta = 0.01$.  The approximate solution changes color when\nswitching between different subdomains.  \nPanel (d) shows the time series for the solutions of Eq.~\\eqref{threeNodeMM} (top) and the PL system (bottom), corresponding to (a)-(c). Panel (h) shows the time series for the solutions of Eq.~\\eqref{threeNodeMM} (top) and the PL system (bottom), corresponding to (e)-(g). \\label{fig:threeDimension}\n}}\n\\end{figure}\n\n\n\n\n\n\n\\section{General reduction  of the model system}\\label{GeneralTheory}\n\nThe approximations described in the previous section can be extended to the more general model given in Eq.~\\eqref{eqn:ProblemEquation}: \n\\begin{align}\\notag\n\t\\frac{du_i}{dt} = A_i\n\\frac{1-u_i}{J_{i}^A+1-u_i}-I_i\\frac{u_i}{J_{i}^I+u_i},\n\\end{align}\nwhere $J_i^A,J_i^I$ are some positive constants.  Here $A_i$ and $I_i$ are\nactivation\/inhibition functions that capture the impact of other variables on\nthe evolution of $u_i$.   The initial conditions are assumed to satisfy $u_i(0)\n\\in [0,1]$ for all $i$.\n\n\n We assume that the activation and inhibition functions are both\naffine~[\\cite{NovakPatakiCilibertoTyson2001,de02}],\n \\begin{equation} \\label{activation\/Inhibition}\n  A_i := \\sum_{j=1}^N{w_{ij}^+u_j} +b_i^+,\n\\quad\n   I_i := \\sum_{j=1}^N{w_{ij}^-u_j} +b_i^-,\n \\end{equation}\t\n where  we use the convention $x^+ = \\text{max}\\{x,0\\}$ and $x^- =\n\\text{max}\\{-x,0\\}$.\nThe  $N\\times N$ matrix, $W = [w_{ij}]$ and the  $N \\times 1$ vector  $b = [\\\nb_1 \\  b_2 \\  ...\\  b_N\\ ]^t$ capture the connectivity and external input to the\nnetwork, respectively. In particular,  $w_{ij}$\ngives  the contribution  of the $j^{\\text{th}}$ variable to the growth rate of\n$i^{\\text{th}}$ variable. If $w_{ij} > 0 $, then $w_{ij}$  appears in the\nactivation function for $u_i$; and if $w_{ij}<0$ then $-w_{ij}$  appears in the\ninhibition function for $u_i$.\nThe intensity of the external input to the $i^{\\text{th}}$ element is $|b_i|$,\nand it contributes to the activation or the inhibition function, depending on\nwhether $b_i > 0$ or $b_i < 0$,  respectively. \n\n\\begin{proposition}\\label{prop:invariantCube}\nThe cube $[0,1]^N$ is invariant for Eq.~\\eqref{eqn:ProblemEquation}.\n\\end{proposition}\n\\begin{proof}\nIt will be enough to show that the vector field at any point on the boundary is not\ndirected outward. Since, $A_i \\geq 0$ and $I_i \\geq 0$, for any $i$,\n\\begin{align*}\n\t\\frac{du_i}{dt}\\bigg|_{u_i = 0} = A_i\\frac{1}{J_{i}^A+1} \\ge 0,\n\\quad\n\\text{and}\n\\quad\n\t\\frac{du_i}{dt}\\bigg|_{u_i = 1} = -I_i\\frac{1}{J_{i}^I+1} \\le 0.\n\\end{align*}\n\\end{proof}\n\n\n\\subsection{The PL approximation}\nTo obtain a solvable reduction of Eq.~\\eqref{eqn:ProblemEquation} we follow the \nprocedure outlined in  Section~\\ref{ExampleProblems}.  We first present the results, and provide the mathematical justification in the next section.  For\nnotational convenience we let $J_i^A = J_i^I = J$, with  $0 < J \\ll 1$.   \nThe general case is equivalent.  We will use  $\\delta = \\delta(J) > 0 $  to define the thickness of the boundary layers.  We start with the subdivision of the\n$N$-dimensional cube, $[0,1]^N$.\n\nLet $T$ and $S$ be two disjoint subsets of  $\\{1,2,...,N\\}$, and let\n\\begin{align} \\label{Rdef} \n\\mathcal{R}^{T}_{S}:=\n\\Big\\{\n(u_1,u_2,...,u_N) \\in [0,1]^N  \\,\\Big|\\,\n u_{s} \t\t <  \\delta \\text{ for all }   s \\in S; \\quad\n    &u_{t}\t \t> 1-\\delta\\text{ for all }   t \\in T; \\notag \\\\\n\\text{and }\n \\delta \\le  &u_k\\le 1-\\delta\n  \\text{ for all }  \\quad k  \\notin S \\cup T\n\\Big \\}.\n\\end{align}\nWe extend the convention used in Table~\\ref{table:2d}, and in Eqs.~(\\ref{R00})\nand (\\ref{R20}) so that  $\\mathcal{R}^{T} :=\\mathcal{R}^{T}_{S}$ when $S$ is\nempty;  $\\mathcal{R}_S :=\\mathcal{R}^{T}_{S}$ when $T$ is empty; and\n $\\mathcal{R} :=\\mathcal{R}^{T}_{S}$  when $T$, $S$ are both  empty.\n\nWithin each subdomain $\\mathcal{R}_S^T$,  Eq.~\\eqref{eqn:ProblemEquation} can be\napproximated by a\n different linear differential--algebraic system. Following the reduction from\nEq.~\\eqref{EquationForToggleSwitch} to Eq.~\\eqref{edgeLinear2d}, for $i \\notin S\n\\cup T$ we obtain the linear system\n\\begin{subequations}\\label{equationInRST}\n\\begin{equation} \n \\frac{du_i}{dt} = \\sum_{j = 1}^N w_{ij}u_j +  b_i. \\label{four}\n\\end{equation}\nFor $s \\in S$ one of the nonlinear terms remains and we  obtain\n\\begin{equation} \n\\frac{du_s}{dt} = \\left(\\sum_{j = 1}^N \nw_{sj}^+u_j +  b_s^+\\right)-\\left(\\sum_{j= 1}^N \nw_{sj}^-u_j +  b_s^-\\right)\\frac{u_s}{J+u_s}, \\label{five}\n\\end{equation}\nwhile for $t \\in T$ we will have\n\\begin{equation} \n\\frac{d u_t}{dt} = \\left(\\sum_{j = 1}^N\nw_{tj}^+u_j + b_t^+\\right)\\frac{1-u_t}{J+1-u_t}\n-\\left(\\sum_{j = 1}^N w_{tj}^-u_j +  b_t^-\\right). \\label{six}\n\\end{equation}\n\\end{subequations}\nEq.~\\eqref{equationInRST} is simpler than Eq.~\\eqref{eqn:ProblemEquation}, but it is\nnot solvable yet.  Following the reduction from Eq.~\\eqref{edgeLinear2d} to\nEq.~\\eqref{smallu},  we now further reduce Eqs.(\\ref{five}--\\ref{six}).\nFirst we use the approximations $u_s \\approx 0$ and $u_t \\approx 1$ in the\nactivation and inhibition functions appearing in Eq.~\\eqref{equationInRST}.\nSecond, we assume\nthat $u_s$ for $s \\in S$ and $u_t$ for $t \\in T$ are in steady state.\n\nUnder these assumptions we obtain the reduction of Eq.~\\eqref{eqn:ProblemEquation}\nwithin any subdomain $\\mathcal{R}_S^T$\n\\begin{subequations}\\label{eqn:mainReduced_RST}\n\\begin{align}\n\\frac{du_i}{dt} &=  \\sum_{j  \\notin S \\cup T}w_{ij}u_j + \\sum_{j \\in T}w_{ij} \n+b_i\n& i \\notin S \\cup T;  \\label{eqn:mainReduced_RSTa} \\\\\n0 &=  \\sum_{j \\notin S \\cup T}w_{sj}^+u_j + \\sum_{t \\in T}w_{st}^+\n  +b_s^+ -\\left( \\sum_{j \\notin S \\cup T }w_{sj}^-u_j + \\sum_{t \\in T}w_{st}^- +\nb_s^-\\right)\\frac{u_s}{J+u_s},\n& s \\in S; \\label{eqn:mainReduced_RSTb} \\\\\n0 &= -\\left( \\sum_{j \\notin S \\cup T }w_{tj}^+u_j + \\sum_{j \\in\nT}w_{tj}^++b_t^+\\right)\\frac{1-u_t}{J+1-u_t}\n+\\sum_{j \\notin S \\cup T}w_{tj}^-u_j + \\sum_{j \\in T}w_{tj}^- + b_t^-,\n& t \\in T. \\label{eqn:mainReduced_RSTc}\n\\end{align}\n\\end{subequations}\n\n Eq.~(\\ref{eqn:mainReduced_RST}) is  solvable since Eq.~(\\ref{eqn:mainReduced_RSTa}) is\ndecoupled, and   Eqs.(\\ref{eqn:mainReduced_RSTb}) and\n(\\ref{eqn:mainReduced_RSTc}) are solvable for ${u}_s$ and ${u}_t$, respectively, as\nfunctions of the solution of Eq.~(\\ref{eqn:mainReduced_RSTa}).\n\n\nNote that in the singular limit $J=0$ we obtain the $0$-th order approximations:\n\\begin{subequations}\\notag\n\\begin{align}\n\\frac{du_i}{dt} &= \\sum_{j\\notin S \\cup T}w_{ij}u_j + \\sum_{j \\in T}w_{ij}+b_i\n& i \\notin S \\cup T;  \\notag \\\\\nu_s &=  0, & s \\in S; \\notag \\\\\nu_t &= 1, & t \\in T. \\notag\n\\end{align}\n\\end{subequations}\n\n\\subsection{Boolean approximation}\n\nTo obtain the Boolean approximation we follow the process described in Section \\ref{ExampleProblems}.  We consider the chambers determined by the complement of the union of the $N$ hyperplanes $\\sum_{j=1}^N w_{ij} u_j +b_i= 0$ (restricted to $[0,1]^N$)  where $i=1,\\ldots, N$. We denote with $\\Omega$ the set of all chambers $\\Omega:=\\{\\mathcal{C}: \\mathcal{C} \\textrm{ is a chamber}\\}$; alternatively, $\\Omega$ is the set of connected components of $[0,1]^N\\setminus \\cup_{i=1}^N\\{u:\\sum_{j=1}^N w_{ij} u_j +b_i=0\\}$. We assume that $\\sum_{j=1}^N w_{ij} x_j +b_i\\neq 0$ for all $i=1,\\ldots, N$ and for all $x\\in\\{0,1\\}^N$. This guarantees that each corner of $[0,1]^N$ belongs to a chamber. The set of parameters excluded by this assumption has measure zero. \n\n\n\nLet $S$ and $T$ be two disjoint subsets of $\\{1, 2, \\ldots, N\\}$ such that $S\\cup T=\\{1, 2, \\ldots, N\\}$ and let $x\\in\\{0,1\\}^N$ be the corner that belongs to the corner subdomain $\\mathcal{R}_S^T$. Note that $x_i=0$ for $i\\in S$ and $x_i=1$ for $i\\in T$. The chamber $\\mathcal{C} \\in \\Omega$ that contains the corner in subdomain $\\mathcal{R}_S^T$\nwill be denoted by  $\\mathcal{C}_S^T$.  We do not name the remaining chambers.\n\nIn general, the chambers can be more complex than in the examples of  Section \\ref{ExampleProblems}. Chambers do not have to be hypercubes,  different corners  may belong to the same chamber, and some chambers may not even contain a corner of $[0,1]^N$, as illustrated in Fig.~\\ref{fig:extra_examples_hyperplanes}.  In the first example, $(0,1)$ and $(1,1)$ belong to the same chamber, that is, $\\mathcal{C}_1^2=\\mathcal{C}^{12}$, and neither  $\\mathcal{C}_{12}$ containing $(0,0),$ nor $\\mathcal{C}_2^1$ containing $(1,0)$ are rectangles. Also, $\\Omega$ has three elements: $\\mathcal{C}_{12}$, $\\mathcal{C}_2^1$, and $\\mathcal{C}_1^2=\\mathcal{C}^{12}$. In the second example, two chambers do not contain any corner of $[0,1]^2$ and are not named. Hence, $\\Omega$ has four elements: $\\mathcal{C}_{12}=\\mathcal{C}_1^2$, $\\mathcal{C}_2^1=\\mathcal{C}^{12}$, and two unnamed chambers that contain no corners. \n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale = .5]{extra_examples_hyperplanes}\n\\parbox{.9\\textwidth}{\n\\caption[fig]{ \\footnotesize\nChambers for $W=\\left[\\begin{array}{rr} -1 & -1 \\\\ 0 & -1 \\end{array}\\right]$ and \n$b=\\left[\\begin{array}{r} -0.5 \\\\ -0.6 \\end{array}\\right]$ (left);  \n$W=\\left[\\begin{array}{rr} 2 & -1 \\\\ 2 & 1 \\end{array}\\right]$ and\n$b=\\left[\\begin{array}{r} -0.5 \\\\ -1.5 \\end{array}\\right]$ (right).  \n\\label{fig:extra_examples_hyperplanes}\n}} \n\\end{center}\n\\end{figure}\n\nTo define the BN, $h=(h_1,\\ldots,h_N):\\{0,1\\}^N\\rightarrow \\{0,1\\}^N$ at $x\\in\\{0,1\\}^N$, we need to find the signs of the components of the vector field $Wu+b$ on the chamber that contains $x$. Consider  $x \\in \\mathcal{R}_S^T$. Within $\\mathcal{C}_S^T$ the signs of the components of $Wu+b$ do not change and are equal to the signs of the components of $Wx+b$. If the sign of the $i$-th component is negative, we let $h_i(x)=0,$ and if the sign is positive we let $h_i(x)=1$. In general, we can write \n\\begin{equation}\\label{eqn:mainReduced_BN}\nh_i(x) =  H\\left(\\sum_{j=1}^N w_{ij}x_j+b_i\\right),   \\qquad \\textrm{ or in vector form}  \\qquad\nh(x)=H(Wx+b).\n\\end{equation}\nHence the value of the BN at a corner $x\\in\\mathcal{R}_S^T$ is given by the  Heaviside function, applied entrywise  to $Wu+b$. Note that corners that are in the same chamber get mapped to the same point. \n\n\nImportantly, using Eq.~\\eqref{eqn:mainReduced_BN} we can compute the BN directly from Eq.~\\eqref{eqn:ProblemEquation}. For example, for Eq.~\\eqref{EquationForToggleSwitch} we have  $h(x_1,x_2)=H(0.5-x_2,0.5-x_1)$; and for  Eq.~\\eqref{threeNodeMM} we have $h(x_1,x_2,x_3)=H(0.6-x_3,0.4-x_1,0.3-x_2)$.\n\nBelow we show that up to a set of small measure, the BN preserves information about the steady states of the original system.  We will also show that under some conditions, ``regular'' trajectories of a BN correspond to trajectories in the original system.\n\n\\section{Mathematical justification}\\label{sec:math_justification}\n\n We next justify the different approximations made above: In Section \\ref{sec:math_PL} we use Geometric Singular Perturbation Theory (GSPT) to justify the PL approximation. In Section \\ref{sec:math_BN} we show that steady state information is preserved by the BN and that, under certain conditions, the BN also provides qualitative information about the global dynamics of the original system.\n\n\\subsection{Piecewise linear approximation}\\label{sec:math_PL}\n\nTo obtain the reduced equations at the boundary of $[0,1]^N$, we define the following rescaled variables \n\\begin{equation} \\label{seven}\n \\tilde{u}_s := \\frac{u_s}{J}  \\quad   \\text{ for }  s \\in S, \\text{ and } \n\\qquad\n \\tilde{u}_t  := \\frac{1-u_t}{J} \\,\\,\\quad \\text{ for }  t \\in T.\n\\end{equation}\nUsing Eq.~ (\\ref{seven}) in Eq.~(\\ref{equationInRST}) we get for $i \\notin S\n\\cup T$\n\\begin{subequations}\\label{equationInRST_scaled}\n\\begin{equation} \\label{eight}\n\\frac{du_i}{dt} =  \\sum_{j  \\notin S \\cup T}w_{ij}u_j + \\sum_{j \\in T}w_{ij}\n  + J\\left(\\sum_{s \\in S}w_{is}\\tilde{u}_s-\\sum_{t \\in\nT}w_{it}\\tilde{u}_t\\right) +b_i,\n\\end{equation}\nand for $s \\in S$,\n\\begin{align} \\label{nine}\nJ\\frac{d\\tilde{u}_s}{dt} = & \\sum_{j \\notin S \\cup T}w_{sj}^+u_j + \\sum_{t \\in\nT}w_{st}^+\n  +J\\left(\\sum_{j \\in S}w_{sj}^+\\tilde{u}_j-\\sum_{t \\in\nT}w_{st}^+\\tilde{u}_t\\right)+b_s^+ \\nonumber\\\\\n&-\\left( \\sum_{j \\notin S \\cup T }w_{sj}^-u_j + \\sum_{t \\in T}w_{st}^- +\nb_s^-\\right)\\frac{\\tilde{u}_s}{1+\\tilde{u}_s}\n  -J\\left(\\sum_{j \\in S}w_{sj}^+\\tilde{u}_j-\\sum_{t \\in\nT}w_{st}^+\\tilde{u}_t5\\right)\\frac{\\tilde{u}_s}{1+\\tilde{u}_s},\n\\end{align}\nand similarly, for $t \\in T$,\n\\begin{align} \\label{ten}\nJ\\frac{d\\tilde{u}_t}{dt} = &-\\left( \\sum_{j \\notin S \\cup T }w_{tj}^+u_j +\n\\sum_{j \\in T}w_{tj}^++b_t^+\\right)\\frac{\\tilde{u}_t}{1+\\tilde{u}_t}\n -J\\left(\\sum_{s \\in S}w_{ts}^+\\tilde{u}_s-\\sum_{j \\in\nT}w_{tj}^+\\tilde{u}_j\\right)\\frac{\\tilde{u}_t}{1+\\tilde{u}_t} \\nonumber \\\\\n&+\\sum_{j \\notin S \\cup T}w_{tj}^-u_j + \\sum_{j \\in T}w_{tj}^- + b_t^-\n+J\\left(\\sum_{s \\in S}w_{ts}^+\\tilde{u}_s-\\sum_{j \\in\nT}w_{tj}^+\\tilde{u}_j\\right).\n\\end{align}\n\\end{subequations}\n\n\nWhen $J$ is small, we can apply Geometric Singular Perturbation Theory (GSPT) to Eq. (\\ref{equationInRST_scaled}) ~[\\cite{Hek2010,Kaper1998}].  The GSPT posits that, under a normal hyperbolicity condition which we verify below, Eq.~(\\ref{equationInRST_scaled}) can be further simplified by assuming that $J = 0$. \nThis yields a \ndifferential-algebraic system\n\\begin{subequations}\\label{mainReduced}\n\\begin{align}\n\\frac{du_i}{dt} &=  \\sum_{j  \\notin S \\cup T}w_{ij}u_j + \\sum_{j \\in T}w_{ij} \n+b_i,\n& i \\notin S \\cup T;  \\label{eleven} \\\\\n0 &=  \\sum_{j \\notin S \\cup T}w_{sj}^+u_j + \\sum_{t \\in T}w_{st}^+\n  +b_s^+ -\\left( \\sum_{j \\notin S \\cup T }w_{sj}^-u_j + \\sum_{t \\in T}w_{st}^- +\nb_s^-\\right)\\frac{\\tilde{u}_s}{1+\\tilde{u}_s},\n& s \\in S; \\label{twelve} \\\\\n0 &= -\\left( \\sum_{j \\notin S \\cup T }w_{tj}^+u_j + \\sum_{j \\in\nT}w_{tj}^++b_t^+\\right)\\frac{\\tilde{u}_t}{1+\\tilde{u}_t}\n+\\sum_{j \\notin S \\cup T}w_{tj}^-u_j + \\sum_{j \\in T}w_{tj}^- + b_t^-,\n& t \\in T. \\label{thirteen}\n\\end{align}\n\\end{subequations}\nwhich is equivalent to Eq.~\\eqref{eqn:mainReduced_RST} after rescaling.  This\nconclusion\nwill be justified if the manifold defined by Eqs.~(\\ref{twelve}) and\n(\\ref{thirteen}) is normally hyperbolic and\nstable~[\\cite{Fenichel1979,Kaper1998,Hek2010}].  We verify this condition next.\n\nLet $ \\hat{u} =  \\{ u_{i_1},...,u_{i_m} \\}$ where $\\{ i_1,...,i_m \\} = \\{\n1,2,...,N \\}\\backslash (S \\cup T) $, be the coordinates of $u$ which are away\nfrom the boundary, and denote the right hand side of Eq.~(\\ref{twelve}) by\n$F_s(\\hat{u}, \\tilde{u}_{i_s})$, for all  $s \\in S$, so that\n\\begin{align*}\n F_s(\\hat{u}, \\tilde{u}_{i_s}) :=  \\sum_{j \\notin S \\cup T}w_{sj}^+u_j + \\sum_{t\n\\in T}w_{st}^+\n  +b_s^+ -\\left( \\sum_{j \\notin S \\cup T }w_{sj}^-u_j + \\sum_{t \\in T}w_{st}^- +\nb_s^-\\right)\\frac{\\tilde{u}_s}{1+\\tilde{u}_s},\n\\end{align*}\nand\n\\begin{align*}\n \\frac{\\partial F_s}{\\partial \\tilde{u}_{i_s}} =  -\\left( \\sum_{j \\notin S \\cup\nT }w_{sj}^-u_j + \\sum_{t \\in T}w_{st}^- + b_s^-\\right) \\left(\n\\frac{1}{1+\\tilde{u}_s}\\right)^2 \n< 0,\n\\end{align*}\nfor all $s \\in S$.\nSimilarly, by denoting the right hand side of Eq.~(\\ref{thirteen}) by\n$G_t(\\hat{u}, \\tilde{u}_{i_t})$, for all  $t \\in T$. \\emph{i.e.}\n\\begin{align*}\n G_t(\\hat{u}, \\tilde{u}_{i_t}) := -\\left( \\sum_{j \\notin S \\cup T }w_{tj}^+u_j +\n\\sum_{j \\in T}w_{tj}^++b_t^+\\right)\\frac{\\tilde{u}_t}{1+\\tilde{u}_t}\n+\\sum_{j \\notin S \\cup T}w_{tj}^-u_j + \\sum_{j \\in T}w_{tj}^- + b_t^-,\n\\end{align*}\nwe see that\n\\begin{align*}\n \\frac{\\partial G_t}{\\partial \\tilde{u}_{i_t}} = -\\left( \\sum_{j \\notin S \\cup T\n}w_{tj}^+u_j + \\sum_{j \\in T}w_{tj}^++b_t^+\\right) \\left(\n\\frac{\\tilde{u}_t}{1+\\tilde{u}_t} \\right)^2\n< 0.\n\\end{align*}\nHence, the manifold defined by Eqs.~(\\ref{twelve}) and (\\ref{thirteen}) is\nnormally hyperbolic and stable.  This completes the proof that the reduction of\nthe non-linear system~(\\ref{eqn:ProblemEquation}) to the solvable system given in Eq.~(\\ref{eqn:mainReduced_RST}) is justified for small $J$.\n\n\\subsection{Boolean approximation}\\label{sec:math_BN}\n\nHere we formally show that the steady states of the BN given in Eq.~\\eqref{eqn:mainReduced_BN} are in a one-to-one correspondence with the steady states of the system  given in Eq.~\\eqref{eqn:ProblemEquation}. We also show that under some conditions trajectories in the BN correspond to trajectories of the  system  Eq.~\\eqref{eqn:ProblemEquation}.\n\n\\subsubsection{Steady state equivalence at the corner subdomains}\\label{sec:math_BN_ss_corner}\n\nFirst we prove the one-to-one correspondence only at the corner subdomains using the PL approximation. We do this by showing that Eq.~\\eqref{eqn:mainReduced_RST} has a steady state at a corner subdomain $\\mathcal{R}_S^T$ if and only if the BN has a steady state at the corner $x\\in\\{0,1\\}^N$ contained in $\\mathcal{R}_S^T$.\n\nWe proceed from Eq.~(\\ref{eqn:mainReduced_RST}) for a corner subdomain $\\mathcal{R}_S^T$ so that $S\\cup T=\\{1,\\ldots,N\\}$. We obtain the equations\n\n\\[0=\\sum_{t \\in T}w_{st}^+  +b_s^+ -\\left( \\sum_{t \\in T}w_{st}^- +b_s^-\\right)\\frac{u_s}{J+u_s},\\qquad  s \\in S \\] and \n\\[0= -\\left( \\sum_{j \\in T}w_{tj}^++b_t^+\\right)\\frac{1-u_t}{J+1-u_t} +\\sum_{j \\in T}w_{tj}^- + b_t^-, \\qquad  t \\in T.\\]\n\nFor the sets $S$ and $T$, consider $x\\in\\{0,1\\}^N$ such that $x_s=0$ for all $s\\in S$ and $x_t=1$ for all $t\\in T$. Then, we can write the equations above in a more compact form\n\n\\[0=A_s(x) -I_s(x)\\frac{u_s}{J+u_s}, \\quad s \\in S, \\qquad \\text{and,} \\qquad\n0=-A_t(x)\\frac{1-u_t}{J+1-u_t} +I_t(x), \\quad t \\in T.\\]\n\nSolving these equations for $u_s$ and $u_t$, respectively, we obtain\n\\begin{equation} \\label{E:fp_equation}\nu_s=-\\frac{A_s(x)J}{A_s(x)-I_s(x)}, \\quad s \\in S, \\qquad  \\text{and,}  \n\\qquad \nu_t=1-\\frac{I_t(x)J}{A_t(x)-I_t(x)}, \\quad t \\in T.\n\\end{equation}\n\nNow, let $\\epsilon>0$ small such that $\\left|\\frac{I_t(x)J}{A_t(x)-I_t(x)}\\right|, \\left| \\frac{I_t(x)J}{A_t(x)-I_t(x)} \\right|\\leq 1$. \nFor all $J$ such that $0<J<\\epsilon$, Eq.~\\eqref{E:fp_equation} has a solution inside $[0,1]^N$ if and only if \n$A_s(x)-I_s(x)<0, s \\in S,$ and $A_t(x)-I_t(x)>0, t \\in T$,\nor more compactly if and only if\n$H(A_i(x)-I_i(x))=x_i \\textrm{ for all } i=1,\\ldots,N.$\nThus, a steady state appears in the corner subdomain corresponding to $x$ if and only if $x$ is a steady state of the BN $h=(h_1,\\ldots,h_N):\\{0,1\\}^N\\rightarrow\\{0,1\\}^N$ given by Eq.~\\eqref{eqn:mainReduced_BN}. \nWe have proved,\n\n\\begin{theorem}\\label{thm:localss}\nThere is an $\\epsilon>0$ such that for all $0<J<\\epsilon$,  Eq.~\\eqref{eqn:mainReduced_RST} has a steady state at a corner subdomain containing $x\\in\\{0,1\\}^N$ if and only if the BN in Eq.~\\eqref{eqn:mainReduced_BN} has a steady state at $x$.\n\\end{theorem}\n\nAlso, since we showed that the PL system given by  Eq.~\\eqref{eqn:mainReduced_RST} is a valid approximation of the full system given in Eq.~\\eqref{eqn:ProblemEquation}, we have the following corollary.\n\n\\begin{corollary}\\label{cor:localss}\nThere is an $\\epsilon>0$ such that for all $0<J<\\epsilon$, the system in Eq.~\\eqref{eqn:ProblemEquation} has a steady state at a corner subdomain containing $x\\in\\{0,1\\}^N$ if and only if the BN in Eq.~\\eqref{eqn:mainReduced_BN} has a steady state at $x$.\n\\end{corollary}\n\n\n\n\\subsubsection{Global equivalence of steady states}\n\\label{sec:math_BN_ss_global}\n\nWe proved that, for $J$ small, the steady states at the corner subdomains of the system in Eq.~(\\ref{eqn:ProblemEquation}) are in a one-to-one correspondence with the steady states of the BN. However, the corner subdomains only cover a small portion of $[0,1]^N$.  We next show that the steady state correspondence is global.\n\nRecall that $\\Omega=\\{\\mathcal{C}: \\mathcal{C} \\textrm{ is a chamber}\\}$ and that  each chamber is  a connected component of $[0,1]^N\\setminus \\cup_{i=1}^N\\{u:\\sum_{j=1}^N w_{ij} u_j +b_i=0\\}$. The chambers are bounded convex open subsets of $[0,1]^N$ (with the topology inherited from $[0,1]^N$). Also note that $[0,1]^N\\setminus \\cup_{\\mathcal{C} \\in\\Omega}\\mathcal{C}$ is a finite union of hyperplanes and hence has measure zero. We denote by $\\mathcal{C}(x)$ the chamber that contains the point $x \\in [0,1]^N$. \n\n\n\\begin{theorem}\\label{thm:one2one}\nLet $K$ be a compact subset of $\\cup_{\\mathcal{C}\\in\\Omega}\\mathcal{C}$ such that $K\\cap \\mathcal{C}$ is convex for any $\\mathcal{C}\\in\\Omega$, and such that $K\\cap \\mathcal{C}(x)$ contains a neighborhood of $x$ for each $x\\in \\{0,1\\}^N$. \n\nThen, there is an $\\epsilon_K>0$ such that for all $0<J<\\epsilon_K$,  \nif $x^*\\in\\{0,1\\}^N$ is a steady state of the BN in Eq.~\\eqref{eqn:mainReduced_BN}, then the ODE in Eq.~\\eqref{eqn:ProblemEquation} has a steady state $u^*\\in \\mathcal{C}(x^*)$; and if $u^*\\in K$ is a steady state of the ODE, then $u^*\\in \\mathcal{C}(x^*)$ for some steady state $x^*$ of the BN. \n\nFurthermore, if $x^*$ is a steady state of the BN in Eq.~\\eqref{eqn:mainReduced_BN}, then the steady state of the ODE in Eq.~\\eqref{eqn:ProblemEquation} is unique in $K\\cap \\mathcal{C}(x^*)$, converges to $x^*$ as $J\\rightarrow 0$, and is asymptotically stable.\n\\end{theorem}\n\\begin{proof}\nSee Appendix.\n\\end{proof} \n\nWe can make the set $K$ in Theorem \\ref{thm:one2one} as close to $[0,1]^N$ as desired. For example, for each chamber $\\mathcal{C}$ and for $r>0$, denote $K_\\mathcal{C}:=\\{u\\in\\mathcal{C}:|\\sum_{j=1}^N w_{ij}u_j+b_i|\\geq r, \\forall i\\}$. By using $r$ small, and denoting Lebesgue measure by $\\mu$, we can make $\\mu([0,1]^N\\setminus \\cup_{\\mathcal{C}\\in\\Omega}K_{\\mathcal{C}})=\\mu( \\cup_{\\mathcal{C}\\in\\Omega} (\\mathcal{C}\\setminus K_{\\mathcal{C}}))= \\sum_{\\mathcal{C}\\in\\Omega} \\mu(\\mathcal{C}\\setminus K_{\\mathcal{C}})$  as small as desired. Hence, we have the following corollary.\n\n\n\\begin{corollary}\\label{cor:one2one}\nFor any $\\epsilon>0$, there is a set $K\\subseteq [0,1]^N$ satisfying $\\mu([0,1]^N\\setminus K)<\\epsilon$ and a number $\\epsilon_K$ such that for all $0<J<\\epsilon_K$, there is a one-to-one correspondence between the steady states of the BN in Eq.~\\eqref{eqn:mainReduced_BN} and the steady states in $K$ of the ODE in Eq.~\\eqref{eqn:ProblemEquation}. Furthermore, the steady states of the ODE in $K$ are asymptotically stable.\n\\end{corollary}\n\n \nNote that the set $K$ does not include the nullclines.  Hence,  steady states  outside $K$ are possible, and they could be stable. Such steady states can be studied using the PL approximation in Eq.~\\eqref{eqn:mainReduced_RSTa}.\n\n\\subsubsection{Equivalence of trajectories}\n\\label{sec:math_BN_trajectories}\n\nWe next examine under which conditions  the trajectories of the BN in Eq.~(\\ref{eqn:mainReduced_BN}) correspond trajectories of\nthe ODE given in Eq.~(\\ref{eqn:ProblemEquation}). The main assumption in the rest of this section is that the hyperplanes divide $[0,1]^N$ into $2^N$ chambers, and each chamber contains a corner. \n\nWe say that the solutions of the system in Eq.~(\\ref{eqn:ProblemEquation})  \\emph{transition from $K\\subseteq[0,1]^N$ to $K'\\subseteq[0,1]^N$} if for any solution of the system, $u(t)$, with initial condition $u(0)\\in K$, there exists $\\hat{t}$ such that $u(\\hat{t})\\in K'$. The \\emph{Hamming distance} between $x,y\\in\\{0,1\\}^N$ is defined as $d(x,y):=|\\{i\\in\\{1,\\ldots,N\\}:x_i\\neq y_i\\}|$. We denote a transition $h(x)=y$ (i.e. $H(Wx+b)=y$) with $x\\rightarrow y$. A trajectory is a sequence of transitions and is denoted similarly.\nWe call a transition $x\\rightarrow y$ \\emph{regular} if either (1) $x=y$ or (2) $d(x,y)=1$ and $h_j(y)=y_j$ for the index $j$ for which $x_i\\neq y_i$. \nIn other words, a transition  $x\\rightarrow y$ in the BN is regular if $x$ is a steady state or if $x$ transitions to $y$ by changing only one coordinate and this coordinate does not change back when transitioning from $y$ to $h(y)$. For example, if $000\\rightarrow 100\\rightarrow 111$, then $000\\rightarrow 100$ is a regular transition ($j=1$); on the other hand, if $000\\rightarrow 100\\rightarrow 010$, then $000\\rightarrow 100$ is not a regular transition. A trajectory is regular if each component transition is regular.\n\n\n\\begin{theorem}\\label{thm:transition}\nSuppose that the hyperplanes divide $[0,1]^N$ into $2^N$ chambers and consider a regular transition of the BN in Eq.~\\eqref{eqn:mainReduced_BN}, $x\\rightarrow h(x)$. Then, there is a neighborhood $K$ of $x$, and an $\\epsilon_K>0$ such that for all $0<J<\\epsilon_K$ the solutions of the ODE in Eq.~\\eqref{eqn:ProblemEquation} transition from $K$ to $\\mathcal{C}(h(x))$. Also, if $x$ is a steady state, the neighborhood $K$ can be chosen to be invariant.\n\\end{theorem}\n\n\\begin{proof}\nSee Appendix.\n\\end{proof}\n\nNext for a steady state $x$ define $B_1(x)=\\{y\\in\\{0,1\\}^N:h(y)=x\\textrm{ and } d(x,y)\\leq 1\\}$; that is, $B_1(x)$ is the set of states in the basin of attraction of $x$ with Hamming distance at most 1 from $x$. The following corollary of Theorem \\ref{thm:transition} states that this part of the basin of attraction of $x$ corresponds to part of the basin of attraction of the steady state in the ODE in Eq.~(\\ref{eqn:ProblemEquation})  corresponding to $x$.\n\\begin{corollary}\nSuppose that $x$ is a steady state of the BN in Eq.~\\eqref{eqn:mainReduced_BN}. Then, for every $y\\in B_1(x)$, there is a neighborhood $K$ of $y$  and $\\epsilon_K>0$, such that for all $0<J<\\epsilon_K$  the solutions of the ODE in Eq.~\\eqref{eqn:ProblemEquation} transition from $K$ to  $\\mathcal{C}(x)$.\n\\end{corollary}\n\n\nNote that Theorem \\ref{thm:transition} implies that for a regular trajectory of the BN in Eq.~(\\ref{eqn:mainReduced_BN}),  $x\\rightarrow h(x)\\rightarrow h^2(x)\\rightarrow \\ldots \\rightarrow h^m(x)$, the solutions of the ODE in Eq.~(\\ref{eqn:ProblemEquation}) will transition from a neighborhood of $x$ to $\\mathcal{C}(h(x))$, from a neighborhood of $h(x)$ to $\\mathcal{C}(h^2(x))$ and so on. \nTo guarantee that a neighborhood of $x$ will reach a neighborhood of $h^m(x)$ (that is, to guarantee that the result is ``transitive''), we need the additional assumption that each hyperplane is orthogonal to some coordinate axis. Note that the example given in Section \\ref{sec:3nodes} satisfies this condition.\n\n\\begin{theorem}\\label{thm:trajectory}\nSuppose that  each hyperplane is orthogonal to some coordinate axis and let  $x\\rightarrow h(x)\\rightarrow \\ldots\\rightarrow h^m(x)$ be a regular trajectory of the BN in Eq.~(\\ref{eqn:mainReduced_BN}). Then, for any compact set $K\\subset \\mathcal{C}(x)$ there is $\\epsilon_{K}>0$ such that for all $0<J<\\epsilon_{K}$, the solutions of the ODE in Eq.~\\eqref{eqn:ProblemEquation} transition from $K$ to $\\mathcal{C}(h^m(x))$ following the order of the regular trajectory.\n\\end{theorem}\n\n\\begin{proof}\nSee Appendix.\n\\end{proof}\n\nFor a steady state of the BN define $B(x)=\\{y:\\textrm{ there is a regular trajectory from $y$ to $x$}\\}$. The following corollary of Theorem~\\ref{thm:trajectory} implies that some states in the basin of attraction of a steady state of the BN in Eq.~(\\ref{eqn:mainReduced_BN}) correspond to chambers in the basin of attraction of a steady state of the ODE in Eq.~(\\ref{eqn:ProblemEquation}).\n\n\n\\begin{corollary} \nSuppose that  each hyperplane is orthogonal to some coordinate axis and let $x$ be a steady state of the BN in Eq.~\\eqref{eqn:mainReduced_BN}. Consider $y\\in B(x)$. Then, for any compact set $K\\subseteq\\mathcal{C}(y)$, there exists $\\epsilon_{K}>0$ such that for all $0<J<\\epsilon_{K}$, the solutions of the ODE in Eq.~\\eqref{eqn:ProblemEquation} transition from $K$ to $\\mathcal{C}(x)$.\n\\end{corollary}\n\nSimilarly, we obtain the following corollary for oscillatory behavior. \n\\begin{corollary}\\label{cor:oscilation}\nSuppose that  each hyperplane is orthogonal to some coordinate axis and let  $x^1\\rightarrow x^2\\rightarrow \\ldots\\rightarrow x^p\\rightarrow x^1$ be a regular periodic orbit of the BN in Eq.~(\\ref{eqn:mainReduced_BN}). Then, for any compact set $K\\subseteq\\mathcal{C}(x^1)$ and any positive integer $m$, there exists $\\epsilon_{K,m}>0$ such that for all $0<J<\\epsilon_{K,m}$, the solutions of the ODE in Eq.~\\eqref{eqn:ProblemEquation} transition between the chambers (starting at $K$) in the order $\\mathcal{C}(x^1), \\mathcal{C}(x^2),\\ldots,\\mathcal{C}(x^p),\\mathcal{C}(x^1)$, $m$ times.\n\\end{corollary}\n\n\nNote that the example in Section \\ref{sec:3nodes} satisfies the hypothesis of this last corollary. In general, Corollary \\ref{cor:oscilation} does not guarantee that the solution is periodic.\n\n\nFinally, we note that the requirement that there are $2^N$ chambers, each containing a corner is necessary.  Even if we have a transition where only one variable changes (e.g. $h(1,0)=(0,0)$), having an intermediate chamber can change the behavior of the solutions before they reach the chamber predicted by the BN. In the example shown in Fig.~\\ref{fig:twoDimension_counterexample_regular} the signs of the vector field of the approximating linear system imply that the BN transitions from $(1,0)$ to $(0,0)$.  However, solutions can transition from the chamber that contains $(1,0)$ to the bottom middle chamber and never reach the chamber that contains $(0,0)$. In summary, even having a transition where only one variable changes may not be sufficient to guarantee that the Boolean transition corresponds to a similar transition in the original system.\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[scale = .35]{twoDimension_counterexample_regular}\n\\parbox{.9\\textwidth}{\n\\caption[]{\\footnotesize\nChambers and signs of vector field for the linear system given by $W=\\left[\\begin{array}{rr} -2 & -1 \\\\ -2 & 1 \\end{array}\\right]$ and $b=\\left[\\begin{array}{r} 1.5 \\\\ 0.5 \\end{array}\\right]$.\n\\label{fig:twoDimension_counterexample_regular}}\n}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\section{Discussion}\\label{Discussion}\n\nModels of biological systems are frequently nonlinear and difficult to analyze mathematically. In addition, accurate models \nfrequently contain numerous parameters whose exact values are not known. Thus, studying which parameters have a large impact on dynamics, and how a  model can be simplified, is crucial in finding the features of biological systems that determine their behavior and responses. Reduction techniques that preserve key dynamical properties are essential in this endeavor.\n\nWe studied a special class of non-linear differential equation  models of biological networks where interactions between nodes are described using Hill functions. When the Michaelis-Menten constants are sufficiently small, the behavior of the  system is captured by an approximate piecewise linear system and a Boolean Network. \nIn this case the domain of the full system naturally decomposes into nested hypercubes.  These hypercubes define subdomains within which a solvable linear--algebraic system approximates the original system. The Boolean Network is obtained from a decomposition of the domain into chambers and describes how solutions evolve between them. \n\n\nThe proposed reductions have a  number of advantages: The piecewise linear approximation is not only  easier to solve than the original system analytically, but also numerically (the original system becomes stiff for small $J$). When  one is interested in qualitative behavior such as steady state analysis, the Boolean approximation can be very useful, especially when the dimension of the system is large. Also, the Boolean framework has been used to model many biological systems where it is assumed that interactions are switch-like and variables can be discretized. It is therefore important to know when, and in what sense such reduced systems can be justified, and in particular, what dynamical properties of the full system are capture by a reduction. \n\nAlthough the case of large exponent $n$ in the Hill function  has been studied in the past [\\cite{GlassKauffman1973,Snoussi1989, thomasbook,mendoza2006, DavitichBornholdt2008, Wittmann2009, Franke2010, Veliz-CubaArthurHochstetlerKlompsKorpi2012}], the case of small $J$ has been studied only recently and heuristically [\\cite{DavitichBornholdt2008}]. In this manuscript we have shown that the PL and the BN approximations are also valid for the case of small $J$, and have given explicit formulas for their computation. The BN approximation preserves steady state behavior and under further restrictions, it can also be used to  infer the basins of attractions and oscillations in the original system.  Note that the Boolean functions in the Boolean approximation are threshold functions, as used in earlier models [\\cite{Li_cc_2004,DavitichBornholdt2008b}].  Our results show that such BN scan indeed appear when  approximating more detailed models, such as those described by Eq.~\\eqref{eqn:ProblemEquation}. \nIn summary, our results for the limit $J\\rightarrow 0$ complement previous results for the limit $n\\rightarrow \\infty$,  providing a useful framework for reducing nonlinear systems to PL systems and BNs.\n\n\n\nA potential limitation in our arguments is that we have an approximation \nvalid only in an asymptotic limit.  It is unknown when and how the approximation breaks down.\nHowever, the approximation is still valid as $J$ increases until we reach a bifurcation point, which can happen for relative large $J$ or for values of $J$ that are biologically relevant. Also, we have not provided a systematic relationship between the thickness of the boundary, $\\delta$, and the Michaelis-Menten constant, $J$. Numerical tests suggest that the relationship is between $J = \\mathcal{O}(\\delta)$ and $J = \\mathcal{O}(\\delta^2)$. Another limitation of our analysis is that, in the general case, it is not known when the transitions or cycles in a BN will correspond to similar transitions or cycles in the original ODE.\n\n\\section*{Acknowledgment}\n\nWe thank Matthew Bennett for help in preparing the manuscript. This work was funded by the NIH, through the joint NSF\/NIGMS Mathematical Biology Program Grant No. R01GM104974.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nTorsion pairs in abelian categories were introduced by Dickson~\\cite{dickson66}\n(see the introduction to~\\cite{br} for further details). They play an important\nrole in the study of localisation (see e.g.~\\cite{br}) and in tilting theory\n(see e.g.~\\cite{ass,br}). Also, torsion pairs in the case of triangulated\ncategories, as defined in~\\cite[2.2]{iy}, have been considered recently by a\nnumber of authors, e.g.~\\cite{hjr,iy,kz,nakaoka,ng,zz}.\n\nThe main object of study of this article is the collection of torsion pairs in\na tube category, which is a hereditary abelian category.\nSuch categories arise as full subcategories of module\ncategories over tame hereditary algebras, and are so-called because their\nAuslander-Reiten quivers have the shape of a tube.\n\nA geometric model for tube categories has been given in~\\cite{bama,warkentin}\n(see also~\\cite{bz}). The indecomposable objects are parametrized by a\ncollection of oriented arcs in an annulus with $n$ marked points on one of its\nboundary components. The dimension of the $\\operatorname{Ext}$-group between two\nindecomposable objects coincides with the negative intersection number of\nthe corresponding pair of arcs.\n\nIn this article, we classify the torsion pairs in a tube category $\\mathbf T$. We build\non results in~\\cite{bk1} giving a bijection between torsion pairs in a tube category and\ncotilting objects in the category obtained by taking arbitrary direct limits of modules in the tube.\nWe show that all torsion pairs in the tube category arise in this way or via a dual construction.\nThus they are in bijection with maximal rigid objects in the category $\\overline{\\mathbf T}$\nobtained from the tube by taking arbitrary direct or inverse limits of objects in the tube.\nWe give an explicit description of this bijection and its inverse.\n\nWe further show that the geometric model referred to above can be extended to the indecomposable\nobjects in $\\overline{\\mathbf T}$, i.e.\\ to include Pr\\\"{u}fer and adic modules associated to the tube. These\nextra objects are represented by certain infinite arcs in the annulus which spiral in towards the\ninner boundary component. The result concerning the dimensions of $\\operatorname{Ext}$-groups extends to this case also.\nThis enables us to give a characterisation of maximal rigid objects in $\\mathbf T$ in the geometric model. We also\ngive a characterisation of torsion\npairs in the geometric model in terms of certain closure properties corresponding to closure properties\nof the subcategories in a torsion pair. In particular, the collection of arcs corresponding to a\nsubcategory in a torsion pair must form an \\emph{oriented Ptolemy diagram}, an oriented version in the\nannulus case of the Ptolemy diagrams appearing in~\\cite{hjr,ng}, as well as satisfying additional criteria.\nWe also give a geometric description of the above bijection and its inverse.\n\nIn order to give the geometric interpretation, we first give a similar model\nfor the linearly oriented quiver in type A\n(note that M. Warkentin~\\cite{warkentin} also suggests such a model) and\nshow how to interpret tilting modules and torsion pairs in this model.\n\nWe remark that, in independent work, T. Holm, P. J\\o rgensen and\nM. Rubey~\\cite{holmjorgensenrubey} have classified the torsion pairs in the\ncluster category associated to a tube (as opposed to the tube itself,\nwhich we study here). The torsion pairs in the cluster case are different,\nalthough we note that unoriented Ptolemy diagrams play a role in the cluster\ntube case, while oriented Ptolemy diagrams appear here.\n\nIn Section 1, we recall the definition and some of the properties of torsion pairs in abelian categories.\nIn Section 2, we discuss the type A case. In Section 3, we recall the geometric model of the tube and show how it can be extended to include Pr\\\"{u}fer and\nadic modules. We also discuss a certain reflection map (from~\\cite{bk1})\non the indecomposable objects of the tube, and its properties.\nIn Section 4 we classify the torsion pairs in the tube and prove that\nthey are in bijection with maximal rigid objects in $\\overline{\\mathbf T}$,\ngiving an explicit description of the bijection and its inverse.\nIn Section 5, we give a geometric interpretation of maximal rigid objects\nin $\\mathbf T$ and torsion pairs in $\\mathbf T$ and the bijection between them.\n\n\\section{Preliminaries} \\label{s:preliminaries}\n\nWe shall adopt the convention throughout that all subcategories considered\nare strictly full (i.e.\\ full and closed under isomorphism). We shall also\nconsider modules up to isomorphism.\nLet $\\mathcal A$ be an abelian category. If $\\mathcal X$ is a subcategory of $\\mathcal A$,\nwe define ${}^{{\\perp}_{\\scriptscriptstyle H}} \\mathcal X$ (respectively, ${}^{{\\perp}_{\\scriptscriptstyle E}} \\mathcal X$) to be\nthe additive subcategory of $\\mathcal A$ consisting of the objects\n$Y$ satisfying $\\operatorname{Hom}_{\\mathcal A}(Y,X)=0$ (respectively,\n$\\operatorname{Ext}^1_{\\mathcal A}(Y,X)$=0) for all $X$ in $\\mathcal X$.\nFor an object $X$ of $\\mathcal A$ we define ${}^{{\\perp}_{\\scriptscriptstyle H}}\nX={}^{{\\perp}_{\\scriptscriptstyle H}} \\operatorname{add} X$, and ${}^{{\\perp}_{\\scriptscriptstyle E}} X={}^{{\\perp}_{\\scriptscriptstyle E}} \\operatorname{add} X$,\nwhere $\\operatorname{add} X$ denotes the additive subcategory of $\\mathcal A$ whose\nobjects are finite direct sums of direct summands of $X$. We\nsimilarly define $\\mathcal X^{{\\perp}_{\\scriptscriptstyle H}}$, $\\mathcal X^{{\\perp}_{\\scriptscriptstyle E}}$, $X^{{\\perp}_{\\scriptscriptstyle H}}$ and\n$X^{{\\perp}_{\\scriptscriptstyle E}}$. For additive subcategories $\\mathcal X_i$, $i=1,2, \\ldots ,m$,\nof $\\mathcal A$, we write $\\coprod_{i=1}^m \\mathcal X_i$ for the smallest additive\nsubcategory of $\\mathcal A$ containing the $\\mathcal X_i$.\n\nIf $\\mathcal X$ is an additive subcategory of $\\mathcal A$, we write $\\operatorname{Gen} \\mathcal X$ for the\nsubcategory consisting of objects which are quotients of objects of $\\mathcal X$ and,\nfor an object $X$, $\\operatorname{Gen} X=\\operatorname{Gen}(\\operatorname{add} X)$. Similarly, we write $\\operatorname{Cogen} \\mathcal X$ for\nthe subcategory consisting of objects which are subobjects of objects of $\\mathcal X$\nand $\\operatorname{Cogen} X=\\operatorname{Cogen}(\\operatorname{add} X)$. We write $\\operatorname{ind} \\mathcal X$ for the subcategory of $\\mathcal X$\nwhose objects are the indecomposable objects in $\\mathcal X$.\n\nWe next recall some of the theory of torsion pairs. Recall that\na pair $(\\mathcal T,\\mathcal F)$ of subcategories of\n$\\mathcal A$ is said to be a \\emph{torsion pair} \\cite[Sect.\\ 1]{dickson66}\nif $\\operatorname{Hom}(T,F)=0$ for all objects $T$ in $\\mathcal T$ and $F$ in $\\mathcal F$ and\nfor every object $A$ in $\\mathcal A$ there is a short exact sequence\n\\begin{equation}\n0 \\rightarrow {T} \\rightarrow A \\rightarrow {F} \\rightarrow 0\n\\label{e:linkingsequence}\n\\end{equation}\nwith $T$ in $\\mathcal T$ and $F$ in $\\mathcal F$.\nWe say that $\\mathcal T$ is the \\emph{torsion} part and $\\mathcal F$ is the\n\\emph{torsion-free} part of the pair.\n\nRecall that an abelian category $\\mathcal A$ is said to be \\emph{finite\nlength} if it is skeletally small and every object in it is artinian\nand noetherian. Equivalently, it is skeletally small and every object\nhas a finite length composition series.\n\nThus, for example, the category $\\operatorname{mod} A$ of finite dimensional modules over a finite-dimensional algebra is a finite length category.\nParts (a), (b) of the following can be proved using arguments as\nin~\\cite[Thm.\\ 2.1]{dickson66} (see also~\\cite[Lemma 1.7]{bk1}); part (c) is\na well-known corollary.\n\n\\begin{lemma} \\label{l:tpclosure}\nSuppose that $\\mathcal A$ is a finite length abelian category.\n\\begin{enumerate}\n\\item[(a)]\nLet $\\mathcal T$ be a subcategory of $\\mathcal A$ and let $\\mathcal F=\\mathcal T^{{\\perp}_{\\scriptscriptstyle H}}$.\nThen $(\\mathcal T,\\mathcal F)$ is a torsion pair in $\\mathcal A$ if and only if\n$\\mathcal T$ is closed under quotients and extensions.\n\\item[(b)]\nLet $\\mathcal F$ be a subcategory of $\\mathcal A$ and let $\\mathcal T={}^{{\\perp}_{\\scriptscriptstyle H}}\\mathcal F$.\nThen $(\\mathcal T,\\mathcal F)$ is a torsion pair in $\\mathcal A$ if and only if $\\mathcal F$ is closed under\nsubobjects and extensions.\n\\item[(c)] A pair $(\\mathcal T,\\mathcal F)$\nof subcategories of $\\mathcal A$ is a torsion pair if and only if\n$\\mathcal T^{{\\perp}_{\\scriptscriptstyle H}}=\\mathcal F$ and ${}^{{\\perp}_{\\scriptscriptstyle H}}\\mathcal F=\\mathcal T$.\n\\end{enumerate}\n\\end{lemma}\n\nRecall that a finite length abelian category is said to be \\emph{serial} if\nevery object is a direct sum of indecomposable uniserial objects. We recall\nthe following well-known result.\n\n\\begin{lemma} \\label{l:uniserialclosure}\nLet $\\mathcal A$ be a serial finite length abelian category. Then\n\\begin{enumerate}\n\\item[(a)] A morphism from an indecomposable object of $\\mathcal A$ to a direct sum of indecomposable\nobjects is a monomorphism if and only if at least one of its components is a monomorphism.\n\\item[(b)] Let $\\mathcal X$ be a subcategory of $\\operatorname{ind} \\mathcal A$.\nThen we have $\\operatorname{ind} \\operatorname{Gen}(\\operatorname{add} \\mathcal X)=\\operatorname{ind} \\operatorname{Gen} \\mathcal X$ and $\\operatorname{ind} \\operatorname{Cogen}(\\operatorname{add} \\mathcal X)=\\operatorname{ind} \\operatorname{Cogen} \\mathcal X$.\n\\end{enumerate}\n\\end{lemma}\n\nNote that the dual of Lemma~\\ref{l:uniserialclosure}(a) also holds.\nWe fix an algebraically-closed field $K$ and denote by $D$ the vector space duality $\\operatorname{Hom}_K(-,K)$.\nNext, consider a finite dimensional $K$-algebra $\\Lambda$. We denote\nby $\\operatorname{Mod} \\Lambda$ the category of all left $\\Lambda$-modules, and by\n$\\operatorname{mod} \\Lambda$ the subcategory of all finitely generated $\\Lambda$-modules.\nRecall that a module ${U}$ in $\\operatorname{mod} \\Lambda$ is called \\emph{tilting} if\n\\begin{itemize}\n\\item[(a)] The projective dimension of ${U}$ is at most 1;\n\\item[(b)] $\\operatorname{Ext}^1({U},{U}) = 0$;\n\\item[(c)] There is a short exact sequence\n$$0\\rightarrow \\Lambda \\rightarrow {U}_0 \\rightarrow {U}_1 \\rightarrow 0$$\nwith ${U}_0$ and ${U}_1$ in $\\operatorname{add} {U}$.\n\\end{itemize}\nCotilting modules in $\\operatorname{mod} \\Lambda$ are defined dually. We only\nconsider basic tilting or cotilting modules, i.e.\\ we assume that\n${U}= \\amalg {U}_i$, with ${U}_i$ indecomposable and ${U}_i \\not\\simeq {U}_j$\nfor $i \\neq j$.\n\n\\section{Dynkin type A}\n\\label{s:Dynkin}\n\nWe consider a linearly oriented quiver $Q$ of type $\\text{A}_n$:\n$$\n\\xymatrix{\n1 & 2 \\ar[l] & 3 \\ar[l] & \\cdots\\ar[l] & n-1 \\ar[l] & n \\ar[l]\n}\n$$\nWe write $S_i$, $i=1,\\dots,n$, for the simple $KQ$-modules.\nLet $M_{ij}$ denote the indecomposable $KQ$-module with composition factors\n$S_{i+1},\\dots,S_{j-1}$ (starting from the socle).\nIf $i$ lies in $\\{j-1,j\\}$ we regard $M_{ij}$ as zero.\nNote that $\\operatorname{mod} KQ$ is serial.\n\n\\subsection{Geometric model}\n\\label{s:geometricmodel}\n\nWe now describe a geometric model for the indecomposable $KQ$-modules. Note that\na similar such model is also suggested in~\\cite[Remark 4.28]{warkentin}.\nHere we will indicate how this model can incorporate torsion theories\nin $\\operatorname{mod} KQ$ --- later we will indicate how this can be done for tubes. We\nalso give some more explicit information as preparation for this.\nWe consider a line segment $\\ell_n$ with marked points $0,1,\\dots,n+1$:\n\\[\n\\begin{picture}(220.00,20.00)\n\\drawline(10.00,15.00)(210.00,15.00)\n\\put(10.00,15.00){\\makebox(0,0)[cc]{\\tiny $\\bullet$}}\n\\put(10.00,05.00){\\makebox(0,0)[cc]{\\tiny $0$}}\n\\put(40.00,15.00){\\makebox(0,0)[cc]{\\tiny $\\bullet$}}\n\\put(40.00,05.00){\\makebox(0,0)[cc]{\\tiny $1$}}\n\\put(70.00,15.00){\\makebox(0,0)[cc]{\\tiny $\\bullet$}}\n\\put(70.00,05.00){\\makebox(0,0)[cc]{\\tiny $2$}}\n\\put(150.00,15.00){\\makebox(0,0)[cc]{\\tiny $\\bullet$}}\n\\put(150.00,05.00){\\makebox(0,0)[cc]{\\tiny $n-1$}}\n\\put(180.00,15.00){\\makebox(0,0)[cc]{\\tiny $\\bullet$}}\n\\put(180.00,05.00){\\makebox(0,0)[cc]{\\tiny $n$}}\n\\put(210.00,15.00){\\makebox(0,0)[cc]{\\tiny $\\bullet$}}\n\\put(210.00,05.00){\\makebox(0,0)[cc]{\\tiny $n+1$}}\n\\end{picture}\n\\]\nand associate the module $M_{ij}$ with the arc $[i,j]$ above $\\ell_n$\nfrom $i$ to $j$ oriented towards $j$, $0\\le i<j-1\\le n$.\nThis gives a bijection between indecomposable $KQ$-modules and the\nset $\\mathcal A(\\ell_n)$ of arcs up to isotopy between marked points of\n$\\ell_n$, above $\\ell_n$, which are not isotopic to boundary arcs.\n\nIt is easy to check that, for $[i,j]$, $[i',j']\\in \\mathcal A(\\ell_n)$,\nwe have $\\operatorname{Ext}^1(M_{ij},M_{i'j'})\\cong K$ if there is a negative crossing between\n$[i,j]$ and $[i',j']$ (see Figure~\\ref{fig:neg}) and is zero otherwise.\nIn the former case, the non-split extension takes the form:\n$$\n0\\to M_{i'j'}\\to \\begin{array}{c} M_{i'j} \\\\ \\amalg \\\\ M_{ij'} \\end{array} \\to M_{ij} \\to 0\n$$\nup to equivalence, if $j'>i+1$, while if $j'=i+1$, it takes the form\n$$\n0\\to M_{i'j'}\\to M_{i'j} \\to M_{ij} \\to 0,\n$$\nThis is interpreted geometrically in Figure~\\ref{fig:split-sum}\nwhere the indecomposable summands of the middle term\nof the short exact sequence are indicated by dotted lines.\n\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{i}{\\tiny $i$}\n\\psfrag{i'}{\\tiny $i'$}\n\\psfrag{j}{\\tiny $j$}\n\\psfrag{j'}{\\tiny $j'$}\n\\includegraphics[scale=.5]{neg.eps}\n\\caption{A negative crossing between $[i,j]$ and $[i',j']$.}\\label{fig:neg}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{i}{\\tiny $i$}\n\\psfrag{i'}{\\tiny $i'$}\n\\psfrag{j}{\\tiny $j$}\n\\psfrag{j'}{\\tiny $j'$}\n\\psfrag{j'=i+1}{\\tiny $j'=i+1$}\n\\subfigure[Case $j'>i+1$]{\n\\includegraphics[scale=.6]{splitsum.eps}}\n\\ \\ \\ \\ \\ \\\n\\subfigure[Case $j'=i+1$]{\n\\includegraphics[scale=.6]{splitsum2.eps}}\n\\caption{Non-split extension}\\label{fig:split-sum}\n\\end{figure}\n\nThe AR-translation moves an arc one step to the left (or gives zero\nif this is not defined). Furthermore, the indecomposable quotients\nof a module correspond to moving the starting point weakly to the right,\nwhile submodules correspond to moving the ending point weakly to the left.\nWe call the former arcs {\\em left-shortenings} of $[i,j]$ and the latter arcs\n{\\em right-shortenings} of $[i,j]$.\n\nBy~\\cite{bo}, a $KQ$-module is tilting if and only if it is\na maximal rigid $KQ$-module.\nSince $M_{0,n+1}$ is projective-injective, it is a summand of every tilting module.\nIt follows that in the above model, tilting modules are in bijection with\ntriangulations of a polygon with $n+2$ sides.\n\nMotivated by~\\cite{ng,hjr} and the above description of extensions,\nwe call a collection of arcs in $\\mathcal A(\\ell_n)$ an {\\em oriented Ptolemy\ndiagram} if, whenever $[i,j]$ and $[i',j']$ lie in the collection, with\n$i'<i<j'<j$, we have that the arcs $[i,j']$ (provided $j'>i+1$) and $[i',j]$\nalso lie in the collection.\n\n\\subsection{Torsion pairs}\n\nBy Lemma~\\ref{l:tpclosure} and Lemma~\\ref{l:uniserialclosure},\na collection of arcs in $\\mathcal A(\\ell_n)$ corresponds to the torsion (respectively, torsion-free) part of a torsion pair in $\\operatorname{mod} KQ$ if and only if it forms an oriented Ptolemy diagram and is closed under left-shortening (respectively, right-shortening).\n\nGiven a tilting $KQ$-module ${U}$, the pair\n$(\\operatorname{Gen} {U}, \\operatorname{Cogen}\\tau {U})=({U}^{{\\perp}_{\\scriptscriptstyle E}},{U}^{{\\perp}_{\\scriptscriptstyle H}})$\nis known to form a torsion pair (see~\\cite[VI.2]{ass}). A torsion pair arises\nin this way if and only if $\\mathcal T$ contains all the indecomposable injective\nmodules, if and only if $\\mathcal F$ contains no non-zero injective module (see~\\cite[VI.6]{ass}\nfor the first equivalence; the second is easy to check). The first equivalence\nholds for an arbitrary finite dimensional hereditary algebra of finite representation type, and the\nequivalence of the second two statements holds for any finite dimensional hereditary algebra.\n\nNoting that a $KQ$-module is tilting if and only if it is cotilting, we have:\n\n\\begin{corollary}\\label{cor:Gen-Cogen}\nThe map:\n$$\n{U} \\mapsto (\\operatorname{Gen} {U},\\operatorname{Cogen}\\tau {U})=({U}^{{\\perp}_{\\scriptscriptstyle E}},{U}^{{\\perp}_{\\scriptscriptstyle H}})\n$$\ngives a bijection between tilting $KQ$-modules and\ntorsion pairs $(\\mathcal T,\\mathcal F)$ for which $\\mathcal T$ contains all the\nindecomposable injective modules. The map:\n$$\n{U} \\mapsto (\\operatorname{Gen}\\tau^{-1}{U},\\operatorname{Cogen} {U})=({}^{{\\perp}_{\\scriptscriptstyle H}}{U},{}^{{\\perp}_{\\scriptscriptstyle E}}{U})\n$$\ngives a bijection between tilting $KQ$-modules and\ntorsion pairs $(\\mathcal T,\\mathcal F)$ for which for which $\\mathcal F$ contains all\nthe indecomposable projective modules.\n\\end{corollary}\n\nUsing Lemma~\\ref{l:uniserialclosure}, the first map\nin Corollary~\\ref{cor:Gen-Cogen} can be interpreted in the geometric model:\n$\\operatorname{Gen} {U}$ is obtained from ${U}$ by closure under left shortening and\n$\\operatorname{Cogen}\\tau {U}$ is obtained from ${U}$ by shifting to the left one step (deleting arcs starting at $0$) and then closing under right shortening (in both cases we then\ntake the additive closure).\n\nConversely, if $(\\mathcal T,\\mathcal F)$ is a torsion pair of the kind considered in\nCorollary~\\ref{cor:Gen-Cogen}, then ${U}$ can be recovered as the direct\nsum of the indecomposable Ext-projectives in $\\mathcal T$, i.e.\\ the objects\n\\begin{equation}\n\\{X\\in \\operatorname{ind}(\\mathcal T): \\operatorname{Ext}^1(X,{T})=0\\ \\text{\\ for all\\ } {T}\\in \\mathcal T\\},\n\\label{e:extprojectives}\n\\end{equation}\nby~\\cite[VI.2.5]{ass}.\nGeometrically, this means taking all of the arcs $X$ in\n$\\operatorname{ind} \\mathcal T$ which do not have a negative crossing with an arc in $\\operatorname{ind} \\mathcal T$.\n\nThere is also a geometric description of the second map in Corollary~\\ref{cor:Gen-Cogen}\nand its inverse. We leave the details to the reader.\n\n\\section{Tubes}\n\n\\subsection{Categorical description}\n\nFix an integer $n\\geq 1$. Consider the quiver $Q$\n\\begin{equation}\\label{quivertilde}\n\\xymatrix@R=0.3cm{\n& 2 \\ar[dl] & 3 \\ar[l] & \\cdots \\ar[l] & n \\ar[l] & \\\\\n1 & & & & & n+1 \\ar[lllll] \\ar[ul]\n}\n\\end{equation}\nof Euclidean type $\\widetilde{\\text{A}}_{1,n}$. The path algebra $\\Lambda= KQ$ is tame hereditary, and the module category $\\operatorname{mod} KQ$ has an\nextension closed subcategory $\\mathbf T_n$, which can be realized as the extension closure of the modules $L, S_2, \\dots, S_{n}$,\nwhere $S_i$ denotes the simple corresponding to vertex $i$, and $L$ denotes the unique indecomposable module\nwith composition factors $S_1$ and $S_{n+1}$. The category $\\mathbf T_n$ is called a {\\em tube} of rank $n$. Note that the indecomposables of $\\mathbf T$ form a standard component\n(see e.g.~\\cite{ass}) of the AR-quiver of $KQ$.\n\nActually $\\mathbf T_n$ is a hereditary finite length abelian category with\n$n$ simple objects, and equivalent categories appear in various\nsettings in representation theory and algebraic geometry.\nFor each pair of objects $X,Y$ in $\\mathbf T_n$, the\nspaces $\\operatorname{Hom}(X,Y)$ and $\\operatorname{Ext}^1(X,Y)$ have finite $K$-dimension, and\nthere is an autoequivalence $\\tau$ on $\\mathbf T_n$, induced by the\nAuslander-Reiten translate on $\\operatorname{mod} \\Lambda$, with the property\nthat $\\operatorname{Hom}(Y, \\tau X) \\simeq D\\operatorname{Ext}^1(X,Y)$. Let\n$\\sigma:\\mathbb{Z}\\rightarrow \\mathbb{Z}$ be the map taking $i$ to\n$i+n$. From now on we denote the simples in $\\mathbf T = \\mathbf T_n$ by\n$M_{i,i+2}$ for $i=0, \\dots, n-1$, in such a way that $\\tau\nM_{i,i+2} = M_{i-1,i+1}$,\nwhere we regard $M_{\\sigma^k(i),\\sigma^k(j)}$ as equal to $M_{i,j}$\nfor any integer $k$.\nThe category $\\mathbf T_n$ is serial; thus the indecomposable objects in\n$\\mathbf T_n$ are uniserial and uniquely determined by their simple socle\nand length (in $\\mathbf T_n$).\nWe denote by $M_{i,i+l+1}$ an indecomposable with socle $M_{i,i+2}$\nand length $l$. Then $\\tau M_{i,i+l+1}  = M_{i-1,i+l}$,\nand the AR-quiver of the tube $\\mathbf T_n$ is as in Figure~\\ref{fig:ARquiverTn}\n(with the columns on the left- and right-hand sides identified).\nNote that each indecomposable object has a unique name $M_{ij}$\n(with $j-i\\geq 2$) if we insist that $i$ lies in $\\{0,1,\\ldots ,n-1\\}$.\n\n\\begin{figure}[ht]\n$$\\xymatrix@R=5pt@C=4pt{\n&&&&&&&&&& \\\\\n&& \\vdots && && \\vdots && \\vdots  \\\\\nM_{n-1,3} \\ar@{--}[rr] \\ar@{.}[dd] \\ar@{.}[uu] \\ar[dr] && M_{0,4} && \\cdots && M_{n-3,1} \\ar@{--}[rr] \\ar[dr] && M_{n-2,2} \\ar@{--}[rr] \\ar[dr] && M_{n-1,3} \\ar@{.}[dd] \\ar@{.}[uu] \\\\\n& M_{0,3} \\ar@{--}[l] \\ar@{--}[r] \\ar[ur] \\ar[dr] && && && \\ar@{--}[l] M_{n-2,1} \\ar@{--}[rr] \\ar[ur] \\ar[dr] && M_{n-1,2} \\ar@{--}[r] \\ar[ur] \\ar[dr] & \\\\\nM_{0,2} \\ar@{--}[rr] \\ar[ur] && M_{1,3} && \\cdots && M_{n-2,0} \\ar@{--}[rr] \\ar[ur] && M_{n-1,1} \\ar@{--}[rr] \\ar[ur] && M_{0,2}\n}$$\n\\caption{The AR-quiver of $\\mathbf T_n$}\n\\label{fig:ARquiverTn}\n\\end{figure}\n\nAn additive subcategory of $\\mathbf T$ is said to be of {\\em finite type} if it contains only finitely many indecomposable objects.\nOtherwise it is said to be of {\\em infinite type}.\nSome particular subcategories of $\\mathbf T$ are important\nfor this paper.\nFor each fixed $i$ in $\\{0, \\dots n-1 \\}$, we consider the additive subcategory\nwhose objects are the indecomposable objects $M_{i,i+t}$, for all $t>1$.\nThis is called a {\\em ray}, and is denoted $\\mathcal R_i$.\nDually, for each  $i$ in $\\{0, \\dots n-1\\}$, we consider the\nadditive subcategory whose indecomposable objects are all indecomposable\nobjects $M_{i-u,i}$, for all $u>1$. This is called a {\\em coray}, and\ndenoted $\\mathcal C_i$.\n\nFor each $i$ in $\\{0, \\dots, n-1\\}$, and each\n$t>1$, the {\\em wing} $\\mathcal W_{i,i+t}$ is the\nadditive subcategory of $\\mathbf T$ whose indecomposable objects are the\n$M_{j,j+u}$ with $u\\geq 2$, $i \\leq j \\leq i+t-2$ and\n$j+u \\leq i+t$. It contains a unique indecomposable object $M_{i,i+t}$\nof maximal length. The objects $M_{i,i+2}$ and $M_{i+t-2,i+t}$ lie at the\nbottom left and bottom right corners of the wing, respectively, in the\ncollection of vertices corresponding to the indecomposable objects of the wing in the\nAR-quiver of $\\mathbf T$.\nFor $t \\leq 1$ we let $\\mathcal W_{i,i+t}$ be the zero subcategory.\nWe also denote the wing of an indecomposable object $X$ by $\\mathcal W_X$.\n\nDue to the following well-known fact, we can apply\nresults from the previous section in our analysis of $\\mathbf T$.\n\n\\begin{lemma}\\label{wingsareA}\nFor $u \\leq n+1$ the wing $\\mathcal W_{i,i+u}$ in $\\mathbf T_n$ is equivalent to\nthe module category $\\operatorname{mod} KQ$, where $Q$ is a linearly\noriented quiver of Dynkin type $A_{u-1}$. If $u\\leq n$ the equivalence\nis exact.\n\\end{lemma}\n\nNote that if $u=n+1$, the object corresponding to the projective-injective\nindecomposable module in $\\operatorname{mod} KQ$ under the above equivalence is not $\\operatorname{Ext}$-projective in the wing.\n\nWe denote by $\\operatorname{Mod} \\Lambda$ the category of all left\n$\\Lambda$-modules. Let $\\varinjlim \\mathbf T$ be the subcategory of $\\operatorname{Mod} \\Lambda$\nwhose objects are direct limits of filtered direct systems of objects in $\\mathbf T$.\nNote that $\\varinjlim \\mathbf T$ contains the  {\\em Pr\\\"{u}fer modules}, i.e.\\ the modules $M_{i,\\infty}$, $i=0,2,\\ldots n-1$ obtained as direct limits\nof the indecomposable objects in the rays, i.e.\\\n$$M_{i,\\infty} = \\varinjlim (M_{i,i+2} \\to M_{i,i+3} \\to M_{i,i+4} \\to \\cdots).$$\nLet $\\varprojlim \\mathbf T$ be the subcategory of $\\operatorname{Mod} \\Lambda$\nwhose objects are inverse limits of filtered inverse systems of objects in $\\mathbf T$.\nThis category contains the {\\em adic} modules, which are\nobtained as inverse limits along a coray:\n$$M_{-\\infty,i} = \\varprojlim (\\cdots \\to M_{i-4,i} \\to M_{i-3,i} \\to\nM_{i-2,i} ).$$\n\nLet $\\overline{\\mathbf T}$ be the subcategory of $\\operatorname{Mod} \\Lambda$ whose\nobjects are all filtered direct limits or filtered inverse limits of\nobjects in $\\mathbf T$. This category clearly contains $\\varinjlim \\mathbf T$\nand $\\varprojlim \\mathbf T$.\nWe extend the definition of $\\sigma$ to Pr\\\"{u}fer and adic modules\nwith the convention that $\\sigma(\\pm \\infty)=\\pm \\infty$: note that any Pr\\\"{u}fer\nmodule has a unique name $M_{i,\\infty}$ if we take $i$ in $\\{0,1,\\ldots ,n-1\\}$, and\nsimilarly for adic modules.\n\nRecall that a $\\Lambda$-module $M$ is \\emph{pure-injective} if\nthe canonical map $M \\rightarrow DDM$ is a split monomorphism.\nFor background on pure-injective modules, and other definitions,\nsee e.g.~\\cite{jensenlenzing89} or~\\cite{prest09}.\nIt can be shown (see \\cite{bk1}) that the category $\\overline{\\mathbf T}$\nhas the following properties:\n\n\\begin{itemize}\n\\item[$\\bullet$] All objects are pure-injective as $\\Lambda$-modules.\n\\item[$\\bullet$] Any object is determined by its indecomposable direct summands.\n\\item[$\\bullet$] The indecomposables in $\\overline{\\mathbf T}$ are exactly the indecomposables\n  $M_{ij}$ in $\\mathbf T$, the Pr{\\\"u}fer modules $M_{i,\\infty}$ and the\n  adic modules $M_{-\\infty,i}$.\n\\end{itemize}\n\nA module $M$ in $\\overline{\\mathbf T}$ is called {\\em rigid} if\n$\\operatorname{Ext}^1_{\\Lambda}(M,M) = 0$. Note that since all objects in\n$\\overline{\\mathbf T}$ are pure-injective, this definition is equivalent to having\n$\\operatorname{Ext}^1_{\\Lambda}(M',M'') = 0$ for all indecomposable direct summands $M',M''$ of\n$M$. Now, a rigid module $M$ in $\\overline{\\mathbf T}$ is called {\\em maximal\n  rigid}, if $\\operatorname{Ext}^1_{\\Lambda}(M \\amalg X,M \\amalg X) = 0$ for an\nindecomposable $X$ in $\\overline{\\mathbf T}$ implies\nthat $X$ is isomorphic to a direct summand in $M$.\n\n\\subsection{Geometric model}\\label{cm}\n\nWe now give a geometric model for $\\overline{\\mathbf T}$. This extends the\nknown geometric model for $\\mathbf T$~\\cite{bama,warkentin}.\nConsider an annulus $\\mathbb A(n)$ with $n$ marked points on the outer boundary.\nThe points are labelled $0, 1, \\dots, n-1$, and arranged anticlockwise\n(see Figure~\\ref{fig:annulus}).\n\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{0}{$\\scriptstyle 0$}\n\\psfrag{1}{$\\scriptstyle 1$}\n\\psfrag{2}{$\\scriptstyle 2$}\n\\psfrag{n-1}{$\\scriptstyle n-1$}\n\\includegraphics[width=2.5cm]{annulus.eps}\n\\caption{An annulus with $n$ marked points on its outer boundary.}\n\\label{fig:annulus}\n\\end{figure}\n\nLet $\\mathbb U(n)$ denote the universal cover of $\\mathbb A(n)$, with marked\npoints corresponding to $\\mathbb{Z}$ (and with $0,1, \\dots, n-1$\nlying in a fundamental domain). See Figure~\\ref{fig:universalcover}.\n\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{-1}{$\\scriptstyle -1$}\n\\psfrag{0}{$\\scriptstyle 0$}\n\\psfrag{1}{$\\scriptstyle 1$}\n\\psfrag{2}{$\\scriptstyle 2$}\n\\psfrag{n-1}{$\\scriptstyle n-1$}\n\\psfrag{n}{$\\scriptstyle n$}\n\\includegraphics[height=2cm]{universalcover.eps}\n\\caption{The universal cover of the annulus in Figure~\\ref{fig:annulus}.}\n\\label{fig:universalcover}\n\\end{figure}\n\nFor integers $i,j$ with $i+2 \\leq j$, let $[i,j]$ denote the arc in\n$\\mathbb U(n)$ with starting point $i$ and ending point $j$, oriented from $i$ to $j$.\nWe also allow arcs which have only one end-point: arcs of the form $[i,\\infty]$\n(respectively, $[-\\infty,j]$ which start\nat $i$ (respectively, end at $j$) and are oriented in the positive $x$ direction.\nSee Figure~\\ref{fig:infinitearc1}.\n\n\\begin{figure}\n\\psfragscanon\n\\psfrag{i}{$i$}\n\\psfrag{iinf}{$[i,\\infty]$}\n\\psfrag{i-inf}{$[-\\infty,i]$}\n\\includegraphics[width=12cm]{infinitearc1.eps}\n\\caption{Infinite arcs in $\\mathbb U(n)$.}\n\\label{fig:infinitearc1}\n\\end{figure}\n\nLet $\\pi_n([i,j])$ denote the corresponding arc in $\\mathbb A(n)$ and\nlet $\\widetilde{A}=\\widetilde{\\mathcal A}(\\mathbb A(n))$ denote the set of (isotopy classes\nof) such arcs. It contains the set $\\mathcal A=\\mathcal A(\\mathbb A(n))$ of arcs of the form $\\pi_n([i,j])$\nwith $i,j$ finite. The map $\\psi:\\widetilde{\\mathcal A}\\rightarrow \\operatorname{ind} \\overline{\\mathbf T}$\nsending $\\pi_n([i,j])$ to $M_{ij}$ is a bijection.\nThe infinite arcs are displayed in Figure~\\ref{fig:infinitearc2}.\n\n\\begin{figure}\n\\psfragscanon\n\\psfrag{i}{$i$}\n\\psfrag{iinf}{$\\pi_n([i,\\infty])$}\n\\psfrag{i-inf}{$\\pi_n([-\\infty,i])$}\n\\includegraphics[height=3cm]{infinitearc2.eps}\n\\caption{Infinite arcs in $\\mathbb A(n)$.}\n\\label{fig:infinitearc2}\n\\end{figure}\n\nDefine a quiver with vertices given by the elements in $\\mathcal A$ and arrows:\n$$\\pi_n([i,j]) \\to \\pi_n([i,j+1])$$ and $$\\pi_n([i,j]) \\to\n\\pi_n([i+1,j]) \\text{ (if $j \\neq i+2)$} $$\nDefining a translate using the formula $\\tau(\\pi_n([i,j])) = \\pi_n([i-1,j-1])$,\nthis becomes a translation quiver. We call this the \\emph{(translation) quiver\nof $\\mathcal A(\\mathbb A(n))$}.\n\nBy~\\cite[Lemma 2.5]{bama},~\\cite[4.18]{warkentin} (or, using unoriented arcs,~\\cite[\\S3.4]{bz},~\\cite{gehrig}), we have:\n\n\\begin{proposition}\nThe restriction of $\\psi$ to $\\mathcal A$ gives an isomorphism between\nthe translation quiver of $\\mathcal A(\\mathbb A(n))$ and the AR-quiver of $\\mathbf T_n$.\n\\end{proposition}\n\nNote that the convention that $M_{\\sigma^k(i),\\sigma^k(j)}=M_{ij}$ corresponds exactly\nto the fact $\\pi_n([\\sigma^k(i),\\sigma^k(j)])=\\pi_n([i,j])$ for any integer $k$.\n\nFor arcs $\\alpha, \\beta$ in $\\mathcal A(\\mathbb A(n))$, let\n$I(\\alpha, \\beta)$ be the minimum number of intersections between arcs in the isotopy classes $\\alpha$ and\n$\\beta$, not allowing non-transverse or multiple intersections. Similarly we let $I^+(\\alpha, \\beta)$ (resp. $I^-(\\alpha, \\beta)$) denote the number of\npositive (resp. negative) crossings between $\\alpha$ and $\\beta$ (see Figure~\\ref{fig:neg} for an example of a negative crossing).\nWe will now prove the following result:\n\n\\begin{theorem}\\label{exts}\nGiven indecomposable objects $M_{ij}$ and $M_{i'j'}$ in\n$\\overline{\\mathbf T}$. Then:\n$$\\operatorname{Ext}^1(M_{ij},M_{i'j'}) \\cong \\prod_{I^-(\\pi_n([i,j]),\n  \\pi_n([i',j']))} K .$$\n\\end{theorem}\n\nIn the case where $i,i',j,j'$ are all finite, the result is proved in\n\\cite[Thm.\\ 3.7]{bama},~\\cite[Thm. 4.23]{warkentin}.\nSee also \\cite{bz} for further results in this direction.\nWe first recall some results we will need:\n\n\\begin{lemma}\\label{limits}\nLet $X,Y$ be arbitrary $\\Lambda$-modules,\n$(X_j)_j$ an arbitrary filtered direct system of modules and\n$(Y_j)_j$ an arbitrary filtered inverse system of modules.\n\\begin{itemize}\n\\item[(a)] $\\operatorname{Hom}(\\varinjlim X_j, Y) \\simeq \\varprojlim \\operatorname{Hom}(X_j,Y)$.\n\\item[(b)] If $X$ is finitely generated, then $\\operatorname{Hom}(X,\\varinjlim Y_j)\\simeq \\varinjlim \\operatorname{Hom}(X,Y_j)$\n\\item[(c)] If the $Y_j$ are finitely generated, then\n$\\varprojlim Y_j\\simeq D\\varinjlim DY_j$.\n\\item[(d)] If $Y$ is pure-injective, then\n$\\operatorname{Ext}^1(\\varinjlim X_j, Y) \\simeq \\varprojlim \\operatorname{Ext}^1(X_j,Y)$.\n\\item[(e)] If the $Y_j$ are finitely generated, then\n$\\operatorname{Ext}^1(X, \\varprojlim Y_j) \\simeq \\varprojlim \\operatorname{Ext}^1(X,Y_j)$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof} For (a) see, for example,~\\cite{trlifaj03}.\nFor (b), see~\\cite[Lemma 1.6]{krausesolberg03}\nor~\\cite[Sect. 1.5]{crawleyboevey98}.\nFor (c), we have, using part (a):\n\\begin{align*}\nD\\varinjlim DY_j &= \\operatorname{Hom}(\\varinjlim \\operatorname{Hom}(Y_j,K),K) \\\\\n&\\simeq \\varprojlim \\operatorname{Hom}(\\operatorname{Hom}(Y_j,K),K)\n\\simeq \\varprojlim DDY_j \\simeq \\varprojlim Y_j,\n\\end{align*}\nas required.\nPart (d) is proved in~\\cite[Prop.\\ I.10.1]{auslander78}.\nFor (e), we recall that $\\operatorname{Ext}^1(X,DY)\\simeq \\operatorname{Ext}^1(Y,DX)$ for all modules\n$X$ and $Y$. Using parts (c) and (d)\nand the fact~\\cite[Prop.\\ 4.3.29]{prest09} that $DX$ is pure-injective for\nany module $X$, we have:\n\\begin{align*}\n\\operatorname{Ext}^1(X,\\varprojlim Y_j) &\\simeq \\operatorname{Ext}^1(X,D(\\varinjlim DY_j))\n\\simeq \\operatorname{Ext}^1(\\varinjlim DY_j,DX) \\\\\n&\\simeq \\varprojlim \\operatorname{Ext}^1(DY_j,DX)\n\\simeq \\varprojlim \\operatorname{Ext}^1(X,DDY_j) \\\\\n&\\simeq \\varprojlim \\operatorname{Ext}^1(X,Y_j),\n\\end{align*}\nand (e) is shown.\n\\end{proof}\n\nWe also need the following (see e.g.~\\cite[Sect.\\ 3.1]{crawleyboevey98}).\n\n\\begin{lemma} \\label{ARformula}\nFor modules $X$ and $Y$ with $X$ finitely generated, we have\n$D\\operatorname{Ext}^1(X,Y)\\cong \\operatorname{Hom}(Y,\\tau X)$ and $\\operatorname{Ext}^1(Y,X)\\cong D\\operatorname{Hom}(\\tau^{-1}X,Y)$.\n\\end{lemma}\n\nNote that if $X,Y$ are finitely generated, then the first formula can also\nbe written $\\operatorname{Ext}^1(X,Y)\\cong D\\operatorname{Hom}(Y,\\tau X)$. We recall the following\n(see, for example,~\\cite[p46]{ringel00}).\n\n\\begin{lemma} \\label{homvanishing}\n\\begin{itemize}\n\\item[(a)] If $X$ is a Pr\\\"{u}fer module and $Y$ is a finitely\ngenerated module then $\\operatorname{Hom}(X,Y)=0$.\n\\item[(b)] If $X$ is a finitely generated module and $Y$ is an adic\nmodule then $\\operatorname{Hom}(X,Y)=0$.\n\\end{itemize}\n\\end{lemma}\n\nWith arguments as in \\cite{bama}, the crossing numbers can now be computed as follows.\nRecall that $\\sigma \\colon \\mathbb{Z} \\to  \\mathbb{Z}$ is the function $i \\mapsto i+n$.\n\n\\begin{proposition}\\label{crossingnumbers}\nWe have the following:\n\\begin{itemize}\n\\item[(a)]  $I^-(\\pi_n([i,\\infty]), \\pi_n([a,b])) = \\mid \\{m \\in\n  \\mathbb{Z}  \\colon a < \\sigma^m(i)< b \\} \\mid$;\n\\item[(b)] $I^-(\\pi_n([a,b]), \\pi_n([i,\\infty])) = I^+(\\pi_n([i,\\infty]), \\pi_n([a,b])) = 0$;\n\\item[(c)] $I^-(\\pi_n([-\\infty,j]), \\pi_n([a,b]))= 0$;\n\\item[(d)] $I^- (\\pi_n([a,b]),\\pi_n([-\\infty,j])) =\nI^+(\\pi_n([-\\infty,j]), \\pi_n([a,b]))\n= \\mid \\{m \\in  \\mathbb{Z}  \\colon a < \\sigma^m(i)< b \\} \\mid$;\n\\item[(e)] $I^-(\\pi_n([i,\\infty]), \\pi_n([-\\infty, i'])) = \\aleph_0$,\n  for all $i,i'$ in $\\{0,\\dots, n-1\\}$;\n\\item[(f)] $ I^-(\\pi_n([i,\\infty]),\n    \\pi_n([i',\\infty])) = 0$ for all $i,i'$ in $\\{0,\\dots, n-1\\}$;\n\\item[(g)] $I^-(\\pi_n([-\\infty, i]), \\pi_n([i',\\infty])) =  0$ for all $i,i'$ in $\\{0,\\dots, n-1\\}$;\n\\item[(h)] $I^-(\\pi_n([-\\infty,\ni]), \\pi_n([-\\infty, i'])) = 0$ for all  $i,i'$ in $\\{0,\\dots, n-1\\}$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}[Proof of Theorem \\ref{exts}]\nWe need to compute $\\operatorname{Ext}^1(X,Y)$ for all pairs of indecomposables\n$X,Y$ in $\\overline{\\mathbf T}$, and compare these with the crossing-numbers\nfrom Proposition \\ref{crossingnumbers}.\nWe first determine $\\operatorname{Ext}^1(M_{i,\\infty}, M_{ab})$. By Lemma~\\ref{ARformula},\n$$\\operatorname{Ext}^1(M_{i,\\infty},M_{ab})\\cong D\\operatorname{Hom}(M_{a+1,b+1},M_{i,\\infty}).$$\nAny morphism $f:M_{a+1,b+1}\\rightarrow M_{i,\\infty}$ must factor through the\nunique submodule of $M_{i,\\infty}$ of length $b-a-1$, which is isomorphic to\n$M_{i,i+b-a}$.\nHence $\\operatorname{Hom}(M_{a+1,b+1},M_{i,\\infty})\\cong \\operatorname{Hom}(M_{a+1,b+1},M_{i,i+b-a})$.\nThus $\\operatorname{Ext}^1(M_{i,\\infty},M_{ab})$ equals the dimension of this last space,\ni.e.\\ the number of times the simple top $M_{b-1,b+1}$ of $M_{a+1,b+1}$\nappears as a composition factor in $M_{i,i+b-a}$, which, using arguments\nas in~\\cite{bama}, is given by\n$$|\\{n\\in\\mathbb{Z}\\,:\\,a<\\sigma^n(i)<b\\}|.$$\n\nUsing a dual argument, we obtain:\n$$\\dim \\operatorname{Ext}^1(M_{i,-\\infty}, M_{ab}) =\n\\mid \\{n \\in \\mathbb{Z} \\colon a < \\sigma^n(i) <b \\} \\mid.$$\n\nBy Lemma~\\ref{ARformula},\n$$D\\operatorname{Ext}^1(M_{ab}, M_{i, \\infty}) \\simeq \\operatorname{Hom}(M_{i,\\infty},M_{a-1, b-1}).$$\nWe see that $\\operatorname{Ext}^1(M_{ab},M_{i,\\infty})=0$ by Lemma~\\ref{homvanishing}(a).\n\nSimilarly, by Lemma~\\ref{ARformula}, we have that\n$$\\operatorname{Ext}^1(M_{-\\infty,i}, M_{ab}) \\simeq D\\operatorname{Hom}(\\tau^{-1}M_{ab},M_{-\\infty,i})=0,$$\nusing Lemma~\\ref{homvanishing}(b).\n\nWe next compute $\\operatorname{Ext}^1(M_{i,\\infty}, M_{-\\infty,i'})$. Using Lemma\n\\ref{limits}(e), we have that\n$$\\operatorname{Ext}^1(M_{i,\\infty}, M_{-\\infty,i'}) = \\operatorname{Ext}^1(M_{i,\\infty},\n\\varprojlim M_{j,i'}) \\simeq \\varprojlim \\operatorname{Ext}^1(M_{i,\\infty}, M_{j,i'}),$$\nwhere $j$ is the dummy variable in the limit.\nThe maps\n$$\\operatorname{Ext}^1(M_{i,\\infty}, M_{j,i'}) \\to \\operatorname{Ext}^1(M_{i,\\infty},\nM_{j-1,i'})$$\nare surjective. As $j$ tends to $-\\infty$, the dimension of\n$\\operatorname{Ext}^1(M_{i,\\infty}, M_{j,i'})$ is unbounded (by the above formula for it).\nTherefore, this limit evaluates to $\\prod_{\\aleph_0} K$.\n\nIt is shown in \\cite[Lemma 2.7]{bk1}, that, for all $i,i'$\nin $\\{0,\\dots ,n-1\\}$, all of \\linebreak\n$\\operatorname{Ext}^1(M_{-\\infty, i}, M_{i',\\infty})$, \n$\\operatorname{Ext}^1(M_{i,\\infty},M_{i',\\infty})$ and $\\operatorname{Ext}^1(M_{-\\infty, i}, M_{-\\infty,i'})$\nvanish.\n\nComparing the crossing numbers with the dimensions of the corresponding\nExt-groups concludes the proof of the theorem.\n\\end{proof}\n\n\\subsection{Reflection}\n\nRecall (see \\cite{bk1}) that there is a map  $M \\mapsto M^{\\vee}$, which gives a bijection\non the indecomposable objects in $\\overline {\\mathbf T}$. With our notation,\nthe map is given by $M_{ij} \\mapsto M_{-j,-i}$ for $i,j$ in\n$\\mathbb{Z} \\cup \\{ \\pm \\infty \\}$.\n\nFor a subcategory $\\mathcal X$ of $\\overline{\\mathbf T}$, we let $\\mathcal X^{\\vee}$ denote the\nsubcategory $\\{X^{\\vee} \\mid X \\in \\mathcal X \\}$. The map has the following properties.\n\\begin{lemma}\\label{properties}\n\\begin{itemize}\n\\item[(a)] $\\dim \\operatorname{Hom}(X,Y) = \\dim \\operatorname{Hom}(Y^{\\vee}, X^{\\vee})$ for\n  $X,Y$ in $\\overline{\\mathbf T}$;\n\\item[(b)] $\\operatorname{Ext}^1(X,Y) = 0$ if and only if $\\dim \\operatorname{Ext}^1(Y^{\\vee},\n  X^{\\vee}) = 0$ for\n  $X,Y$ in $\\overline{\\mathbf T}$;\n\\item[(c)]$(\\tau X)^{\\vee} = \\tau^{-1} X^{\\vee}$ for all\n  indecomposables $X$ in $\\mathbf T$.\n\\item[(d)] For each object $X$ in $\\overline \\mathbf T$, we have\n$(X^{{\\perp}_{\\scriptscriptstyle H}})^{\\vee} = {^{{\\perp}_{\\scriptscriptstyle H}}(X^{\\vee})}$ and\n$(X^{\\vee})^{{\\perp}_{\\scriptscriptstyle E}} = (^{{\\perp}_{\\scriptscriptstyle E}}X)^{\\vee}$.\n\\end{itemize}\n\\end{lemma}\n\nWe will also need the following.\n\n\\begin{lemma} \\label{l:twistgen}\nFor a maximal rigid object $U$ in $\\overline{\\mathbf T}$, we have\n\\begin{itemize}\n\\item[(a)] $(\\operatorname{Gen} {U}\\cap \\mathbf T)^{\\vee} = \\operatorname{Cogen}({U}^{\\vee})\\cap \\mathbf T$;\n\\item[(b)] $(\\operatorname{Cogen} {U}\\cap \\mathbf T)^{\\vee} = \\operatorname{Gen}({U}^{\\vee})\\cap \\mathbf T$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nThis follows directly from the definition of the map using\nLemma~\\ref{homvanishing} and the fact that\nan indecomposable $X$ in $\\mathbf T$ is generated by a maximal\nrigid object ${U}$ if and only if it is generated by an indecomposable direct\nsummand in ${U}$ (see also \\cite{bk2}).\n\\end{proof}\n\n\\section{Torsion pairs and maximal rigid objects}\n\nIn this section we give an improvement of a result from\n\\cite{bk1}, and discuss a combinatorial interpretation.\nThe idea is to link maximal rigid objects in $\\overline{\\mathbf T}$ with\ntorsion pairs in $\\mathbf T$.\n\n\\subsection{A bijection}\nA rigid object in $\\overline{\\mathbf T}$ is said to be of {\\em Pr\\\"{u}fer type}\nif it has a Pr\\\"{u}fer module as a direct summand, and it is said to be\nof {\\em adic type} if it has an adic module as a direct summand.\n\nAn object ${U}$ in $\\varinjlim \\mathbf T$ is maximal rigid if and only if it is\ncotilting in the category $\\varinjlim \\mathbf T$, in the sense of Colpi~\\cite{colpi99}.\nThis is proved in~\\cite{bk2} (see~\\cite[1.10]{bk2} and note that,\nby~\\cite[Lemma 1.2]{bk2}, a finitely presented $\\tilde{\\Lambda}_n$-module\nis maximal rigid if and only if it is a tilting module, where\n$\\tilde{\\Lambda}_n$ denotes the completion of the path algebra of an\noriented $n$-cycle).\n\nWe consider equivalence classes of maximal rigid\nobjects in $\\overline{\\mathbf T}$, where two maximal rigid objects are considered to\nbe equivalent if they have the same indecomposable direct summands.\nThe following is proved in~\\cite{bk1}.\n\n\\begin{proposition}\\label{propbk1}\nLet ${U}$ be a rigid object in $\\overline{\\mathbf T}$.\n\\begin{itemize}\n\\item[(a)] ${U}$ is maximal rigid in $\\overline{\\mathbf T}$ if and only if it has $n$ pairwise nonisomorphic indecomposable direct summands.\n\\item[(b)] If ${U}$ is maximal rigid, it is either of Pr{\\\"u}fer type or of adic type, but not both.\n\\item[(c)] A rigid object of Pr{\\\"u}fer type lies in the subcategory\n$\\varinjlim \\mathbf T \\subset \\overline{\\mathbf T}$.\n\\end{itemize}\n\\end{proposition}\n\nWe will give a parallel result concerning torsion pairs in $\\mathbf T$.\nA torsion pair $(\\mathcal T,\\mathcal F)$ in $\\mathbf T$ is said to be of {\\em ray type} if\n$\\mathcal F$ contains at least one ray of $\\mathbf T$.  It is said to be of {\\em coray type} if\n$\\mathcal T$ contains at least one coray of $\\mathbf T$.\nA class of objects (or subcategory) $\\mathcal X$ of a category $\\mathcal C$ is said to be {\\em generating},\nif for each map $f$ in $\\mathcal C$, there is an object $X$ in $\\mathcal X$ with\n$\\operatorname{Hom}(X,f) \\neq 0$. Dually, one can define {\\em cogenerating} classes\nof objects, i.e.\\ $\\mathcal X$ is cogenerating if, for each map $f$ in $\\mathcal C$,\nthere is an object $X$ in $\\mathcal X$ with $\\operatorname{Hom}(f,X)\\neq 0$.\n\n\\begin{lemma}\\label{onefinite}\nLet $(\\mathcal T,\\mathcal F)$ be a torsion pair in $\\mathbf T$. Then it is not the case that\nboth $\\mathcal T$ and $\\mathcal F$ are of finite type.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\mathcal T$ and $\\mathcal F$ are both of finite type.\nBy the definition of a torsion pair, every object $M$ in $\\mathbf T$\nmust be the middle term of an exact sequence\n\\begin{equation*}\n0 \\to T \\to M \\to F \\to 0,\n\\end{equation*}\nwhere $T\\in \\mathcal T$ and $F\\in \\mathcal F$.\nSince $\\mathbf T$ is serial, if $M$ is indecomposable then so are\n$T$ and $F$. It follows that $\\mathbf T$ itself is of finite\ntype, a contradiction.\nHence, we cannot have that both $\\mathcal T$ and $\\mathcal F$ are of finite type.\n\\end{proof}\n\nWe next prove that the torsion pairs with $\\mathcal T$ of infinite type\nare exactly those of coray type. For an indecomposable object $X$ in $\\mathbf T$\nwe write $\\mathcal C_X$ for the coray containing $X$ and $\\mathcal R_X$ for the ray containing\n$X$.\n\n\\begin{lemma}\\label{coray-type}\nLet $(\\mathcal T,\\mathcal F)$ be a torsion pair in $\\mathbf T$, where $\\mathbf T$ has rank $n$.\nAssume $\\mathcal T$ is of infinite type. Then the following hold:\n\\begin{itemize}\n\\item[(i)] $\\mathcal T$ contains an indecomposable object $X$, with $l(X) = n$.\n\\item[(ii)] $\\mathcal F$ is of finite type.\n\\item[(iii)] $\\mathcal T$ contains the coray $\\mathcal C_X$.\n\\item[(iv)] $\\mathcal T$ cogenerates $\\mathbf T$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nFirst note that since $\\mathcal T$ is of infinite type, there is no limit on the length of the indecomposable objects\nin $\\mathcal T$. Since $\\mathcal T$ is closed under factor objects, it must therefore contain an indecomposable object with $l(X) = n$.\nHence (i) holds.\nSince $\\mathcal T$ is closed under factor objects, the\nindecomposable objects in the coray $\\mathcal C_X$ below $X$ are contained in\n$\\mathcal T$; more precisely $\\operatorname{ind}(\\mathcal C_X) \\cap \\{Y \\mid l(Y) \\leq n \\} \\subset \\mathcal T$.\nAssume first that $n>1$.\nLet $X'$ be the (uniquely defined) indecomposable object in $\\mathbf T$ such that there is an irreducible\nmonomorphism $X' \\to X$.\nSince $l(X) = n$, we have that $\\operatorname{Hom}(X,Y) = 0$ for an indecomposable object\n$Y$ in $\\mathbf T$ if and only if $Y$ is in the wing\n$\\mathcal W_{X'}$. By the definition of a torsion pair, we have that\n$\\mathcal F$ is contained in $\\mathcal W_{X'}$ so is of finite type.\nIf $n=1$, we have $\\operatorname{Hom}(X,Y)\\neq 0$ for any indecomposable\nobject $Y$ in $\\mathbf T$, so $\\mathcal F$ is the zero subcategory and (ii) holds.\nBy Lemma~\\ref{l:tpclosure}, (iii) holds, while (iv) is a direct consequence of (iii).\n \\end{proof}\n\nWe state the dual version of Lemma \\ref{coray-type}.\n\n\\begin{lemma}\\label{ray-type}\nLet $(\\mathcal T,\\mathcal F)$ be a torsion pair in $\\mathbf T$, where $\\mathbf T$ has rank $n$.\nAssume $\\mathcal F$ is of infinite type. Then the following hold:\n\\begin{itemize}\n\\item[(i)] $\\mathcal F$ contains an indecomposable object $X$, with $l(X) = n$.\n\\item[(ii)] $\\mathcal T$ is of finite type.\n\\item[(iii)] $\\mathcal F$ contains the ray $\\mathcal R_X$.\n\\item[(iv)] $\\mathcal F$ generates $\\mathbf T$.\n\\end{itemize}\n\\end{lemma}\n\nCombining Lemma \\ref{onefinite}, \\ref{coray-type} and  \\ref{ray-type}, we obtain the following direct consequence.\n\n\\begin{corollary}\\label{cortorsion}\nLet $(\\mathcal T,\\mathcal F)$ be a torsion pair in $\\mathbf T$.\n\\begin{itemize}\n\\item [(a)]  The following are equivalent\n\\begin{itemize}\n\\item[(i)] $\\mathcal T$ is of infinite type;\n\\item[(ii)] $\\mathcal T$ contains a coray;\n\\item[(iii)] $\\mathcal T$ cogenerates $\\mathbf T$;\n\\item[(iv)] $\\mathcal F$ is of finite type.\n\\end{itemize}\n\\item[(b)]\n$({\\mathcal T, \\mathcal F})$ is either of ray or of coray type (and not both).\n\\end{itemize}\n\\end{corollary}\n\nMoreover, by Lemma~\\ref{l:tpclosure} and a direct application of Lemma \\ref{properties},\nwe obtain the following.\n\n\\begin{lemma}\nThe map $M \\to M^{\\vee}$,  maps a torsion pair $(\\mathcal T, \\mathcal F)$ to a\ntorsion pair $(\\mathcal F^{\\vee}, \\mathcal T^{\\vee})$. Moreover if  $(\\mathcal T, \\mathcal F)$ is of\nray-type, then  $(\\mathcal F^{\\vee}, \\mathcal T^{\\vee})$ is of coray-type and vice-versa.\n\\end{lemma}\n\nFor a maximal rigid object ${U}$ of Pr{\\\"u}fer type, consider the\nsubcategory $\\mathcal F_{U} = {^{{\\perp}_{\\scriptscriptstyle E}}{U}} \\cap \\mathbf T$. We set $\\mathcal T_{U} =\n{^{{\\perp}_{\\scriptscriptstyle H}}(\\mathcal F_{U})} \\cap \\mathbf T.$\nFor a maximal rigid object ${U}$ of\nadic type, we define\n$\\mathcal T_{U} ={U} ^{{\\perp}_{\\scriptscriptstyle E}} \\cap \\mathbf T$ and $\\mathcal F_{U} =\n(\\mathcal T_{U})^{{\\perp}_{\\scriptscriptstyle H}} \\cap \\mathbf T$.\nWe have the following reformulation of a result of~\\cite{bk2}:\n\n\\begin{theorem}\\label{propbk2}\nThe map ${U} \\mapsto (\\mathcal T_{U}, \\mathcal F_{U})$ gives\na one-to-one correspondence between equivalence classes of\nmaximal rigid objects in $\\varinjlim \\mathbf T$ and\ntorsion pairs in $ \\mathbf T$ with the property that $\\mathcal F$ generates $\\mathbf T$.\n\\end{theorem}\n\nNow, using Lemma \\ref{properties}, we obtain a commutative square (*):\n$$\n\\xymatrix{\n\\{\\text{maximal rigid objects of Pr{\\\"u}fer type in } \\overline{\\mathbf T} \\}\n\\ar[r]  \\ar[d] & \\{\\text{torsion pairs of ray-type in } \\mathbf T \\} \\ar[d] \\\\\n\\{\\text{maximal rigid objects of adic type in } \\overline{\\mathbf T} \\}\n\\ar[r]  & \\{\\text{torsion pairs of coray-type in } \\mathbf T  \\}\n}\n$$\nwhere the horizontal maps are given by ${U} \\mapsto (\\mathcal T_{U}, \\mathcal F_{U})$ and\nthe vertical maps are induced by $M \\mapsto M^{\\vee}$.\n\nAs a direct consequence of Lemma~\\ref{properties} (as\nalso observed in \\cite{bk1}), we have that the left vertical map in (*)\nis a bijection. We have already observed that the right vertical map\nis a bijection. The upper horizontal map is bijective by Theorem~\\ref{propbk2},\ncombining with Proposition~\\ref{propbk1} and Lemma~\\ref{ray-type},\nand it follows that the lower horizontal map is bijective.\n\nCombining the commutative diagram of bijections (*) with Corollary \\ref{cortorsion}\nwe obtain the following improvement of Theorem \\ref{propbk2}.\n\n\\begin{theorem} \\label{t:mainbijection}\nThe map ${U} \\mapsto (\\mathcal T_{U}, \\mathcal F_{U})$ gives a bijection between\n\\begin{itemize}\n\\item[$\\bullet$] Equivalence classes of maximal rigid objects in $\\overline{\\mathbf T}$\n\\item[$\\bullet$] Torsion pairs $(\\mathcal T,\\mathcal F)$ in $\\mathbf T$, and\n\\end{itemize}\n\\end{theorem}\n\n\\begin{corollary}\nThe number of torsion pairs in $\\mathbf T$ is $2\\binom{2n-1}{n-1}$.\n\\end{corollary}\n\n\\begin{proof}\nBy~\\cite[2.4, 5.2 and B.1]{bk2},\nthe number of cotilting objects in $\\varinjlim \\mathbf T$\nis $\\binom{2n-1}{n-1}$. The result then follows from Proposition~\\ref{propbk1},\nthe above diagram of maps, and Theorem~\\ref{t:mainbijection}.\n\\end{proof}\n\nNote that the above results hold in the case $n=1$: in this case there are two maximal\nrigid objects: the unique Pr\\\"{u}fer and adic modules, and two corresponding torsion pairs,\n$(0,\\mathbf T)$ and $(\\mathbf T,0)$ respectively.\n\n\\subsection{Alternative and explicit descriptions of $\\mathcal T_{U}$ and $\\mathcal F_{U}$}\n\nWe give alternative and more explicit descriptions for the subcategories\n$\\mathcal T_{U}$ and $\\mathcal F_{U}$ corresponding to a maximal rigid object ${U}$ in\n$\\overline{\\mathbf T}$.\n\nThe following observation is useful.\n\n\\begin{lemma}\\label{perps}\nLet $\\mathbf T$ be a tube of rank $n$. Then we have:\n\\begin{itemize}\n\\item[(a)]$^{{\\perp}_{\\scriptscriptstyle E}} M_{i,\\infty} \\cap \\mathbf T = \\mathbf T$;\n\\item[(b)]$M_{i,\\infty}^{{\\perp}_{\\scriptscriptstyle E}} \\cap \\mathbf T = \\mathcal W_{i,i+n}$\n\\end{itemize}\n\\end{lemma}\n\nWe now give an explicit description of the torsion pair $(\\mathcal T_{U},\\mathcal F_{U})$\ncorresponding to a  maximal rigid module ${U}$. We first of\nall give a combinatorial lemma concerning wings, which is easy to\ncheck.\n\n\\begin{lemma} \\label{l:wings}\nLet $0 \\leq i_0 < i_1 < \\cdots < i_{k-1} \\leq n-1$ be integers.\nThen\n\\begin{itemize}\n\\item[(i)] We have:\n$$\\bigcap_{r=0}^{k-1} \\mathcal W_{i_r,i_r + n} = \\coprod_{r=0}^{k-1} \\mathcal W_{i_r,i_{r+1} }$$\n(where the subscripts are interpreted modulo $k$ and, for $r=k-1$, we\ninterpret $i_{r+1}=i_0$ as $i_0+n$).\n\\item[(ii)] The wing $\\mathcal W_{i_r,i_{r+1} }$ is zero if and only if\n$i_{r+1} - i_{r}= 1$.\n\\item[(iii)] The wings $\\mathcal W_{i_r,i_{r+1}}$ do not overlap,\ni.e.\\ each indecomposable object in $\\mathbf T$ belongs to at most one $\\mathcal W_{i_r,i_{r+1}}$.\n\\item[(iv)] If $X$ and $Y$ lie in different wings among the\n  $\\mathcal W_{i_r,i_{r+1} } $, then $\\operatorname{Ext}^1(X,Y) = 0 = \\operatorname{Ext}^1(Y,X)$.\n\\end{itemize}\n\\end{lemma}\n\nFor an example illustrating Lemma~\\ref{l:wings}(i),\nwith $n=10$, $k=4$, $i_0=0$, $i_1=4$, $i_2=7$ and $i_3=8$, see\nFigure~\\ref{fig:wingsexample}.\n\n\\begin{figure}[hb]\n$$\\xymatrix@!0@=0.5cm{\n&&&&&&&&&& &&&&&&&&&& \\\\\n&& \\scriptstyle M_{7,17} && \\scriptstyle M_{8,18} &&&& \\scriptstyle M_{0,10} && &&&&&& \\scriptstyle M_{4,14} &&&& \\\\\n\\circ && *=0{\\circ} \\ar@{-}[rrrrrrrrdddddddd]+0 && *=0{\\circ} \\ar@{-}[rrrrrrrrdddddddd]+0 && \\circ && *=0{\\circ} \\ar@{-}[rrrrrrrrdddddddd]+0 && \\circ && \\circ && \\circ && *=0{\\circ} \\ar@{-}[rrrrdddd]+0 && \\circ && \\circ \\\\\n& \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ \\\\\n*=0{\\circ} \\ar@{-}[rruu]+0 && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ \\\\\n& \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ \\\\\n*=0{\\circ} \\ar@{-}[rrrruuuu]+0 \\ar@{-}[rrrrdddd]+0 && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ \\\\\n& \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\\\\n\\circ && \\bullet & \\scriptstyle M_{0,4} & \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ && \\circ \\\\\n& \\bullet && \\bullet && \\circ && \\circ && \\bullet & \\scriptstyle M_{4,7} & \\circ && \\circ && \\circ && \\circ && \\circ && \\\\\n*=0{\\bullet} \\ar@{.}[uuuuuuuuuu]+0 \\ar@{-}[rrrrrrrruuuuuuuu]+0 && \\bullet && \\bullet && \\circ && *=0{\\bullet} \\ar@{-}[rrrrrrrruuuuuuuu]+0 && \\bullet && \\circ && *=0{\\circ} \\ar@{-}[rrrrrruuuuuu]+0 &&\n*=0{\\bullet} \\ar@{-}[rrrruuuu]+0 && \\circ && *=0{\\bullet} \\ar@{.}[uuuuuuuuuu]+0 \\\\\n\\scriptstyle M_{0,2} && \\scriptstyle M_{1,3} && \\scriptstyle M_{2,4} && \\scriptstyle M_{3,5} && \\scriptstyle M_{4,6} && \\scriptstyle M_{5,7} && \\scriptstyle M_{6,8} && \\scriptstyle M_{7,9} && \\scriptstyle M_{8,10} && \\scriptstyle M_{9,11} && \\scriptstyle M_{0,2}\n}$$\n\\caption{The indecomposable objects in the intersection of the four wings\n$\\mathcal W_{0,10},\\mathcal W_{4,14},\\mathcal W_{7,17}$ and $\\mathcal W_{8,18}$ in a tube of rank\n$10$ corresponding to a maximal rigid object of Pr\\\"{u}fer\ntype. The objects lying in the intersection are indicated by filled-in circles.\nThe intersection coincides with $\\mathcal W_{0,4}\\amalg \\mathcal W_{4,7}\\amalg \\mathcal W_{8,10}$\n(note that the wing $\\mathcal W_{i_3,i_4}=\\mathcal W_{7,8}$ is zero).}\n\\label{fig:wingsexample}\n\\end{figure}\n\nIn the following proposition and the sequel, we adopt the same convention\nfor the wings as in Lemma~\\ref{l:wings}(i).\n\n\\begin{proposition} \\label{p:ftgt}\nLet ${U}$ be maximal rigid in $\\overline{\\mathbf T}$.\n\\begin{itemize}\n\\item[(a)] If ${U}$ is of Pr{\\\"u}fer type, then\n$$\n(\\mathcal T_{U}, \\mathcal F_{U})  = ( {^{{\\perp}_{\\scriptscriptstyle H}}{U}} \\cap \\mathbf T, {^{{\\perp}_{\\scriptscriptstyle E}}{U}} \\cap \\mathbf T)\n  = (\\tau^{-1} (\\operatorname{Gen} {U} \\cap \\mathbf T), \\operatorname{Cogen} {U} \\cap \\mathbf T).\n$$\n\\item[(b)]\nAssume ${U}$ is of Pr{\\\"u}fer type with Pr{\\\"u}fer summands\n$M_{i_r,\\infty}$\nfor $r=0, \\dots, k-1$ where $0 \\leq i_0 < i_1 < \\cdots < i_{k-1}\n\\leq n-1$. Then $$(\\mathcal T_{U}, \\mathcal F_{U}) = (\\coprod_{r=0}^{k-1}\n  \\mathcal T_r, (\\coprod_{r=0}^{k-1}\\mathcal F_r) \\amalg \\mathcal F_{\\infty})),$$ where\n$(\\mathcal T_r, \\mathcal F_r)$ is a torsion pair in $\\mathcal W_{i_r, i_{r+1}+1}$ with\n$\\mathcal F_r$ containing all of the projective objects in $\\mathcal W_{i_r,i_{r+1}+1}$\nand $\\mathcal F_{\\infty} = \\coprod_{r=0}^{k-1} \\mathcal R_{i_r}$.\n\n\\item[(c)]  If ${U}$ is of adic type, then\n$$(\\mathcal T_{U}, \\mathcal F_{U})  = ({U}\n  ^{{\\perp}_{\\scriptscriptstyle E}} \\cap \\mathbf T, {U} ^{{\\perp}_{\\scriptscriptstyle H}} \\cap \\mathbf T) =\n  (\\operatorname{Gen} {U} \\cap \\mathbf T, \\tau(\\operatorname{Cogen} {U} \\cap \\mathbf T)).$$\n\\item[(d)] Assume ${U}$ is of adic type with adic summands $M_{i_r,\\infty}$\n  for $r=0, \\dots, k-1$,  where $0 \\leq i_0 < i_1 < \\cdots < i_{k-1}\n\\leq n-1$. Then $$(\\mathcal T_{U}, \\mathcal F_{U}) = ((\\coprod_{r=0}^{k-1}\n  \\mathcal T_r) \\amalg \\mathcal T_{\\infty}, \\coprod_{r=0}^{k-1}\\mathcal F_r),$$ where\n$(\\mathcal T_r, \\mathcal F_r)$ is a torsion pair in $\\mathcal W_{i_r, i_{r+1}+1}$ with\n$\\mathcal T_r$ containing all of the injective objects in $\\mathcal W_{i_r,i_{r+1}+1}$ and\n$\\mathcal T_{\\infty} = \\coprod_{r=0}^{k-1} \\mathcal C_{i_{r+1}+1}$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nWe give the details for (a) and (b), while statements (c) and (d) can\nbe proved similarly (or using Lemmas~\\ref{properties} and ~\\ref{l:twistgen}).\nLet ${U}$ be maximal rigid of Pr{\\\"u}fer type in $\\overline{\\mathbf T}$.\nOur aim is to compute $\\mathcal F_{U}={}^{{\\perp}_{\\scriptscriptstyle E}} {U}\\cap \\mathbf T$ and then\n$\\mathcal T_{U}={}^{{\\perp}_{\\scriptscriptstyle H}}\\mathcal F_{U}\\cap \\mathbf T$. We use the Pr\\\"{u}fer direct summands\nof ${U}$ in order to compute $\\mathcal F_{U}$ more precisely in terms of a set\nof wings in $\\mathbf T$. We then use this to compute $\\mathcal T_{U}$ using the theory of\ntorsion pairs in type A (see Section~\\ref{s:Dynkin}).\n\nLet ${U}_{\\mathbf T}$ be the direct sum of all indecomposable\ndirect summands of ${U}$ which are finitely generated.\nLet $M_{i_0,\\infty},\\ldots ,M_{i_{k-1},\\infty}$\nwith $0\\leq i_0<i_1<\\cdots <i_{k-1}\\leq n-1$ be the\nindecomposable Pr{\\\"u}fer summands of ${U}$.\n\nBy Lemma \\ref{perps}, we then have that the finite part ${U}_{\\mathbf T}$\nlies in $\\cap_{r=0}^{k-1} \\mathcal W_{i_r,i_r + n}$, which coincides with\n$\\coprod_{r=0}^{k-1} \\mathcal W_{i_r,i_{r+1}}$ by Lemma~\\ref{l:wings}(i).\nWe draw attention to the fact that the wings\nin the statement of the proposition are slightly larger than these ---\nthis will become clearer later in the proof.\n\nHence, we consider the decomposition ${U}_{\\mathbf T} =\\amalg_{r=0}^{k-1} {U}_r$, where each ${U}_r$ is\nrigid in $\\mathcal W_{i_r,i_{r+1}}$ (note that some of the ${U}_r$ then\nmight be $0$, i.e.\\ in the case when $i_{r+1}-i_r=1$).\nThen it follows easily from Lemma~\\ref{l:wings}(iv) that ${U}_r$ is\nmaximal rigid in $\\mathcal W_{i_r,i_{r+1}}$.\nSince ${U}_r$ is maximal rigid, its restriction to the abelian\ncategory $\\mathcal W_{i_r,i_{r+1}}$ is also cotilting. Our aim is to use this\ndecomposition of ${U}$ to compute $\\mathcal F_{U}$.\n\nIf $\\mathcal W_{i_r,i_{r+1}}$ is nontrivial, $M_{i_r,i_{r+1}}$\nis necessarily an indecomposable direct summand of ${U}_r$.\nIt is then straightforward to check that\n$$^{{\\perp}_{\\scriptscriptstyle E}}(\\amalg_{r=0}^{k-1} M_{i_r,i_{r+1}} \\cap \\mathbf T) =\n(\\coprod_{r=0}^{k-1}\\mathcal W_{i_r,i_{r+1}}) \\amalg \\coprod_{r=0}^{k-1} \\mathcal R_{i_r},$$\nand from this it follows that $\\mathcal F \\subset (\\coprod_{r=0}^{k-1}\\mathcal W_{i_r,i_{r+1}})\n\\amalg (\\coprod_{r=0}^{k-1} \\mathcal R_{i_r})$.\n\nWe also have that\n$^{{\\perp}_{\\scriptscriptstyle E}}{U}_{\\mathbf T}  \\cap \\mathcal W_{i_r,i_{r+1}} = {^{{\\perp}_{\\scriptscriptstyle E}}{U}_r}  \\cap \\mathcal W_{i_r,i_{r+1}}$,\nwhich gives us the following description of $\\mathcal F$:\n$$\\mathcal F = (\\coprod_{r=0}^{k-1} (^{{\\perp}_{\\scriptscriptstyle E}}{U}_r)\\cap \\mathbf T)\n\\amalg (\\coprod_{r=0}^{k-1} \\mathcal R_{i_r}).$$\nSince ${U}_r$ is cotilting in $\\mathcal W_{i_r,i_{r+1}}$, we have that\n$^{{\\perp}_{\\scriptscriptstyle E}}{U}_r  \\cap \\mathcal W_{i_r,i_{r+1}} = \\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}}} {U}_r$.\nFrom this it follows that\n$$\\mathcal F = (\\coprod_{r=0}^{k-1} \\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}}} {U}_r)\n\\amalg \\coprod_{r=0}^{k-1} \\mathcal R_{i_r}.$$\nNoting that $\\operatorname{Cogen} M_{i,\\infty} \\cap \\mathbf T = \\mathcal R_i$,\nwe see that $\\mathcal F = \\operatorname{Cogen} {U} \\cap \\mathbf T$ as claimed in (a).\n\nNext, we consider slightly larger wings, $\\mathcal W_{i_r,i_{r+1}+1}$ in order to\nobtain the description of $\\mathcal F$ in (b).\nFor each $r$, define $\\widetilde{{U}}_r = {U}_r \\amalg M_{i_r, i_{r+1}+1}$. Then\n$\\widetilde{{U}}_r$ is a cotilting module in $\\mathcal W_{i_r, i_{r+1}+1}$.\nNote that if $U$ has exactly one Pr\\\"{u}fer summand, we need to\nregard $\\mathcal W_{i_r,i_{r+1}+1}=\\mathcal W_{i_0,i_0+n+1}$ as being a type $A$ category non-exactly embedded in the tube, since the image of the projective-injective object is not rigid.\n\nNote that\n$$\\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}+1}}\\widetilde{{U}}_r=(\\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}}}{U}_r)\\amalg\n\\operatorname{add} M_{i_r,i_{r+1}}.$$\nWe can thus rewrite $\\mathcal F$ as follows:\n$$\\mathcal F = (\\coprod_{r=0}^{k-1} \\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}+1}} \\widetilde{{U}}_r)\n\\amalg (\\coprod_{r=0}^{k-1} \\mathcal R_{i_r}).$$\nSo, setting $\\mathcal F_r=\\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}+1}} \\widetilde{{U}}_r$, we obtain a\ndescription of $\\mathcal F$ as claimed in (b).\n\nNext, we compute $\\mathcal T$.\nBy definition, we have that $\\mathcal T= {^{{\\perp}_{\\scriptscriptstyle H}}\\mathcal F} \\cap \\mathbf T$. Furthermore,\n$$^{{\\perp}_{\\scriptscriptstyle H}}( \\amalg_{r=0}^{k-1} M_{i_r,i_{r+1}}) \\cap \\mathbf T= (\\coprod_{r=0}^{k-1}\n\\tau^{-1} \\mathcal W_{i_r,i_{r+1}}) \\amalg (\\coprod_{r_0}^{k-1} \\mathcal C_{i_{r+1} +1}).$$\n\nNoting that\n$\\operatorname{Hom}(M_{l,i_{r+1}+1}, M_{i_r,i_{r+1}+1}) \\not= 0$ for $l \\leq i_r$, we see\nthat\n$$\\mathcal T \\subset \\coprod_{r=0}^{k-1} \\tau^{-1} (\\mathcal W_{i_r,i_{r+1}}) \\subset \\coprod_{r=0}^{k-1} \\mathcal W_{i_r,i_{r+1}+1}.$$\nFrom this, it follows that\n$\\mathcal T = \\coprod_{r=0}^{k-1}(^{{\\perp}_{\\scriptscriptstyle H}}\\mathcal F_r \\cap \\mathcal W_{i_r,i_{r+1}+1})$, noting\nthat $$\\mathcal F_r= \\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}+1}} \\widetilde{{U}}_r.$$\nUsing the fact that $\\widetilde{{U}}_r$ is cotilting in\n$\\mathcal W_{i_r,i_{r+1}+1}$, combined with Corollary~\\ref{cor:Gen-Cogen},\nwe see that\n$\\mathcal T = \\coprod_{r=0}^{k-1}\\mathcal T_r$, where\n$$\\mathcal T_r = {^{{\\perp}_{\\scriptscriptstyle H}}\\widetilde{{U}}_r} \\cap\n\\mathcal W_{i_r,i_{r+1}+1} = \\operatorname{Gen}_{\\mathcal W_{i_r, i_{r+1}+1}} \\tau_{\\mathcal W_{i_r,i_{r+1}+1}}^{-1} \\widetilde{{U}}_r$$\nis the torsion part of the torsion pair in $\\mathcal W_{i_r,i_{r+1}+1}$ with torsion-free\npart $\\mathcal F_r$.\nWe have that $^{{\\perp}_{\\scriptscriptstyle H}}(\\amalg_{r=0}^{k-1} M_{i_r, \\infty}) =\n\\coprod_{r=0}^{k-1} \\mathcal W_{i_r, i_{r+1}}$, and therefore $\\mathcal T= {^{{\\perp}_{\\scriptscriptstyle H}}{U}} \\cap\n\\mathbf T$.\n\nSince $\\operatorname{Gen} M_{i, \\infty} \\cap \\mathbf T$ is zero for all $i$ and\nsince we have\n$\\tau_{\\mathcal W_{i_r,i_{r+1}+1}}^{-1} \\widetilde{{U}}_r  = \\tau_{\\mathcal W_{i_r,i_{r+1}+1}}^{-1} {U}_r=\n\\tau^{-1} {U}_r$, we see that\n$\\mathcal T = \\tau^{-1} {\\operatorname{Gen} {U} \\cap \\mathbf T}$.\n\nSee Figure~\\ref{fig:wing} for an illustration of one of the wings\n$\\mathcal W_{i_r,i_{r+1}+1}$. We have that $\\mathcal F_r=\\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}+1}}\\widetilde{{U}}_r$.\nThe indecomposable objects in $\\mathcal F_r$ apart from $M_{i_r,i_{r+1}+1}$ lie\nin the left hand shaded triangle. Also, $\\mathcal T_r=\\operatorname{Gen}_{\\mathcal W_{i_r,i_{r+1}+1}}\\tau_{\\mathcal W_{i_r,i_{r+1}+1}}^{-1}\\widetilde{{U}}_r$,\nand the indecomposable objects in $\\mathcal T_r$ lie in the right hand shaded triangle.\n\\begin{figure}\n\\psfragscanon\n\\psfrag{M1}{$\\scriptstyle M_{i_r,i_r+2}$}\n\\psfrag{M2}{$\\scriptstyle M_{i_r+1,i_r+3}$}\n\\psfrag{M3}{$\\scriptstyle M_{i_{r+1}-2,i_{r+1}}$}\n\\psfrag{M4}{$\\scriptstyle M_{i_{r+1}-1,i_{r+1}+1}$}\n\\psfrag{M5}{$\\scriptstyle M_{i_r,i_{r+1}}$}\n\\psfrag{M6}{$\\scriptstyle M_{i_r,i_{r+1}+1}$}\n\\psfrag{M7}{$\\scriptstyle M_{i_r,i_{r+1}+1}$}\n\\includegraphics[width=7cm]{wing.eps}\n\\caption{A wing $\\mathcal W_{i_r,i_{r+1}+1}$.}\n\\label{fig:wing}\n\\end{figure}\n\\end{proof}\n\n\\subsection{The maximal rigid module corresponding to a torsion pair}\nIn this section we give an explicit description of the inverse of the bijection\n${U}\\mapsto (\\mathcal T_{U},\\mathcal F_{U})$ in Theorem~\\ref{t:mainbijection} between maximal rigid\nobjects in $\\overline{\\mathbf T}$ and torsion pairs in $\\mathbf T$.\n\n\\begin{lemma} \\label{l:limG}\nLet $(\\mathcal T,\\mathcal F)$ be a torsion pair in $\\mathbf T$. Then an indecomposable object\nin $\\varinjlim(\\operatorname{ind} \\mathcal F)$ either lies in $\\operatorname{ind}\\mathcal F$\nor it is a Pr\\\"{u}fer module which is the direct limit along a ray in\n$\\operatorname{ind} \\mathcal F$.\n\\end{lemma}\n\\begin{proof}\nIt is clear that any indecomposable in $\\mathcal F$ or Pr\\\"{u}fer module which\nis the direct limit along a ray in $\\operatorname{ind} \\mathcal F$ lies in\n$\\operatorname{ind}(\\varinjlim \\mathcal F)$.\nSo suppose that $X=\\varinjlim X_j$ is an indecomposable\nin $\\overline{\\mathbf T}$, where each $X_j$ is in $\\operatorname{ind} \\mathcal F$.\nFirstly note that $X$ cannot be an adic module, since the adic modules\ndo not lie in $\\varinjlim \\mathbf T$.\nNext, for any object ${T}$ in $\\mathcal T$, we have\n$$\\operatorname{Hom}({T},X)=\\operatorname{Hom}({T},\\varinjlim X_j)\\simeq \\varinjlim \\operatorname{Hom}({T},X_j)=0,$$\nusing Lemma~\\ref{limits}(b), since each $X_j$ lies in $\\mathcal F$.\nIt follows that if $X$ is in $\\operatorname{ind} \\mathbf T$, we have that $X$ lies in $\\operatorname{ind}\\mathcal F$.\nThe only other possibility is if $X$ is a Pr\\\"{u}fer module. If $Y$ is\nany indecomposable object in the corresponding ray in $\\mathbf T$,\nwe have an embedding $Y\\rightarrow X$. Hence $\\operatorname{Hom}({T},Y)=0$ for any object\n${T}$ in $\\mathcal T$, so $Y$ lies in $\\mathcal F$. The result follows.\n\\end{proof}\n\n\\begin{proposition} \\label{p:TfromFG}\n\\begin{enumerate}\n\\item[(a)] Let $(\\mathcal T,\\mathcal F)$ be a torsion pair of ray type. Then $(\\mathcal T,\\mathcal F)$ can\nbe written in the form $(\\mathcal T_{U},\\mathcal F_{U})$, where ${U}$\nis the direct sum of the indecomposable objects in\n\\begin{equation}\n\\varinjlim (\\operatorname{ind} \\mathcal F) \\cap (\\varinjlim (\\operatorname{ind} \\mathcal F))^{{\\perp}_{\\scriptscriptstyle E}}.\n\\label{e:rayinverse}\n\\end{equation}\nCompare~\\cite[Thm.\\ 1.5]{bk1}.\n\\item[(b)] Let $(\\mathcal T,\\mathcal F)$ be a torsion pair of coray type. Then $(\\mathcal T,\\mathcal F)$ can\nbe written in the form $(\\mathcal T_{U},\\mathcal F_{U})$, where ${U}$\nis the direct sum of the indecomposable objects in\n\\begin{equation}\n\\varprojlim (\\operatorname{ind} \\mathcal T) \\cap {}^{{\\perp}_{\\scriptscriptstyle E}}(\\varprojlim (\\operatorname{ind} \\mathcal T)).\n\\label{corayinverse}\n\\end{equation}\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nWe only consider (a); the proof of (b) is similar.\nBy Theorem~\\ref{t:mainbijection} and Proposition~\\ref{p:ftgt}(b) and its proof,\nthere is a maximal rigid object ${U}$ in $\\overline{\\mathbf T}$ of Pr\\\"{u}fer type such\nthat $(\\mathcal T,\\mathcal F)=(\\mathcal T_{U},\\mathcal F_{U})$ and $\\mathcal T_{U}$, $\\mathcal F_{U}$ have the following description.\nLet the Pr{\\\"u}fer summands of ${U}$ be $M_{i_r,\\infty}$\nfor $r=0, \\dots, k-1$ where $0 \\leq i_0 < i_1 < \\cdots < i_{k-1}\n\\leq n-1$. Then $$(\\mathcal T_{U}, \\mathcal F_{U}) = (\\coprod_{r=0}^{k-1}\n  \\mathcal T_r, (\\coprod_{r=0}^{k-1}\\mathcal F'_r) \\amalg \\mathcal F_{\\infty}),$$ where\n$\\mathcal T_r,\\mathcal F_{\\infty}$ are as in Proposition~\\ref{p:ftgt}(b) and\n$\\mathcal F'_r=\\operatorname{Cogen}_{\\mathcal W_{i_r,i_{r+1}}} {U}_r$, where ${U}|_{\\mathbf T}=\\amalg_{r=0}^{k-1} {U}_r$\nwith ${U}_r$ in $\\mathcal W_{i_r,i_{r+1}}$.\n\nBy~\\eqref{e:extprojectives} in Section~\\ref{s:Dynkin},\n\\begin{align*}\n\\operatorname{ind} {U}_r &=  \\{X\\in \\operatorname{ind}\\mathcal F'_r\\,:\\, \\operatorname{Ext}^1(Y,X)=0\\text{\\ for all\\ } Y\\in \\mathcal F'_r\\} \\\\\n&= \\{X\\in \\operatorname{ind}\\mathcal F'_r \\,:\\, \\operatorname{Ext}^1(Y,X)=0\\text{\\ for all\\ } Y\\in \\operatorname{ind}\\mathcal F\\}.\n\\end{align*}\nNote that $\\mathcal F_{\\infty}=\\coprod_{r=0}^{k-1} \\mathcal R_{i_r} \\subset \\operatorname{ind}\\mathcal F$.\nIt follows that any indecomposable object $X$ in $\\mathbf T$ which satisfies $\\operatorname{Ext}^1(Y,X)=0$\nfor all $Y$ in $\\operatorname{ind} \\mathcal F$ must lie inside some wing $\\mathcal W_{i_r,i_{r+1}}$,\nand hence in some $\\mathcal F'_r$. Therefore\n\\begin{align*}\n\\operatorname{ind} {U}|_{\\mathbf T} &= \\operatorname{ind}(\\amalg_{r=0}^{k-1} {U}_r) \\\\\n&= \\{X\\in \\operatorname{ind} \\mathcal F\\,:\\,\\operatorname{Ext}^1(Y,X)=0\\text{\\ for all\\ } Y\\in \\operatorname{ind}\\mathcal F\\}.\n\\end{align*}\n\nNext, by Lemma~\\ref{l:limG}, an indecomposable object in\n$\\varinjlim(\\operatorname{ind} \\mathcal F)$ either lies in $\\operatorname{ind}\\mathcal F$ or is the direct limit\nalong a ray in $\\mathcal F$, i.e.\\ it is a Pr\\\"{u}fer summand $M_{i_r,\\infty}$ of ${U}$.\nIf $Y$ is one of the latter summands and $X$ lies in $\\operatorname{ind}\\mathcal F$, we have\n$D\\operatorname{Ext}^1(Y,X)\\simeq \\operatorname{Hom}(X,\\tau Y)=0$,\nby Lemmas~\\ref{ARformula} and~\\ref{homvanishing}. Since the Pr\\\"{u}fer summands\nthemselves satisfy\n$\\operatorname{Ext}^1(M_{i_r,\\infty},M_{i_{r'},\\infty})=0$\nfor all $r,r'$, the result follows.\n\\end{proof}\n\n\\section{Geometric interpretation in the tube case}\n\\label{s:tubegeometric}\n\\subsection{Short exact sequences}\n\nLet $[i,j],[i',j']$ be arcs in $\\mathcal A(\\mathbb U(n))$, giving\nrise to arcs $\\pi_n([i,j]),\\pi_n([i',j'])$ in $\\mathcal A(\\mathbb A(n))$ with corresponding\nindecomposable objects $M_{ij},M_{i'j'}$ in ${U}$.\nSuppose that $I^-_{\\mathbb U(n)}([i,j],[i'+kn,j'+kn])$ (i.e.\\ the intersection\nnumber between arcs $[i,j]$ and $[i'+kn,j'+kn]$ in $\\mathbb U(n)$) is equal to $1$ for some\n$k$ in $\\mathbb{Z}$. Then, if $j'+kn>i+1$, we have four objects in the Auslander-Reiten quiver of $\\mathbf T$ as shown in Figure~\\ref{fig:diamond}(a); if $j'+kn=i+1$, we have three\nobjects as shown in Figure~\\ref{fig:diamond}(b).\n\n\\begin{figure}\n\\centering\n\\subfigure[Case $j'+kn>i+1$.]{\n$$\\begin{xy} *!D\\xybox{\n\\xymatrix@!0@=1.2cm{\n& M_{i'+kn,j} \\ar[dr] & \\\\\nM_{i'j'}=M_{i'+kn,j'+kn} \\ar[dr] \\ar[ur] && M_{ij} \\\\\n& M_{i,j'+kn} \\ar[ur]\n}}\\end{xy}$$}\n\\subfigure[Case $j'+kn=i+1$.]{\n$$$$\\begin{xy} *!D\\xybox{\n\\xymatrix@!0@=1.2cm{\n& M_{i'+kn,j} \\ar[dr] & \\\\\nM_{i'j'}=M_{i'+kn,j'+kn} \\ar[ur] && M_{ij} \\\\\n&\n}}\\end{xy}$$}\n\\caption{Objects and paths in the Auslander-Reiten quiver of $\\mathbf T$ corresponding\n to an intersection in $\\mathbb U(n)$ between arcs $[i,j]$ and $[i'+kn,j'+kn]$.}\n\\label{fig:diamond}\n\\end{figure}\n\nIn the case $j'+kn>i+1$, this corresponds to a non-split short exact sequence:\n\\begin{equation}\n0 \\rightarrow M_{i',j'} \\overset{f}{\\longrightarrow} M_{i'+kn,j}\\amalg M_{i,j'+kn}\n\\overset{g}{\\rightarrow} M_{ij}\\rightarrow 0,\n\\label{e:diamondses}\n\\end{equation}\nand in the case $j'+kn=i+1$, it corresponds to a short exact sequence:\n\\begin{equation}\n0 \\rightarrow M_{i',j'} \\overset{f}{\\longrightarrow} M_{i'+kn,j}\\overset{g}{\\rightarrow} M_{ij}\\rightarrow 0\n\\label{e:diamondsesb}\n\\end{equation}\nin $\\mathbf T$. As in~\\cite[Remark 4.25]{warkentin},\nthese can be interpreted geometrically in $\\mathbb U(n)$: see Figure~\\ref{fig:tubesplit-sum}.\n\n\\begin{figure}\n\\psfragscanon\n\\psfrag{i'+kn}{$\\scriptstyle i'+kn$}\n\\psfrag{i}{$\\scriptstyle i$}\n\\psfrag{j'+kn}{$\\scriptstyle j'+kn$}\n\\psfrag{j}{$\\scriptstyle j$}\n\\subfigure[Case $j'+kn>i+1$.]{\\includegraphics[width=3.2cm]{tubesplitsum.eps}}\n\\quad \\quad \\quad\n\\subfigure[Case $j'+kn=i+1$.]{\\includegraphics[width=3.25cm]{tubesplitsum2.eps}}\n\\caption{Non-split extensions in $\\mathbf T$ represented geometrically.}\n\\label{fig:tubesplit-sum}\n\\end{figure}\n\nIf $j'+kn>i+1$, write $f=\\begin{pmatrix} f_1 \\\\ f_2 \\end{pmatrix}$ and\n$g=\\begin{pmatrix} g_1 & g_2 \\end{pmatrix}$.\nThen $f_1$ and $g_2$ are monomorphisms and $f_2$ and $g_1$ are epimorphisms.\nThe $f_i$ and $g_i$ are uniquely determined up to a choice\nof scalars. If $j'+kn=i+1$, $f$ is a monomorphism and $g$ is an epimorphism,\nagain uniquely determined up to a choice of scalars.\n\n\\begin{lemma}\nAny non-split short exact sequence with first term $M_{i'j'}$ and last\nterm $M_{ij}$ has the same form as~\\eqref{e:diamondses} or~\\eqref{e:diamondsesb}.\n\\end{lemma}\n\n\\begin{proof}\nLet\n\\begin{equation}\n0\\rightarrow M_{i'j'} \\overset{u}{\\rightarrow} E \\overset{v}{\\rightarrow} M_{ij} \\rightarrow 0\n\\label{e:typicalses}\n\\end{equation}\nbe an arbitrary non-split short exact sequence with first term $M_{i'j'}$ and\nlast term $M_{ij}$.\nBy Lemma~\\ref{l:uniserialclosure} we may write $E=E_1\\amalg E_2$ where $E_1$ is\nindecomposable and such that we have that, decomposing $u=\\begin{pmatrix} u_1 \\\\ u_2 \\end{pmatrix}$ and $v=(v_1,v_2)$, we have that $u_1$ a monomorphism.\nSince the sequence is not split, $u_1$ is not an isomorphism, so, denoting the length\nof an object $M$ in $\\mathbf T$ by $\\ell(M)$, we have\n$$\\ell(E_2)=\\ell(M_{ij})+\\ell(M_{i'j'})-\\ell(E_1)<\\ell(M_{ij}).$$\nSince $v_2$ is not an epimorphism, $v_1$ must be an epimorphism, again using\nLemma~\\ref{l:uniserialclosure}.\n\n\\emph{Case (i)}: Suppose first that $v_1u_1\\not=0$.\nIf an integer $k$ is such that $i'+kn\\leq i$ and $i+2\\leq j'+kn\\leq j$,\nthere is a homomorphism in $\\mathbf T$ from $M_{i'j'}$ to $M_{ij}$ obtained by\ncomposing an epimorphism from $M_{i'j'}$ to $M_{i,j'+kn}$ with a monomorphism\nfrom $M_{i,j'+kn}$ to $M_{ij}$ (i.e.\\ the two maps in the lower edges of the\ndiamond in Figure~\\ref{fig:diamond}).\n\nIt is easy to check that the homomorphisms of this kind (allowing $k$ to vary)\nform a basis of $\\operatorname{Hom}(M_{i'j'},M_{ij})$. Since $vu=0$ and $v_1u_1\\not=0$ and\n$v_1u_1$ is a scalar multiple of such a basis element, there must be an indecomposable\nsummand $X$ of $E_2$ such that $v_Xu_X$ is a scalar multiple of $v_1u_1$\n(where $u_X$, $v_X$ are the corresponding components of $u_2$, $v_2$).\nBut\n\\begin{equation}\n\\ell(X)\\leq \\ell(M_{ij})+\\ell(M_{i'j'})-\\ell(E_1),\n\\label{e:lowpoint}\n\\end{equation}\nand there is a unique path from $M_{i'j'}$ to $M_{ij}$ through\nsuch an $X$ giving rise to $g_1f_1$\n(i.e.\\ with $X=M_{i,j'+kn}$) from which it follows that\nwe have equality in~\\eqref{e:lowpoint} and thus that $E_2=X$ is\nindecomposable and $u_2$ is an epimorphism and $v_2$ is a monomorphism.\nIt follows that the short exact sequence~\\eqref{e:typicalses} is of\nthe form~\\eqref{e:diamondses} up to a choice of scalars.\n\n\\emph{Case (ii)}: Now assume that $v_1u_1=0$. This implies that\n$\\ell(E_1)\\geq \\ell(M_{ij})+\\ell(M_{i'j'})$,\nbut we also have $\\ell(E_1)\\leq \\ell(E_1\\amalg E_2)=\\ell(M_{ij})+\\ell(M_{i'j'})$,\nso we must have equality and $E=E_1$ is indecomposable.\nIt follows that~\\eqref{e:typicalses} is of the form~\\eqref{e:diamondsesb} up\nto a choice of scalars. The proof is complete.\n\\end{proof}\n\n\\begin{definition}\nWe call a collection $\\SS$ of arcs in $\\mathcal A(\\mathbb A(n))$ an \\emph{oriented Ptolemy\ndiagram} (in $\\mathbb A(n)$) if, whenever $\\pi_n([i,j])$ and $\\pi_n([i',j'])$ lie in\n$\\SS$ with $i'<i<j'<j$ then $\\pi_n([i,j'])$ (when $j'>i+1$) and $\\pi_n([i',j])$\nalso lie in $\\SS$ (see related definitions in Section~\\ref{s:geometricmodel} and~\\cite{ng,hjr}).\n\\end{definition}\n\nWe note that the additive closure of a collection of indecomposable objects in $\\mathbf T$ is closed\nunder extensions if and only if the corresponding collection of arcs is an oriented Ptolemy diagram in $\\mathbb A(n)$. It is also easy to check that if $\\pi_n([i,j])$\nis an arc in $\\mathcal A(\\mathbb A(n))$, then the indecomposable quotients of $M_{ij}$ are the $M_{i'j}$ where\n$i\\leq i'\\leq j-2$, i.e.\\ arcs in $\\mathbb A(n)$ corresponding to arcs in $\\mathbb U(n)$ with the same ending point and with starting point weakly to the right of $i$.\nCall these the \\emph{left-shortenings} of $\\pi_n([i,j])$. Similarly, submodules\nare given by right-shortenings.\n\nWe make the following remark, which follows from Proposition~\\ref{crossingnumbers}.\nRecall that $\\widetilde{\\mathcal A}(\\mathbb A(n))$ denotes $\\mathcal A = \\mathcal A(\\mathbb A(n))$ extended to include the homotopy classes of the arcs $\\pi_n([i,\\infty])$ and $\\pi_n([-\\infty,i])$.\n\n\\begin{remark}\nThe bijection $\\psi:\\pi_n([i,j])\\mapsto M_{ij}$ between $\\widetilde{\\mathcal A}(\\mathbb A(n))$ and $\\operatorname{ind}(\\overline{\\mathbf T})$\ninduces a bijection between maximal noncrossing collections of arcs in $\\mathbb A(n)$ (including\nthe infinite arcs) and maximal rigid objects in $\\overline{\\mathbf T}$.\n\\end{remark}\n\nWe can now describe the conditions on collections of arcs appearing in torsion\npairs in $\\mathbf T$, using the above and Lemmas~\\ref{l:tpclosure} and~\\ref{l:uniserialclosure}.\n\n\\begin{proposition}\n\\begin{enumerate}\n\\item[(a)]\nA collection $\\SS$ of arcs in $\\mathcal A(\\mathbb A(n))$ corresponds to the torsion part\nof a torsion pair in $\\mathbf T$ if and only if\n$\\SS$ is an oriented Ptolemy diagram in $\\mathbb A(n)$ and $\\SS$ is closed under left-shortening.\n\\item[(b)]\nA collection $\\SS$ of arcs in $\\mathcal A(\\mathbb A(n))$ corresponds to the torsion-free part\nof a torsion pair in $\\mathbf T$ if and only if $\\SS$ is an oriented Ptolemy diagram in $\\mathbb A(n)$\nand $\\SS$ is closed under right-shortening.\n\\end{enumerate}\n\\end{proposition}\n\nWe remark that, if ${U}$ is of Pr\\\"{u}fer type, by Proposition~\\ref{p:ftgt},\n$\\psi(\\operatorname{ind}\\mathcal T_{U})$ can be obtained by taking the closure of the set of arcs\ncorresponding to finitely generated indecomposable summands of ${U}$ under left\nshortening and rotating all resulting arcs one step to the right.\nWe obtain $\\psi(\\operatorname{ind}\\mathcal F_{U})$ by taking the closure of the set of arcs\ncorresponding to \\emph{all} of the summands of ${U}$ under right shortening.\n\nSimilarly, by Proposition~\\ref{p:TfromFG}, if $(\\mathcal T,\\mathcal F)$ is a torsion pair where\n$\\mathcal F$ generates $\\mathbf T$ (i.e.\\ of ray type), then $\\psi(\\varinjlim(\\operatorname{ind}\\mathcal F))$ can be obtained from\n$\\psi(\\operatorname{ind}\\mathcal F)$ by first adding any infinite arc $\\pi_n([i,\\infty])$ for which all\narcs $\\pi_n([i,j])$ for $j\\geq a$ for some $a$ lie in $\\psi(\\operatorname{ind}\\mathcal F)$.\nThen ${U}$ is the direct sum of the indecomposable objects corresponding to the\narcs $\\alpha$ in $\\psi(\\varinjlim(\\operatorname{ind}\\mathcal F))$ such that the pair\n$(\\beta,\\alpha)$ of arcs has no negative intersections for all\n$\\beta$ in $\\psi(\\varinjlim(\\operatorname{ind}\\mathcal F))$.\n\nSimilar descriptions can be given in the adic\/coray type case.\n\nFinally, we give an example in a tube of rank $n=14$ to illustrate Proposition~\\ref{p:ftgt} and the results in this section.\nThe arcs corresponding to the indecomposable direct summands of ${U}$ are\ndisplayed in Figure~\\ref{fig:tubeex}\n(only the beginnings of the infinite arcs are shown). Note that the Pr\\\"{u}fer modules\nwhich are indecomposable summands of ${U}$ are $M_{0,\\infty},M_{6,\\infty},M_{10,\\infty}$ and\n$M_{13,\\infty}$, so $i_0=0$, $i_1=6$, $i_2=10$ and $i_3=13$.\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{0}{$0$}\n\\psfrag{1}{$1$}\n\\psfrag{2}{$2$}\n\\psfrag{3}{$3$}\n\\psfrag{4}{$4$}\n\\psfrag{5}{$5$}\n\\psfrag{6}{$6$}\n\\psfrag{7}{$7$}\n\\psfrag{8}{$8$}\n\\psfrag{9}{$9$}\n\\psfrag{10}{$10$}\n\\psfrag{11}{$11$}\n\\psfrag{12}{$12$}\n\\psfrag{13}{$13$}\n\\includegraphics[width=10cm]{tubeex.eps}\n\\caption{The maximal rigid object ${U}$}\n\\label{fig:tubeex}\n\\end{figure}\nThe arcs corresponding to the indecomposable objects in $\\mathcal T_{U}$ are displayed in\nFigure~\\ref{fig:tubeF}.\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{0}{$0$}\n\\psfrag{1}{$1$}\n\\psfrag{2}{$2$}\n\\psfrag{3}{$3$}\n\\psfrag{4}{$4$}\n\\psfrag{5}{$5$}\n\\psfrag{6}{$6$}\n\\psfrag{7}{$7$}\n\\psfrag{8}{$8$}\n\\psfrag{9}{$9$}\n\\psfrag{10}{$10$}\n\\psfrag{11}{$11$}\n\\psfrag{12}{$12$}\n\\psfrag{13}{$13$}\n\\includegraphics[width=10cm]{tubeF.eps}\n\\caption{The torsion part, $\\mathcal T_{U}=\\tau^{-1}(\\operatorname{Gen} {U}\\cap \\mathbf T)$, of the torsion pair corresponding to ${U}$.\nThe arcs corresponding to indecomposable objects not in $\\tau^{-1}(\\operatorname{add} {U}\\cap \\mathbf T)$ are drawn in blue.}\n\\label{fig:tubeF}\n\\end{figure}\n\nThe arcs corresponding to the indecomposable objects in the $\\mathcal F_{U}\\cap \\mathcal W_{i_r,i_{r+1}+1}$\nare displayed in Figure~\\ref{fig:tubeG}\n(with dotted arcs indicating the indecomposable\nsummands of ${U}$ which are not in $\\mathbf T$ (or $\\operatorname{ind}\\mathcal F$)).\nNote that there are infinitely many additional arcs not displayed, corresponding to indecomposables in $\\mathcal F_{\\infty}$ but not in any of the $\\mathcal F_r$.\nThe missing arcs are $\\pi_{14}([0,j])$ for $j\\geq 8$, $\\pi_{14}([6,j])$ for $j\\geq 12$,\n$\\pi_{14}([10,j])$ for $j\\geq 15$ and $\\pi_{14}([13,j])$ for $j\\geq 16$.\nNote that, as indicated by Proposition~\\ref{p:ftgt}, the intersections of\n$\\mathcal T_{U}$ and $\\mathcal F_{U}$ with the wings $\\mathcal W_{i_r,i_{r+1}+1}$ (which are\n$\\mathcal W_{0,7},\\mathcal W_{6,11},\\mathcal W_{10,14}$ and $\\mathcal W_{13,15}$) are torsion pairs.\n\n\\begin{figure}[ht]\n\\psfragscanon\n\\psfrag{0}{$0$}\n\\psfrag{1}{$1$}\n\\psfrag{2}{$2$}\n\\psfrag{3}{$3$}\n\\psfrag{4}{$4$}\n\\psfrag{5}{$5$}\n\\psfrag{6}{$6$}\n\\psfrag{7}{$7$}\n\\psfrag{8}{$8$}\n\\psfrag{9}{$9$}\n\\psfrag{10}{$10$}\n\\psfrag{11}{$11$}\n\\psfrag{12}{$12$}\n\\psfrag{13}{$13$}\n\\includegraphics[width=10cm]{tubeG.eps}\n\\caption{The torsion-free part, $\\mathcal F_{U}=\\operatorname{Cogen} {U}\\cap \\mathbf T$, of the torsion pair corresponding to ${U}$.\nThe arcs not in $\\operatorname{add} {U}$ are drawn in red. The dashed arcs are the indecomposable summands of ${U}$\nwhich are not of finite length (and thus not in $\\mathcal F_{U}$).\nThe arcs $\\pi_{14}([0,j])$ for $j\\geq 8$, $\\pi_{14}([6,j])$ for $j\\geq 12$,\n$\\pi_{14}([10,j])$ for $j\\geq 15$ and $\\pi_{14}([13,j])$ for $j\\geq 16$ have been omitted\nfor clarity.}\n\\label{fig:tubeG}\n\\end{figure}\nIn Figure~\\ref{fig:tubeAR}, we show the indecomposable summands of $\\mathbf T$ and the indecomposable objects in $\\mathcal T_{U}$ and $\\mathcal F_{U}$ in the AR-quiver of the tube.\n\\begin{figure}[ht]\n$$\\xymatrix@!0@=0.35cm{\n&& \\ast &&&&&& \\ast && \\ast &&&&&&&&&&&& \\ast &&&&&& \\\\\n&&&&&&&&&&&&&&&&&&&&&&&&&&&& \\\\\n\\ar@{--}[uurr] &&&&&&&&&&&&&&&&&&&&&&&&&&&& \\\\\n&\\cdot && \\cdot && \\circ \\ar@{--}[uuurrr] && \\circ \\ar@{--}[uuurrr] && \\cdot && \\cdot && \\cdot && \\cdot && \\cdot && \\circ \\ar@{--}[uuurrr] && \\cdot && \\cdot && \\cdot && \\circ \\ar@{-}[ur] & \\\\\n\\cdot && \\cdot && \\circ && \\circ && \\cdot && \\cdot && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\\\\n&\\cdot && \\circ && \\circ && \\cdot && \\cdot && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\cdot && \\cdot && \\circ && \\cdot & \\\\\n\\cdot && \\circ && \\bullet && \\Box && \\cdot && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\cdot && \\\\\n& \\circ && \\circ && \\cdot && \\Box && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\cdot && \\cdot && \\circ && \\cdot && \\cdot & \\\\\n\\circ && \\circ && \\cdot && \\bullet && \\Box && \\cdot && \\cdot && \\bullet && \\Box && \\cdot && \\cdot && \\circ && \\cdot && \\cdot && \\circ && \\\\\n&\\circ && \\cdot && \\circ && \\cdot && \\Box && \\cdot && \\circ && \\bullet && \\Box && \\cdot && \\bullet && \\Box && \\cdot && \\circ & \\\\\n*=0{\\bullet}  \\ar@{.}[uuuuuuuuu]+0 && \\Box && \\bullet && \\Box && \\bullet && \\Box && \\circ && \\bullet && \\Box && \\Box && \\bullet && \\Box && \\Box && \\circ && *=0{\\bullet} \\ar@{.}[uuuuuuuuu]+0 &&\n}$$\n\\caption{The AR-quiver of the tube, showing the indecomposable summands of ${U}$ ($\\bullet$ or $\\ast$), $\\operatorname{ind}(\\mathcal T_{U})$ ($\\Box$) and $\\operatorname{ind}(\\mathcal F_{U})$ ($\\circ$ or $\\bullet$). The Pr\\\"{u}fer direct summands of ${U}$ are shown (symbolically) at the top of the diagram as asterisks.}\n\\label{fig:tubeAR}\n\\end{figure}\n\n\\noindent \\textbf{Acknowledgements}\nAll three authors would like to thank the referee for helpful comments on an\nearlier version of this manuscript, and the Mathematics Research Institute in\nOberwolfach for its support during a conference in February 2011.\nABB would also like to thank Karin Baur and the FIM at ETH for their support\nand kind hospitality during a visit in May 2011.\nKB would like to thank Aslak Buan and the NTNU for their kind hospitality\nduring a visit in December 2010.\nRJM would like to thank Karin Baur and the FIM at the ETH, Zurich, for their\nsupport and kind hospitality during a visit in Spring 2011,\nand would like to thank Andrew Hubery for a helpful conversation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}