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+{"text":"\\section{Introduction: Cantor Ternary Set.}\nThe Cantor  ternary set $C$ iis probably the best known example of a {\\em perfect nowhere-dense} set in the real line. It was constructed by George Cantor in 1883, see \\cite{cantor}.\\\\\n\n$C$ is obtained from the closed interval $[0,1]$ by a sequence of deletions of open intervals known as ''middle thirds\".\nWe begin with the interval $[0,1]$, let us call it $C_0$, and remove the middle third, leaving us with leaving us with the union of two closed intervals of length $1\/3$\n$$C_1= \\left[0,\\frac{1}{3}\\right]  \\cup \\left[\\frac{2}{3}, 1\\right] .$$ Now we remove the middle third from each of these intervals, leaving us with the union of four closed intervals of length $1\/9$\n\\begin{equation*}\nC_2=\\left[0,\\frac{1}{9}\\right] \\cup \\left[\\frac{2}{9},\\frac{1}{3}\\right] \\cup\\left[\\frac{2}{3},\\frac{7}{9}\\right] \\cup\\left[\\frac{8}{9},1\\right] .\n\\end{equation*}\nThen we remove the middle third of each of these intervals leaving us with eight intervals of length $1\/27$,\n\\begin{equation*}\nC_{3}= [0,\\frac{1}{27}]\\cup[\\frac{2}{27}, \\frac{1}{9}]\\cup[\\frac{2}{9}, \\frac{7}{27}]\\cup [ \\frac{8}{27},\\frac{1}{3}] \\cup [\\frac{2}{3},\\frac{19}{27}]\\cup[\\frac{20}{27},\\frac{7}{9}]\\cup[\\frac{8}{9},\\frac{25}{27}]\\cup[\\frac{26}{27},1].\n\\end{equation*}\n\nWe continue this process inductively, then for each $k=1,2, 3\\cdots $ we get a set $C_k$  which is the union of $2^k$ closed intervals of length $1\/3^k$. This iterative construction is illustrated in the following figure, for the first four steps:\n\\begin{center}\n\\includegraphics[width=3in]{outputs-figure0.pdf}\n\\end{center}\nFinally, we define the {\\em Cantor ternary set} $C$ as the intersection\n\\begin{equation}\nC= \\bigcap_{n=0}^\\infty C_n.\n\\end{equation}\n\nClearly $C \\neq \\emptyset $, since  trivially $0,1 \\in C$. Moreover  $C$ is a a closed set, being the countable intersection of closed sets, and  trivially bounded, since it is a subset of $[0,1]$. Therefore, by the Heine-Borel theorem $C$ is a {\\em compact set}. Moreover, observe by the construction that if $y$ is the end point of some closed subinterval of a given  $C_n$ then it is also the end point of some of the subintervals of $C_{n+1}$. Because at each stage, endpoints are never removed, it follows that $y \\in C_n$ for all $n$. Thus $C$ contains all the end points of all the intervals that make up each of the sets $C_n$ (or alternatively,  the endpoints to the intervals removed)  all of which are rational ternary numbers in $[0,1]$, i.e. numbers of the form $k\/3^n$. But $C$ contains much more than that; actually it is an uncountable set since it is a {\\em perfect set} \\footnote{A perfect set $P$ is a set that is closed and every point  $x \\in P$ is a limit point i.e there is a sequence $\\{x_n\\} \\subset P$, $x_n \\neq x$ and $x_n \\rightarrow x$.}. To prove that simply observe that every point of $C$ is approachable arbitrarily closely by the endpoints of the intervals removed (thus for any $x\\in C$ and for each $n \\in \\mathbb{N}$ there is an endpoint, let us call it $y_n \\in C_n$, such that $|x-y_n| < 1\/3^n$).\n\nThere is an alternative characterization of $C$, the {\\em ternary expansion characterization}.\nConsider the ternary representation for  $x \\in [0,1],$ \\footnote{Observe, for the ternary rational  numbers $k\/3^n$ there are two possible ternary expansions, since\n$$ \\frac{k}{3^n} = \\frac{k-1}{3^n} + \\frac{1}{3^n}  =  \\frac{k-1}{3^n} + \\sum_{k=n+1}^\\infty \\frac{2}{3^k}.$$ \nSimilarly, for the dyadic rational numbers $k\/2^n$ there are two possible dyadic expansions as\n$$ \\frac{k}{2^n} = \\frac{k-1}{2^n} + \\frac{1}{2^n}  =  \\frac{k-1}{2^n} + \\sum_{k=n+1}^\\infty \\frac{1}{2^k}.$$ \nThus for the uniqueness of the dyadic and the ternary representations we will take  the infinite expansions representations for the dyadic and ternary rational numbers.}\n \n\\begin{equation}\\label{ternaryexp}\n x = \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{3^k}, \\quad \\varepsilon_k(x)=0, 1, 2 \\quad \\mbox{for all} \\,k = 1, 2, \\cdots \n\\end{equation}\n\nObserve that removing the elements where at least one of the \\(\\varepsilon_{k}\\) is equal to one is the same as removing the middle third in the iterative construction, thus \nthe Cantor ternary set is the set of numbers in $[0,1]$ that can be written in base 3 without using the digit 1, i.e.\n\\begin{equation}\\label{ternaryChar}\n C=\\left\\{x\\in [0,1] : x = \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{3^k}, \\quad \\varepsilon_k(x)=0, 2 \\quad \\mbox{for all} \\; k = 1, 2, \\cdots\n \\right\\}.\n\\end{equation}\n\nUsing this characterization  of  $C$ we can get a direct proof that it is uncountable. Define \nthe mapping $f: C \\rightarrow [0,1]$ for $  x=\\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{3^k} \\in C$, as \n\\begin{equation}\\label{cantorfun}\nf(x) =    \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)\/2}{2^k}= \\frac{1}{2}    \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{2^k}.\n\\end{equation}\n\nIt is clear that $f$ is one-to-one correspondence from $C$ to $[0,1]$ (observe that as $\\varepsilon_k=0, 2$ then $\\varepsilon_k\/2=0, 1$). \nalso, the uncoantablity of $C$ can be obtained from the fact that $C$ is perfect, see Abbott \\cite{abot}, page 90. \\\\\n\n$C$ is a {\\em nowhere-dense} set, that is, there are no intervals included in $C$. One way to prove that is using the Taking two arbitrary points in $C$ we can always find a number between them that requires the digit 1 in its ternary representation, and therefore there are no intervals included in $C$, thus $C$ is a nowhere dense set. Alternatively, we can prove this simply by contradiction. Assuming that there is a interval $I=[a,b] \\subset C, \\; a <b$. Then $I=[a,b] \\subset C_n$ for all $n$ but as $|C_n| \\rightarrow 0$ as $n \\rightarrow \\infty$ then $|I| = b-a =0.$\\\\\n\n\n$C$ has measure zero, since\n$$ m(C) = m([0,1]) - m(C^c) = 1 - \\sum_{n=1}^\\infty \\frac{2^{n-1}}{3^n} = 1 -\\frac{1}{3}  \\sum_{n=0}^\\infty (\\frac{2}{3})^{n} = 1 -  \\frac{1\/3}{1- 2\/3}=1-1=0.$$\\\\\n\nSo far we have study the main properties of the Cantor set: $C$ is, non-empty (moreover uncountable) compact, perfect , nowhere dense set, with measure zero. Also we have seen that $C$ can then be obtained by any of the following constructions:\n\\begin{enumerate}\n\\item [i)] {\\em Proportional (fractral) construction}: by removing a fixed proportion (one third) of each subinterval in each of the iterative steps.\n\\item[ii)] {\\em Power construction}: by removing the length $1\/3^k$ from the center of each subinterval in the $k^\\text{th}-$step.\n\\item [iii)] {\\em Ternary expansion characterization} : by removing  the digit one in each position of the ternary expansion.\n\\end{enumerate}\n\nOf course these three constructions are equivalent. Nevertheless, the first construction, removing a fixed proportion of each subinterval in each of the iterative steps, give us another important property of the Cantor set, its self-similarity (fractal characteristic) across scales, this is illustrated in the following figure:\n\\begin{center}\n\\includegraphics[width=3in]{outputs-figure1.pdf}\n\\end{center}\n\nIn order to compute the Hausdorff dimension of $C$, S. Abbott in his book uses a nice little trick (avoiding the technicalities of the definition of Hausdoff dimension) by dilating $C_1$ by a the factor $3$ (see \\cite{abot} page 77) obtaining\n$$ [0,1] \\cup [2, 3].$$\nThus if we continue the iterations we will get two Cantor-like sets, then it can be concluded that\n$$3^d= 2, $$\nand therefore the Hausdorff dimension of $C$ is\n $$d= \\frac{\\ln 2}{\\ln 3}=0.630930.$$\n \n The Cantor ternary set $C$ was not the first perfect nowhere-dense set  in the real line to be constructed. The first construction was done by the a British mathematician Henry J. S. Smith in 1875, but not many mathematician where awarw of Smith's construction. Vito Volterra, still a graduate student in Italy, also showed how to construct such a set  in 1881, but he published his result in an italian journal not widely read.In 1883 Cantor rediscover this construction himself, and due to Cantor's prestige, then the Cantor ternary set was  (and still is) the typical example of a perfect nowhere-dense set. Following D. Bresoud \\cite{bres} we will refer as the Smith-Volterra-Cantor sets or $SVC(n)$ sets to the family of examples of perfect, nowhere-dense sets exemplified by the work of Smith, Voterra and Cantor, by removing in the $k$-iteration, an open interval of length $1\/n^k$ from the center of the remaining closed intervals. Observe that $C = SVC(3)$.\n \nAdditionally, the Cantor set can also be obtained using an Iterated Function System (IFS), see \\cite{hutch}. Consider the maps:\n\\begin{eqnarray}\n\\omega_0(x) = x\/3, \\quad \\omega_1(x) = x\/3 + 2\/3,\n\\end{eqnarray}\nthen\n$$ C = \\omega_0([0,1]) \\cup \\omega_1([0,1]),$$\nwhere this is a disjoint union. $C$ is the only set in the real line that satisfies this relation. This is a very interesting construction with important applications in the study of $C$, but we will not consider it in further details in this article.\n\n\\section{Variations of the Cantor ternary set:  Cantor-like sets.}\n\nIs there something special with the number $3$? The answer is actually yes and no!. As we have already mentioned  $C$ can then be obtained by removing a fixed proportion (one third) of each subinterval in each of the iterative steps, by removing the length $1\/3^k$ from each subinterval in the $k^\\text{th}-$step, or removing  the digit one of the ternary expansion. What happen if we choose $n=2$ instead of $3$?\nUsing the proportional construction, removing the open ``middle half\" from each component, we get at the end of the iterative process the set that we will denote as $C^{1\/2}$.  Here are the first 3 iterative steps,\n\n\\begin{eqnarray*}\nC^{1\/2}_{1}&=& [0,\\frac{1}{4}]\\cup [\\frac{3}{4},1],\\\\\nC^{1\/2}_{2}&=& [0,\\frac{1}{16}]\\cup[\\frac{3}{16}, \\frac{1}{4}]\\cup [\\frac{3}{4},\\frac{13}{16}]\\cup[\\frac{15}{16},1],\\\\\n\\end{eqnarray*}\n\\begin{eqnarray*}\nC^{1\/2}_{3}&=& [0,\\frac{1}{64}]\\cup[\\frac{3}{64}, \\frac{1}{16}]\\cup[\\frac{3}{16}, \\frac{13}{64}]\\cup [ \\frac{15}{64},\\frac{1}{4}] \\cup [\\frac{3}{4},\\frac{49}{64}]\\cup[\\frac{51}{64},\\frac{13}{16}]\\cup[\\frac{15}{16},\\frac{61}{64}]\\cup[\\frac{63}{64},1].\n\\end{eqnarray*}\nThen, continuing this process inductively , then for each $k=1,2, \\cdots $ we get a set $C^{1\/2}_k$  which is the union of $2^k$ closed intervals of length $1\/4^k$, and \n$$ C^{1\/2} = \\bigcap_{k=1}^\\infty C^{1\/2}_{k}.$$\n$C^{1\/2}$ share the same properties as $C$, i.e. it is a perfect, nowhere dense set in the real line. Also $C^{1\/2}$ has measure zero, as \n\\begin{eqnarray*}\n1- \\frac{1}{2} -2 \\frac{1}{8}-4\\frac{1}{32} +\\cdots &=& 1- \\frac{1}{2}[1+ \\frac{1}{2}+\\frac{1}{4} +\\cdots ]\\\\\n&=& 1- \\frac{1}{2}  \\sum_{k=0}^\\infty (\\frac{1}{2})^k =  1- \\frac{1}{2}  \\frac{1}{1-\\frac{1}{2}} =  1- \\frac{1}{2} \\frac{1}{\\frac{1}{2}} = 1- 1=0.\n\\end{eqnarray*}\nIn order to compute the Hausdorff dimension of this Cantor-like set, following the same argument as before, if we dilate $C_1^{1\/2}$ by $4$ we obtain\n$$ [0,1] \\cup [3, 4].$$\nThus, if we continue the iterations we will get at the end two Cantor-like sets, therefore\n$$4^d= 2,\\; \\mbox{so}\\; d= \\frac{\\ln 2}{\\ln 4}= \\frac{\\ln 2}{2\\ln 2} = \\frac{1}{2}= 0.5,$$\n so for this Cantor-like set the{\\em dust} obtained  is {\\em more sparse} than the one generated by $C$.\nAlso observe that, we can get a {\\em expansion characterization} of $C^{1\/2}$ but with base $n=4$, \n\\begin{equation*}\n C^{1\/2}=\\left\\{x\\in [0,1] : x = \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{4^k}, \\quad \\varepsilon_k(x)=0, 3 \\quad \\mbox{for all} \\; k = 1, 2, \\cdots\n \\right\\}.\n\\end{equation*}\n\nOn the other hand, what happen if we use the power construction? i.e.what happen if we remove the length $1\/2^k$ from the center of each subinterval in the $k^\\text{th}-$step?.  The first step is the same as before, obtaining the set\n$$ [0,\\frac{1}{4}]\\cup [\\frac{3}{4},1],$$\nbut next we need to remove two open intervals of length $1\/2^2 = 1\/4$, so we get only four points,\n$$\\{0, 1\/4, 3\/4,1\\}$$\nso the process stops there, at $k=2$, since there are no more intervals to remove form. Clearly this is not the same as removing  open ``middle half\" from each component.\\\\\n\nWhat happen if we consider now $n=4$? In this case both the proportional and the power construction can be iterated an infinity number of times:\n\n\\begin{enumerate}\n\\item [i)] {\\em Proportional construction}: If we repeat the Cantor set's construction starting with the interval [0,1], removing the open ``middle fourth\" from the center of the remaining closed intervals, we get at the end of the iterative process a set that we will denote as $ C^{1\/4}$. Here are the first three iterations\n\\begin{eqnarray*}\nC^{1\/4}_1 &=& [0,\\frac{3}{8}] \\cup [\\frac{5}{8}, 1]\\\\\nC^{1\/4}_2 &=& [0,\\frac{9}{64}]\\cup [\\frac{15}{64}, \\frac{3}{8}]\\cup[\\frac{5}{8},\\frac{49}{64}]\\cup[\\frac{55}{64},1]\\\\\nC^{1\/4}_3 &=& [0,\\frac{27}{512}]\\cup [\\frac{45}{512}, \\frac{9}{64}]\\cup[\\frac{15}{64},\\frac{147}{512}]\\cup[\\frac{165}{512},\\frac{3}{8}]\\cup[\\frac{5}{8},\\frac{347}{512}]\\cup[\\frac{365}{512},\\frac{49}{64}]\\cup[\\frac{55}{64},\\frac{467}{512}]\\cup[\\frac{485}{512},1].\n\\end{eqnarray*}\nThen, continuing this process inductively , then for each $k=1,2, \\cdots $ we get a set $C^{1\/4}_k$  which is the union of $2^k$ closed intervals of length $(3\/8)^k$. \n$$ C^{1\/4} = \\bigcap_{k=1}^\\infty C^{1\/4}_{k}.$$\n\\begin{center}\n\\includegraphics[width=3in]{C025-proportions.pdf}\n\\end{center}\n$C^{1\/4}$ share the same properties as $C$, i.e. it is a perfect, nowhere dense set in the real line.\nAlso $C^{1\/4}$ has measure zero, as \n\\begin{eqnarray*}\n1- \\frac{1}{4} -2 \\frac{3}{32}-4\\frac{9}{256} +\\cdots &=& 1- \\frac{1}{4}[1+ \\frac{3}{4}-\\frac{9}{16} +\\cdots ]\\\\\n&=& 1- \\frac{1}{4} \\frac{1}{1-\\frac{3}{4}} =  1- \\frac{1}{4} \\frac{1}{\\frac{1}{4}} = 1-1 =0.\n\\end{eqnarray*}\n\nIn order to compute the Hausdorff dimension of this Cantor-like set, following the same argument as before, if we dilate $C^{1\/4}_1$ by $8\/3$ we obtain\n$$ [0,1] \\cup [\\frac{5}{3}, \\frac{8}{3}].$$\nThus if we continue the iterations we will get two Cantor-like sets, thus\n$$(\\frac{8}{3})^d= 2, \\mbox{i.e} \\quad d= \\frac{\\ln 2}{\\ln 8\/3} = \\frac{\\ln 2}{\\ln 8- \\ln3}= .706695.$$\nObserve that if we do the same argument in the second or third step we get the same result,\n$$(\\frac{64}{9})^d= 4, \\mbox{i.e} \\quad d= \\frac{\\ln 4}{\\ln 64\/9} = \\frac{\\ln 2}{\\ln 8- \\ln3}= .706695,$$\n$$(\\frac{512}{27})^d= 8, \\mbox{i.e} \\quad d= \\frac{\\ln 8}{\\ln 512\/27} = \\frac{\\ln 2}{\\ln 8- \\ln3}= .706695.$$\nThus, this Cantor-like  set is less sparse than $C$.\\\\\n\n\\item [ii)] {\\em Power construction}: if we repeat the Cantor construction starting with the interval [0,1], removing, in the $k$-iteration, an open interval of length $1\/4^k$ from the center of the remaining closed intervals, we get \n\\begin{eqnarray*}\nSVC(4)_1 &=& [0,\\frac{3}{8}] \\cup [\\frac{5}{8}, 1]\\\\\nSVC(4)_2 &=& [0,\\frac{5}{32}]\\cup [\\frac{7}{32}, \\frac{3}{8}]\\cup[\\frac{5}{8},\\frac{25}{32}]\\cup[\\frac{27}{32},1]\\\\\nSVC(4)_3 &=& [0,\\frac{9}{128}]\\cup [\\frac{11}{128}, \\frac{5}{32}]\\cup[\\frac{7}{32},\\frac{37}{128}]\\cup[\\frac{39}{128},\\frac{3}{8}]\\cup[\\frac{5}{8},\\frac{89}{128}]\\cup[\\frac{91}{128},\\frac{25}{32}]\\cup[\\frac{27}{32},\\frac{59}{64}]\\cup[\\frac{119}{128},1].\n\\end{eqnarray*}\nThen\n$$ SVC(4) = \\bigcap_{k=1}^\\infty SVC(4) _{k}.$$\n\\begin{center}\n\\includegraphics[width=3in]{SVC4-proportions.pdf}\n\\end{center}\n The total length of $SVC(4)$ can be obtained after removing all the open intervals in constructing this set,\n\\begin{eqnarray*}\n1- \\frac{1}{4} -2 \\frac{1}{16}-4\\frac{1}{64} +\\cdots &=& 1- \\frac{1}{4}[1+ \\frac{1}{2}-\\frac{1}{4} +\\cdots ]\\\\\n&=& 1- \\frac{1}{4} \\frac{1}{1-\\frac{1}{2}} =  1- \\frac{1}{4} \\frac{1}{\\frac{1}{2}} = 1-\\frac{1}{2} = \\frac{1}{2}.\n\\end{eqnarray*}\\\\\n\nThus the set $SVC(4)$ must have positive measure $1\/2$. Cantor-like sets with positive measure are called {\\em fat-Cantor sets}.\\footnote{Moreover, observe that the only $SCV(n)$ set that has measure zero is $C$, since the total length of what is remove is\n$$ 1 - \\frac{1}{n} -\\frac{2}{n^2} - \\frac{4}{n^3}- \\cdots = 1 - \\frac{1}{n}[1 -\\frac{2}{n} - \\frac{2^2}{n^2}- \\cdots ] =1 - \\frac{1}{n} \\frac{1}{1- 2\/n} = \\frac{n-3}{n-2}.$$}\\\\\n\nIn this case if we try to compute the Hausdorff dimension, since the sets $SVC(4)_k$ are not proportional,  a similar  argument as  before fails,  since if we dilate $SVC(4)_1$ by $8\/3$ we obtain\n$$ [0,1] \\cup [\\frac{5}{3}, \\frac{8}{3}].$$\nThus if we continue the iterations we will get two Cantor-like sets, thus\n$$(\\frac{8}{3})^d= 2, \\mbox{i.e.} \\quad d= \\frac{\\ln 2}{\\ln 8\/3} = \\frac{\\ln 2}{\\ln 8- \\ln3}= .706695.$$\nBut now if we do the same argument in the second or third step we get the different  results,\n$$(\\frac{32}{5})^d= 4, \\mbox{i.e.} \\quad d= \\frac{\\ln 4}{\\ln 32\/5} = \\frac{2\\ln 2}{\\ln 32- \\ln 5}= .746806,$$\n$$(\\frac{128}{9})^d= 8, \\mbox{i.e.} \\quad d= \\frac{\\ln 8}{\\ln 128\/9} = \\frac{3\\ln 2}{\\ln 128- \\ln 9}= .783274.$$\nSince $$\\lim_{k\\rightarrow \\infty} \\frac{k \\ln 2}{ \\ln (2 \\cdot 4^k) - \\ln (1+ 2^k)} = 1,$$ \nwe  can conclude that the Hausdorff dimension of $SVC(4)$  is actually one.\\\\\n$SVC(4)$ is called the {\\em Volterra set}. This is the set that was considered in 1881 by V.  Volterra to construct his famous counter-example of a function with bounded derivative that exists everywhere but the derivative is not Riemann integrable in any closed bounded interval i.e the Fundamental Theorem of Calculus fails!.\\\\\nObserve that in this case there is not a expansion characterization of the set $SVC(4)$.\\\\\n\\end{enumerate}\n\nNow, if we repeat the Cantor set's {\\em proportional construction} starting with the interval [0,1], removing the open ``middle three fourth\" from the center of the remaining closed intervals, then we get in the first three iterations,\n\\begin{eqnarray*}\nC^{3\/4}_1 &=& [0,\\frac{1}{8}] \\cup [\\frac{7}{8}, 1]\\\\\nC^{3\/4}_2 &=& [0,\\frac{1}{64}]\\cup [\\frac{7}{64}, \\frac{1}{8}]\\cup[\\frac{7}{8},\\frac{57}{64}]\\cup[\\frac{63}{64},1]\\\\\nC^{3\/4}_3 &=& [0,\\frac{1}{512}]\\cup [\\frac{7}{512}, \\frac{1}{64}]\\cup[\\frac{7}{64},\\frac{57}{512}]\\cup[\\frac{63}{512},\\frac{1}{8}]\\cup[\\frac{7}{8},\\frac{449}{512}]\\cup[\\frac{455}{512},\\frac{57}{64}]\\cup[\\frac{63}{64},\\frac{505}{512}]\\cup[\\frac{511}{512},1].\n\\end{eqnarray*}\nThen, continuing this process inductively , then for each $k=1,2, \\cdots $ we get a set $C^{3\/4}_k$  which is the union of $2^k$ closed intervals of length $(3\/8)^k$, and\n$$ C^{3\/4} = \\bigcap_{k=1}^\\infty C^{3\/4}_{k}.$$\n$C^{3\/4}$ share the same properties as $C$, i.e. it is a perfect, nowhere dense set in the real line.\nAlso $C^{3\/4}$ has measure zero, as \n\\begin{eqnarray*}\n1- \\frac{3}{4} -2 \\frac{3}{32}-4\\frac{3}{256} +\\cdots &=& 1- \\frac{3}{4}[1+ \\frac{1}{4}-\\frac{1}{16} +\\cdots ]\\\\\n&=& 1- \\frac{3}{4} \\frac{1}{1-\\frac{1}{4}} =  1- \\frac{3}{4} \\frac{1}{\\frac{3}{4}} = 1-1 =0.\n\\end{eqnarray*}\n\nIn order to compute the Hausdorff dimension of this Cantor-like set, following the same argument as before, if we dilate $C^{3\/4}_1$ by $8$ we obtain\n$$ [0,1] \\cup [7, 8].$$\nThus if we continue the iterations we will get two Cantor-like sets, thus\n$$8^d= 2, \\mbox{i.e} \\quad d= \\frac{\\ln 2}{\\ln 8} = \\frac{\\ln 2}{3\\ln 2}= \\frac{1}{3}= .666666.$$\n so this Cantor-like  set is less sparse than $C$. Also observe that in this case there is no {\\em power construction.}\n \nFinally, it is possible  an {\\em expansion characterization} of that $C^{3\/4}$ for $n=8$,\n\\begin{equation}\\label{Cant2}\n C^{3\/4}=\\left\\{x\\in [0,1] : x = \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{8^k}, \\quad \\varepsilon_k(x)=0, 7 \\quad \\mbox{for all} \\; k = 1, 2, \\cdots\n \\right\\}.\n\\end{equation}\n \\\\\n\nOn the other hand, using the idea behind the ternary expansion characterization of the Cantor ternary set \\ref{ternaryChar}, we can get a general method to generalize $C$. For instance, take the base $n=5$, and \nconsider\n$$C^5= \\left\\{x\\in [0,1] : x = \\sum_{k=1}^\\infty \\frac{\\varepsilon_k(x)}{5^k}, \\quad \\varepsilon_k(x)=0, 2, 4 \\quad \\mbox{for all} \\; k = 1, 2, \\cdots\n \\right\\}.$$\n\nThis characterization is equivalent, since, in the step $k \\geq 1$ we are removing all the numbers in $[0,1]$ such that they have in its $5$-adic expansion the $k$-th digit is  $1$ or $3$, to the following proportional contraction:\nStart again with the unit interval $[0,1]$, partition the interval into $5$  intervals of equal length and remove the second and  the fourth open intervals of the partition obtaining $C^5_{1}$. \n$$C^5_{1}= [0,\\frac{1}{5}]\\cup [\\frac{2}{5},\\frac{3}{5}]\\cup [\\frac{4}{5},1]$$\nIn the second iteration, partition each of the remain $3$ intervals in $C^5_{1}$ into $5$ partitions of equal length and remove the same $2$ open intervals of the partition corresponding to the intervals removed in the first iteration, obtaining $C^k_{2}$, \n\\begin{eqnarray*}\nC^5_{2}&=& [0,\\frac{1}{25}]\\cup[\\frac{2}{25}, \\frac{3}{25}]\\cup [\\frac{4}{25},\\frac{1}{5}]\\cup[\\frac{2}{5},\\frac{11}{25}]\\cup[\\frac{12}{25},\\frac{13}{25}]\\cup[\\frac{14}{25},\\frac{3}{5}],\\\\\n&& \\quad \\cup [\\frac{4}{5},\\frac{21}{25}]\\cup[\\frac{22}{25}, \\frac{23}{25}]\\cup [\\frac{24}{25},1]\\\\\n\\end{eqnarray*}\nOne more iteration give us $C^5_{3}$,\n\\begin{eqnarray*}\nC^5_{3}&=& [0,\\frac{1}{125}]\\cup[\\frac{2}{125}, \\frac{3}{125}]\\cup [\\frac{4}{125},\\frac{1}{25}]\\cup[\\frac{2}{5},\\frac{11}{125}]\\cup[\\frac{12}{125},\\frac{13}{25}]\\cup[\\frac{14}{25},\\frac{3}{25}],\\\\\n&& \\quad \\cup [\\frac{4}{25},\\frac{21}{125}]\\cup[\\frac{22}{125}, \\frac{23}{125}]\\cup [\\frac{24}{125},\\frac{1}{5}]\\ \\cup [\\frac{2}{5},\\frac{51}{125}]\\cup[\\frac{52}{125}, \\frac{53}{125}]\\cup [\\frac{54}{125},\\frac{11}{25}]\\\\\n&& \\quad \\cup [\\frac{12}{25},\\frac{61}{125}]\\cup[\\frac{62}{125}, \\frac{63}{125}]\\cup [\\frac{64}{125},\\frac{13}{125}]\\ \\cup [\\frac{14}{25},\\frac{71}{125}]\\cup[\\frac{72}{125}, \\frac{73}{125}]\\cup [\\frac{74}{125},\\frac{3}{5}]\\\\\n&& \\quad \\cup [\\frac{4}{5},\\frac{101}{125}]\\cup[\\frac{102}{125}, \\frac{103}{125}]\\cup [\\frac{104}{125},\\frac{21}{25}]\\ \\cup [\\frac{22}{25},\\frac{111}{125}]\\cup[\\frac{112}{125}, \\frac{113}{125}]\\cup [\\frac{114}{125},\\frac{23}{52}]\\\\\n&& \\quad \\cup [\\frac{24}{25},\\frac{121}{125}]\\cup[\\frac{122}{125}, \\frac{123}{125}]\\cup [\\frac{124}{125},1].\n\\end{eqnarray*}\nContinue partitioning each subinterval in $5$ subinterval and removing $2$ of them in the same pattern.  Observe that for each $k$,  $C^{5}_{k}$ is the union of $3^{k-1}$ closed intervals of length $1\/5^k$. Then\n$$C^5=\\bigcap_{k=i}^{\\infty} C^5_{k}.$$\nBy construction $ C^{5} $ is also perfect nowhere set. The total length after removing all the open intervals in constructing this set is\n\\begin{eqnarray*}\n1- 2\\frac{1}{5} -6 \\frac{1}{25}-18\\frac{1}{125} +\\cdots &=& 1- \\frac{2}{5}[1+ \\frac{3}{5}+\\frac{9}{25} +\\cdots ]\\\\\n&=& 1- \\frac{2}{5}  \\sum_{k=0}^\\infty (\\frac{3}{5})^k = 1- \\frac{2}{5}  \\frac{1}{1-\\frac{3}{5}} \\\\\n&=&   1- \\frac{2}{5} \\frac{1}{\\frac{2}{5}} = 1- 1=0.\n\\end{eqnarray*}\nThus $ C^{5}$ must have measure $0$.\n\nIn order to compute the Hausdorff dimension of this Cantor-like set, following the same argument as before, if we dilate $C_1^{5}$ by $5$ we obtain\n$$ [0,1] \\cup [2, 3] \\cup [4, 5].$$\nThus if we continue the iterations we will get three Cantor-like sets, thus\n$$5^d= 3, \\; \\mbox{i.e} \\; d= \\frac{\\ln 3}{\\ln 5}=0.6826,$$\nso this Cantor-like  set is even less sparse than $C$.\\\\\n\nOf course this construction can be generalized for any $n \\in \\mathbb{N}$ with $n>2$, $1<p<n$, and start again with the unit interval $[0,1]$.  Partition the interval into $n$  intervals of equal length and remove $p$ open intervals of the partition making sure to leave the first and last intervals\\footnote{ if not, the set obtained would not be perfect since the construction would produce isolated points.} obtaining $C^n_{1}$. In the second iteration, partition each of the remain $n-p$ intervals in $C^n_{1}$ into $n$ partitions of equal length and remove the same $p$ open intervals of the partition corresponding to the intervals removed in the first iteration, obtaining $C^n_{2}$.  Continue partitioning each subinterval in $n$ subinterval and removing $p$ of them in the same pattern.  Then\n$$C^k=\\bigcap_{k=i}^{\\infty} C^k_{k}.$$\nObserve that in this construction in each step $p \\geq 1$ subintervals are removed from each closed subinterval in  $C^n_{k}$.\\\\\n   \nClearly, the $n$-adic Cantor  set $C^n$,  has the {\\em expansion characterization} \n\\begin{eqnarray*}\n&& C^n = \\{ x \\in [0,1] :  x=\\sum_{k=1}^{\\infty}\\frac{\\epsilon_{k}}{n^{k}}\\,  \\text{where }\\epsilon_{k} \\in \\{ \\eta_0, \\eta_1, \\cdots, \\eta_p\\} \\subseteq \\{0,1,\\cdots,n-1\\}, \\\\\n&& \\hspace{8cm} \\,  \\eta_0=0 \\text{ and } \\;\\eta_p=n-1\\} \n\\end{eqnarray*}\n\nObserve that in this case, using the dimension trick, we get\n$$ n^d = p,$$\nand therefore the Hausdorff dimension is\n$$ d =\\frac{\\ln p}{\\ln n}.$$\\\\\n\nSeveral other constructions are possible. For instance,  take $ 0< \\lambda \\leq 1$ and repeat the construction of the Cantor set such that at the $k$-th step, instead of taking out an open interval of length $1\/3^k$ from the center of each of the $2^{k-1}$ intervals of equal length, take out an open interval of length $\\lambda\/3^k$ to obtain $C_{\\lambda,k}$, then, the first 3 iterative steps $C_{\\lambda,1}$, $C_{\\lambda,2}$, $C_{\\lambda,3}$ are\n\n\\begin{eqnarray*}\nC_{\\lambda,1}&=& [0,\\frac{3-\\lambda}{6}]\\cup [\\frac{3+\\lambda}{6},1],\\\\\nC_{\\lambda,2}&=& [0,\\frac{9-5\\lambda}{36}]\\cup[\\frac{9-\\lambda}{36}, \\frac{3-\\lambda}{6}]\\cup [\\frac{3+\\lambda}{6},\\frac{27+\\lambda}{36}]\\cup[\\frac{27+5\\lambda}{36},1],\\\\\n\\end{eqnarray*}\n\\begin{eqnarray*}\nC_{\\lambda,3}&=& [0,\\frac{27-19\\lambda}{216}]\\cup[\\frac{27-11\\lambda}{216}, \\frac{9-5\\lambda}{36}]\\cup[\\frac{9-\\lambda}{36}, \\frac{81-25\\lambda}{216}]\\cup [ \\frac{81-17\\lambda}{216},\\frac{3-\\lambda}{6}]\\\\\n&& \\cup [\\frac{3+\\lambda}{6},\\frac{135+17\\lambda}{216}]\\cup[\\frac{135+25\\lambda}{216},\\frac{27+\\lambda}{36}]\\cup[\\frac{27+5\\lambda}{36},\\frac{189+11\\lambda}{216}]\\cup[\\frac{189+19\\lambda}{216},1].\n\\end{eqnarray*}\nThen,\n$$ C_{\\lambda} = \\bigcap_{k=1}^\\infty C_{\\lambda,k}$$\n$ C_{\\lambda}$ is, again, a perfect, nowhere dense set with positive measure, as\n\\begin{eqnarray*}\n1- \\frac{\\lambda}{3} -2 \\frac{\\lambda}{9}-4\\frac{\\lambda}{27} +\\cdots &=& 1- \\frac{\\lambda}{3}[1+ \\frac{2}{3}-\\frac{4}{9} +\\cdots ]\\\\\n&=& 1- \\frac{\\lambda}{3}  \\sum_{k=0}^\\infty (\\frac{2}{3})^k =  1- \\frac{\\lambda}{3}  \\frac{1}{1-\\frac{2}{3}} =  1- \\frac{\\lambda}{3} \\frac{1}{\\frac{1}{3}} = 1- \\lambda,\n\\end{eqnarray*}\nThus $C_{\\lambda}$ is a {\\em fat Cantor-like} set. In order to compute the Hausdorff dimension, if we dilate $C_{\\lambda,1}$ by $6\/(3-\\lambda)$ we obtain\n$$ [0,1] \\cup [\\frac{3+\\lambda}{3-\\lambda}, \\frac{6}{3-\\lambda}].$$\nThus if we continue the iterations we will get two Cantor-like sets, thus\n$$(\\frac{6}{3-\\lambda})^d= 2, \\,\\mbox{i.e.} \\; d= \\frac{\\ln 2}{\\ln [6\/(3-\\lambda)]} = \\frac{\\ln 2}{\\ln 6- \\ln(3-\\lambda)}.$$\n\nIt is worth noting that, unlike the other previously explored constructions, $C^n$ need not be symmetric with respect to 1\/2. As long as the first and last intervals of the {\\em expansion characterization} partition remain, in order to avoid isolated points, any choice of removed partitions can be made. Neither the measure nor the dimension of $C^n$ is affected by the choice to remove intervals asymmetrically.\\\\\n\nA more systematic study of the possible generalizations of the Cantor ternary set can be found in \\cite{dimartinourb}.\\\\\n\nFinally, a nice application of fat Cantor sets is the following. It is mention in almost anywhere that if we consider the space of Riemann integrable functions  it turns out that is not a complete space, this is one of the reasons why the Lebesgue integral is needed. Unfortunately it is hard to find a concrete counterexample for that statement. Using fat Cantor set it is easy to construct such a counterexample. First of all remember that a function is Riemann integrable if and only if the set of discontinuities has measure zero. Now let us take a fat Cantor set, let us take $SVC(4)$, the Volterra set, $ C^{1\/4} = \\bigcap_{k=1}^\\infty C^{1\/4}_{k}.$ We have proved that $SVC(4)$ has measure $1\/2>0$. If we consider \n$$K= [0,1] - SVC(4) = (\\frac{3}{8}, \\frac{5}{8}) \\cup ( \\frac{5}{32}, \\frac{7}{32})\\cup (\\frac{25}{32},  \\frac{27}{32})\\cup ( \\frac{9}{128}, \\frac{11}{128}) \\cup \\cdots =\\bigcup_{i=1}^\\infty E_i.$$\nLet $f_n$ be the indicator function of the set $\\bigcup_{i=1}^n E_i$, and $f$ be the indicator function of $K$. Then $f_n$ is Riemann integrable on $[0,1]$ since it is equal to one (i.e. continuous) for all but a finite number of points in [0,1]. Also trivially by construction $f_n (x)\\rightarrow f(x),$ for any $x \\in [0,1]$ and moreover,\n$$ \\int_0^1 |f_n(x) - f(x) | dx = \\int_{\\bigcup_{i=n+1}^\\infty E_i} dx = \\sum_{i=n+1}^\\infty |E_i| \\rightarrow 0, $$\nas $n \\rightarrow \\infty.$ But $f$ is not Riemann integrable on $[0,1]$ since its set of  discontinuities is precisely $SVC(4)$ that has positive measure.\\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction.} \n\nIn the book \\cite{FOOO}, the authors studied the moduli spaces \nof bordered stable \nmaps of genus 0 with Lagrangian boundary condition \nin a systematic way and constructed \nthe filtered $A_{\\infty}$-algebra associated to Lagrangian \nsubmanifolds.  \nSince our construction depends on various auxiliary choices, \nwe considered the {\\it canonical model} of filtered \n$A_{\\infty}$-algebras, which is unique up to filtered \n$A_{\\infty}$-isomorphisms.  \nThe aim of this note is to explain the construction of the canonical \nmodel and to apply such an argument to obtain the filtered \n$A_{\\infty}$-structure to the Morse complex on the Lagrangian \nsubmanifold.  \nThe resulting filtered $A_{\\infty}$-operations are described \nby the moduli \nspaces of certain configurations consisting of pseudo-holomorphic \ncurves and gradient flow lines.   \nNote that the first named author \\cite{Fuk94} studied \nthe quantization of Morse homotopy based on \nthe moduli spaces of certain configurations consisting \nof pseudo-holomorphic discs and gradient flow lines of {\\it multiple} \nMorse functions, \nsee \\cite{Oh2} for monotone case and Theorem A4.28 in \\S A 4 in \n\\cite{2000}.  \nSuch configurations are also studied in monotone case by \nBuhovsky \\cite{Buk} and Biran and Cornea \\cite{BC}.  \nWe follow Chapter 5 in \\cite{FOOO} to explain the algebraic aspect \nof canonical models and use the geometric construction in Chapter 7 \nin \\cite{FOOO}.  \n\nWe briefly review the background of our study.  \nFloer \\cite{Fl} invented a new theory, which is now called \nFloer (co)homology for Lagrangian intersections.  \nVery roughly speaking, it is an analog of Morse theory for \nthe action functional on the space of paths with end points \non Lagrangian submanifolds.  \nFor a transversal pair of Lagrangian submanifolds $L_0, L_1$, \nthe cochain complex is generated by the intersection points \nof $L_0$ and $L_1$.  The coboundary operator is defined by \ncounting connecting orbits joining the intersection points.  \nThe theory was extended by the second named author \\cite{Oh} \nto the class of \nmonotone Lagrangian submanifolds with the minimal Maslov number \nbeing at least 3.  \nIn general, however, there arise obstructions to constructing the Floer \ncochain complex caused by the bubbling-off of pseudo-holomorphic \ndiscs in the moduli space of connecting orbits.  \nWe started a systematic study of the moduli spaces of \npseudo-holomorphic discs with Lagrangian boundary condition \nand formulated the obstructions in terms of the Mauer-Cartan \nequation on the filtered $A_{\\infty}$-algebra associated to \nthe Lagrangian submanifold \\cite{FOOO}. \nIn order to give consistent orientations on the moduli spaces, \nwe introduced the notion of relative spin structure and \ncosidered relative spin Lagrangian submanifolds.  \nFor a relative spin pair $(L_0,L_1)$ of Lagrangian submanifolds, \nwe constructed a filtered $A_{\\infty}$-bimodule over \nthe $A_{\\infty}$-algbras associated to $L_0$ and $L_1$.  \nIf each $L_i$, $i=0,1$, admits a solution $b_i$ of \nthe Maurer-Cartan equation, we can rectify the Floer operator \nto obtain a coboundary operator $\\delta^{b_0,b_1}$.  \nHence the Floer complex \n$(CF^{\\bullet}(L_0,L_1), \\delta^{b_0,b_1})$ is obtained.  \nWe also considered the case that the Lagrangian submanifolds \nadmit solutions of the Maurer-Cartan equation modulo multiples of \nthe fundamental class $[L_i]$ ({\\it weak solution}).  \nFor a weak solution $b_i$, we assign the potential \n${\\mathfrak{PO}}(b_i)$. \nIf ${\\mathfrak{PO}}(b_0)={\\mathfrak{PO}}(b_1)$, we can construct \nthe Floer complex $(CF(L_0,L_1),\\delta^{b_0,b_1})$ deformed by \n$b_0, b_1$.  \nThis extension with the weak bounding cochains plays crutial role \nin our study of Floer theory on compact toric manifolds \\cite{FOOO2}.   \n\nWe firstly constucted the filtered $A_{\\infty}$-algebra \nmentioned above on suitable subcomplex of \nthe singular cochain complex of $L_i$ using systematic \nmulti-valued perturbation of Kuranshi maps describing \nthe moduli spaces.  \nWe briefly review these constructions in subsequent section.  \nThus the resulting filtered $A_{\\infty}$-algebra depends \non various choices, i.e., the choice of the subcomplex, \nthe choice of systematic multi-valued perturbation, etc.  \nIn order to make the construction canonical, we introduced \nthe notion of the {\\it canonical model}.  \nSince the structure constants of the filtered $A_{\\infty}$-algebra \ndepends on these choices, it is appropriate to work with \nthe canonical model when we make practical computation of \nthe structure constants.  \nWhen we consider ${\\mathfrak{PO}}(b)$ as a function \non the set of weak solutions of the Maurer-Cartan equation, \nwe call it the {\\it potential function}.  \nThe canonical model provides an appropriate domain of \nthe definition of the potential function.  \nThe canonical models also play a role in the convergence of a certain \nspectral sequence, see Chapter 6 in \\cite{FOOO}.  \n(In \\cite{2000}, we used another kind of \nfinitely generated complex to ensure the weak finiteness property \nof the filtered $A_{\\infty}$ algebras.)    \nIt may be also worth mentioning that we rely on canonical models in \nsome places in \\cite{FOOO}, since the degree of the ordinary \ncohomology is bounded, though the degree of the singular complex is \nnot bounded above.  \n\nWe also developed \nan algebraic theory for filtered $A_{\\infty}$-algebras, bimodules, \nin particular, \nthe homotopy theory of the filtered $A_{\\infty}$-algebras, \nbimodules and proved that the homotopy type \nof the resulting algebraic object does not depend on such choices.  \nWe can also reduce the filtered $A_{\\infty}$-structure to \nappropriate free subcomplexes of the original complex.  \nIn particular, if we work over the ground coefficient field, \nwe obtain the filtered $A_{\\infty}$-structure on the {\\it classical} \n(co)homology of the complex.  \nIn this note, we review the construction of the canonical models \nof filtered $A_{\\infty}$-algebras and filtered $A_{\\infty}$-bimodules \nand explain its implication in a geometric setting.  \n\nIn section 5, we induce the filtered $A_{\\infty}$-algebra structure \non Morse complex based on the argument in the construction of \ncanonical models.  \nWe choose a Morse function $f$ on $L$, which is adapted to \na triangulation of $L$ (see section 5).   \n\\smallskip\n\n\\noindent\n{\\bf Theorem \\ref{Morse}}\n{\\it \nLet $L$ be a relatively spin Lagrangian submanifold in a closed \nsymplectic manifold $(M,\\omega)$ and $f$ a Morse function on $L$ \nas above.  \nThen Morse complex $CM^*(f)\\otimes \\Lambda_{nov}$ \ncarries a structure of a filtered $A_{\\infty}$-algebra, \nwhich is homotopy equivalent to the filtered $A_{\\infty}$-algebra \nassociated to $L$ constructed in \\cite{FOOO}.  \n}\n\n\\smallskip\n\nThis note is based on the lecture at Yashafest and we thank  \nthe organizers for the invitation.  \n\n\\section{Flitered $A_{\\infty}$-algebras, bimodules} \nIn this subsection, we recall the definition of (filtered) \n$A_{\\infty}$-algebras, bimodules, homomorphisms \nand prepare necessary notations.  \nThen we explain the notion of homotopy between (filtered) \n$A_{\\infty}$-homomorphisms.  \nIn fact, the notion of homotopy between \n$A_{\\infty}$-homomorphisms can be found in the literature, \ne.g., \\cite{Smi}.  \nFor differential graded algebras, homotopy theory was studied \nin rational homotopy theory, see \\cite{Su}, \\cite{G-M}.  \nIn order to make clear the relation among such notions, \nwe introduced the notion of the model of \n$[0,1] \\times \\overline{C}$ ($[0,1] \\times C$) for \na (filtered) $A_{\\infty}$-algebra $C$ ($\\overline{C}$) \nand defined the notion of homotopy using such a model.  \n\n\\smallskip\n\\noindent\n{\\bf 2.1) Unfiltered $A_{\\infty}$-algebras, homomorphisms, bimodules, \nbimodule homomorphisms.}\n\nLet $R$ be a commutative ring, e.g., $\\Z$, $\\Q$.  \nLet $\\overline{C}^{\\bullet}$ be a cochain complex over $R$.  \nWe assume that $\\overline{C}^k = 0$ for $k < 0$.  \nDenote by $\\overline{\\mathfrak m}_1$ its differential.  \nSet $\\overline{C}[1]^k=\\overline{C}^{k+1}$ and denote the shifted degree by \n$\\deg' x = \\deg x -1$, where $\\deg$ is the original degree \nof $\\overline{C}^{\\bullet}$.  \nIn this section, we use only shifted degrees.  \nConsider a series of $k$-ary operations, $k=1,2, \\dots,$ \n\n\\[\n\\overline{\\mathfrak m}_k : \n(\\overline{C}[1]^{\\bullet})^{\\otimes k} \\to \\overline{C}[1]^{\\bullet}\n\\]\nof degree $1$ with respect to the shifted degrees.  \n\nBefore giving the definition of $A_{\\infty}$-algebras, we explain \nthe case of differential graded algebras.  \nLet $(\\overline{C}^{\\bullet}, d, \\cdot)$ be a differential \ngraded algebra.  \nDefine $\\overline{\\mathfrak m}_1(x)=(-1)^{\\deg x} da$ and \n$\\overline{\\mathfrak m}_2(x \\otimes y) = \n(-1)^{\\deg x \\cdot (\\deg y +1)} x \\cdot y$.  \nThen we find that \n\n\\begin{align}\n&\\overline{\\mathfrak m}_1 \\circ \\overline{\\mathfrak m}_1 (x)= 0,\n\\nonumber \\\\\n&\\overline{\\mathfrak m}_1 \\circ \\overline{\\mathfrak m}_2(x \\otimes y) \n+ \\overline{\\mathfrak m}_2 (\\overline{\\mathfrak m}_1(x_1) \\otimes x_2) \n+ (-1)^{\\deg' x_1} \\overline{\\mathfrak m}_2 (x_1 \\otimes \n\\overline{\\mathfrak m}_1(x_2))=0, \n\\nonumber \\\\\n&\\overline{\\mathfrak m}_2(\\overline{\\mathfrak m}_2(x_1 \\otimes x_2) \\otimes x_3) + (-1)^{\\deg' x_1} \\overline{\\mathfrak m}_2(x_1 \\otimes \n\\overline{\\mathfrak m}_2(x_2 \\otimes x_3))=0, \\nonumber\n\\end{align}\nwhich follow from the facts that $d$ is a differential, \nthe multiplication and the differential $d$ satisfies Leibniz' rule \nand the multiplication is associative.  \n\nThere are some geometric situations where multiplicative structures \nare defined but not exactly associative.  \nA typical example is the composition in based loop spaces.  \nIn fact, Stasheff \\cite{St} introduced the notion of \n$A_{\\infty}$-structure on topological spaces in order to characterize \nthe homotopy types of based loop spaces.  \nHe also defined the $A_{\\infty}$-structure in algebraic setting.  \nFor instance, a multiplicative structure is said to be associative \nup to homotopy, if there exists \n$\\overline{\\mathfrak m}_3: (\\overline{C}[1]^{\\bullet})^{\\otimes 3} \\to \n\\overline{C}[1]^{\\bullet}$ such that \n\n\\begin{align}\n& \\overline{\\mathfrak m}_2(\\overline{\\mathfrak m}_2(x_1 \\otimes x_2) \\otimes x_3) + (-1)^{\\deg' x_1} \\overline{\\mathfrak m}_2(x_1 \\otimes \n\\overline{\\mathfrak m}_2(x_2 \\otimes x_3)) \\nonumber \\\\ \n+ & \\overline{\\mathfrak m}_1 \\circ \\overline{\\mathfrak m}_3\n(x_1 \\otimes x_2 \\otimes x_3) + \n\\overline{\\mathfrak m}_3(\\overline{\\mathfrak m}_1(x_1) \n\\otimes x_2 \\otimes x_3) \\nonumber \\\\\n+ & (-1)^{\\deg' x_1} \\overline{\\mathfrak m}_3\n(x_1 \\otimes \\overline{\\mathfrak m}_1(x_2) \\otimes x_3)  \n+ (-1)^{\\deg' x_1 + \\deg' x_2} \\overline{\\mathfrak m}_3 \n(x_1 \\otimes x_2 \n\\otimes \\overline{\\mathfrak m}_1(x_3)) \\nonumber \\\\\n= & 0.  \\nonumber\n\\end{align}\n\nNote that it coincides with the relation corresponding to the \nassociativity, if $\\overline{\\mathfrak m}_3=0$.  \nWe can continue higher homotopies in a similar way:  \n$$\n\\sum_{k_1+k_2=k+1} \\sum_i (-1)^{\\sum_{j=1}^{i-1} \\deg' x_j} \n\\overline{\\mathfrak m}_{k_1}\n(x_1, \\dots, \\overline{\\mathfrak m}_{k_2}\n(x_i, \\dots, x_{i+k_2-1}), \\dots, x_k) = 0.\n$$  \nHere $k_1, k_2$ are positive integers.  \nFor a concise description of relations among higher homotopies, \nwe introduce the bar complex of $\\overline{C}^{\\bullet}$, which is \ndefined by \n\\[\nB(\\overline{C}[1]^{\\bullet})=\\bigoplus_{k=0}^{\\infty} B_k(\\overline{C}[1]^{\\bullet}), \\ \\ \nB_k(\\overline{C}[1]^{\\bullet})=\\bigoplus_{m_1,\\dots, m_k} \n\\overline{C}[1]^{m_1} \\otimes \\cdots \\otimes \\overline{C}[1]^{m_k}, \n\\]\nwhich we consider as a tensor coalgebra.  \nThe comultiplication is given by \n$$\n\\Delta (x_1 \\otimes \\dots \\otimes x_k) \n= \\sum_{i=0}^k (x_1 \\otimes \\dots \\otimes x_i) \\otimes (x_{i+1} \\otimes \n\\dots \\otimes x_k),\n$$ \nwhere $x_1 \\otimes \\dots \\otimes x_i, x_{i+1} \\otimes \\dots \\otimes \nx_k \\in B(\\overline{C}[1]^{\\bullet})$ and \nthe former with $i=0$ and the latter with $i=k$ are understood as \n$1 \\in B_0(\\overline{C}[1]^{\\bullet})$.  \nExtend $\\overline{\\mathfrak m}_k$ to the graded coderivation \n$\\widehat{\\overline{\\mathfrak m}_k}$ on $B(\\overline{C}[1]^{\\bullet})$. \nNamely, \n\n\\begin{align}\n& \\widehat{\\overline{\\mathfrak m}}_k(x_1 \\otimes \\dots \\otimes x_N) \n\\nonumber \\\\\n= & \\sum_{i=1}^{N-k+1} (-1)^{\\sum_{j=1}^{i-1} \\deg' x_j} \nx_1 \\otimes \\dots \\otimes x_{i-1} \\otimes \n\\overline{\\mathfrak m}_k (x_i \\otimes \\dots \\otimes x_{i+k-1}) \\otimes \nx_{i+k} \\otimes \\dots \\nonumber \\\\\n& \\hspace{0.5in} \\cdots \\otimes x_N. \\nonumber\n\\end{align}\n\nWe call $(\\overline{C}^{\\bullet}, \\{\\overline{\\mathfrak m}_k\\})$ \nan $A_{\\infty}$-algebra, if \n\\[\n\\widehat{\\overline{d}}=\\sum_k \\widehat{\\overline{\\mathfrak m}}_k:\nB(\\overline{C}[1]^{\\bullet}) \\to B(\\overline{C}[1]^{\\bullet})\n\\] \nsatisfies $\\widehat{\\overline{d}} \\circ \\widehat{\\overline{d}} = 0$.  \nIn the case that $\\overline{\\mathfrak m}_k=0$ for $k > 2$, \nthis condition \nis equivalent to the notion of differential graded algebras.  \n\nFor a collection $\\{\\overline{\\mathfrak f}_k:B_k(\\overline{C}[1]^{\\bullet}) \\to \n\\overline{C}'[1]^{\\bullet}\\}_{k=1}^{\\infty}$ of degree $0$, \nwe extend it to a homomorphism as tensor coalgebras\n\\[\n\\widehat{\\overline{f}}(x_1 \\otimes \\cdots \\otimes x_k)  \n= \n\\sum_{k_1 + \\cdots + k_n=k} \n\\overline{\\mathfrak f}_{k_1}(x_1 \\otimes \\cdots \\otimes x_{k_1}) \n\\otimes \\cdots \\otimes \n\\overline{\\mathfrak f}_{k_n}(x_{k+1-k_n} \\otimes \\cdots \\otimes x_k).\n\\]\nWe call $\\{\\overline{\\mathfrak f}_k\\}$ an $A_{\\infty}$-homomorphism, \nif $\\widehat{\\overline{f}}$ satisfies \n$\\widehat{\\overline{d}}_{\\overline{C}'} \\circ \\widehat{\\overline f} = \n\\widehat{\\overline f} \\circ \\widehat{\\overline{d}}_{\\overline{C}}.$  \n\nIn terms of the components $\\overline{\\mathfrak m}_k$'s and \n$\\overline{\\mathfrak f}_k$'s, \nthis is equivalent to \n\n\\begin{align} \n& \\sum_{i_1+ \\dots + i_k=n} \\overline{\\mathfrak m}_k(\n\\overline{\\mathfrak f}_{i_1}(x_1 \\otimes \\dots \\otimes x_{i_1}) \n\\otimes \\dots \n\\otimes \\overline{\\mathfrak f}_{i_k}(x_{i_1+ \\dots + i_{k-1} +1} \n\\otimes \n\\dots \\otimes x_{n})) \\nonumber \\\\\n= & \n\\sum_{j_1+ \\dots + j_{\\ell}=n} \\sum_{p=1}^{\\ell} \n(-1)^{\\sum_{i=1}^{j_1+ \\dots j_{p-1}} \\deg' x_i}\n\\overline{\\mathfrak f}_{j_1}(x_1 \\otimes \\dots \\otimes x_{j_1}) \\otimes \n\\dots \\otimes \\nonumber \\\\\n& \\overline{\\mathfrak m}_{j_p}(x_{j_1+ \\dots + j_{p-1}+1} \n\\otimes \\dots \\otimes \nx_{j_1+ \\dots + j_p}) \\otimes \\dots \\otimes \n\\overline{\\mathfrak f}_{j_{\\ell}}(x_{j_1+ \\dots j_{\\ell -1} +1} \n\\otimes \\dots \\otimes x_n) \\nonumber\n\\end{align}\n\nLet $(\\overline{C}^{\\bullet}_i, \\{\\overline{\\mathfrak m}_k^{(i)}\\})$, \n$i=0,1$, \nbe $A_{\\infty}$-algebras, $\\overline{D}^{\\bullet}$ a graded module and \n$\\overline{\\mathfrak n}_{k_1,k_0}:B_{k_1}(\\overline{C}_1[1]^{\\bullet}) \\otimes \n\\overline{D}[1]^{\\bullet} \\otimes B_{k_0}(\\overline{C}_0[1]^{\\bullet}) \\to \n\\overline{D}[1]^{\\bullet}$ homomorphisms of degree $1$.  \nWe call $(\\overline{D}^{\\bullet},\\{\\overline{\\mathfrak n}_{k_1,k_0}\\})$ \nan $A_{\\infty}$-bimodule, if \n\\[\n\\widehat{\\overline{d}}_{\\overline{\\mathfrak n}} \\circ \n\\widehat{\\overline{d}}_{\\overline{\\mathfrak n}} = 0, \n\\]\nwhere $\\widehat{\\overline{d}}_{\\overline{\\mathfrak n}}$ is \ndefined on \n$\nB(\\overline{C}_1[1]^{\\bullet}) \\otimes \\overline{D}[1]^{\\bullet} \\otimes \nB(\\overline{C}_0[1]^{\\bullet})\n$ as follows:  \n\n\\begin{align}\n& \\widehat{\\overline{d}}_{\\overline{\\mathfrak n}}\n(x_{1,1} \\otimes \\dots \\otimes x_{1,k_0} \\otimes y \\otimes \nx_{0,1} \\otimes \\dots \\otimes x_{0,k_0}) \\nonumber \\\\\n= & \\widehat{\\overline{d}}^{(1)}(x_{1,1} \\otimes \\dots \n\\otimes x_{1,k_1}) \n\\otimes y \\otimes x_{0,1} \\otimes \\dots \\otimes x_{0,k_0} \\nonumber \\\\\n& + \\sum_{k'_1 \\leq k_1, k'_0 \\leq k_0} \n(-1)^{\\sum_{i=1}^{k_1-k'_1} \\deg' x_i}\nx_{1,1} \\otimes \\dots x_{k_1-k'_1} \\nonumber \\\\\n& \\otimes {\\overline{\\mathfrak n}}_{k'_1,k'_0}(x_{1,k_1-k'_1+1} \n\\otimes \\dots x_{k_1} \\otimes y \\otimes x_{0,1} \n\\otimes \\dots \n\\otimes x_{0,k'_0}) \\otimes x_{0,k_0+1} \\otimes \\dots \\otimes x_{0,k_0} \n\\nonumber \\\\\n& + (-1)^{\\sum_{i=1}^{k_1} \\deg' x_i + \\deg' y}\nx_{1,1} \\otimes \\dots x_{1,k_1} \\otimes y \\otimes \\widehat{\\overline{d}}^{(0)}(x_{0,1} \\otimes \\dots \\otimes x_{0,k_0}). \\nonumber\n\\end{align}\n\nThe condition for $(\\overline{D},\\{\\overline{\\mathfrak n}_{k_1,k_0}\\})$ \nto be an $A_{\\infty}$-bimodules over $\\overline{C}_i$, $i=0,1$ is \nequivalent to the identity \n \n\\begin{align}\n& \\overline{\\mathfrak n}_{*,*}(\\widehat{\\overline{d}}^{(1)}(\nx_{1,1} \\otimes \\dots \\otimes x_{1,k_1}) \\otimes y \\otimes \nx_{0,1} \\otimes \\dots \\otimes x_{0,k_0}) \\nonumber \\\\\n+ & \\sum_{k'_1 \\leq k_1, k'_0 \\leq k_0} (-1)^{\\sum_{i=1}^{k_1-k'_1} \n\\deg' x_i} \n\\overline{\\mathfrak n}_{k_1-k'_1,k_0-k'_0}\n(x_{1,1} \\otimes \\dots \\otimes \nx_{k_1-k'_1} \\otimes \\nonumber \\\\\n& \\overline{\\mathfrak n}_{k'_1,k'_0}\n(x_{1,k_1-k'_1+1} \\otimes \\dots \\otimes x_{1,k_1} \\otimes y \n\\otimes x_{0,1} \\otimes x_{0,k'_0}) \\otimes x_{0,k'_0+1} \\otimes \n\\dots \\otimes x_{0,k_0}) \\nonumber \\\\\n+ & (-1)^{\\sum_{i=1}^{k_1} \\deg' x_i + \\deg' y} \n\\overline{\\mathfrak n}_{*,*} (x_{1,1} \\otimes \\dots \\otimes \nx_{1,k_1} \\otimes y \\otimes \\widehat{\\overline{d}}^{(0)} \n(x_{0,1} \\otimes \\dots \\otimes x_{0,k_0})) \\nonumber \\\\\n= & 0. \\nonumber\n\\end{align}\n\nHere $\\overline{\\mathfrak n}_{*,*}:B(\\overline{C}_1[1]^{\\bullet}) \n\\otimes \\overline{D}[1] \\otimes B(\\overline{C}_0[1]^{\\bullet}) \n\\to \\overline{D}[1]$ is defined to be \n$\\overline{\\mathfrak n}_{k_1,k_0}$ on \n$B_{k_1}(\\overline{C}_1[1]^{\\bullet}) \n\\otimes \\overline{D}[1] \\otimes B_{k_0}(\\overline{C}_0[1]^{\\bullet})$.  \n\nLet $\\{\\overline{\\mathfrak f}_k^{(i)}:\\overline{C}_i[1]^{\\bullet} \\to \n\\overline{C}_i'[1]^{\\bullet}\\}$, $i=0,1$, be $A_{\\infty}$-homomorphisms \nand $\\overline{D}^{\\bullet}$, resp. $\\overline{D'}^{\\bullet}$ \nan $A_{\\infty}$-bimodules over \n$\\overline{C}_i$, resp. $\\overline{C}_i'$, $i=0,1$.  \nFor a collection $\\{\\overline{\\phi}_{k_1,k_0}\\}:\nB_{k_1}(\\overline{C}_1[1]^{\\bullet}) \\otimes \\overline{D}[1]^{\\bullet} \n\\otimes \nB_{k_0}(\\overline{C}_0[1]^{\\bullet}) \n\\to \\overline{D}'[1]^{\\bullet}$ of degree $0$, we define \n\\[\n\\widehat{\\overline{\\phi}}:B(\\overline{C}_1[1]^{\\bullet}) \\otimes \\overline{D}[1]^{\\bullet} \n\\otimes B(\\overline{C}_0[1]^{\\bullet}) \\to \nB(\\overline{C}_1'[1]^{\\bullet}) \\otimes \\overline{D}'[1]^{\\bullet} \\otimes \nB(\\overline{C}_0'[1]^{\\bullet})\n\\]\nas the homomorphism determined by $\\{\\overline{\\mathfrak f}_k^{(i)}\\}$, \n$i=0,1$ and $\\{\\overline{\\phi}_{k_1,k_0}\\}$.  \nWe call $\\{\\overline{\\phi}_{k_1,k_0}\\}$ a homomorphism of \n$A_{\\infty}$-bimodules, if \n\\[\n\\widehat{\\overline{d}}_{\\overline{\\mathfrak n}'} \\circ \n\\widehat{\\overline{\\phi}} = \\widehat{\\overline{\\phi}} \\circ \n\\widehat{\\overline{d}}_{\\overline{\\mathfrak n}}.\n\\]\n \n\\smallskip\n\\noindent \n{\\bf 2.2) The universal Novikov ring and the energy filtration.}\n\nTo explain the notion of filtered $A_{\\infty}$-algebras,   \nwe introduce the universal Novikov ring.  \nLet $e$ and $T$ be formal variables of degree $2$ and $0$, \nrespectively.  \nSet  \n\\begin{align}\n \\Lambda_{nov} & = \\{ \\sum_i a_i e^{\\mu_i} T^{\\lambda_i} \\ \n\\vert \\ a_i \\in R, \n\\ \\mu_i \\in \\Z, \\ \\lambda_i \\in \\R, \\ \n\\lambda_i \\to +\\infty (i \\to +\\infty)\\} \\nonumber \\\\\n \\Lambda_{0,nov} & = \n\\{ \\sum_i a_i e^{\\mu_i} T^{\\lambda_i} \\in \\Lambda_{nov} \n\\ \\vert \\ \\lambda_i \\geq 0 \\}. \\nonumber \n\\end{align}\n\nSet \n$$C^{\\bullet}=\n\\{\\sum c_i e^{\\mu_i} T^{\\lambda_i} \\vert \nc_i \\in \\overline{C}^{\\bullet}, \\mu_i \\in \\Z, \\lambda_i \\in \\R, \n\\lambda_i \\to +\\infty (i \\to +\\infty)\\},\n$$\nwhich is the completion of the graded tensor product \n$\\overline{C}^{\\bullet} \\otimes_R \\Lambda_{0,nov}$ with respect to \nthe energy filtration given below. \nWe define the filtration defined by \n\\[\nF^{\\lambda}C^{\\bullet}= \\{ \\sum_i x_i e^{\\mu_i} T^{\\lambda_i} \\in C^{\\bullet}\n\\ \\vert \\ x_i \\in \\overline{C}^{\\bullet}, \\lambda_i \\geq \\lambda \\} \n\\]\non $C^{\\bullet}$ and denote by \n$\nF^{\\lambda}(C[1]^{m_1} \\otimes \\cdots \\otimes \nC[1]^{m_k})\n$ \nthe submodule of $\\C[1]^{m_1} \\otimes \\cdots \\otimes C[1]^{m_k}$ \nspanned by \n\\[\nF^{\\lambda_1}(C[1]^{m_1}) \\otimes \\cdots \\otimes \nF^{\\lambda_k}(C[1]^{m_k}), \\ \\ \n\\sum_{i=1}^k \\lambda_i = \\lambda.\n\\]  \nDefine the bar complex of $C[1]^{\\bullet}$ by \nthe completion with respect to the energy filtration and denote it by  \n$B_k(C[1]^{\\bullet})$.\n\n\\smallskip\n\\noindent\n{\\bf 2.3) Filtered case and $G$-gapped conditions.}\n\nConsider the $k$-ary operations, $k=0,1,2, \\dots$,  \n\\[\n{\\mathfrak m}_k:B_k(C[1]^{\\bullet}) \\to C[1]^{\\bullet}\n\\]\nsuch that \n\\[\n{\\mathfrak m}_k(F^{\\lambda_1}C[1]^{\\bullet} \\otimes \\cdots \n\\otimes F^{\\lambda_k} C[1]^{\\bullet}) \\subset F^{\\lambda_1 + \\cdots \n\\lambda_k} C[1]^{\\bullet}\n\\]\nand \n\\[\n{\\mathfrak m}_0(1) \\in F^{\\lambda'}C[1]^{\\bullet} \n\\text{ for some } \\lambda'>0.\n\\]\n\nWe used the induction on the energy level in various arguments \nin \\cite{FOOO}, see also section 3 in this note.  \nFor such purposes, we introduced the \n$G$-gapped condition, which we assume from now on, as follows.  \nNote that the $G$-gapped condition follows from Gromov's compactness \ntheorem in the case of symplectic Floer theory.  \nLet $G \\subset \\R_{\\geq 0} \\times 2\\Z$ be a monoid such that \n${\\rm pr}_1^{-1}([0,c])$ is finite for any $c \\geq 0$ and \n\\[\n{\\rm pr}_1^{-1} (0) = \\{{\\mathbf 0}=(0,0)\\}.\n\\]\nHere ${\\rm pr}_i$ is the projection to the $i$-th factor, $i=1,2$.  \nThe filtered $A_{\\infty}$-algebra is said to be $G$-gapped, if \nthere exist \n\\[\n{\\mathfrak m}_{k,\\beta_i}:B_k(\\overline{C}[1]^{\\bullet}) \\to \n\\overline{C}[1]^{\\bullet}\n\\]\nfor $\\beta_i=(\\lambda_i,\\mu_i) \\in G \\subset \\R_{\\geq 0} \\times 2\\Z$ \nsuch that \n${\\mathfrak m}_{0,{\\mathbf 0}}=0$ and \n\\[\n{\\mathfrak m}_k = \\sum_{i} \nT^{\\lambda_i} e^{\\mu_i\/2} {\\mathfrak m}_{k, \\beta_i}.  \n\\] \nExtend ${\\mathfrak m}_k$ to the graded coderivation \n$\\widehat{\\mathfrak m}_k$ on $B(C[1]^{\\bullet})$.  \nWe call $(C^{\\bullet},\\{{\\mathfrak m}_k\\})$ a filtered \n$A_{\\infty}$-algebra, if \n\\[\n\\widehat{d} = \\sum_k \\widehat{\\mathfrak m}_k:B(C[1]^{\\bullet}) \\to B(C[1]^{\\bullet})\n\\] \nsatisfies $\\widehat{d} \\circ \\widehat{d} = 0$.  \nIn other words, \n$$\n\\sum_{k_1+k_2=k+1} \\sum_i (-1)^{\\sum_{j=1}^{i-1} \\deg' x_j} \n{\\mathfrak m}_{k_1}\n(x_1, \\dots, {\\mathfrak m}_{k_2}\n(x_i, \\dots, x_{i+k_2-1}), \\dots, x_k) = 0.\n$$  \nHere $k_1$ is a positive integer and $k_2$ is a non-negative \nintegers.  \nWhen $k_2=0$, ${\\mathfrak m}_{k_2}(x_i, \\dots, x_{i+k_2-1})$ \nis understood as ${\\mathfrak m}_0(1)$.  \n\nFor a filtered $A_{\\infty}$-algebra \n$(C^{\\bullet},\\{{\\mathfrak m}_k\\})$,\nset $\\overline{\\mathfrak m}_k= {\\mathfrak m}_{k,{\\bf 0}}$, \n${\\bf 0}=(0,0) \\in \\R_{\\geq 0} \\times 2\\Z$.  \nThen $(\\overline{C}^{\\bullet},\\{\\overline{\\mathfrak m}_k\\})$ is an \n$A_{\\infty}$-algebra.  \nWe call $(C^{\\bullet},\\{{\\mathfrak m}_k\\})$ a deformation of \n$(\\overline{C}^{\\bullet},\\{\\overline{\\mathfrak m}_k\\})$.  \n\nNote that ${\\mathfrak m}_1 \\circ {\\mathfrak m}_1$ may not be zero \nand we have \n$$\n{\\mathfrak m}_1 \\circ {\\mathfrak m}_1 (x) + {\\mathfrak m}_2(\n{\\mathfrak m}_0(1), x) + (-1)^{\\deg' x} {\\mathfrak m}_2(x,\n{\\mathfrak m}_0(1)) = 0.\n$$\nWe set \n$$\ne^b=1+b+b\\otimes b + b \\otimes b \\otimes b + \\dots, \n$$\nfor $b \\in {\\mathcal F}^{\\lambda}(C[1]^{0})$ with $\\lambda > 0$ \nand consider the {\\it Maurer-Cartan} equation: \n$$\n\\widehat{d}(e^b) = 0,\n$$\nwhich is equivalent to \n$$\n{\\mathfrak m}_0(1)+{\\mathfrak m}_1(b)+{\\mathfrak m}_2(b,b)+ \n{\\mathfrak m}_3(b,b,b) + \\dots = 0.\n$$\nFor a given $b$, we define a coalgebra homomorphism\n$$\n\\Phi^b(x_1 \\otimes x_2 \\otimes \\dots \\otimes x_k) \n=e^b \\otimes x_1 \\otimes e^b \\otimes x_2 \\otimes e^b \n\\otimes \\dots \\otimes e^b \\otimes x_k \\otimes e^b.\n$$\nThen define \n$$\n{\\mathfrak m}_k^b(x_1 \\otimes \\dots \\otimes x_k)\n={\\mathfrak m}_* \\circ \\Phi^b(x_1 \\otimes \\dots \\otimes x_k),\n$$\nwhere \n${\\mathfrak m}_*:B(C[1]^{\\bullet}) \\to C[1]^{\\bullet}$ is defined \nby ${\\mathfrak m}_*\\vert_{B_k(C[1]^{\\bullet})}={\\mathfrak m}_k$.  \nThen, for a solution $b$ of the Maurer-Cartan equation, we find that \n${\\mathfrak m}_0^b(1)=0$, hence ${\\mathfrak m}_1^b \\circ \n{\\mathfrak m}_1^b=0$.  \nNamely, the original ${\\mathfrak m}_1$ is rectified to \na coboundary operator ${\\mathfrak m}_1^b$ using a solution of \nthe Maurer-Cartan equation, which we also call a bounding cochain.  \n\n\nFor a collection $\\{ {\\mathfrak f}_k:B_k(C[1]^{\\bullet}) \\to C'[1]^{\\bullet} \\}_{k=0}\n^{\\infty}$ of degree $0$, we define \n\\[\n\\widehat{\\mathfrak f}(x_1 \\otimes \\cdots \\otimes x_k) \n=  \n\\sum_{k_1 + \\cdots k_n=k} \n{\\mathfrak f}_{k_1}(x_1 \\otimes \\cdots \\otimes x_{k_1}) \n\\otimes \\cdots \\otimes \n{\\mathfrak f}_{k_n}(x_{k+1-k_n} \\otimes \\cdots \\otimes x_k), \n\\]\nfor $k>0$ and \n\\[\n\\widehat{\\mathfrak f}(1)=1 + {\\mathfrak f}_0(1) + \n{\\mathfrak f}_0(1) \\otimes {\\mathfrak f}_0(1) + \\cdots,\n\\]\nwhere $1 \\in \\Lambda_{0,nov}=B_0(C[1]^{\\bullet})$.  \nWe assume the $G$-gapped condition, i.e., \nthere exist \n\\[\n{\\mathfrak f}_{k,\\beta_i}:B_k(\\overline{C}[1]^{\\bullet}) \\to \n\\overline{C}'[1]^{\\bullet}\n\\] \nfor $\\beta_i=(\\lambda_i,\\mu_i) \\in G$ \nwith $\\lambda_i \\to +\\infty$ as \n$i \\to +\\infty$ \nsuch that \n\\[\n{\\mathfrak f}_k = \\sum_i T^{\\lambda_i} e^{\\mu_i\/2} \n{\\mathfrak f}_{k,\\beta_i}.\n\\]\nIn particular, $\\widehat{\\mathfrak f}$ preserves the energy \nfiltration.  Namely, \n\\[\n\\widehat{\\mathfrak f}(F^{\\lambda}B(C[1]^{\\bullet})) \\subset F^{\\lambda}C'[1]^{\\bullet},\n\\]\nwhere $\\{F^{\\lambda}B(C[1]^{\\bullet})\\}$ is the filtration derived from \nthe filtration $F^{\\lambda}$ on $C[1]^{\\bullet}$.  \nWe call $\\{ {\\mathfrak f}_k \\}$ a $G$-gapped \nfiltered $A_{\\infty}$-homomorphism, \nif \n$\\widehat{d}_{C}' \\circ \\widehat{\\mathfrak f} = \\widehat{\\mathfrak f} \n\\circ \\widehat{d}_{C}$.  \nWhen we do not specify the monoid $G$, we call gapped filtered $A_{\\infty}$-algebras, gapped filtered $A_{\\infty}$-homomorphisms, etc.  \n\n\\smallskip\n\\noindent\n{\\bf 2.4) Homotopy theory.}\n\nIn \\cite{FOOO}, we introduced the notion of models of \n$[0,1] \\times C^{\\bullet}$ (Definition 15.1) and gave two constructions. Using this notion, we developed \nthe homotopy theory of filtered\n$A_{\\infty}$-algebras and filtered-$A_{\\infty}$ bimodules.\nOur formulation has an advantage to clarify \nequivalence of various definitions of homotopy\nof $A_{\\infty}$ algebras appearing in the literature\neven for the unfiltered cases.\n\nLet $C$ be the completion of $\\overline{C} \\otimes \\Lambda_0$, which is \na filtered $A_{\\infty}$-algebra. \n\n\\begin{defn}\nLet ${\\mathfrak C}$ be the completion of $\\overline{\\mathfrak C} \\otimes \\Lambda_0$, which is \na filtered $A_{\\infty}$-algebra together with \nfiltered $A_{\\infty}$-homomorphisms.  \n\\[\n{\\rm Incl}:C \\to {\\mathfrak C}, \\ \n{\\rm Eval}_{s=i}:{\\mathfrak C} \\to C, i=0,1.\n\\]  \nWe call $\\mathfrak C$ a model of $[0,1] \\times C$, if \nthe following conditions are satisfied:  \n\n\\begin{itemize}\n\\item\n${\\rm Incl}_{k,\\beta}$ and ${\\rm Eval}_{s=i}, i=0,1$ are \nzero unless $(k,\\beta)=(1,\\beta_0)$.  \n\\item\n${\\rm Incl}_{1,\\beta_0}, ({\\rm Eval}_{s=0})_{1,\\beta_0}$ \nare cochain homotopy equivalences between \n$\\overline{C}$ and $\\overline{\\mathfrak C}$.  \n\\item\n${\\rm Eval}_{s=0} \\circ {\\rm Incl} = {\\rm Eval}_{s=1} \\circ {\\rm Incl} \n= {\\rm id.}$  \n\\item\n${\\rm Eval}_{s=0} \\oplus {\\rm Eval}_{s=1}:{\\mathfrak C} \\to C \\oplus C$ \nis surjective.  \n\\end{itemize}\n\\end{defn}\n\nWe quote here one of constructions of the models of $[0,1] \\times C$ \nfor reader's convenience.  \n\nSet $$C^{[0,1]}=C \\oplus C[-1] \\oplus C,$$\nand define ${\\mathfrak I}_0, {\\mathfrak I}_1:C \\to C^{[0,1]}$ of \ndegree 0 and \n${\\mathfrak I}:C \\to C^{[0,1]}$ of degree 1 by \n$$\n{\\mathfrak I}_0(x)=(x,0,0), {\\mathfrak I}_1(x)=(0,0,x), \n{\\mathfrak I}_1(x)=(0,x,0).\n$$\nWe extend ${\\mathfrak I}_0,{\\mathfrak I}_1$ to \n$B(C[1]) \\to B(C^{[0,1]}[1])$ and denote them by the same symbol.  \nDefine \n\n\\begin{align}\n({\\rm Eval}_{s=0})_1(x,y,z)=x, \\ ({\\rm Eval}_{s=1})_1(x,y,z)=z, \n\\nonumber \\\\\n({\\rm Incl})_1(x)={\\mathfrak I}_0(x)+{\\mathfrak I}_1(x)=(x,0,x). \n\\nonumber\n\\end{align}\n\nWe define the filtered $A_{\\infty}$-structure $\\{{\\mathfrak M}_k\\}$.  \n\nFor ${\\mathfrak M}_0$, ${\\mathfrak M}_1$, we set  \n\\begin{align}\n{\\mathfrak M}_0(1) & \n= ({\\rm Incl})_1({\\mathfrak m}_0(1)), \\nonumber \\\\\n{\\mathfrak M}_1({\\mathfrak I}_0(x)) & \n={\\mathfrak I}_0({\\mathfrak m}_1(x)) \n+ (-1)^{\\deg' x} {\\mathfrak I}(x), \\nonumber \\\\\n{\\mathfrak M}_1({\\mathfrak I}_1(x)) & \n={\\mathfrak I}_1({\\mathfrak m}_1(x)) - (-1)^{\\deg' x} {\\mathfrak I}(x), \n\\nonumber \\\\\n{\\mathfrak M}_1({\\mathfrak I}(x)) & \n={\\mathfrak I}({\\mathfrak m}_1(x)).\n\\nonumber\n\\end{align}\n\nWe define ${\\mathfrak M}_k$, $k \\geq 2$ as follows.  \nFor ${\\mathbf x} \\in B_k(C[1])$, $y \\in C$ and \n${\\mathbf z} \\in B_{\\ell}(C[1])$, we set \n$$\n{\\mathfrak M}_{k+\\ell + 1}({\\mathfrak I}_0({\\mathbf x}), {\\mathfrak I}(y), {\\mathfrak I}_1({\\mathbf z})) = \n(-1)^{\\deg' {\\mathbf z}} {\\mathfrak I}({\\mathfrak m}_{k + \\ell + 1} \n({\\mathbf x}, y, {\\mathbf z}),\n$$\nand \n$$\n{\\mathfrak M}_k({\\mathfrak I}_0({\\mathbf x}))=\n{\\mathfrak I}_0({\\mathfrak m}_k({\\mathbf x})), \n{\\mathfrak M}_{\\ell}({\\mathfrak I}_1({\\mathbf z}))=\n{\\mathfrak I}_1({\\mathfrak m}_{\\ell}({\\mathbf z})), \\ \\ \n{\\text{ for }} k, \\ell \\geq 2.\n$$  \nHere the order of \n${\\mathfrak I}_0({\\mathbf x}), {\\mathfrak I}(y), \n{\\mathfrak I}_1({\\mathbf z})$ is important.  \nWe define operators ${\\mathfrak M}_k$ on $C^{[0,1]}$ other than \nthose defined above to be zero.  \n\n\nModels of $[0,1] \\times C$ are not unique, but we proved the following:\n\n\\begin{thm}[Theorem 15.34 in \\cite{FOOO}]\nLet $C_1,C_2$ be gapped filtered $A_{\\infty}$-algebras and \n${\\mathfrak C}_1, {\\mathfrak C}_2$ any models for $[0,1] \\times C_1, \n[0,1] \\times C_2$, respectively.  \nLet ${\\mathfrak f}:C_1 \\to C_2$ be a gapped filtered \n$A_{\\infty}$-homomorphism.  \nThen there exists a gapped filtered $A_{\\infty}$-homomorphism \n${\\mathfrak F}:{\\mathfrak C}_1 \\to {\\mathfrak C}_2$ such that \n$$\n{\\rm Eval}_{s=s_0} \\circ {\\mathfrak F} = {\\mathfrak f} \\circ \n{\\rm Eval}_{s=s_0}, \\ s_0=0,1\n$$\nand \n$$\n{\\rm Incl} \\circ {\\mathfrak f} = {\\mathfrak F} \\circ {\\rm Incl}.\n$$\n\\end{thm}\n\nWe define two filtered $A_{\\infty}$-homomorphisms \n${\\mathfrak f}_i:C_1 \\to C_2, i=0,1$ are homotopic, if \nthere is a model ${\\mathfrak C}_2$ of $C_2$ and \na filtered $A_{\\infty}$-homomorphism \n${\\mathfrak F}:C_1 \\to {\\mathfrak C}_2$ such that \n${\\mathfrak f}_i={\\rm Eval}_{s=i} \\circ {\\mathfrak F}$.  \nAlthough the definition literally depends on the choice of the model \n${\\mathfrak C}_2$, \nwe can show that the notion of homotopy between ${\\mathfrak f}_i$ \ndoes not depend on the choice of the model and the homotopy \nis, in fact, an equivalence relation, see Chapter 4 in \\cite{FOOO}.  \n\nNote that the notion of homotopy between $A_{\\infty}$-homomorphisms \nin the unfiltered case appeared in literaturure, e.g., \\cite{Smi}.  \nBy taking a suitable model, we can find that \nour definition above coincides with such a definition.  \nIt also implies that various definitions which appear in the literature \nare equivalent to one another.  \nWe think that the notion of models clarifies arguments and is \nalso useful when we consider the gauge equivalence between \nsolutions of the Maurer-Cartan equation.  \nNamely, two solutions $b, b'$ of the Maurer-Cartan equation is \ngauge equivalent, if there is a model ${\\mathfrak C}$ of \n$[0,1] \\times C$ and a solution $\\widetilde{b}$ of the Maurer-Cartan \neqution on ${\\mathfrak C}$ such that \n${\\rm Eval}_{s=0}(\\widetilde{b})=b$ and \n${\\rm Eval}_{s=1}(\\widetilde{b})=b'$.  \nFor details, see section 16 in \\cite{FOOO}.  \n\nAmong other things, we proved the Whitehead type theorem as follows.  \nAn $A_{\\infty}$-homomorphism $\\{\\overline{\\mathfrak f}_k\\}$ from \n$\\overline{C}^{\\bullet}$ to $\\overline{C}^{' \\bullet}$ \nis called a weak homotopy equivalence, if \n$\\overline{f}_1:\\overline{C}^{\\bullet} \\to \\overline{C}^{' \\bullet}$ \nis a cochain homotopy equivalence \nbetween $\\overline{\\mathfrak m}_1$-complexes. \nA filtered $A_{\\infty}$-homomorphism $\\{{\\mathfrak f}_k\\}$ from \n$C[1]^{\\bullet}$ to $C'[1]^{\\bullet}$ is called \na weak homotopy equivalence, if \n$\\overline{\\mathfrak f}_1={\\mathfrak f}_{1,{\\bf 0}}$ is a cochain \nhomotopy equivalence between \n$\\overline{\\mathfrak m}_1={\\mathfrak m}_{1,{\\bf 0}}$-complexes.  \n\n\\begin{thm}[Theorem 15.45 in \\cite{FOOO}]\\label{whitehead}\n(1) A weak homotopy equivalence of $A_{\\infty}$-algebras is \na homotopy equivalence.  \n\n\\noindent\n(2) A gapped weak homotopy equivalence between gapped filtered \n$A_{\\infty}$-algebras is a homotopy equivalence.  \nThe homotopy inverse of a strict weak homotopy equivalence can be taken \nto be strict.  \n\\end{thm}\n\nNote that the above theorem does not hold in the realm of \ndifferential graded algebras.  The notion of $A_{\\infty}$-homomorphism \nis much wider than that of homomorphisms as differential graded \nalgebras.  \n\n\\smallskip\n\\noindent\n{\\bf 2.5) Filtered $A_{\\infty}$-bimodules.} \n\nLet $(C_i^{\\bullet},\\{{\\mathfrak m}_k^{(i)}\\})$, $i=0,1$, be filtered \n$A_{\\infty}$-algebras and $\\overline{D}^{\\bullet}$ a graded module.  \nWrite \n\\[\nD[1]^{\\bullet}=\\overline{D}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}\n\\]\nand \n\\[\n\\widetilde{D}[1]^{\\bullet}=\\overline{D}[1]^{\\bullet} \\otimes \\Lambda_{nov}.\n\\]\nLet \n${\\mathfrak n}_{k_1,k_0}:B_{k_1}(C_1[1]^{\\bullet}) \\otimes D[1]^{\\bullet} \\otimes \nB_{k_0}(C_0[1]^{\\bullet}) \\to D[1]^{\\bullet}$, $k_1,k_0=0,1,2,\\dots$, \nbe $\\Lambda_{0,nov}$-module homomorphisms of degree $1$.  \nWe also denote its extension to $\\widetilde{D}[1]^{\\bullet}$ by the same symbol \n${\\mathfrak n}_{k_1,k_0}$.  \nWe call $(D^{\\bullet},\\{{\\mathfrak n}_{k_1,k_0}\\})$ and \n$(\\widetilde{D}^{\\bullet},\\{{\\mathfrak n}_{k_1,k_0}\\})$ a filtered \n$A_{\\infty}$-bimodule over $(C_i^{\\bullet},\\{{\\mathfrak m}_k\\})$, if \n\\[\n\\widehat{d}_{\\mathfrak n} \\circ \\widehat{d}_{\\mathfrak n} = 0,\n\\]\nwhere $\\widehat{d}_{\\mathfrak n}$ is the coderivation \non $B(C_1[1]^{\\bullet}) \\otimes D[1]^{\\bullet} \\otimes B(C_0[1]^{\\bullet})$ determined by \n$\\{{\\mathfrak m}_k^{(i)}\\}$, $i=0,1$, and \n$\\{{\\mathfrak n}_{k_1,k_0}\\}$.  \nThe $G$-gapped condition is defined in a similar way to \nthe case of filtered $A_{\\infty}$-algebras: \n\\[\n{\\mathfrak n}_{k_1,k_0} = \\sum_{\\beta \\in G} T^{{\\rm pr}_1(\\beta)} \ne^{{\\rm pr}_2(\\beta)\/2} {\\mathfrak n}_{k_1,k_0,\\beta}, \n\\]\nwhere \n\\[\n{\\mathfrak n}_{k_1,k_0,\\beta}:B_{k_1}(\\overline{C}_1[1]^{\\bullet}) \\otimes \n\\overline{D}[1]^{\\bullet} \\otimes \nB_{k_0}(\\overline{C}_0[1]^{\\bullet}) \\to \\overline{D}[1]^{\\bullet}.  \n\\]\nFor $\\lambda \\in \\R$, we set \n\\[\n{\\mathcal F}^{\\lambda}\\widetilde{D}^{\\bullet}=T^{\\lambda} \\cdot \nD^{\\bullet}.\n\\]  \nFor $\\lambda \\geq 0$, \n${\\mathcal F}^{\\lambda}\\widetilde{D}^{\\bullet} \\subset D^{\\bullet}$, \nhence we obtain  \na filtration on $D^{\\bullet}$.  \nIt is clear that \n\\[\n\\widehat{d}_{\\mathfrak n}(F^{\\lambda_1}B(C_1[1]^{\\bullet}) \\otimes \n{\\mathcal F}^{\\lambda_2}\\widetilde{D}^{\\bullet} \n\\otimes F^{\\lambda_3}B(C_0[1]^{\\bullet})) \n\\subset {\\mathcal F}^{\\lambda_1+\\lambda_2+\\lambda_3}\\widetilde{D}[1]^{\\bullet}.  \n\\]\n\nNote that ${\\mathfrak n}_{0,0} \\circ {\\mathfrak n}_{0,0}$ may not \nbe zero and we have \n$$\n{\\mathfrak n}_{0,0} \\circ {\\mathfrak n}_{0,0}(y) \n+ {\\mathfrak n}_{1,0}({\\mathfrak m}_0^{(1)}(1), y) \n+(-1)^{\\deg' y} {\\mathfrak n}_{0,1}(y,{\\mathfrak m}_0^{(0)}(1)) = 0.\n$$\nFor $b_i \\in {\\mathfrak F}^{\\lambda^{(i)}}(C_i[1]^{\\bullet})$ \nwith $\\lambda^{(i)} > 0$, we define \n$$\n{\\mathfrak n}^{b_0,b_1}_{k_1,k_0}({\\mathbf x} \\otimes y \\otimes \n{\\mathbf z})={\\mathfrak n}_{*,*}(\\Phi^{b_1}({\\mathbf x}) \n\\otimes y \\otimes \\Phi^{b_0}({\\mathbf z})), \n$$\nfor ${\\mathbf x} \\in B_{k_1}(C_1[1]^{\\bullet})$ and \n${\\mathbf z} \\in B_{k_0}(C_0[1]^{\\bullet})$.  \nIn particular, \n$$\n{\\mathfrak n}_{0,0}^{b_0,b_1}(y)=\n{\\mathfrak n}_{*,*}(e^{b_1} \\otimes y \\otimes e^{b_0}).\n$$\nIf $b_0, b_1$ are solutions of the Maurer-Cartan equations \nin the filtered $A_{\\infty}$-algebras $C_0, C_1$, respectively, \nwe find that \n$${\\mathfrak n}_{0,0}^{b_0,b_1} \\circ \n{\\mathfrak n}_{0,0}^{b_0,b_1} = 0.$$  \nNamely, we can rectify the original ${\\mathfrak n}_{0,0}$ \nto a coboundary operator ${\\mathfrak n}_{0,0}^{b_0,b_1}$ on $D[1]$.  \n\nLet $\\{{\\mathfrak f}_k^{(i)}\\}$, $i=0,1$, be \nfiltered $A_{\\infty}$-homomorphisms from $C_i^{\\bullet}$ to \n$C_i^{' \\bullet}$ and \n$(\\widetilde{D}^{\\bullet},\\{{\\mathfrak n}_{k_1,k_0}\\})$, resp. \n$(\\widetilde{D}^{' \\bullet},\\{{\\mathfrak n}_{k_1,k_0}'\\})$, \nfiltered $A_{\\infty}$-bimodules over \n$(C_i^{\\bullet},\\{{\\mathfrak m}_k^{(i)}\\})$, resp. \n$(C_i^{' \\bullet},\\{{\\mathfrak m}_k{'(i)}\\})$.  \nSuppose that there exist a real number $c$ and \n$\\Lambda_{nov}$-homomorphisms \n${\\phi}_{k_1,k_0}:B_{k_1}(C_1[1]^{\\bullet}) \\otimes \\widetilde{D}[1]^{\\bullet} \n\\otimes B_{k_0}(C_0[1]^{\\bullet}) \n\\to \\widetilde{D}'[1]^{\\bullet}$, $k_1,k_0=0,1,2,\\dots$, \nsuch that \n\\[\n{\\phi}_{k_1,k_0}(F^{\\lambda_1}B_{k_1}(C_1[1]^{\\bullet}) \\otimes \n{\\mathcal F}^{\\lambda_2} \\widetilde{D}[1]^{\\bullet} \\otimes \nF^{\\lambda_3}B_{k_0}(C_0[1]^{\\bullet})) \\subset \n{\\mathcal F}^{\\lambda_1+\\lambda_2+\\lambda_3 -c}\\widetilde{D}'[1]^{\\bullet}.\n\\]\nWe call such $c$ the energy loss of $\\{{\\phi}_{k_1,k_0}\\}$.  \nFor such a collection $\\{{\\phi}_{k_1,k_0}\\}$, \nwe define \n\\[\n\\widehat{\\mathfrak \\phi}:B(C_1[1]^{\\bullet}) \\otimes \\widetilde{D}[1]^{\\bullet} \\otimes \nB(C_0[1]^{\\bullet}) \\to B(C_1'[1]^{\\bullet}) \\otimes \\widetilde{D}'[1]^{\\bullet} \\otimes B(C_0'[1]^{\\bullet}) \n\\]\nas the homomorphism determined by \n$\\{{\\mathfrak f}_k^{(i)}\\}$, $i=0,1$, and $\\{{\\phi}_{k_1,k_0}\\}$.  \nWe call $\\phi=\\{{\\phi}_{k_1,k_0}\\}$ a weakly filtered \n$A_{\\infty}$-homomorphism of filtered $A_{\\infty}$-bimodules, if \n\\[\n\\widehat{d}_{{\\mathfrak n}'} \\circ \\widehat{\\phi} = \n\\widehat{\\phi} \\circ \\widehat{d}_{\\mathfrak n}.  \n\\]\nWhen we can take $c=0$ , $\\phi=\\{{\\phi}_{k_1,k_0}\\}$ is called \na filtered $A_{\\infty}$-homomorphism.  \nSuppose that $C_i^{\\bullet},C_i^{' \\bullet}$ are $G$-gapped.  \nLet $G' \\subset \\R \\times 2\\Z$ be a $G$-set such that \n${\\rm pr}_1|_G^{-1} ((-\\infty,\\lambda])$ is finite \nfor any $\\lambda \\in \\R$ \nand ${\\rm pr}_1(G)$ is bounded from below.   \nWe say that $\\phi=\\{{\\phi}_{k_1,k_0}\\}$ is $G'$-gapped, if \n\\[\n\\phi_{k_1,k_0}=\\sum_{\\beta' \\in G'} T^{{\\rm pr}_1(\\beta')} \ne^{{\\rm pr}_2(\\beta')\/2} \\phi_{k_1,k_0,\\beta'},\n\\]\nwhere \n\\[\n\\phi_{k_1,k_0,\\beta'}:B_{k_1}(\\overline{C}_1[1]^{\\bullet}) \\otimes \n\\overline{D}[1]^{\\bullet} \\otimes B_{k_0}(\\overline{C}_0[1]^{\\bullet}) \n\\to \\overline{D}'[1]^{\\bullet}.\n\\]\n\nThe homotopy theory between filtered $A_{\\infty}$-homomorphisms of \nfiltered $A_{\\infty}$-bimodules is also developed in \\cite{FOOO}.  \n\nWe also proved the Whitehead theorem for (filtered) \n$A_{\\infty}$-bimodules.  \n\n\\begin{thm}[Theorem 21.35 \\cite{FOOO}]\nLet $\\phi:D^{\\bullet} \\to D^{' \\bullet}$ be a gapped \nfiltered $A_{\\infty}$-bimodule \nhomomorphism over $({\\mathfrak f}^{(1)},{\\mathfrak f}^{(0)})$, where \n${\\mathfrak f}^{(i)}:C_i^{\\bullet} \\to C_i^{' \\bullet}$ \nare homotopy equivalences.  \nSuppose that $\\phi_{(0,0,{\\mathbf 0})}$ is a chain homotopy equivalence.  \nThen $\\phi$ is a homotopy equivalence of filtered \n$A_{\\infty}$-bimodules.\n\\end{thm}\n\n\\begin{rem}\nHere we require $\\phi$ is a filtered $A_{\\infty}$-homomorphism.  \nSince a weakly filtered $A_{\\infty}$-homomorphism, \n$\\phi_{(0,0,{\\mathbf 0})}$ may not induce a chain map with respect to \n${\\mathfrak n}_{0,0,{\\mathbf 0}}$ and \n${\\mathfrak n}_{0,0,{\\mathbf 0}}'$.\n\\end{rem}\n\n\\smallskip\n\\noindent\n{\\bf 2.6) Filtered $A_{n,K}$ structure.}\n\nFor a later argument in section 4, \nwe recall the notion of filtered $A_{n,K}$-algebras.  \nLet $G \\subset \\R{\\geq 0} \\times 2\\Z$ be a monoid as above and \n$\\beta_0={\\mathbf 0} \\in G$.    \nFor $\\beta \\in G$, we define \n\n\\[\n\\parallel \\beta \\parallel = \\left\\{\n\\begin{array}{ll} \n\\sup \\{ n | \\exists \\beta_i \\in G \\setminus \\{\\beta_0\\}, \\ \n\\sum_{i=1}^n \\beta_i = \\beta \\} + [{\\rm pr}_1(\\beta)] -1 & \n\\text{ if } \\beta \\neq \\beta_0 \\\\\n-1 & \\text{ if } \\beta = \\beta_0,   \n\\end{array}\n\\right.\n\\]\n\nThen we introduce a partial order on $(G \\times \\Z_{\\geq 0})  \n\\setminus \\{(\\beta_0,0)\\}$ by \n$(\\beta_1,k_1) \\succ (\\beta_2,k_2)$ if and only if \neither \n\\[\n\\parallel \\beta_1 \\parallel + k_1 > \n\\parallel \\beta_2 \\parallel + k_2\n\\] or \n\\[\n\\parallel \\beta_1 \\parallel + k_1 = \n\\parallel \\beta_2 \\parallel + k_2, \\text{ and } \n\\parallel \\beta_1 \\parallel > \\parallel \\beta_2 \\parallel.\n\\]\nWe write $(\\beta_1,k_1) \\sim (\\beta_2,k_2)$, when \n\\[\n\\parallel \\beta_1 \\parallel + k_1 = \n\\parallel \\beta_2 \\parallel + k_2, \\text{ and } \n\\parallel \\beta_1 \\parallel = \\parallel \\beta_2 \\parallel.\n\\]\nWe define $(\\beta_1,k_1) \\succsim (\\beta_2,k_2)$ if \neither $(\\beta_1,k_1) \\succ (\\beta_2,k_2)$ or $(\\beta_1,k_1) \\sim \n(\\beta_2,k_2)$.  \n\nWe also write $(\\beta,k) \\prec (n',k')$, when \n$\\parallel \\beta \\parallel + k < n' + k'$ or \n$\\parallel \\beta \\parallel + k = n' + k'$ and \n$\\parallel \\beta \\parallel < n'$.  \n\nLet $\\overline{C}^{\\bullet}$ be a cochain complex over $R$ and \n$C^{\\bullet}=\\overline{C}^{\\bullet}\\otimes \\Lambda_{0,nov}$.  \nSuppose that there are \n\\[\n{\\mathfrak m}_{k,\\beta}:B(\\overline{C}[1]^{\\bullet}) \\to \\overline{C}[1]^{\\bullet}\n\\]\nfor $(\\beta,k) \\in (G \\times \\Z) \\setminus \\{(\\beta_0,0)\\}$ \nwith $(\\beta,k) \\prec (n,K)$.  \nWe also suppose that ${\\mathfrak m}_{1,\\beta_0}$ is \nthe boundary operator of the cochain complex $C^{\\bullet}$.  \nWe call $(C^{\\bullet},\\{{\\mathfrak m}_{k,\\beta}\\})$ \na $G$-gapped filtered $A_{n,K}$-algebra, if the following holds\n\\[\n\\sum_{\\beta_1+\\beta_2=\\beta, k_1+k_2=k+1} \\sum_i \n(-1)^{\\deg'{\\mathbf x}_i^{(1)}} {\\mathfrak m}_{k_2,\\beta_2}\n\\bigl({\\mathbf x}_i^{(1)}, \n{\\mathfrak m}_{k_1,\\beta_1}({\\mathbf x}_i^{(2)}), \n{\\mathbf x}_i^{(3)} \\bigr) = 0\n\\] \nfor all $(\\beta,k) \\prec  (n,K)$, \nwhere \n\\[\n\\Delta^2({\\mathbf x})=\\sum_i{\\mathbf x}_i^{(1)} \\otimes \n{\\mathbf x}_i^{(2)} \\otimes {\\mathbf x}_i^{(3)}.\n\\]\nHere $\\Delta$ is the coproduct of the tensor coalgebra.  \n\nWe also have the notion of filtered $A_{n,K}$-homomorphisms, \nfiltered $A_{n,K}$-homotopy equivalences in a natural way.  \nIn \\cite{FOOO}, we proved the following:\n\n\\begin{thm}[Theorem 30.72 in \\cite{FOOO}]\\label{ext(n,K)}\nLet $C_1^{\\bullet}$ be a filtered $A_{n,K}$-algebra and \n$C_2^{\\bullet}$ a filtered $A_{n',K'}$-algebra such that \n$(n,K) \\prec (n',K')$.  \nLet ${\\mathfrak h}:C_1^{\\bullet} \\to C_2^{\\bullet}$ \nbe a filtered $A_{n,K}$-homomorphism.  \nSuppose that ${\\mathfrak h}$ is a filtered $A_{n,K}$-homotopy \nequivalence.  \nThen there exist a filtered $A_{n',K'}$-algebra structure on \n$C_1^{\\bullet}$ \nextending the given filtered $A_{n,K}$-algebra structure \nand a filtered $A_{n',K'}$-homotopy equivalence \n$C_1^{\\bullet} \\to C_2^{\\bullet}$ extending the given \nfiltered $A_{n,K}$-homotopy equivalence ${\\mathfrak h}$.  \n\\end{thm}\n\n\n\\section{Canonical models}\nIn this section, we give the notion of canonical models and \nexplain their construction after section 23, Chapter 5 of \\cite{FOOO}.  \nThe unfiltered version of such a result goes back to Kadeishvili \n\\cite{Kad}.  \nThere are two methods to construct canonical models in unfiltered \ncase.  \nOne is based on obstruction theory due to Kadeishvili and \nthe other uses the summation over trees due to Kontsevich and \nSoibelman \\cite{KS}.  \nOur argument is an adaptation of the latter argument and we also \nconstructed the canonical models for filtered case.  \n\nWhen the ground coefficent ring $R$ is a field, we have the \nfollowing:  \n\n\\begin{thm}[Theorem 23.1, Theorem 23.2 in \\cite{FOOO}]\\label{canmodel}\n(1) Any unfiltered $A_{\\infty}$-algebra \n$(\\overline{C}^{\\bullet},\\{\\overline{\\mathfrak m}_k\\})$ is homotopy equivalent to an $A_{\\infty}$-algebra \n$(\\overline{C'}^{\\bullet},\\{\\overline{\\mathfrak m}'_k\\})$ with \n$\\overline{\\mathfrak m}'_1=0$.  \n\n\\noindent\n(2) Any gapped filtered $A_{\\infty}$-algebra \n$(C^{\\bullet},\\{{\\mathfrak m}_k\\})$ \nis homotopy equivalent to a gapped filtered $A_{\\infty}$-algebra \n$(C^{' \\bullet},\\{{\\mathfrak m}'_k\\})$ \nwith $\\overline{\\mathfrak m}'_1=0$.  \nMoreover, the homotopy equivalence can be taken as a gapped \n$A_{\\infty}$-homomorphism.  \n\\end{thm}\n \nAn $A_{\\infty}$-algebra is called {\\it canonical}, if \n$\\overline{\\mathfrak m}_1=0$.  \nA canonical model of an $A_{\\infty}$-algebra is \na canonical $A_{\\infty}$-algebra homotopy equivalent to \nthe original one.  \nThe statement (1) is Kadeishvili's theorem and implies that the \n$\\overline{\\mathfrak m}_1$-cohomology has a structure of an \n$A_{\\infty}$-algebra.  \nNote that, in general,  \nwe do not have ${\\mathfrak m}_1$-cohomology, \nsince ${\\mathfrak m}_1 \\circ {\\mathfrak m}_1$ may not be zero.  \nA filtered $A_{\\infty}$-algebra is called {\\it canonical}, if \n${\\mathfrak m}_{1,0}=\\overline{\\mathfrak m}_1=0$.  \nA canonical model of a filtered $A_{\\infty}$-algebra is \na canonical filtered $A_{\\infty}$-algebra homotopy equivalent to \nthe original one.  \n \nPick a submodule \n${\\mathcal H}^{\\bullet} \\stackrel{\\iota}{\\hookrightarrow} \n\\ker \\overline{\\mathfrak m}_1 \\cap \\overline{C}^{\\bullet}$ such that \n$\\iota_*: {\\mathcal H}^k \\cong \n{\\rm H}^k(\\overline{C}^{\\bullet},\\overline{\\mathfrak m}_1)$, \nand $\\Pi^k:\\overline{C}^k \\to {\\mathcal H}^k \\subset \\overline{C}^k$ \nsuch that $\\Pi^k \\circ \\Pi^k = \\Pi^k$ and \n$\\Pi^k \\circ \\overline{\\mathfrak m}_1 = 0$.  \nWe will construct a structure of a filtered $A_{\\infty}$-algebra on \n${\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}$ and \na filtered $A_{\\infty}$-homomorphism from \n${\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}$ \nto $C[1]^{\\bullet}$, which is a weak homotopy equivalence.  \nSince $R$ is a field, any cochain homomorphism inducing an isomorphism \non cohomologies is a weak homotopy equivalence.  \nFirstly, we observe the following:  \n\n\\begin{lem}\\label{H}\nThere exist $G^k:\\overline{C}^k \\to \\overline{C}^{k-1}$, \n$k=0,1,\\dots, n$, such that \n\\begin{align}\nid - \\Pi^k & =  -(\\overline{\\mathfrak m}_1 \\circ G^k + G^{k+1} \\circ \\overline{\\mathfrak m}_1), \\label{htpy} \\\\\nG^{k} \\circ G^{k+1} & =  0. \\label{sqzero} \n\\end{align}\n\\end{lem}\n\nFrom now on, let ${\\mathcal H}^{\\bullet} \n\\stackrel{\\iota}{\\hookrightarrow} \\overline{C}^{\\bullet}$ \nbe a subcomplex and $\\Pi:\\overline{C}^k \\to \n\\overline{C}^k$ be a projection to ${\\mathcal H}^k$ such that \nthere exist $G^k:\\overline{C}^k \\to \\overline{C}^{k-1}$ \nsatisfying (\\ref{htpy}), (\\ref{sqzero}) in Lemma \\ref{H}.  \nWe do not assume that $\\overline{\\mathfrak m}_1|_{\\mathcal H}=0$.  \nThus ${\\mathcal H}^{\\bullet}$ is not necessarily isomorphic \nto the cohomology $H^{\\bullet}(\\overline{C}^{\\bullet})$.  \nBut the condition (\\ref{htpy}) implies that \n$\\iota_*: {\\rm H}^{\\bullet}({\\mathcal H},\\overline{\\mathfrak m}_1|_{{\\mathcal H}}) \\cong \n{\\rm H}^{\\bullet}(\\overline{C}^{\\bullet},\\overline{\\mathfrak m}_1)$.  \nTheorem \\ref{canmodel} follows from the following:\n\n\\begin{thm}\\label{reduction}\n(1) There exists a structure \n$\\{\\overline{\\mathfrak m}_k'\\}_{k=1}^{\\infty}$ of \nan $A_{\\infty}$-algebra on ${\\mathcal H}$ with \n${\\mathfrak m}_1'={\\mathfrak m}_1|_{\\mathcal H}$.  \nThe inclusion $\\iota$ extends to an $A_{\\infty}$-homomorphism \n$\\{\\overline{\\mathfrak f}_k\\}_{k=1}^{\\infty}$ \nwith $\\overline{\\mathfrak f}_1 = \\iota$.  \n\n\\noindent\n(2) There exists a structure \n$\\{{\\mathfrak m}_k\\}_{k=0}^{\\infty}$ of a filtered $A_{\\infty}$-algebra \non ${\\mathcal H} \\otimes \\Lambda_{0,nov}$.  \nThe inclusion $\\iota$ extends to a filtered $A_{\\infty}$-homomorphism \n$\\{{\\mathfrak f}\\}_{k=0}^{\\infty}$ with \n${\\mathfrak f}_{1,0}=\\iota$.  \n\\end{thm}\n\n\nLet $G$ be a monoid as in section 2 and ${\\rm pr}_1(G)=\\{\\lambda_{(i)}\\}$ \nsuch that \n\\[\n0=\\lambda_{(0)} < \\lambda_{(1)} < \\lambda_{(2)} < \\cdots \\to +\\infty, \n\\]\nunless $G=\\{(0,0)\\}$.  \nWe write \n\\[\n{\\mathfrak m}_{k,i}=\n\\sum_{\\beta \\in G \\ \\ {\\rm pr}_1(\\beta)=\\lambda_{(i)}} \ne^{{\\rm pr}_2(\\beta)\/2} {\\mathfrak m}_{k,\\beta}\n\\]\nand\n\\[\n{\\mathfrak m}_{k,i}^{\\circ}=T^{\\lambda_{(i)}}{\\mathfrak m}_{k,i}.\n\\]\nThus ${\\mathfrak m}_k= \\sum_i {\\mathfrak m}_{k,i}^{\\circ}$. \nHere ${\\mathfrak m}_{k,i}$ is considered as \n\\[\n{\\mathfrak m}_{k,i}:B_k(\\overline{C}[1]^{\\bullet}) \\otimes R[e,e^{-1}] \n\\to \\overline{C}[1]^{\\bullet} \\otimes R[e,e^{-1}].\n\\]  \nBy an abuse of notation, we also denote by $\\Pi^k$, $G^k$ \nthe extensions thereof \nto $\\overline{C}^k \\otimes R[e,e^{-1}]$ as a $R[e,e^{-1}]$-module \nhomomorphism.  \n\nIn order to define a $G$-gapped filtered $A_{\\infty}$-structure \non ${\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}$ and \na $G$-gapped $A_{\\infty}$-homomorphism from ${\\mathcal H}[1]^{\\bullet} \n\\otimes \\Lambda_{0,nov}$ to $C[1]^{\\bullet}$, we introduce some notation.  \n\nA decorated planar rooted tree is a quintet  \n$\\Gamma=(T,i,v_0,V_{tad}, \\eta)$, which consists of \n\\begin{itemize}\n\\item $T$ is a tree, \n\\item $i:T \\to D^2$ is an embedding, \n\\item $v_0$ is the root vertex, \n\\item $V_{tad}=\\{ \\text{vertices of valency } 1\\} \n\\setminus C^0_{ext}(T)$,\n\\item $\\eta:C^0_{int}(T)=C^0(T) \\setminus C^0_{ext}(T) \\to \\{0,1,2,\\dots\\}.$\n\\end{itemize}\nHere $C^0(T)$ is the set of vertices of the tree $T$, \n$C^0_{ext}(T)=i^{-1}(\\partial D^2)$ is the set of exterior vertices \nand $C^0_{int}(T)$ is the set of interior vertices.  \nNote that the root vertex $v_0$ is an exterior vertex and \n$V_{tad} \\subset C^0_{int}(T)$.  \nLet ${G}_k^+$ be the set of $\\Gamma=(T,i,v_0,V_{tad},\\eta)$ \nsuch that $\\# C^0_{ext} = k$ and $\\eta(v) > 0$ if $v \\in C^0_{int}(T)$ \nis a vertex of valency $1$ or $2$.  \nWe set $E(\\Gamma)=\\sum_{v \\in C^0_{int}(T)} \\lambda_{(\\eta(v))}$.  \n\nFor each $\\Gamma \\in {G}_{k+1}^+$, we construct \n\\[\n{\\mathfrak m}_{\\Gamma}:B_k({\\mathcal H}[1]^{\\bullet}) \\otimes R[e,e^{-1}] \\to \n{\\mathcal H}[1]^{\\bullet} \\otimes R[e,e^{-1}],\n\\] \nwhich is of degree $1$ and \n\\[\n{\\mathfrak f}_{\\Gamma}:B_k({\\mathcal H}[1]^{\\bullet}) \\otimes R[e,e^{-1}] \\to \n\\overline{C}[1]^{\\bullet} \\otimes R[e,e^{-1}],\n\\]\nwhich is of degree $0$.  \nThen we define \n\\[\n{\\mathfrak m}_k'=\\sum_{\\Gamma \\in {G}_{k+1}^+} \nT^{E(\\Gamma)} {\\mathfrak m}_{\\Gamma}:\nB_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov} \\to \n{\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}\n\\]\nand \n\\[\n{\\mathfrak f}_k=\\sum_{\\Gamma \\in {G}_{k+1}^+}\nT^{E(\\Gamma)} {\\mathfrak f}_{\\Gamma}:\nB_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov} \\to \n\\overline{C}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}.\n\\]\nWe will show that \n$({\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}, \\{{\\mathfrak m}_k'\\})$ \nis a $G$-gapped filtered $A_{\\infty}$-algebra and \n${\\mathfrak f}=\\{{\\mathfrak f}_k\\}$ is a $G$-gapped \n$A_{\\infty}$-homomorphism, which is a weak homotopy equivalence.  \nThen the Whitehead type theorem implies that ${\\mathfrak f}$ is \na homotopy equivalence.    \n\n\\smallskip \\noindent\n{\\bf Step 1.} \\ \\ The case that $\\# C^0_{int}(T)=0$.\n\nSuch a $T$ consists of two exterior vertices and an edge \njoining them.  \nTherefore, there is unique element $\\Gamma_0$, which belongs to \n$G_2^+$.  \nWe define \n\\[\n{\\mathfrak m}_{\\Gamma_0}=\n\\overline{\\mathfrak m}_1|_{{\\mathcal H}[1]^{\\bullet}}\n\\] \nand \n\\[\n{\\mathfrak f}_{\\Gamma_0}:{\\mathcal H}[1]^{\\bullet} \\otimes R[e,e^{-1}] \\to \n\\overline{C}[1]^{\\bullet} \\otimes R[e,e^{-1}]\n\\]\nto be the inclusion $\\iota$.  \n\n\\smallskip \\noindent\n{\\bf Step 2.} \\ \\ The case that $\\# C^0_{int}(T)=1$.\n\nFor any $k=0,1,2, \\dots$, \nthere is a unique planar tree with $\\#C^0_{ext}(T)=k+1$ and \n$\\#C^0_{int}(T)=1$.   \nLet $\\Gamma_{k+1} \\in G_{k+1}^+$ be a decorated planar tree with \none interior vertex $v$, see Figure \\ref{elmdectree}.  \n\n\\begin{figure}\n\\begin{center}\n\\include{elmdectree}\n\\end{center}\n\\caption{}\n\\label{elmdectree}\n\\end{figure}\n\nWe define \n\\[\n{\\mathfrak m}_{\\Gamma_{k+1}}=\\Pi \\circ {\\mathfrak m}_{k,\\eta(v)}:\nB_k({\\mathcal H}[1]^{\\bullet}) \\otimes R[e,e^{-1}] \\to \n{\\mathcal H}[1]^{\\bullet} \\otimes R[e,e^{-1}]\n\\]\nand \n\\[\n{\\mathfrak f}_{\\Gamma_{k+1}}=G \\circ {\\mathfrak m}_{k,\\eta(v)}: \nB_k({\\mathcal H}[1]^{\\bullet}) \\otimes R[e,e^{-1}] \\to \n\\overline{C}[1]^{\\bullet} \\otimes R[e,e^{-1}].\n\\]\nSince the degree of $\\Pi$, resp. $G$, is $0$, resp. $-1$, \n${\\mathfrak m}_{\\Gamma_{k+1}}$, resp. ${\\mathfrak f}_{\\Gamma_{k+1}}$, \nis of degree $1$, resp. $0$.  \n\n\\smallskip \\noindent\n{\\bf Step 3.} \\ \\ General case. \n\nLet $v_1$ is the vertex closest to the root vertex $v_0$.  \nCut the decorated planar tree at $v_1$, then $\\Gamma$ is decomposed \ninto decorated planar trees $\\Gamma^{(1)}, \\dots, \\Gamma^{({\\ell})}$ \nand an interval toward $v_0$ in counter-clockwise order, see \nFigure \\ref{gendectree}.  \n\n\\begin{figure}\n\\begin{center}\n\\include{gendectree}\n\\end{center}\n\\caption{}\n\\label{gendectree}\n\\end{figure}\n\nThen we define \n\\[\n{\\mathfrak m}_{\\Gamma}=\\Pi \\circ {\\mathfrak m}_{\\ell, \\eta(v_1)} \n\\circ ({\\mathfrak f}_{\\Gamma^{(1)}} \\otimes \\cdots \\otimes {\\mathfrak f}_{\\Gamma^{({\\ell})}})\n\\]\nand \n\\[\n{\\mathfrak f}_{\\Gamma}=G \\circ {\\mathfrak m}_{\\ell, \\eta(v_1)} \n\\circ ({\\mathfrak f}_{\\Gamma^{(1)}} \\otimes \\cdots \\otimes {\\mathfrak f}_{\\Gamma^{({\\ell})}}).\n\\]\n\n\\smallskip\n\nFinally we define \n\\[\n{\\mathfrak m}'_k=\\sum_{\\Gamma \\in G_{k+1}^+} \nT^{E(\\Gamma)} {\\mathfrak m}_{\\Gamma}\n\\] \nand \n\\[\n{\\mathfrak f}_k=\\sum_{\\Gamma \\in G_{k+1}^+} \nT^{E(\\Gamma)} {\\mathfrak f}_{\\Gamma}.\n\\] \nAs in \\S 2.1, we obtain a graded coderivation \n\\[\n\\widehat{d}'=\\sum_k \\widehat{\\mathfrak m}'_k:B({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov} \\to B({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}\n\\] \nand a (formal) coalgebra homomorphism \n\\[\n\\widehat{\\mathfrak f}:B({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov} \\to \nB(C[1]^{\\bullet}).\n\\] \n\nWe will show the following:\n\n\\begin{prop}\\label{mainprop}\n\\[\n\\widehat{\\mathfrak f} \\circ \\widehat{d}' = \\widehat{d} \\circ \n\\widehat{\\mathfrak f},\n\\]\nwhere $\\widehat{d}=\\sum_k \\widehat{\\mathfrak m}_k:B(C[1]^{\\bullet}) \\to B(C[1]^{\\bullet})$. \n\\end{prop}\n\nSince $\\overline{\\mathfrak f}_1={\\mathfrak f}_{\\Gamma_0}$ is the \ninclusion, we find that $\\widehat{\\mathfrak f}$ is injective \nusing the energy filtration and the number filtration on the bar \ncomplex.  \nThen $\\widehat{d}' \\circ \\widehat{d}' = 0$ follows from \n$\\widehat{d} \\circ \\widehat{d} = 0$.  \nHence we obtain the following:\n\n\\begin{cor}\n(1) $({\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}, \\{{\\mathfrak m}'_k\\})$ \nis a $G$-gapped filtered $A_{\\infty}$-algebra.  \n\n\\noindent\n(2) $\\widehat{\\mathfrak f}$ is a $G$-gapped $A_{\\infty}$-homomorphism \nfrom $({\\mathcal H}[1]^{\\bullet} \\otimes \\Lambda_{0,nov}, \\{{\\mathfrak m}'_k\\})$ \nto $(C[1]^{\\bullet}, \\{{\\mathfrak m}_k\\})$.  \n\\end{cor}\n\nThe rest of this section is devoted to the proof of \nProposition \\ref{mainprop}, \nwhich is equivalent to that \n\\[\n{\\mathfrak f} \\circ \\widehat{d}' = {\\mathfrak m} \\circ \n\\widehat{\\mathfrak f}  \n\\]\nas maps $B({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov} \\to C[1]^{\\bullet}$, \nwhere \n\\[\n{\\mathfrak f}:B({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov} \n\\stackrel{\\widehat{\\mathfrak f}}{\\to} B(C[1]^{\\bullet}) \\stackrel{\\rm pr}{\\to} \nC[1]^{\\bullet},\n\\]\nand \n\\[\n{\\mathfrak m}:B(C[1]^{\\bullet}) \\stackrel{\\widehat{d}}{\\to} B(C[1]^{\\bullet}) \n\\stackrel{\\rm pr}{\\to} C[1]^{\\bullet}.\n\\]\nNamely, ${\\mathfrak f}\\vert_{B_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}} \n= {\\mathfrak f}_k$, \n${\\mathfrak m}\\vert_{B_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}} \n= {\\mathfrak m}_k$.  \n\nWe introduce an order on $\\{(k,i) \\vert k,i =0,1,2, \\dots \\}$ by \n$(k_1,i_1) \\prec (k_2,i_2)$ if \neither $i_1 < i_2$ or $i_1=i_2$ and $k_1 < k_2$.  \nWe show the following claim by the induction on $(k,i)$.  \n\n\\begin{claim}\n\\[\n{\\mathfrak f} \\circ \\widehat{d}' \\equiv {\\mathfrak m} \\circ \n\\widehat{\\mathfrak f}  \\mod T^{\\lambda_{(i+1)}}\\cdot C[1]^{\\bullet} \\text{ \\rm on } \nB_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}.\n\\]\n\\end{claim}\n\nThe key ingredients in the proof are the following relations \npresented in Figures \\ref{green}, \\ref{ainfty0}. \n\n\\begin{figure}\n\\begin{center}\n\\include{green}\n\\end{center}\n\\caption{}\n\\label{green}\n\n\\begin{center}\n\\include{ainfty0}\n\\end{center}\n\\caption{}\n\\label{ainfty0}\n\\end{figure}\n\n\nFirstly, we consider the case that $i=0$.  \nClaim $(0,0)$ follows from the gapped condition.  \nBy the choice of ${\\mathcal H}$, Claim (1,0) holds clearly.  \nSuppose that Claim $(\\ell,0)$ holds for $\\ell < k$.  \nNote that \n\\[\n\\overline{\\mathfrak f}_k={\\mathfrak f}_{k,0}^{\\circ}  \n= \\bigl(\\sum_{1<\\ell\\leq k} G \\circ \\overline{\\mathfrak m}_{\\ell} \\circ \n\\widehat{\\overline{\\mathfrak f}} + \n\\delta_{k1} {\\mathfrak f}_{\\Gamma_0}\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}\n\\] \nand \n\\[\n\\overline{\\mathfrak m}'_k={\\mathfrak m}_{k,0}^{\\circ}\n= \\bigl(\\sum_{1<\\ell\\leq k} \\Pi \\circ \\overline{\\mathfrak m}_{\\ell} \n\\circ \\widehat{\\overline{\\mathfrak f}} + \n\\delta_{k1} {\\mathfrak m}_{\\Gamma_0}\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}.\n\\]\nHere $\\delta_{ij}$ is Kronecker's delta.  \nRecall that ${\\mathfrak m}_{\\Gamma_0}={\\mathfrak m}_{1,0}|_{\\mathcal H}$ and \n${\\mathfrak f}_{\\Gamma_0}$ is the inclusion.  \nNote also that the restriction of $\\widehat{\\overline{\\mathfrak f}}$ \nto $B_k({\\mathcal H}[1]^{\\bullet})$ in the right hand sides is determined by \n$\\overline{\\mathfrak f}_1, \\dots, \\overline{\\mathfrak f}_{k-1}$.  \n\nThus we have \n\n\\begin{align}\n\\overline{\\mathfrak m} \\circ \\widehat{\\overline{\\mathfrak f}}\n|_{B_k({\\mathcal H}[1]^{\\bullet})} \n& = \n\\bigl(\\overline{\\mathfrak m}_1 \\circ \\widehat{\\overline{\\mathfrak f}} + \n\\sum_{1< \\ell \\leq k} \\overline{\\mathfrak m}_{\\ell} \\circ \n\\widehat{\\overline{\\mathfrak f}}\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}  \\nonumber \\\\\n& =  \n\\bigl(\\sum_{1< \\ell \\leq k} \\overline{\\mathfrak m}_1 \\circ G \\circ \n\\overline{\\mathfrak m}_{\\ell} \\circ \\widehat{\\overline{\\mathfrak f}} + \n\\delta_{k1} \\overline{\\mathfrak m}_1 \\circ {\\mathfrak f}_{\\Gamma_0} \n+ \\sum_{1< \\ell \\leq k} \\overline{\\mathfrak m}_{\\ell} \n\\circ \\widehat{\\overline{\\mathfrak f}}\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}.  \n\\nonumber \\\\\n& = \n\\bigl(\\sum_{1< \\ell \\leq k} \\Pi \\circ \\overline{\\mathfrak m}_{\\ell} \n\\circ \\widehat{\\overline{\\mathfrak f}} - \n\\sum_{1< \\ell \\leq k} G \\circ \\overline{\\mathfrak m}_1 \\circ \n\\overline{\\mathfrak m}_{\\ell} \\circ \\widehat{\\overline{\\mathfrak f}}\n+\\delta_{k1} \\overline{\\mathfrak m}_1 \\circ {\\mathfrak f}_{\\Gamma_0} \n\\bigr)\n|_{B_k({\\mathcal H}[1]^{\\bullet})}\n\\nonumber \\\\\n& =  \n\\bigl({\\mathfrak f}_{\\Gamma_0} \\circ \\overline{\\mathfrak m}'_k - \n\\sum_{1< \\ell \\leq k} G \\circ \\overline{\\mathfrak m}_1 \\circ \n\\overline{\\mathfrak m}_{\\ell} \\circ \\widehat{\\overline{\\mathfrak f}}\n\\bigr)\n|_{B_k({\\mathcal H}[1]^{\\bullet})} \n\\nonumber \\\\\n& = \n\\bigl({\\mathfrak f}_{\\Gamma_0} \\circ \\overline{\\mathfrak m}'_k + \n\\sum_{1< \\ell' \\leq k} G \\circ \\overline{\\mathfrak m}_{\\ell'} \\circ \n\\widehat{\\overline{d}} \\circ \\widehat{\\overline{\\mathfrak f}}\\bigr)\n|_{B_k({\\mathcal H}[1]^{\\bullet})}. \n\\nonumber\n\\end{align}\n\nHere we used the fact that \n$\\overline{\\mathfrak m}_1 \\circ G + G \\circ \n\\overline{\\mathfrak m}_1 = \\Pi - id$ and the $A_{\\infty}$-relation \n$\\widehat{\\overline{d}} \\circ \\widehat{\\overline{d}} = 0$.   \nSince we assumed Claim $(\\ell,0)$ for $\\ell < k$, i.e., \n\\[\n\\overline{\\mathfrak m} \\circ \\widehat{\\overline{\\mathfrak f}} = \n\\overline{\\mathfrak f} \\circ \\widehat{\\overline{d}'} \n\\text{ on } B_{\\ell}({\\mathcal H}[1]^{\\bullet}),\n\\]  \nwe have \n\\[\n\\widehat{\\overline{d}} \\circ \\widehat{\\overline{\\mathfrak f}} \\equiv \n\\widehat{\\overline{\\mathfrak f}} \\circ \\widehat{\\overline{d}'} \n\\mod \\overline{C}[1]^{\\bullet}=B_1(\\overline{C}[1]^{\\bullet})\n\\text{ on } B_k({\\mathcal H}[1]^{\\bullet}).\n\\]\nTherefore we find that \n\\[\n(\\sum_{1< \\ell' \\leq k} G \\circ \\overline{\\mathfrak m}_{\\ell'} \\circ \n\\widehat{\\overline{d}} \\circ \\widehat{\\overline{\\mathfrak f}})\n|_{B_k({\\mathcal H}[1]^{\\bullet})}\n= \n(\\sum_{1< \\ell' \\leq k} G \\circ \\overline{\\mathfrak m}_{\\ell'} \\circ \n\\widehat{\\overline{\\mathfrak f}} \\circ \\widehat{\\overline{d}'})\n|_{B_k({\\mathcal H}[1]^{\\bullet})}.  \n\\]\nHence we showed Claim $(k,0)$, i.e.,   \n\\[\n\\overline{\\mathfrak m} \\circ \\widehat{\\overline{\\mathfrak f}} \n= {\\overline{\\mathfrak f}} \\circ \\widehat{\\overline{d}'}\n\\] \non $B_k({\\mathcal H}[1]^{\\bullet})$.  \n\nNext we assume that Claim $(k,i)$ holds for all $k=0,1,2, \\dots$.  \nWe prove Claim $(k,i+1)$ by the induction on $k$.  \nNote that Case 3-1 below does not occur in the case that $k=0$.  \n\nFirst of all, we recall from the definition of \n$G_{k+1}^+$ that \n\\[\n{\\mathfrak f}_k=\\sum_{\\Gamma \\in G_{k+1}^+} T^{E(\\Gamma)} \n{\\mathfrak f}_{\\Gamma} \n= \\sum_{(\\ell, j) \\neq (1,0)} G \\circ {\\mathfrak m}_{\\ell,j}^{\\circ} \n\\circ \\widehat{\\mathfrak f} |_{B_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}} + \\delta_{k1} \n{\\mathfrak f}_{\\Gamma_0}.\n\\]\nThen we have \n\n\\begin{align}\n{\\mathfrak m} \\circ \\widehat{\\mathfrak f}\n|_{B_k({\\mathcal H}[1]^{\\bullet})} \n= &\n\\bigl({\\mathfrak m}_{1,0} \\circ \\widehat{\\mathfrak f} + \n\\sum_{(\\ell, j) \\neq (1,0)} {\\mathfrak m}_{\\ell,j}^{\\circ} \\circ \n\\widehat{\\mathfrak f}\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}  \\nonumber \\\\\n= & \n\\bigl(\\sum_{(\\ell,j) \\neq (1,0)} {\\mathfrak m}_{1,0} \\circ G \\circ \n{\\mathfrak m}_{\\ell,j}^{\\circ} \\circ \\widehat{\\mathfrak f} + \n\\delta_{k1} {\\mathfrak m}_{1,0} \\circ {\\mathfrak f}_{\\Gamma_0} \n\\nonumber \\\\\n& \n+ \\sum_{(\\ell,j) \\neq (1,0)} {\\mathfrak m}_{\\ell,j}^{\\circ} \n\\circ \\widehat{\\mathfrak f}\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}.  \n\\nonumber \\\\\n= & \n\\bigl(\\sum_{(\\ell,j) \\neq (1,0)} \\Pi \\circ \n{\\mathfrak m}_{\\ell,j}^{\\circ} \\circ \\widehat{\\mathfrak f} - \n\\sum_{(\\ell,j) \\neq (1,0)} G \\circ {\\mathfrak m}_{1,0} \\circ \n{\\mathfrak m}_{\\ell,j}^{\\circ} \\circ \\widehat{\\mathfrak f} \n\\nonumber \\\\\n& \n+\\delta_{k1} {\\mathfrak m}_{1,0} \\circ {\\mathfrak f}_{\\Gamma_0} \n\\bigr)|_{B_k({\\mathcal H}[1]^{\\bullet})}\n\\nonumber \\\\\n= & \n\\bigl({\\mathfrak f}_{\\Gamma_0} \\circ {\\mathfrak m}'_k - \n\\sum_{(\\ell,j) \\neq (1,0)} G \\circ {\\mathfrak m}_{1,0} \\circ \n{\\mathfrak m}_{\\ell,j}^{\\circ} \\circ \\widehat{\\mathfrak f}\n\\bigr)\n|_{B_k({\\mathcal H}[1]^{\\bullet})} \n\\nonumber \\\\\n= & \n\\bigl({\\mathfrak f}_{\\Gamma_0} \\circ {\\mathfrak m}'_k + \n\\sum_{(\\ell',j') \\neq (1,0)} G \\circ \n{\\mathfrak m}_{\\ell',j'}^{\\circ} \n\\circ \\widehat{d} \\circ \\widehat{\\mathfrak f}\\bigr)\n|_{B_k({\\mathcal H}[1]^{\\bullet})}. \n\\nonumber\n\\end{align}\nIn the third equality, we used the fact that \n${\\mathfrak m}_{1,0} \\circ G + G \\circ \n{\\mathfrak m}_{1,0} = \\Pi - id$.   \n\nWe will show that \n\\[\n\\sum_{(\\ell',j') \\neq (1,0)} G \\circ {\\mathfrak m}_{\\ell',j'}^{\\circ} \n\\circ \\widehat{d} \\circ \\widehat{\\mathfrak f} \\equiv \n\\sum_{(\\ell',j') \\neq (1,0)} G \\circ {\\mathfrak m}_{\\ell',j'}^{\\circ} \n\\circ \\widehat{\\mathfrak f} \\circ \\widehat{d'} \\mod \nT^{\\lambda_{(i+2)}},\n\\]\nwhich implies that \n\\[\n{\\mathfrak m} \\circ \\widehat{\\mathfrak f} \\equiv \n\\widehat{\\mathfrak f} \\circ \\widehat{d'} \\mod T^{\\lambda_{(i+2)}}.\n\\]\n\n\\smallskip \\noindent\n{\\bf Case 1: $\\ell'=0$.} \\ \\ \nNote that the $B_0(C[1]^{\\bullet})=\\Lambda_{0,nov}$-components of \n${\\mathrm{Im}} \\ \\widehat{d} \\circ \\widehat{\\mathfrak f}$ and \n${\\mathrm{Im}} \\ \\widehat{\\mathfrak f} \\circ \\widehat{d}'$ are zero.  \nHence we have\n\\[\n{\\mathfrak m}_{0,j'}^{\\circ} \n\\circ \\widehat{d} \\circ \\widehat{\\mathfrak f} =\n{\\mathfrak m}_{0,j'}^{\\circ} \n\\circ \\widehat{\\mathfrak f} \\circ \\widehat{d'} = 0.  \n\\]\n\n\\smallskip \\noindent\n{\\bf Case 2: $\\ell'=1$.} \\ \\ \nFor $j' \\neq 0$, ${\\mathfrak m}_{1,j'}^{\\circ} \\equiv 0 \n\\mod T^{\\lambda_{(1)}}$.  \nBy the induction hypothesis, we have \n\\[\n\\widehat{\\mathfrak f} \\circ \\widehat{d'} \\equiv \n\\widehat{d} \\circ \\widehat{\\mathfrak f} \\mod T^{\\lambda_{i+1}}.\n\\]\nSince $\\lambda_{(i+2)} \\leq \\lambda_{(i+1)} + \\lambda_{(1)}$, \nwe obtain \n\\[\n{\\mathfrak m}_{1,j'}^{\\circ} \\circ \\widehat{\\mathfrak f} \\circ \n\\widehat{d'} \\equiv \n{\\mathfrak m}_{1,j'}^{\\circ} \\circ \\widehat{d} \\circ \n\\widehat{\\mathfrak f} \\mod T^{\\lambda_{(i+2)}}.\n\\] \n\n\\smallskip \\noindent\n{\\bf Case 3: $\\ell' \\geq 2$.} \\ \\ \nLet ${\\mathbf x} \\in B_k({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}$.  \nWrite \n\\[\n\\Delta^{\\ell'-1} {\\mathbf x} = \\sum_a {\\mathbf x}_{1,a} \\otimes \n\\cdots \\otimes {\\mathbf x}_{\\ell',a},\n\\]\nwhere $\\Delta$ is the coproduct and ${\\mathbf x}_{i,a} \\in \nB_{k_{i,a}}({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}$.  \nThen we have \n\n\\begin{align}\n\\widehat{\\mathfrak f} \\circ \\widehat{d'} ({\\mathbf x}) \n= & \\sum_a \\sum_j (-1)^{\\deg' {\\mathbf x}_{1,a}+\\cdots + \n\\deg'{\\mathbf x}_{j-1,a}}\n{\\mathfrak f}_{k_{1,a}}({\\mathbf x}_{1,a}) \\otimes \\cdots \\otimes \n{\\mathfrak f}_{k_{j,a}}(\\widehat{d'}({\\mathbf x}_{j,a})) \\otimes \n\\cdots \\nonumber \\\\\n& \\hspace{0.3in} \\cdots \\otimes {\\mathfrak f}_{k_{\\ell',a}}({\\mathbf x}_{\\ell',a}). \n\\nonumber\n\\end{align}\n\n\\smallskip \\noindent\n{\\bf Case 3-1: $k_{j,a} < k$.} \nIn this case, we have \n\\[\n{\\mathfrak f}_{k_{j,a}}(\\widehat{d'}({\\mathbf x}_{j,a}) \\equiv \n{\\mathfrak m} \\circ \\widehat{\\mathfrak f}({\\mathbf x}_{j,a}) \n\\mod T^{\\lambda_{(i+2)}}\n\\]\nby the induction hypothesis.  \nHence \n\n\\begin{align}\n& \n{\\mathfrak f}_{k_{1,a}}({\\mathbf x}_{1,a}) \\otimes \\cdots \\otimes \n{\\mathfrak f}_{k_{j,a}}(\\widehat{d'}({\\mathbf x}_{j,a})) \\otimes \n\\cdots \\otimes {\\mathfrak f}_{k_{\\ell',a}}({\\mathbf x}_{\\ell',a}) \n\\nonumber \\\\ \n\\equiv & \n{\\mathfrak f}_{k_{1,a}}({\\mathbf x}_{1,a}) \\otimes \\cdots \\otimes \n{\\mathfrak m} \\circ \\widehat{\\mathfrak f}({\\mathbf x}_{j,a}) \\otimes \n\\cdots \\otimes {\\mathfrak f}_{k_{\\ell',a}}({\\mathbf x}_{\\ell',a}) \n \\ \n\\mod T^{\\lambda_{(i+2)}}. \\nonumber \n\\end{align}\n\n\\smallskip \\noindent\n{\\bf Case 3-2: $k_{j,a} = k$.} \nIn this case, $k_{j',a} = 0$ for $j' \\neq j$, i.e., \n${\\mathbf x}_{j',a} \\in B_0({\\mathcal H}[1]^{\\bullet}) \\otimes \\Lambda_{0,nov}$.  \nWithout loss of generality, we may assume that ${\\mathbf x}_{j',a}=1$ \nfor $j' \\neq j$.  \n\nBy the induction hypothesis, we have \n\\[\n{\\mathfrak f}(\\widehat{d'}({\\mathbf x}_{j,a})) \\equiv \n{\\mathfrak m}(\\widehat{\\mathfrak f}({\\mathbf x}_{j,a})) \n\\mod T^{\\lambda_{(i+1)}},\n\\]\nwhich implies that \n\\[\n{\\mathfrak f}_0(1) \\otimes \\cdot \\otimes {\\mathfrak f}\n(\\widehat{d'}({\\mathbf x}_{j,a})) \\otimes \\cdots \\otimes \n{\\mathfrak f}_1(1)  \\equiv  \n{\\mathfrak f}_0(1) \\otimes \\cdot \\otimes {\\mathfrak m} \n(\\widehat{\\mathfrak f}({\\mathbf x}_{j,a})) \\otimes \\cdots \\otimes \n{\\mathfrak f}_1(1) \n\\mod T^{\\lambda_{(i+2)}}.\n\\]\nHere we used ${\\mathfrak f}_0(1) \\equiv 0 \\mod T^{\\lambda_{(1)}}$. \n\nIn sum, we obtain Claim $(k,i+1)$ for all $k$.  \n\nBy the construction, $\\overline{\\mathfrak f}_1$ is a chain homotopy \nequivalence ($\\Pi$ is a homotopy inverse).  \nTherefore, Theorem \\ref{whitehead} implies that \n$\\{{\\mathfrak f}_k\\}$ is a homotopy equivalence of \nfiltered $A_{\\infty}$-algebras.  \n\n\\section{Filtered $A_{\\infty}$-algebra associated to Lagrangian \nsubmanifolds}\n\nLet $(M,\\omega)$ be a closed symplectic manifold and \n$L$ a Lagrangian submanifold.  \nWe only consider the case that $L$ is an embedded compact \nLagrangian submanifold without boundary equipped with a \nrelative spin structure, see \\S 44 in \\cite{FOOO}.   \nWe constructed a filtered $A_{\\infty}$-algebra \nassociated to $L$ in $(M,\\omega)$.  \nAs we explained in section 2, the framework of (filtered) \n$A_{\\infty}$-algebras, bimodules, etc. is adequate to formulate \nthe condition under which Floer complex is obtained.  \n\nIn this section, we briefly recall the way of constructing \nfiltered $A_{\\infty}$-algebra associated to $L$.  \nAlthough the readers may find Proposition 4.1 below too technical, \nwe present it precisely so that we can explain how to modify it \nfor the purpose of section 5.  \n\nA naive idea of the construction is to use the moduli space \nof pseudo-holomorphic discs to {\\it deform} \nthe intersection products of chains in $L$ \nin a similar way to the quantum cohomology, where the intersection \nproduct on (co)homology is deformed by the moduli space \nof pseudo-holomorphic spheres, more precisely, stable maps of genus 0.  \nHere appears a difference: while the moduli spaces of stable maps of \ngenus 0 are (virtual) cycles, the moduli spaces of stable \nbordered stable maps are, in general, not (virtual) cycles, \nbut with codimension 1 boundary (in the sense of Kuranishi structure).  \nTherefore, we cannot restrict ourselves to cycles and forced to \nwork with chains.  \nHowever, the intersection product is not defined \nin chain level, e.g., the self intersection of chains.  \nWe start with a subcomplex of the singular \nchain complex such that the inclusion induces an isomorphism \non homology.   \nThen take {\\it perturbed} intersection product \nof generators of the subcomplex and add them to get a larger \nsubcomplex such that the inclusion induces an isomorphism on homology.  \nOnce we get such nested subcomplexes, we apply the argument \nin the proof of Theorem 3.3 to define the operation\n$\\overline{\\mathfrak m}_2$ on a fixed subcomplex.\nThis multiplicative structure is not associative, but associative \nup to homotopy.  \nSo we proceed to constructed other operations \n$\\overline{\\mathfrak m}_k$ in a similar way, see \nCorollary 30.89 in section 30.6, \\cite{FOOO} for a detailed argument.  \nIn this way, we obtain an $A_{\\infty}$-algebra.  \n\nFor the construction of the filtered $A_{\\infty}$-algebra, \nwe include the effect from the moduli space of bordered stable \nmaps.  \nWe need to take perturbations of the moduli spaces to define \nthe operations not only perturbation in the intersection product \nmentioned above.  \nOur strategy is to construct an $A_{n,K}$-algebra on \n$C_{(g)}(L)$, which is generated by $\\chi_{(g)}$ in \nProposition \\ref{transv}, for a sufficiently large $g$.  \nThen we use the obstruction theory to extend a filtered \n$A_{n,K}$-structure to a filtered $A_{n',K'}$-structure \n($(n,K) \\precsim (n',K')$).  \nThe resulting filtered $A_{\\infty}$-structure is unique up to \nhomotopy, see \\S 30 in Chapter 7, \\cite{FOOO}.  \n\nLet $\\mu_L \\in {\\rm H}^2(M,L;\\Z)$ be the Maslov class of the \nLagrangian submanifold $L$.  \nWe introduce an equivalence relation $\\sim$ on ${\\rm H}_2(M,L;\\Z)$ \nby $\\beta_1 \\sim \\beta_2$ if and only if \n$\\omega(\\beta_1) = \\omega(\\beta_2)$ and \n$\\mu_L(\\beta_1)=\\mu_L(\\beta_2)$.  \n\nPick an almost complex structure $J$ compatible with $\\omega$.  \nDenote by ${\\mathcal M}(\\beta;L,J)$ the moduli space \nof bordered stable maps $u:(\\Sigma, \\partial \\Sigma) \\to (M,L)$ \nof genus $0$ representing $\\beta$ and \nby ${\\mathcal M}_{k+1}(\\beta;L,J)$ be the moduli space of \nbordered stable maps in the class $\\beta$ of genus $0$ \nwith $k+1$ marked points \n$z_0,z_1, \\dots, z_k$ on the regular part of $\\partial \\Sigma$.  \nDenote by ${\\mathcal M}_{k+1}^{\\rm main}(\\beta;L,J)$ the component, \non which the marked points $z_0, z_1, \\dots, z_k$ respect \nthe {\\it counter-clockwise} cyclic order on the boundary of bordered \nsemi-stable curve of genus 0 with connected boundary.  \nLet ${\\mathfrak G}(L)$ be \nthe monoid contained in $\\Pi(M,L)$ generated by \n$\\beta$ with ${\\mathcal M}(\\beta;L,J) \\neq \\emptyset$.  \nWe write $\\beta_0=0 \\in {\\mathfrak G}(L)$.  \n\nOur basic idea is as follows.  \nFor singular simplices $P_1, \\dots, P_k$ in $L$, \nwe consider the fiber product in the sense of Kuranishi structure \n\\[\n{\\mathcal M}_{k+1}^{\\rm main}(\\beta;P_1, \\dots, P_k) ={\\mathcal M}_{k+1}\n^{\\rm main}(\\beta;L,J)_{\\mathbf{ev}} \\times_{L^k} (P_1 \\times \\dots \\times P_k), \n\\]\nwhere ${\\mathbf{ev}}=(ev_1, \\dots , ev_k)$ is the evaluation map at \n$z_1, \\dots , z_k$.  \n(For the orientation issue, see Chapter 9 \\cite{FOOO}.)  \nThen we would like to define \n\\[\n{\\mathfrak m}_{k,\\beta}(P_1, \\dots, P_k) = (ev_0: \n{\\mathcal M}_{k+1}^{\\rm main}(\\beta;P_1, \\dots, P_k) \\to L),\n\\]\nwhere $ev_0$ is the evaluation at $z_0$.  \n\nNote that ${\\mathcal M}_{k+1}^{\\rm main}(\\beta)$ is not necessarily \na manifold or an orbifold and that \n$ev_i$ are not necessarily submersions even if \n${\\mathcal M}_{k+1}^{\\rm main}(\\beta)$ is such a nice space.  \nIn order to deal with this issue, we introduced the notion of \nKuranishi structure \\cite {FO}, see also Appendix in \\cite{FOOO}.  \nHere is a digression on Kuranishi structure.  \n\nLet $X$ be a compact Hausdorff space.  \nA Kuranishi structure on $X$ consists of \na covering of $X$ by Kuranishi neighborhoods of the same \n{\\it virtual} dimension and coordinate changes among them.  \nA Kuranishi neighborhood around $p \\in X$ is a quintet  \n $(V_p,E_p,\\Gamma_p, s_p, \\psi_p)$, where \n\\begin{itemize}\n\\item\n$V_p$ is a smooth manifold of finite dimension,\n\\item\n$E_p$ is a real vector bundle over $V_p$ of finite rank, \n\\item\n$\\Gamma_p$ is a finite group acting smoothly and effectively \non $V_p$ and $E_p$ such that $E_p \\to V_p$ is a $\\Gamma_p$-equivariant \nvector bundle, \n\\item\n$s_p$ is a $\\Gamma_p$-equivariant section of $E_p \\to V_p$, \n\\item \n$\\psi_p$ is a homeomorphism from $s_p^{-1}(0)\/\\Gamma_p$ to \na neighborhood of $p$ in $X$.\n\\end{itemize}\nThe vector bundle $E_p \\to V_p$ is called the {\\it obstruction} bundle \nand the section $s_p$ the Kuranishi map.  \nWe have coordinate changes among Kuranishi neighborhoods, see \n\\cite{FO}, \\cite{FOOO}.  \nWe require that $\\dim V_p - {\\rm rank}~ E_p$ does not depend on \n$p \\in X$ and call it the {\\it virtual} dimension of the space $X$ \nequipped with Kuranishi structure.  \n\nThe moduli spaces of stable maps, bordered stable maps carry \nKuranishi structures, hence we can locally describe the moduli space \nas $s_p^{-1}(0)\/\\Gamma_p$ in the definition of Kuranishi neighborhoods. \nIf $s_p$ is transversal to the zero section, the moduli space is \nlocally an orbifold.  \nIn general, we cannot perturb $s_p$ to \na $\\Gamma_p$-equivariant section $s'_p$, \nwhich is transversal to the zero section.  \nInstead of single valued sections,  \nwe consider perturbations by \n$\\Gamma_p$-equivariant {\\it multi-valued} sections, \neach branch of which is transversal to the zero section.  \nThen we arrange them compatible under the coordinate change.  \nIn this way, we obtain {\\it perturbed} moduli spaces.  \n\nWe take a multi-valued perturbation $\\mathfrak s$ of \nKuranishi maps for \n${\\mathcal M}_{k+1}^{\\rm main}(\\beta;P_1, \\dots, P_k)$ such that \neach branch of $\\mathfrak s$ is transversal to the zero section.  \nAfter taking a triangulation of \nthe perturbed zero locus \n${\\mathcal M}_{k+1}^{\\rm main}(\\beta;P_1, \\dots, P_k)^{\\mathfrak s}$ \nof $\\mathfrak s$, we obtain a {\\it virtual} chain \n$$\nev_0: {\\mathcal M}_{k+1}^{\\rm main}(\\beta;P_1, \\dots, P_k)^{\\mathfrak s}\n\\to L.\n$$  \n\nTo make this argument rigorous, we build a sequence of \nsubcomplexes of the singular chain complex of $L$ and \na series of operations ${\\mathfrak m}_{k,\\beta}^{geo}$.    \nFor details, see Chapter 7 in \\cite{FOOO}.  \nHere we briefly recall a part of it, in particular, \nthe construction of a series of subcomplexes of singular \nchain complex of $L$.  \nIn section 5, we explain how to arrange this construction \nin relation with the Morse theory.  \n\nIn \\S 30 in \\cite{FOOO}, we constructed countable sets $\\chi_g(L)$ \nof singular $C^{\\infty}$-simplices on $L$.  \nFor a simplex $P \\in \\chi_g(L)$, we call $g$ the generation of $P$.   \nWrite \n\\[\n\\chi_{(g)}=\\bigcup_{g' \\leq g} \\chi_{g'}(L)\n\\] \nand denote by $C_{(g)}(L;R)$ the $R$-vector space generated by \n$\\chi_{(g)}(L)$.  \nLet $S(L;R)$ be the singular $C^{\\infty}$-chain complex of $L$ \nwith coefficients in $R$.  \n\n\n\\smallskip\n\\noindent\n{\\bf Condition 1.} \\ \\  Any face of $P \\in \\chi_g(L)$ belongs to \n$\\chi_{(g)}(L)$.  \n\n\\smallskip\n\\noindent\n{\\bf Condition 2.} \\ \\ The inclusion $C_{(g)}(L) \\to S(L;R)$ \ninduces an isomorphism on homology.  \n\n\\smallskip\n\nFor $\\beta \\in {\\mathfrak G}(L)$, we define \n\n\\[\n\\parallel \\beta \\parallel = \n\\left\\{ \n\\begin{array}{ll}\n\\sup\\{n|\\exists \\beta_1,\\dots,\\beta_n \\in {\\mathfrak G}(L) \\setminus \n\\{\\beta_0\\}, \\sum_{i=1}^n \\beta_i = \\beta \\} \n+[\\omega(\\beta)] -1 & \n\\text{ if } \\beta \\neq \\beta_0 \\\\\n-1 & \\text{ if } \\beta = \\beta_0\n\\end{array}\n\\right.\n\\]\nHere $[\\omega(\\beta)]$ is the largest integer not greater than \n$\\omega(\\beta)$.  \n\nBy Gromov's compactness, \nthe number of $\\beta \\in {\\mathfrak G}(L)$ with \n$\\parallel \\beta \\parallel \\leq C$ is finite for any $C$.  \n\nNext we introduce an additional data \n${\\mathfrak d}:\\{1, \\dots ,k\\} \\to \\Z_{\\geq 0}$, which is called \na decoration.  \nFor a pair $({\\mathfrak d},\\beta)$ such that \n${\\mathcal M}_{k+1}^{\\rm main}(\\beta) \\neq \\emptyset$, we define \n\n\\[\n\\parallel ({\\mathfrak d},\\beta) \\parallel = \n\\left\\{ \n\\begin{array}{ll} \n\\max_{i \\in \\{1, \\dots, k\\}} {\\mathfrak d}(i) + \n\\parallel \\beta \\parallel + k & \\text{ if } k \\neq 0 \\\\\n\\parallel \\beta \\parallel & \\text{ if } k = 0.  \n\\end{array}\n\\right.\n\\]\n\nWe will take the fiber product of \n${\\mathcal M}_{k+1}^{\\rm main}(\\beta)$ and singular simplices \n$P_i$ in $L$.  \nThe decoration $\\mathfrak d$ is introduced in order to \ninclude the generations of singular simplices $P_i$ into the data.  \nWhen we emphasize that the decoration $\\mathfrak d$ \nis equipped with the moduli space \n${\\mathcal M}_{k+1}^{\\rm main}(\\beta)$, we denote it by \n${\\mathcal M}_{k+1}^{{\\rm main}, {\\mathfrak d}}(\\beta)$.  \n\n\\begin{prop}[Proposition 30.35 in \\cite{FOOO}]\\label{transv}\nFor any $\\delta > 0$ and ${\\mathcal K} > 0$, there exist \n$\\chi_{(g)}(L)$, $g=0,\\dots,{\\mathcal K}$, and multisections \n${\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}$ for \n$\\parallel ({\\mathfrak d},\\beta) \\parallel \\leq {\\mathcal K}$ \nwith the following properties: \n\n\\begin{itemize}\n\\item $\\chi_{(g)}(L)$ satisfies Conditions 1 and 2 above.  \n\\item Let $P_i \\in \\chi_{{\\mathfrak d}(i)}(L), i=1,\\dots,k$.  \nWe put \n\\[\n{\\mathcal M}_{k+1}^{{\\rm main},{\\mathfrak d}}(\\beta;P_1,\\dots,P_k)\n= \n{\\mathcal M}_{k+1}^{{\\rm main},{\\mathfrak d}}(\\beta)\\times_{L^k} \n\\prod P_i\n\\]\nand define a multisection \n${\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}$ thereof.   \n${\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}$ is transversal to \nthe zero section.  \n\\item If $g=\\parallel ({\\mathfrak d}, \\beta) \\parallel$, then \n\\[\nev_{0*}\\bigl({\\mathcal M}_{k+1}^{{\\rm main},{\\mathfrak d}}\n(\\beta;P_1,\\dots,P_k)^{{\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}} \n\\bigr)\n\\]\nis decomposed into elements of $\\chi_{(g)}(L)$.  \nHere and henceforth we denote \n\\[\n{\\mathcal M}_{k+1}^{{\\rm main},{\\mathfrak d}}\n(\\beta;P_1,\\dots,P_k)^{{\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}}\n:= {\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}^{-1}(0).\n\\]\n\\item The multisections ${\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}$ \nsatisfy certain compatibility conditions.\n\\item The zero locus ${\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}^{-1}\n(0)$ is in a $\\delta$-neighborhood of the zero locus of the original \nKuranishi map.  \n\\end{itemize}\n\\end{prop}\n\nFor the compatibility conditions in the above statement, see \nConditions 30.38 and 30.44 in \\cite{FOOO}.  \n\nNow we explain the way of constructing the filtered \n$A_{\\infty}$-algebra associated to $L$.  \n\nWe put \n\\[\n{\\mathfrak m}_{k,\\beta}^{geo}(P_1,\\dots,P_k)\n=(ev_0:{\\mathcal M}_{k+1}^{{\\rm main},{\\mathfrak d}}(\\beta;P_1,\\dots,\nP_k)^{{\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}} \\to L),\n\\]\nwhen $P_i \\in \\chi_{({\\mathfrak d}(i))}$, $i=1,\\dots, k$.  \nThen ${\\mathfrak m}_{k,\\beta}^{geo}(P_1,\\dots,P_k)$ is decomposed \ninto elements of $\\chi_{(g)}$, \nwhere $g = \\parallel ({\\mathfrak d},\\beta) \\parallel$.  \nUsing the idea in section 3, we showed the following:  \n\n\\begin{prop}[Proposition 30.78 in \\cite{FOOO}]\\label{geo}\nFor any $g_0, n, K$, there exists $g_1 > g_0$ and a filtered \n$A_{n,K}$-structure ${\\mathfrak m}_{k,\\beta}$ on $C_{(g_1)}(L) \n\\otimes \\Lambda_{0,nov}$ such that \n\\[\n{\\mathfrak m}_{k,\\beta}(P_1,\\dots,P_k)\n={\\mathfrak m}_{k,\\beta}^{geo}(P_1,\\dots,P_k),\n\\]\nif $P_i \\in \\chi_{(g_0)}(L)$.  \n\\end{prop}\n\nCombining Theorem \\ref{ext(n,K)} and Proposition \\ref{geo}, \nwe can construct a filtered $A_{\\infty}$-algebra associated to \n$L$, for details see \\cite{FOOO}.  Hence we obtain \n\n\\begin{thm}[Theorem 10.11 in \\cite{FOOO}]\nLet $L$ be a relatively spin Lagrangian submanifold.  \nThen there exist a countably generated subcomplex $C(L)$ of \nthe singular chain complex and a filtered $A_{\\infty}$-algebra \nstructure on $C(L)\\otimes \\Lambda_{0,nov}$.  \n\\end{thm}\n\nWe also proved that the homotopy type of \nthe filtered $A_{\\infty}$-algebra is unique.  \n\nApplying the construction of canonical models in section 3, \nwe obtain a filtered $A_{\\infty}$-algebra structure on \n${\\rm H}(L) \\otimes \\Lambda_{0,nov}$.  \n\nLet $(L_0,L_1)$ be a relative spin pair of Lagrangian submanifolds.  \nAssume that $L_0$ and $L_1$ intersect transversely.  \nThen we have the following:\n\n\\begin{thm}\nLet $D^{\\bullet}$ be a free $\\Lambda_{0,nov}$-module \ngenerated by $L_0 \\cap L_1$.  \nThen there exists a filtered $A_{\\infty}$-bimodule structure \nover filtered $A_{\\infty}$-algebras associated to $L_i$, $i=0,1$.  \n\\end{thm}\n\n\\section{Canonical models and Morse complexes}\n\nIn this section, we apply Theorem \\ref{reduction} \nand reduce the filtered $A_{\\infty}$-structure \non $C^{\\bullet}(L) \\otimes \\Lambda_{0,nov}$ to the Morse complex \n$CM^{\\bullet}(f) \\otimes \\Lambda_{0,nov}$.   \n\n\n\nWe pick a specific Morse function \nas follows.  \nChoose and fix a triangulation ${\\mathfrak T}$ of $L$.  \nWe may assume that the triangulation is sufficiently fine \nby taking subdivision.  \nPick a Morse function $f:L \\to \\R$ with the following property.  \nCritical points of $f$ are in one-to-one correspondence with \nbarycenters of simplices.  \nMoreover, the Morse index of a critical point \nis equal to the dimension of the corresponding simplex.  \nThen we can take a gradient-like vector field $X$ such that \nthe unstable manifold $W^u(p)$ at each critical point $p$ is \nthe interior of the corresponding simplex.  \nDenote by $\\{\\rho_t\\}$ the flow generated by $X$.  \n(The function $f$ increases along the orbits of $\\{\\rho_t\\}$.)  \n\nNow we prove the following:\n\n\\begin{thm}\\label{Morse}\nLet $L$ be a relatively spin Lagrangian submanifold in a closed \nsymplectic manifold $(M,\\omega)$ and $f$ a Morse function on $L$ \nas above.  \nThen Morse complex $CM^*(f)\\otimes \\Lambda_{0,nov}$ \ncarries a structure of a filtered $A_{\\infty}$-algebra, \nwhich is homotopy equivalent to the filtered $A_{\\infty}$-algebra \nassociated to $L$ constructed in \\cite{FOOO}.  \n\\end{thm} \nThe proof occupies the rest of this section.  \nWe explain how to choose $\\chi_g(L)$ in section 4. \nFirstly, we choose and fix a linear order on the set of vertices \nin ${\\mathfrak T}$.  \nThen we regard each $T_i \\in {\\mathfrak T}$ as \na singular simplex by the affine parametrization \n$\\sigma_i:\\Delta_{k_i} \\to T_i$ \nrespecting the order of the vertices.  \nIn particular, all simplices are oriented, hence \nthe unstable manifolds $W^u(p)$.  \nFor our construction, we have to start with the following \nset of singular simplices.  \nSet $\\chi_{\\mathfrak T}(L)=\\{\\sigma_i\\}$ \nand identify the Morse complex $CM^{\\bullet}(f)$ \nwith $C_{\\mathfrak T}(L)$, which is a subcomplex of the singular \nchain complex of $L$ generated by $\\chi_{\\mathfrak T}(L)$.  \nNote that $\\chi_{\\mathfrak T}(L)$ satisfies Conditions 1 and 2 \ngiven in section 4.  \n\nWe define $\\chi_g(L) \\supset \\chi_{\\mathfrak T}(L)$ \nin an inductive way as follows.  \nFor $g=-1$, we set $\\chi_{-1}(L)=\\chi_{\\mathfrak T}(L)$.\nFor $g=0,1,\\dots$, suppose that \nwe constructed $\\chi_{g'}(L)$, $g' < g$.  \n\nWe can choose the perturbations \n${\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}$ in \nProposition \\ref{transv} with the following property.  \n\n\\smallskip\n\nEach face $\\tau$ of any simplex in the triangulation of \n\\[\nev_{0*}\\bigl({\\mathcal M}_{k+1}^{{\\rm main},{\\mathfrak d}}\n(\\beta;P_1,\\dots,P_k)^{{\\mathfrak s}_{{\\mathfrak d},k,\\beta,\\vec{P}}} \n\\bigr)\n\\] \nwith $g=\\parallel ({\\mathfrak d}, \\beta) \\parallel$\nis transversal to the stable manifold $W^s(p)$ \nat any $p \\in {\\rm Crit}(f)$.  \nMoreover, for each $\\tau$ of dimension at most \n$\\dim L$, there exists at most one \n$p=p(\\tau) \\in {\\rm Crit}(f)$ such that the stable submanifold \n$W^s(p)$ is of complementary dimension to $\\tau$ and \n$W^s(p)$ and $\\tau$ intersect at a unique point.  \nDenote by $T(p) \\in {\\mathfrak T}$ the simplex containing $p$.  \nLet $\\chi_g^{\\circ}(L)$ be the set of these singular simplices \n$\\tau$.\n\n\\smallskip\n\nWe have to add $\\chi_g^{\\circ}(L)$ to previous \n$\\bigcup_{g' < q}\\chi_{g'}(L)$.  \nIn order to guarantee Condition 2, we further add the following \nsingular simplices to $\\chi_g^{\\circ}(L)$ and obtain \n$\\chi_g(L)$.  \nDenote by $\\sigma^{\\tau} \\in \\chi_{\\mathfrak T}$ \nthe singular simplex corresponding to $T(p(\\tau))$.  \nDefine $\\Pi(\\tau)=\\epsilon \\sigma^{\\tau}$, where \n$\\epsilon = \\pm 1$ is given by the following equation.  \n\\[\n\\tau \\cap W^s(p(\\tau)) = \\epsilon W^u(p(\\tau)) \\cap W^s(p(\\tau)),\n\\]\nif there exists a unique stable manifold $W^s(p(\\tau))$, \nwhich intersects $\\tau$ transversely at a unique point. \nOtherwise, we define $\\Pi(\\tau)=0$.  \nIn particular, if $\\tau > \\dim L$, $\\Pi(\\tau)=0$.  \nFor each $\\tau$ as above, we will find a singular chain $G(\\tau)$ \nsuch that \n$$\n\\Pi (\\tau) - \\tau = \\overline{\\mathfrak m}_1 G(\\tau) \n+ G (\\overline{\\mathfrak m}_1 \\tau),\n$$\nwhere $\\overline{\\mathfrak m}_1=(-1)^{\\dim L} \\partial$.  \nWe can find such $G(\\tau)$ by induction on dimension of $\\tau$.  \nIn our case, we construct $G(\\tau)$ using the gradient-like \nflow $\\{\\rho_t\\}$.  \nSet \n\\[\n{\\mathfrak r}({\\mathrm{Im}} \\tau)=\\bigcup_{t\\leq 0} \\rho_t(\n{\\mathrm{Im}} \\tau).\n\\]\nBy the choice of our perturbations above, \nthe closure of ${\\mathfrak r}({\\mathrm{Im}} \\tau)$ can be \ntriangulated in a compatible way with $\\tau$ and $\\Pi(\\tau)$.  \nPick such a triangulation and then define $G(\\tau)$ the corresponding \nsingular chain.  \nFor the chain $G(\\tau)$, we define $G(G(\\tau))=0$.   \n\n\nNote that $\\Pi:C_{(g)}(L) \\to C_{(0)}(L)$ and \n$G:C_{(g)}(L) \\to C_{(g)}(L)$ satisfy the conditions in Lemma \n3.2, hence $C_{(g)}(L)$ satisfies Condition 2.  \nTherefore we can apply Theorem \\ref{reduction} to reduce \nthe filtered $A_{\\infty}$-structure on $C^{\\bullet}(L; \n\\Lambda_{0,nov})$ to $CM^{\\bullet}(f) \\otimes \\Lambda_{0,nov}$ \nand obtain a filtered $A_{\\infty}$-algebra \n$(CM^{\\bullet}(f) \\otimes \\Lambda_{0,nov},\\{{\\mathfrak m}'_k\\})$, \nwhich is homotopy equivalent to $(C_{(g)}(L) \\otimes \\Lambda_{0,nov},\n\\{{\\mathfrak m}_k\\})$.  \nTheorem \\ref{Morse} is proved.   \n\nIn the proof of Theorem 3.3, we constructed the operator \n${\\mathfrak m}'_k$ from ${\\mathfrak m}_{\\Gamma}$, \n$\\Gamma \\in G^+_{k+1}$.  \nThe geometric meaning of ${\\mathfrak m}_{\\Gamma}$ \nis as follows.  \nRecall that $G(\\tau)$ assigns the closure of the union of \nflow lines arriving at $\\tau$.  \nWe assign $G$ to the interior edges.   \nThe interior vertices correspond to $J$-holomorphic discs, \nmore precisely, bordered stable maps of genus 0.  \nIn order to describe the operation ${\\mathfrak m}_{\\Gamma}$,  \nwe need only rigid configuration of $\\tau_i \\in \n\\chi_{\\mathfrak T}(L)$ (the barycenters of $\\tau_i$ are inputs), \n$J$-holomorphic discs, (broken) negative flow lines of $X$ and \n$W^s(q)$ ($q$ is the output).  \nWe choose the perturbation ${\\mathfrak s}$ generically so that \nthe moduli spaces of holomorphic discs and the flow $\\{\\rho_t\\}$ \nare in general position so that \nthe inner edges correspond to negative flow lines of $X$.  \nHence the ${\\mathfrak m}_{\\Gamma}$ is defined by using \nthe configuration of pseudo-holomorphic discs and \nMorse negative gradient trajectories according to the decorated tree \n$\\Gamma \\in \\cup_k G^+_{k+1}$.  \n\nFor a decorated tree $\\Gamma \\in G_{k+1}^+$, \neach edge is oriented in the direction \nfrom the $k$ input vertices to the root vertex.  \nWe denote by $v^{\\pm}(e)$ the vertices such that \n$e$ is an oriented edge from $v^-(e)$ to $v^+(e)$.  \nConsider \nthe moduli space ${\\mathcal M}_{\\Gamma}(h;p_1, \\dots. p_k, q)$ \nconsisting of the configuration of the following \n\n\\begin{itemize}\n\\item \nfor each interior vertex $v \\in \\Gamma$, a bordered \nstable map $u_v$ representing the class $\\beta_{\\eta(v)}$ with \n$\\ell (v)$ boundary marked points, \nwhere $\\ell(v)$ is the valency of $v$, \n(we denote by $p(e,v)$ the marked point corresponding to the  \nedge $e$ attached to $v$)\n\\item\nthe $i$-th input edge $e_i$ corresponds to a broken negative \ngradient flow line $\\gamma_i$ starting from the critical point $p_i$ \nto $u_{v^+(e_i)}(p(e_i,v^+(e_i)))$, \n\\item\nthe output edge corresponds to a broken negative gradient flow line \n$\\gamma_0$ from $u_{v^-(e_0)}(p(e_0,v^-(e_0)))$ \nending at the critical point $q$,\n\\item\nan interior edge $e$ corresponds to a broken negative gradient flow line $\\gamma_e$ from $u_{v^-(e)}(p(e,v^-(e)))$ to $u_{v^+(e)}(p(e,v^+(e)))$. \n\\end{itemize}\n\nCounting the weighted order of the moduli spaces of vitual \ndimension $0$, we get\n$$\n{\\mathfrak m}_{\\Gamma}(p_1 \\otimes \\dots \\otimes p_k) \n= \\sum_q  \\# {\\mathcal M}_{\\Gamma}(h;p_1, \\dots, p_k,q) \\cdot \ne^{\\sum_{v} \\mu(\\beta_{\\eta(v)})\/2}q\n$$\nand \n$${\\mathfrak m}_k= \\sum_{\\Gamma \\in G^+_{k+1}} \nT^{E(\\Gamma)} {\\mathfrak m}_{\\Gamma}.\n$$\n\n\nFor example, \nwe obtain the configuration as in Figure \\ref{configuration} associated \nto the decorated planar tree $\\Gamma$ \nwith inputs $T(p),T(p'),T(p'') \\in \n\\chi_{\\mathfrak T}(L)$  as in Figure \\ref{modelconfiguration}.\n\n\n\\begin{figure}\n\\begin{center}\n\\include{modelconfiguration}\n\\end{center}\n\\caption{}\n\\label{modelconfiguration}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\include{configuration}\n\\end{center}\n\\caption{}\n\\label{configuration}\n\\end{figure}\n\nThis is essentially the configuration introduced in \\cite{Fuk94}.  \nNote that the first named author \\cite{Fuk94} took multiple \nMorse functions to achieve transversality.  \nHere we use one Morse function and apply the argument in section 3 \nto squeeze the filtered \n$A_{\\infty}$-algebra structure to the Morse complex.   \nWe emphasize that this becomes possible only after working out \nthe chain level intersection theory in detail, which we explained \nin section 4.  \nTo find an appropriate perturbation of \n${\\mathcal M}_{\\Gamma}(h;p_1, \\dots, p_k,q)$ directly without using \nthe argument in section 4 (or section 30 in \\cite{FOOO}) seems \nextremely difficult.  \n\nThe use of multiple Morse functions enables to construct the \ntopological (or partial) filterd $A_{\\infty}$-category of \nMorse functions on $L$ in the case that ${\\mathfrak m}_0 = 0$.  \nNote that, in a topological (or partial) filtered $A_{\\infty}$-category \n${\\mathcal A}$, \nthe set $Ob_{\\mathcal A}$ of objects is a topological space and \nthe set $Mor_{\\mathcal A}(a,b)$ of morphisms is defined \nfor $(a,b)$ in an open dense subset of $Ob_{\\mathcal A} \\times \nOb_{\\mathcal A}$.  \nWhen ${\\mathcal A}$ is a filtered $A_{\\infty}$-category, each \nobject $a$ is equipped with the filtered $A_{\\infty}$-algebra \n$Mor_{\\mathcal A}(a,a)$.   \nIn our case, the filtered $A_{\\infty}$-algebra on Morse complex \n$CM^{\\bullet}(f)$ \ncorresponds to the filtered $A_{\\infty}$-algebra associated to \nthe object $f$.  \nNote that, in the construction of this paper and in Theorem 5.1, \nwe do not need to assume that ${\\mathfrak m}_0=0$ \nin our construction. \n\nFor a relative spin pair $(L_0,L_1)$ of Lagrangian submanifolds, \nwhich intersect transversely, we obtain \nthe filtered $A_{\\infty}$-bimodule over the filtered \n$A_{\\infty}$-algebras on $CM^{\\bullet}(f_i) \\otimes \\Lambda_{0,nov}$, \nwhere $f_i:L_i \\to \\R$, \n$i=0,1$, are Morse functions.  \nWhen $L_0$ and $L_1$ are of clean intersection, \nthere exists a certain local system $\\Theta$ on $L_0 \\cap L_1$ \nand we can reduce the filtered $A_{\\infty}$-bimodule structure on \n$C^{\\bullet}(L_0 \\cap L_1;\\Theta) \\otimes \\Lambda_{0,nov}$ to \n$CM^{\\bullet}(h;\\Theta) \n\\otimes \\Lambda_{0,nov}$ \nover the filtered $A_{\\infty}$-algebras on $CM^{\\bullet}(f_i) \\otimes \n\\Lambda_{0,nov}$.  \nHere $h$ is a Morse function on $L_0 \\cap L_1$, which may be \ndisconnected with various dimensions.  \nFor the canonical models of filtered $A_{\\infty}$-bimodules, \nsee \\cite{FOOO}.  \n\n\\smallskip \\noindent\n{\\bf Acknowledgement.}  We thank Otto van Koert for his kind \ninstruction of making figures in this article.  \n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nEccentric gaseous disks arise in a surprisingly wide variety of astrophysical\ncontexts. A number of mechanisms can explain the existence of eccentric disks,\nthe most commonly invoked one being external perturbation. In eccentric\nbinaries, secular gravitational interaction endows forced and free\neccentricities upon circumbinary and circumobject disks\n\\citep[e.g.,][]{2000ssd..book.....M}; in circular binaries, tidal forces couple\nto circumobject disks through the $3:1$ mean motion resonance and allow free\neccentricity to grow exponentially \\citep{1991ApJ...381..259L}. Another\npossibility is that disks become more eccentric over time. Viscous\noverstability \\citep{1978MNRAS.185..629K}, which amplifies small-scale\neccentric perturbations in isolated disks \\citep[e.g.,][]{1994MNRAS.266..583L,\n2001MNRAS.325..231O}, is often cited in this connection. A third option is for\ndisks to be born eccentric. Outgassing from planetesimals can create eccentric\ndisks \\citep{2021MNRAS.505L..21T}, and so can the tidal disruption of stars\n\\citep{2015ApJ...804...85S, 2015ApJ...806..164P, 2017MNRAS.467.1426S} and\nmolecular clouds \\citep[e.g.,][]{2008Sci...321.1060B} by supermassive black\nholes. On the phenomenological side, eccentric disks are sometimes invoked to\nexplain asymmetric lines in white dwarfs \\citep[e.g.,][]{2006Sci...314.1908G},\nas well as asymmetric broad emission lines in active galactic nuclei\n\\citep[e.g.,][]{1995ApJ...438..610E, 2021MNRAS.506.6014T} and \\acp{TDE}\n\\citep[e.g.,][]{2014ApJ...783...23G, 2017MNRAS.472L..99L}.\n\nBecause ideal \\acp{MHD} is supported even by low levels of ionization\n\\citep{1994ApJ...421..163B, 1996ApJ...457..355G}, we expect magnetic fields to\nplay a role in many of the eccentric disks enumerated above. The presence of\nmagnetic fields changes the way disks evolve because of the \\ac{MRI}\n\\citep{1991ApJ...376..214B, 1991ApJ...376..223H}. Simply put, in a disk whose\ninner parts rotate faster than the outer parts, differential rotation can latch\nonto horizontal bits of the magnetic field, stretch them out, and amplify them.\nThe gas connected to one of these bits on the inside is pulled back by magnetic\ntension, loses angular momentum, migrates inward, and picks up orbital speed.\nIn the meantime, the gas on the outside is dragged forward, gains angular\nmomentum, drifts outward, and slows down. The rising velocity difference across\nthe horizontal magnetic field in turn enhances its stretching, precipitating an\ninstability.\n\nAnalytic calculations for circular disks show that a perturbation can grow by\norders of magnitude per orbit in the linear stage \\citep{1991ApJ...376..214B},\nmaking the \\ac{MRI} among the most vigorous \\ac{MHD} instabilities. The\ninitially exponential amplification eventually enters the nonlinear stage and\nbreaks down into \\ac{MHD} turbulence. Orbital shear enforces a correlation\nbetween the radial and azimuthal components of the turbulent velocity, and\nbetween the same components of the magnetic field. Turbulent stresses transport\nangular momentum outward; gas robbed of angular momentum sinks to smaller\nradii, and the disk accretes.\n\nThe saturation process is amenable only to numerical investigation. Previous\nsimulations of circular disks, numbering in the hundreds, are divided into\nshearing-box simulations, which consider a small neighborhood of the disk as\nrepresentative of the whole \\citep[e.g.,][]{1992ApJ...400..595H,\n1995ApJ...440..742H, 1996ApJ...464..690H, 1995ApJ...446..741B,\n1996ApJ...463..656S}, and global simulations, which take in the entire disk\n\\citep[e.g.,][]{1991ApJ...376..223H, 1998ApJ...501L.189A, 1999ASSL..240..195M,\n2000ApJ...528..462H, 2003ApJ...592.1060D}. This large body of work converged on\nthe consensus that, irrespective of the circumstances simulated, \\ac{MHD}\nturbulence in circular disks saturates within 10 to 20 orbits, and the stresses\nat saturation correspond to a \\citet{1973A&A....24..337S} alpha parameter\nbetween \\numrange[range-phrase={ and }]{0.01}{1}. For circular disks around\nblack holes, the alpha parameter may change substantially near their inner\nedges at the \\ac{ISCO}. In disks with gas-dominated pressure, the alpha\nparameter can increase very rapidly as matter approaches and crosses the\n\\ac{ISCO} \\citep{2010ApJ...711..959N}; alternatively, in super-Eddington\nradiation-dominated disks, it may exhibit a sharp peak at a radius a short\ndistance outside the \\ac{ISCO} \\citep{2014ApJ...796..106J}.\n\nThere is no reason to expect the \\ac{MRI} and the associated \\ac{MHD} stresses\nto be absent from eccentric disks, though their character and the exact manner\nin which they approach saturation may differ from circular disks. Very little\nheed, however, has hitherto been paid to any aspect of the role of magnetic\nfields in eccentric disks. We were the first to establish that eccentric disks\nare susceptible to the \\ac{MRI} \\citep{2018ApJ...856...12C}. Compared to the\ncircular \\ac{MRI}, the growth rate of the eccentric \\ac{MRI} is smaller at the\norder-unity level and the range of unstable wavelengths is wider. That work,\nhowever, is incomplete because it examined only linear stability. It remains an\nopen question whether the robust growth of \\ac{MHD} stresses in the linear\nstage would, as the \\ac{MRI} turns nonlinear, translate to saturated stress\nlevels significant enough to affect disk evolution. There are, however, two\nchallenges to the numerical work requisite to follow the nonlinear process.\n\nThe first problem stems from the fact that existing Newtonian \\acp{MHD} codes\nare capable of handling only Cartesian, cylindrical, and spherical coordinate\nsystems. If a moderately eccentric disk were simulated in one of these\ncoordinate systems, the streamlines would be oblique to the grid. Numerical\nartifacts reflecting grid symmetry would creep in; at the same time, excessive\nnumerical dissipation would prevent the disk from maintaining its shape over a\nlong time. These drawbacks limited previous simulations of isolated eccentric\ndisks, all dealing with pure hydrodynamics, to eccentricities small enough to\nbe implementable as a $m=1$ perturbation to the initial velocity\n\\citep[e.g.,][]{2005A&A...432..757P}. Here we demonstrate that Newtonian\nsimulations can be performed in arbitrary coordinate systems with \\acp{GRMHD}\ncodes, provided that the metric is judiciously chosen. Employing a coordinate\nsystem molded to the shape of moderately eccentric disks significantly\nsuppresses the numerical errors arising from ordinary cylindrical coordinates.\n\nThe second difficulty is with the simulation setup. One may think that the\nsetting of localized perturbations in \\citet{2018ApJ...856...12C} lends itself\nnaturally to shearing-box simulations \\citep[e.g.,][]{2014MNRAS.445.2621O,\n2018MNRAS.477.4838W}. Drawing inspiration from circular disks, one may imagine\nshearing boxes in eccentric disks to have edges running along curves of\nconstant semilatus rectum and constant azimuth. However, the very notion of an\neccentric shearing box is suspect. Circular shearing boxes assume that the\ndisk, being homogeneous, is equivalent to a tiling of the shearing box; this\njustifies periodic azimuthal and shift-periodic radial boundary conditions. The\nassumption breaks down for eccentric disks. \\citet{2018ApJ...856...12C} showed\nthat a perturbation grows differently at different positions along the orbit,\ndepending on the local orbital shear. This means the conditions at the leading\nedge of an eccentric shearing box are different from the trailing edge, and\nthey also vary along each of the other two edges, so boundary conditions that\ndirectly copy one edge to the other would be inappropriate. We can avoid these\nquestions about the eccentric shearing box by performing global simulations\ninstead. It is worth noting that the first simulations of the circular \\ac{MRI}\nwere also global \\citep{1991ApJ...376..223H}.\n\nWe recount our simulation setup in \\cref{sec:methods}. The results from the\nsimulations are presented in \\cref{sec:results} (see\n\\href{https:\/\/youtube.com\/playlist?list=PLlhsZldWhMs6OIvfFxf5DZy9UbpimeGGt}{movies})\nand discussed in \\cref{sec:discussion}. Our concluding remarks are gathered in\n\\cref{sec:conclusions}.\n\n\\section{Methods}\n\\label{sec:methods}\n\nWe outline our simulation strategy in \\cref{sec:overall strategy}. We continue\nwith the details of the simulation setup in the subsequent subsections and in\nthe Appendix; readers uninterested in the technicalities may skip to the\nresults in \\cref{sec:results}.\n\n\\subsection{Overall strategy}\n\\label{sec:overall strategy}\n\nOur goal is to simulate the nonlinear evolution of the eccentric \\ac{MRI} in a\npurely Newtonian setting. The only reason we turn to a \\acp{GRMHD} code is\nbecause numerical issues force us to tailor the coordinate system to the\neccentric disk shape, but existing Newtonian codes lack the facility to deal\nwith bespoke coordinate systems. The \\acp{GRMHD} code we use for this purpose\nis Athena++ \\citep{2016ApJS..225...22W, 2020ApJS..249....4S}.\n\nThere can be drawbacks to solving Newtonian problems with a \\acp{GRMHD} code,\nprincipally the large truncation error potentially created by the smallness of\nthe typical kinetic and internal energies compared to the rest energy. This\nerror can, however, be mitigated by careful design of the simulation setup, as\ndescribed in later subsections.\n\nA major difference of \\acp{GRMHD} codes compared with Newtonian codes is that\ngravity enters not as explicit momentum and energy source terms, but through\nthe metric. Our choice of the metric in \\cref{sec:coordinates,sec:potential}\nensures that orbits are closed ellipses that do not apsidally precess, allowing\nour simulations to closely approximate Newtonian behavior.\n\nWe simulate a suite of disks. The initial hydrodynamic configuration of the\ndisks follow the common prescription in \\cref{sec:hydrodynamic initial\ncondition}: Gas is placed within a limited radial range, so that the inner and\nouter edges of the disk are well-separated from the inner and outer boundaries,\nrespectively, of the simulation domain. All disks are tracked for 15 orbits so\nturbulence may have enough time to reach saturation. The individual disks\ncomprising the suite are designed with contrast in mind. They are classified\nalong two orthogonal dimensions: circular versus eccentric, unmagnetized versus\nmagnetized.\n\nComparison between circular and eccentric disks gives us an idea whether the\nsaturation level of the \\ac{MRI} depends on eccentricity. The eccentric disks\nhave a moderate eccentricity of 0.5, so that the character of the \\ac{MRI}\nspecific to eccentric disks can reveal itself without being overwhelmed by\nhydrodynamic effects.\n\nComparison between unmagnetized and magnetized disks helps us disentangle\n\\ac{MHD} effects from hydrodynamic effects. Magnetized disks are seeded with an\ninitial magnetic field as described in \\cref{sec:magnetic initial condition}.\nTwo magnetic topologies are considered, vertical- and dipolar-field, because\ntopology can influence the saturated \\ac{MHD} turbulence\n\\citep[e.g.,][]{1995ApJ...440..742H, 1996ApJ...464..690H, 2004ApJ...605..321S,\n2013ApJ...767...30B}.\n\n\\subsection{Equations}\n\nWe employ natural units, which means the length and velocity units are the\ngravitational radius and the speed of light, respectively. The sign convention\nis $(-,+,+,+)$, Greek indices range over $\\{0,1,2,3\\}$, Latin indices range\nover $\\{1,2,3\\}$, and Einstein summation is implied. \n\nThe equations of \\acp{GRMHD} are\n\\begin{alignat}{3}\n& \\partial_t[(-g)^{1\/2}\\rho u^t\n  &]& +\\partial_j[(-g)^{1\/2}\\rho u^j\n  &]&= 0, \\\\\n& \\partial_t[(-g)^{1\/2}T^t_\\mu\n  &]& +\\partial_j[(-g)^{1\/2}T^j_\\mu\n  &]&= (-g)^{1\/2}T^\\nu_\\sigma\\Gamma^\\sigma_{\\mu\\nu}, \\\\\n& \\partial_t[(-g)^{1\/2}\\astF^{i0}\n  &]& +\\partial_j[(-g)^{1\/2}\\astF^{ij}\n  &]&= 0.\n\\end{alignat}\nHere $t$ is the coordinate time, $\\rho$ is the comoving mass density, $u^\\mu$\nis the velocity, $g$ is the determinant of the metric $g_{\\mu\\nu}$, and\n$\\Gamma^\\sigma_{\\mu\\nu}$ is the Christoffel symbol of the second kind. From the\nHodge dual of the electromagnetic tensor $\\astF^{\\mu\\nu}$ we obtain the\nmagnetic field $B^i=\\astF^{i0}$ and the projected magnetic field\n$b^\\mu=u_\\nu\\astF^{\\nu\\mu}$. Lastly, the stress--energy tensor is\n\\begin{equation}\nT^{\\mu\\nu}=\\biggl(p+\\frac12b_\\sigma b^\\sigma\\biggr)g^{\\mu\\nu}\n  +\\biggl(\\rho+\\frac\\gamma{\\gamma-1}p+b_\\sigma b^\\sigma\\biggr)u^\\mu u^\\nu\n  -b^\\mu b^\\nu,\n\\end{equation}\nwhere $p$ and $\\gamma$ are the gas pressure and adiabatic index, respectively.\n\n\\subsection{Orbital coordinate system}\n\\label{sec:coordinates}\n\nWe solve the \\acp{GRMHD} equations in an orbital coordinate system\n\\citep{2001MNRAS.325..231O}. It is similar to the cylindrical coordinate\nsystem, except that circular coordinate surfaces are replaced by axially\naligned elliptical ones, chosen such that their cross sections along the\nmidplane match the initial disk streamlines. Adapting the coordinate system to\nthe disk reduces numerical artifacts and dissipation in our simulations. The\neccentricities and orientations of the coordinate surfaces can in principle\nvary from one elliptical cylinder to the next, but here we specialize to the\ncase in which both are spatially uniform.\n\nLet $(t,R,\\varphi,z)$ be cylindrical coordinates. We work with gravity weak\nenough to be well-described by a quasi-Newtonian potential $\\Phi(R,z)$; the\nnonzero components of the metric in this limit are\n\\begin{align}\ng_{tt} &= -[1+2\\Phi(R,z)], \\\\\ng_{RR} &= 1, \\\\\ng_{\\varphi\\varphi} &= R^2, \\\\\ng_{zz} &= 1.\n\\end{align}\nLet $(t,\\log\\lambda,\\phi,z)$ be orbital coordinates specialized for use in our\nsimulations; they are related to cylindrical coordinates by\n\\begin{align}\nR &= \\lambda\/(1+e\\cos\\phi), \\\\\n\\varphi &= \\phi.\n\\end{align}\nHere $e$ is the eccentricity of our orbital coordinates, set to 0 for circular\ndisks and 0.5 for eccentric disks. Coordinate surfaces of constant $\\lambda$\nare elliptical cylinders of semilatus rectum $\\lambda$, or equivalently,\nsemimajor axis $a=\\lambda\/(1-e^2)$. We opt for $\\log\\lambda$ instead of\n$\\lambda$ in order to generate a logarithmic grid, but for ease of\nunderstanding we continue to label that coordinate by $\\lambda$ and express\nresults in terms of $\\lambda$. We reuse $t$ and $z$ without risk of ambiguity\nbecause these two coordinates do not participate in the coordinate\ntransformation from cylindrical to orbital. The metric and connection in both\ncoordinates are provided in \\cref{sec:metric}.\n\n\\subsection{Gravitational potential and orbits}\n\\label{sec:potential}\n\nWe ignore vertical gravity in these first simulations of the eccentric\n\\ac{MRI}, so the potential depends only on $R$. In this sense, our simulations\nresemble earlier simulations of unstratified circular disks\n\\citep[e.g.,][]{1991ApJ...376..223H, 1995ApJ...440..742H, 1996ApJ...464..690H,\n2004ApJ...605..321S, 2007A&A...476.1113F, 2007A&A...476.1123F,\n2007MNRAS.378.1471L, 2009ApJ...690..974S, 2009ApJ...694.1010G,\n2011ApJ...739...82B}. However, instead of the point-mass gravitational\npotential $\\Phi(R,z)=-1\/R$, we adopt\n\\begin{equation}\n\\Phi(R,z)=-1\/(R+2)\n\\end{equation}\nbecause, as proven in \\cref{sec:potential derivation}, this potential admits\nclosed eccentric orbits at all distances. The modification matters because\ngeneral-relativistic apsidal precession, albeit small at large distances,\naccumulates over the tens of orbits during which the \\ac{MRI} saturates.\n\nThe properties of orbits in our potential are also derived in\n\\cref{sec:potential derivation}; here we repeat the parts that support our\nexposition. For an orbit along a coordinate curve of semilatus rectum\n$\\lambda$ or semimajor axis $a=\\lambda\/(1-e^2)$, the orbital period is\n\\begin{equation}\\label{eq:orbital period}\nT=2\\pi a^{3\/2}(1+2\/a).\n\\end{equation}\nThe specific energy and angular momentum conserved with respect to our metric\nare\n\\begin{align}\n\\label{eq:orbit energy}\nE &= (1+1\/a)^{-1\/2}, \\\\\n\\label{eq:orbit angular momentum}\nL &= \\lambda^{1\/2}E.\n\\end{align}\nThe specific energy includes the rest energy; $E=1$ corresponds to marginally\nbound material, and $E\\to0$ as material becomes more bound. The nonzero\ncomponents of the orbital velocity are\n\\begin{align}\n\\label{eq:orbit velocity 0} u^t &= E\/[1+2\\Phi(R,z)], \\\\\n\\label{eq:orbit velocity 2} u^\\phi &= L\/R^2,\n\\end{align}\nso the physical velocity at pericenter is\n\\begin{equation}\nv_\\su p(\\lambda)=\\frac{Ru^\\phi}{u^t}=\n  \\frac{\\lambda^{1\/2}(1+e)}{\\lambda+2(1+e)}.\n\\end{equation}\n\nConsider a collection of such orbits nested within each other, all following\ncoordinate curves. The velocity field thus generated has finite divergence:\n\\begin{equation}\n(-g)^{-1\/2}\\partial_\\mu[(-g)^{1\/2}u^\\mu]=\n  \\frac{u^\\phi}{R+2}\\frac{e\\sin\\phi}{1+e\\cos\\phi}.\n\\end{equation}\nConsequently, if the motion of a gas without pressure is described by this\nvelocity field, the density along a streamline of constant $\\lambda$ cannot be\nuniform, but must instead vary with $\\phi$ in proportion to\n\\begin{equation}\nd(\\lambda,\\phi)\\eqdef\n  \\biggl[\\frac{\\lambda+2(1+e\\cos\\phi)}{\\lambda+2(1+e)}\\biggr]^{1\/2}.\n\\end{equation}\nSo as to guarantee an initial condition that is a genuine hydrodynamical steady\nstate, we take this modulation into account in \\cref{sec:hydrodynamic initial\ncondition} even though the modulation amplitude is tiny for our disk\nparameters.\n\n\\subsection{Hydrodynamic initial condition}\n\\label{sec:hydrodynamic initial condition}\n\nBecause we ignore vertical gravity, our initial disk is translationally\nsymmetric along the $z$\\nobreakdash-direction. It can be described as an\nelliptical annular cylinder, each shell of which orbits the coordinate axis\nalong a coordinate surface of constant $\\lambda$ with a velocity as given by\n\\cref{eq:orbit velocity 0,eq:orbit velocity 2}. The scale of the disk is\ncharacterized by its fiducial orbit, whose semilatus rectum is the geometric\nmean of the semilatera recta of its inner and outer edges.\n\nThe disk should be large enough that orbital velocities are non-relativistic,\nand small enough that severe truncation errors do not arise from the evolution\nof the total energy as a result of the smallness of the kinetic energy with\nrespect to the rest energy. In light of the fact that the linear growth rate of\nthe \\ac{MRI} is inversely proportional to the orbital period\n\\citep{2018ApJ...856...12C}, we additionally require that runs of different $e$\nhave the same semimajor axis and thus orbital period. We settle on a semilatus\nrectum of $\\lambda_*=200(1-e^2)$ for the fiducial orbit, and we report time in\nunits of the orbital period at this orbit.\n\nThe initial density profile is\n\\begin{equation}\\label{eq:initial density}\n\\rho(\\lambda,\\phi)=\\rho_*m(\\lambda,\\phi),\n\\end{equation}\nwith $\\rho_*$ the density at the pericenter of the fiducial orbit,\n$(\\lambda,\\phi)=(\\lambda_*,0)$. To give the disk edges that are not too sharp\nand to build in a numerical vacuum, the density is modulated spatially as\n\\begin{equation}\nm(\\lambda,\\phi)=\n  (1-f_\\su v)d(\\lambda,\\phi)h(l_\\su{gb},l_\\su{gt};2q^\\lambda)+f_\\su v,\n\\end{equation}\nwhere\n\\begin{equation}\nh(a,s;x)\\eqdef\n  \\frac14\\biggl(1+\\tanh\\frac{x+a}s\\biggr)\\biggl(1+\\tanh\\frac{a-x}s\\biggr)\n\\end{equation}\nis a smoothed top-hat function and\n\\begin{equation}\nq^\\lambda\\eqdef\n  \\frac{\\log\\lambda-\\log\\lambda_*}{\\max(\\log\\lambda)-\\min(\\log\\lambda)}\n\\end{equation}\nis the fractional $(\\log\\lambda)$\\nobreakdash-position within the simulation\ndomain. The modulation is governed by three parameters: $f_\\su v=0.01$ is the\nvacuum-to-disk density ratio, and $l_\\su{gb}=0.5$ and $l_\\su{gt}=0.1$ are the\nfractional $(\\log\\lambda)$\\nobreakdash-extents of the disk body and the\ndisk--vacuum transition, respectively.\n\nBecause the most unstable mode of the circular \\ac{MRI} is incompressible, the\nthermodynamic properties of the gas is expected to have little bearing on the\ngrowth of the eccentric \\ac{MRI}. Even so, we would like the pressure to be\nsmall enough initially that the configuration above is close to equilibrium,\nand the internal energy to be large enough at all times that catastrophic\ntruncation errors are avoided. We balance these competing desires by setting\nthe Mach number at the fiducial pericenter to $M_*=30$. Additionally, we adopt\nan adiabatic index of $\\gamma=1+10^{-5}$, corresponding to a nearly isothermal\ngas, so that the internal energy is larger at fixed pressure. The initial\npressure is therefore\n\\begin{equation}\np_*=\\rho_*[v_\\su p(\\lambda_*)\/M_*]^2\/\\gamma\n\\end{equation}\nat the fiducial pericenter and\n\\begin{equation}\np(\\lambda,\\phi)=p_*[m(\\lambda,\\phi)]^\\gamma\n\\end{equation}\neverywhere else. This initial condition is not strictly hydrostatic; transient\noutgoing waves are launched from the inner edge as the disk seeks force\nbalance. Moreover, our use of a soft equation of state means that density\nperturbations are stronger and pressure gradients have a lesser effect on disk\nevolution than if an adiabatic equation of state were used.\n\n\\subsection{Magnetic initial condition}\n\\label{sec:magnetic initial condition}\n\nThe magnetized runs are initialized with the two kinds of magnetic topologies\nillustrated in \\cref{fig:topology}. The vertical-field topology refers to a\nmagnetic field with one nonzero component\n\\begin{equation}\nB^z\\propto d(\\lambda,\\phi)h(l_\\su{mb},l_\\su{mt},2q^\\lambda),\n\\end{equation}\nwhere $l_\\su{mb}=0.4$ and $l_\\su{mt}=0.1$ are the fractional\n$(\\log\\lambda)$\\nobreakdash-extents of the magnetized disk body and the\ntransition from the magnetized disk body to the unmagnetized disk edges,\nrespectively. The net vertical magnetic flux persists throughout the\nsimulation. Thanks to its simplicity and its ability to generate the\nfastest-growing instabilities, the vertical-field topology was considered in\nthe first studies of the circular \\ac{MRI} \\citep{1991ApJ...376..214B,\n1991ApJ...376..223H} and in our analytic study of the eccentric \\ac{MRI}\n\\citep{2018ApJ...856...12C}.\n\n\\begin{figure}\n\\includegraphics{topology}\n\\caption{Poloidal slices showing with arrows the initial magnetic field for our\ntwo magnetic topologies. Background colors plot the initial density on the same\nscale as \\cref{fig:density}. The ordinate is more stretched than the abscissa,\nand the vectors are arbitrarily scaled.}\n\\label{fig:topology}\n\\end{figure}\n\nThe dipolar-field topology is derived from a magnetic potential with one\nnonzero\ncomponent\n\\begin{equation}\nA_\\phi\\propto\n  (-g)^{1\/2}\\cos^2(\\min(\\tfrac12,\\abs{q^\\lambda\/l_\\su{mb}})\\pi)\\cos^2(q^z\\pi),\n\\end{equation}\nwhere\n\\begin{equation}\nq^z\\eqdef\\frac z{\\max(z)-\\min(z)}\n\\end{equation}\nis the fractional $z$\\nobreakdash-position within the simulation domain and\n$l_\\su{mb}=0.4$. The inclusion of the metric determinant makes the\nmagnetic-field strength more uniform over azimuth. The dipolar-field topology\nsees frequent application in global simulations of the circular \\ac{MRI}\n\\citep[e.g.,][]{2000ApJ...528..462H}.\n\nFor both topologies, the initial plasma beta, defined as the initial volume\nintegral of gas pressure to that of magnetic pressure, is 100. The pressure due\nto the magnetic field is subtracted from the gas to preserve the total. The gas\npressure in magnetized regions is perturbed at the 0.01 level to seed the\n\\ac{MRI}.\n\n\\subsection{Simulation domain, boundary conditions, and other numerical\nconcerns}\n\nThe simulation domain spans\n$[\\exp(-2)\\lambda_*,\\exp(1)\\lambda_*]\\times[-\\pi,\\pi]\\times[-10,10]$ in\n$(\\lambda,\\phi,z)$. The lower end of the $\\lambda$\\nobreakdash-range ensures\nvelocities are never close to the speed of light, and the asymmetry of the\n$\\lambda$\\nobreakdash-range gives the infalling disk more room to evolve freely\nbefore hitting the inner boundary. The resolution is $240\\times240\\times60$ in\nthe three directions.\n\nPeriodic boundary conditions apply to the $\\phi$\\nobreakdash-direction. They\nare also employed in the $z$\\nobreakdash-direction, in accordance with our\nneglect of vertical gravity. Outflow boundary conditions are used in the\n$\\lambda$\\nobreakdash-direction: We copy all quantities to the ghost zone, zero\nthe $\\lambda$\\nobreakdash-component of the velocity if it points into the\nsimulation domain, and zero the $\\phi$- and $z$\\nobreakdash-components of the\nmagnetic field always. This last step reduces unphysical influences from the\nboundaries.\n\nTo prevent numerical issues, we require that the pressure in the simulation\ndomain always satisfies $(\\gamma p\/\\rho)^{1\/2}\\ge\\num{2e-4}$. In addition,\nwhenever the recovery of primitive variables fails, the primitive variables\nfrom the previous time step are carried forward.\n\nAs the simulation progresses, different parts of the disk may evolve\ndifferently in eccentricity and orientation, so the disk could eventually\nbecome misaligned with the grid, resulting in greater numerical dissipation.\nThe disk also occupies a wider range of semilatera recta due to pressure\ngradients and outward angular momentum transport, bringing the now differently\nshaped disk into contact with the boundaries. The inner boundary poses a lower\nlimit on the pericenter, but this restriction is arguably physical because\nthere are indeed radii an accretion flow cannot return from. If the central\nobject is a star, the disk cannot extend inside the star or its magnetosphere.\nIf the central object is a black hole, \\cref{fig:effective potential} tells us\nthat material with sufficiently low angular momentum and high energy can plunge\ndirectly into the black hole; low angular momentum and high energy are, of\ncourse, the hallmarks of an eccentric orbit. By contrast, the interaction with\nthe outer boundary is unphysical and can lead to numerical artifacts;\ntherefore, we restrict our attention to the first 15 orbits at $a=200$, before\nthe disk starts running into the outer boundary and numerical artifacts appear.\nSteady state appears to obtain at this time for the plasma beta and the alpha\nparameter, despite the continual evolution in disk shape.\n\n\\begin{figure}\n\\includegraphics{effpotl}\n\\caption{Effective potential $V_\\su{eff}=-1\/r+\\tfrac12(L^2\/r^2)(1-2\/r)$ in\nSchwarzschild spacetime as a function of spherical radius $r$ and specific\nangular momentum $L$. The specific energy $E$ must satisfy\n$\\tfrac12(E^2-1)>V_\\su{eff}$, and marginally bound trajectories have $E=0$.\nMaterial accretes by losing angular momentum; thus, its trajectory is described\nby potentials of decreasing $L$. In circular disks, trajectories have the\nlowest energy allowed by the stable potential well; such trajectories evolve\nalong the sequence of dots downward until they arrive at the smallest radius\nthat supports circular orbits, the \\ac{ISCO}. In eccentric disks, trajectories\nhave energies above the bottom of the potential well, which allows them to make\nradial excursions as suggested by the double-headed arrow. The smallest radius\na bound trajectory can reach without falling in is the marginally bound orbit;\nsuch a trajectory has $E=0$ and $L^2=16$. Because trajectories energetic enough\nto overcome the centrifugal barrier have energies greater than \\iac{ISCO}\norbit, material plunging into the black hole on these trajectories have less\nenergy available for radiation compared to material accreting on circular\ntrajectories.}\n\\label{fig:effective potential}\n\\end{figure}\n\n\\section{Results}\n\\label{sec:results}\n\n\\subsection{Overview}\n\n\\Cref{fig:density} tells us how much the disks have changed by the end of the\nsimulations. The most conspicuous contrast between unmagnetized and magnetized\ndisks, of whatever eccentricity, is that unmagnetized disks remain smooth while\nmagnetized disks develop large density fluctuations. This, of course, is due to\nthe \\ac{MRI} creating \\ac{MHD} turbulence in the magnetized disks. The\nfluctuations are larger in vertical-field disks than in dipolar-field disks.\n\nComparison can also be made between circular and eccentric disks regardless of\nmagnetic topology. Unlike the circular disks, which stay circular despite the\n\\ac{MHD} turbulence, the inner parts of eccentric disks grow more eccentric and\nprecess prograde. As the inner parts of eccentric disks shrink, their\npericenters move inside the inner boundary, and their material accretes across\nthe boundary while retaining its eccentricity. This loss of material from the\nmost eccentric orbits is the reason why there is a sparsely populated region\nbetween the inner parts and the inner boundary, visible in the late-time\neccentric disks in \\cref{fig:density}.\n\n\\begin{figure}\n\\includegraphics{density}\n\\caption{Midplane slices of density, in units of the fiducial density $\\rho_*$\nfrom \\cref{eq:initial density}. The top row shows two initial disks with\ndifferent eccentricities. The panels under each top-row panel are the outcomes\nof imposing various magnetic topologies on an initial disk and evolving it for\n15 orbits. The boundaries of the simulation domain are traced by thin ellipses\nin order to better distinguish low-density regions from regions not covered by\nthe simulations. Short lines from the origin indicate the approximate\norientations of the inner parts of the eccentric disks.}\n\\label{fig:density}\n\\end{figure}\n\nThere are also differences among the eccentric disks, concerning chiefly\neccentricity evolution and to a lesser degree precession. The unmagnetized and\nvertical-field disks in \\cref{fig:density} have largely preserved their initial\neccentricity, even though the outer parts of the unmagnetized disk have become\nsomewhat rounder, and the inner parts of the vertical-field disk have precessed\nslightly more. The dipolar-field disk features the steepest eccentricity\ngradient and a much higher degree of precession.\n\nThe eccentricity gradient can be quantified by computing the instantaneous\neccentricity $\\bar e$, which is the eccentricity of the orbit material would\nfollow given its instantaneous velocity $u^\\mu$ if only gravitational forces\nact; this orbit is also known as the osculating orbit. We calculate $\\bar e$\nfrom $u^\\mu$ using \\cref{eq:orbit velocity 0,eq:orbit velocity\n2,eq:eccentricity}. \\Cref{fig:instantaneous eccentricity} contains plots of the\nmass-weighted vertical average of $\\bar e$:\n\\begin{equation}\\label{eq:instantaneous eccentricity map}\n\\mean{\\bar e}_{z;\\rho}\\eqdef\\int dz\\,\\rho\\bar e\\bigg\/\\int dz\\,\\rho.\n\\end{equation}\nThe instantaneous eccentricity deviates little from its initial value in the\nunmagnetized and vertical-field disks, but it develops a clear gradient in the\ndipolar-field disk, with the eccentricity higher than its initial value in the\ninner parts and lower in the outer parts. The implications of the eccentricity\ngradient will be discussed in \\cref{sec:inner edge}. It is also apparent in\nthis figure that the prograde precession is largest in the dipolar-field disk.\n\n\\begin{figure}\n\\includegraphics{eccntrcy}\n\\caption{Mass-weighted vertical average of the instantaneous eccentricity, as\ndefined in \\cref{eq:instantaneous eccentricity map}. The top-left panel shows\nthe eccentric initial disk. The other panels are the outcomes of imposing\nvarious magnetic topologies on the initial disk and evolving it for 15 orbits.\nThe boundaries of the simulation domain are traced by thin ellipses.}\n\\label{fig:instantaneous eccentricity}\n\\end{figure}\n\n\\subsection{Plasma beta and the alpha parameter}\n\\label{sec:magnetic field}\n\nThe top half of \\cref{fig:magnetic field map} portrays the mass-weighted\nvertical average of the plasma beta at the end of the simulations, defined as\n\\begin{equation}\\label{eq:plasma beta map}\n\\mean\\beta_{z;\\rho}\\eqdef\n  \\int dz\\,\\rho\\frac{2p}{b_\\mu b^\\mu}\\bigg\/\\int dz\\,\\rho.\n\\end{equation}\nThe plasma beta is \\num{\\sim10} in vertical-field disks and \\num{\\sim100} in\ndipolar-field disks; the variation within a disk is about one order of\nmagnitude. Comparable levels of plasma beta are witnessed in circular and\neccentric disks with the same magnetic topology, suggesting that the \\ac{MRI}\nis unimpeded by eccentricity. The stronger magnetic fields in the\nvertical-field topology accord with simulations of circular disks in the\nliterature \\citep[e.g.,][]{1995ApJ...440..742H, 1996ApJ...464..690H,\n2004ApJ...605..321S, 2013ApJ...767...30B}.\n\n\\begin{figure}\n\\includegraphics{magmap}\n\\caption{Mass-weighted vertical averages of the plasma beta in the top half and\nMaxwell-only alpha parameter in the bottom half, as defined in \\cref{eq:plasma\nbeta map,eq:alpha parameter map}, respectively. The panels are the outcomes of\nimposing various magnetic topologies on an initial disk and evolving it for 15\norbits. Regions with negligible levels of magnetic field have exceedingly large\nvalues of plasma beta. The boundaries of the simulation domain are traced by\nthin ellipses.}\n\\label{fig:magnetic field map}\n\\end{figure}\n\nWe can also examine the plasma beta along a one-dimensional profile running\nfrom the inside of the disk to the outside at different times during the\nsimulations. Considering that the disk evolves in eccentricity and orientation,\nit makes little sense to look at profiles over semilatus rectum; instead, we\nconstruct profiles over cylindrical radius. The mass-weighted average plasma\nbeta at cylindrical radius $R$ is given by\n\\begin{equation}\\label{eq:plasma beta profile}\n\\mean\\beta_{t\\varphi z;\\rho}\\eqdef\n  \\int dS\\,\\rho\\frac{2p}{b_\\mu b^\\mu}\\bigg\/\\int dS\\,\\rho,\n\\end{equation}\nwhere the hypersurface element is\n\\begin{equation}\\label{eq:hypersurface element}\ndS\\eqdef(-g)^{1\/2}\\,dt\\,d(\\log\\lambda)\\,d\\phi\\,dz\n  \\mathop\\delta\\biggl(\\frac\\lambda{1+e\\cos\\phi}-R\\biggr).\n\\end{equation}\nTemporal and spatial averaging smooths out turbulent fluctuations. Temporal\naveraging is performed over two intervals, each lasting one-third of the\nsimulation duration; comparison between the intervals gives us an idea how\nclose the \\ac{MRI} is to saturation. Spatial averaging is limited to the\ncylindrical shell of radius $R$ picked out by the delta-function. The results\nare plotted in the top half of \\cref{fig:magnetic field profile}, and the\nlegend lists the intervals of temporal averaging. The relatively small\ndifference between the two intervals at radii $100\\lesssim R\\lesssim200$\nsuggests that steady state is achieved to some degree at those radii, despite\nthe relatively short simulation duration. We also reach similar conclusions as\nwe did with \\cref{fig:magnetic field map}: The plasma beta is quite uniform\nover the disk, it is not significantly modified by the introduction of\neccentricity, but it is one to two orders of magnitude lower in vertical-field\ndisks than in dipolar-field disks.\n\n\\begin{figure}\n\\includegraphics{magprof}\n\\caption{Mass-weighted radial profiles of the plasma beta in the top half and\nMaxwell-only alpha parameter in the bottom half, as defined in \\cref{eq:plasma\nbeta profile,eq:alpha parameter profile}, respectively. The profiles are\ntime-averaged over the two intervals in the legend. The two rows in the lower\nhalf have different vertical scales. The magnetic field strength varies weakly\nwith eccentricity, but is much stronger in vertical-field disks than in\ndipolar-field disks.}\n\\label{fig:magnetic field profile}\n\\end{figure}\n\nThe alpha parameter is conventionally taken to be the sum of Reynolds and\nMaxwell stresses divided by the gas pressure. However, it is difficult to\ndetermine the mean flow and departures from it in a disk whose inner and outer\nparts evolve differently in eccentricity and orientation. We therefore consider\nonly the Maxwell, not Reynolds, contribution to the alpha parameter, working\nunder the assumption that the Maxwell stress dominates the total, as is\nuniformly the case for circular disks \\citep[e.g.,][]{1995ApJ...440..742H}. The\nMaxwell stress is defined in terms of the projected magnetic field $b^\\mu$,\nsimilar to the stress--energy tensor. To make the stress more physically\ninterpretable, we measure $b^\\mu$ in a local orthonormal basis whose basis\nvectors are those of cylindrical coordinates, but with lengths normalized to\nunity:\n\\begin{align}\n\\label{eq:physical magnetic field 1}\nb^{\\hat R}\n  &= R\\biggl[b^\\lambda+b^\\phi\\frac{e\\sin\\phi}{1+e\\cos\\phi}\\biggr], \\\\\n\\label{eq:physical magnetic field 2}\nb^{\\hat\\varphi}\n  &= Rb^\\phi, \\\\\n\\label{eq:physical magnetic field 3}\nb^{\\hat z}\n  &= b^z.\n\\end{align}\nWe then define the Maxwell stress to be $b^{\\hat\\varphi}b^{\\hat R}$. A factor\nof $R$ is attached to $b^\\lambda$ in \\cref{eq:physical magnetic field 1}\nbecause our orbital coordinates use $\\log\\lambda$, not $\\lambda$.\nOrthonormality guarantees that\n\\begin{equation}\ng_{\\lambda\\lambda}b^\\lambda b^\\lambda+2g_{\\lambda\\phi}b^\\lambda b^\\phi\n  +g_{\\phi\\phi}b^\\phi b^\\phi=\n  b^{\\hat R}b^{\\hat R}+b^{\\hat\\varphi}b^{\\hat\\varphi}.\n\\end{equation}\n\nThe bottom half of \\cref{fig:magnetic field map} depicts the mass-weighted\nvertical average of the Maxwell-only alpha parameter at the end of the\nsimulations:\n\\begin{equation}\\label{eq:alpha parameter map}\n\\mean{\\alpha_\\su m}_{z;p}\\eqdef\n  -\\int dz\\,b^{\\hat\\varphi}b^{\\hat R}\\bigg\/\\int dz\\,p.\n\\end{equation}\nThe alpha parameter is positive almost everywhere in circular disks, as\nrequired for outward angular momentum transport. By contrast, the alpha\nparameter is uniformly positive in the lower half of the disk where material\nfalls to pericenter, but it is consistently \\textit{negative} in certain\nsectors of the upper half where material flies out to apocenter. In the\nvertical-field disk, positive sectors occupy significantly more area than\nnegative sectors. The magnitude of the alpha parameter, whether positive or\nnegative, varies from \\numrange{\\sim0.2}{5}; however, if we construct\narea-weighted histograms of the magnitude separately for positive and negative\nsectors, we find that the positive histogram is shifted by a factor of\n\\num{\\sim2} toward larger values relative to its negative counterpart. In the\ndipolar-field disk, the total area of positive sectors is only slightly larger\nthan that of negative sectors. In addition, the magnitude of the alpha\nparameter has a narrower distribution, \\numrange{\\sim0.3}{3}; the positive\nhistogram is again displaced by a factor of \\num{\\sim2} compared to the\nnegative one. We shall speculate about why the alpha parameter switches sign in\n\\cref{sec:stresses}.\n\nThe net effect of angular momentum transport is revealed by integrating the\nMaxwell stress over an orbit. When doing so by eye on the basis of\n\\cref{fig:magnetic field map}, it is important to take into account the\nrelative areas and alpha-parameter ranges of the positive and negative sectors.\nMore quantitatively, we construct in \\cref{fig:magnetic field profile}\nmass-weighted radial profiles of the alpha parameter at different times using\nthe prescription\n\\begin{equation}\\label{eq:alpha parameter profile}\n\\mean{\\alpha_\\su m}_{t\\varphi z;p}\\eqdef\n  -\\int dS\\,b^{\\hat\\varphi}b^{\\hat R}\\bigg\/\\int dS\\,p,\n\\end{equation}\nwith $dS$ the same hypersurface element from \\cref{eq:hypersurface element}.\nThe alpha parameter is comparable in circular and eccentric disks of the same\nmagnetic topology. In vertical-field disks, the alpha parameter is\n\\numrange{\\sim0.5}{1}; in dipolar-field disks, it is still positive, but only\n\\numrange{\\sim0.05}{0.15}. Stronger stress for vertical than dipolar magnetic\nfield agrees with previous simulations of circular disks\n\\citep[e.g.,][]{1995ApJ...440..742H, 1996ApJ...464..690H, 2004ApJ...605..321S,\n2013ApJ...767...30B}.\n\n\\subsection{Specific angular momentum and binding energy}\n\\label{sec:angular momentum and binding energy}\n\nTo investigate the effect of internal stresses, we examine how the specific\nangular momentum squared $L^2$ and binding energy $E_\\su b=1-E$ evolve. The\nmass-weighted vertical averages of the two quantities at the end of the\nsimulations are\n\\begin{align}\n\\label{eq:angular momentum squared map}\n\\mean{L^2}_{z;\\rho} &\\eqdef\n  \\int dz\\,\\rho(Ru^{\\hat\\varphi})^2\\bigg\/\\int dz\\,\\rho, \\\\\n\\label{eq:binding energy map}\n\\mean{E_\\su b}_{z;\\rho} &\\eqdef\n  \\int dz\\,\\rho(1+u_t)\\bigg\/\\int dz\\,\\rho.\n\\end{align}\nHere $u^{\\hat\\varphi}=Ru^\\phi$ is the velocity measured in the local\northonormal cylindrical basis, defined analogously to $b^{\\hat\\varphi}$ in\n\\cref{eq:physical magnetic field 2}, and the covariant velocity component\n$u_t=-E$ is a conserved quantity of our time-independent metric.\n\n\\Cref{fig:angular momentum squared and binding energy} displays the specific\nangular momentum squared and binding energy, normalized to their initial values\nat the inner edge. Our focus is on the eccentric disks. Because of pressure\ngradients, even the inner edge of the unmagnetized disk experiences a reduction\nin specific angular momentum squared by \\SI{\\sim25}{\\percent} from its initial\nvalue. The decrease is greater in magnetized disks where the Maxwell stress\nalso contributes: the dipolar-field disk reports a drop by\n\\SI{\\sim33}{\\percent}, and the vertical-field disk, which has a smaller plasma\nbeta and larger alpha parameter than the dipolar-field disk, records a\nsuppression by \\SI{\\sim50}{\\percent}. It should be noted that the range of\nspecific angular momentum is constrained by the geometry of the simulation\ndomain: Once material loses enough angular momentum that its pericenter recedes\ninside the inner boundary, it leaves the simulation domain.\n\nThe specific binding energy at the inner edge changes by rather less. It is\nalmost identical to its initial value in the unmagnetized disk,\n\\SI{\\sim30}{\\percent} higher in the vertical-field disk, and\n\\SI{\\sim10}{\\percent} \\textit{lower} in the dipolar-field disk. The smaller\nfractional changes suggest that the torques transporting angular momentum\noutward, regardless of whether they are hydrodynamic or magnetic in nature,\noccur preferentially near apocenter where the attendant work done is smaller.\n\n\\begin{figure*}\n\\includegraphics{angmenrg}\n\\caption{Mass-weighted vertical averages of the specific angular momentum\nsquared in the left half and specific binding energy in the right half, as\ndefined in \\cref{eq:angular momentum squared map,eq:binding energy map},\nrespectively. The top row shows two initial disks with different\neccentricities. The panels under each top-row panel are the outcomes of\nimposing various magnetic topologies on an initial disk and evolving it for 15\norbits. In all panels, the inner edge is arbitrarily defined to be where the\ndensity is $0.1\\,\\rho_*$, with $\\rho_*$ the fiducial density from\n\\cref{eq:initial density}, and regions less dense than that are left blank.\nFurthermore, the specific angular momentum squared or binding energy in each\npanel is normalized by its value at the inner edge of the corresponding initial\ndisk, which is why the inner edge appears yellow in the top row. The boundaries\nof the simulation domain are traced by thin ellipses. The fractional changes in\nthe specific angular momentum squared and binding energy together determine the\nchange in eccentricity.}\n\\label{fig:angular momentum squared and binding energy}\n\\end{figure*}\n\n\\subsection{Quality factors}\n\nWe close this section by examining how well our magnetized disks resolve the\n\\ac{MRI}. The figures of merit for circular disks are the quality factors,\ndefined as the ratio of the characteristic wavelength of the \\ac{MRI} to cell\nsizes in different directions \\citep{2011ApJ...738...84H}. We generalize their\nmass-weighted vertical averages to eccentric disks as\n\\begin{align}\n\\label{eq:quality 3}\n\\mean{Q_z}_{z;\\rho} &\\eqdef \\int dz\\,\\rho\\frac{\\abs{b^{\\hat z}}T}\n  {\\rho^{1\/2}\\Delta z}\\bigg\/\\int dz\\,\\rho, \\\\\n\\label{eq:quality 2}\n\\mean{Q_\\phi}_{z;\\rho} &\\eqdef \\int dz\\,\\rho\\frac{\\abs{b^{\\hat\\varphi}}T}\n  {\\rho^{1\/2}R\\Delta\\phi}\\bigg\/\\int dz\\,\\rho,\n\\end{align}\nwhere $T$ is the orbital period from \\cref{eq:orbital period}, and $\\Delta z$\nand $\\Delta\\phi$ are the cell sizes in the $z$- and\n$\\phi$\\nobreakdash-directions, respectively. In keeping with work on circular\ndisks, we base our quality factors on the physical, cylindrical components\n$b^{\\hat\\mu}$ of the projected magnetic field $b^\\mu$, given by\n\\cref{eq:physical magnetic field 2,eq:physical magnetic field 3}.\n\n\\Cref{fig:quality} showcases the quality factors at the end of the simulations.\nFor ease of comparison with the criteria that indicate adequate resolution for\ncircular disks, to wit, $\\mean{Q_z}\\gtrsim15$ and $\\mean{Q_\\phi}\\gtrsim20$\n\\citep{2013ApJ...772..102H}, the color scales of the figure are centered on\nthese values. In terms of $\\mean{Q_z}$, the vertical-field disks are extremely\nwell-resolved everywhere, but the same is true for the dipolar-field disks only\nfor a limited range of semilatera recta. In terms of $\\mean{Q_\\phi}$, all disks\nare marginally under-resolved. Interestingly, $\\mean{Q_\\phi}$ is noticeably\nhigher in the pericentric half of the disks; this is because the strong orbital\nshear near pericenter amplifies $b^{\\hat\\varphi}$, not $b^{\\hat z}$.\n\n\\begin{figure}\n\\includegraphics{quality}\n\\caption{Mass-weighted vertical averages of the vertical quality factor in the\ntop half and azimuthal quality factor in the bottom half, as defined in\n\\cref{eq:quality 3,eq:quality 2}, respectively. The panels are the outcomes of\nimposing various magnetic topologies on an initial disk and evolving it for 15\norbits. The boundaries of the simulation domain are traced by thin ellipses.}\n\\label{fig:quality}\n\\end{figure}\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\\subsection{Stresses in circular and eccentric disks}\n\\label{sec:stresses}\n\nOur earlier investigation into the linear stage of the \\ac{MRI} found that the\n\\ac{MRI} grows in both circular disks and eccentric disks; the growth rate in\neccentric disks is about half that in circular disks\n\\citep{2018ApJ...856...12C}. The present simulations suggest that the nonlinear\nstage of the \\ac{MRI} is also not so different in circular and eccentric disks,\nin that it saturates to comparable levels of plasma beta and alpha parameter in\nthe two kinds of disks. Thus, the \\ac{MRI} functions in much the same capacity\nin eccentric disks as in circular disks, mediating outward angular momentum\ntransport.\n\nThat being said, the \\ac{MRI} in eccentric disks exhibits two intriguing\nfeatures not witnessed in circular disks. The first one is the sign flip in the\nmagnetic-only alpha parameter. In circular disks, the alpha parameter is\npositive almost everywhere; by contrast, in eccentric disks, the alpha\nparameter can be consistently negative in some sectors of the disk even though\nthe azimuthal integral of the Maxwell stress remains positive.\n\nTo understand the basic principle underlying this surprising behavior, we break\ntemporarily from the general-relativistic treatment used in the rest of the\narticle and work in the Newtonian limit. We denote the Newtonian velocity and\nmagnetic field by $\\vec{\\tilde v}$ and $\\vec{\\tilde B}$, respectively, and\ntheir components measured against the local orthonormal cylindrical basis by\n$\\tilde v_i$ and $\\tilde B_i$. We conflate $\\tilde B^i$ with $b^{\\hat i}$\nelsewhere in the text, so the Maxwell stress is simply $-\\tilde B_R\\tilde\nB_\\varphi$. Our starting point is the induction equation:\n\\begin{equation}\n\\pd{\\vec{\\tilde B}}t=\\curl(\\vec{\\tilde v}\\cross\\vec{\\tilde B}).\n\\end{equation}\nTo track the time-evolution of the magnetic field at a point comoving with the\nflow, we consider\n\\begin{equation}\n\\od{\\vec{\\tilde B}}t\n  =\\pd{\\vec{\\tilde B}}t+(\\vec{\\tilde v}\\cdot\\grad)\\vec{\\tilde B}\n  =(\\vec{\\tilde B}\\cdot\\grad)\\vec{\\tilde v}\n  -\\vec{\\tilde B}(\\divg\\vec{\\tilde v}).\n\\end{equation}\nWe set $\\divg\\vec{\\tilde v}=0$ because the \\ac{MRI} is largely incompressible,\nand we take $\\pds{}z=0$ because we ignore vertical gravity in our simulations.\nConsequently,\n\\begin{equation}\\label{eq:Maxwell stress evolution}\n\\od{}t(-\\tilde B_R\\tilde B_\\varphi)=\n  -\\tilde B_R^2\\pd{\\tilde v_\\varphi}R\n  -\\tilde B_\\varphi^2\\biggl(\\frac1R\\pd{\\tilde v_R}{\\varphi}\\biggr)\n  -(-\\tilde B_R\\tilde B_\\varphi)\\frac{\\tilde v_R}R.\n\\end{equation}\n\nIn circular disks, the time- and azimuth-averaged $\\pds{\\tilde v_\\varphi}R$ is\nnegative, the averaged $(1\/R)(\\pds{\\tilde v_R}\\varphi)$ is zero, and the\naveraged $\\tilde v_R$ is also negative, albeit with a very small magnitude.\nBecause $-\\tilde B_R\\tilde B_\\varphi>0$ nearly everywhere, all terms on the\nright-hand side of \\cref{eq:Maxwell stress evolution} are therefore on average\nzero or positive, so $-\\tilde B_R\\tilde B_\\varphi$ grows over time until\nlimited by dissipation.\n\nIn eccentric disks, the time-averaged $\\pds{\\tilde v_\\varphi}R$ is also\nnegative everywhere. However, the averaged $(1\/R)(\\pds{\\tilde v_R}\\varphi)$ is\nno longer zero: it changes from negative to positive shortly before pericenter\nand back after pericenter. The averaged $\\tilde v_R$ also changes sign, from\nnegative to positive at pericenter and back at apocenter. The second and third\nterms therefore have azimuth-dependent signs. When orbits are significantly\neccentric, the averaged $(1\/R)(\\pds{\\tilde v_R}\\varphi)$ is comparable in\nmagnitude to the averaged $\\pds{\\tilde v_\\varphi}R$, and the averaged $\\tilde\nv_R$ is comparable to the averaged $\\tilde v_\\varphi$, so all three terms can\nbe important.\n\nConsider material starting from apocenter with positive Maxwell stress, that\nis, $-\\tilde B_R\\tilde B_\\varphi>0$. Initially all three terms work together to\nmake $-\\tilde B_R\\tilde B_\\varphi$ more positive. As the material swings toward\npericenter and its velocity becomes more azimuthal than radial,\n$(1\/R)(\\pds{\\tilde v_R}\\varphi)$ turns positive while $\\abs{\\tilde v_R}$ drops,\nflipping the sign of the second term and reducing the magnitude of the third\nterm. If $\\abs{\\tilde B_R}\\lesssim\\abs{\\tilde B_\\varphi}$, which is marginally\nsatisfied at these azimuths, the second term can dominate, pulling $-\\tilde\nB_R\\tilde B_\\varphi$ toward negative values. For some distance past pericenter,\nthe second term continues to be negative; at the same time, $\\tilde v_R>0$ and\n$\\ods{\\tilde v_R}t>0$, so the third term now tends to reduce the magnitude of\n$-\\tilde B_R\\tilde B_\\varphi$ at an increasing rate regardless of its sign. The\nsecond and third terms combined make $-\\tilde B_R\\tilde B_\\varphi<0$ at some\npoint near pericenter. Further beyond pericenter, the second term changes back\nto positive and the third term decreases in magnitude. By the time the material\nreturns to apocenter, the first and second terms have restored $-\\tilde\nB_R\\tilde B_\\varphi$ to positive, transporting angular momentum outward.\n\nBoth magnetic topologies share these qualitative features, but they differ in\nthe quantitative details. The dipolar-field topology yields a very close\nalignment between the local directions of velocity and magnetic field, whereas\nthe vertical-field topology produces merely a correlation between the two\ndirections that has much more scatter. In addition, as we can see in\n\\cref{fig:magnetic field map}, the sectors of the dipolar-field disk with\n$-\\tilde B_R\\tilde B_\\phi<0$ coincide with the region where $\\tilde v_R>0$,\nwhereas the vertical-field disk sectors with $-\\tilde B_R\\tilde B_\\phi<0$ take\nup only a small part of the $\\tilde v_R>0$ region, and their boundary is much\nmore irregular. We speculate that both of these contrasts are due to the larger\namplitude of turbulence the \\ac{MRI} drives in the presence of a vertical\nmagnetic field, an effect well-documented in circular disks\n\\citep[e.g.,][]{1995ApJ...440..742H, 1996ApJ...464..690H, 2004ApJ...605..321S,\n2013ApJ...767...30B}. Because the third term of \\cref{eq:Maxwell stress\nevolution} is sensitive to correlations between fluctuations in velocity and\nmagnetic field, large-amplitude turbulence may change the stress evolution.\nThis supposition is corroborated by the fact that the vertical root-mean-square\nof $v_z$, a good proxy for the magnitude of turbulent fluctuations in\nunstratified simulations, in the vertical-field disk is \\num{\\sim4} times that\nin the dipolar-field disk.\n\nThe other curiosity is that, whereas \\ac{MHD} stresses in circular disks\ntransport angular momentum and energy at such rates as to keep shrinking orbits\ncircular, the two rates can be independent of each other in eccentric disks.\nCircular disks have $E_\\su bL^2\\approx\\tfrac12$ at all radii, according to\n\\cref{eq:Newtonian eccentricity}; by contrast, $E_\\su bL^2$ varies spatially\nover our eccentric disks because $E_\\su b$ and $L^2$ change by different\nfractional amounts, and the mismatch dictates how the eccentricity evolves. The\ninner edge of the vertical-field disk is the least eccentric of all three\neccentric disks because the fractional changes in $E_\\su b$ and $L^2$ are\nnearly equal and opposite; conversely, the inner edge of the dipolar-field disk\nis the most eccentric because the two changes are in the same direction.\n\n\\subsection{Dynamics and energy dissipation at the inner edge}\n\\label{sec:inner edge}\n\nAs demonstrated in \\cref{sec:angular momentum and binding energy}, \\ac{MHD}\nstresses in eccentric disks are typically more effective at moving angular\nmomentum than energy. Because angular momentum constrains the size of the inner\nedge, and because the binding energy at the inner edge determines the amount of\nenergy available for radiation, the different transport rates of these two\nquantities could have observable effects on eccentric disks.\n\nTo explore these effects, let us first consider how disks behave around a star.\nFor a circular disk, the inner edge is located at the larger of the stellar or\nAlfv\\'en radius. The total energy a given amount of material dissipates in the\ndisk is its binding energy on a circular orbit at that radius. Additional\ndissipation happens in the boundary layer at the inner edge as the material\ncomes into corotation with the star or its magnetosphere. If the star-regulated\nrotation speed in this boundary layer is small compared to the orbital speed,\nthe additional dissipation is equal to the dissipation that took place in the\ndisk itself.\n\nFor an eccentric disk, the stellar or Alfv\\'en radius likewise defines the\ninner edge, but in this case, it is through a match to the pericenter of the\ninner edge. Because the corresponding semimajor axis is larger than the stellar\nor Alfv\\'en radius, material dissipates less energy in the disk proper. The\ntotal energy dissipated, however, is exactly the same as in the circular case\nbecause the eccentric orbit of the material ultimately transforms into a\ncircular one at the stellar or Alfv\\'en radius. What makes eccentric disks\ndifferent is that dissipation in the boundary layer accounts for a larger\nfraction of the total. Energy can be dissipated there in a variety of ways,\ndepending on how the inner edge interacts with the star or its magnetosphere.\nIn one extreme, the material immediately comes into corotation. The fast-moving\nmaterial dissipates large amounts of energy in a small region, potentially\nproducing hard radiation. In the other extreme, the material loses a tiny part\nof its kinetic energy each time it grazes past the star or its magnetosphere,\nand the material migrates inward gradually on orbits of decreasing semimajor\naxes and eccentricities. Dissipation in this case could be more spatially\ndistributed, happening both in the grinding encounters and on the outer parts\nof the circularizing orbits; if so, the eccentric disk may resemble a circular\none in appearance.\n\nWe now turn to disks around black holes. As shown in \\cref{fig:effective\npotential}, material on eccentric orbits can reach the event horizon even if it\nhas more angular momentum than that of \\iac{ISCO} orbit provided that it has\nenough energy to overcome the centrifugal barrier inside the \\ac{ISCO} radius;\nequivalently, it can do so if its eccentricity is high enough. Moreover, when\nthe angular momentum is at least that of \\iac{ISCO} orbit, the peak of the\ncentrifugal barrier is always above the energy of \\iac{ISCO} orbit.\nConsequently, the amount of energy available for such material to radiate is\nalways smaller than the binding energy at the \\ac{ISCO}, the conventional\nestimate for radiative efficiency in circular disks \\citep{1973blho.conf..343N,\n1974ApJ...191..499P}. The energy available to eccentric material is even\nsmaller when we take into account extra energy extraction by the Maxwell stress\nin circular disks \\citep{1974ApJ...191..507T, 1999ApJ...515L..73K,\n1999ApJ...522L..57G, 2002ApJ...573..754K, 2009ApJ...692..411N,\n2016ApJ...819...48S, 2016MNRAS.462..636A, 2021ApJ...922..270K}. Unlike\neccentric disks around stars, the energy retained by the plunging material\ncannot be recovered for radiation in some boundary layer.\n\nThe diminution of the radiative efficiency of eccentric disks around black\nholes could be particularly relevant to \\acp{TDE}, in which the energy radiated\nis considerably below that expected from the accretion of a reasonable fraction\nof a stellar mass of material onto a black hole through a conventional circular\ndisk \\citep{2015ApJ...806..164P}. \\Citet{2017MNRAS.467.1426S} suggested that\nthis \\textquote{inverse energy problem} could be resolved by internal stresses\nthat transport angular momentum preferentially and cause the most bound debris\nto plunge; our results provide quantitative support to that qualitative\nargument.\n\n\\subsection{Effects of vertical gravity}\n\nOur simulations of the nonlinear development of the \\ac{MRI} in eccentric disks\nignored vertical gravity. We did so for two reasons. The first is to isolate\nthe effects of orbital-plane motions, the chief driver of the \\ac{MRI}, when\nthey are azimuthally modulated in eccentric disks. The second is to produce a\nbaseline for comparison with future simulations that do include vertical\ngravity.\n\nBecause vertical gravity is a form of tidal gravity, it is much weaker than\nradial gravity as long as the disk is thin. For this reason, the modifications\nit introduces are likely minor and unable to quench the \\ac{MRI}. Here we\nexamine what these modifications may be.\n\nOne effect of vertical gravity that is discernible in circular disks carries\nover to eccentric disks: it regulates the buoyant eviction of accumulated\nmagnetic flux that may determine the saturation level of the \\ac{MRI}\n\\citep[e.g.,][]{1992MNRAS.259..604T, 2010ApJ...713...52D, 2011MNRAS.416..361B,\n2011ApJ...732L..30H, 2015ApJ...809..118B, 2018ApJ...861...24H}.\n\nThe variation of vertical gravity around an eccentric orbit causes the disk to\nexpand in height as it travels from pericenter to apocenter and to collapse on\nthe way back. This \\textquote{breathing} can have large amplitudes even for\nmildly eccentric disks \\citep{2014MNRAS.445.2621O, 2021MNRAS.500.4110L}. For\nthe thickest and most eccentric disks, breathing can even become non-adiabatic:\nextreme compression can create shocks that eject material vertically\n\\citep{2021ApJ...920..130R}.\n\nVertical gravity can also influence the eccentric \\ac{MRI} more subtly, without\nshocks. Vertical motion opens up more avenues of energy exchange between the\norbit and the magnetic field. Compression and expansion can alter the Alfv\\'en\nspeed, hence the critical wavelength for stability against the \\ac{MRI}, and it\ncan also modulate the wavelengths of advected perturbations. All these\nvariations are periodic, raising the possibility of instability through\nparametric resonance \\citep{2005A&A...432..743P}.\n\nIt bears repeating that the changes due to vertical gravity should be small for\nthin, moderately eccentric disks, so we expect our results here to remain\ngenerally valid.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn circular disks, the \\ac{MRI} stirs up correlated \\ac{MHD} turbulence,\nturbulent stresses transport angular momentum outward, and the disk accretes.\nOur simulations demonstrate that much the same process operates in eccentric\ndisks, and with comparable efficiency. Like circular disks, the quantitative\nlevel of Maxwell stress achieved in eccentric disks depends on the magnetic\ntopology, but eccentric and circular disks with the same magnetic topology\nreach comparable levels of plasma beta and alpha parameter.\n\nAlthough the mass-weighted, disk-averaged Maxwell stress in an eccentric disk\nproduces an outward angular momentum flux similar in magnitude to that in a\ncircular disk with the same magnetic topology, the Maxwell stress in an\neccentric disk can cause \\textit{inward} angular momentum transport in certain\ndisk sectors. This behavior is seen only in eccentric disks likely because the\nradial velocity and the azimuthal gradient of the radial velocity are both\nnonzero and have signs that vary over azimuth.\n\nBy removing proportionately more angular momentum than energy from the inner\nparts of the disk, \\ac{MHD} stresses can promote accretion of highly eccentric\nmaterial. This material has higher energy than a circular orbit with the same\nangular momentum, so the radiative efficiency of a disk around a black hole can\nbe suppressed relative to a circular disk. These findings corroborate earlier\nsuggestions about how the power output from eccentric disks may explain why the\ntotal observed radiated energy in many \\acp{TDE} is one to two orders of\nmagnitude lower than expected from circular accretion of a stellar mass of\nmaterial \\citep{2017MNRAS.467.1426S}.\n\n\\begin{acknowledgments}\nThe authors thank Scott Noble for useful discussions. CHC and TP were supported\nby ERC Advanced Grant \\textquote{TReX}. CHC was additionally supported by NSF\ngrant AST-1908042. JHK was partially supported by NSF grants AST-1715032 and\nAST-2009260. The simulations were performed on the Rockfish cluster at the\nMaryland Advanced Research Computing Center (MARCC).\n\\end{acknowledgments}\n\n\\software{Athena++ \\citep{2016ApJS..225...22W, 2020ApJS..249....4S}, NumPy\n\\citep{2020Natur.585..357H}, SymPy \\citep{10.7717\/peerj-cs.103}, Matplotlib\n\\citep{2007CSE.....9...90H}}\n\n\\begin{appendices}\n\n\\section{Metric components and Christoffel symbols}\n\\label{sec:metric}\n\nIn cylindrical coordinates $(t,R,\\varphi,z)$, the nonzero metric components are\n\\begin{align}\ng_{tt} &= -\\symsf P, \\\\\ng_{RR} &= 1, \\\\\ng_{\\varphi\\varphi} &= R^2, \\\\\ng_{zz} &= 1, \\\\\ng^{tt} &= -1\/\\symsf P, \\\\\ng^{RR} &= 1, \\\\\ng^{\\varphi\\varphi} &= 1\/R^2, \\\\\ng^{zz} &= 1,\n\\end{align}\nthe metric determinant is\n\\begin{equation}\ng=-R^2\\symsf P,\n\\end{equation}\nand the nonzero Christoffel symbols of the second kind are\n\\begin{align}\n\\Gamma^t_{tR}=\\Gamma^t_{Rt}\n  &= \\partial_R\\Phi\/\\symsf P, \\\\\n\\Gamma^t_{tz}=\\Gamma^t_{zt}\n  &= \\partial_z\\Phi\/\\symsf P, \\\\\n\\Gamma^R_{tt}\n  &= \\partial_R\\Phi, \\\\\n\\Gamma^R_{\\varphi\\varphi}\n  &= -R, \\\\\n\\Gamma^\\varphi_{R\\varphi}=\\Gamma^\\varphi_{\\varphi R}\n  &= 1\/R, \\\\\n\\Gamma^z_{tt}\n  &= \\partial_z\\Phi,\n\\end{align}\nwhere\n\\begin{equation}\n\\symsf P\\eqdef1+2\\Phi(R,z).\n\\end{equation}\n\nPerforming a coordinate transformation to orbital coordinates\n$(t,\\log\\lambda,\\phi,z)$, we find that the nonzero metric components are\n\\begin{align}\ng_{tt} &= -\\symsf P, \\\\\ng_{\\lambda\\lambda} &= R^2, \\\\\ng_{\\lambda\\phi}=g_{\\phi\\lambda} &= R^2\\symsf Q, \\\\\ng_{\\phi\\phi} &= R^2(1+\\symsf Q^2), \\\\\ng_{zz} &= 1, \\\\\ng^{tt} &= -1\/\\symsf P, \\\\\ng^{\\lambda\\lambda} &= (1+\\symsf Q^2)\/R^2, \\\\\ng^{\\lambda\\phi}=g^{\\phi\\lambda} &= -\\symsf Q\/R^2, \\\\\ng^{\\phi\\phi} &= 1\/R^2, \\\\\ng^{zz} &= 1,\n\\end{align}\nthe metric determinant is\n\\begin{equation}\ng=-R^4\\symsf P,\n\\end{equation}\nand the nonzero Christoffel symbols of the second kind are\n\\begin{align}\n\\Gamma^t_{t\\lambda}=\\Gamma^t_{\\lambda t}\n  &= (\\partial_R\\Phi)R\/\\symsf P, \\\\\n\\Gamma^t_{t\\phi}=\\Gamma^t_{\\phi t}\n  &= (\\partial_R\\Phi)R\\symsf Q\/\\symsf P, \\\\\n\\Gamma^t_{tz}=\\Gamma^t_{zt}\n  &= \\partial_z\\Phi\/\\symsf P, \\\\\n\\Gamma^\\lambda_{tt}\n  &= \\partial_R\\Phi\/R, \\\\\n\\Gamma^\\lambda_{\\lambda\\lambda}\n  &= 1, \\\\\n\\Gamma^\\lambda_{\\phi\\phi}\n  &= -R\/\\lambda, \\\\\n\\Gamma^\\phi_{\\lambda\\phi}=\\Gamma^\\phi_{\\phi\\lambda}\n  &= 1, \\\\\n\\Gamma^\\phi_{\\phi\\phi}\n  &= 2\\symsf Q, \\\\\n\\Gamma^z_{tt}\n  &= \\partial_z\\Phi,\n\\end{align}\nwhere\n\\begin{equation}\n\\symsf Q\\eqdef e\\sin\\phi\/(1+e\\cos\\phi).\n\\end{equation}\n\n\\section{Precession-free gravitational potential}\n\\label{sec:potential derivation}\n\nSchwarzschild spacetime in general relativity produces prograde apsidal\nprecession, while an extended gravitating mass in Newtonian mechanics causes\nretrograde apsidal precession. It is natural to ask if one can be made to\ncancel the other exactly.\n\nWe work in cylindrical coordinates $(t,R,\\varphi,z)$. The metric in the\nweak-gravity limit is given in \\cref{sec:metric}, but here we assume $\\Phi$ is\nnot yet decided. The velocity of a particle is $u^\\mu$; without loss of\ngenerality we restrict the particle to the midplane, so $z=0$ and $u^z=0$.\nBecause $t$ and $\\varphi$ are ignorable in the metric, we immediately have two\nintegrals of motion\n\\begin{align}\n\\label{eq:conserved energy}\nE &= u^t\\symsf P, \\\\\n\\label{eq:conserved angular momentum}\nL &= R^2u^\\varphi.\n\\end{align}\nVelocity normalization requires\n\\begin{equation}\\label{eq:velocity normalization}\n(u^R)^2=-1+E^2\/\\symsf P-L^2\/R^2.\n\\end{equation}\nDividing the equation by $R^4(u^\\varphi)^2=L^2$, changing variable to\n$\\xi\\eqdef1\/R$, and differentiating with respect to $\\xi$ yields\n\\begin{equation}\\label{eq:inverse radius equation}\n\\odd\\xi\\varphi+\\xi=-\\frac{E^2}{2L^2\\symsf P^2}\\od{\\symsf P}\\xi.\n\\end{equation}\nClosed eccentric orbits exist for all values of $E$ and $L$ if and only if\n\\begin{equation}\n-\\frac1{2\\symsf P^2}\\od{\\symsf P}\\xi=C_1,\n\\end{equation}\nwhich has the solution\n\\begin{equation}\n\\symsf P=1\/(2C_1\\xi+C_2),\n\\end{equation}\nwhere $C_1$ and $C_2$ are constants. We take $C_1=C_2=1$ so that $\\symsf\nP=1+2\\Phi\\approx1-2\\xi$ for $\\xi\\ll1$, as befitting point-mass gravity. The\npotential $\\Phi=-1\/(R+2)$ describes softened gravity, so there are no\ncoordinate or physical singularities.\n\nOur next step is to determine the velocity $u^\\mu$ at any point along an\neccentric orbit. The solution to \\cref{eq:inverse radius equation} is\n\\begin{equation}\nR=\\bar\\lambda\/[1+\\bar e\\cos(\\varphi+C_3)],\n\\end{equation}\nwhere $\\bar e$ and $C_3$ are constants and\n\\begin{equation}\\label{eq:semilatus rectum}\n\\bar\\lambda\\eqdef L^2\/E^2.\n\\end{equation}\nThe embellishments on $\\bar e$ and $\\bar\\lambda$ serve to distinguish them from\n$e$ and $\\lambda$ defining our orbital coordinates. \\Cref{eq:semilatus rectum}\ndescribes an ellipse of eccentricity $\\bar e$, semilatus rectum $\\bar\\lambda$,\nand semimajor axis $\\bar a=\\bar\\lambda\/(1-\\bar e^2)$. We pick $C_3=0$ so that\nthe pericenter of the ellipse is at $\\varphi=0$, and we fix $\\bar e$ by solving\n\\cref{eq:velocity normalization} at pericenter, where $u^R=0$:\n\\begin{equation}\\label{eq:eccentricity}\n\\bar e^2=1+(E^2-1)L^2\/E^4.\n\\end{equation}\nThis expression reduces to its Newtonian equivalent\n\\begin{equation}\\label{eq:Newtonian eccentricity}\n\\bar e^2\\approx1-2E_\\su bL^2\n\\end{equation}\nwhen $\\abs{E_\\su b}=\\abs{1-E}\\ll1$. Once we know $\\bar e$ and $\\bar\\lambda$ for\nan eccentric orbit, we can solve for $E$ and $L$ using \\cref{eq:semilatus\nrectum,eq:eccentricity}; the results are \\cref{eq:orbit energy,eq:orbit angular\nmomentum}. These can then be substituted into \\cref{eq:conserved\nenergy,eq:conserved angular momentum,eq:velocity normalization} to yield the\nvelocity $u^\\mu$ in cylindrical coordinates. The velocity $u^\\mu$ in orbital\ncoordinates, given in \\cref{eq:orbit velocity 0,eq:orbit velocity 2}, follows a\ncoordinate transformation.\n\nThe final missing piece is the analogue of Kepler's equation for our potential.\nWe introduce the eccentric anomaly $\\symcal E$, defined by\n\\begin{equation}\nR=\\bar a(1-\\bar e\\cos\\symcal E).\n\\end{equation}\nUsing a result from geometry,\n\\begin{equation}\n(1-\\bar e)^{1\/2}\\tan\\tfrac12\\varphi=\n  (1+\\bar e)^{1\/2}\\tan\\tfrac12\\symcal E,\n\\end{equation}\nwe can express $t$ along the orbit as a function of $\\varphi$:\n\\begin{equation}\\label{eq:time}\nt=\\int d\\symcal E\\,\\od\\varphi{\\symcal E}\\frac{u^t}{u^\\varphi}\n  =\\bar a^{3\/2}[(1+2\/\\bar a)\\symcal E-\\bar e\\sin\\symcal E].\n\\end{equation}\nThe integration constant is set to zero for simplicity. As $\\symcal E$\nincreases by $2\\pi$, $t$ also increases by an amount equal to the orbital\nperiod:\n\\begin{equation}\nT=2\\pi\\bar a^{3\/2}(1+2\/\\bar a).\n\\end{equation}\nMultiplying both sides of \\cref{eq:time} by $2\\pi\/T$ furnishes us with the\nanalogue of Kepler's equation:\n\\begin{equation}\n\\symcal M=\\symcal E-\\frac{\\bar e\\sin\\symcal E}{1+2\/\\bar a},\n\\end{equation}\nwhere $\\symcal M=2\\pi t\/T$ is the mean anomaly.\n\n\\end{appendices}\n\n\\ifapj","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Preliminaries}\\label{sec:intro}\n\nA {\\em polymatroid $(f,M)$} is a non-negative, monotone and submodular\nfunction $f$ defined on the collection of non-empty subsets of the finite\nset $M$. Here $M$ is the {\\em ground set}, and $f$ is the {\\em rank function}.\nThe polymatroid is {\\em integer} if all ranks are integer. An integer\npolymatroid is a {\\em matroid}, if the rank of singletons are either zero or\none. Matroids are combinatorial objects which generalize the properties of\nlinear dependence among a finite set of vectors. For an introduction to\nmatroids, see \\cite{oxley}; and about polymatroids consult \\cite{Lov,fmadhe}. The\nrank function $f$ can be identified with a $(2^{|M|}-1)$-dimensional real vector,\nwhere the indices are the non-empty subsets of $M$. \nIn this paper he {\\em distance} of two polymatroids $f$ and $g$ on the same ground\nset is measured as the usual Euclidean distance of the corresponding vectors, and is\ndenoted as $\\|f-g\\|$.\n\nFollowing the usual practice, ground sets and their subsets are denoted by\ncapital letters, their elements by lower case letters. The union sign\n$\\cup$ is frequently omitted as well as the curly brackets around singletons,\nthus $Aab$ denotes the set $A\\cup\\{a,b\\}$. For a function $f$\ndefined on the subsets of the finite set $M$ (such as the rank function of a\npolymatroid) the usual information-theoretical abbreviations are used. Here\n$I$, $J$, $K$ are disjoint subsets of the ground set:\n\\begin{align*}\n   f(I,J|K) &= f(IK)+f(JK)-f(IJK)-f(K),\\\\\n   f(I,J)   &= f(I,J|\\emptyset) = f(I)+f(J)-f(IJ)-f(\\emptyset),\\\\\n   f(I|K)   &= f(IK)-f(K).\n\\end{align*}\nWhen $f$ is a rank function, $f(\\emptyset)$ is considered to be zero.\nIn cases when the function $f$ is clear from the context, even $f$ is\nomitted. \nAdditionally, the {\\em Ingleton\nexpression} \\cite{ingleton} is abbreviated as\n$$\n   f[I,J,K,L] = -f(I,J)+f(I,J|K)+f(I,J|L)+f(K,L).\n$$\nObserve that it is symmetrical for swapping $I$ and\n$J$ as well as swapping $K$ and $L$. \n\nVectors corresponding to polymatroids on the ground set $M$ form the pointed polyhedral cone \n$\\Gamma_M$ \\cite{yeungbook}. \nIts facets are the hyperplanes determined by the\nbasic submodular inequalities $(i,j|K)\\ge 0$ with distinct $i,j\\in M{-}K$\nand $K\\subseteq M$ ($K$ can be empty), and the monotonicity\nrequirements $(i|M{-}i)\\ge 0$, see \\cite[Theorem 2]{fmadhe}. Much less is known\nabout the extremal rays of this cone. They have been computed for ground sets\nup to five elements \\cite{studeny-kocka}, without indicating any structural\nproperty.\n\n\\subsection{Entropic, linear, and modular polymatroids}\n\nAn important class of polymatroids describes the entropy structure of the\nmarginals of finitely many discrete random variables. Assume\n$\\{\\xi_i:i\\in M\\}$ is a collection of (jointly distributed) random\nvariables. For $A\\subseteq M$ let $\\mathbf{H}(\\xi_A)$ be the usual Shannon entropy \nof the marginal distribution $\\xi_A = \\{ \\xi_i: i\\in A\\}$. The function $f(A) =\n\\mathbf{H}(\\xi_A)$ is a polymatroid \\cite{fuji}. Such polymatroids are called {\\em\nentropic}, and the collection of entropic polymatroids is\n$\\Gamma^*_M\\subseteq \\Gamma_M$ \\cite{yeungbook}. The closure of $\\Gamma^*_M$ \n(in the usual\nEuclidean topology) is the collection of {\\em almost entropic} or\n{\\em aent} polymatroids.\nStudying\npolymatroids is motivated partly by the difficult task of understanding the\nentropic region as well as solving problems arising in secret sharing\n\\cite{padro1,seymour}, network coding \\cite{acy}, and other areas.\n\nAnother important subclass is the linear polymatroids. \n$(f,M)$ is {\\em linearly representable} if there is a vector\nspace $V$ over some finite field, linear subspaces $V_i\\subseteq V$ for\neach $i\\in M$, such that $f(A)$ is the dimension of the linear subspace\nspanned by the vectors in $\\bigcup\\{ V_i:i\\in A\\}$. Linearly\nrepresentable polymatroids are integer.\nA polymatroid is {\\em\nlinear} if it is in the conic hull of linearly representable polymatroids,\nnamely, it can be written as a non-negative linear combination of such\npolymatroids. Linear polymatroids are almost entropic, see\n\\cite{entreg,M.twocon,padro}.\n\nThe polymatroid $(f,M)$ is {\\em modular} if $f(I|J)=0$ for any two\ndisjoint non-empty subsets $I,J\\subset M$, or, equivalently, if \nfor all $A\\subseteq M$ we have\n$$\n    f(A) = {\\textstyle\\sum}\\, \\{ f(i)\\,:\\, i\\in A\\}.\n$$\nModular polymatroids are entropic and linear \\cite{fmadhe}.\n\nIt is well known that all polymatroids on two or three \nelements are linear, moreover a polymatroid $f$ on the four element set\n$abcd$ is linear if and only if it satisfies all six instances of the\nIngleton inequality::\n\\begin{align*}\n   f[a,b,c,d]\\ge 0, ~~~ f[a,c,b,d]\\ge 0, ~~~ f[a,d,b,c]\\ge 0,\\\\\n   f[b,c,a,d]\\ge 0, ~~~ f[b,d,a,c]\\ge 0, ~~~ f[c,d,a,b]\\ge 0,\n\\end{align*}\nsee \\cite{matus-studeny}.\nLinear polymatroids on a five element set can also be characterized by \nsome finite set of linear inequalities \\cite{dougherty5}. Polymatroids on\nground set of size at most five have the following {\\em simultaneous\napproximation property}, which will be used in Section \\ref{sec:sticky-2}.\n\\begin{proposition}\\label{prop:pm-approx}\nLet $|M|\\le 5$, and\nlet $f_1$ and $f_2$ be linear polymatroids on $M$.\nFor each large enough vector space $V$ and positive $\\varepsilon$ there is a\n$\\lambda>0$ and integer polymatroids $\\ell_1$ and $\\ell_2$ on $M$\nlinearly representable over $V$, such that $\\|f_i-\\lambda\\ell_i\\|<\\varepsilon$,\nadditionally $\\ell_1(A)=\\ell_2(A)$ whenever $f_1(A)=f_2(A)$ ($A\\subseteq\nM$).\n\\end{proposition}\n\\begin{proof}\nOn ground set $|M|\\le5$ linear polymatroids form a polyhedral cone.\nMoreover, for every large enough vector space $V$, extremal rays of this\ncone contain polymatroids linearly representable over $V$, see\n\\cite{dougherty5,matus-studeny}. On one hand, non-negative\nrational combinations of these polymatroids form a dense subset of\nlinear polymatroids. On the other, integer combinations of linearly\nrepresentable polymatroids over the same space are linearly\nrepresentable. From these facts the claim follows.\n\\end{proof}\n\n\n\\subsection{Amalgam and adhesive extension}\\label{subsec:adhesive}\n\nLet $M$, $X$, and $Y$ be disjoint sets. \nPolymatroids $f_X$ and $f_Y$ on the ground sets $M\\cup X$ and $M\\cup Y$,\nrespectively, with joint restriction on $M$,\nhave an {\\em amalgam}, or {\\em can be glued together}, if there is a\npolymatroid $f$ on $M\\cup X\\cup Y$ extending both $f_X$ and $f_Y$\n\\cite{oxley}. This\nextension is {\\em modular} if, in addition, $X$ and $Y$ are independent over\n$M$, that is, $f(X,Y|M)=0$. If $f_X$ and $f_Y$ have such a modular\nextension $f$, then $f_X$ and $f_Y$ are {\\em adhesive}, and $f$ is an {\\em\nadhesive extension}. Adhesive extensions were defined and studied by\nF.~Mat\\'u\\v s in \\cite{fmadhe}. The main observation is that restrictions of\nan almost entropic polymatroid are adhesive \\cite[Lemma 2]{fmadhe}. In this\npaper we investigate adhesive extensions on their own right.\n\nWhen speaking about amalgam, or adhesive extension, the polymatroids are\ntacitly assumed to have the same restriction on the intersection of their\nground sets.\n\n\\smallskip\n\nWe have defined the amalgam of $f_X$ and $f_Y$ as a {\\em polymatroid} extending\nboth $f_X$ and $f_Y$. The amalgam of two matroids is traditionally required to be a\nmatroid. It is an interesting problem to decide whether the two different\nnotions of amalgam coincide.\n\\begin{problem}\\label{problem:1}\\rm\nSuppose the matroids $f_X$ and $f_Y$ on $M\\cup X$ and $M\\cup Y$,\nrespectively, have a polymatroid amalgam on $M\\cup X\\cup Y$.\nIs it true that then they have a matroid amalgam as well?\n\\end{problem}\n\\noindent\nIt is easy to see that if the extension polymatroid is integer valued then\nit is a matroid; and if there is an extension at all, then there is one with\nrational values.\n\n\\smallskip\n\nWhether two matroids have an amalgam is a combinatorial question; the\nsame question about polymatroids is a {\\em geometrical} one. Polymatroids\n$f_X$ and $f_Y$ have an amalgam if and only if the point $(f_X,f_Y)$ (merged\nalong coordinates corresponding to subsets of $M$) is in the {\\em\ncoordinatewise projection} of the polymatroid cone $\\Gamma_{MXY}$ to the\nsubspace with coordinates $I\\subseteq MXY$ where $I\\subseteq MX$ or\n$I\\subseteq MY$.\nThe projection is a polyhedral cone whose bounding\nhyperplanes correspond to (homogeneous) linear inequalities on the projected\ncoordinates. Thus $f_X$ and $f_Y$ have an amalgam if and only if the\nvector $(f_X,f_Y)$ satisfies all of these inequalities. While\ntheoretically simple, in practice it is unclear how to calculate\nthe facets of the projection efficiently.\n\nThe same reasoning applies to the adhesive extensions. Such an extension\nsatisfies the additional constraint $f(X,Y|M)=0$, thus the modular extensions\nform a subcone of dimension one less: the intersection of\n$\\Gamma_{MXY}$ and the hyperplane $f(XM)+f(YM)-f(XYM)-f(Y)=0$. $f_X$ and\n$f_Y$ have an adhesive extension if an only if the point $(f_X,f_Y)$ is in\nthe projection of this restricted cone.\n\n\\smallskip\nThe polymatroid $h$ is {\\em sticky} if any two extensions\nof $h$ have an amalgam. Modular polymatroids are sticky, the proof in\n\\cite[Theorem 12.4.10]{oxley} works in the polymatroid case as well, see\nalso \\cite[Theorem 1]{fmadhe}. The ``sticky matroid conjecture'' asserts that all\nsticky matroids are modular \\cite{bonin}. The same conjecture is stated here\nfor polymatroids.\n\\begin{polymatroidconjecture}\nSticky polymatroids are modular.\n\\end{polymatroidconjecture}\nFactors of sticky polymatroids are sticky, and the collection of sticky\npolymatroids on a given ground set forms a closed cone, thus to settle the\nabove conjecture it is enough to consider polymatroidal extensions of a\nmatroid.\n Consequently, if the the answer to Problem \\ref{problem:1} is {\\em yes} and\nthe sticky matroid conjecture is true, then so is the\nsticky polymatroid conjecture.\n\n\\medskip\n\nTo state some of our results we need one more definitions.\nThe polymatroid $(h,M)$ is {\\em $k$-$\\ell$-sticky}, if any two of its extensions\n$(f_X,MX)$ and $(f_Y,MY)$ with $|X|\\le k$ and $|Y|\\le\\ell$ have an amalgam.\nA polymatroid is {\\em $k$-sticky}, if it is $k$-$k$-sticky.\nSticky polymatroids on small ground sets are discussed in Sections\n\\ref{sec:sticky-2} and \\ref{sec:sticky-3}.\n\n\n\\subsection{New polymatroids from old ones}\\label{subsec:constructions}\n\nEach polymatroid can be decomposed as a sum of a modular and a tight\npolymatroid as described in Lemmas \\ref{lemma:down} and \n\\ref{lemma:tightening}; it is\na generalization of \\cite[Lemma 2]{entreg}. Lemma \\ref{lemma:excess}\ndiscusses how one can extend a polymatroid adding a new element to the base set.\nThe method will be used in later sections to create several extensions.\n\n\\medskip\n\nFor a subset $A\\subset M$ define the function $\\r_A$ on (non-empty)\nsubsets of $M$ as follows:\n$$\n    \\r_A(I) = \\begin{cases} 1 & \\mbox{if $A\\cap I$ is not empty,}\\\\\n                            0 & \\mbox{otherwise}.\n              \\end{cases}\n$$\nClearly $(\\r_A,M)$ is a matroid. In Information Theory it corresponds \nto the atoms in the $I$-measure defined in \\cite{yeungbook}.\n\\begin{lemma}\\label{lemma:down}\nLet $(f,M)$ be a polymatroid and $A\\subset M$. Suppose the real number $\\lambda$ satisfies the\nfollowing conditions:\n$$\\begin{array}{r@{\\;\\le\\;}ll}\n\\lambda & f(x,y|B)    &\\mbox{for {\\em different} $x,y\\in A$ and all $B\\subseteq\nM{-}A$; and}\\\\[3pt]\n\\lambda & f(x|M{-}A)  &\\mbox{for every $x\\in A$.}\n\\end{array}$$\nThen $(f-\\lambda \\r_A,M)$ is a polymatroid.\n\\end{lemma}\n\\noindent\nObserve that if $A$ has a single member $a$, then the first condition\nholds vacuously, and the second condition simplifies to $\\lambda \\le\nf(a|M{-}a)$.\n\\begin{proof}\nThe claim clearly holds when $\\lambda\\le 0$, so assume $\\lambda>0$, and\nlet $f^*=f-\\lambda \\r_A$. If $I$ and $A$ are disjoint, then $f(I)-f^*(I)=0$,\nin the other cases this difference is $\\lambda$. One has to check the monotonicity\nfor the special case $f^*(Cx)-f^*(C)\\ge 0$, $Cx\\subseteq M$ only. This difference equals to\n$f(Cx)-f(C)$ except when $A$ and $C$ are disjoint and $x\\in A$. But then\n\\begin{align*}\n   f^*(Cx)-f^*(C)&=f(Cx)-f(C)-\\lambda \\\\\n               {}&=f(x|C)-\\lambda \\ge f(x|M{-}A)-\\lambda\\ge 0\n\\end{align*}\nby assumption.\n\nTo check submodularity, observe that $f^*(x,y|B)=f(x,y|B)$ except when $A$\nand $B$ are disjoint and both $x$ and $y$ are in $A$. In the latter case\n$f^*(x,y|B)=f(x,y|B)-\\lambda$, which is non-negative by the fist assumption.\n\\end{proof}\n\nLet $(f,M)$ be any polymatroid and $a\\in M$. Choosing $\\lambda$ to be the\nmaximal $f(a|M{-}a)$,\nthe polymatroid $f-\\lambda\\r_a$ is denoted by\n$f\\down a$, and called {\\em tightening of $f$ at} (or on) $a$. $f$ is\n{\\em tight at $a$} if $f=f\\down a$, that is, if $f(a|M{-}a)=0$.\nNote that\n$(f\\down a)\\down a = f\\down a$, thus $f\\down a$ is tight at $a$; moreover \n$(f\\down a)\\down b = (f\\down\nb)\\down a$. Thus one can define the {\\em tight part of $f$} at\n$A=\\{a_1,\\dots,a_k\\}$ as $f\\down a_1\\down\\cdots\\down a_k$. $f$ is {\\em tight\non $A$}, if $f=f\\down A$, and is {\\em tight} if $f=f\\down M$. The next\nlemma summarizes the properties of tightening used in this paper.\n\n\\begin{lemma}\\label{lemma:tightening}\nLet $(f,M)$ be a polymatroid and $A\\subseteq M$. \\begin{itemize}\n\\item $f$ is tight on $A$ if and only if it is tight on all elements of $A$.\n\\item $f\\down A$ is tight on $A$.\n\\item $f-f\\down A$ is a modular polymatroid.\n\\item $f\\down M$ is tight, and $f=f\\down M + (f-f\\down M)$ is the unique\ndecomposition of $f$ into the sum of a tight and modular part. \\qed\n\\end{itemize}\n\\end{lemma}\n\n\n\\medskip\n\nIn the last part of this section we investigate how to extend the\npolymatroid $(f,M)$ to the ground set $Mx$ using the {\\em excess\nfunction} $e(A)=f(xA)- f(A)$ defined for all subsets $A\\subseteq M$ including\nthe empty set. In agreement with the previous notation,\n$e(a,b|A)$ abbreviates $e(aA)+e(bA)-e(abA)-e(A)$, in particular,\n$e(a,b)=e(a,b|\\emptyset)=e(a)+e(b)-e(ab)-e(\\emptyset)$.\n\n\\begin{lemma}\\label{lemma:excess}\nSuppose $x$ is not in the ground set $M$ of the polymatroid $f$. Extend\n$f$ to the subsets of $Mx$ by $f_x(Ax) = f(A)+e(A)$. Then $f_x$ is a\npolymatroid on $Mx$ if and only if the following conditions hold:\n\\begin{enumerate}\n\\item[\\rm 1.] $e$ is non-negative and non-increasing: $e(A)\\ge e(B) \\ge 0$\nfor $A\\subseteq B\\subseteq M$;\n\\item[\\rm 2.] $e(a|M{-}a)+f(a|M{-}a)\\ge 0$ for all $a\\in M$;\n\\item[\\rm 3.] $e(a,b|A)+f(a,b|A)\\ge 0$ for all $abA\\subseteq M$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\nAn easy case by case checking.\n\\end{proof}\n\\begin{example}\\label{example:excess-1}\nLet $f$ be a polymatroid on $M$ and $\\,0\\le u,t$. Define the excess function\n$e_x$ by\n$$\n   e_x(A) = \\left\\{\\begin{array}{ll}\n         u+t & \\mbox{ if $A=\\emptyset$,}\\\\\n         u & \\mbox{ otherwise}.\n          \\end{array}\\right.\n$$\nIf $t\\le f(a,b)$ for all pairs $a,b\\in M$, then $f_x$ is a polymatroid .\n\\end{example}\n\\begin{proof}\nConditions 1 and 2 of Lemma \\ref{lemma:excess} trivially hold. As for\nCondition 3,\n$e_x(a,b|A)$ is zero except when $A=\\emptyset$, and then $e_x(a,b)=-t$. Thus\nit also holds by the assumption on $t$.\n\\end{proof}\n\\noindent\nAn easy calculation shows that for this extension $f_x$, for all pairs\n$a,b\\in M$ and non-empty $A\\subseteq M{-}a$ we have $f_x(x,a|A)=0$, \nand $f_x(a,b|x)=f(a,b)-t$.\n\\begin{example}\\label{example:excess-2}\nLet $c\\in M$ and $\\,0\\le u,t$. Define the excess function $e_x$ by\n$$\n    e_x(A) =\\left\\{\\begin{array}{ll}\n         u+t & \\mbox{ if $A=\\emptyset$ or $A=\\{c\\}$,}\\\\\n         u & \\mbox{ otherwise}.\n          \\end{array}\\right.\n$$\nIf $t\\le f(a,b)$ and $t\\le f(a,b|c)$ for all pairs $a,b\\in M{-}c$,\nthen $f_x$ is a polymatroid.\n\\end{example}\n\\begin{proof}\nSimilar to the previous Example. Conditions 1 and 2 hold, moreover\n$e_x(a,b|A)$ is either zero or $-t$, and the latter case holds when $A=\\emptyset$\nand $a,b\\in M{-}c$, or when $A=\\{c\\}$. Thus in all cases Condition 3 holds\nas well.\n\\end{proof}\n\n\n\\section{Adhesivity versus amalgam}\\label{sec:adh-eq-amalgam}\n\nAs defined in Section \\ref{subsec:adhesive}, polymatroids $f_X$ and $f_Y$ on\nground set $MX$ and $MY,$ respectively, have an {\\em amalgam} if there is a\npolymatroid\non $MXY$ extending both $f_X$ and $f_Y$. The same polymatroids are\n{\\em adhesive} if, in addition, they have a modular extension.\nWhen $Y$ has a single element $y$, then the polymatroid on $My$ will be\ndenoted by $f_y$. In this special case adhesivity of $f_X$ and $f_y$ is\nequivalent to the existence of the amalgam of closely related\npolymatroids. Recall that $f_y$ is {\\em tight on $y$} if $f_y(y|M)=0$,\nand by tightening $f_y$ on $y$ one gets the (tight) polymatroid\n$$\n    f_y\\down y = f_y - f_y(y|M)\\cdot\\r_y.\n$$\n\n\\begin{theorem}\\label{thm:amalgam-Xy}\nPolymatroids $f_X$ and $f_y$ are adhesive if and only if $f_X$ and\n$f_y\\down y$ have an amalgam.\n\\end{theorem}\n\\begin{proof}\nFirst let $g$ be the modular extension of $f_X$ and $f_y$, that is\n$g(X,y|M)=0$. This equality rewrites to\n$$\n   g(y|MX) = g(XMy)-g(MX)=g(My)-g(M)=f_y(My)-f_y(M)=f_y(y|M).\n$$\nLet $g^*=g\\down y$. The above equality means that restricting $g^*$ to $My$\none gets $f_y\\down y$, and, as $g$ and $g^*$ on $MX$ are the same, \nrestricting $g^*$ to $MX$ one gets\n$f_X$. Consequently $g^*$ is the required amalgam of $f_X$ and $f_y\\down y$.\n\nConversely, let $g^*$ be an amalgam of $f_X$ and $f_y\\down y$. Then using\nthat $f_y\\down y$ is tight on $y$, \n$g^*(My)=f_y\\down y(My) = f_y\\down y(M) = g^*(M)$, thus\n$$\n  g^*(XMy)-g^*(XM)\\le g^*(My)-g^*(M)=0,\n$$\nwhich means that $g^*(X,y|M)=0$. Let\n$g=g^*+\\lambda\\r_y$ with $\\lambda=f_y(y|M)$. Then $g$ extends $f_X$ (as\n$g\\restr MX=g^*\\restr MX=f_X$), and $f_y$ (as $g\\restr My=\ng^*\\restr My +\\lambda\\r_y = (f_y-\\lambda\\r_y)+\\lambda\\r_y$). Finally,\n$g(X,y|M)=g^*(X,y|M)=0$, as required.\n\\end{proof}\n\nThe last step in the proof works in a more general setting.\n\n\\begin{proposition}\\label{prop:tight-XY}\nSuppose $f_X\\down X$ and $f_Y\\down Y$ have an amalgam. Then $f_X$ and $f_Y$\nhave an amalgam as well. \n\\end{proposition}\n\\begin{proof}\nIf $g$ is an amalgam of $f_X\\down X$ and $f_Y\\down Y$, then\n$g+(f_X-f_X\\down X) + (f_Y-f_Y\\down Y)$ is an amalgam of $f_X$ and $f_Y$.\n\\end{proof}\n\nIn particular, to show that $f$ is sticky, it is enough to consider\nextensions $f_X$ and $f_Y$ which are tight on $X$ and $Y$, respectively. The\ncondition stated in Proposition \\ref{prop:tight-XY} is sufficient but not \nnecessary. Polymatroids $f_x$ and $f_y$ in\nExample \\ref{example:ex1} have an amalgam but are not adhesive. Thus, by\nTheorem \\ref{thm:amalgam-Xy}, $f_x\\down x$ and $f_y\\down y$ have no\namalgam.\n\n\n\\section{One-element extensions}\\label{sec:sticky-2}\n\nThis section starts with an alternative proof for a result of F.~Mat\\'u\\v s \n\\cite{fmadhe} which claims, using our terminology, that polymatroids on two\nelement sets are 1-sticky. A similar proof to this one will be given for\nTheorem \\ref{thm:nonsticky-2}.\nTheorem \\ref{thm:amalgam3} gives a sufficient and necessary condition for\na pair of one-element extensions of a polymatroid on three elements \nto have an amalgam. \nUsing Theorem \\ref{thm:amalgam-Xy}, this\nis turned into sufficient and necessary conditions for such\npolymatroid pairs to be adhesive, which, in turn, yields new 5-variable \nnon-Shannon entropy inequalities stated in Corollary \\ref{corr:5ineq}.\n\nThe section concludes with several examples. The first one specifies two\nlinearly representable (entropic) polymatroids which have an amalgam, but\nare not adhesive. Put in other words, they have a polymatroid\nextension, but no almost entropic (or linear) extension. Finally, there are\ntwo general examples for 1-sticky and not 1-sticky polymatroids on three\nelements.\n\n\\def\\cite[Corollary 2]{fmadhe}{\\cite[Corollary 2]{fmadhe}}\n\\begin{theorem}[\\cite[Corollary 2]{fmadhe}]\\label{thm:1-1-sticky}\nAll Polymatroids $f_x$ and $f_y$ on the ground set $abx$ and $aby$ with\ncommon restriction to $ab$\nare adhesive. In particular, such polymatroids have \nan amalgam, thus every polymatroid on a two element set is 1-sticky.\n\\end{theorem}\n\\begin{proof}\nAs discussed in Section \\ref{subsec:adhesive}, adhesive polymatroid pairs\n$(f_x,f_y)$ form a polyhedral cone. Consequently, $(f_x,f_y)$ is adhesive\nif and only if $(\\lambda f_x,\\lambda f_y)$ is adhesive for some (or all)\npositive $\\lambda$. The adhesive cone is closed, thus to show that a\nparticular pair $(f_x,f_y)$ is\nadhesive, it is enough to find, for each positive $\\varepsilon$, some adhesive pair\n$(\\ell_x,\\ell_y)$ such that $\\|f_x-\\lambda\\ell_x\\|<\\varepsilon$, and\n$\\|f_y-\\lambda\\ell_y\\|<\\varepsilon$. In this particular case $\\ell_1$ and $\\ell_2$\nwill be the linearly representable polymatroids guaranteed by\nProposition \\ref{prop:pm-approx}. Thus $\\ell_1$ and $\\ell_2$ are represented\nover the same vector space $V$,\n$\\lambda\\ell_x$ and $\\lambda\\ell_y$ are \n$\\varepsilon$-close to $f_x$ and $f_y$, respectively, and the linear subspaces\nin both representations corresponding to\nsubsets of $\\{ab\\}$ have the same dimensions: $\\ell_x(a)=\\ell_y(a)$,\n$\\ell_x(b)=\\ell_y(b)$ and $\\ell_x(ab)=\\ell_y(ab)$ as these\nequalities are true for polymatroids $f_x$ and $f_y$. To conclude\nthe claim of the theorem it is enough to show that $(\\ell_x,\\ell_y)$ is an\nadhesive pair.\n\nThe dimensions of subspaces spanned by $V_a$, $V_b$, and $V_a\\cup V_b$ are\nthe same in both representations. Choose a base in the first representation\nwhich can be partitioned to $B^x_x\\cup B^x_a \\cup B^x_b\\cup B^x_{ab}$ such\nthat $\\ell_x(a)=|B^x_a\\cup B^x_{ab}|$, $\\ell_x(b)=|B^x_b\\cup B^x_{ab}|$, and\n$\\ell_x(ab)=|B^x_a\\cup B^x_b\\cup B^x_{ab}|$, and similarly for $\\ell_y$.\nIdentify $B^x_a$ and $B^y_a$, $B^x_b$ and $B^y_b$, $B^x_{ab}$ and\n$B^y_{ab}$, and take the vector space with base $B^x_x\\cup B_a\\cup B_b\\cup\nB_{ab}\\cup B^y_y$ (that is, glue the representations of $\\ell_x$ and\n$\\ell_y$ along their common part). It will be a linear representation of a\npolymatroid on $abxy$, where $x$ and $y$ are independent given $ab$.\nConsequently $\\ell_x$ and $\\ell_y$ have an adhesive extension, which\nconcludes the proof.\n\\end{proof}\n\nNow we turn to the case of one-point extensions of polymatroids on\nthree-element sets. If not mentioned otherwise, all polymatroids on this\nsection are extensions of a fixed polymatroid on $M=\\{a,b,c\\}$.\n\n\\begin{theorem}\\label{thm:amalgam3}\nPolymatroids $f_x$, $f_y$ on the ground sets $abcx$ and $abcy$ have an amalgam if \nand only if the following inequalities and their permutations\n(permuting $a,b,c$ and $x,y$) hold:\n\\begin{align}\\label{eq:amalgam3}\n  & ~~~ f_x(a,x|c)+f_x(a,b|x)+f_y(a,b|y)+f_y(c,y)+{}\\\\\n  &+\\twoline{f_x(b,x|ac)}{f_y(b,y|ac)} + \\twoline{f_x(c,x|ab)}{f_y(c,y|ab)} + \n    2\\twoline{f_x(x|abc)}{f_y(y|abc)} \\ge f_x(a,b).\\nonumber\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nIt is clear that all terms are defined over one of\nthe polymatroids $f_x$ and $f_y$. Also, these inequalities hold for\nany polymatroid with ground set $abcxy$, which can easily be checked\nusing an automated entropy checker, thus they {\\em must} hold when $f_x$ and\n$f_y$ have an amalgam. Actually, inequalities in (\\ref{eq:amalgam3}) rewrite to\n\\begin{align}\n   & ~~~ (a,b|xy)+(x,y|a)+(x,y|b)+(c,y|x)+(a,x|cy)+{}\\label{eq:amalgamproof}\\\\\n   &+\\twoline{(b,x|acy)}{(b,y|acx)}+\\twoline{(c,x|aby)}{(c,y|abx)} +\n    2\\twoline{(x|abcy)}{(y|abcx)} \\ge 0,\\nonumber\n\\end{align}\nwhich evidently holds for any polymatroid on five elements.\n\nThe sufficiency can be checked by the method indicated in Section\n\\ref{subsec:adhesive}. Polymatroids $f_x$ and $f_y$ determine 23 out of the\n31 coordinates of the polymatroid on $N=abcxy$. The missing \n8 variables are indexed by subsets of the form $Axy$ with $A\\subseteq\n\\{abc\\}$.\n\nThe facets of the polymatroid cone $\\Gamma_N$ are determined by the basic submodular\ninequalities $(i,j|K)\\ge 0$ and by the monotonicity requirements $(i|N{-}i)\\ge\n0$. The strong duality of linear programming says that the facet equations of the\nprojection are non-negative linear combinations of these inequalities in\nwhich the combined coefficients of the projected (dropped) variables are zero. Let $\\mathcal\nM_N$ denote the matrix whose columns are indexed by the non-empty subsets of\n$N$, and whose rows contain the coefficients of the bounding facets of\n$\\Gamma_N$. In each row there are two, three, or four non-zero entries only.\nWhen $\\mathcal M$ is restricted to the eight columns labeled by $xyA$,\n\\begin{table}[!htb]\n\\begin{center}\\begin{tabular}{|cccccccc|ll|}\n\\hline\n\\rule{0pt}{11pt}$xy$&$axy$&$bxy$&$cxy$&$abxy$&$acxy$&$bcxy$&$abcxy$&&\\\\\n\\hline\n\\rule{0pt}{12pt}\n-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & $(x,y)$&\\\\[2pt]\n1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & $(c,x|y)$&  $(c,y|x)$\\\\[2pt]\n-1 & 1 & 1 & 0 & -1 & 0 & 0 & 0 & $(a,b|xy)$&\\\\[2pt]\n0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & $(x,y|a)$&\\\\[2pt]\n0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & $(x,y|b)$&\\\\[2pt]\n0 & 0 & 0 & 1 & 0 & -1 & 0 & 0 & $(a,x|cy)$&  $(a,y|cx)$\\\\[2pt]\n0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & $(c,x|aby)$&  $(c,y|abx)$\\\\[2pt]\n0 & 0 & 0 & 0 & 0 & 1 & 0 & -1 & $(b,x|acy)$&  $(b,y|acx)$\\\\[2pt]\n0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & $(x|abcy)$& $(y|abcx)$\\\\[2pt]\n\\hline\n\\end{tabular}\\end{center}\\vskip -5pt\n\\caption{A submatrix of $\\mathcal M_{abcxy}$}\\label{table:3-facets}\n\\end{table}\n27 different non-zero rows remain. Table \\ref{table:3-facets} shows some of\nthem with the corresponding facet equations (one or two). There are 154\nextremal non-negative linear combinations of the 27 rows which give zero for\nall eight columns. One of them is the combination taking each but the\nfirst and last row from Table \\ref{table:3-facets} once, and take the last\nrow twice. Eight of the corresponding facet combinations give the\ninequalities in (\\ref{eq:amalgamproof}). The other $3\\cdot 8$ combinations,\nwhen one takes $(c,x|y)$ instead of $(c,y|x)$, or $(a,y|cx)$ instead of\n$(a,x|cy)$, or both, yield supporting hyperplanes to the projected cone, but\nnot facets as they are consequences of the basic inequalities for $abcx$ and\n$abcy$. In other words, these hyperplanes do not cut into the cones\n$\\Gamma_{abcx}$ and $\\Gamma_{abcy}$.\n\nThe extremal combinations of the 27 rows were generated by a computer\nprogram. From that list the possible facets of the projection were generated\nand checked whether they are really facets of the projection. This search\nresulted in the statement of the Theorem.\n\\end{proof}\n\n\\begin{corollary}\\label{corr:3-adhesive}\nPolymatroids $f_x$ and $f_y$ on the ground sets $abcx$ and $abcy$ are\nadhesive if and only if the following inequalities and their permutations\nhold:\n\\begin{align}\\label{eq:adhesive3}\n  & ~~~ f_x(a,x|c)+f_x(a,b|x)+f_y(a,b|y)+f_y(c,y) +{}\\\\\n  &+\\twoline{f_x(b,x|ac)}{f_y(b,y|ac)} + \\twoline{f_x(c,x|ab)}{f_y(c,y|ab)} \\ge\nf_x(a,b).\\nonumber\n\\end{align}\n\\end{corollary}\n\\begin{proof}\nBy Theorem \\ref{thm:amalgam-Xy}, $f_x$ and $f_y$ are adhesive if and only if\n$f_x\\down x$ and $f_y\\down y$ have an amalgam. All terms in\n(\\ref{eq:amalgam3})\nare the same for $f_x$ and $f_x\\down x$ ($f_y$ and\n$f_y\\down y$) except for $(f_x\\down x)(x|abc)=0$ and \n$(f_y\\down y)(y|abc)=0$.\n\\end{proof}\n\n\\begin{corollary}\\label{corr:5ineq}\nThe following are five-variable non-Shannon information inequalities, that\nis, they hold in every entropic polymatroid on  at least five elements:\n\\begin{align*}\n  & ~~~ (a,x|c)+(a,b|x)+(a,b|y)+(c,y) +{}\\\\\n  &+\\twoline{(b,x|ac)}{(b,y|ac)} + \\twoline{(c,x|ab)}{(c,y|ab)} \\ge\n(a,b).\n\\end{align*}\n\\end{corollary}\n\\begin{proof}\nAs observed in \\cite{fmadhe}, restrictions of an entropic polymatroid are\nadhesive, consequently the inequalities (\\ref{eq:adhesive3}) in Corollary\n\\ref{corr:3-adhesive} must hold.\n\\end{proof}\n\n\\subsection{Examples}\\label{subsec:examples}\n\n\\begin{example}\\label{example:ex1}\nThere are linearly representable polymatroids $f_x$ and $f_y$ on $abcx$ and\n$abcy$ which have an amalgam but are not adhesive.\n\\end{example}\n\\begin{proof}\nPolymatroids $f_x$ and $f_y$ will be extensions of the uniform polymatroid\n$$\n   f(A) = \\left\\{\\begin{array}{ll}\n            4 & \\mbox{ if $|A|=1$},\\\\\n            6 & \\mbox{ otherwise},\n          \\end{array}\\right. ~~~~ A\\subseteq \\{abc\\}.\n$$\nClearly $f(i,j)=f(i,j|k)=2$ for\nall distinct $i$, $j$, $k$. The excess functions defining $f_x$ and $f_y$ are\n$$\n e_x(A)=\\left\\{\\begin{array}{ll} \n            3 & \\mbox{ if $A=\\emptyset$, }\\\\\n            1 & \\mbox{ otherwise},\n        \\end{array}\\right.\n~~ \\mbox{ and } ~~\n e_y(A)=\\left\\{\\begin{array}{ll} \n            3 & \\mbox{ if $A=\\emptyset$ or $A=\\{c\\}$,}\\\\\n            1 & \\mbox{ otherwise}.\n        \\end{array}\\right.\n$$\n\\newcommand\\mf[1]{\\mathbf{#1}}%\nBy Examples \\ref{example:excess-1} and \\ref{example:excess-2} both\n$f_x$ and $f_y$ are polymatroids. They are {\\em not} adhesive, as all terms on the\nleft hand side of (\\ref{eq:adhesive3}) are zero, while $f_x(a,b)=f(a,b)=2$.\nTo show that they have an amalgam, one can check that all conditions of\nTheorem \\ref{thm:amalgam3} hold. The polymatroid $f_{xy}$ specified in \nTable \\ref{table:ex1} gives such an extension explicitly.\n\\begin{table}[!htb]\\begin{center}\\begin{tabular}{|ccc@{\\qquad}ccc@{\\qquad}ccc@{\\qquad}ccc|}\n\\hline\\multicolumn{3}{|c@{\\qquad}}{\\rule{0pt}{12pt}$A$}&\n      \\multicolumn{3}{@{}c@{\\qquad}}{$Ax$ }&\n      \\multicolumn{3}{@{}c@{\\qquad}}{$Ay$ }&\n      \\multicolumn{3}{@{}c|}{$Axy$ }\\\\[2pt]\n\\hline\\rule{0pt}{13pt}\n    & 6 &   &    & 7 &   &    & 7 &   &    & 7 &     \\\\[2pt]\n~ 6 & 6 & 6 &  7 & 7 & 7 &  7 & 7 & 7 &  7 & 7 & 7~  \\\\[2pt]\n~ 4 & 4 & 4 &  5 & 5 & 5 &  5 & 5 & 7 &  6 & 6 & 7~  \\\\[2pt]\n    & 0 &   &    & 3 &   &    & 3 &   &    & 5 &     \\\\[2pt]\n\\hline\n\\end{tabular}\\end{center}\\vskip -5pt\n\\caption{The polymatroid $f_{xy}$ for $A\\subseteq \\{abc\\}$}\\label{table:ex1}\n\\end{table}\nThe four groups contain the values for the subsets indicated at the top line\nwhere $A$ runs over all subsets of $abc$. The values are arranged\nin four lines (from bottom to top) for $A=\\emptyset$, one-element\nsubsets $a$, $b$, $c$, two-element subsets $ab$, $ac$, $bc$, and $abc$ at\nthe top.\n\nFinally, the polymatroids $f_x$ and $f_y$ are linearly representable over\nany field. Choose seven independent vectors $\\mf s_1$, $\\mf s_2$, $\\mf u_1$,\n$\\mf u_2$, $\\mf v_1$, $\\mf v_2$, and $\\mf r$. Subspaces assigned to the\nground elements are are the ones spanned by the vectors listed below:\n$$\\begin{array}{rl}\n\\multicolumn{2}{c}{\\mbox{$abcx$ is linear}}\\\\[3pt]\na: & \\mf s_1,\\mf s_2, \\mf u_1,\\mf u_2 \\\\[2pt]\nb: & \\mf s_1, \\mf s_2, \\mf v_1, \\mf v_2\\\\[2pt]\nc: & \\mf s_1,\\mf s_2,\\mf u_1+\\mf v_1, \\mf u_2+\\mf v_2 \\\\[2pt]\nx: & \\mf s_1,\\mf s_2,\\mf r\n\\end{array}\n\\qquad\n\\begin{array}{rl}\n\\multicolumn{2}{c}{\\mbox{$abcy$ is linear}}\\\\[3pt]\na: & \\mf s_1,\\mf s_2, \\mf u_1,\\mf u_2 \\\\[2pt]\nb: & \\mf s_1, \\mf s_2, \\mf v_1, \\mf v_2\\\\[2pt]\nc: & \\mf u_1,\\mf u_2,\\mf v_1, \\mf v_2 \\\\[2pt]\ny: & \\mf s_1,\\mf s_2,\\mf r\n\\end{array}\n$$\nIt is easy to check that all generated subspaces have the right dimension.\nNote that while the dimensions of the subspaces corresponding to subsets of \n$abc$ are the same, the subspace arrangements are {\\em not} isomorphic.\n\\end{proof}\n\nIt is easy to check that $f_x\\down x$ and $f_y\\down y$ are also linearly\nrepresentable. As $f_x$ and $f_y$ are not adhesive, according to Theorem\n\\ref{thm:amalgam-Xy}, $f_x\\down x$ and $f_y\\down y$ have no amalgam.\n\n\\medskip\n\nTheorem \\ref{thm:amalgam3} can be used to characterizes 1-sticky polymatroids on\nthree-element sets. The following examples show some particular cases.\n\n\\begin{example}\\label{example:nonsticky-1}\nLet $f$ be a polymatroid on $\\{abc\\}$. If $f(a,b)$,\n$(a,b|c)$ are positive, $(a,b)\\le (a,c)$, $(a,b)\\le (b,c)$,\nthen $f$ is not 1-sticky.\n\\end{example}\n\\begin{proof}\nWe specify two extensions $f_x$ and $f_y$ so that one of the\ninequalities in Theorem \\ref{thm:amalgam3} fails.\nLet $t=(a,b)>0$, and $u=\\min\\{ (a,b),~ (a,b|c) \\}>0$.\nDefine the excess functions $e_x$, $e_y$ by\n$$\ne_x(A) = \\left\\{\n\\begin{array}{rl}\n    t & \\mbox{ if $A=\\emptyset$,}\\\\\n    0 & \\mbox{ otherwise},\n\\end{array}\\right.\n~~~\\mbox{ and } ~~~\ne_y(A)=\\left\\{\n\\begin{array}{rl}\n    u & \\mbox{ if $A=\\emptyset$ or $A=\\{c\\}$,} \\\\\n    0 & \\mbox{ otherwise}.\n\\end{array}\\right.\n$$\nAccording to Examples \\ref{example:excess-1} and \\ref{example:excess-2},\n$f_x$ and $f_y$ are polymatroids. In this case\n$f_x(a,x|c)=f_x(b,x|ac)=f_x(c,x|ab)=0$, $f_x(x|abc)=0$, and\n$f_x(a,b|x)=f(a,b)-t=0$, see the remark following Example\n\\ref{example:excess-1}. Similarly, we have \n$f_y(a,b|y)=f(a,b)-u=t-u$, $f_y(c,y)=0$, thus the\nleft hand side of the top line in (\\ref{eq:amalgam3}) is\n\\begin{align*}\n  & f_x(a,x|c)+(f_x(a,b|x)+f_y(a,b|y)+f_y(c,y)+{}\\\\\n  +& f_x(b,x|ac)+f_x(c,x|ab)+2f_x(x|abc) = t-u,\n\\end{align*}\nwhile the right hand side is $f(a,b)=t$. Thus no amalgam of $f_x$ and $f_y$\nexists.\n\\end{proof}\n\n\\begin{example}\\label{example:is-sticky}\nSuppose $(a|bc)=(b|ac)=(c|ab)=0$, and at least one of $(a,b|c)$,\n$(a,c|b)$, $(b,c|a)$ is zero. Then $f$ is 1-sticky.\n\\end{example}\n\\begin{proof}\nLet $f_x$ and $f_y$ be two extensions of $f$. Our goal is to show that all\ninstances of the inequalities in Theorem \\ref{thm:amalgam3} hold. From the\nassumptions it follows that for $|A|\\ge 2$ we have $f(A)=f(abc)=t$; moreover\nat least one of $f(a)$, $f(b)$, $f(c)$ also equals $t$. Suppose\n$f_x$ and $f_y$ are specified by the excess functions $e_x$ and\n$e_y$. By Proposition \\ref{prop:tight-XY} we can assume that $f_x$ is tight\non $x$ and $f_y$ is tight on $y$, which gives $e_x(M)=e_y(M)=0$. where\n$M=\\{abc\\}$. In our case\n$f(i|M{-}i)=0$, thus we must also have $e_x(i|M{-}i)=e_y(i|M{-}i)=0$, thus\n$e_x(A)=e_y(A)=0$ for all two-element subsets of $M$. This means\n$$\n   f_x(i,x|M{-}i) = f_x(x|M)= 0, ~~~ f_y(i,y|M{-}i)=f_y(y|M)=0,\n$$\nthus all terms in the second line of (\\ref{eq:amalgam3}) are zero.\nConsequently we only need to\nshow that\n$$\n  f_x(a,x|c)+f_x(a,b|x)+f_y(a,b|y)+f_y(c,y)\\ge f(a,b),\n$$\nwhich rewrites to\n\\begin{equation}\\label{eq:is-sticky}\n  f(a,b)+e_x(a)+e_x(b)+e_x(c)-e(x) + e_y(a)+e_y(b)-e_y(c)\\ge 0.\n\\end{equation}\nThe condition that one of $f(a)$, $f(b)$, $f(c)$ equals $t$ was not used\nyet.\nIf $f(c)=t$, then $e_x(c)=e_y(c)=0$, and then (\\ref{eq:is-sticky})\nfollows from\n$$\n    f(a,b)+e_x(a,b)+e_y(a)+e_y(b) \\ge 0,\n$$\nwhich holds by Lemma \\ref{lemma:excess}, Condition 3. When $f(a)=t$ (or,\nsymmetrically, $f(b)=t$), then $e_x(a)=e_y(a)=0$,\n$f(a,b)=f(b)=f(b,c)+f(a,b|c)$, and (\\ref{eq:is-sticky}) rewrites to\n$$\n   f(b,c)+e_x(b,c)+f(a,b|c)+e_y(a,b|c)+e_y(b)\\ge 0,\n$$\nwhich, again, holds by Lemma \\ref{lemma:excess}.\n\\end{proof}\n\n\n\n\\section{Two-element extensions}\\label{sec:sticky-3}\n\nUsing similar techniques necessary and sufficient conditions for the\nexistence of an amalgam of polymatroids on larger sets can be obtained.\nTheorem \\ref{thm:sticky-2-1} is\nsuch an example. It is a consequence of \\cite[Remark 6]{fmadhe} and Theorem\n\\ref{thm:amalgam-Xy}; we sketch a direct proof. The result is used to\nget a characterization of 2-sticky polymatroids on two-element sets.\n\n\\begin{theorem}\\label{thm:sticky-2-1}\nPolymatroids $f_X$ and $f_y$ on $abx_1x_2y$ and $aby$, respectively, have an amalgam if and\nonly if the following inequalities and all of their permutations (permuting\n$a$ and $b$, and $x_1$ and $x_2$) hold:\n$$\n   \\twoline{f_X[a,b,x_1,x_2]}{f_X[a,x_1,b,x_2]}\n   + f_y(y,a|b)+f_y(y,b|a)+f_y(a,b|y)\n +3f_y(y|ab)\\ge 0.\n$$\n\\end{theorem}\n\\begin{proof}\nThe inequalities rewrite to\n\\begin{align*}\n&~~~\\twoline{(x_2,y|b)+(x_1,x_2|y)+(a,b|x_2y)+(a,y|x_1x_2)}\n            {(x_2,b|y)+(x_2,y|x_1)+(a,y|bx_2)+(a,x_1|x_2y)}+{}\\\\\n&+ (x_1,y|a)+(x_2,y|a)+(x_1,y|b)+(a,b|x_1y)+{}\\\\\n&+(y|abx_1)+(y|abx_2)+(y|ax_1x_2) \\ge 0,\n\\end{align*}\nthus if $f_X$ and $f_y$ have an amalgam, then the expressions must be\nnon-negative.\n\nThe sufficiency can be checked similarly as in Theorem \\ref{thm:amalgam3} by\ncomputing the facets of the projection of the cone $\\Gamma_{\\{abx_1x_2y\\}}$ \nto the coordinates which are subsets of $abx_1x_2$ and $aby$. There are 12\ndropped coordinates: $x_1y$, $x_2y$, $x_1x_2y$, \\dots, $abx_1y$, $abx_2y$,\n$abx_1x_2y$. Restricting the matrix\n$\\mathcal M_{\\{abx_1x_2y\\}}$ to these columns, one gets 48 different non-zero\nrows, some of them are shown in Table \\ref{table:proj-facets}.\n\\begin{table}[!htb]\n\\begin{center}\\begin{tabular}{|ccc|ccc|ccc|ccc|l|}\n\\hline\n\\rule{0pt}{11pt}$x_1y$&$x_2y$&$x_1x_2y$&\\multicolumn{3}{|c}{$a$}&\\multicolumn{3}{|c}{$b$}&\\multicolumn{3}{|c|}{$ab$}&\\\\\n\\hline\n\\rule{0pt}{12pt}\n  1&0&0& -1&0&0& 0&0&0& 0&0&0  &$(a,x_1|y)$, $(a,y|x_1)$\\\\[2pt]\n-1&0&0& 1&0&0& 1&0&0& -1&0&0   &$(a,b|x_1y)$\\\\[2pt]\n 1&1&-1&0&0&0& 0&0&0&  0&0&0   &$(x_1,x_2|y)$\\\\[2pt]\n 0&-1&0& 0&1&0& 0&1&0& 0&-1&0  &$(a,b|x_2y)$\\\\[2pt]\n 0&0&1&  0&0&-1&0&0&0& 0&0&0   &$(a,y|x_1x_2)$\\\\[2pt]\n 0&0&0&-1&0&0& 0&0&0& 0&0&0    &$(x_1,y|a)$\\\\[2pt]\n 0&0&0& 0&-1&0&0&0&0& 0&0&0    &$(x_2,y|a)$\\\\[2pt]\n 0&0&0& 0&0&1 &0&0&0& 0&0&-1   &$(b,y|ax_1x_2$\\\\[2pt]\n 0&0&0& 0&0&0&-1&0&0& 0&0&0    &$(x_1,y|b)$\\\\[2pt]\n 0&0&0& 0&0&0& 0&-1&0&0&0&0    &$(x_2,y|b)$\\\\[2pt]\n 0&0&0& 0&0&0& 0&0&0& 1&0&-1   &$(x_2,y|abx_1)$\\\\[2pt]\n 0&0&0& 0&0&0& 0&0&0& 0&1&-1   &$(x_1,y|abx_2)$\\\\[2pt]\n 0&0&0& 0&0&0& 0&0&0& 0&0&1    &$(y|abx_1x_2)$\\\\[2pt]\n\\hline\n\\end{tabular}\\end{center}\\vskip -5pt\n\\caption{A submatrix of $\\mathcal M_{\\{abuxy\\}}$}\\label{table:proj-facets}\n\\end{table}\nOne facet of the projection is generated by the linear combination taking\nall but the first and last rows once, and the last row three times. There are\n6938 extremal non-negative linear combinations \ngiving zeros for the 12 projected variables. From\nthis list the possible facets of the projection were generated and checked\nif they are really facets. This search confirmed the claim.\n\\end{proof}\n\n\\begin{theorem}\\label{thm:nonsticky-2}\nThe polymatroid on the two-element set $ab$ is 2-sticky if and only if\none of the following cases hold: $(a,b)=0$ (it is modular); $(a|b)=0$, or\n$(b|a)=0$ (one of them determines the other).\n\\end{theorem}\n\\begin{proof}\nFirst we show that these polymatroids are 2-2-sticky. Modular polymatroids are sticky\nwithout any restriction, so suppose, e.g., that $f(a|b)=0$. Let $f_X$ be an\nextension on $abx_1x_2$.\nAll six\nIngleton expressions for $f_X$ are non-negative using the following\nequalities and their symmetric versions:\n\\begin{align*}\n   [a,b,x_1,x_2] &+(a|b)=(a,x_1|b)+(a,x_2|b)+(x_1,x_2|a)+(a|x_1x_2);\\\\\n   [a,x_1,b,x_2] &+(a|b)=(a,x_1|b)+(b,x_2|a)+(a,x_2|x_1)+(a|bx_2);\\\\\n   [b,x_1,a,x_2]\n   &+(a|b)=(a,x_1|b)+(a,x_2|b)+(a,x_1|x_1)+(b,x_1|ax_2)+(a|bx_1x_2);\\\\\n   [x_1,x_2,a,b]\n   &+(a|b)=(a,x_1|b)+(a,x_2|x_1)+(a,b|x_2)+(x_1,x_2|ab)+(a|bx_1x_2).\n\\end{align*}\nIt means that $f_X$ is linear, and the same is true for $f_Y$. As in the\nproof of Theorem \\ref{thm:1-1-sticky}, using Proposition \\ref{prop:pm-approx}\nwe may assume that $f_X$ and $f_Y$ are linearly representable over the same\nfield, and the\ndimensions of the subspaces corresponding to the common subsets $a$, $b$\nand $ab$ are the same in both representations. \nChoose maximal independent set of vectors in both representations\nwhich span these subspaces in an equivalent way. Extend this set to be a\nbase in both representations. Glue the \ntwo vector spaces together along the equivalent set of base vectors. \nThis gives an amalgam (even an adhesive extension) of $f_X$ and $f_Y$, as\nrequired.\n\nIn the other direction first we show that given $f$ with $f(a,b)>0$, $f(a|b)>0$,\n$f(b|a)>0$, it can be extended to a polymatroid $f_X$ on $abx_1x_2$\nso that $f_X[a,b,x_1,x_2]<0$. For the construction we recall the {\\em \nnatural coordinates} of polymatroids on four elements from \\cite{entreg}. \nThis coordinate system has the additional advantage\nthat points with natural coordinates in the non-negative orthant $\\R_{\\ge 0}^{15}$\nare polymatroids.\nLet us recall these coordinates below:\n\\begin{align*}\n   & -[a,b,x_1,x_2], \\\\\n   & (a,b|x_1),~ (a,b|x_2), ~ (a,x_1|b), ~ (b,x_1|a), ~ (a,x_2|b), ~ (b,x_2|a),\\\\\n   & (x_1,x_2|a), ~ (x_1,x_2|b), ~ (x_1,x_2), ~(a,b|x_1x_2), \\\\\n   & (a|bx_1x_2), ~ (b|ax_1x_2), ~ (x_1|abx_2), (x_2|abx_1).\n\\end{align*}\nFrom these coordinates the values $f_X(a)$, $f_X(b)$, and $f_X(ab)$ can be \nexpressed as\nfollows, where only the coefficients of the coordinates in the above order\nare shown:\n\\def\\sb#1{\\mathbf #1}\n\\begin{align*}\n   f_X(a) &= (\\sb2,~ 1,1,1,0,1,0,~ \\sb1,\\sb0,\\sb1,1,~ 1,0,0,0)\\\\\n   f_X(b) &= (\\sb2,~ 1,1,0,1,0,1,~ \\sb0,\\sb1,\\sb1,1,~ 0,1,0,0)\\\\\n  f_X(ab) &= (\\sb3,~ 1,1,1,1,1,1,~ \\sb1,\\sb1,\\sb1,2,~ 1,1,0,0).\n\\end{align*}\nChoose the coordinates first and from eighth to tenth (typeset in bold)\nto have the positive values $\\varepsilon$, $f(a|b)-\\varepsilon$, $f(b|a)-\\varepsilon$, and \n$f(a,b)-\\varepsilon$, respectively, for some small enough $\\varepsilon$; set all other\ncoordinates to zero. With this choice $f_X$ will be a polymatroid which\nextends the one given on $ab$ as, e.g.,\n$f_X(a)=2\\varepsilon+f(a,b)-\\varepsilon+f(a|b)-\\varepsilon=f(a)$,\nmoreover the Ingleton value $f_X[a,b,x_1,x_2]$, as given by the first coordinate, \nis $-\\varepsilon$, which is negative.\n\nDefine the other extension $f_y$ by the excess function\n$$\n    e_y(A) = \\left\\{\\begin{array}{ll}\n               f(a,b) & \\mbox{ if $A=\\emptyset$,}\\\\\n               0     & \\mbox{ otherwise}.\n             \\end{array}\\right.\n$$\nBy the remark after Example \\ref{example:excess-1}, $f_y$ is a polymatroid\nand $f_y(a,b|y)=f_y(a,y|b)=f_y(b,y|a)=0$ as well as $f_y(y|ab)=0$.\nAccording to Theorem \\ref{thm:sticky-2-1} if $f_X$ and $f_y$ have an amalgam, \nthey must satisfy\n$$\n   f_X[a,b,x_1,x_2] + f_y(a,b|y)+f_y(a,y|b)+f_y(b,y|a)+3f_y(y|ab)\\ge 0.\n$$\nThis value, however, is negative, which proves the theorem.\n\\end{proof}\n\n\n\n\\section*{Acknowledgment}\nThe author would like to thank the generous support of \nthe Institute of Information Theory and Automation of the CAS, Prague.\nThe research reported in this paper has been supported by GACR project\nnumber 19-04579S, and partially by the Lend\\\"ulet program of the HAS.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Discussion}\n\\label{sec:discussion}\n\nWe have presented a reanalysis of a cosmological simulation designed\nto study the intergalactic medium with the aims to both better understand the systematics of the methods used to identify absorbers and better understand the gas comprising the warm-hot intergalactic medium (WHIM; $10^5$~K$<T<10^7$~K) by analyzing the physical environments associated with the absorbers. We have devised three different methods for\nidentifying QSO absorption line systems in simulation data-- the cut\nmethod, the contour method, and the spectral\nmethod. The cut method slices the line of sight into regions of equal\nredshift. The contour method identifies continuous regions above a\nnumber density cutoff. The spectral method fits absorption spectra\ngenerated for each line of sight. Often these methods give different\nresults, so it is important to choose the method with care.\n\n\\subsection{Absorber Identification Method Comparison}\n\nCreating a one-to-one correspondence of absorbers between the two methods\nallows us to get a sense of how well the two methods agree on column\ndensity for an arbitrary absorber.  However, this method of matching is\nimperfect and does not always match two absorbers that correspond to the same\nphysical structure. One way we combat this problem is by considering both\nspectral absorber matching of individual line components as well as total absorbing\ncomplexes. We do this to account for the cases where the spectral method\nfinds substructure that the contour method is unable to identify. \n\nOur spectral fitting routine does a good job of finding\nabsorbers in a manner that is consistent with the contour method for both\nLy$\\alpha$\\ and \\ion{O}{6}. Some fraction of Ly$\\alpha$\\ lines with column densities in the\nneighborhood of $N_{HI}\\sim 10^{16}$~cm$^{-2}$ are difficult\ndifficult to fit accurately due to line saturation. \n\nResults from the $d{\\cal N}\/dz$\\ comparison for Ly$\\alpha$\\ absorbers show very good\nagreement outside the $N_{HI}=10^{16}$~cm$^{-2}$ range between the\nthree methods. \\ion{O}{6}\\ absorbers show considerable \ndifference with observations, however, the $d{\\cal N}\/dz$\\ values produced \nby the contour and spectral methods are quite consistent with each other. We hypothesize\nthat differences between simulated and observed \\ion{O}{6}\\ absorption line statistics \nare due to ionization and feedback choices made in the underlying simulation, \nrather than in the methods we use to create and determine the processes of \nsynthetic absorbers.\n\nAs the contour method does not take line of sight velocities into account\nwhen identifying absorbers\nwe do not expect the b-values found by the spectral method to\ninstrinsically correspond to the b-values from the contour method. Despite\nthis difference, the two methods find comparable b-value distributions\nto both each other and observations \\citep{tripp2008} for Ly$\\alpha$, especially\nat lower column densities ($N_{HI}\\leq 10^{15}$~cm$^{-2}$).  \n\nThere is a larger discrepancy between the contour method and the \nspectral method in \\ion{O}{6}\\ which may be an indication of line of sight \nvelocities having a larger effect or a systematic underfitting\nby the spectral method. The slight systematic increase in the contour\nmethod b-values at high column densities is consistent with observations \\citep{tripp2008}\nand some simulations \\citep{2012ApJ...753...17C}, but contrast others\nthat find a much flatter distribution \\citep{teppergarcia2011}. The b-value distribution \nof \\ion{O}{6}\\ is also much more sensitive to the method of feedback used in \nthe simulation indicating, so it is not instructive to read too much into\nsuch a comparison.\n\nOne interesting result that comes from our comparison of the two\nmethods is the linear size-column density relation for both \\ion{H}{1}\\ and\n\\ion{O}{6}.  We find (as can be seen in Figure~\\ref{fig:size_Var}) that in\nour calculations absorber size is a weak function of column density,\nwith $L \\propto N^{-0.12}$ for both \\ion{H}{1}\\ and \\ion{O}{6}.  This is in\ndisagreement with analytic predictions\n\\citep[e.g.,][]{2001ApJ...559..507S}, though does not appear to\ndisagree with observations.  We speculate that this may be due to\nsimulation resolution or choice of cooling model, but also note that\nthe analytic models make the assumption of hydrostatic equilibrium,\nwhich is not necessarily accurate.\n\n\\subsection{Understanding the WHIM}\n\nThe physical conditions of the absorbers showed roughly the\nsame results as found by \\citet{britton2011}. The same general trends held\nfor mean \\ion{H}{1}\\ number density, metallicity, and \\ion{O}{6}\\ fraction as a\nfunction of column density. \n\nWe also find a similar phase distribution of absorbers as \\cite{britton2011}\nwith $69\\%$ in the WHIM phase, $30\\%$ in the warm phase, and\n$1\\%$ in the condensed phase. We do not, however, see strong evidence for a \ndistinct bimodality in temperature of absorbers as \\cite{britton2011} found for the phase\ndistribution of the total gas; instead we see a smoothly varying distribution across the temperature range. This does not indicate that there cannot\nbe two distinct collisionally ionized and photoionized populations, only that they are not represented in these absorber statistics. Such a population could still exist in the very rarest, highest column absorbers. It is also possible that these populations exist as part of a multi-phase absorber that would in effect, hide such a bimodality from observations. Further analysis could be done to examine such a distribution of gas within a single absorbing feature.\n\nIn effort to distinguish the characteristics of absorbers found in the WHIM\nphase versus the warm phase, we segment the absorber population by temperature\ncuts, with a temperature greater than $10^5$~K indicating that the\nabsorber is in the WHIM phase. These absorbers have similar \ncolumn density distributions as the rest of the population, but b-values that are\nmuch higher on average. There are no \\ion{H}{1}\\ (\\ion{O}{6}) absorbers in the WHIM phase with\nb-values less than 40~km~s$^{-1}$ (10~km~s$^{-1}$) by definition, but above those \nlimits the fraction of total absorbers is strongly dominated by WHIM absorbers.\nThese high b-value, low column \\ion{H}{1}\\ absorbers correspond to the population \nof broad Ly$\\alpha$\\ (BLA) absorbers identified through COS \\citep{2010ApJ...710..613D}\nand STIS \\citep{2004ApJS..153..165R,2006A&A...451..767R}.\n\nThere has been substantial recent debate about the origin of \\ion{O}{6}\\ absorption line systems, with some work suggesting that it comes entirely from collisional ionization, and other work suggesting that it comes from a combination of collisional and photo ionization.  As part of this debate, there has been controversy about the temperature of the plasma where \\ion{O}{6}\\ absorption lines predominantly occur. In this work, we investigate the nature of \\ion{O}{6}\\ absorbers in the intergalactic medium, but take a slightly different approach: we break the absorber population into absorbers dominated by collisional ionization and those dominated by photo ionization, and compare their thermal phase-space properties and column density distributions.  We find that the two populations have similar ranges of column densities, but (perhaps unsurprisingly) have bimodal distributions in both temperature and b-value.\n\nTo investigate the correlation between \\ion{H}{1}\\ and \\ion{O}{6}\\ and the phases of gas\nthat they trace, we find the associated \\ion{O}{6}\\ column densities for regions\nidentified as \\ion{H}{1}\\ absorbers. This shows a marked distinction between the\ncolumn of associated \\ion{O}{6}\\ by column density and b-value of \\ion{H}{1}\\ for the\ntwo phases, WHIM and warm, defined as having temperatures greater than and less thatn $10^5$~K respectively. The WHIM phase has a much higher median column density of associated \\ion{O}{6}\\ in all regimes except the highest column \\ion{H}{1}\\ absorbers. This is \nconsistent with the idea that the WHIM is effectively traced by low column, high b-value Ly$\\alpha$\\ absorbers (BLAs), as well as by \\ion{O}{6}.\n\nFinally, in consideration of the product $(Z\/Z_\\odot)f_{\\mathrm{OVI}}$, we find a\nmuch lower average value than the literature standard, comparable with that\nfound by \\cite{shull2012}. This may significantly affect predictions of IGM metallicity that have been primarily based on \\ion{O}{6}\\ absorption.\n\n\\section{Introduction}\n\nIt is now well accepted that a large portion of all baryons at $z =\n0$, up to $\\sim60$\\%, exist in a phase that is extremely difficult to\ndetect.  Unlike at high redshift ($z \\gtrsim 4$) where nearly all\nbaryons are accountable through Ly$\\alpha$\\ absorption \n\\citep{1997ApJ...489....7R, 2009and..book..419P}, only $\\sim30$\\% are\nobserved to be associated with the cool, photoionized Lyman alpha\nforest today \\citep{1992MNRAS.258P..14P, 1994MNRAS.267...13B,\n1998ApJ...503..518F, 2004ApJ...616..643F, danforth2008, shull2012},\nwith only an additional $\\sim10$\\% in galaxies.  However, it has been\nshown by cosmological simulations \\citep[e.g.][]{1999ApJ...514....1C,\n1999ApJ...511..521D, 2001ApJ...552..473D, britton2011} that most of these ``missing\nbaryons'' still exist in the IGM, but have simply been heated through\na combination of accretion shocks from structure formation and\nfeedback from galaxies to temperatures too hot ($\\sim 10^{5-7}$ K) to\nbe easily traceable with neutral hydrogen.  Because of these high\ntemperatures, this so-called warm\/hot intergalactic medium (WHIM) can\nonly be traced through absorption lines of highly ionized metals, such\nas \\ion{O}{6}\\ and \\ion{Ne}{8}\\ at ultraviolet wavelengths, and \\ion{O}{7}\\ and\n\\ion{O}{8}\\ in the X-ray.  To date, the \\ion{O}{6}\\ doublet at\n$\\lambda\\lambda$1032, 1038~\\AA\\ has proven to be the most fruitful of\nthese detection methods, with numerous large samples having been\ncompiled \\citep{2005ApJ...624..555D, danforth2008,\n2006ApJ...640..716D, tripp2008, 2008ApJS..179...37T,\n2008ApJ...683...22T, tilton2012, 2014arXiv1402.2655D, 2014ApJS..212....8S}, providing an excellent benchmark for\nsimulations.\n\nNumerous simulation studies have succeeded at reproducing the observed\nnumber density per unit redshift $d{\\cal N}\/dz$\\ of \\ion{O}{6}\\ absorbers\n\\citep[][]{2006ApJ...650..573C, 2009MNRAS.395.1875O,\nbritton2011, 2011ApJ...731...11C, teppergarcia2011,\n2012MNRAS.420..829O,2012ApJ...753...17C}, despite making significantly different\npredictions as to the physical conditions of the associated gas.  For\nexample, \\citet{2009MNRAS.395.1875O} and \\citet{2012MNRAS.420..829O}\nfind that an overwhelming majority of \\ion{O}{6}\\ absorbers are associated\nwith cool, photoionized gas with a mean temperature of $\\sim15,000$ K.\nOn the other hand, \\citet{britton2011} and \\citet{shull2012} find that \\ion{O}{6}\\ absorbers come\nfrom gas in two distinct temperature regimes -- a photoionized\ncomponent at $T \\sim 30,000$ K and a hotter, collisionally ionized\ncomponent at $T \\sim 300,000$ K, comprising $\\sim35$\\% and $\\sim55$\\%\nof all \\ion{O}{6}\\ absorption, respectively (with the remaining 10\\%\ncontained in collapsed structures).  \\citet{teppergarcia2011} find a\nsimilar distribution of \\ion{O}{6}\\ as \\citet{britton2011} in their\nsimulated volume, but \\ion{O}{6}\\ absorption is\nbiased towards detecting the hotter, collisionally ionized phase.  Some of these\ndisagreements may arise from differences in the simulations, such as\nthe methods by which metals are injected and mixed into the\nintergalactic medium by the feedback scheme, but\nanother potential source of difference is the method in which\nsynthetic \\ion{O}{6}\\ absorbers are created and analyzed in these works.\n\\citet{2009MNRAS.395.1875O}, \\citet{teppergarcia2011}, and\n\\citet{2012MNRAS.420..829O} create synthetic absorption spectra which\nare then analyzed in a fashion similar to actual observations.\nHowever, \\citet{britton2011} create absorbers simply by summing the\ntotal column density along a small section of a randomly oriented line\nof sight in their simulation box.\n\nIn this paper we present a re-analysis of a simulation from\n\\citet{britton2011}, with the intent being to better understand the\ncorrespondence between absorption lines and the physical conditions\nwith which they are associated, and to better understand possible systematics\nbetween the usage of different means of extracting absorber data\nfrom simulations.  To accomplish this, we create a\nsophisticated pipeline for fitting absorption-line spectra that can be\nused for both synthetic and actual absorption spectra.  Our primary\ngoal is to understand the systematic differences between quantities\ninferred from absorption lines in synthetic spectra and quantities \nderived more directly from the simulation outputs.\nIn Section~\\ref{sec:sim}, we describe\nbriefly the simulation upon which our analysis is focused.  In\nSection~\\ref{sec:methods}, we detail three different methods of\nvarying sophistication that we use to create synthetic absorber\nsamples.  In Section~\\ref{sec:results}, we examine a range of\nproperties of the synthetic Ly$\\alpha$\\ and \\ion{O}{6}\\ absorbers created using\neach of these three methods, comparing each method both to the other\ntwo and to observations of the statistical properties of real quasar\nabsorption line systems.  We provide a discussion of our results in\nSection~\\ref{sec:discussion}, and summarize in\nSection~\\ref{sec:summary}.\n\n\\subsection{Choosing an Appropriate Contour Cut-off Density}\n\\label{sec:cutoff}\nAs discussed in Section~\\ref{sec:contour}, for each spatially\ncontiguous region of cells above the mean number density\nthe characteristic density is calculated as some fraction \nof the peak number density of the region. Absorbers are then \ndetermined by spatially contiguous cells above the characteristic\ndensity. Thus, lowering or raising the characteristic density \nfraction could affect the number of absorbers\nand the extent of individual absorbers. As the effects of a poor\nchoice in characteristic density fraction are not immediately obvious,\nappropriate care must be taken to select a reasonable value.\n\nTo check that the characteristic density approach is reasonable\nat all we examine Figure\n\\ref{fig:spatial}, where we average the number densities for a set\nof absorbers with low column densities ($N_{HI}=10^{12}-10^{13}$~cm$^{-2}$,\n$N_{OVI}=10^{13}-10^{13.5}$~cm$^{-2}$) over physical distance along\nthe line of sight from the peak\nnumber density for the region. A total of 5692 and 1319 absorbers were\nused for Ly$\\alpha$\\ and \\ion{O}{6}\\ respectively. The solid line indicates\nthe average number density, and the shaded region shows $1\\sigma$\ndeviations. The averages were taken by creating bins of physical\ndistance and, for every absorber, assigning the lixel with the most\nclosely corresponding physical distance from peak number density to\nthe appropriate bin. We then fit the resulting distribution with a\npower law at small radii and a constant at larger radii (i.e., we fit \n$n(r) = r^\\alpha + b$ for $r<d$ and $n(r)=n_o$ for $r>d$). The\nslope of the power law, $\\alpha$,  allows us to\nassess the extent to which an absorber will be dominated by its peak, and\ncorrespondingly, how much a change in characteristic\ndensity fraction could affect column\ndensity weighted properties for absorbers of a given species. The\nconstant fit for the tail of the distribution, $b$, reflects the average ion \nnumber density of the regions \\textit{between} absorbers. This number\ndensity is adopted as the mean number density of the species, but again \nthis mean refers to the mean of the background between absorbers, not\nthe mean number density of the entire volume or a given sightline. We note\nthat the constant background value is a very low \\ion{O}{6}\\ number\ndensity.  Examination of the raw simulation data shows that these values correspond to cells in the\nsimulation volume with baryon densities of $10^{-3} - 10^{-1}$ of the cosmic\nmean, metallicities of $10^{-4} - 10$~Z$_\\odot$, and temperatures of\n$10^5$~K or below (resulting in \\ion{O}{6}\\ fractions of $10^{-7}$ or\nbelow at the densities in question when the metagalactic ultraviolet\nbackground is taken into consideration).\n\n\\begin{figure*}\n\\begin{center}\n    \\includegraphics{spatial.pdf}\n    \\end{center}\n  \\caption{Average number density as a function of distance along the\n    line of sight from the center of an absorber.  The shaded region\n    shows $1\\sigma$ deviations. All absorbers with column density in\n    range ($10^{12}-10^{13}$~cm$^{-2}$) for \\ion{H}{1}\\ (left) and ($10^{12}-10^{14}$~cm$^{-2}$)for \\ion{O}{6}\\ (right) identified with the contour method. The center of\n    the absorber is defined as the lixel with the highest number\n    density. The dashed line indicates a power-law fit of the\n    average number density.  We adopt cutoff densities of\n    $10^{-12}$~cm$^{-3}$ (\\ion{H}{1}) and $10^{-13}$~cm$^{-3}$ (\\ion{O}{6}). }\n  \\label{fig:spatial}\n\\end{figure*}\n     \n\n\n\nWe recognize that due to the shallowness of the power law for \\ion{H}{1}\\\nnumber density ($\\alpha_{HI} = -2.4$), varying the characteristic density \nfraction may affect the properties of the absorbers, especially if a very low\ncharacteristic density fraction is chosen. In this case, contour-identified absorbers will fail to\nreflect actual structures, and quantities that should be dominated by peak values\nwill begin to deviate due to the large excess of sampled\nmaterial. \\ion{O}{6}\\ should be much less sensitive to the choice in cutoff\ngiven the steepness of the power law ($\\alpha_{OVI} = -6.3$).\n\nThis intuition is reflected in Figure \\ref{fig:char_frac_tests}, where we show\nnumber of absorbers, line of sight size, and $b$ as a function of \ncolumn density for a variety of characteristic density fractions.\nFor each cutoff, a corresponding set of absorbers is found independently. \nThis analysis is performed for a subset of the total light rays; however,\nthe results are representative of the total sample. We find as expected that\nmost quantities are fairly robust to changing characteristic \ndensity fraction, as displayed by \nthe consistency of the b-value results.  The linear size of absorbers, which is not weighted by number\ndensity, increases dramatically when lowering the characteristic density\nfraction.  We further investigate the size as a function of column density in \nsection \\ref{sec:size}. Given the later results we chose a\n  characteristic density that is 0.5 times the absorber's maximum baryon density.\n\n\\begin{figure*}[ht!]\n\\begin{center}\n    \\includegraphics[width=.9\\textwidth]{char_frac_tests.pdf}\n  \\caption{ Absorbers were identified using the contour method for a variety of characteristic density fractions (\\emph{colors}). For each set of identified absorbers number of absorbers (\\emph{Top}), average b-value (\\emph{Middle}), and average size (\\emph{Bottom}) were found as a function of column density. \\emph{Left:} \\ion{H}{1}\\ absorbers, \\emph{Right: } \\ion{O}{6}\\ absorbers.}\n  \\label{fig:char_frac_tests}\n\\end{center}\n\\end{figure*}\n\n\n\\subsubsection{Absorber Size}\n\\label{sec:size}\n\nAs absorber size was the most clearly varying parameter with characteristic \ndensity fraction, it is a useful metric to assess the choice of said \nfraction. Analytic estimates \\citep{2001ApJ...559..507S} give an expected \nsize-column density relation of $L\\propto N^{-1\/3}$ for \\ion{H}{1}\\ absorbers. \nIn Figure \\ref{fig:size_Var} we plot mean absorber size as a function of \ncolumn density for \\ion{H}{1}\\ and \\ion{O}{6}. We also overplot a line of best fit for \nthe size-column relation obtained through a least squares fit of the median \nvalue in each column density bin weighted by the number of absorbers in the \nbin. We find a best-fit power law of $-0.12$ for \\ion{H}{1}, which is substantially \nshallower than the expected slope of $-1\/3$.\nFor \\ion{O}{6}\\ we find a power law of $-0.12$.\n\n\\begin{figure*}\n\\begin{center}\n    \\includegraphics{size_Var.pdf}\n  \\caption{Median size of absorbers binned by column density using . Error bars \n    given by 1st and 4th quartiles. Black dashed line shows line of best \n    fit for median values weighted by number of absorbers in each bin, \n    with fit given in the legend. Grey dashed line in left hand panel shows \n    expected analytic relation, normalized to a 100~kpc absorber at \n    a column density $N_{\\mathrm{HI}}=10^{14}$~cm$^{-2}$. \n    Dotted line indicates the resolution limit of the simulation.\n    \\emph{Left:} \\ion{H}{1}\\ absorbers. \\emph{Right:} \\ion{O}{6}\\\n    absorbers.}\n  \\label{fig:size_Var}\n\\end{center}\n\\end{figure*}\n\n\\subsection{Comparison of Methods}\n\n\\subsubsection{Comparison of Contour and Cut Methods}\n\\label{sec:cutcontour}\nAlthough we do not expect \\textit{a priori} that the cut and contour methods \nwill give identical results,  we give a short comparison of these\nmethods. It is important to show comparison to the cut method despite\nits lack of physical motivation, as it was used\nfor a previous analysis of these simulations in \\citet{britton2011}. In\norder to correlate the absorbers, each absorber from the contour\nmethod is matched by redshift with an absorber from the cut method.\n\nMatching absorbers between methods is accomplished by finding an\nabsorber from the contour method within a given redshift ($\\delta z =\n10^{-6}$) of an absorber from the cut method. This process is then\nrepeated several times after increasing the tolerance by an order of\nmagnitude until $\\delta z = 10^{-3}$. As the redshift range we consider ($0\\leq\nz\\leq 0.4$) is quite small in comparison to the range of redshift windows we\nconsider, we do not bother to use a window of size $\\delta z\/(1+z)$. We have chosen to match the absorbers\nin this way to create a robust way of matching as many absorbers as\npossible, while still ensuring that the matches are as accurate as\npossible. Changing the start and end tolerances by an order of\nmagnitude has minimal effect on the final matches, as long as the\nmatching is accomplished using a series of monotonically increasing\nwindow sizes.\n\nThe column densities of each method for the appropriately matched\nand then binned by contour column density. Median contour and cut\ncolumn densities and first and fourth quartiles were found for\neach bin. This comparison, shown in the lower panels of Figure\n\\ref{fig:NVar_cutcontour}, indicates no obvious systematic difference \nat any column density. This is indicative of the absorbers being dominated\nby the sharp peaks in number density.  \n\nAdditionally, the total number of Ly$\\alpha$\\ absorbers found at low column\ndensities, shown in the upper panel of Figure \\ref{fig:NVar_cutcontour}, is\nslightly larger when using the cut method. This is evidence of\nthe splitting effect, as more absorbers of low column will be found\nwhen a single physical feature is inappropriately identified as two\nabsorbers. \n\n\nGiven that these results show a comparable column density result, the\ncontour method is clearly an adequate substitute in analyzing the\ndirect output. Furthermore, since the contour method finds fewer total\nabsorbers than the cut method (as it effectively combines multiple\nartificially-segmented absorbers into a single, physically meaningful object), as well as absorbers that directly\ncorrelate to the actual cosmological structure in the simulation\nvolume, it is straightforward to correlate absorbers found with the\ncontour method with those found using the spectral method. Thus, all\nof the analysis in the following sections of this paper will be\ncompleted using values found through the contour method.\n\n\\begin{figure*}\n\\begin{center}\n  \\includegraphics[width=.95\\textwidth]{NVar_cut_contour.pdf}\n  \\caption{\\emph{Top: } Total number of absorbers found in each column\ndensity range for contour method (\\emph{light blue}) and cut method\n(\\emph{dark blue}). \\emph{Bottom: } The contour absorbers were binned by\ncolumn density and the median column density of the matching cut\nabsorbers was plotted against the median column density of the\ncontour absorbers for that bin with vertical error bars showing\nfirst and fourth quartiles in cut column density. Dashed line\nshows $\\rm{N}_{\\rm{contour}}=\\rm{N}_{\\rm{cut}}$.  \\emph{Left: } \\ion{H}{1}\\\nabsorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers.}\n  \\label{fig:NVar_cutcontour}\n\\end{center}\n\\end{figure*}\n\t\n\\subsubsection{Comparison of Contour and Spectral Methods}\n\\label{sec:NVar}\n\nWe perform an analysis similar to Section~\\ref{sec:cutcontour} in\ncomparing the contour and spectral methods. After matching absorbers\nfrom the contour method with the column density determined by\na single component line fit using the spectral method technique,\nthe contour absorbers were then binned by\ncolumn density. The median spectral and contour column densities for\nthe absorbers in each of these bins was then plotted with error\nbars showing the corresponding first and fourth quartiles for each\nbin. These matches were determined using the same\nincreasing redshift tolerance process as described in\nSection~\\ref{sec:cutcontour}. \n        \n\\begin{figure*}\n\\begin{center}\n    \\includegraphics[width=.95\\textwidth]{NVar_split.pdf}\n  \\caption{ The contour\n    absorbers were binned by column density and the median column\n    density  of the matching spectral absorbers was plotted against\n    the median column density of the contour absorbers for that bin\n    with vertical error bars showing first and fourth quartiles in spectral\n    column density and horizontal error bars showing first and fourth \n    quartiles in contour column density. Dashed line shows\n    $\\rm{N}_{\\rm{contour}}=\\rm{N}_{\\rm{spectral}}$. The spectral method \n    treats each component found in multi-component fits of\n    a line complex separately.  \\emph{Left: } \\ion{H}{1}\\\n    absorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers. }\n  \\label{fig:NVar_split}\n\\end{center}\n\\end{figure*}\n        \nWe find that for Ly$\\alpha$\\ the pipeline has significant trouble\nappropriately matching corresponding absorbers at all column\ndensities and significantly underpredicts the results from the contour method.\nFor \\ion{O}{6}\\ the contour method finds higher column densities\nby approximately a factor of $2$ for low column densities\n($N_{OVI}=10^{12}-10^{14}$~cm$^{-2}$) and increasing to an order of\nmagnitude for higher column densities\n($N_{OVI}=10^{14}-10^{16}$~cm$^{-2}$).\n\nThis is expected as the spectral method is not biased against using\nmultiple component fits if the $\\chi^2$ error is lower in using more \ncomponents. In order to negate this effect, we sum together components\nidentified in the same complex. A complex is defined as a contiguous region \nwith a flux less than $F\/F_{continuum}=0.99$. The total column density \nof the region is thus $N_{total} = \\sum N_i$. The redshift of the\ncomplex is slightly more difficult to define, but we use a column\ndensity weighted average to assign a single value of redshift to \nthe entire complex. Performing the analysis in this way loses some\ninformation about physical structure from line of sight velocities \nthat the spectral method is able to identify, but allows us to check \nif the two methods track similar amounts of total material\n\n\nAs seen in Figure \\ref{fig:NVar_combine}, matching the contour method\nabsorbers to a line complex rather than a line component\ngives more comparable column density results for Ly$\\alpha$\\ and \\ion{O}{6}\\ at\nall column densities.\nThere is a larger scatter about the mean for absorbers with higher\ncolumn densities ($\\mathrm{N_{HI}}\\geq 10^{16}$~cm$^{-2}$,\n$\\mathrm{N_{OVI}}\\geq 10^{14.5}$~cm$^{-2}$). This may be indicative\nof the difficulty of fitting saturated lines. \n  We can now be reasonably certain that the spectral\nand contour methods identify the same absorbing structures, and we can\nmatch them reasonably effectively. The spectral method may find different underlying\nIGM substructure than the contour method due to line of sight velocity effects, but the two methods find roughly\nequivalent bulk material.\n\n        \n\\begin{figure*}  \n\\begin{center}\n    \\includegraphics[width=.95\\textwidth]{NVar_combine.pdf}\n  \\caption{The contour\n    absorbers were binned by column density and the median column\n    density  of the matching spectral absorbers was plotted against\n    the median column density of the contour absorbers for that bin\n    with vertical error bars showing first and fourth quartiles in spectral\n    column density. Dashed line shows\n    $\\rm{N}_{\\rm{contour}}=\\rm{N}_{\\rm{spectral}}$. The spectral method \n    combines each component found in multi-component fits of\n    a line complex into a single absorber whose column density is the sum of\n    the components' column densities.   \\emph{Left: } \\ion{H}{1}\\\n    absorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers.} \n  \\label{fig:NVar_combine}\n\\end{center}\n\\end{figure*}\n\nIn Figure \\ref{fig:sliced_b_distribution} we compare the distribution of b-values for \\ion{H}{1}\\ and \\ion{O}{6}\\ absorbers over slices in column density. The slices were chosen to give a relative idea of of how the methods compare for absorbers of interest; as such, we do not show higher column density slices for \\ion{H}{1}, despite the fact that higher column absorbers are fit. We find that for \\ion{H}{1}, the distribution of b-values is shifted for the spectral method relative to the contour method at low column ($10^{13}\\leq N \\leq 10^{13.5}$~cm$^{-2}$) with the peak of the spectral distribution at 25~km~s$^{-1}$ compared to 20~km~s$^{-1}$ for the contour method. For increasing column density slices this distinction shifts with spectral the spectral distribution peaking around 30~km~s$^{-1}$ and the contour distribution peaking around 25~km~s$^{-1}$. The \\ion{O}{6}\\ distributions look quite distinct for all column density slices. It seems that compared to the contour method, the spectral method identifies an overabundance of low column absorbers with very low b-values as well as a relative lack of high column aborbers with higher b-values. \n        \n\\begin{figure*}  \n\\begin{center}\n    \\includegraphics[width=0.95\\textwidth]{sliced_b_distribution.pdf}\n  \\caption{ Histogram of b-values for absorbers with column densities in ranges $10^{13}<N<10^{13.5}$~cm$^{-2}$ (\\emph{Top row}), $10^{13.5}<N<10^{14}$~cm$^{-2}$ (\\emph{2nd row}), $10^{14}<N<10^{14.5}$~cm$^{-2}$ (\\emph{3rd row}), and  $10^{14.5}<N<10^{15}$~cm$^{-2}$ (\\emph{Bottom row}). Red bars indicate spectral absorbers, blue bars indicate contour absorbers, and purple indicates an overlap in the distributions. \\emph{Left: } \\ion{H}{1}\\ absorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers.} \n  \\label{fig:sliced_b_distribution}\n\\end{center}\n\\end{figure*}\n\n\n        \n\n\\section{Approaches to Identifying IGM Absorbers}\n    \\label{sec:methods}\n\nUsing the simulations described in Section \\ref{sec:sim} and the\n\\texttt{yt} analysis toolkit\\footnote{\\url{http:\/\/yt-project.org}}\n\\citep{yt}, we generate 2000 pencil-beam\ndata arrays meant to be representative of QSO sight lines. Each line\nof sight extends from $z=0$ to $z=0.4$ and has a random starting\nposition in $(x,y,z)$ and random orientation in $(\\theta,\\phi)$. As\nthe physical distance corresponding to \n$\\Delta z = 0.4$ is much larger than the extent of the simulation box\nsize at low redshifts, the line of sight generation tool stacks rays from multiple\nepochs together. This is accomplished by pre-calculating the number of\ndata outputs and the corresponding position in redshift space needed\nto traverse the full length of the desired sample.  All pertinent\ninformation is recorded for each cell intersected by a line of sight,\nincluding baryon density, metallicity, ion densities, ion fractions,\nand temperature.  The pathlength $dl$ of the ray as it passes through\nthe cell is also recorded; this length is not constant given the\nrandom orientation of the sight line, but is on the order of the resolution of the simulation.\n\n The methods with which observers find structure in the universe is\nintrinsically different than the ways simulators\napproach the same problem. Even after generating lines of sight,\nsimulators still have access to all the information retained by the\npixels along the line of sight, or `lixels' as per \\citet{britton2011},\nfrom temperature to specific number densities. Observationally, we are\nlimited to interpreting these quantities through indirect methods such\nas fitting absorption lines, apparent optical depth, or pixel optical\ndepth. Each of these approaches is susceptible to its own systematic\neffects that may make it difficult to compare them. By performing both\nan observationally motivated analysis technique as well as a computational\none on the same lines of sight we can better correlate\nthe products of simulations with actual observations. Thus, in the\nfollowing sections we will detail several methods with which we\nanalyzed our synthetic lines of sight; one, the `spectral' method,\nthat uses only absorption line fits of synthetic spectra \nand two approaches, the `contour' and\n`cut' methods, which utilize cell-by-cell output. A sample of the\nresults of identification of absorbers using these methods can be seen\nin Figure~\\ref{fig:methods}. The relevant number densities ($\\mathrm{n_{HI}}$,\n$\\mathrm{n_{OVI}}$) are computed in both photo- and collisional ionization equilibrium, with the former assuming a uniform \\citet{2001cghr.confE..64H} radiation background. \n\n\nFor each method, after generating a full set of absorbers we filter by column density and only include \\ion{H}{1}\\ absorbers with column densities N (cm$^{-2}$) in the range $12.5\\leq \\log \\mathrm{N_{HI}} \\leq 17.2$ and \\ion{O}{6}\\ absorbers with column densities in the range $12.5\\leq \\log \\mathrm{N_{OVI}} \\leq 15.2$. We restrict our column density range on the higher end to reflect the absence of radiative transfer in our simulation. On the low end we restrict our column density range to reflect observable limits.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{schematic_methods.pdf}\n\\caption{Schematic example of three different methods for\n  finding absorbers in mock QSO lines of sight. In this figure\n  we examine \\ion{H}{1}; this applies equally well to any\n  species. \\emph{Top row:} The ``spectral method.'' A synthetic spectrum is\n  generated using the region of number density as a function of\n  redshift  (with density fields shown in the bottom two rows). Normalized flux plotted against\n  redshift appears shifted in redshift relative to peaks in\n  number density due to line-of-sight\n  velocities. \\emph{Middle row:} The ``cut method.''  \\ion{H}{1}\\ number density as\n  a function of redshift along the line of sight. Each shaded\n  region of fixed redshift width is a separate absorber.\n  \\emph{Bottom row:}  The ``contour method.''  The same region of \\ion{H}{1}\\ number\n  density as a function of redshift as above, but regions are identified as spatially contiguous cells above the mean number density of the species. The mean number density is given by the solid line at $\\mathrm{n_{HI}}=10^{-14}$~cm$^{-3}$. The dashed line is the characteristic density for each region above the mean number density. The spatially contiguous cells identified with number densities above the characteristic densities (shaded in blue) are then identified as absorbers. }\n\\label{fig:methods}\n\\end{center}\n\\end{figure}\n        \n\\subsection{Spectral Method}\n\nIn order to mimic true observations with the data generated via\nsimulation, a synthetic QSO absorption spectrum was created and\nthen fitted for each line of sight. The fitted components then\ndetermined the absorber -- in particular, the absorber's column density and\ndoppler-value. This method will henceforth be referred to as the\n``spectral method.'' \n\n       \n\\subsubsection{Creating Absorption Spectra}\n\nFor each of the 2000 simulated QSO sight lines we create a corresponding\nabsorption spectrum by considering the effects of absorption for each \nlixel of gas. The optical depth for a cloud of gas with column density N of a\nwavelength $\\lambda$ is given by $\\tau_i(\\lambda) = C_i N a H(a,x(\\lambda))$,\nwhere $x=(\\lambda-\\lambda_o)\/\\lambda_o$, $a=\\Gamma_i\/(4\\pi \\Delta \\nu_D)$, and $C_i$\nis a transition-dependent constant. Here $\\Delta  \\nu_D = b_{th}\/(c\\lambda_i)$ and \n$\\Gamma_i$ is the reciprocal of the mean lifetime of the transition. The Voigt-Hjerting function, $H$, is given by \n\n\\begin{equation}\n  H(a,x) = \\frac{a}{\\pi}\\int_{-\\infty}^{\\infty}\\frac{e^{-y^2}}{(x-y)^2+a^2}dy\\;,\n  \\label{eq:voigt_long}\n\\end{equation}\n\nwhere $y=v\/b_{th}$. The central wavelength $\\lambda_o$ is given by \n\n\\begin{equation}\n  \\lambda_o = \\lambda_i(1+z)(1+v_{los}\/c) \\;,\n\\end{equation}\n\nwhere $\\lambda_i$ is the rest wavelength of the transition for species $i$,\n$z$ is the redshift of the lixel generating the profile, and $v_{los}$ is\nthe peculiar velocity of the lixel generating the profile. \n\nBy defining a Doppler parameter as in Equation \\eqref{eq:bth}, we\ncan define and $\\Delta \\lambda_D = (b_{th}\/c)\\lambda_i$ \nwhere $\\lambda_i$ is the resonant wavelength\nof the corresponding transition given by $\\lambda_i = hc\/E_i$. The\nthermal Doppler parameter $b_{th}$ quantifies the width of a\nsingle line:\n\n\\begin{equation}\n  b_{th}=\\sqrt{2 k_B T\/m_i} \\;.\n  \\label{eq:bth}\n\\end{equation}\n\nThe resulting normalized flux is then given by $F(\\tau) =\ne^{-\\tau}$. The absorption feature generated in this fashion is\nknown as the Voigt profile. One of these profiles is generated for\neach lixel using the algorithm detailed by \\citet{armstrong1967}. \n\nThe spectra were generated at a resolving power of\n$R=\\lambda\/\\Delta\\lambda=10^{5}$. This is substantially higher\nresolution than a typical IGM observation using Hubble's Cosmic\nOrigins Spectrograph (COS), which has resolving power of $R=2,000$ (G140L) and\n$18,000$ (G130M); however, the simulated spectra can easily be degraded\nto any desired lower resolution, and noise can be added.\n\nAs each lixel generates its own Voigt profile, large scale\nfeatures in the spectra are typically composed of contributions\nfrom multiple lixels and their corresponding column densities.\nThis effect can be seen in Figure~\\ref{fig:methods} where, despite\na wide range of \\ion{H}{1}\\ number densities over a number of cells, only four\nsignificant spectral features are visible. This causes a\ndegeneracy in our notion of b-value, since broadening can occur as a\nresult of disordered line-of-sight motion within a cloud of gas (as a\nresult of, e.g., turbulence) in addition to\nthe broadening due to thermal motion. Cells with different\nvelocities create Voigt profiles at slightly different wavelengths\ndue to the Doppler effect and, as a result, the overall line to\nlooks broader. This broadening is typically indistinguishable from the\nbroadening due to thermal motion.\n\nIn generation of the spectrum as many ions as desired\ncan be included. Fitting a single ion at a time removes any\nambiguity of what ion is being analyzed, so all further analysis\nwas completed in this manner. Our spectra were also fit without\nthe effects of noise as an initial test of our methods, though we\nnote that the method works with noisy spectra as well. \n\n\\subsubsection{Fitting Absorption Spectra}     \nDue to the challenge of fitting a large number of synthetic\nspectra, we developed an automated line-fitting pipeline. This\nsystem takes the synthetic spectra and fits a series of Voigt\nprofiles to each spectrum in a completely automated\nfashion. To automatically identify absorbers, each contiguous\nregion with normalized flux below an adjustable cut-off\n($F\/F_{cont}=0.99$) is identified. This cut-off was chosen\nprimarily to underestimate the observable limit due to noise\nlimitations, without over-fitting very shallow and hard-to-detect lines. Lowering this value by a few percent does not\nsignificantly affect results. Raising the value begins to\nsubstantially increase computational cost as regions become\nlarger and a typical region contains more lines; however, the\nresults are not typically affected as a region is generally\nconsidered successfully fit regardless of the inclusion of\nthe smaller fluctuations.\n\nExceptionally large complexes ($\\Delta \\lambda >50\\,\\r{A}$) with multiple lines\nare broken up at points of minimum absorption, if that minimum\npoint has a flux within a few percent of the adjustable\ncutoff. This is done to prevent very large regions from\ndramatically slowing down the fitting procedure due to the\ndifficulty of optimizing the fits of many lines\nsimultaneously.\n\nEach region is then taken in turn from minimum to maximum\nwavelength and fit by iteratively adding and adjusting Voigt\nprofile parameters (column density, broadening value, and\nredshift) using the least squares method until the total\n$\\chi^2$ error is smaller than an acceptable fractional error threshold\n($10^{-4}$) multiplied by the number of points in the region\nin units of normalized flux. We attempt to fit up to 8\nindividual components simultaneously. If this is\nunsuccessful or if the error begins to increase upon adding\nmore components, the fit is accepted if the total error\nis within two orders of magnitude of the desired error. Given\nthe stringent conditions used for our standard fitting procedure, these fits are\ntypically quite good, and would certainly be acceptable if noise\nwere included in our synthetic spectra (which we defer to a later paper).\nChanging the value of the acceptable fractional error threshold \nby an order of magnitude has little effect on the overall results.\n\nIn cases of ions that create more than one line (as in the\n\\ion{O}{6}\\ doublet), the lower wavelength line is fit to\nthe region as, in the case of Li-like ions (2s-2p), this is the stronger\nline. We then attempt to fit the higher wavelength line\nwith the parameters calculated for the lower wavelength\ncounterpart and if the resulting total fit has a low enough\nerror then it is accepted. This allows some amount of leeway\nfor blanketed line identification where a large line occurs in\nthe same wavelength space as another smaller line, effectively\nhiding the smaller line.\n\nAn example of this fitting procedure is shown in Figure~\\ref{fig:fitex}. \nOnce the region is identified, a single line\nis optimized to fit the whole region. Given the structure of\nthe region, a single line is insufficient to constrain the\nregion and the difference between the fit and the data is\nstill larger than allowed by the average error per point. Thus,\nanother line is added and the region is fit using two lines,\nwith three free parameters, N, $b$, and $z$. This fit is again not\ngood enough so this procedure is repeated now with three lines for\na total of nine free parameters. The three-line fit then satisfies the\naverage error per point bound and the region is considered\nsuccessfully fit.\n\n\n\\begin{figure}\n  \\includegraphics[width=0.45\\textwidth]{multifit.pdf}\n  \\caption{An example Ly$\\alpha$\\ fit for a region of wavelength space where\nthe best fit consists of multiple line components.  The x-axis\nindicates wavelength difference measured from an arbitrary reference\nwavelength, and the y-axis indicates normalized flux.  The solid blue\nline indicates the total normalized flux generated from a range of\n\\ion{H}{1}\\ number density along a line of sight, while the dashed lines\nindicate components found for the fit.}\n  \\label{fig:fitex}\n\\end{figure}\n        \nIn regions where the regular fitting tools fail to perform\nproperly and the line in consideration is Ly$\\alpha$, a more\nrobust approach is used. Absorbers with low temperatures\n($T\\leq10^4$ K) and relatively high column densities\n($N_{HI}\\sim 10^{18}$ cm$^{-2}$) generate Voigt profiles\nwith damping wings. Typical parameters that\nare appropriate for the large majority of absorbers often fail\nto converge to a good fit for absorbers with these properties.\nWhen such a region is identified, a separate set of widely\nvaried initial temperatures and column densities are tried,\nwhich allows for a more accurate fit at the expense of\nsubstantially increased computational cost. In\nthis paper we primarily examine \\ion{H}{1}\\ and \\ion{O}{6}\\ absorption. Most\nof the challenging fits are of \\ion{H}{1}\\ absorbers, due to their more\nvaried (and higher) column densities and environments. \n\n\n\n\\subsection{Cut Method}\n\\label{sec:cut}\nInstead of creating absorption spectra, simulators have the\nopportunity to examine exactly how number density and other\nphysical quantities vary with redshift along a line of sight.\nHowever, in order to compare with observation, it is necessary\nto break up this continuous variation into discrete absorbing\nstructures. A simple approach, henceforth referred to as the\n`cut method,' was employed by  \\citet{britton2011}\nin a previous analysis of the simulations examined in this\npaper. In this method, a given line of sight is cut into\npieces of constant length in redshift space where the cells\nbetween each cut contribute to that absorber. The sampling\nresolution ($\\lambda\/\\delta\\lambda$) is set at 5000; this\nallows smoothing of the region over multiple lixels while\nmaintaining a constant resolution in the range of\nobservation. A change of resolution by an order of magnitude \nin either direction does not significantly affect\nthe results. \n\nThe column density for ion x is given by the sum over cells\n$N_x=\\sum dl_i n_{x,i}$, where $dl_i$ is the pathlength and\n$n_{x,i}$ is the number density of ion x in a given absorber species in a given cell.\nThe b-value is comprised of a thermal and a non-thermal\ncomponent for a total b-value given by\n$b=\\sqrt{b_{th}^2+(\\sigma_{v})^2}$.  The thermal component is\nthe same as the spectral thermal Doppler parameter,\n$b_{th}=\\sqrt{2kT\/m_i}$. To approximate the non-thermal\ncomponent of broadening we find the variance in the line of\nsight velocities weighted by column density of each cell. All\nother relevant quantities such as ion fraction, metallicity,\nand temperature are found via column density-weighted average for all cells contributing to the\nabsorber. \n\nThis method is simple because it requires no additional\nanalysis beyond dividing the line of sight properties into\nequal redshift bins; however, it does not identify\nphysical structures at all. Large absorbers may happen to fall\nalong the boundary of two redshift bins as seen at $z\\sim0.085$\nin Figure~\\ref{fig:methods}. This may cause a single region to\nbe interpreted as multiple absorbers of lower column density\nand distort the inferred physical conditions associated with\nthe absorber. Although properties dominated by the sharp peaks\nin number density may still hold, this method cannot be\nexpected to give good intuition for the physical nature of a\nstructure because, quite simply, the method of defining\nabsorbers in not based on physics, but rather numerically\nconvenient quantities.\n\n        \n\\subsection{Contour Method}\n\\label{sec:contour}\nThe `contour method' of defining IGM absorbers was developed\nin response to the limitations of the cut method described in\nSection~\\ref{sec:cut}. Instead of cutting blindly by redshift,\nregions are identified where groups of spatially contiguous cells\nall have number densities above the mean number density of a given \nspecies. A discussion of identifying the mean number density of each\nspecies is given in Section~\\ref{sec:cutoff}, but the values adopted\nfor the remained of this work are $\\mathrm{n_{HI} = 10^{-14}~cm^{-3}}$ \nand $\\mathrm{n_{OVI} = 10^{-22}~cm^{-3}}$.\nFor each region, a peak species number density is \ndetermined. An absorber is then quantified as spatially contiguous \ncells within the original region  with number densities \nabove a characteristic density, set by some\nfraction of the peak number density. For most of the following analysis\nwe chose the characteristic density as 0.5 times the\nthe peak number density, but the effects of varying this fraction are \ninvestigated in section~\\ref{sec:cutoff}. This process is illustrated\nin Figure~\\ref{fig:schematic_contour}. Multiple absorbers can be identified \nin a single region as illustrated in Figure~\\ref{fig:methods} at \n$z \\simeq 0.086$. By setting a characteristic density for each region \nrather than a single number density cutoff for the whole species we \nensure that only the cells that maximally contribute to a given absorbing \nfeature as selected as part of the absorber, while still identifying low \ncolumn absorbers. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.45\\textwidth]{schematic_contour.pdf}\n\\caption{Schematic example of the ``contour method''. Initially, the light blue region is identified as a spatially contiguous group of cells above the mean number density (\\emph{Bottom Line}). Within this region the maximum number density is identified. The characteristic number density (\\emph{Top line}) of this region is then calculated as some fraction of the maximum number density. The absorber for this region is then identified as the spatially contguous subset of cells with number densities greater than the characteristic number density.}\n\\label{fig:schematic_contour}\n\\end{center}\n\\end{figure}\n\nAll properties of these absorbers are found using the \nsame methodology as was used for the cut method. The identification \nof the regions in this way isolates large structures in a \nphysically-motivated way that preserves overall structure. \n\n\n\\section{Results}\n\\label{sec:results}\n\n\\input{methods-compare}\n\\input{obvs-compare}\n\\input{physical-conditions}\n\n\\input{discussion}\n\\input{summary}\n\n\n\\bibliographystyle{apj}\n\n\\subsection{Comparison to Observations}\nWe now compare our synthetic absorber population with observed absorbers to assess any systematic differences, attributable to our simulation or methods. \n\n\\subsubsection{Differential and Integral dN\/dz}\n\nA common test of the accuracy of simulations is to recreate the\nobserved number density of \\ion{O}{6}\\ absorbers per unit redshift\n\\citep{tilton2012,danforth2008}, as well as the cumulative number\ndensity of \\ion{O}{6}\\ absorbers per unit redshift\n\\citep[e.g.,][]{fang2001,cen2006,oppenheimer2009,teppergarcia2011,britton2011}.\nIt is thus quite useful to understand how generating this statistic\nfrom simulations using an observationally-motivated method\nintrinsically differs from generating this statistic from absorbers\nfound using analysis of cell-by-cell output.\n\nFigure~\\ref{fig:dndz} shows this analysis for \\ion{H}{1}\\ and \\ion{O}{6}\\ absorbers\nfound in our simulation, along\nwith observational data from \\citet{tilton2012} and\n\\citet{danforth2008} for comparison for \\ion{H}{1}\\ and \\ion{O}{6}\\, respectively. We consider each component\nof a multi-component fit of a complex separately because, although\nsingle-component fits were favored in both observational cases,\ncomplexes with clear evidence of substructure were fit in a\nmulti-component fashion with each component listed separately in the\ncalculation of $d \\mathcal{N}\/dz$. In the top row of this Figure we\nshow the cumulative line number density of \\ion{H}{1}\\ and \\ion{O}{6}\\ shown in the\nstandard way -- i.e., we show the number of\nabsorbers above a given species column density at the mean redshift of\nthe simulation outputs, normalized by the total redshift interval\n$\\Delta z$ of the synthetic observations.  The bottom row displays the\ndifferential line number density normalized by the total redshift\ninterval $\\Delta z$, also known as the column density distribution function.\n\nThe two methods agree quite well at all column densities for both\nLy$\\alpha$\\ and \\ion{O}{6}\\ as seen in both the differential and integral forms.\nThere is a systematic under-prediction relative to the observed\nabsorber frequency using either simulated method in \\ion{O}{6}; however, this was also\nfound in the initial analysis presented by \\citet{britton2011}, and is\nlikely to be an intrinsic property of the simulation rather than a\nfeature of our method of determining absorber properties.\n\nIt should be noted that we consider absorbers above a column\ndensity of $10^{12.5}$~cm$^{-2}$ for the contour method, and fit below\nthe observable limit for both \\ion{H}{1}\\ and \\ion{O}{6}\\ with the spectral method using noiseless\nspectra. As a result, no significant completion correction is required\nfor low column absorbers in our simulated results, which may result in\nsystematic differences from the cited observational results.\n        \n\\begin{figure*}\n\\begin{center}\n  \\includegraphics[width=.95\\textwidth]{dndz.pdf}\n  \\caption{\\emph{Top: } Number of\n    absorbers ($\\mathcal{N}$) with a column density greater than N\n    per redshift z (i.e., cumulative line number density). \\emph{Bottom:} Number of absorbers ($\\mathcal{N}$) per column\n    density ($N$) per redshift ($z$) (i.e., line number density). \\emph{Left:} \\ion{H}{1}\\ column densities with\n    observational points and error bars from \\citet{tilton2012}. \\emph{Right:}\n    \\ion{O}{6}\\ column densities, with observational points and error bars\n    from \\citet{danforth2008}.  In all panels, the red line\n    corresponds to the spectral method, the blue line to the contour\n    method, and the black line to observations.}\n  \\label{fig:dndz}\n\\end{center}\n\\end{figure*}\n\n\n\\subsubsection{Broadening Value}\n\nThe other parameter that is found directly using the spectral method\nis the Doppler parameter, or `b-value.'  This parameter determines the width of a line and\ntypically has contributions from both thermal and non-thermal broadening.\n\nWhile observationally these components are very challenging to distinguish and\nrequire simultaneous fits of multiple species with different masses,\nin a simulation we have perfect knowledge\nof the thermal and kinematic behavior of the plasma everywhere in our\nvolume. To that end, a more detailed and precise study is\npossible. Figure~\\ref{fig:bVar} shows the median b-value plotted against\ncolumn density for both the contour and spectral methods, as well as\nthe squared fraction of total b-value due to thermal motion as a function of\ncolumn density for the contour method. We look at the squared fraction as\nthe thermal and non-thermal components of the total b-value are added in\nquadrature.\n\nThe Ly$\\alpha$\\ values agree reasonably well between methods at lower column\ndensities and but diverge slightly at high column\ndensities ($N_{\\mathrm{HI}}\\geq 10^{16}$~cm$^{-2}$). The values increase\nslightly with column to $N_{\\mathrm{HI}}\\sim10^{14.5}$~cm$^{-2}$. The\nb-values from the contour method then begin to slightly decrease again\nwhile the values derived from the spectral method remain fairly constant.\nObservational results from \\citet{tripp2008} show a similar pattern as \nthe contour method.\n\nFor \\ion{O}{6}, the median b-value increases stays roughly constant at \n$b\\sim 20 $~km s$^{-1}$ using the contour method. The spectral method \nstays roughly constant at  $b\\sim 15$~km s$^{-1}$. \nWe do not see the trend indicated by \\citet{tripp2008} of increasing\nb-value with column density for either method.\nWe note that the biggest divergence between the two\nmethods occurs where the thermal component of the b-value becomes less\ndominant; however, this does not cause a systematic under or over\nprediction of b-value by either method across both species. \n\nWe note that here we only consider the b-values of each individual\nspectral component in this analysis. Although combining complexes was\nuseful in an attempt to correlate spectral structures with contour\nstructures as in Sections \\ref{sec:NVar}, this method obscures the\nunderlying structure that can be determined with the aid of the\nb-values. We thus do not expect the contour and spectral b-value\ndistributions to be the same, as they simply do not correspond to the\nsame quantities; the contour method takes the variation in bulk motion\nof separate features and gives a higher non-thermal motion component of\nthe b-value, whereas the spectral method divides up the same region of\nphysical space into separate absorbing components. \n\n\n\\begin{figure*}\n\\begin{center}\n    \\includegraphics[width=0.95\\textwidth]{bVar.pdf}\n  \\caption{\\emph{Top: } Squared fraction (due to component b-value \n    addition in quadrature) of b-value due to thermal motion as a\n    function of column density for the contour method. \\emph{Bottom: }Mean b-value vs Column\n    density for contour (\\emph{blue}) and spectral (\\emph{red})\n    methods. Observational points (\\emph{black}) from\n    \\citet{tilton2012}. Error bars show $1\\sigma$\n    deviations. \\emph{Left:} \\ion{H}{1}\\ absorbers. \\emph{Right:} \\ion{O}{6}\\\n    absorbers.}\n  \\label{fig:bVar}\n\\end{center}\n\\end{figure*}\n\n\\subsection{Physical Conditions of Absorbers}\nIn order to extend our understanding of the WHIM and its relationship to observations, we examine the physical environment associated with the absorbers identified with the contour method.\n\n\n\\subsubsection{Median Quantities Of Absorber Systems}\n\nFigure~\\ref{fig:param} details the physical condition of the gas as a\nfunction of absorber column density. \nGas temperature appears to increase and then decrease with increasing column\ndensity for \\ion{H}{1}\\  absorbers, while\nthe temperature of \\ion{O}{6}\\ absorbers decreases with increasing\ncolumn. The temperature\nbehavior for \\ion{H}{1}\\  also typically has much less variance at a given column density. \nMetallicity, defined as the total mass in elements heavier than helium relative \nto the total gas mass normalized by the Solar metal fraction, increases \nsteadily with increasing \\ion{O}{6}\\ column density.\nThere is slight increase with increasing \\ion{H}{1}\\ column density, albeit \nwith an increasing variance at low $\\mathrm{N_{HI}}$, suggesting that \nlow $\\mathrm{N_{HI}}$ systems may trace a large variety of environments. \nClear trends in \\ion{O}{6}\\ fraction \ncan be seen as a function of \\ion{H}{1}\\ column density, with a peak at \nN$_{HI} \\simeq 10^{14}$~cm$^{-2}$ and lower fractions at larger and smaller columns.\nThe median number density of H increases steadily for both \\ion{H}{1}\\ \nand \\ion{O}{6}\\ \nabsorbers, although over a wider range with smaller variance for \\ion{H}{1}.  The fraction \nof \\ion{H}{1}\\ ionization steadily increases as a function of increasing \\ion{H}{1} column density. The \\ion{H}{1}\\\nfraction also seems to increase with \\ion{O}{6}\\ column density, but the variance in this\ntrend is far larger. These trends appear largely consistent with those initially\nprovided by \\citet{britton2011}. \n\n\\begin{figure*}\n\\begin{center}\n    \\includegraphics[width=.95\\textwidth]{pVar.pdf}  \n  \\caption{Median temperature, metallicity, \\ion{O}{6}\\ fraction, total H number density,\n    and \\ion{H}{1}\\ fraction plotted against column density for absorbers found using the contour method. Error bars show first and\n    fourth quartiles. \\emph{Left:} Ly$\\alpha$\\ absorbers. \\emph{Right:} \\ion{O}{6}\\\n    absorbers.}\n  \\label{fig:param}\n\\end{center}\n\\end{figure*}\n            \n\\subsubsection{Thermal State of \\ion{O}{6}\\ Absorbers}\nOne of the primary purposes of these simulations was to investigate the \nutility of \\ion{O}{6}\\ as a tracer of the WHIM. The initial analysis presented by\n\\citet{britton2011} showed a bimodality in the temperature distribution\nof \\ion{O}{6}\\ also seen in \\citet{teppergarcia2011}. The bimodality was centered\nat $T\\sim10^5$ K, with 57\\% of \\ion{O}{6}\\ absorbers found\naround temperatures of $10^{5.5}$ K in the WHIM phase and 37\\% found\nat temperatures of $10^{4.5}$ K in the warm phase (and the remaining 6\\%\nfound at higher densities in what \\citet{britton2011} defined as the\n`condensed' phase, with a baryon overdensity of $\\Delta_b \\geq 1000$). Such a\nbimodality suggests that both collisionally ionized and photo-ionized \\ion{O}{6}\\ are present in significant amounts in the IGM.\n\nWe then must ask if these statistics are proportionally recreated when\nlooking at absorbers found using the methods presented in this\npaper. \\citet{britton2011} found there to be no bias between the true phase distribution of \\ion{O}{6}\\ in the simulation and that inferred from absorbers created with the cut method,\nwhereas \\citet{teppergarcia2011} found \\ion{O}{6}\\ absorption to be biased towards higher temperatures. Figure~\\ref{fig:phase} shows the baryon overdensity and\ntemperature for each absorber found using the spectral method.  Baryon\nproperties are determined by using the mean overdensity and\ntemperature of the gas in the same absorber found with the contour method, \nas determined using column density-weighted averages of all cells in the absorber.\nThe distribution shows no evidence of a strong temperature bimodality, but \ninstead shows a roughly smooth distribution over the temperature range \n$4.5 \\leq \\log (T \/ K) \\leq 5.5$. If a bimodality\ndoes exist it is only present in the very highest column absorbers. There are no\nsignificant discrepancies between the absorbers identified here with\nthe contour method versus those identified initially with the cut\nmethod. We also find similar overall phase fractions, with $69\\%$ of\n\\ion{O}{6}\\ absorbers in the WHIM phase, $30\\%$ of absorbers in the warm\nphase, and $1\\%$ of absorbers in the condensed phase.\n            \n\\begin{figure*}\n  \\begin{center}\n  \\includegraphics[width=0.85\\textwidth]{phase.pdf}\n  \\caption{Thermal-state distribution of \\ion{O}{6}\\ absorbers.  Each absorber is plotted\n    as a function of its mean baryon overdensity\n    (defined as $\\rho\/\\rho_o$) and temperature,\n    with points colored according to absorber column density. The dashed line\n    indicates where collisional ionization equals photoionization (collisional\n    ionization dominates above the curve).\n    Histograms along the top and right side of the scatter plot show\n    overdensity and\n    temperature, respectively, for all \\ion{O}{6}\\ absorbers with column\n    densities greater than $10^{15}$~cm$^{-2}$\n    (light blue), 10$^{14.5}$~cm$^{-2}$ (dark blue), $10^{14}$~cm$^{-2}$ (dark green),\n     and $10^{13}$~cm$^{-2}$ (light green).}\n  \\label{fig:phase}\n  \\end{center}\n\\end{figure*}\n\n\\subsubsection{Cool vs. Warm IGM}\n\\label{sec:temp_bimodality}\n\nThe WHIM is often defined as gas with temperatures in the range \n$10^5-10^7$ K \\citep{1999ApJ...514....1C,2001ApJ...552..473D}. \nUsing temperatures derived via the contour method, we attempt to \ndetermine the characteristics of observables for absorbers in this \ntemperature range and assess any systematic differences between \nabsorbers with lower temperatures. For this analysis we henceforth\ndefine a WHIM asborber as an absorber with $\\mathrm{T>10^5~K}$ and\na warm absorber as one with $\\mathrm{T<10^5~K}$.\nFigure \\ref{fig:temperature_bimodality} \nshows column density and $b$-value histogrammed separately for \nWHIM and warm absorbers. The warm \\ion{H}{1}\\ absorbers have a roughly \nlinear distribution by column density in logspace, while the WHIM \n\\ion{H}{1}\\ absorbers fall off sharply after $\\mathrm{N_{HI}}\\sim 10^{14}$~cm$^{-2}$. \nThe warm and WHIM \\ion{O}{6}\\ absorbers show very similar distributions \nby column density, with the WHIM absorbers dominating slightly below \n$N_{OVI}\\sim 10^{14}$~cm$^{-2}$. The $b$-value histograms show \ntwo distinct distributions for the WHIM and warm absorbers \nfor both \\ion{H}{1}\\ and \\ion{O}{6}. \\ion{H}{1}\\ warm absorbers have a peak $b$-value \nof $~15$~km~s$^{-1}$ whereas WHIM absorbers peak at around \n45~km~s$^{-1}$. In \\ion{O}{6}, warm absorbers peak at 10~km~s$^{-1}$ while \nWHIM absorbers peak at 20~km~s$^{-1}$.\n\n\n\\begin{figure*}  \n\n\\begin{center}\n  \\includegraphics[width=0.95\\textwidth]{temperature_bimodality.pdf}\n  \\caption{  Absorbers binned by column density (\\emph{top}) and b-value (\\emph{bottom}). WHIM absorbers are in blue, warm absorbers are in purple.\\emph{Left: } \\ion{H}{1}\\ absorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers. All absorbers have been identified using the contour method.}\n  \\label{fig:temperature_bimodality}\n\\end{center}\n\\end{figure*}\n\nAfter establishing the relative distributions of absorber observables \nin the warm and WHIM phases, we examine the relationship between the \nobservables. In Figure \\ref{fig:temperature_bVar} we show median \nb-value plotted over column density bins. We find a general decrease \nin the b-value of an absorber with column density for WHIM phase \\ion{H}{1}\\ \nabsorbers. A absorber with column density in the last bin appears to \nreverse the trend, but this point is not statistically significant. \nFor \\ion{H}{1}\\ absorbers in the warm phase, b-values increase with column density \nuntil roughly N$_{HI}=10^{15}$~cm$^{-2}$ and then begin to decrease again \nslightly. For \\ion{O}{6}\\ absorbers both the warm and WHIM absorbers have \nslightly increasing b-values with column density, but the median b-value \nof WHIM absorbers is typically 5-10~km~s$^{-1}$ higher than the median \nb-value of the warm absorbers for a given column density bin.\n\n\n\\begin{figure*}  \n\n\\begin{center}\n    \\includegraphics[width=.95\\textwidth]{temperature_bVar.pdf}\n  \\caption{  Median b-value vs column\n    density for WHIM (\\emph{blue}) and warm (\\emph{purple}) \n    absorbers. Error bars show 1st and 4th quartiles. \\emph{Left: } \\ion{H}{1}\\\n    absorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers.} \n  \\label{fig:temperature_bVar}\n\\end{center}\n\\end{figure*}\n\n\nIn Figure \\ref{fig:temperature_2dhist} we show the fraction of \nWHIM absorbers out of total absorbers histogrammed two-dimensionally \nalong column density and $b$-value. We find that above $b$-values of \n40 and 15 km~s$^{-1}$ for \\ion{H}{1}\\ and \\ion{O}{6}, respectively, the fraction \nis dominated by WHIM absorbers. These sharp cutoffs are to be expected \nbecause for a temperature of at least $10^5$ K there is a minimum \n$b$-value given by Equation \\ref{eq:bth}. We also see that the higher \n$b$-values are associated with lower column densities, although this \ncorrelation is stronger for \\ion{H}{1}\\ absorbers.\n\n\\begin{figure*}\n\\begin{center}\n    \\includegraphics[width=.95\\textwidth]{temperature_2dhist.pdf}\n  \\caption{ Fraction of total absorbers that are WHIM absorbers placed into two-dimensional histograms of column density and b-value. The color shows the fraction of total absorbers where red indicates no absorbers in the WHIM and white indicates no absorbers of any phase. \\emph{Left: } \\ion{H}{1}\\ absorbers. \\emph{Right: } \\ion{O}{6}\\ absorbers.} \n  \\label{fig:temperature_2dhist}\n\\end{center}\n\\end{figure*}\n\n\n\n\\subsubsection{Collisionally Ionized vs. Photoionized \\ion{O}{6}}\n\nIn Figure \\ref{fig:phase} we plotted a dotted line indicating where \ncollisional ionization begins to dominate over photoionization for \\ion{O}{6}, given by \n\n\\begin{equation}\n  \\begin{split}\n    \\log \\rho \/ \\bar{\\rho}_b = \n    7.68  \\left( \\frac{1+z}{1.2} \\right)^{-3}  T^{-1\/2} \\left(1 + \\frac{T}{1.32 \\times 10^7}\\right)\\\\\n    \\times  e^{1.32 \\times 10^6 \/ T}\n  \\end{split}\n\\end{equation}\n\nderived in \\citet{shull2012}, where $\\bar{\\rho}_b$ is the\nuniversal mean\nbaryon density such that $\\rho\/\\bar{\\rho}_b = \\Delta_b$.  This\nrelation is valid in the density regime where the IGM is optically\nthin to the metagalactic ionizing background, which includes all of\nthe absorbers shown in Figure~\\ref{fig:phase}.  In this section we \nperform a similar analysis as was done in section \\ref{sec:temp_bimodality}, \nbut instead of making a temperature cut to distinguish between WHIM \nand warm absorbers, we differentiate between absorbers where collisional \nionization dominates and where photoionization dominates. As this \ndifferentiation is only appropriate for \\ion{O}{6}, we do not show results \nfor \\ion{H}{1}\\ absorbers here. \n\nFigure \\ref{fig:ionization_bimodality} shows column density, $b$-value, \nand temperature histogrammed for photoionization dominated and \ncollisional ionization dominated absorbers. The distribution of \ncolumn density shows no significant difference for the two absorber \npopulations. The $b$-value distributions both appear roughly exponential \nbut the photoionization dominated population's distribution peaks at \n$b$ $\\sim 15$~km~s$^{-1}$. while the collisional ionization dominated \npopulation peaks at $b\\sim 20$~km~s$^{-1}$. Similarly, both temperature \ndistributions appear roughly gaussian with photoionization dominated \npeaking at T$\\sim 10^5$~K and collisionally ionization dominated at \nT$\\sim 10^{5.5}$~K. \n\n\\begin{figure*}  \n\\begin{center}\n  \\includegraphics[width=0.95\\textwidth]{ionization_bimodality.pdf}\n  \\caption{\\ion{O}{6}\\ absorbers binned by column density (\\emph{left}), b-value (\\emph{center}), and temperature (\\emph{right}). Collisional ionization dominated absorbers are in blue, photoionization dominated absorbers are in purple. }\n  \\label{fig:ionization_bimodality}\n\\end{center}\n\\end{figure*}\n\n\nIn an approach similar to the one in Figure \\ref{fig:temperature_2dhist}, \nFigure \\ref{fig:ionization_2dhist} shows the fraction of collisional \nionization-dominated absorbers out of total absorbers in a two \ndimensional histogram of column density and $b$-value. The results are \nqualitatively similar to those seen in the \\ion{O}{6}\\ panel of Figure \n\\ref{fig:temperature_2dhist}, but with a slightly smoother transition \nalong the $b$-value axis.\n\n\\begin{figure}\n\\begin{center}\n    \\includegraphics[width=0.45\\textwidth]{ionization_2dhist.pdf}\n  \\caption{Fraction of total \\ion{O}{6}\\ absorbers that are collisional ionization dominated placed into two-dimensional histograms of column density and b-value. The color shows the fraction of total absorbers where red indicates no absorbers in the WHIM and white indicates no absorbers of any phase.  } \n  \\label{fig:ionization_2dhist}\n\\end{center}\n\\end{figure}\n\n\n\\subsubsection{Relating \\ion{H}{1}\\ and \\ion{O}{6}}\n\nOne advantage of using the contour method is that all information \nassociated with a single grid cell is also associated with a given \nlixel and consequently associated with a given absorber. This \nallows us to identify \\ion{H}{1}\\ absorbers and then automatically \nidentify the associated \\ion{O}{6}\\ content by summing the \\ion{O}{6}\\ number \ndensities along the lixels in the absorber. Thus, after identifying \n\\ion{H}{1}\\ absorbers using the contour method we find the column density \nof \\ion{O}{6}\\ associated with this \\ion{H}{1}\\ identified set of cells. We plot \nthis associated \\ion{O}{6}\\ column density as a function of the original \n\\ion{H}{1}\\ number density and b-value in Figure \\ref{fig:OVI_in_HI}. \n\nWe find that the associated \\ion{O}{6}\\ column shows very different \nbehavior for WHIM absorbers ($T>10^5~K$) versus warm absorbers. In \nnearly all cases but the highest column \\ion{H}{1}\\ absorbers, the \nassociated \\ion{O}{6}\\ column density is significantly greater for WHIM \nabsorbers than for warm absorbers. As a function of \\ion{H}{1}\\ column \ndensity the \\ion{O}{6}\\ column for WHIM absorbers increases and then \ndecreases, with the peak at around $10^{14}$~cm$^{-2}$. The \nmedian \\ion{O}{6}\\ column for warm absorbers increases fairly steadily \nas a function of \\ion{H}{1}\\ column density. \n\nThe median \\ion{O}{6}\\ column density increases steadily as a function \nof b-value starting with the minimum possible b-value for WHIM \nabsorbers. Warm absorbers show an initially sharp increase, but after \n$b\\sim 60$~km~s$^{-1}$ the points become statistically insignificant. \n\n\n\\begin{figure*}  \n\\begin{center}\n  \\includegraphics[width=0.95\\textwidth]{OVI_in_HI.pdf}\n  \\caption{  Median associated \\ion{O}{6}\\ column density versus \\ion{H}{1}\\ column density (\\emph{Left}) and \\ion{H}{1}\\ b-value (\\emph{Right}). Plotted for both WHIM (\\emph{blue}) and warm (\\emph{purple}) absorbers. Error bars show 1st and 4th quartiles. Dashed line indicates minimum b-value for a WHIM absorber, as given by Equation \\eqref{eq:bth}.}\n  \\label{fig:OVI_in_HI}\n\\end{center}\n\\end{figure*}\n\n\n\n\\subsubsection{Metallicity Ionization Fraction Product}\n\nThe product $(Z\/Z_\\odot)\\times f_{OVI}$, where $(Z\/Z_\\odot)$ is the gas\nmetallicity in solar units and $f_{OVI}$ is the fraction of oxygen\nin the \\ion{O}{6}~ionization state, is of key importance for constraining\nthe budget of baryons traced by \\ion{O}{6},  as given by \n\n\\begin{equation}\n  \\Omega_b^{\\mathrm{OVI}}=\n  \\left[\n  \\frac{\\mu_bH_o}{c\\rho_{cr}(O\/H)_\\odot}\\right] \\int_{N_{min}}^{N_{max}}\n  \\left(\\frac{\\partial^2\\mathcal{N}}{\\partial z\\partial N}\\right)\\frac{N}{Z_O(N)f_{\\mathrm{OVI}}(N)}dN\n  \\label{eq:baryonOVI}\n\\end{equation}\n\nwhere $\\mu_b$ is the mean baryon mass per hydrogen, $\\rho_{cr}$ is\nthe cosmic closure density, and $(O\/H)_\\odot$ is the solar oxygen abundance.\n\nPreviously the product $(Z\/Z_\\odot)\\times f_{OVI}$ has been\nassumed to be constant, with typical estimates of $Z\/Z_\\odot=0.1$ and\n$f_{OVI}=0.2$, giving a product of $0.02$. \\citet{teppergarcia2011} first\nexploited the advantage of simulations in calculating $\\Omega_{\\mathrm{OVI}}$\ntaking into account the ionisation fraction and metallicity of each\nindividual absorber, and \nprevious analysis of our simulations using the cut method \ncontinued this study to find that\nthis product varied in a power-law distribution\nproportional to $(N_{OVI})^\\gamma,\\, \\gamma={0.7}$. Using this variation with an\nobservational fit of $\\partial\\mathcal{N}\/\\partial z \\propto N^{-\\beta},\\, \\beta=2.0$ \\citep{danforth2008}, one\ncan constrain the baryon budget as follows: \n\\begin{equation}\n  \\Omega_b^{\\mathrm{OVI}}\\propto \\int_{N_{min}}^{N_{max}}\n  N^{1-\\beta-\\gamma}dN  \\; .\n\\end{equation}\n\nAs can be seen in Figure~\\ref{fig:zzf}, we find a similar power-law \ndistribution when using our more physically-motivated contour method\nto find absorbers.  The product of ($Z\/Z_{\\odot}$) and $f_{OVI}$ is\nproportional to $(N_{OVI})^{0.7}$. In an effort to make this result applicable\nto observational surveys, we limit the absorbers we consider for the\naverage and fit to those with \\ion{O}{6}\\ column densities between\n$10^{13}$~cm$^{-2}$ and $10^{15}$~cm$^{-2}$. We then weight \nthe product by $(N_{\\mathrm{OVI}})^{1-\\beta},\\, \\beta=2$ \\citep{danforth2008} to estimate the\naverage contribution to the estimate \n$(Z\/Z_\\odot)f_{\\mathrm{OVI}}=0.007$, which is once again a significantly lower value\nthan previous literature suggests.  As before, we only use absorbers within the\ncolumn density range $10^{13}$~cm$^{-2}\\leq N_{\\mathrm{OVI}}\\leq 10^{15}$~cm$^{-2}$ to mimic\nobservational results. For further analysis of the implications of a lower value of the product $(Z\/Z_\\odot)\\times f_{OVI}$ , we direct readers to \\citet{shull2012}.\n\n\\begin{figure}\n  \\includegraphics[width=0.45\\textwidth]{zzf.pdf}\n  \\caption{Values of the product $(Z\/Z_\\odot) \\times f_{OVI}$ over column\n    density.   The vertical gray dotted lines show limits of\n    points that were included in averages and fit, representative of\n    range of typical \\ion{O}{6}\\ surveys \\citep{danforth2008}. The\n    horizontal blue dashed\n    line shows the previously assumed value of 0.02\n    obtained by using the standard values of $Z\/Z_\\odot=0.1$ and\n    $f_{OVI}=0.2$. The horizontal red dashed line\n    shows the average of column-limited points\n    weighted by $(N_{OVI})^{-1}$. The horizontal green dashed line \n    shows value previously obtained in a similar $\\log(N_{OVI})$\n    weighted average by \\citet{shull2012}.  The solid yellow line\n    indicates a power law fit of the column-limited sample of\n    points given\n    by $(Z\/Z_\\odot)\\times f_{OVI} = 0.021\\times[N_{OVI}\/(10^{14}\n    \\mathrm{cm}^{-2})]^{0.715}$. }\n  \\label{fig:zzf}\n\\end{figure}\n\t\t\n\n\t\t\t\n\n\\section{Simulation}\n\\label{sec:sim}\n\nIn this work, we analyze a simulation performed with the open source\ncosmological adaptive mesh refinement + N-body code,\n\\texttt{Enzo}\\footnote{\\url{http:\/\/enzo-project.org}}.\n\\citep{2013arXiv1307.2265T, 2004astro.ph..3044O, 2007arXiv0705.1556N}.\nThe simulation, which is run 50\\_1024\\_2 from \\citet{britton2011}, has\na comoving box size of 50 Mpc\/$h$ with 1024$^{3}$ grid cells and dark\nmatter particles, corresponding to a comoving spatial resolution of 49\nkpc\/$h$ and a dark matter particle mass of $7\\times10^{6}\\ M_{\\odot}$.\nThe simulation includes the metallicity dependent radiative cooling\nmethod of \\citet{2008MNRAS.385.1443S}, modified to include a\nmetagalactic UV background and a modified version of the star\nformation and feedback method of \\citet{1992ApJ...399L.113C} that\ninjects stellar feedback into a $3\\times3\\times3$ cube of grid cells\ncentered on the star particle (referred to in \\citet{britton2011} as\nthe ``distributed'' method). Note that the distributed feedback method\ndoes not take into account a physical scale, it suffers from the same\noverproduction of stars and metals as its predecessor when used with higher \nresolution simulations. However,  \\citet{britton2011} have shown that this\nsimulation is able to reasonably reproduce both the observed global\nstar formation history \\citep{0004-637X-651-1-142} and the number density of \\ion{O}{6}\\ absorbers per\nunit redshift ($d{\\cal N}\/dz$) in the redshift range $0 \\le z \\le 0.4$.\n\\citet{britton2011} performed simulations using two different stellar\nfeedback models.  However, since the goal of this work is to\nunderstand how the statistics of synthetic absorbers are affected by\nthe method with which they are produced, we choose to focus on just the\nsingle simulation from \\citet{britton2011} that best matched\nobservations.\n\n\\section{Summary}\n\\label{sec:summary}\nIn this paper, we compare two primary methods for finding\nIGM absorbers along lines of sight cast through a simulation box.  One\nmethod (the ``spectral method'') uses synthetic absorption line\nspectra, and is meant to directly correspond to observational attempts\nto find structure through fitting Voigt profiles to variations in\nflux.  The other primary method (the ``contour method'') relies on\ndefining absorbers by associating contiguous regions along the line of\nsight based on a threshold number density of the species of interest.\nAfter comparing these methods to each other, we compare to observational data of \\ion{H}{1}\\ and \\ion{O}{6}\\ absorption\nline systems.  The key results of this paper are as follows:\n\n\\begin{enumerate}\n\n\\item The two methods give comparable column densities for a given\nabsorber, although there is some difficulty in creating a one-to-one\ncorrespondence of absorbers between the two methods. The primary issue\nfor comparing absorbers generated with the two methods appears to be how \none decides whether an absorber is a single coherent structure or a complex.\n\n\\item The number of \\ion{H}{1}\\ and \\ion{O}{6}\\  absorbers per column density per unit redshift, or $d{\\cal N}\/dz$, traced by the two\nmethods give similar results, indicating that the two methods find\nsimilar amounts of overall baryons in these ionization states regardless of the ability to match each\nindividual absorber. \\ion{H}{1}\\ $d{\\cal N}\/dz$\\ compares favorably to observation,\nwhile the \\ion{O}{6}\\ $d{\\cal N}\/dz$\\ in our simulation underpredicts the observational results.\nThis is likely a shortcoming of the simulation itself, as the abundance \nof OVI depends sensitively on the assumptions of star formation, feedback, and metal transport.\n\n\\item The distribution of Doppler parameters (or b-values) by column\ndensity in our simulations are similar using the contour and spectral\nmethods in both Ly$\\alpha$\\ and \\ion{O}{6}.\nBoth methods of extracting b-values from our simulation match\nobservations of Ly$\\alpha$\\ systems as a function of column density. Neither method\nprovides a particularly good fit for the Doppler parameters measured in observed \\ion{O}{6}\\ systems; we speculate that this \nis due to our choice stellar feedback algorithms.\n\n\\item The distribution of \\ion{O}{6}\\ absorbers over baryon overdensity and\ntemperature was found to be similar to a previous analysis of the same simulation data,\nbut we do not see evidence for a bimodality in absorber distribution by temperature.\nInstead, we see that \\ion{O}{6}\\ absorbers are distributed smoothly in temperature-space \nfrom $4.5 < \\log (T \/ K) < 5.5$.\n\n\\item Using the contour method, we find that the relationship between linear size and column density for a given absorber scales as L~$\\propto$~N$^{-0.12}$, rather than the analytically-predicted exponent $-1\/3$.  The reason for this is unclear, though we speculate that it may be related to simulation resolution or physics.\n\n\\item We examine the individual properties of warm vs. WHIM tracing \\ion{H}{1}\\ and \\ion{O}{6}\\ absorbers (warm vs. WHIM having temperatures greater and less than $10^5$~K respectively) as well as photo- vs. collisional ionization dominated OVI absorbers.  \\ion{H}{1}\\ warm\/WHIM-tracing absorbers show slightly different column density distributions with number of WHIM absorbers of a given column density falling off much more sharply than number of warm absorbers after $\\mathrm{N_{HI}}\\sim10^{14}$~cm$^{-2}$. \\ion{O}{6}\\ show similarly shaped column density distributions, albeit with different normalizations.  \\ion{H}{1}\\ and \\ion{O}{6}\\ absorbers associated with WHIM gas have systematically higher b-values.   \\ion{H}{1}\\ and \\ion{O}{6}\\ absorbers associated with warm gas have b-value distributions centered around 12 and 10~km~s$^{-1}$, respectively, while \\ion{H}{1}\\ and \\ion{O}{6}\\ absorbers associated with the WHIM have b-value distributions centered around 45 and 20~km~s$^{-1}$.  Dividing \\ion{O}{6}\\ absorbers into photoionized  and collisionally ionized populations shows a similar results to the warm\/WHIM division.\n\n\\item We investigate the association of warm and WHIM-tracing \\ion{H}{1}\\ absorbers with \\ion{O}{6}\\ absorbers.  For \\ion{H}{1}\\ absorbers emanating from warm gas, there is a positive correlation between \\ion{H}{1}\\ column density and column density of associated \\ion{O}{6}.  Higher column density warm \\ion{H}{1}\\ absorbers tend to be associated with higher column density \\ion{O}{6}\\ absorbers.  Comparatively, low column density WHIM-tracing \\ion{H}{1}\\ absorbers are associated with higher column density \\ion{O}{6}\\ absorbers than are the warm-tracing \\ion{H}{1}.  However, there does not exist such a trend of increasing \\ion{O}{6}\\ column density with increasing \\ion{H}{1}\\ column density.  Instead, the average associated \\ion{O}{6}\\ column density peaks at $\\sim10^{13}$~cm$^{-2}$ at an \\ion{H}{1}\\ column density of $\\sim 10^{14}$~cm$^{-2}$.\n\n\\item Finally, we study the relation between the column density of \\ion{O}{6}\\ absorbers and the value of  $(Z\/Z_\\odot)\\times\nf_{\\mathrm{OVI}}$, finding that $f_{\\mathrm{OVI}}\\,\\propto\\,\n[N_{\\mathrm{OVI}}\/(10^{14}\\,\\mathrm{cm}^{-2})]^{0.7}$.  Over the column density range $13\\leq \\log N_{OVI}\/ \\mathrm{cm}^{-2} \\leq 15$ this yields an average value of $(Z\/Z_\\odot)\\times\nf_{\\mathrm{OVI}} = 0.007$, in reasonable agreement with \\citet{shull2012}, but nearly a factor of three lower than earlier estimates.  This would imply that \\ion{O}{6}\\ traces roughly triple the number of baryons previously thought.\n\n\\end{enumerate}\n\nIn subsequent papers we hope to expand our analysis of the systematic\ndifferences in the contour and spectral methods, and use the nearly\none-to-one correlation between the two sets of absorbers to better\nunderstand the correspondence between properties of absorption line\nsystems to features in physical structure. We also intend to look at\nthe effects of noise, as well as line blanketing when fitting multiple\nions together.\n\n\\acknowledgments\n\nThis work was supported by NASA through grants NNX09AD80G,\nNNX12AC98G, NNX08AC14G, and NNX07AG77G,  and by the NSF through AST grant 0908819.  The simulations\npresented here were performed and analyzed on the NICS Kraken and\nNautilus supercomputing resources under XSEDE allocations TG-AST090040\nand TG-AST120009.  We thank Charles Danforth for helpful discussions\nduring the course of preparing this manuscript.  BWO was supported in\npart by the MSU Institute for Cyber-Enabled Research and by Hubble Theory Grant  HST-AR-13261.01-A.  HE was\nsupported in part by the MSU Hantel Fellowship and the MSU College of\nNatural Sciences.  \\texttt{Enzo} and \\texttt{yt} are developed by a\nlarge number of independent research from numerous institutions around\nthe world.  Their commitment to open science has helped make this\nwork possible.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}